Chemical Reaction Engineering Model Library Manual

VERSION 4.4 Chemical Reaction Engineering Module Model Librar y Manual C o n t a c t I n f o r ma t i o n Visit the Contact COMSOL page at www.comsol.com/contact to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at www.comsol.com/contact/offices for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at www.comsol.com/support/case. Other useful links include: • Support Center: www.comsol.com/support • Product Download: www.comsol.com/support/download • Product Updates: www.comsol.com/support/updates • COMSOL Community: www.comsol.com/community • Events: www.comsol.com/events • COMSOL Video Center: www.comsol.com/video • Support Knowledge Base: www.comsol.com/support/knowledgebase Part number: CM021606 C h e mi c a l R e a c t i o n E n g i n e e r i n g M o d e l L i b r a r y M a n u a l © 1998–2013 COMSOL Protected by U.S. Patents 7,519,518; 7,596,474;7,623,991; 8,219,373; and 8,457,932. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/sla) and may be used or copied only under the terms of the license agreement. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/tm. Version: November 2013 COMSOL 4.4 Solved with COMSOL Multiphysics 4.4 1 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N De t e r mi ni ng Ar r he ni us Pa r a me t e r s us i ng Pa r a me t e r Es t i ma t i on Introduction This model shows how to use the Parameter Estimation feature in the Reaction Engineering physics interface to find the Arrhenius parameters of a first order reaction. Note: This model requires the Optimization Module. Inspiration for this example is taken from Ref. 1. Model Definition Benzene diazonium chloride in the gas phase decomposes to benzene chloride and nitrogen according to: (1) The reaction is first order with the rate: (2) where the temperature dependent rate constant given by: (3) Above, A is the frequency factor (1/s) and E is the activation energy (J/mol). In order to evaluate the Arrhenius parameters, A and E, a set of experiments were conducted using a perfectly mixed isothermal batch system. The concentration of benzene diazonium chloride was monitored as function of time for the temperatures; T = 313 K, 319 K, 323 K, 328 K, and 333 K. N N 2 + N Cl Cl k r kc PhN2Cl = k A E R g T ----------- – \ . | | exp = Solved with COMSOL Multiphysics 4.4 2 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N The five experimental data sets are available as comma separated value files (csv-files) together with the model file for this example. Results and Discussion Parameter estimation calculations give the values A = 1.1·10 16 (1/s) and E = 116 (kJ/mol) for the frequency factor and activation energy, respectively. Plots of the model results and the associated experimental data points are shown below. Figure 1: Model results and experimental data for PhN 2 Cl concentration as function of time. Solved with COMSOL Multiphysics 4.4 3 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N Notes About the COMSOL Implementation The parameter estimation solver will be more efficient in finding an optimal parameter set if the model experiences similar sensitivity with respect to changes in parameter values. In this problem we therefore define a parameter A ex that is to be estimated together with the activation energy E, such that the rate constant is written as (4) The frequency factor A is then evaluated as: (5) The data indicates that the rate constant is of the order ~1·10 -3 (1/s) at T = 323 K. Taking this into account and using an initial guess for the activation energy of 150 kJ/ mol, an initial guess is set for A ex = 49. Reference 1. H.S. Fogler, Elements of Chemical Reaction Engineering 4th ed., p. 95, Prentice Hall, 2005. Model Library path: Chemical_Reaction_Engineering_Module/ Optimization/activation_energy Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. k A ex ( ) E R g T ----------- – \ . | | exp · exp = A A ex ( ) ln = Solved with COMSOL Multiphysics 4.4 4 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N 5 In the tree, select Preset Studies>Time Dependent. 6 Click the Done button. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: R E A C T I O N E N G I N E E R I N G Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type PhN2Cl=>PhCl+N2. Species: PhN2Cl 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: PhN2Cl. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 1000. Now, add a Parameter Estimation feature, define parameters and set initial values. Parameter Estimation 1 1 On the Physics toolbar, click Global and choose Parameter Estimation. 2 In the Parameter Estimation settings window, locate the Control Variables section. 3 In the Control variables table, enter the following settings: Create separate Experiment features for the data collected at different temperatures (T_iso). Name Expression Value Description T_iso 313[K] 313.00 K temerature Variable Initial value Lower bound Upper bound Aex 49 E 150e3 Solved with COMSOL Multiphysics 4.4 5 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N Experiment 1 1 Right-click Component 1>Reaction Engineering>Parameter Estimation 1 and choose Experiment. 2 In the Experiment settings window, locate the Experimental Data section. 3 Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file activation_energy_experiment313K.csv. 5 Click the Import button. 6 In the table, enter the following settings: 7 Locate the Experimental Parameters section. Click Add. 8 In the table, enter the following settings: Experiment 2 1 Right-click Parameter Estimation 1 and choose Experiment. 2 In the Experiment settings window, locate the Experimental Data section. 3 Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file activation_energy_experiment319K.csv. 5 Click the Import button. 6 In the table, enter the following settings: 7 Locate the Experimental Parameters section. Click Add. Data column Use Model variables time \ t conc PhN2Cl (313K) \ c_PhN2Cl Parameter names Parameter expressions T_iso 313 Data column Use Model variables time \ t conc PhN2Cl (319K) \ c_PhN2Cl Solved with COMSOL Multiphysics 4.4 6 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N 8 In the table, enter the following settings: Experiment 3 1 Right-click Parameter Estimation 1 and choose Experiment. 2 In the Experiment settings window, locate the Experimental Data section. 3 Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file activation_energy_experiment323K.csv. 5 Click the Import button. 6 In the table, enter the following settings: 7 Locate the Experimental Parameters section. Click Add. 8 In the table, enter the following settings: Experiment 4 1 Right-click Parameter Estimation 1 and choose Experiment. 2 In the Experiment settings window, locate the Experimental Data section. 3 Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file activation_energy_experiment328K.csv. 5 Click the Import button. 6 In the table, enter the following settings: Parameter names Parameter expressions T_iso 319 Data column Use Model variables time \ t conc PhN2Cl (323K) \ c_PhN2Cl Parameter names Parameter expressions T_iso 323 Data column Use Model variables time \ t conc PhN2Cl (328K) \ c_PhN2Cl Solved with COMSOL Multiphysics 4.4 7 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N 7 Locate the Experimental Parameters section. Click Add. 8 In the table, enter the following settings: Experiment 5 1 Right-click Parameter Estimation 1 and choose Experiment. 2 In the Experiment settings window, locate the Experimental Data section. 3 Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file activation_energy_experiment333K.csv. 5 Click the Import button. 6 In the table, enter the following settings: 7 Locate the Experimental Parameters section. Click Add. 8 In the table, enter the following settings: Next, return to the Reaction Engineering feature and introduce the parameters to the reaction model. 9 In the Model Builder window, click Reaction Engineering. 10 In the Reaction Engineering settings window, locate the General section. 11 In the T edit field, type T_iso. 1: PhN2Cl=>PhCl+N2 1 In the Model Builder window, under Component 1>Reaction Engineering click 1: PhN2Cl=>PhCl+N2. 2 In the Reaction settings window, locate the Rate Constants section. 3 Select the Use Arrhenius expressions check box. 4 In the A f edit field, type exp(Aex). Parameter names Parameter expressions T_iso 328 Data column Use Model variables time \ t conc PhN2Cl (333K) \ c_PhN2Cl Parameter names Parameter expressions T_iso 333 Solved with COMSOL Multiphysics 4.4 8 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N 5 In the E f edit field, type E. S T U D Y 1 Step 1: Time Dependent 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type range(0,50,5000). Optimization 1 On the Study toolbar, click Optimization. 2 In the Optimization settings window, locate the Optimization Solver section. 3 From the Method list, choose Levenberg-Marquardt. Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Optimization Solver 1 node, then click Time-Dependent Solver 1. 3 In the Time-Dependent Solver settings window, click to expand the Output section. 4 From the Times to store list, choose Specified values. 5 Click to expand the Absolute tolerance section. Locate the Absolute Tolerance section. In the Tolerance edit field, type 1e-5. 6 From the Method list, choose Unscaled. 7 In the Tolerance edit field, type 1e-5. 8 On the Home toolbar, click Compute. R E S U L T S Experiment 1 Group 1 In the Model Builder window, expand the Results>Experiment 1 Group node, then click Global 1. 2 In the Global settings window, locate the Data section. 3 From the Parameter selection (T_iso) list, choose From list. 4 In the Parameter values (T_iso) list, choose 313, 319, 323, 328, and 333. This restricts the plot to the results associated with T_iso = 313 K. Alternatively, you could have chosen From list and then selected the proper entries. Solved with COMSOL Multiphysics 4.4 9 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N 5 Locate the y-axis data section. Click Concentration (comp1.re.c_PhN2Cl) in the upper-right corner of the section. Locate the x-axis data section. Click Time (t) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Experiment 2 Group 1 In the Model Builder window, expand the Results>Experiment 2 Group node, then click Global 1. 2 In the Global settings window, locate the Data section. 3 From the Parameter selection (T_iso) list, choose From list. 4 In the Parameter values (T_iso) list, select 319. 5 Locate the y-axis data section. Click Concentration (comp1.re.c_PhN2Cl) in the upper-right corner of the section. Locate the x-axis data section. Click Time (t) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Experiment 3 Group 1 In the Model Builder window, expand the Results>Experiment 3 Group node, then click Global 1. 2 In the Global settings window, locate the Data section. 3 From the Parameter selection (T_iso) list, choose From list. 4 In the Parameter values (T_iso) list, select 323. 5 Locate the y-axis data section. Click Concentration (comp1.re.c_PhN2Cl) in the upper-right corner of the section. Locate the x-axis data section. Click Time (t) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Experiment 4 Group 1 In the Model Builder window, expand the Results>Experiment 4 Group node, then click Global 1. 2 In the Global settings window, locate the Data section. 3 From the Parameter selection (T_iso) list, choose From list. 4 In the Parameter values (T_iso) list, select 328. 5 Locate the y-axis data section. Click Concentration (comp1.re.c_PhN2Cl) in the upper-right corner of the section. Locate the x-axis data section. Click Time (t) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Experiment 5 Group 1 In the Model Builder window, expand the Results>Experiment 5 Group node, then click Global 1. Solved with COMSOL Multiphysics 4.4 10 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N 2 In the Global settings window, locate the Data section. 3 From the Parameter selection (T_iso) list, choose From list. 4 In the Parameter values (T_iso) list, select 333. 5 Locate the y-axis data section. Click Concentration (comp1.re.c_PhN2Cl) in the upper-right corner of the section. Locate the x-axis data section. Click Time (t) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. In the last step, output the estimated parameter to a table. Derived Values On the Results toolbar, click Evaluate All. Taking scaling into account E is found to be 1.16e5 and Aex is evaluated to 36.94. Solved with COMSOL Multiphysics 4.4 1 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N Boa t Re a c t or f or L ow Pr e s s ur e Che mi c al V apor De pos i t i on Introduction Chemical vapor deposition (CVD) is an important step in the process of manufacturing microchips. A common application is the deposition of silicon on wafers at low pressure. Low-pressure reactors are used to get a high diffusivity of the gaseous species, which results in a uniform deposition thickness, because the process becomes limited by the deposition kinetics (Ref. 1 and Ref. 2). Figure 1: Schematic of a boat reactor and the principle of the deposition process. This example models the momentum and mass transport equations coupled to the reaction kinetics for the deposition process. It treats a low-pressure boat reactor, where the goal of the simulation is to describe the rate of deposition as a function of the fluid mechanics and kinetics in the system. The gas, in this case silane (SiH 4 ), enters the reactor from the left and reacts on the wafers to form hydrogen and silicon. The remaining mixture leaves the reactor through the outlet on the right. The deposition of silicon on the surface of the wafers depends on the local concentration of silane, which is determined by the operating conditions for the reactor. More details about this example can be found in Elements of Chemical Reaction Engineering by H. Scott Fogler (Ref. 1). Solved with COMSOL Multiphysics 4.4 2 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N Model Definition First assume that the density of the gas is constant throughout the reactor. This implies that the reacting gas is either diluted in a carrier gas or that the conversion in the reactor is small. Moreover, only account for the mass transport of the reactant gas, in this case silane, and assume constant temperature in the reactor. In the wafer bundle convection transport can be neglected, so that the reacting gas can only be transported through diffusion. To save time and computational memory, also simplify the geometrical description of the wafer bundle by modeling it as an anisotropic medium. To this end, because silane cannot diffuse through the physical wafers, assume that the axial diffusivity in the wafer bundle is zero. Furthermore, correct the diffusivity in the radial direction according to the degree of packing in the bundle. Finally, neglect the influence of the support boat on the transport process that holds the wafer bundle in place. The structure of the boat reactor means that the 3D geometry can be reduced to a 2D axisymmetric model. The modeling domain is shown in Figure 2. Figure 2: The model geometry showing the domain and boundary labels. The chemical reaction accounted for in this example is: Axial symmetry Wafer bundle domain Free flow domain Walls Inlet Outlet Solved with COMSOL Multiphysics 4.4 3 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N (1) The rate of this reaction depends on the partial pressure of silane and the temperature in the reactor. The assumptions mentioned above in combination with the chemical reaction for the deposition process make it possible to define an equation system. The momentum equations and the continuity equations for laminar flow in cylindrical coordinates read (2) Here q (SI unit: kg/(m·s)) denotes the viscosity; µ (SI unit: kg/m 3 ) is the density of the gas; u and v (SI unit: m/s) refer to the velocity vector’s r- and z-components, respectively; and p (SI unit: Pa) is the pressure. The mass transport in the free-fluid domain is given by the following equation, expressed in cylindrical coordinates: (3) Here D denotes the diffusivity (SI unit: m 2 /s) and c is the concentration of silane (SI unit: mol/m 3 ). You obtain the corresponding mass transport equation for the wafer bundle domain by neglecting transport by convection and adding a reaction-rate term for the dissociation of silane: (4) Because you neglect diffusion in the axial direction, the effective diffusivity tensor, D eff , only has an rr-component. In the equation above, k (SI unit: m/s) denotes the rate constant for the reaction, and S a (SI unit: m 2 /m 3 ) refers to the specific surface area. You solve the system of equations defined above by using the proper boundary conditions. For laminar flow, no-slip conditions apply at the reactor-wall surface and between the free channel and the wafer bundle: SiH 4 g ( ) Si s ( ) 2H 2 g ( ) + ÷ r c c qr u c r c ------ – \ . | | z c c qr u c z c ------ – \ . | | µr u u c r c ------ v u c z c ------ + \ . | | r p c r c ------ + + + 0 = in O ff r c c qr v c r c ------ – \ . | | z c c qr v c z c ------ – \ . | | µr u v c r c ------ v v c z c ------ + \ . | | r p c z c ------ + + + 0 = in O ff r u c r c ------ v c z c ------ + \ . | | v + 0 = in O ff r c c Dr c c r c ----- – \ . | | z c c Dr c c z c ----- – \ . | | r u c c r c ----- r v c c z c ----- + + + 0 in O ff = r c c D eff rr , r c c r c ----- – \ . | | r – kS a c in O wb = Solved with COMSOL Multiphysics 4.4 4 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N (5) At the symmetry axis, the radial velocity component vanishes: (6) The last three conditions for the momentum equations and continuity equation are (7) For the mass transport Equation 3 and Equation 4, the boundary conditions are (8) where D i represents the diffusivity in O ff or O wb depending on to which boundary segment you apply the equation. This equation implies that there is no flux perpendicular to these boundaries. At the inlet, the composition of the gas is known, which yields: (9) At the outlet, assume that the transport of species takes place mainly by convection and neglect the concentration gradients perpendicular to this boundary: (10) It remains to discuss the material parameters appearing in Equation 4: D eff, rr , S a , and k. First, calculate the specific surface area (that is, the area per unit volume) of the wafer bundle, S a , by assuming a certain pitch between the wafers; see Figure 3. u v , ( ) 0 0 , ( ) = at O c wall , O c iw ff , , and O c wb ff , u 0 at O c sym = u 0 = at O c in v v 0 = at O c in p p 0 = at O c out q Vu Vu ( ) T + ( )n 0 = at O c out D i rr , c c r c ----- D i zz , c c z c ----- – , – \ . | | n · 0 = at O c wall , O c sym , and O iw c c c 0 at O c in = D c c r c ----- D c c z c ----- – , – \ . | | n · at 0 O c out = Solved with COMSOL Multiphysics 4.4 5 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N Figure 3: Calculation of the specific surface area. Furthermore, to estimate the effective diffusivity in the radial direction inside the wafer bundle, multiply the diffusivity in the free-fluid domain by the ratio of the contact area between the free gas and the wafer bundle to the total lateral surface area of the wafer-bundle domain: (11) The rate constant, k (SI unit: m/s), is a function of the partial pressure of silane. At 600 °C and a total system pressure of 25 Pa, Ref. 2 provides the value k = 8.06·10 ÷3 m/s. A crucial characteristic of the reactor’s performance is the silicon deposition rate, A Si , which expresses the growth rate of the silicon layer on the wafers. The amount of silicon deposited on the wafers, expressed in mass per unit area per unit time, is the product of the rate constant, k, the silane concentration, c, and the molar mass of silicon, M Si (SI unit: kg/mol). Dividing the so obtained quantity by the density of silicon, µ Si (SI unit: kg/m 3 ), gives the deposition rate: (12) In this model, you study the radial and axial distribution of A Si inside the wafer bundle. D eff rr , 1 d w d cc -------- – \ . | | D in O wb = A Si kcM Si µ Si ----------------- (nm/min) = Solved with COMSOL Multiphysics 4.4 6 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N Results and Discussion Figure 4 shows the concentration distribution in the boat reactor, indicating that the conversion is quite small. Solved with COMSOL Multiphysics 4.4 7 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N Figure 4: Concentration distribution in the reactor. Figure 5 shows the flow distribution in the reactor. Figure 5: Flow distribution in the reactor. The surface color and the arrows both represent the velocity. Solved with COMSOL Multiphysics 4.4 8 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N The plots in Figure 6 display the deposition rate for the inlet velocities 1 m/s (top panel) and 2 m/s (bottom panel). Figure 6: The deposition rate in the wafer bundle for the inlet velocities 1 m/s (top) and 2 m/s (bottom). Solved with COMSOL Multiphysics 4.4 9 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N In both cases, the highest deposition rate is obtained near the reactor inlet and close to the free-fluid channel. The difference in deposition rate between the center and periphery of the wafers is approximately 0.1 nm/min (or roughly 2.5%), and that along the length of the reactor approximately 0.5 nm/min (roughly 12.5%). Thus, as desired, the variations in the deposition rate inside the reactor are rather small. Moreover, comparing the plots it is evident that the deposition rate changes only marginally when the gas inlet velocity is doubled, showing that convection does not have a major influence on reactors of this type. References 1. H. Scott Fogler, Elements of Chemical Reaction Engineering, 3rd ed., Prentice Hall, 1999. 2. A.T. Voutsas and M.K. Hatalis, “Structure of As-Deposited LPCVD Silicon Films at Low Deposition Temperatures and Pressures,” J. Electrochem. Soc., vol. 139, no. 9, pp. 2659–2665, 1992. Model Library path: Chemical_Reaction_Engineering_Module/ Surface_Reactions_and_Deposition_Processes/boat_reactor Modeling Instructions MO D E L WI Z A R D 1 Go to the Model Wizard window. 2 Click the 2D axisymmetric button. 3 Click Next. 4 In the Add physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf). 5 Click Add Selected. 6 In the Add physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 7 Click Add Selected. 8 Click Next. Solved with COMSOL Multiphysics 4.4 10 | B O A T R E A C T O R F O R L O W P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N 9 Find the Studies subsection. In the tree, select Preset Studies for Selected Physics>Stationary. 10 Click Finish. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, Model Builder window, click Parameters. 2 In the Parameters settings window, locate the Parameters section. Click Load from File, and double click the file boat_reactor_parameters_1.txt in the models model library folder. D E F I N I T I O N S Variables 1 1 In the Model Builder window, under Model 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: G E O ME T R Y 1 Create the geometry. To simplify this step, insert a prepared geometry sequence: 1 On the Geometry toolbar, click Import/Export and choose Insert Sequence. 2 Browse to the model’s Model Library folder and double-click the file boat_reactor.mph. Then click Build all on the Geometry toolbar. D E F I N I T I O N S Explicit 1 1 In the Model Builder window, under Model 1 right-click Definitions and choose Selections>Explicit. 2 Right-click Explicit 1 and choose Rename. 3 Go to the Rename Explicit dialog box and type wafers in the New name edit field. 4 Click OK. 5 Select Domain 2 only. Name Expression Description Delta_Si c*k*M_Si/rho_Si Silicon deposition rate Solved with COMSOL Multiphysics 4.4 11 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N Explicit 2 1 In the Model Builder window, right-click Definitions and choose Selections>Explicit. 2 Right-click Explicit 2 and choose Rename. 3 Go to the Rename Explicit dialog box and type reactor in the New name edit field. 4 Click OK. 5 Select Domain 1 only. G L O B A L D E F I N I T I O N S Parameters 1 On the Hone toolbar, Model Builder window, click Parameters. 2 In the Parameters settings window, locate the Parameters section. Click Load from File, and double click the file boat_reactor_parameters_2.txt in the models model library folder. L A MI N A R F L OW 1 In the Laminar Flow settings window, locate the Domain Selection section. 2 From the Selection list, choose reactor. Fluid Properties 1 1 In the Model Builder window, expand the Laminar Flow node, then click Fluid Properties 1. 2 In the Fluid Properties settings window, locate the Fluid Properties section. 3 From the µ list, choose User defined. In the associated edit field, type rho. 4 From the µ list, choose User defined. In the associated edit field, type eta. Inlet 1 1 In the Model Builder window, right-click Laminar Flow and choose Inlet. 2 Select Boundary 2 only. 3 In the Inlet settings window, locate the Velocity section. 4 In the U 0 edit field, type v0. Outlet 1 1 Right-click Laminar Flow and choose Outlet. 2 Select Boundary 9 only. Solved with COMSOL Multiphysics 4.4 12 | B O A T R E A C T O R F O R L O W P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N TR A N S P O R T O F D I L U T E D S P E C I E S Convection and Diffusion 1 1 In the Model Builder window, expand the Model 1>Transport of Diluted Species node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Diffusion section. 3 In the D c edit field, type D. 4 Locate the Model Inputs section. From the u list, choose Velocity field (spf/fp1). Convection and Diffusion 2 1 In the Model Builder window, right-click Transport of Diluted Species and choose Convection and Diffusion. 2 In the Convection and Diffusion settings window, locate the Domain Selection section. 3 From the Selection list, choose wafers. 4 Locate the Diffusion section. From the symmetry property list, choose Diagonal. 5 In the D c table, enter the following settings: Reactions 1 1 Right-click Transport of Diluted Species and choose Reactions. 2 In the Reactions settings window, locate the Domain Selection section. 3 From the Selection list, choose wafers. 4 Locate the Reactions section. In the R c edit field, type -(k*S_a)*c. Inflow 1 1 Right-click Transport of Diluted Species and choose Inflow. 2 Select Boundary 2 only. 3 In the Inflow settings window, locate the Concentration section. 4 In the c 0,c edit field, type c0. Outflow 1 1 Right-click Transport of Diluted Species and choose Outflow. 2 Select Boundary 9 only. D_eff 0 0 0 Solved with COMSOL Multiphysics 4.4 13 | B O A T R E A C T O R F O R L OW P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N ME S H 1 1 In the Model Builder window, under Model 1 click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Element size list, choose Extra fine. 4 Click the Build All button. S T U D Y 1 Step 1: Stationary 1 In the Model Builder window, under Study 1 click Step 1: Stationary. 2 In the Stationary settings window, click to expand the Study Extensions section. 3 Select the Continuation check box. 4 Click Add. 5 In the table, enter the following settings: 6 In the Model Builder window, right-click Study 1 and choose Compute. R E S U L T S Velocity (spf) 1 In the Model Builder window, under Results click Velocity (spf). 2 In the 2D Plot Group settings window, locate the Data section. 3 From the Parameter value (v0) list, choose 1. 4 In the Model Builder window, under Results>Velocity (spf) click Surface 1. 5 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Transport of Diluted Species>Species c>Concentration (c). 6 Click the Plot button. 7 Click the Zoom Extents button on the Graphics toolbar. 8 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Laminar Flow>Velocity magnitude (spf.U). 9 In the Model Builder window, right-click Velocity (spf) and choose Arrow Surface. 10 In the Arrow Surface settings window, locate the Arrow Positioning section. Continuation parameter Parameter value list v0 1 1.5 2 Solved with COMSOL Multiphysics 4.4 14 | B O A T R E A C T O R F O R L O W P R E S S U R E C H E M I C A L V A P O R D E P O S I T I O N 11 Find the z grid points subsection. In the Points edit field, type 25. 12 Click the Plot button. 13 Click the Zoom Extents button on the Graphics toolbar. Data Sets 1 In the Model Builder window, under Results right-click Data Sets and choose Solution. 2 In the Solution settings window, locate the Solution section. 3 From the Solution list, choose Solver 1. 4 Right-click Results>Data Sets>Solution 2 and choose Add Selection. 5 In the Selection settings window, locate the Geometric Entity Selection section. 6 From the Geometric entity level list, choose Domain. 7 Select Domain 2 only. 2D Plot Group 6 1 In the Model Builder window, right-click Results and choose 2D Plot Group. 2 Right-click 2D Plot Group 6 and choose Surface. 3 In the Surface settings window, locate the Data section. 4 From the Data set list, choose Solution 2. 5 From the Parameter value (v0) list, choose 1. 6 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Definitions>Silicon deposition rate (Delta_Si). 7 Click the Plot button. 8 Click the Zoom In button on the Graphics toolbar. 9 Locate the Data section. From the Parameter value (v0) list, choose 2. 10 Click the Plot button. Solved with COMSOL Multiphysics 4.4 1 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Car bon De pos i t i on i n He t e r oge ne ous C a t a l y s i s Introduction Carbon deposition onto the surface of solid catalysts is commonly observed in hydrocarbon processing. Carbon deposits can affect both the activity of catalysts as well as the flow of gas through a catalyst bed. This example investigates the thermal decomposition of methane into hydrogen and solid carbon. In the first model you look at the isothermal process occurring in an ideal reactor, simulated with the 0D Reaction Engineering interface. The influence of carbon deposition on catalyst activity is also considered. In the second model, you study the effect that the carbon deposits have on the fluid flow. The second simulation takes both time and space dependencies into account. Model Definition C H E MI S T RY Methane decomposes over a Ni/Al 2 O 3 catalyst according to the overall chemical reaction: (1) Under atmospheric pressure, temperate ranging from 490 to 590 °C and volume fraction of hydrogen between 0 and 40%, the following reaction rate expression has been reported in the literature (Ref. 1): (2) where (3) CH 4 C 2H 2 + ÷ r k P CH 4 P H 2 2 K P ---------- – 1 k H P H 2 + ( ) 2 --------------------------------------- · = k k 0 20.492 104200 R g T -------------------- – \ . | | exp · = Solved with COMSOL Multiphysics 4.4 2 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S and k 0 in Equation 3 is 2.31·10 ÷5 (mol/(m 3 ·s)).The unit for pressure in Equation 2 is bar. I D E A L R E A C T O R MO D E L This model treats the isothermal decomposition of methane (Figure 2) in a perfectly mixed reactor with constant volume. The species mass balances are summarized by The rate term, R i (mol/(m 3 ·s)), takes into account the reaction stoichiometry, v i , the reaction rate, r (mol/(m 3 ·s)), and the catalyst activity, a: The mass balances of the reacting species are then The time dependence of the catalytic activity is expressed by (4) where k H 163200 R g T -------------------- 22,426 – \ . | | exp = K p 5.088 10 5 91200 R g T ---------------- – \ . | | exp · · = dc i dt -------- R i = R i v i ra = dc CH 4 dt ---------------- ra – = dc C dt ---------- ra = dc H 2 dt ------------ 2ra = da dt ------- k a r 2 c C c a – = k a k a0 135600 R g T -------------------- 32.007 – \ . | | exp · = Solved with COMSOL Multiphysics 4.4 3 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Where k a0 is 8.324·10 6 ((m 3 /mol) 3 ·s). Solving the mass balances provides the evolution of the species concentrations over time. The fact that carbon is in the solid phase is taken into account by removing its effect on gas phase physical properties. The pressure in the reactor is a function of only the methane and hydrogen concentrations: S PA C E - A N D T I ME - D E P E N D E N T MO D E L The second model takes both fluid flow and the chemical reaction into account.It is created by Generate Space-Dependent Model submenu under Reaction Engineering interface and solved in COMSOL Multiphysics. The flow reactor is set up in 2D, as illustrated below: Figure 1: A flow reactor is set up in 2D. Methane enters from the left and reacts in the porous catalytic bed in the mid-section of the geometry. MO ME N T U M B A L A N C E S The flow in the free channel section is described by the Navier-Stokes equations: (5) where r denotes density (kg/m 3 ), u represents the velocity (m/s), q is the dynamic viscosity (Pa·s)), and p refers to the pressure (Pa). In the porous domain, the Brinkman equations govern the flow: (6) Here c p is the porosity and k denotes permeability (m 2 ) of the porous medium. As you can see in Equation 4 and Equation 5, the momentum-balance equations are closely related. The term on the right-hand side of the Navier-Stokes formulation corresponds P R g T c CH 4 c H 2 + ( ) = wall wall Outlet Inlet Porous catalytic bed µ t c c u V q u V u V ( ) T + ( ) – pI + ( ) · + µ u V · ( )u – = V u · 0 = µ c P ----- t c cu V q c P ----- Vu Vu ( ) T + ( ) – pI + \ . | | · + q k ---u – = V u · 0 = Solved with COMSOL Multiphysics 4.4 4 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S to momentum transported by convection in free flow. In the Brinkman formulation, this term is replaced by a contribution associated with the drag force experienced by the fluid as it flows through a porous medium. COMSOL Multiphysics automatically combines free and porous-media flow to solve the equations simultaneously. The boundary conditions for the flow are: inlet walls outlet Mass transport in the reactor is described by the diffusion-convection equations: where D i denotes the diffusion coefficient (m 2 /s) and c i is the species concentration (mol/m 3 ). The term R i (mol/(m 3 ·s)) corresponds to the species’ net reaction rate. In the free channel, the inlet condition is set as (concentrations) At the outlet, use the convective flux condition All other boundaries, use the insulating or symmetry condition B A L A N C E F O R VO I D F R A C T I O N The void fraction of the catalytic bed decreases as carbon is deposited. This, in turn, affects the flow through the reactor. A balance for the void fraction, or porosity, of the bed is given by: Where k por is constant, M C (kg/mol) is carbon molar weight and µ soot (kg/m 3 ) is deposited carbon density. This equation can be implemented in the Domain ODEs and DAEs physics interface of COMSOL Multiphysics, resulting in porosity distribution u n · u 0 = u 0 = p 0 = t c cc i V D i Vc i – c i u + ( ) · + R i = c c in = n DVc – ( ) · 0 = n DVc – cu + ( ) · 0 = dc dt ------ k pos crM C µ soot --------------- · – = Solved with COMSOL Multiphysics 4.4 5 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S across the catalytic bed as a function of time. The initial porosity of the bed is assumed to be c = 0.4. The porosity is related to the permeability of the porous domain by the expression (Ref. 2): (7) In this way, the porosity balance couples the mass and momentum balances describing the reacting system. Results and Discussion I D E A L R E A C T O R MO D E L Figure 2 shows the concentration transients of methane, hydrogen and deposited carbon as methane decomposes over a Ni/Al 2 O 3 catalyst. Figure 2: Concentration transients of methane decomposition over a Ni/Al 2 O 3 catalyst. k k 0 c c 0 ----- \ . | | 3.55 = Solved with COMSOL Multiphysics 4.4 6 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Figure 3 shows the deactivation of catalyst during methane decomposition. The activity of catalyst decreases rapidly at the early stage of reaction, then decreases slowly. Figure 3: Change of activity of catalyst with reacting time. The reactor pressure increases (see Figure 4) with the proceeding of decomposition due to the gas expanding according to the reaction (Equation 1). The effect of activity of catalyst on decomposition is shown in Figure 5. The reaction rate with activity of catalyst constant is obvious larger than that with deactivation of catalyst. Solved with COMSOL Multiphysics 4.4 7 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Figure 4: Reactor pressure during methane decomposition. Figure 5: Comparison of concentration transients under two conditions of catalyst: 1) deactivation; 2) constant activity (1). Solved with COMSOL Multiphysics 4.4 8 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S S PA C E - D E P E N D E N T MO D E L This model concerns the stationary space-dependent model in which the Free and Porous Media Flow physics interface is used. Figure 6 shows the parabolic velocity profile (2D) in the reactor. The flow velocity of reacting gas is reduced from 0.45 mm/s to about 0.30 mm/s due to the limited permeability in the porous domain while it is almost 0.45 mm/s in other two free channel sections (Figure 7). As the same reason as that for velocity, there is moderate pressure drop along the catalytic bed (see Figure 8 and Figure 9). Figure 6: Velocity flow field in the 2D reactor under stationary state. Solved with COMSOL Multiphysics 4.4 9 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Figure 7: The velocity distribution along the center line of reactor under stationary state. Figure 8: Distribution of pressure in the reactor for non-reacting gas passing through a clean catalyst bed. Solved with COMSOL Multiphysics 4.4 10 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Figure 9: Pressure drop along the packed catalyst bed. S PA C E - A N D T I ME - D E P E N D E N T MO D E L The following results concern a space- and time-dependent model simulated in a fully coupled model consisting of Reaction Engineering, Transport of Diluted Species, Free and Porous Media Flow, and Domain ODEs and DAEs physics interfaces. The reacting gas passes through the catalytic bed whose permeability is correlated to its porosity as in Figure 6. The fully coupled model is solved by using the result from stationary Free and Porous Media Flow as the initial values. Figure 10 shows the 2D profile for the concentration of methane at reacting time 4000 s. Figure 11 shows the comparison of concentration distributions for methane and hydrogen at different reacting times. At reacting time 4000 s, the methane and hydrogen concentrations are 12.5 and 5 mol/m 3 at the end of catalytic bed, respectively. The average residence time in the catalytic bed is 400 mm/0.3 (mm/s) which equals 1333 s. In Figure 2, the methane and hydrogen concentrations are also about 12.5 and 5 mol/m 3 at the reacting time 1333 s. This means the contribution of diffusion to the mass transfer is negligible compared to convection. Solved with COMSOL Multiphysics 4.4 11 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Figure 12 shows the velocity field. It is very similar to the velocity field under stationary state (Figure 6). Figure 13 shows the pressure distribution in reactor. The pressure drop is obvious larger than that under stationary state (Figure 8). In the simulation condition, the effect of carbon deposition on porosity is significant (Figure 14). Figure 15 shows the permeability distribution along the catalytic bed at different reacting times. Figure 10: Concentration distribution of methane as a function of the bed position at reacting time 4000 s. Solved with COMSOL Multiphysics 4.4 12 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Figure 11: Concentration distribution of CH 4 and H 2 along the center line of reactor under fully coupled physics interfaces at different reacting times. Figure 12: Velocity flow field in the 2D reactor at reacting time = 4000 s. Solved with COMSOL Multiphysics 4.4 13 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Figure 13: Distribution of pressure in the reactor for reacting gas passing through a clean catalyst under transient state at reacting time 4000 s. Figure 14: Porosity distribution in the 2D reactor after reacting time 4000 s. Solved with COMSOL Multiphysics 4.4 14 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Figure 15: Permeability distribution along the center line of reactor at different reacting times References 1. S.G. Zavarukhin and G.G. Kuvshinov, “The kinetic model of formation of nanofibrous carbon from CH 4 –H 2 mixture over a high-loaded nickel catalyst with consideration for the catalyst deactivation,” J. Appl. Catal. A, vol. 272, pp. 219–227, 2004. 2. E.A. Borisova and P.M. Adler, “Deposition in porous media and clogging on the field scale,” Phys. Rev. E, vol. 71, p. 016311-1, 2005. Model Library path: Chemical_Reaction_Engineering_Module/ Heterogeneous_Catalysis/carbon_deposition Modeling Instructions From the File menu, choose New. Solved with COMSOL Multiphysics 4.4 15 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Time Dependent. 6 Click the Done button. R E A C T I O N E N G I N E E R I N G Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type CH4=>C+2H2. Species 1 Add a special species representing the catalyst activity. 1 On the Physics toolbar, click Global and choose Species. 2 In the Species settings window, locate the Species Formula section. 3 In the edit field, type a. When a new species is created a mass balance equation is set up along with it. In this case: the left-hand side is defined internally in the software and the right-hand side corresponds to the expression given in the R edit field. Note also that you can remove the effect of catalyst activity from your model by selecting the Lock concentration/ activity check box.This removes the species mass balance and sets the concentration of the species to the value entered in the Initial concentration edit field. G L O B A L D E F I N I T I O N S Load the model parameters from a text file. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. Solved with COMSOL Multiphysics 4.4 16 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S 4 Browse to the model’s Model Library folder and double-click the file carbon_deposition_parameters.txt. D E F I N I T I O N S Load the model variables from a text file. Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file carbon_deposition_variables.txt. R E A C T I O N E N G I N E E R I N G 1: CH4=>C+2H2 1 In the Model Builder window, under Component 1>Reaction Engineering click 1: CH4=>C+2H2. 2 In the Reaction settings window, locate the Reaction Rate section. 3 From the Reaction rate list, choose User Defined. 4 In the r edit field, type c_a*k*(p_CH4-p_H2^2/Kp)/(1+kH*sqrt(p_H2))^2. Species: CH4 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: CH4. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type c_CH4in. Species: H2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: H2. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type c_H2in. Species: a 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: a. Solved with COMSOL Multiphysics 4.4 17 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 1. 4 From the Rate expression list, choose User Defined. 5 In the R edit field, type -ka*r_1^2*c_C*c_a. 6 In the Model Builder window, click Reaction Engineering. 7 In the Reaction Engineering settings window, locate the General section. 8 In the T edit field, type 850[K]. 9 In the p edit field, type R_const*T*(c_CH4+c_H2). S T U D Y 1 Step 1: Time Dependent 1 In the Model Builder window, under Study 1 click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type range(0,400,4000). 4 On the Home toolbar, click Compute. R E S U L T S Concentration (re) 1 In the Model Builder window, under Results right-click Concentration (re) and choose Rename. 2 Go to the Rename 1D Plot Group dialog box and type Concentration (CH4, C, H2, re) in the New name edit field. 3 Click OK. Concentration (CH4, C, H2, re) 1 In the Model Builder window, expand the Results>Concentration (CH4, C, H2, re) node, then click Global 1. 2 In the Global settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Reaction Engineering>Concentration (comp1.re.c_CH4). 3 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Reaction Engineering>Concentration (comp1.re.c_C). 4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Reaction Engineering>Concentration (comp1.re.c_H2). Solved with COMSOL Multiphysics 4.4 18 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S 5 On the 1D plot group toolbar, click Plot. 1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Model Builder window, under Results right-click 1D Plot Group 2 and choose Rename. 3 Go to the Rename 1D Plot Group dialog box and type Activity(a, re) in the New name edit field. 4 Click OK. Activity(a, re) 1 On the 1D plot group toolbar, click Global. 2 In the Global settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Reaction Engineering>Concentration (comp1.re.c_a). 3 Locate the y-Axis Data section. In the table, enter the following settings: 4 Click to expand the Legends section. Clear the Show legends check box. 5 On the 1D plot group toolbar, click Plot. Plot the pressure in the reactor versus reacting time. 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Model Builder window, under Results right-click 1D Plot Group 3 and choose Rename. 3 Go to the Rename 1D Plot Group dialog box and type Pressure(re) in the New name edit field. 4 Click OK. Pressure(re) 1 On the 1D plot group toolbar, click Global. 2 In the Global settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Reaction Engineering>Pressure (comp1.re.p). 3 Click to expand the Legends section. Clear the Show legends check box. Expression Unit Description comp1.re.c_a mol/m^3 Activity Solved with COMSOL Multiphysics 4.4 19 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S 4 On the 1D plot group toolbar, click Plot. R E A C T I O N E N G I N E E R I N G Now study the reaction when the catalyst activity is held constant (initial value). Species: a 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: a. 2 In the Species settings window, locate the General Expressions section. 3 Select the Lock concentration/activity check box. R O O T On the Home toolbar, click Add Study. A D D S T U D Y 1 Go to the Add Study window. 2 Find the Studies subsection. In the tree, select Preset Studies>Time Dependent. 3 In the Add study window, click Add Study. S T U D Y 2 Step 1: Time Dependent 1 In the Model Builder window, under Study 2 click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type range(0,400,4000). 4 On the Home toolbar, click Compute. R E S U L T S Concentration (re) Compare the concentrations between locked and unlocked species a. 1 In the Model Builder window, under Results right-click Concentration (re) and choose Rename. 2 Go to the Rename 1D Plot Group dialog box and type Concentration comparison (re) in the New name edit field. 3 Click OK. Solved with COMSOL Multiphysics 4.4 20 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Concentration comparison (re) 1 In the Model Builder window, expand the Results>Concentration comparison (re) node, then click Global 1. 2 In the Global settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Concentration (comp1.re.c_CH4). 3 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Concentration (comp1.re.c_C). 4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Concentration (comp1.re.c_H2). 5 Locate the Legends section. From the Legends list, choose Manual. 6 In the table, enter the following settings: 7 Right-click Results>Concentration comparison (re)>Global 1 and choose Duplicate. 8 In the Global settings window, locate the Data section. 9 From the Data set list, choose Solution 1. 10 Click to expand the Title section. From the Title type list, choose Manual. 11 In the Title text area, type Concentration comparison. 12 Locate the Legends section. In the table, enter the following settings: 13 On the 1D plot group toolbar, click Plot. R E A C T I O N E N G I N E E R I N G Synchronize the Reaction Engineering physics with mass and momentum transport physics interfaces to generate a time- and space-dependent model. Legends CH4: locked C: locked H2: locked Legends CH4: unlocked C: unlocked H2: unlocked Solved with COMSOL Multiphysics 4.4 21 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Species: a 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: a. 2 In the Species settings window, locate the General Expressions section. 3 Clear the Lock concentration/activity check box. Generate Space-Dependent Model 1 1 In the Model Builder window, right-click Reaction Engineering and choose Generate Space-Dependent Model. 2 In the Generate Space-Dependent Model settings window, locate the Geometry Settings section. 3 From the Geometry to use list, choose 2D: New. 4 Locate the Physics Interfaces section. From the Momentum balance list, choose Free and Porous Media Flow: New. 5 Select the Create inflow and outflow features check box. 6 Locate the Study Type section. From the Study type list, choose Time dependent. 7 Locate the Space-Dependent Model Generation section. Click the Create/Refresh button. C O MP O N E N T 2 Set up the geometry as a union of two rectangles. G E O ME T R Y 1 In the Model Builder window, expand the Component 2 node. Rectangle 1 1 Right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Height edit field, type 0.1. Rectangle 2 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 0.4. 4 In the Height edit field, type 0.1. 5 Locate the Position section. In the x edit field, type 0.4. Solved with COMSOL Multiphysics 4.4 22 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S 6 Click the Build All Objects button. C O MP O N E N T 2 Add the Domain ODE and DAE interface for modeling the porosity change in the porous domain. On the Home toolbar, click Add Physics. A D D P HY S I C S 1 Go to the Add Physics window. 2 In the Add physics tree, select Mathematics>ODE and DAE Interfaces>Domain ODEs and DAEs (dode). 3 In the Add physics window, click Add to Component. D O MA I N O D E S A N D DA E S 1 In the Model Builder window, under Component 2 click Domain ODEs and DAEs. 2 Select Domain 2 only. 3 In the Domain ODEs and DAEs settings window, click to expand the Dependent variables section. 4 Locate the Dependent Variables section. In the Field name edit field, type por. 5 In the Dependent variables table, enter the following settings: Distributed ODE 1 1 In the Model Builder window, under Component 2>Domain ODEs and DAEs click Distributed ODE 1. 2 In the Distributed ODE settings window, locate the Source Term section. 3 In the f edit field, type -k_por*por*root.comp1.re.r_1*M_C/rho_soot. Initial Values 1 1 In the Model Builder window, under Component 2>Domain ODEs and DAEs click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the por edit field, type por0. por Solved with COMSOL Multiphysics 4.4 23 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S TR A N S P O R T O F D I L U T E D S P E C I E S 1 Convection and Diffusion 1 1 In the Model Builder window, expand the Component 2>Transport of Diluted Species 1 node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Diffusion section. 3 In the D cCH4 edit field, type D_CH4. 4 In the D cH2 edit field, type D_H2. Reactions 1 In the Model Builder window, under Component 2>Transport of Diluted Species 1 click Reactions. 2 Select Domain 2 only. Inflow 1 1 In the Model Builder window, under Component 2>Transport of Diluted Species 1 click Inflow 1. 2 Select Boundary 1 only. Outflow 1 1 In the Model Builder window, under Component 2>Transport of Diluted Species 1 click Outflow 1. 2 Select Boundary 10 only. Convection and Diffusion 2 1 On the Physics toolbar, click Domains and choose Convection and Diffusion. 2 Select Domain 2 only. 3 In the Convection and Diffusion settings window, locate the Model Inputs section. 4 From the u list, choose Velocity field (fp/fp1). 5 Locate the Diffusion section. In the D cCH4 edit field, type k_eff*D_CH4. 6 In the D cH2 edit field, type k_eff*D_H2. F R E E A N D PO R O U S ME D I A F L OW 1 1 In the Model Builder window, under Component 2 click Free and Porous Media Flow 1. 2 In the Free and Porous Media Flow settings window, locate the Physical Model section. 3 From the Compressibility list, choose Compressible flow (Ma<0.3). Solved with COMSOL Multiphysics 4.4 24 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Fluid Properties 1 1 In the Model Builder window, expand the Free and Porous Media Flow 1 node, then click Fluid Properties 1. 2 In the Fluid Properties settings window, locate the Fluid Properties section. 3 From the µ list, choose User defined. In the associated edit field, type rho. 4 From the µ list, choose User defined. In the associated edit field, type eta. Inlet 1 1 In the Model Builder window, under Component 2>Free and Porous Media Flow 1 click Inlet 1. 2 Select Boundary 1 only. 3 In the Inlet settings window, locate the Boundary Condition section. 4 From the Boundary condition list, choose Laminar inflow. 5 Locate the Laminar Inflow section. In the U av edit field, type u_in. Outlet 1 1 In the Model Builder window, under Component 2>Free and Porous Media Flow 1 click Outlet 1. 2 Select Boundary 10 only. Porous Matrix Properties 1 1 On the Physics toolbar, click Domains and choose Porous Matrix Properties. 2 Select Domain 2 only. 3 In the Porous Matrix Properties settings window, locate the Porous Matrix Properties section. 4 From the c p list, choose User defined. In the associated edit field, type por. 5 From the k list, choose User defined. In the associated edit field, type kappa. D E F I N I T I O N S Variables 2 1 In the Model Builder window, under Component 2 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. Solved with COMSOL Multiphysics 4.4 25 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S 3 In the table, enter the following settings: ME S H 1 Build a mesh with Boundary Layers. Size 1 In the Model Builder window, under Component 2 right-click Mesh 1 and choose Boundary Layers. 2 In the Size settings window, locate the Element Size section. 3 From the Calibrate for list, choose Fluid dynamics. Boundary Layer Properties 1 In the Model Builder window, under Component 2>Mesh 1>Boundary Layers 1 click Boundary Layer Properties. 2 Select Boundaries 2–6, 8, and 9 only. 3 In the Boundary Layer Properties settings window, locate the Boundary Layer Properties section. 4 In the Number of boundary layers edit field, type 4. 5 Click the Build All button. S T U D Y 3 Set up a stationary study for the Free and Porous Media Flow physics. The result is also used later as initial values for the fully coupled model. Step 2: Stationary 1 On the Study toolbar, click Study Steps and choose Stationary>Stationary. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Name Expression Unit Description kappa kappa0*(por/ por0)^3.55 m² permeability Physics Solve for Discretization Reaction Engineering × physics Transport of Diluted Species 1 × physics Domain ODEs and DAEs × physics Solved with COMSOL Multiphysics 4.4 26 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Step 2: Time Dependent 1 Right-click Study 3>Step 2: Stationary and choose Move Up. 2 In the Model Builder window, under Study 3 right-click Step 2: Time Dependent and choose Disable. 3 On the Home toolbar, click Compute. R E S U L T S Velocity (fp1) 1 In the Model Builder window, under Results right-click Velocity (fp1) and choose Rename. 2 Go to the Rename 2D Plot Group dialog box and type Stationary velocity (fp1) in the New name edit field. 3 Click OK. Stationary velocity (fp1) Plot an arrow surface of the velocity field. 1 Right-click Results>Velocity (fp1) and choose Arrow Surface. 2 In the Arrow Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Free and Porous Media Flow 1>Velocity field (u,v). 3 Click to expand the Title section. From the Title type list, choose None. 4 Locate the Arrow Positioning section. In the Points edit field, type 10. 5 Locate the Coloring and Style section. Select the Scale factor check box. 6 In the associated edit field, type 100. 7 On the 2D plot group toolbar, click Plot. Data Sets Create a dataset for 1D plotting. 1 On the Results toolbar, click Cut Line 2D. 2 In the Cut Line 2D settings window, locate the Line Data section. 3 In row Point 1, set y to 0.05. 4 In row Point 2, set y to 0.05. 1D Plot Group 7 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. Solved with COMSOL Multiphysics 4.4 27 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S 2 In the Model Builder window, under Results right-click 1D Plot Group 7 and choose Rename. 3 Go to the Rename 1D Plot Group dialog box and type Velocity(center line, fp1) in the New name edit field. 4 Click OK. 5 In the 1D Plot Group settings window, locate the Data section. 6 From the Data set list, choose Cut Line 2D 1. Velocity(center line, fp1) 1 On the 1D plot group toolbar, click Line Graph. 2 In the Line Graph settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Free and Porous Media Flow 1>Velocity magnitude (fp.U). 3 On the 1D plot group toolbar, click Plot. 1D Plot Group 8 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Model Builder window, under Results right-click 1D Plot Group 8 and choose Rename. 3 Go to the Rename 1D Plot Group dialog box and type Pressure(center line, fp1) in the New name edit field. 4 Click OK. 5 In the 1D Plot Group settings window, locate the Data section. 6 From the Data set list, choose Cut Line 2D 1. 7 Click to expand the Axis section. Select the Manual axis limits check box. 8 In the x minimum edit field, type 0.4. 9 In the x maximum edit field, type 0.8. 10 In the y minimum edit field, type 0. 11 In the y maximum edit field, type 1.2. Pressure(center line, fp1) 1 On the 1D plot group toolbar, click Line Graph. 2 In the Line Graph settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Free and Porous Media Flow 1>Pressure (p). 3 On the 1D plot group toolbar, click Plot. Solved with COMSOL Multiphysics 4.4 28 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S A D D S T U D Y Now study the fully coupled model consisting of the Reaction Engineering, Transport of Diluted Species, Free and Porous Media Flow, Domain ODEs and DAEs interfaces. 1 Go to the Add Study window. 2 Find the Studies subsection. In the tree, select Preset Studies>Time Dependent. 3 In the Add study window, click Add Study. S T U D Y 4 Step 2: Stationary 1 On the Study toolbar, click Study Steps and choose Stationary>Stationary. 2 In the Model Builder window, under Study 4 right-click Step 2: Stationary and choose Move Up. 3 In the Stationary settings window, locate the Physics and Variables Selection section. 4 In the table, enter the following settings: Step 2: Time Dependent 1 In the Model Builder window, under Study 4 click Step 2: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type range(0,400,4000). Solver 4 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solver 4 node, then click Dependent Variables 2. 3 In the Dependent Variables settings window, locate the Scaling section. 4 From the Method list, choose Manual. 5 In the Scale edit field, type 10. 6 On the Home toolbar, click Compute. Physics Solve for Discretization Reaction Engineering × physics Transport of Diluted Species 1 × physics Domain ODEs and DAEs × physics Solved with COMSOL Multiphysics 4.4 29 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S R E S U L T S Data Sets 1 On the Results toolbar, click Cut Line 2D. 2 In the Cut Line 2D settings window, locate the Data section. 3 From the Data set list, choose Solution 4. 4 Locate the Line Data section. In row Point 1, set y to 0.05. 5 In row Point 2, set y to 0.05. 1D Plot Group 13 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Model Builder window, under Results right-click 1D Plot Group 13 and choose Rename. 3 Go to the Rename 1D Plot Group dialog box and type Concentration 1D (chds1) in the New name edit field. 4 Click OK. 5 In the 1D Plot Group settings window, locate the Data section. 6 From the Data set list, choose Cut Line 2D 2. 7 From the Time selection list, choose Manual. 8 In the Time indices (1-11) edit field, type 2,3,11. 9 Locate the Axis section. Select the Manual axis limits check box. 10 In the x minimum edit field, type 0.4. 11 In the x maximum edit field, type 0.8. 12 In the y minimum edit field, type 0. 13 In the y maximum edit field, type 15. Concentration 1D (chds1) Plot the concentration distributions for CH4 and H2 at different reacting times. 1 On the 1D plot group toolbar, click Line Graph. 2 In the Line Graph settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Transport of Diluted Species 1>Species cCH4>Concentration (cCH4). 3 Click to expand the Legends section. Select the Show legends check box. 4 From the Legends list, choose Manual. Solved with COMSOL Multiphysics 4.4 30 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S 5 In the table, enter the following settings: 6 On the 1D plot group toolbar, click Line Graph. 7 In the Line Graph settings window, locate the Data section. 8 From the Data set list, choose Cut Line 2D 2. 9 From the Time selection list, choose Manual. 10 In the Time indices (1-11) edit field, type 2,3,11. 11 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Transport of Diluted Species 1>Species cH2>Concentration (cH2). 12 Locate the Legends section. Select the Show legends check box. 13 From the Legends list, choose Manual. 14 In the table, enter the following settings: 15 On the 1D plot group toolbar, click Plot. Velocity (fp1) 1 In the Model Builder window, under Results right-click Velocity (fp1) and choose Arrow Surface. 2 In the Arrow Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Free and Porous Media Flow 1>Velocity field (u,v). 3 Locate the Title section. From the Title type list, choose None. 4 Locate the Arrow Positioning section. In the Points edit field, type 10. 5 Locate the Coloring and Style section. Select the Scale factor check box. 6 In the associated edit field, type 100. 7 On the 2D plot group toolbar, click Plot. Legends 400 CH4 800 CH4 4000 CH4 Legends 400 H2 800 H2 4000 H2 Solved with COMSOL Multiphysics 4.4 31 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Pressure (fp1) 1 This is the distribution of pressure drop in porous media section of reactor at reacting time = 4000 s. 1 In the Model Builder window, expand the Results>Pressure (fp1) 1 node. 2D Plot Group 12 This is the porosity distribution in porous media section of reactor at reacting time = 4000 s. 1D Plot Group 14 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Model Builder window, under Results right-click 1D Plot Group 14 and choose Rename. 3 Go to the Rename 1D Plot Group dialog box and type Permeability (center line, ODE and DAE) in the New name edit field. 4 Click OK. 5 In the 1D Plot Group settings window, locate the Data section. 6 From the Data set list, choose Cut Line 2D 2. 7 From the Time selection list, choose Manual. 8 In the Time indices (1-11) edit field, type 2,3,5,6,11. Permeability (center line, ODE and DAE) 1 On the 1D plot group toolbar, click Line Graph. 2 In the Line Graph settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Definitions>permeability (kappa). 3 Locate the Legends section. Select the Show legends check box. 4 From the Legends list, choose Manual. 5 In the table, enter the following settings: 6 On the 1D plot group toolbar, click Plot. Legends 400 s 800 s 1600 s 2000 s 4000 s Solved with COMSOL Multiphysics 4.4 32 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S Solved with COMSOL Multiphysics 4.4 1 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E Homoge ne ous Char ge Compr e s s i on I gni t i on of Me t hane Introduction Homogeneous Charge Compression Ignition (HCCI) engines are being considered as an alternative to traditional spark- and compression-ignition engines. As the name implies, a homogeneous fuel/oxidant mixture is autoignited by compression with simultaneous combustion occurring throughout the cylinder volume. Combustion temperatures under lean burn operation are relatively low, resulting in low levels of NOx emission. Furthermore, the fuel’s homogeneous nature as well as the combustion process itself lead to low levels of particulate matter being produced. Although HCCI combustion shows much promise, the method also suffers from a number of recurring problems, one of the more important being ignition timing. The following model examines the HCCI of methane, investigating ignition trends as a function of initial temperature, initial pressure, and fuel additives. This model solves the mass and energy balances describing the detailed combustion of methane in a variable-volume system. The large amount of kinetic and thermodynamic data required to set up the problem is readily made available by importing relevant files into the Reaction Engineering interface. Model Definition It is difficult to form the uniform mixtures required for HCCI with conventional diesel fuel. Natural-gas fuels, on the other hand, readily produce homogeneous mixtures and have the potential to serve as HCCI fuels. This example considers the combustion of methane, as described by the GRI-3.0 mechanism, incorporating a detailed reaction mechanism of 53 species taking part in 325 reactions. The files describing the reaction kinetics and thermodynamics of the GRI-3.0 mechanism are available on the Internet (Ref. 1), and you can import these files directly into the Reaction Engineering interface VA R I A B L E VO L U ME R E A C T O R This model represents the combustion cylinder with a perfectly mixed batch system of variable volume, a reactor type that is predefined in the Reaction Engineering interface. Figure 1 shows a drawing of an engine cylinder, and it points out parameters Solved with COMSOL Multiphysics 4.4 2 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E relevant for calculating the instantaneous cylinder volume. Figure 1: The volume of a combustion cylinder can be expressed as a function of time with the slider-crank relationship. This diagram shows the key geometric parameters. La is the length of the crank arm, Lc gives the length of the connecting rod, D equals the cylinder diameter, and o is the crank angle. The volume change as a function of time is described by the slider-crank equation: (1) where, V is the cylinder volume (m 3 ), V c gives the clearance volume (m 3 ), CR equals the compression ratio, and R denotes the ratio of the connecting rod to the crank arm (L c /L a ). Further, o is the crank angle (rad), which is also a function of time (2) where N is the engine speed in rpm, and t is the time (s). The engine specifications used in the model are: ENGINE SPECIFICATION VARIABLE NAME VALUE Bore D 13 cm Stroke S 16 cm Connecting rod Lc 26.93 cm Crank arm La 8 cm Engine speed N 1500 rpm Compression ratio CR 15 o La Lc D V V c ------ 1 CR 1 – ( ) 2 ----------------------- R 1 o R 2 o sin ( ) 2 – – cos – + | | + = o 2tN 60 ------------t = Solved with COMSOL Multiphysics 4.4 3 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E Equation 3 includes the clearance volume, V c , which is calculated from (3) V s is the volume swept by the piston during a cycle from the equation (4) Figure 2 shows the calculated cylinder volume as a function of the crank angle. The piston is initially at bottom dead center (BDC), corresponding to a crank angle of ÷180 degrees. Figure 2: Cylinder volume as function of crank angle. The crank angle is defined as being zero at top dead center (TDC). MA S S A N D E N E R G Y B A L A N C E S The mass balances describing a perfectly mixed reactor with variable volume are summarized by (5) V c V s CR 1 – ( ) ----------------------- = V s tD 2 4 -----------S = d Vc i ( ) dt ----------------- VR i = Solved with COMSOL Multiphysics 4.4 4 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E where c i represents the species concentration (mol/m 3 ), and R i denotes the species rate expression (mol/(m 3 ·s)). For an ideal gas mixture, the reactor energy balance is (6) where C p,i is the species molar heat capacity (J/(mol·K)), T is the temperature (K), and p gives the pressure (Pa). In this equation, Q is the heat due to chemical reaction (J/s) (7) where H j is the enthalpy of reaction (J/(mol·K)), and r j equals the reaction rate (mol/ (m 3 ·s)). Q ext denotes heat added to the system (J/s). The model being described assumes adiabatic conditions, that is, Q ext = 0. The kinetic and thermodynamic data for methane combustion is available in the form of data input files. Once the files are imported into the Reaction Engineering interface, the software automatically sets up the mass and energy balances detailed in Equation 5 and Equation 6. To complete the model setup, all that remains is to define the initial conditions. In this model, methane is combusted under lean conditions, that is, supplying more than the stoichiometric amount of oxidizer. The stoichiometric requirement of the oxidizer (air) to combust methane is found from the overall reaction: (8) Assuming that the composition of air is 21% oxygen and 79% nitrogen, the stoichiometric air-fuel ratio is (9) The equivalence ratio relates the actual air-fuel ratio to the stoichiometric requirements (10) This model sets the equivalence ratio to u = 0.5. V r c i C p i , dT dt -------- i ¿ Q Q + ext V r dp dt ------- + = Q V r H j r j j ¿ – = CH 4 + O 2 N 2 CO 2 H 2 O N 2 + ( ) 2 2 7.52 3.76 + + A F ( ) stoic m air m fuel -------------- \ . | | stoic 4.76 2 M air · · 1 M fuel · ------------------------------------ = = u A F ( ) stoic A F ( ) ---------------------------- = Solved with COMSOL Multiphysics 4.4 5 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E From Equation 9 and Equation 10 it is possible to calculate the molar fraction of fuel in the reacting mixture as (11) and subsequently the initial concentration is (12) The initial pressure and the initial temperature are variable model parameters. Results and Discussion Figure 3 shows the cylinder pressure as a function of time when a methane-air mixture is compressed and ignites. The piston starts at bottom dead center (BDC) and reaches top dead center (TDC) after 0.02 s. At BDC the pressure is set to 1.5·10 5 Pa, u is 0.5, and the compression ratio is CR = 15. The initial temperature is varied from 400 K to 800 K. Figure 3: Pressure traces illustrating the compression and ignition of fuel in an engine cylinder. The initial temperature varies between 400 K and 800 K. x fuel 1 4.76 2 · u 1 + ------------------------------------- = c fuel x fuel p init RgT init ------------------------- = Solved with COMSOL Multiphysics 4.4 6 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E Consistent with literature results, methane does not ignite at an initial temperature of 400 K (Ref. 2). Furthermore, the induction delay decreases with increasing initial temperature. The induction delay time can be evaluated from the pressure gradient. For instance, the induction delay is 0.0193 s when T init = 500 K. Figure 4 illustrates the pressure traces as the initial pressure varies from 1·10 5 Pa to 3·10 5 Pa. The initial temperature is 500 K. An increase in pressure means an increase in the species concentrations in the fuel-air mixture, resulting in the expected advance in ignition times. Figure 4: Increased initial gas pressure advances ignition times. As mentioned previously, a significant challenge to the realization of HCCI engines is ignition control. In this regard, combustion at TDC has been suggested as the optimum timing (Ref. 3). The results just discussed show that the inlet temperature of the fuel-air mixture is a potential tuning parameter for ignition. However, relatively high inlet temperatures are often required for proper timing. This adversely affects engine performance because the trapped mass as well as the volumetric efficiency decreases. An alternative that facilitates ignition is to mix small amounts of additives into the fuel-air mixture (Ref. 4). These additives chemically activate the reaction mixture even at relatively low temperatures. This approach alleviates the requirements of high intake temperatures. Figure 5 shows how small amounts of formaldehyde Solved with COMSOL Multiphysics 4.4 7 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E (CH 2 O) cause ignition at an initial temperature of 400 K, a temperature insufficient to induce combustion with a pure methane fuel. Figure 5: Small amounts of formaldehyde stimulate ignition of the fuel-air mixture. The increased reactivity observed in the presence of CH 2 O is explained by the opening of a new chemical pathway leading to the formation of hydroxyl radicals. Specifically, CH 2 O reacts with O 2 to produce H 2 O 2 : H 2 O 2 , in turn, decomposes to reactive OH radicals, which subsequently react violently with the fuel molecules to cause ignition: The results in the following graphs show the species molar fractions of CH 2 O, HO 2 , H 2 O 2 , and OH during the combustion of methane. Figure 6 shows molar fraction plots for the case when 0.13% CH 2 O is added to the fuel; Figure 7 is the equivalent species plots for the case when pure methane is combusted. In each case conditions O 2 + CH 2 O HO 2 CHO + + CH 2 O H 2 O 2 CHO + HO 2 + M OH M + H 2 O 2 2 Solved with COMSOL Multiphysics 4.4 8 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E have been tuned to produce ignition near TDC so as to provide a reference point for comparing the species concentrations. Figure 6: Selected species molar fractions as a function of crank angle. 0.26 molar percent CH 2 O is added to the reacting mixture, which is initially at 400 K and 1.5 bar. Solved with COMSOL Multiphysics 4.4 9 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E Figure 7: Selected species molar fraction as a function of crank angle. Only methane is combusted. The initial temperature is 469 K and the initial pressure 1.5 bar. The implications of the CH 2 O reaction path just outlined are directly visible by comparing Figure 6 and Figure 7; CH 2 O stimulates the production of HO 2 and H 2 O 2 that in turn produce OH radicals in amounts critical to fuel ignition. References 1. G.P. Smith, D.M. Golden, M. Frenklach, N.W. Moriarty, B. Eiteneer, M. Goldenberg, C.T. Bowman, R.K. Hanson, S. Song, W.C. Gardiner, Jr., V. V. Lissianski, and Z. Qin, GRI-MECH 3.0, http://www.me.berkeley.edu/gri_mech/. 2. S.B. Fiveland and D.N. Assanis, SAE Paper 2000-01-0332, 2000. 3. D.L. Flowers, S.M. Aceves, C.K. Westbrook, J.R. Smith, and R.W. Dibble, J. Eng. Gas Turbine Power, vol. 123, p. 433, 2001. 4. M.H. Morsy, Fuel, vol. 86, p. 533, 2007. Solved with COMSOL Multiphysics 4.4 10 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E Model Library path: Chemical_Reaction_Engineering_Module/ Homogeneous_Reactions_and_Catalysis/compression_ignition Notes about the COMSOL Implementation The kinetic and thermodynamic data required for this model are available on the Internet. Find the GRI-Mech 3.0 input files at (Ref. 1): http://www.me.berkeley.edu/gri_mech/version30/text30.html. Download the reaction mechanism and rate coefficient file (grimech30.dat), as well as thermodynamic data file (thermo30.dat) and store them on your computer. Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Time Dependent. 6 Click the Done button. R E A C T I O N E N G I N E E R I N G 1 In the Model Builder window, under Component 1 click Reaction Engineering. 2 In the Reaction Engineering settings window, click to expand the CHEMKIN section. Import the CHEMKIN file, grimech30.dat. 3 Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file grimech30.dat. Solved with COMSOL Multiphysics 4.4 11 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E 5 Click the Import button. Import also the thermodynamic data file, thermo30.dat. 6 Click the Browse button. 7 Browse to the model’s Model Library folder and double-click the file thermo30.dat. 8 Click the Import button. G L O B A L D E F I N I T I O N S Add the parameter and variables to use for this model from text files. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file compression_ignition_parameters.txt. D E F I N I T I O N S Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file compression_ignition_variables.txt. R E A C T I O N E N G I N E E R I N G The model is a batch reactor, with a variable volume. 1 In the Model Builder window, under Component 1 click Reaction Engineering. 2 In the Reaction Engineering settings window, locate the Reactor Settings section. 3 From the Reactor type list, choose Batch. 4 Locate the Mass Balance section. In the V r edit field, type Vol. Energy Balance Add an energy balance equation to the model, set the initial temperature. Solved with COMSOL Multiphysics 4.4 12 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E 1 Right-click Component 1>Reaction Engineering and choose Energy Balance. 2 In the Energy Balance settings window, locate the Energy Balance section. 3 In the T 0 edit field, type T_init. Now set the initial concentrations. Species: O2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: O2. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type c_O2_0. Species: CH2O 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: CH2O. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type c_CH2O_0. Species: CH4 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: CH4. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type c_CH4_0. Species: N2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: N2. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type c_N2_0. Use uniform scaling of the concentration variables to improve computational performance. 4 In the Model Builder window’s toolbar, click the Show button and select Discretization in the menu. 5 In the Model Builder window, click Reaction Engineering. 6 In the Reaction Engineering settings window, click to expand the Discretization section. 7 Select the Uniform scaling of concentration variables check box. Solved with COMSOL Multiphysics 4.4 13 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E S T U D Y 1 Step 1: Time Dependent Set up the time dependent study, modify the default tolerance settings to improve the accuracy of the solution. 1 In the Model Builder window, under Study 1 click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type 0 0.026. 4 Select the Relative tolerance check box. 5 In the associated edit field, type 1e-8. Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solver 1 node, then click Time-Dependent Solver 1. 3 In the Time-Dependent Solver settings window, click to expand the Absolute tolerance section. 4 Locate the Absolute Tolerance section. In the Tolerance edit field, type 1E-9. 5 Clear the Update scaled absolute tolerance check box. 6 On the Home toolbar, click Compute. R E S U L T S 1D Plot Group 3 The following steps create a plot of the pressure versus time. 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, click to expand the Title section. 3 From the Title type list, choose None. 4 On the 1D plot group toolbar, click Global. 5 In the Global settings window, locate the y-axis data section. 6 Click Pressure (comp1.re.p) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. 1D Plot Group 4 The following steps reproduce Figure 6. 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. Solved with COMSOL Multiphysics 4.4 14 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E 2 In the 1D Plot Group settings window, click to expand the Title section. 3 From the Title type list, choose None. 4 Locate the Plot Settings section. Select the x-axis label check box. 5 In the associated edit field, type Crank angle (deg). 6 Select the y-axis label check box. 7 In the associated edit field, type Molar fraction. 8 On the 1D plot group toolbar, click Global. 9 In the Global settings window, locate the y-axis data section. 10 Click Concentration (comp1.re.c_s35) in the upper-right corner of the section. Locate the y-Axis Data section. In the table, enter the following settings: 11 Locate the x-Axis Data section. From the Parameter list, choose Expression. 12 In the Expression edit field, type comp1.crank_angle. 13 Click to expand the Legends section. Find the Include subsection. Select the Description check box. 14 Click the y-Axis Log Scale button on the Graphics toolbar. 15 On the 1D plot group toolbar, click Plot. 16 Right-click Results>1D Plot Group 4>Global 1 and choose Duplicate. 17 In the Global settings window, locate the y-Axis Data section. 18 In the table, enter the following settings: 19 Right-click Results>1D Plot Group 4>Global 2 and choose Duplicate. 20 In the Global settings window, locate the y-Axis Data section. Expression Unit Description comp1.re.c_s35/ (comp1.re.p/ (R_const*comp1.re.T)) 1 xOH Expression Unit Description comp1.re.c_s5/ (comp1.re.p/ (R_const*comp1.re.T)) 1 xH2O2 Solved with COMSOL Multiphysics 4.4 15 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E 21 In the table, enter the following settings: 22 Right-click Results>1D Plot Group 4>Global 3 and choose Duplicate. 23 In the Global settings window, locate the y-Axis Data section. 24 In the table, enter the following settings: 25 In the Model Builder window, click 1D Plot Group 4. 26 In the 1D Plot Group settings window, click to expand the Axis section. 27 Select the Manual axis limits check box. 28 In the x minimum edit field, type -30. 29 In the x maximum edit field, type 30. 30 In the y minimum edit field, type 1e-8. 31 In the y maximum edit field, type 1e-2. 32 On the 1D plot group toolbar, click Plot. The following steps reproduce Figure 7. First change the temperature and the initial CH2O concentration, then resolve. G L O B A L D E F I N I T I O N S Parameters 1 In the Model Builder window, under Global Definitions click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Expression Unit Description comp1.re.c_s12/ (comp1.re.p/ (R_const*comp1.re.T)) 1 xHO2 Expression Unit Description comp1.re.c_s40/ (comp1.re.p/ (R_const*comp1.re.T)) 1 xCH2O Name Expression Value Description T_init 469[K] 469.00 K Initial temperature at BDC x_CH2O 0 0 Initial CH2O mole fraction Solved with COMSOL Multiphysics 4.4 16 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S 1D Plot Group 4 1 On the 1D plot group toolbar, click Plot. To reproduce Figure 3, create a parametric sweep over the initial temperature parameter. S T U D Y 1 Parametric Sweep 1 On the Study toolbar, click Extension Steps and choose Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: 5 In the Model Builder window, click Study 1. 6 In the Study settings window, locate the Study Settings section. 7 Clear the Generate default plots check box. 8 On the Home toolbar, click Compute. R E S U L T S 1D Plot Group 3 1 In the 1D Plot Group settings window, locate the Data section. 2 From the Data set list, choose Solution 2. 3 On the 1D plot group toolbar, click Plot. 4 In the Model Builder window, click 1D Plot Group 3. 5 In the 1D Plot Group settings window, locate the Axis section. 6 Select the Manual axis limits check box. 7 In the x minimum edit field, type 0.012. 8 In the x maximum edit field, type 0.026. Parameter names Parameter value list T_init 400 450 500 600 800 Solved with COMSOL Multiphysics 4.4 17 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E 9 In the y minimum edit field, type 0. 10 In the y maximum edit field, type 1.2e7. 11 On the 1D plot group toolbar, click Plot. G L O B A L D E F I N I T I O N S Parameters To reproduce Figure 4, sweep instead over the initial pressure parameter. Set the initial temperature to 500 K first. 1 In the Model Builder window, under Global Definitions click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: S T U D Y 1 Parametric Sweep 1 In the Model Builder window, under Study 1 click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 In the table, enter the following settings: 4 On the Home toolbar, click Compute. R E S U L T S 1D Plot Group 3 1 In the 1D Plot Group settings window, locate the Axis section. 2 In the y maximum edit field, type 2.5e7. 3 On the 1D plot group toolbar, click Plot. Name Expression Value Description T_init 500[K] 500.00 K Initial temperature at BDC Parameter names Parameter value list p_init {1 1.5 2 3}*1e5 Solved with COMSOL Multiphysics 4.4 18 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E G L O B A L D E F I N I T I O N S Parameters To reproduce Figure 5, sweep instead over the initial CH2O mole fraction. Set the initial temperature to 400 K first. 1 In the Model Builder window, under Global Definitions click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: S T U D Y 1 Parametric Sweep 1 In the Model Builder window, under Study 1 click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 In the table, enter the following settings: 4 On the Home toolbar, click Compute. R E S U L T S 1D Plot Group 3 1 In the 1D Plot Group settings window, locate the Axis section. 2 In the y maximum edit field, type 1.8e7. 3 On the 1D plot group toolbar, click Plot. Name Expression Value Description T_init 400[K] 400.00 K Initial temperature at BDC Parameter names Parameter value list x_CH2O 0 0.001 0.01 0.05 Solved with COMSOL Multiphysics 4.4 1 | S T A R T U P OF A C S T R S t a r t up of a C S T R Introduction The hydrolysis of propylene oxide into propylene glycol is an important chemical process with 400,000 metric tons produced worldwide each year. Propylene glycol finds wide application as a moisturizer in foods, pharmaceuticals, and cosmetics. In this example you model the startup phase of a continuous stirred tank reactor (CSTR) used to produce propylene glycol. The nonisothermal process is described by a set of coupled mass and energy balances that you easily set up and solve in the Chemical Reaction Engineering Module. The model highlights the use of the predefined CSTR reactor type and also shows how to enter the thermodynamic data needed for energy balances. Model Description This model reproduces the results found in Ref. 1. Propylene glycol (PrOH) is produced from the reaction of propylene oxide (PrO) with water (W) in the presence of an acid catalyst: (1) The reaction rate (mol/(m 3 ·s)) is first order with respect to propylene oxide: (2) where the rate constant is temperature dependent according to the expression (3) The Arrhenius parameters in Equation 3 are A 1 = 4.71·10 9 (1/s) and E 1 = 75,358 (J/ mol). H 2 O + O H 2 SO 4 OH HO r 1 k 1 c PrO – = k 1 A 1 E 1 RgT ------------ – \ . | | exp = Solved with COMSOL Multiphysics 4.4 2 | S T A R T U P OF A C S T R The liquid phase reaction takes place in a continuous stirred tank reactor (CSTR) equipped with a heat-exchanger. Methanol (MeOH) is added to the mixture but does not react. It is further assumed that the mixture density is constant over time. Figure 1: A perfectly mixed CSTR for the production of propylene glycol. The CSTR is a predefined reactor type in the Chemical Reaction Engineering Module. The time evolution of the nonisothermal reacting system is given by several coupled balance equations. The species mass balances are (4) In Equation 4, c i is the species molar concentration (mol/m 3 ), V r denotes the reactor volume (m 3 ), R i is the species rate expression (mol/(m 3 ·s)), and v is the volumetric flow rate (m 3 /s). For an incompressible and ideally mixed reacting liquid, the energy balance is (5) where C p,i is the species molar heat capacity (J/(mol·K)), and T is the temperature (K). On the right-hand side, Q represents the heat due to chemical reaction (J/s), and Q ext denotes heat added to the system (J/s), for instance by a heat exchanger. The last term signifies heat added as species flow through the reactor. In this term, h i is the species molar enthalpy (J/mol). This example assumes that the species heat capacities, C p,i , represent an average over the temperature interval. The associated species’ enthalpies are then given by (6) where h i (T ref ) is the standard heat of formation at the reference temperature T ref . d c i V r ( ) dt -------------------- v f i , c f i , vc i R i V r + – = V r c i C p i , dT dt -------- i ¿ Q Q + ext v f i , c f i , h f i , h i – ( ) i ¿ + = h i C p i , T T ref – ( ) h i T ref ( ) + = Solved with COMSOL Multiphysics 4.4 3 | S T A R T U P OF A C S T R The heat of reaction is given by (7) where H j is the enthalpy of reaction (J/mol), and r j denotes the reaction rate (mol/(m 3 ·s)). The heat added by the heat exchanger is given by (8) where F is the molar flow rate (mol/s), U is the overall heat transfer coefficient (J/(K·m 2 ·s)), and A represents the heat exchange area (m 2 ). The subscript x refers to the heat exchanger medium, which in this case is water. T x is the inlet temperature of the heat exchanger medium. The following table summarizes additional parameters describing the reactor setup and process conditions: PARAMETER VALUE DESCRIPTION V r 1.89 m 3 Reactor volume v f 3.47·10 -3 m 3 /s Volumetric flow rate c f,PrO 2903 mol/m 3 Concentration of PrO in feed stream c f,W 36291 mol/m 3 Concentration of W in feed stream c f,MeOH 3629 mol/m 3 Concentration of MeOH in feed stream c 0,W 55273 mol/m 3 Initial concentration of W in the reactor C p,PrO 146.5 J/(mol·K) Heat capacity of PrO C p,W 75.4 J/(mol·K) Heat capacity of W C p,PrOH 192.6 J/(mol·K) Heat capacity of PrOH C p,MeOH 81.6 J/(mol·K) Heat capacity of MeOH C px 75.4 J/(mol·K) Heat capacity of heat exchanger medium h ref,PrO -153.5·10 3 J/mol Enthalpy of formation of PrO at T ref h ref,W -286.1·10 3 J/mol Enthalpy of formation of W at T ref h ref,PrOH -525.6·10 3 J/mol Enthalpy of formation of PrOH at T ref h ref,MeOH -238.6 J/mol Enthalpy of formation of MeOH at T ref T f 297 K Feed stream temperature T 0 297 K Initial reactor temperature Q V r H j r j j ¿ – = Q ext F x C p x , T x T – ( ) 1 UA F x C p x , ------------------ \ . | | exp – · = Solved with COMSOL Multiphysics 4.4 4 | S T A R T U P OF A C S T R The model described here is readily set up and solved using the predefined CSTR reactor type in the Chemical Reaction Engineering Module. Results and Discussion Figure 2 shows the concentration of PrO (mol/m 3 ) as a function of reaction time. Figure 2: Concentrations of reactant PrO (mol/m 3 ) after 4 hours of operation. The corresponding development of the reactor temperature is shown in Figure 3. T ref 293 K Reference temperature T x 289 K Temperature of heat exchanger medium at inlet F x 126 mol/s Heat exchanger medium molar flow UA 8441 J/(s·K) Heat exchange parameter PARAMETER VALUE DESCRIPTION Solved with COMSOL Multiphysics 4.4 5 | S T A R T U P OF A C S T R Figure 3: Reactor temperature (K) after 4 hours of operation (right). Initially both the reactant concentration and the temperature oscillate around their respective steady-state values (491 mol/m 3 and 336 K, respectively). The model predicts that the reactor temperature passes a maximum value higher than the steady-state temperature. From a safety perspective it is therefore relevant to look closer at possible sets of initial conditions to see if process operation limits are violated. In the process modeled here, it is undesirable to exceed a reactor temperature of 355 K to avoid undesirable side reactions and not damage reactor equipment. Figure 4 shows Solved with COMSOL Multiphysics 4.4 6 | S T A R T U P OF A C S T R the concentration-temperature phase plane for three initial condition scenarios: (c PrO = 0, T 0 = 297 K), (c PrO = 0, T 0 = 340 K), and (c PrO = 1400, T 0 = 340 K). Figure 4: Trajectories in the concentration-temperature phase plane for three sets of initial conditions. The plot shows that all investigated initial conditions converge to the same steady state. However, starting with c PrO = 1400 mol/m 3 and T 0 = 340 K leads to violation of the temperature safety limits. Reference 1. H.S. Fogler, Elements of Chemical Reaction Engineering 3rd ed., Prentice Hall PTR, Example 9-4, pp. 553–559, 1999. Model Library path: Chemical_Reaction_Engineering_Module/ Batch_Reactors/cstr_startup Safety limit Solved with COMSOL Multiphysics 4.4 7 | S T A R T U P OF A C S T R Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Time Dependent. 6 Click the Done button. R E A C T I O N E N G I N E E R I N G ( R E ) 1 In the Model Builder window, under Component 1 (comp1) click Reaction Engineering (re). 2 In the Reaction Engineering settings window, locate the Reactor Settings section. 3 From the Reactor type list, choose CSTR, constant volume. 4 From the Mixture list, choose Liquid. 5 Select the Calculate thermodynamic properties check box. Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type PrO+W=>PrOH. 4 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 5 In the A f edit field, type 4.71e9. 6 In the E f edit field, type 75358[J/mol]. 7 Locate the Reaction Rate section. From the Reaction rate list, choose User Defined. 8 In the r edit field, type kf_1*c_PrO. Species 1 1 On the Physics toolbar, click Global and choose Species. 2 In the Species settings window, locate the Species Formula section. Solved with COMSOL Multiphysics 4.4 8 | S T A R T U P OF A C S T R 3 In the edit field, type MeOH. Feed Stream 1 1 On the Physics toolbar, click Global and choose Feed Stream. 2 In the Feed Stream settings window, locate the Feed Stream Properties section. 3 In the v f edit field, type 3.47e-3[m^3/s]. 4 In the T f edit field, type Tfeed. Species: PrO 1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Species: PrO. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type c0_pro. 4 Click to expand the Species feed stream section. Locate the Species Feed Stream section. From the list, choose Feed Stream 1. 5 In the c f edit field, type 2903. 6 In the h f edit field, type hf_pro. 7 Click to expand the Species thermodynamic expressions section. Locate the Species Thermodynamic Expressions section. In the C p edit field, type cp_pro. 8 In the h edit field, type h_pro. Species: W 1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Species: W. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 55273[mol/m^3]. 4 Click to expand the Species feed stream section. Locate the Species Feed Stream section. From the list, choose Feed Stream 1. 5 In the c f edit field, type 36291. 6 In the h f edit field, type hf_w. 7 Click to expand the Species thermodynamic expressions section. Locate the Species Thermodynamic Expressions section. In the C p edit field, type cp_w. 8 In the h edit field, type h_w. Solved with COMSOL Multiphysics 4.4 9 | S T A R T U P OF A C S T R Species: PrOH 1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Species: PrOH. 2 In the Species settings window, click to expand the Species thermodynamic expressions section. 3 Locate the Species Thermodynamic Expressions section. In the C p edit field, type cp_proh. 4 In the h edit field, type h_proh. Species: MeOH 1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Species: MeOH. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. From the list, choose Feed Stream 1. 4 In the c f edit field, type 3629. 5 In the h f edit field, type hf_meoh. 6 Click to expand the Species thermodynamic expressions section. Locate the Species Thermodynamic Expressions section. In the C p edit field, type cp_meoh. 7 In the h edit field, type h_meoh. Energy Balance 1 In the Model Builder window, right-click Reaction Engineering (re) and choose Energy Balance. 2 In the Energy Balance settings window, locate the Energy Balance section. 3 In the Q ext edit field, type Q_xch. 4 In the T 0 edit field, type Tinit. 5 In the Model Builder window, click Reaction Engineering (re). 6 In the Reaction Engineering settings window, locate the Mass Balance section. 7 In the V r edit field, type 1.89. 8 In the v p edit field, type 0[m^3/s]. G L O B A L D E F I N I T I O N S Next, add a set of model parameters by importing their definitions from a data text file provided with the Model Library. Solved with COMSOL Multiphysics 4.4 10 | S T A R T U P O F A C S T R Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file cstr_startup_parameters.txt. D E F I N I T I O N S Similarly, variables for the concentration-dependent and temperature-dependent enthalpies are available in a text file. Variables 1 1 In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file cstr_startup_variables.txt. S T U D Y 1 Step 1: Time Dependent 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type range(0,20,4*3600). 4 On the Home toolbar, click Compute. R E S U L T S Concentration (re) 1 In the Model Builder window, expand the Concentration (re) node, then click Global 1. 2 In the Global settings window, locate the y-axis data section. 3 Click Concentration (comp1.re.c_PrO) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Solved with COMSOL Multiphysics 4.4 11 | S T A R T U P OF A C S T R S T U D Y 1 Next, compute the corresponding solutions for a set of initial temperatures and propylene-oxide concentrations. Parametric Sweep 1 On the Study toolbar, click Extension Steps and choose Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: 5 Click Add. 6 In the table, enter the following settings: 7 On the Home toolbar, click Compute. R E S U L T S Concentration (re) 1 1 In the Model Builder window, expand the Concentration (re) 1 node, then click Global 1. 2 In the Global settings window, locate the y-axis data section. 3 Click Concentration (comp1.re.c_PrO) in the upper-right corner of the section. Locate the x-axis data section. Click Temperature (comp1.re.T) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Parameter names Parameter value list Tinit 297[K] 340[K] 340[K] Parameter names Parameter value list c0_pro 0 0 1400[mol/m^3] Solved with COMSOL Multiphysics 4.4 12 | S T A R T U P O F A C S T R Solved with COMSOL Multiphysics 4.4 1 | S E P A R A T I O N T H R O U G H D I A L Y S I S S e par at i on T hr ough Di al y s i s Introduction Dialysis is a frequently used membrane separation process. An important application is hemodialysis, where membranes are used as artificial kidneys for people suffering from renal failure. Other applications include the recovery of caustic colloidal hemicellulose during viscose manufacturing, and the removal of alcohol from beer (Ref. 1). In the dialysis process, specific components are preferentially transported through a membrane. The process is diffusion-driven, that is, components diffuse through a membrane due to concentration differences between the dialysate and the permeate sides of the membrane. Separation between solutes is obtained as a result of differences in diffusion rates across the membrane arising from differences in molecular size and solubility. This example looks at a process aimed at lowering the concentration of a contaminant component in an aqueous product stream. The dialysis equipment is made of a hollow fiber module, where a large number of hollow fibers act as the membrane. It focuses on the transport of the contaminant in the hollow fiber and through its wall. Figure 1 shows a diagram of the hollow fiber assembly. A large number of hollow fibers are assembled in a module where the dialysate flows on the fibers’ insides while the permeate flows on their outsides in a co-current manner. The contaminant diffuses through the fiber walls to the permeate side due to a concentration gradient, whereas species with a higher molecular weight, those one wants to keep in the dialysate, are retained due to their low solubility and diffusivity in the membrane. Figure 1: Diagram of the hollow fiber module. Solved with COMSOL Multiphysics 4.4 2 | S E P A R A T I O N T H R O U G H D I A L Y S I S Model Definition This example models a piece of hollow fiber through which the dialysate flows with a fully developed laminar parabolic velocity profile. The fiber is surrounded by a permeate, which flows laminarly in the same direction as the dialysate. This example thus models three separate phases: the dialysate, the membrane, and the permeate. The model domain appears in Figure 2. Assume there are no angular gradients, so you can thus use an axisymmetrical approximation. Figure 2: Diagram of the dialysis fiber. The contaminant is transported by diffusion and convection in the two liquid phases, whereas diffusion is the only transport mechanism in the membrane phase. You can formulate the following mass transport equations to describe the system: (1) where c i denotes the concentration of the contaminant (mol/m 3 ) in the respective phases, D denotes the diffusion coefficient (m 2 /s) in the liquid phases, and D m is the diffusion coefficient in the membrane, while u denotes the velocity (m/s) in the respective liquid phase. V D – Vc 1 c 1 u + ( ) · 0 = V D – m Vc 2 ( ) · 0 = V D – Vc 3 c 3 u + ( ) · 0 = in O dialysate in O membrane in O permeate Solved with COMSOL Multiphysics 4.4 3 | S E P A R A T I O N T H R O U G H D I A L Y S I S The fiber is 75 times longer than its radial dimension, in this case 0.28 mm in radius and 21 mm in length. To avoid excessive amounts of elements and nodes you must scale the problem. Therefore introduce a new scaled z-coordinate, , and a corresponding differential for the mass transports: (2) In the mass-transport equations, c is differentiated twice in the diffusion term, which implies that the diffusive flux vector’s z-component must be multiplied by (1/scale) 2 . The convective component is only differentiated once, and therefore must be multiplied by 1/scale. You can introduce the scaling of the diffusive part of the flux as an anisotropic diffusion coefficient where the diffusion in the z direction is scaled by the factor (1/scale) 2 . This gives the following diffusion-coefficient matrix: (3) To obtain the convective part of the flux, assume fully developed laminar flow both inside and outside the hollow fiber. This allows you to introduce the velocity distributions analytically. For the interior, this example uses the following velocity distribution (Ref. 2): (4) where v z is the axial component of the velocity, v max is the maximum velocity in the axial direction, r represents the radial coordinate, and R 1 equals the inner radius of the hollow fiber. The velocity vector must be multiplied by 1/scale to account for the new scaled z-coordinate. z ˆ z ˆ z scale --------------- = dz scale dz ˆ · = D D 0 0 D scale 2 ----------------- = v z dialysate v max 1 r R 1 ------- \ . | | 2 – = Solved with COMSOL Multiphysics 4.4 4 | S E P A R A T I O N T H R O U G H D I A L Y S I S Outside the fiber the velocity profile is more complicated. You can draw a hexagonal-shaped unit cell of the fiber assembly (Figure 3): Figure 3: Hexagonal-shaped unit cell of the fiber assembly. By approximating the hexagon with a circle, you can assume that the circle indicates the permeate’s position of maximum velocity in the axial direction. In order to characterize the flow profile, the model twice integrates a momentum balance over a thin cylindrical shell (Ref. 2) to eventually get the following analytical expression for the permeate velocity distribution: (5) Here A (1/(m·s)) is a constant defined by (6) In these equations, R 2 and R 3 are the radial coordinates of the outer fiber wall and the approximated circle, respectively, q (Pa·s) is the permeate’s dynamic viscosity, and P 0 ÷ P L (Pa) represents the pressure drop over a length L. The contaminant must dissolve into the membrane phase in order to be transported through it. The interface conditions between the liquid and membrane phases for the concentration are described by the dimensionless partition coefficient, K: (7) v z ˆ permeate A r 2 R 2 2 – 2 R 3 2 r R 2 ------- \ . | | ln · · – · = A P 0 P L – 4qL scale · ------------------------------- – = K c 2 d c 1 d ----- c 2 p c 3 p ----- = = Solved with COMSOL Multiphysics 4.4 5 | S E P A R A T I O N T H R O U G H D I A L Y S I S Figure 4 shows a schematic concentration profile. Note that there are discontinuities in the concentration profile at the phase boundaries. Figure 4: Diagram of the concentration profile across the membrane (see Equation 7). To obtain a well-posed problem, you must define an appropriate set of boundary conditions; for the relevant notation, see Figure 5. Figure 5: Boundaries and boundary labels for the modeled system. At the inlet to the model domain, define concentration conditions as: Solved with COMSOL Multiphysics 4.4 6 | S E P A R A T I O N T H R O U G H D I A L Y S I S (8) At the outlet, assume that the convective contribution to the mass transport is much larger than the diffusive contribution: (9) Here n is the normal unit vector to the respective boundary. Further, assume that you have no transport over the symmetry boundaries: (10) Also assume symmetry at the horizontal boundaries of the membrane: (11) You can verify this assumption after solving the model by studying the very small vertical concentration gradient in the membrane. MO D E L D A T A The input data used in this model are listed in the following table: PROPERTY VALUE DESCRIPTION D 10 -9 m 2 /s Diffusion coefficient, liquid phases D m 10 -9 m 2 /s Diffusion coefficient, membrane R 1 0.2 mm Inner radius, hollow fiber R 2 0.28 mm Outer radius, hollow fiber R 3 0.7 mm Approximative radius, unit cell v max 1 mm/s Maximum velocity, dialysate A -2·10 -3 1/(m·s) Permeate velocity prefactor K 0.7 Partition coefficient c 0 1 M Inlet concentration, dialysate M 10 4 m/s Stiff-spring velocity scale 7 Axial coordinate scale factor c 1 c 0 = at cO d, in c 3 0 = at cO p, in D – Vc i c i u + ( ) n · c i u n at cO d,out and cO p, out · = D – Vc i c i u + ( ) n · 0 at cO d, sym and cO p, sym = D m – Vc 2 ( ) n · 0 at cO m, high and cO m, low = Solved with COMSOL Multiphysics 4.4 7 | S E P A R A T I O N T H R O U G H D I A L Y S I S Results The surface plot in Figure 6 visualizes the concentration distribution throughout the three model domains: the dialysate region inside the hollow fiber on the left side, the membrane in the middle, and the permeate to the right. As the plot shows, the concentration inside the hollow fiber decreases markedly over the first 1 mm from the inlet. The figure further shows the developing diffusion layers on both sides of the fiber wall. Figure 6: Concentration in the three domains. The figure also shows the concentration jump that arises at the boundary between the dialysate and the membrane. Also, the maximum concentration in the permeate occurs about 0.35 mm downstream from the inlet. If there is a risk of scaling on the fiber’s outer surface due to high concentration of filtrated species, it is largest at the location of this maximum. Note that this example models only a short piece at the hollow fiber’s inlet end. Using a larger scale factor you can model the fiber’s entire length. Solved with COMSOL Multiphysics 4.4 8 | S E P A R A T I O N T H R O U G H D I A L Y S I S Modeling in COMSOL Multiphysics Because there are discontinuities in the concentration profile at the boundaries between liquid and membrane phases, you must use three separate variables to describe the concentration in the respective phases. To get continuous flux over the phase boundaries, apply a special type of boundary condition using the stiff-spring method. Instead of defining Dirichlet concentration conditions according to the partition coefficient K, which would destroy the continuity of the flux, you can define continuous flux conditions that, at the same time, force the concentrations to the desired values: (12) Here M is a (nonphysical) velocity large enough to let the concentration differences in the brackets approach zero, thereby satisfying Equation 7. These boundary conditions also give a continuous flux across the interfaces provided that M is sufficiently large. References 1. M. Mulder, Basic Principles of Membrane Technology, 2nd ed., Kluwer Academic Publishers, 1998. 2. R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, 1960. Model Library path: Chemical_Reaction_Engineering_Module/ Separation_Processes/dialysis Modeling Instructions From the File menu, choose New. D – Vc 1 c 1 u + ( ) n · M c 2 Kc 1 – ( ) = at cO d/m D m – Vc 2 ( ) n · M Kc 1 c 2 – ( ) = at cO m/d D m – Vc 2 ( ) n · M Kc 3 c 2 – ( ) = at cO m/p D – Vc 3 c 3 u + ( ) n · M c 2 Kc 3 – ( ) = at cO p/m Solved with COMSOL Multiphysics 4.4 9 | S E P A R A T I O N T H R O U G H D I A L Y S I S N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 2D Axisymmetric button. 2 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 3 Click the Add button. 4 In the Concentrations table, enter the following settings: 5 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 6 Click the Add button. 7 In the Concentrations table, enter the following settings: 8 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 9 Click the Add button. 10 In the Concentrations table, enter the following settings: 11 Click the Study button. 12 In the tree, select Preset Studies for Selected Physics>Stationary. 13 Click the Done button. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. c1 c2 c3 Solved with COMSOL Multiphysics 4.4 10 | S E P A R A T I O N T H R O U G H D I A L Y S I S 3 In the table, enter the following settings: G E O ME T R Y 1 1 In the Model Builder window, under Component 1 click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose mm. Rectangle 1 1 Right-click Component 1>Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type R2. 4 In the Height edit field, type H/scale. 5 Click the Build Selected button. Name Expression Value Description R1 0.2[mm] 2.0000E-4 m Inner radius, hollow fiber R2 0.28[mm] 2.8000E-4 m Outer radius, hollow fiber R3 0.7[mm] 7.0000E-4 m Approximate radius, unit cell H 21[mm] 0.021000 m Fiber length scale 7 7.0000 Axial coordinate scale factor h_max 0.1*R1 2.0000E-5 m Maximum mesh element size D 1e-9[m^2/s] 1.0000E-9 m²/s Diffusion constant, liquid phases Dm 1e-9[m^2/s] 1.0000E-9 m²/s Diffusion constant, membrane M 1e4[m/s] 10000 m/s Stiff-spring velocity K 0.7 0.70000 Partition coefficient c0 1[mol/liter] 1000.0 mol/m³ Inlet concentration, dialysate v1_max 1[mm/s] 0.0010000 m/s Maximum velocity, dialysate A -2e3[1/(m*s)] -2000.0 1/(m·s) Permeate velocity prefactor Solved with COMSOL Multiphysics 4.4 11 | S E P A R A T I O N T H R O U G H D I A L Y S I S 6 Click the Zoom Extents button on the Graphics toolbar. Rectangle 2 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type R3-R1. 4 In the Height edit field, type H/scale. 5 Locate the Position section. In the r edit field, type R1. 6 Click the Build Selected button. 7 Click the Zoom Extents button on the Graphics toolbar. Form Union In the Model Builder window, under Component 1>Geometry 1 right-click Form Union and choose Build Selected. D E F I N I T I O N S Explicit 1 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1>Definitions right-click Explicit 1 and choose Rename. 3 Go to the Rename Explicit dialog box and type Dialysate in the New name edit field. 4 Click OK. 5 Select Domain 1 only. Explicit 2 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1>Definitions right-click Explicit 2 and choose Rename. 3 Go to the Rename Explicit dialog box and type Membrane in the New name edit field. 4 Click OK. 5 Select Domain 2 only. Explicit 3 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1>Definitions right-click Explicit 3 and choose Rename. Solved with COMSOL Multiphysics 4.4 12 | S E P A R A T I O N T H R O U G H D I A L Y S I S 3 Go to the Rename Explicit dialog box and type Permeate in the New name edit field. 4 Click OK. 5 Select Domain 3 only. Variables 1 1 In the Model Builder window, right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose Dialysate. 5 Locate the Variables section. In the table, enter the following settings: Variables 2 1 Right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose Membrane. 5 Locate the Variables section. In the table, enter the following settings: Variables 3 1 Right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose Permeate. 5 Locate the Variables section. In the table, enter the following settings: TR A N S P O R T O F D I L U T E D S P E C I E S 1 In the Model Builder window, under Component 1 click Transport of Diluted Species. Name Expression Unit Description c_all c1 mol/m³ Concentration Name Expression Unit Description c_all c2 mol/m³ Concentration Name Expression Unit Description c_all c3 mol/m³ Concentration Solved with COMSOL Multiphysics 4.4 13 | S E P A R A T I O N T H R O U G H D I A L Y S I S 2 In the Transport of Diluted Species settings window, locate the Domain Selection section. 3 From the Selection list, choose Dialysate. Convection and Diffusion 1 1 In the Model Builder window, under Component 1>Transport of Diluted Species click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Diffusion section. 3 From the symmetry property list, choose Diagonal. 4 In the D c1 table, enter the following settings: 5 Specify the u vector as Initial Values 1 1 In the Model Builder window, under Component 1>Transport of Diluted Species click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the c1 edit field, type c0. Symmetry 1 1 On the Physics toolbar, click Boundaries and choose Symmetry. 2 Select Boundary 1 only. Concentration 1 1 On the Physics toolbar, click Boundaries and choose Concentration. 2 Select Boundary 2 only. 3 In the Concentration settings window, locate the Concentration section. 4 Select the Species c1 check box. 5 In the c 0,c1 edit field, type c0. Flux 1 1 On the Physics toolbar, click Boundaries and choose Flux. D 0 0 D/scale^2 0 r v1_max*(1-(r/R1)^2)/scale z Solved with COMSOL Multiphysics 4.4 14 | S E P A R A T I O N T H R O U G H D I A L Y S I S 2 Select Boundary 4 only. 3 In the Flux settings window, locate the Inward Flux section. 4 Select the Species c1 check box. 5 In the N 0,c1 edit field, type M*(c2-K*c1). Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 3 only. TR A N S P O R T O F D I L U T E D S P E C I E S 2 1 In the Model Builder window, under Component 1 click Transport of Diluted Species 2. 2 In the Transport of Diluted Species settings window, locate the Domain Selection section. 3 From the Selection list, choose Membrane. 4 Locate the Transport Mechanisms section. Clear the Convection check box. Diffusion 1 In the Model Builder window, under Component 1>Transport of Diluted Species 2 click Diffusion. 2 In the Diffusion settings window, locate the Diffusion section. 3 From the symmetry property list, choose Diagonal. 4 In the D c2 table, enter the following settings: Flux 1 1 On the Physics toolbar, click Boundaries and choose Flux. 2 Select Boundary 4 only. 3 In the Flux settings window, locate the Inward Flux section. 4 Select the Species c2 check box. 5 In the N 0,c2 edit field, type M*(K*c1-c2). Flux 2 1 On the Physics toolbar, click Boundaries and choose Flux. 2 Select Boundary 7 only. Dm 0 0 Dm/scale^2 Solved with COMSOL Multiphysics 4.4 15 | S E P A R A T I O N T H R O U G H D I A L Y S I S 3 In the Flux settings window, locate the Inward Flux section. 4 Select the Species c2 check box. 5 In the N 0,c2 edit field, type M*(K*c3-c2). TR A N S P O R T O F D I L U T E D S P E C I E S 3 1 In the Model Builder window, under Component 1 click Transport of Diluted Species 3. 2 In the Transport of Diluted Species settings window, locate the Domain Selection section. 3 From the Selection list, choose Permeate. Convection and Diffusion 1 1 In the Model Builder window, under Component 1>Transport of Diluted Species 3 click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Diffusion section. 3 From the symmetry property list, choose Diagonal. 4 In the D c3 table, enter the following settings: 5 Specify the u vector as Initial Values 1 1 In the Model Builder window, under Component 1>Transport of Diluted Species 3 click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the c3 edit field, type 0.1*c0. Flux 1 1 On the Physics toolbar, click Boundaries and choose Flux. 2 Select Boundary 7 only. 3 In the Flux settings window, locate the Inward Flux section. 4 Select the Species c3 check box. 5 In the N 0,c3 edit field, type M*(c2-K*c3). D 0 0 D/scale^2 0 r A*((r^2)-(R2^2)-2*(R3^2)*log(r/R2))/scale z Solved with COMSOL Multiphysics 4.4 16 | S E P A R A T I O N T H R O U G H D I A L Y S I S Concentration 1 1 On the Physics toolbar, click Boundaries and choose Concentration. 2 In the Concentration settings window, locate the Concentration section. 3 Select the Species c3 check box. 4 Select Boundary 8 only. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 9 only. ME S H 1 Size 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Free Triangular. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type h_max. 5 In the Resolution of narrow regions edit field, type 2. 6 Click the Build All button. S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S Concentration (chds) 1 In the Model Builder window, expand the Concentration (chds) node, then click Surface 1. 2 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Definitions>Concentration (c_all). 3 On the 2D plot group toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. 5 In the Model Builder window, right-click Concentration (chds) and choose Arrow Surface. Solved with COMSOL Multiphysics 4.4 17 | S E P A R A T I O N T H R O U G H D I A L Y S I S 6 In the Arrow Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Transport of Diluted Species 3>Species c3>Total flux (chds3.tfluxr_c3,...,chds3.tfluxz_c3). 7 On the 2D plot group toolbar, click Plot. Solved with COMSOL Multiphysics 4.4 18 | S E P A R A T I O N T H R O U G H D I A L Y S I S Solved with COMSOL Multiphysics 4.4 1 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Di s s oc i a t i on i n a T ubul a r Re a c t or Introduction Tubular reactors are often used in continuous large-scale production, for example in the petroleum industry. One key design and optimization parameter is the conversion, or the amount of reactant that reacts to form the desired product. In order to achieve high conversion, process engineers optimize the reactor design: its length, width and heating system. An accurate reactor model is a very useful tool, both at the design stage and in tuning an existing reactor. Figure 1: Dissociation reaction in a tubular reactor. This example deals with a gas-phase dissociation process, species A reacts to form B (see Figure 1). First, the conversion and reaction distribution is modeled as an isothermal tubular reactor under steady-state conditions. Secondly, the temperature dependence of the reaction kinetics are included and the changes in reaction distribution and conversion are studied. The following physics are used • Laminar flow with compressible formulation. This formulation allows for changes in density, which is needed because of the volumetric change caused by the gas dissociation reaction. • Transport of Concentrated Species. This handles the species transport in systems with high concentrations, where Fick’s law is not valid. • Heat transfer in Fluids. A A+B A 2B Solved with COMSOL Multiphysics 4.4 2 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Model Definition K E Y I N S T R U C T I V E E L E ME N T S This model illustrates several attractive features in the Chemical Reaction Engineering Module: • The use of the Transport of Concentrated Species to account for multicomponent diffusion. • How to couple the variable density to a Laminar Flow physics interface. • Implementation of temperature- and composition÷dependent reaction kinetics. • The use of a mapped mesh, which is structured, to discretize a long and thin geometry, typical for tubular reactors • The setup of heat balances and how to couple these to both the mass balances and the velocity field H A N D L I N G E X P A N D I N G F L OW—C O MP R E S S I B L E F L OW F O R MU L A T I O N Each mole of the reactant, A, reacts to form two moles of the product, B: This leads to a volumetric expansion of the gas mixture as the reaction proceeds. The fluid’s change in density influences the gas velocity in the reactor, causing an acceleration as the reaction proceeds. In order to model the flow, use a compressible formulation of the Navier÷Stokes equations, defined according to the following equations: Here µ denotes the solution’s density (kg/m 3 ), u is the velocity vector (m/s), p gives the pressure (Pa), µ represents the solution’s viscosity (kg/(m·s), or Pa·s), and I denotes the identity matrix. The density is expressed through the ideal gas law as a function of pressure, temperature, and composition: A 2B ÷ µ u V · ( )u V pI – µ u u V ( ) T + V ( ) 2 3 ---µ V u · ( )I – + · = V µu ( ) · 0 = µ p RT -------- w A M A w A M A + ( ) = Solved with COMSOL Multiphysics 4.4 3 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R where T denotes temperature (K), w A and w B are the mass fractions of species A and B, while M A and M B denote the molecular weight of A and B (kg/mol), respectively. The model applies the Laminar Flow physics interface, which solves the above equations, describing the momentum balances and the continuity (mass conservation) for fluids with variations in density. C O NV E C T I O N A N D D I F F U S I O N I N MU L T I C O MP O N E N T S Y S T E MS — TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S As the dissociation reaction proceeds, the composition of the mixture changes from pure A at the inlet to a mixture of A and B. The total mass flux is strongly influenced by the flux of each species. In addition, several molecular interactions occur; A interacts with B and other A molecules, B interacts with A and other B molecules. This implies that the simple Fick’s law formulation, with one constant diffusivity for each species is not applicable here. In a concentrated multicomponent mixture you must account for all possible interactions, and the flux is dependent on the fluid’s local composition. Simple Fick diffusivity accounts only for the interaction between solvent and solute. In the Transport of Concentrated Species with the Maxwell-Stefan or Mixture-Averaged diffusion equations, multicomponent diffusivities describe the interactions between all components in the system. Since a change in a gas mixture composition will affect the density, the species transport equation needs to be coupled with the flow equations (Laminar Flow, Navier Stokes in this case). Now consider a mathematical formulation of this discussion. The mass-balance equation for each species is with the source terms given by the reaction kinetics follow the equations t c c µw A ( ) V n A · + R A = t c c µw B ( ) V n B · + R B = R A k f c A M A – = R B 2k f c A M B = Solved with COMSOL Multiphysics 4.4 4 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R where k f denotes the forward reaction rate constant (s -1 ), c A represents the concentration of A (mol/m 3 ), and n i is the mass flux vector for species i (kg/(m 2 ·s)). Because the reaction is a pure dimerization, it is inherent that M B equals half of M A . As mentioned earlier, it is possible to rewrite the mass-balances equations for each species by replacing one of the species’ mass balance with a total mass balance. A solution with two species follows following equation: Because the system consists only of two species, the sum of w A and w B is always unity, and the sum of the reaction terms is zero. The above equation now becomes which is the total mass-balance equation. The reaction rate is described by an Arrhenius law according to where A 0 , the pre-exponential factor, is set to 41.3 s -1 , E a , the activation energy, is set to 30 kJ/mol, R is the gas constant, 8.314 J/(mol·K), and T the temperature (K). The rate of production of species B is thus dependent on both composition and temperature. However, in the first model the gas is assumed to be isothermal, making the rate vary only with composition. G E O ME T R Y The geometry of the tubular reactor is rotationally symmetric, and it is possible to reduce the model from 3D to a 2D axisymmetric problem. This means that you only have to model half of the tube cross section, as illustrated in Figure 2. t c c µ w A w B + ( ) ( ) V n A n B + ( ) · + R A R B + = t c c µ V n A n B + ( ) · + 0 = k f A 0 E a – RT ---------- \ . | | exp = Solved with COMSOL Multiphysics 4.4 5 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Figure 2: Model geometry. B O U N D A R Y C O N D I T I O N S 1 Laminar Flow physics interface The flow in the reactor is driven by a pressure drop. The pressure at the inlet, p in , is slightly higher than that at the outlet, where the pressure is set to atmospheric pressure The walls are represented by no-slip boundary conditions 2 Transport of Concentrated Species physics interface At the inlet, the mass fraction of A is set close to unity(0.99). The outlet boundary condition is a convective flux condition. The convective flux condition implies that diffusive flux for the species is zero perpendicular to the boundary. This is a common assumption when modeling the outlet in tubular reactors. No-flux conditions—referred to as insulation/symmetry in COMSOL Multiphysics— apply at all other boundaries: n · n A = 0. Outlet Inlet r z r=0 p p atm = u 0 = Solved with COMSOL Multiphysics 4.4 6 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R ME S H In this example, a mapped (structured) mesh is a good choice due to the reactor’s regular shape. A mapped mesh gives full control of the elements’ width/length relationship. The use of a structured mesh is especially suitable when the requirements for the mesh density are greater in one direction (radial) than the other (axial). In this example a denser mesh is required in the inlet region to resolve gradients at the inlet. This is achieved by specifying the edge element distribution, as you see in Modeling Instructions section below. Study 1—Results and Discussion for Isothermal Conditions Under isothermal conditions, the Laminar Flow and Transport of Concentrated Species physics interfaces in COMSOL Multiphysics are applied to solve the coupled model of the compressible Navier-Stokes equations and the Maxwell-Stefan convection and conduction equations. Figure 3 shows the velocity magnitude for the isothermal case at different cross-sections of reactor. The velocity increases along the axis direction (z) because of the volume expansion of gas mixture during the proceeding of reaction. The maximum of velocity is found at the center of the tube due to the no-slip on the side surface. Solved with COMSOL Multiphysics 4.4 7 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Figure 3: Velocity magnitude for the isothermal case. Figure 4 shows the mass fraction of species B for the isothermal case at different cross-sections of reactor. The closer to the side surface, the lower is the convective flow velocity, which gives rise to the higher mass fraction of species B towards the tube surface. The average mass fraction of species B at the outlet is 68.7%. Solved with COMSOL Multiphysics 4.4 8 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Figure 4: Mass fraction of species B for the isothermal case. The average conversion rate depends on flow rate profile, density distribution, and velocity field. It is defined as The average conversion rate at the outlet under isothermal conditions is 69.1%. Model Definition—Non-Isothermal Model Now it is time to expand the model by including an energy-balance equation modeling the temperature. In the previous model, the temperature was constant and set to 473 K. Now assume that the gas enters the reactor at room temperature, 293 K, and that the reactor walls are heated to 473 K to accomplish heating of the gas and reaction. This means that the gas is heated by the walls as the gas flows along the reactor. In addition, the heat of reaction is also included, acting as a source term. For ¸ B X B µu n · s d } µu n · s d } ---------------------------------- = Solved with COMSOL Multiphysics 4.4 9 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R the current dimerization reaction the heat of reaction is ÷80 kJ/kg. This means that the fluid mixture is heated as the reaction proceeds. The influence of the temperature on the reaction rate is significant. The rate follows an Arrhenius law, it increases exponentially with temperature. Thus, the reaction rate increases as the fluid flows through the reactor and is heated by the walls and by the heat of reaction. The energy-balance equation is where k is the thermal conductivity (W/(m·K)), C p is the specific heat capacity (J/ (kg·K)), and Q is the heat source term (W/m). The heat source term, Q, is given by the heat of reaction, AH (÷80 000 J/kg), and reaction rate, R: The material properties are specified to be those of the mixture. In this example assume that the heat capacity, C p , and conductivity, k, are similar to those of propane, which is present in the Material Library. The boundary conditions for the energy balance are similar to those of the mass balances. At the inlet, the gas temperature is specified, in this case to 293 K. The default Axial symmetry condition gives a zero temperature gradient at the symmetry boundary: n · VT = 0. At the outlet, apply a convective flux condition: . Model the reactors heated walls by applying a heat flux condition on the wall, using a heat transfer coefficient, h, (50 W/(m 2 ·K)) for the heating fluid, and the heating temperature, T f , which is 473 K. The condition is: Study 2—Results and Discussion for Non÷Isothermal Conditions For the non-isothermal case, add a Heat Transfer in Fluids physics interface to the isothermal model. µC p t c cT V kVT – ( ) · + Q µC p u VT · ( ) – = Q AHR A – = n T V · 0 = n kVT ( ) · h T f T – ( ) = Solved with COMSOL Multiphysics 4.4 10 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Figure 5 shows the velocity magnitude for the non-isothermal case at different cross-sections of reactor. The velocity magnitude for the non-isothermal case is slightly smaller than that for the isothermal case (see Figure 3). This is due to the lower reaction rate caused by the lower temperature. Figure 5: Velocity magnitude for the non-isothermal case. Figure 6 shows the mass fraction of species B for the non÷isothermal case at different cross-sections of reactor. At the region close to the side wall, the mole fraction is much higher than that in the central region due to the higher temperature close to the heating wall. The overall mole fraction is lower than that for the isothermal conditions (see Figure 4) because the low temperature in the reactor. The average mass fraction of species B at the outlet is 29.0%. and the average conversion rate at the outlet under non÷isothermal conditions is 26.7%. Solved with COMSOL Multiphysics 4.4 11 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Figure 6: Mass fraction of species B for the non-isothermal case. Figure 7 shows the temperature distribution under non-isothermal conditions. The temperature is much higher close to the reactor wall. This temperature profile has a significant impact on the reaction rate in the reactor, see Figure 5 Solved with COMSOL Multiphysics 4.4 12 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Figure 7: Temperature distribution for non-isothermal case. Model Library path: Chemical_Reaction_Engineering_Module/ Tubular_Reactors/dissociation Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 2D Axisymmetric button. 2 In the Select physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf). 3 Click the Add button. Solved with COMSOL Multiphysics 4.4 13 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R 4 In the Select physics tree, select Chemical Species Transport>Transport of Concentrated Species (chcs). 5 Click the Add button. 6 Click the Study button. 7 In the tree, select Preset Studies for Selected Physics>Stationary. 8 Click the Done button. G L O B A L D E F I N I T I O N S Load the model parameters from a text file. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file dissociation_parameters.txt. D E F I N I T I O N S Load the variables from a text file. Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file dissociation_variables.txt. G E O ME T R Y 1 The model geometry is simply a rectangle. Rectangle 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type W0. 4 In the Height edit field, type L0. Solved with COMSOL Multiphysics 4.4 14 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R 5 Click the Build All Objects button. 6 Click the Zoom Extents button on the Graphics toolbar. D E F I N I T I O N S Average 1 1 On the Definitions toolbar, click Component Couplings and choose Average. You will use this operator later in the results analysis. With the check box 'Compute integral in revolved geometry' enabled, it automatically performs a surface integration by multiplying by 2*pi*r. 2 In the Average settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 3 only. 5 Locate the Advanced section. Select the Compute integral in revolved geometry check box. L A MI N A R F L OW Fluid Properties 1 1 In the Model Builder window, under Component 1>Laminar Flow click Fluid Properties 1. 2 In the Fluid Properties settings window, locate the Fluid Properties section. 3 From the µ list, choose Density (chcs/cdm1). 4 From the µ list, choose User defined. In the associated edit field, type eta. Inlet 1 1 On the Physics toolbar, click Boundaries and choose Inlet. 2 Select Boundary 2 only. 3 In the Inlet settings window, locate the Boundary Condition section. 4 From the Boundary condition list, choose Pressure, no viscous stress. 5 Locate the Pressure, No Viscous Stress section. In the p 0 edit field, type p_in. Outlet 1 1 On the Physics toolbar, click Boundaries and choose Outlet. 2 Select Boundary 3 only. 3 In the Outlet settings window, locate the Pressure Conditions section. 4 Select the Normal flow check box. Solved with COMSOL Multiphysics 4.4 15 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S 1 In the Model Builder window, under Component 1 click Transport of Concentrated Species. 2 In the Transport of Concentrated Species settings window, locate the Transport Mechanisms section. 3 From the Diffusion model list, choose Maxwell-Stefan. 4 Click to expand the Dependent variables section. Locate the Dependent Variables section. In the Mass fractions table, enter the following settings: 5 Locate the Species section. From the From mass constraint list, choose wB. Convection and Diffusion 1 1 In the Model Builder window, under Component 1>Transport of Concentrated Species click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 From the u list, choose Velocity field (spf/fp1). 4 In the T edit field, type T_atm. 5 From the p list, choose Pressure (spf/fp1). 6 In the p ref edit field, type p_atm. 7 Locate the Density section. In the M wA edit field, type MA. 8 In the M wB edit field, type MB. 9 Locate the Diffusion section. In the D ik table, enter the following settings: Reactions 1 1 On the Physics toolbar, click Domains and choose Reactions. 2 Select Domain 1 only. 3 In the Reactions settings window, locate the Reactions section. 4 In the R wA edit field, type -Ra. Mass Fraction 1 1 On the Physics toolbar, click Boundaries and choose Mass Fraction. wA wB 1 DA DA 1 Solved with COMSOL Multiphysics 4.4 16 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R 2 Select Boundary 2 only. 3 In the Mass Fraction settings window, locate the Mass Fraction section. 4 Select the Species wA check box. 5 In the e 0,wA edit field, type wA_in. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 3 only. ME S H 1 Mapped 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Mapped. Distribution 1 1 In the Model Builder window, under Component 1>Mesh 1 right-click Mapped 1 and choose Distribution. 2 Select Boundaries 1 and 4 only. 3 In the Distribution settings window, locate the Distribution section. 4 In the Number of elements edit field, type 100. Distribution 2 1 Right-click Mapped 1 and choose Distribution. 2 Select Boundary 2 only. 3 In the Distribution settings window, locate the Distribution section. 4 In the Number of elements edit field, type 10. Boundary Layer Properties 1 In the Model Builder window, right-click Mesh 1 and choose Boundary Layers. 2 Select Boundary 2 only. 3 In the Boundary Layer Properties settings window, locate the Boundary Layer Properties section. 4 In the Number of boundary layers edit field, type 3. 5 In the Boundary layer stretching factor edit field, type 1.3. 6 From the Thickness of first layer list, choose Manual. 7 In the Thickness edit field, type 4/500. Solved with COMSOL Multiphysics 4.4 17 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R S T U D Y 1 1 In the Model Builder window, click Study 1. 2 In the Study settings window, locate the Study Settings section. 3 Clear the Generate default plots check box. 4 On the Home toolbar, click Compute. R E S U L T S Derived Values 1 On the Results toolbar, click More Derived Values and choose Average>Line Average. 2 Select Boundary 3 only. 3 In the Line Average settings window, locate the Expression section. 4 Click Mass fraction (wB) in the upper-right corner of the section. Right-click Results>Derived Values>Line Average 1 and choose Evaluate>New Table. 5 On the Results toolbar, click Global Evaluation. 6 In the Global Evaluation settings window, locate the Expression section. 7 In the Expression edit field, type aveop1(w*spf.rho*wB)/aveop1(w*spf.rho). Data Sets 1 Right-click Results>Derived Values>Global Evaluation 1 and choose Evaluate>Table 1 - Line Average 1 (wB). 2 On the Results toolbar, click More Data Sets and choose Revolution 2D. 3D Plot Group 1 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the Model Builder window, under Results right-click 3D Plot Group 1 and choose Slice. 3 In the Slice settings window, locate the Plane Data section. 4 From the Plane list, choose xy-planes. 5 In the Planes edit field, type 10. 6 On the 3D plot group toolbar, click Plot. 7 Click the Zoom Extents button on the Graphics toolbar. 3D Plot Group 2 1 In the Model Builder window, right-click 3D Plot Group 1 and choose Duplicate. 2 In the Model Builder window, expand the 3D Plot Group 2 node, then click Slice 1. Solved with COMSOL Multiphysics 4.4 18 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R 3 In the Slice settings window, locate the Expression section. 4 Click Mass fraction (wB) in the upper-right corner of the section. On the 3D plot group toolbar, click Plot. C O MP O N E N T 1 Add the Heat Transfer in Fluids physics and a corresponding study. On the Home toolbar, click Add Physics. A D D P HY S I C S 1 Go to the Add Physics window. 2 In the Add physics tree, select Heat Transfer>Heat Transfer in Fluids (ht). 3 In the Add physics window, click Add to Component. R O O T On the Home toolbar, click Add Study. A D D S T U D Y 1 Go to the Add Study window. 2 Find the Studies subsection. In the tree, select Preset Studies>Stationary. 3 In the Add study window, click Add Study. H E A T TR A N S F E R I N F L U I D S Heat Transfer in Fluids 1 1 In the Model Builder window, under Component 1>Heat Transfer in Fluids click Heat Transfer in Fluids 1. 2 In the Heat Transfer in Fluids settings window, locate the Model Inputs section. 3 From the p list, choose Pressure (spf/fp1). 4 In the p ref edit field, type p_atm. 5 From the u list, choose Velocity field (spf/fp1). 6 Locate the Heat Conduction, Fluid section. From the k list, choose User defined. In the associated edit field, type k_mix. 7 Locate the Thermodynamics, Fluid section. From the µ list, choose Density (chcs/ cdm1). 8 From the C p list, choose User defined. In the associated edit field, type Cp_mix. 9 From the ¸ list, choose User defined. Solved with COMSOL Multiphysics 4.4 19 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Initial Values 1 1 In the Model Builder window, under Component 1>Heat Transfer in Fluids click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the T edit field, type T_atm. Heat Source 1 1 On the Physics toolbar, click Domains and choose Heat Source. 2 Select Domain 1 only. 3 In the Heat Source settings window, locate the Heat Source section. 4 In the Q edit field, type 80e3[J/kg]*Ra. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 3 only. Temperature 1 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 Select Boundary 2 only. 3 In the Temperature settings window, locate the Temperature section. 4 In the T 0 edit field, type T_atm. Heat Flux 1 1 On the Physics toolbar, click Boundaries and choose Heat Flux. 2 Select Boundary 4 only. 3 In the Heat Flux settings window, locate the Heat Flux section. 4 In the q 0 edit field, type 50[W/m^2/K]*(Tf-T). TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S Convection and Diffusion 1 1 In the Model Builder window, under Component 1>Transport of Concentrated Species click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 From the T list, choose Temperature (ht). S T U D Y 2 1 In the Model Builder window, click Study 2. Solved with COMSOL Multiphysics 4.4 20 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R 2 In the Study settings window, locate the Study Settings section. 3 Clear the Generate default plots check box. 4 On the Home toolbar, click Compute. R E S U L T S Derived Values 1 In the Model Builder window, expand the Results>Derived Values node. 2 Right-click Line Average 1 and choose Duplicate. 3 In the Line Average settings window, locate the Data section. 4 From the Data set list, choose Solution 2. 5 Right-click Results>Derived Values>Line Average 2 and choose Evaluate>Table 1 - Line Average 1 (wB). 6 Right-click Results>Derived Values>Global Evaluation 1 and choose Duplicate. 7 In the Global Evaluation settings window, locate the Data section. 8 From the Data set list, choose Solution 2. Data Sets 1 Right-click Results>Derived Values>Global Evaluation 2 and choose Evaluate>Table 1 - Line Average 1 (wB). 2 On the Results toolbar, click More Data Sets and choose Revolution 2D. 3 In the Revolution 2D settings window, locate the Data section. 4 From the Data set list, choose Solution 2. 3D Plot Group 3 1 In the Model Builder window, under Results right-click 3D Plot Group 1 and choose Duplicate. 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Revolution 2D 2. 4 On the 3D plot group toolbar, click Plot. 5 Click the Zoom Extents button on the Graphics toolbar. 3D Plot Group 2 In the Model Builder window, expand the 3D Plot Group 3 node. 3D Plot Group 4 1 Right-click Results>3D Plot Group 2 and choose Duplicate. Solved with COMSOL Multiphysics 4.4 21 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Revolution 2D 2. 4 On the 3D plot group toolbar, click Plot. 3D Plot Group 5 1 Right-click Results>3D Plot Group 4 and choose Duplicate. 2 In the Model Builder window, expand the 3D Plot Group 5 node, then click Slice 1. 3 In the Slice settings window, locate the Expression section. 4 Click Temperature (T) in the upper-right corner of the section. On the 3D plot group toolbar, click Plot. S T U D Y 1 Study 1 concerns only isothermal conditions. Remove 'Heat Transfer in Fluids' from it so that it can run as expected again. Step 1: Stationary 1 In the Model Builder window, under Study 1 click Step 1: Stationary. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Physics Solve for Discretization Heat Transfer in Fluids × physics Solved with COMSOL Multiphysics 4.4 22 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R Solved with COMSOL Multiphysics 4.4 1 | I B U P R O F E N S Y N T H E S I S I bupr of e n S y nt he s i s Introduction Kinetic analysis of catalytic reactions is essential for understanding rate behavior as well as the reaction mechanism. Developing knowledge of intrinsic reaction kinetics and of rate equations is central to reaction engineering studies aimed at improving reactor design. This example illustrates the reaction kinetics of a complex chemistry occurring in a perfectly stirred tank reactor. The homogeneous catalysis of 1-(4-isobutylphenyl) ethanol into the anti-inflammatory drug ibuprofen serves as the example chemistry. The model determines concentrations of reactants, intermediates, and products as functions of time for the network of chemical reactions. The chemistry in this example involves homogeneous catalysis. As this terminology suggests, the catalyst and the reacting species are in the same phase. Most commonly, a liquid reaction mixture contains a soluble metalorganic complex that affects the catalysis. Organometallic catalysts can often be fine-tuned with respect to reaction activity and selectivity. Because these relatively expensive catalysts produce highly-refined reaction products, they commonly find application in fine chemicals or pharmaceutics. The model focuses on the use of the Chemical Reaction Engineering Module for a kinetics investigation. You easily enter chemical reaction formulas from the keyboard, then the Reaction Engineering interface automatically generates rate expressions and material balances. It solves the equations, and you postprocess results directly in the COMSOL Desktop. Model Description Analyzing chemical kinetics involves solving the set of ordinary differential equations corresponding to individual steps in a network of reactions. This example illustrates the kinetics of ibuprofen synthesis. Figure 1 shows the reaction steps displayed in a Solved with COMSOL Multiphysics 4.4 2 | I B U P R O F E N S Y N T H E S I S catalytic cycle (Ref. 1). Figure 1: Catalytic cycle of ibuprofen synthesis. Prior to entering the cycle, the starting material, 1-(4-isobutylphenyl)ethanol, is first dehydrated to form 4-isobutylstyrene. This species subsequently undergoes the addition of HCl to produce the active substrate 1-(4-isobutylphenyl)ethyl chloride. The palladium catalyst must also go through an initial transformation, from L 2 PdCl 2 (L=triphenylphosphine) to anionic L 2 PdCl, before becoming active. The activated catalyst then assists in the carbonylation and hydrolysis of 1-(4-isobutylphenyl)ethyl chloride, producing ibuprofen. The following reactions represent the catalytic cycle: L 2 PdCl CO H 2 O OH L 2 PdCl 2 Cl L 2 PdCl 2 COClPdL 2 COOH H + H + HCl - - Cl - Solved with COMSOL Multiphysics 4.4 3 | I B U P R O F E N S Y N T H E S I S (1) (2) (3) (4) (5) (6) (7) Reaction 1 involves the dehydration of the reactant alcohol to form the corresponding alkene. Reaction 2 describes the hydrohalogenation of alkene, resulting in the active substrate 1-(4-isobutylphenyl)ethyl chloride. Reaction 3 shows the dehydrohalogenation of the active substrate, assisted by a base, B. Reaction 4 describes the transformation of the pre-catalytic species L 2 PdCl 2 into the active anionic catalyst L 2 PdCl. In Reaction 5 the active substrate undergoes oxidative addition to the L 2 PdCl catalyst. Reaction 6 summarizes the carbonylation, and Reaction 7 describes the hydrolysis of the metalorganic species, leading to the formation of ibuprofen and regeneration of the catalyst. In order to make species notation more manageable, this example uses the following labels: OH + H + k 1 + H 2 O + H + + + H + Cl - Cl k 2 + + H + Cl - + B + B Cl k 3 + CO + H 2 O L 2 PdCl 2 k 4 L 2 PdCl + + Cl - + CO 2 - 2H + L 2 PdCl + k 5 L 2 PdCl 2 Cl - - CO k 6 + Cl - L 2 PdCl 2 + L 2 PdClCO - H 2 O k 7 + H + + L 2 PdCl + L 2 PdClCO COOH - Solved with COMSOL Multiphysics 4.4 4 | I B U P R O F E N S Y N T H E S I S Making use of these notations, the reaction rates corresponding to Reaction 1 through Reaction 7 are: (8) (9) (10) (11) (12) (13) (14) The Chemical Reaction Engineering Module automatically generates these expressions and displays them immediately when you enter the chemical reaction species abbreviation species abbreviation OH L 2 PdCl 2 L 2 PdCl L 2 PdCl 2 Cl L 2 PdClCO - - roh ren rhcl pd1 pd2 pd3 pd4 COOH ibu r 1 k 1 c roh c H = r 2 k 2 c ren c H c Cl = r 3 k 3 c rhcl c B = r 4 k 4 c pd1 c CO c H 2 O = r 5 k 5 c pd2 c rhcl = r 6 k 6 c pd3 c CO = r 7 k 7 c pd4 c H 2 O = Solved with COMSOL Multiphysics 4.4 5 | I B U P R O F E N S Y N T H E S I S formulas. By default, the software assumes that the chemistry takes place isothermally in a perfectly stirred batch reactor. Figure 2: A perfectly stirred batch reactor where the reactant alcohol is carbonylated to form ibuprofen by means of palladium catalysis. With no inflow or outflow from the reactor, the change of species concentrations with time is a function only of the reaction rates. The following set of ODEs result from the reaction rates (Equation 8 to Equation 14) and stoichiometry of the reaction formulas (Reaction 1 to Reaction 7): (15) (16) (17) (18) (19) (20) (21) t d dc roh r 1 = t d dc ren r 1 r 2 – r 3 + = t d dc rhcl r 2 r 3 – r 5 – = t d dc ibu r 7 = t d dc pd1 r – 4 = t d dc pd2 r 4 r 5 – r 7 + = t d dc pd3 r 5 r 6 – = Solved with COMSOL Multiphysics 4.4 6 | I B U P R O F E N S Y N T H E S I S (22) (23) (24) (25) The Chemical Reaction Engineering Module automatically generates and solves these equations. The model investigates two reaction conditions. The first simulation (Case 1) solves Equation 15 through Equation 25. In Case 2 you subsequently modify the reaction network with an additional reaction, altering the simulation results. Assume that the reactant alcohol and product ibuprofen (a carboxylic acid) react reversibly, forming an ester: The results of the two simulations are compared to gain insight in the process implications. t d dc pd4 r 6 r 7 – = t d dc H r – 2 r 3 2r 4 r 7 + + + = t d dc Cl r – 2 r 3 r 4 r 6 + + + = t d dc H2O r 1 r 4 – r 7 – = OH + H + k 8 + H 2 O H + COOH + k 8 O O + r f Solved with COMSOL Multiphysics 4.4 7 | I B U P R O F E N S Y N T H E S I S Results C A S E 1 Figure 3 shows the concentration profiles for reactants and products over time. Figure 3: Species concentrations (mol/m 3 ) as a function of time (s). Clearly, after approximately 2 hours the process has run to completion. Solved with COMSOL Multiphysics 4.4 8 | I B U P R O F E N S Y N T H E S I S C A S E 2 This expansion of the original case adds a reversible reaction between the reactant alcohol and the product ibuprofen to form an ester. Figure 4 shows the concentration transients. Figure 4: Species concentrations (mol/m 3 ) as a function of time (s). In the course of the reaction, ester forms as an intermediary product. In order to achieve the same final concentration of ibuprofen for Case 2 as in Case 1, the process must run for at least 12 hours. In conclusion, this example illustrates the use of the Chemical Reaction Engineering Module for analyzing the kinetics of a complex reaction network. When you enter the chemical-reaction formulas into the physics interface, the Reaction Engineering interface automatically sets up the corresponding rate expressions and material balances. You can modify simulation conditions effortlessly, for instance by activating/ deactivating individual reactions or by changing initial conditions. Reference 1. R.V. Chaudhari, A. Seayad, and S. Jayasree, Catalysis Today, vol. 66, p. 371, 2001. Solved with COMSOL Multiphysics 4.4 9 | I B U P R O F E N S Y N T H E S I S Model Library path: Chemical_Reaction_Engineering_Module/ Homogeneous_Reactions_and_Catalysis/ibuprofen_synthesis Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Time Dependent. 6 Click the Done button. R E A C T I O N E N G I N E E R I N G Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type roh+H+=>ren+H2O+H+. 4 Locate the Rate Constants section. In the k f edit field, type 7.45e-3. Reaction 2 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type ren+H++Cl-=>rhcl. 4 Locate the Rate Constants section. In the k f edit field, type 1.25e-2. Reaction 3 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. Solved with COMSOL Multiphysics 4.4 10 | I B U P R O F E N S Y N T H E S I S 3 In the Formula edit field, type rhcl+B=>ren+H++Cl-+B. 4 Locate the Rate Constants section. In the k f edit field, type 1.6e-3. Reaction 4 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type pd1+CO+H2O=>pd2+Cl-+2H++CO2. 4 Locate the Rate Constants section. In the k f edit field, type 1.5e-1. Reaction 5 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type pd2+rhcl=>pd3. 4 Locate the Rate Constants section. In the k f edit field, type 1.59. Reaction 6 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type pd3+CO=>pd4+Cl-. 4 Locate the Rate Constants section. In the k f edit field, type 2.14e-1. Reaction 7 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type pd4+H2O=>pd2+H++ibu. 4 Locate the Rate Constants section. In the k f edit field, type 9.52e-1. Species: roh 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: roh. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 1.23. Species: H+ 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: H+. 2 In the Species settings window, locate the General Expressions section. Solved with COMSOL Multiphysics 4.4 11 | I B U P R O F E N S Y N T H E S I S 3 In the c 0 edit field, type 0.2. Species: H2O 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: H2O. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 3. Species: Cl- 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: Cl-. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 0.2. Species: B 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: B. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 0.1. Species: pd1 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: pd1. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 1.21e-2. Species: CO 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: CO. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 1.1. S T U D Y 1 Step 1: Time Dependent 1 In the Model Builder window, under Study 1 click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type 0 3600*3. Solved with COMSOL Multiphysics 4.4 12 | I B U P R O F E N S Y N T H E S I S Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Study 1>Solver Configurations node. 3 In the Model Builder window, expand the Solver 1 node, then click Time-Dependent Solver 1. 4 In the Time-Dependent Solver settings window, click to expand the Output section. 5 On the Home toolbar, click Compute. R E S U L T S Concentration (re) The plot in the Graphics window should look like that in Figure 3. R E A C T I O N E N G I N E E R I N G Reaction 8 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type ibu+roh+H+<=>ester+H2O+H+. 4 Locate the Rate Constants section. In the k f edit field, type 0.5. 5 In the k r edit field, type 1e-2. R O O T On the Home toolbar, click Add Study. A D D S T U D Y 1 Go to the Add Study window. 2 Find the Studies subsection. In the tree, select Preset Studies>Time Dependent. 3 In the Add study window, click Add Study. S T U D Y 2 Step 1: Time Dependent 1 In the Time Dependent settings window, locate the Study Settings section. 2 In the Times edit field, type 0 3600*12. 3 On the Home toolbar, click Compute. Solved with COMSOL Multiphysics 4.4 13 | I B U P R O F E N S Y N T H E S I S R E S U L T S Concentration (re) 1 Compare the plot in the Graphics window with that in Figure 4. Solved with COMSOL Multiphysics 4.4 14 | I B U P R O F E N S Y N T H E S I S Solved with COMSOL Multiphysics 4.4 1 | I S O E L E C T R I C S E P A R A T I O N I s oe l e c t r i c S e par at i on This modeling example applies the Transport of Diluted Species interface to model a separation process. A stream containing six different ionic species is divided into pure component streams are by means of migrative transport in an electric field. Introduction Free flow electrophoresis can be used to separate macromolecules such as proteins, based on their mobility perpendicular to the flow of the carrier fluid. If, in addition, a pH gradient is applied across the carrier flow, then molecules can be focused along their isoelectric points. The isoelectric point is the pH at which a the molecule has zero net charge. The concept of isoelectric focusing is illustrated in Figure 1. Figure 1: Ampholytic molecules can be focused around their isoelectric point by means of migrative transport in an electric field. + - Isoelectric point Migrative transport Migrative transport Electric Field pH gradient Solved with COMSOL Multiphysics 4.4 2 | I S O E L E C T R I C S E P A R A T I O N Molecules with a positive net charge will travel in the direction of the electric field, along the pH gradient, until they reach the isoelectric point. At this instance the migrative transport is switched off as the molecules net charge is zero. Similarly, anionic species travel in the direction opposite of the electric field. Model Definition G E O ME T R Y The geometry shown in Figure 2 represents the separation region in an isoelectric focusing chip. A laminar carrier stream transports a mixture of six proteins, injected at the bottom of the cell. Figure 2: Model geometry. MA S S T R A N S P O R T E Q U A T I O N S In addition to transport due to convection and diffusion, the Transport of Diluted Species interface supports ionic species transport by migration in electric fields. The mass balance is given by the transport equation (1) where c i denotes the concentration of species i (SI unit: mol/m 3 ), D i is the diffusion coefficient of species i (SI unit: m 2 /s), u is the fluid velocity (SI unit: m/s), F refers Injection boundary Outlet boundary 0 V 70 V Laminar flow V D i c i z i u m,i Fc i V V – V – ( ) u Vc i · + · 0 = Solved with COMSOL Multiphysics 4.4 3 | I S O E L E C T R I C S E P A R A T I O N to Faraday’s constant (SI unit: s·A/mol), V denotes the electric potential (SI unit: V), z i is the charge number of the ionic species (unitless), and u m,i its ionic mobility (SI unit: s·mol/kg). The velocity can be a computed fluid velocity field from a fluid-flow interface or a specified function of the space variables x, y, and z. This is the case for this model, where a parabolic velocity profile is specified for the laminar flow: (2) Above, v max is the maximum velocity and w is the width of the cell. Results and Discussion Figure 3 shows the electric field inside the cell. Figure 3: Electric potential (surface) and electric field (arrows) inside the focusing cell. v 4v max x w ---- 1 x w – ( ) = Solved with COMSOL Multiphysics 4.4 4 | I S O E L E C T R I C S E P A R A T I O N Figure 4 illustrates how the stream of mixed protein is affected by the electric field, with each of the six components migrating to focus around their respective isoelectric point. Figure 4: Total protein concentration. Solved with COMSOL Multiphysics 4.4 5 | I S O E L E C T R I C S E P A R A T I O N The plot below shows that the six proteins are well resolved as the stream reaches the outlet. Figure 5: Protein components are well resolved at the outlet of the cell. Model Library path: Chemical_Reaction_Engineering_Module/ Separation_Processes/isoelectric_separation Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select AC/DC>Electrostatics (es). 3 Click the Add button. 4 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). Solved with COMSOL Multiphysics 4.4 6 | I S O E L E C T R I C S E P A R A T I O N 5 Click the Add button. 6 In the Number of species edit field, type 6. 7 Click the Study button. 8 In the tree, select Preset Studies for Selected Physics>Stationary. 9 Click the Done button. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description W 3.5[mm] 0.003500 m Channel width H 10[mm] 0.01000 m Channel height a 0.3[mm] 3.000E-4 m Inlet width iep_1 3.5 3.500 Isoelectric point, protein 1 iep_2 4.7 4.700 Isoelectric point, protein 2 iep_3 6.1 6.100 Isoelectric point, protein 3 iep_4 7.5 7.500 Isoelectric point, protein 4 iep_5 9 9.000 Isoelectric point, protein 5 iep_6 10.2 10.20 Isoelectric point, protein 6 D_p 5e-9[m^2/s] 5.000E-9 m²/s Protein diffusion coefficient mob_p 2e-13[s*mol/ kg] 2.000E-13 s·mol/ kg Protein mobility c_in 1000[mol/m^3] 1000 mol/m³ Inlet concentration v_max 2[mm/s] 0.002000 m/s Maximum fluid velocity V0 70[V] 70.00 V Voltage epsilon 80 80.00 Relative permittivity, water Solved with COMSOL Multiphysics 4.4 7 | I S O E L E C T R I C S E P A R A T I O N Step 1 (step1) 1 On the Home toolbar, click Functions and choose Global>Step. 2 In the Step settings window, locate the Parameters section. 3 In the From edit field, type -1. 4 Click to expand the Smoothing section. In the Size of transition zone edit field, type 1. G E O ME T R Y 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose mm. Rectangle 1 (r1) 1 Right-click Component 1 (comp1)>Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type W. 4 In the Height edit field, type H. Point 1 (pt1) 1 In the Model Builder window, right-click Geometry 1 and choose Point. 2 In the Point settings window, locate the Point section. 3 In the x edit field, type (W-a)/2. Point 2 (pt2) 1 Right-click Geometry 1 and choose Point. 2 In the Point settings window, locate the Point section. 3 In the x edit field, type (W+a)/2. Form Union (fin) In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Form Union (fin) and choose Build Selected. D E F I N I T I O N S Create an interpolation function for the pH value versus the x-coordinate. Interpolation 1 (int1) 1 On the Home toolbar, click Functions and choose Global>Interpolation. 2 In the Interpolation settings window, locate the Definition section. 3 In the Function name edit field, type pH. Solved with COMSOL Multiphysics 4.4 8 | I S O E L E C T R I C S E P A R A T I O N 4 In the table, enter the following settings: 5 Locate the Units section. In the Arguments edit field, type mm. 6 In the Function edit field, type 1. Variables 1 1 In the Model Builder window, right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose All domains. 5 Locate the Variables section. In the table, enter the following settings: E L E C T R O S T A T I C S ( E S ) Charge Conservation 1 1 In the Model Builder window, expand the Component 1 (comp1)>Electrostatics (es) node, then click Charge Conservation 1. 2 In the Charge Conservation settings window, locate the Electric Field section. 3 From the c r list, choose User defined. In the associated edit field, type epsilon. Electric Potential 1 1 On the Physics toolbar, click Boundaries and choose Electric Potential. t f(t) 0 2 0.5 3.5 1 4.9 1.5 6.3 2 7.8 2.5 9.2 3 10.5 3.5 12 Name Expression Unit Description c_tot c1+c2+c3+c4+c5+c6 mol/m³ Total protein concentration v_lam v_max*4*x/W*(1-x/W) m/s Laminar velocity profile Solved with COMSOL Multiphysics 4.4 9 | I S O E L E C T R I C S E P A R A T I O N 2 Select Boundary 1 only. Electric Potential 2 1 On the Physics toolbar, click Boundaries and choose Electric Potential. 2 Select Boundary 6 only. 3 In the Electric Potential settings window, locate the Electric Potential section. 4 In the V 0 edit field, type V0. TR A N S P O R T O F D I L U T E D S P E C I E S ( C H D S ) Add the effect of migration to the mass transport equations. 1 In the Model Builder window, under Component 1 (comp1) click Transport of Diluted Species (chds). 2 In the Transport of Diluted Species settings window, locate the Transport Mechanisms section. 3 Select the Migration in electric field check box. Convection, Diffusion, and Migration 1 In the Model Builder window, expand the Transport of Diluted Species (chds) node, then click Convection, Diffusion, and Migration. 2 In the Convection, Diffusion, and Migration settings window, locate the Model Inputs section. 3 Specify the u vector as 4 From the V list, choose Electric potential (es). 5 Locate the Diffusion section. In the D c1 through D c6 edit field, type D_p. 6 Locate the Migration in Electric Field section. From the Mobility list, choose User defined. 7 In the u m,c1 through u m,c6 edit field, type mob_p. 8 In the z c1 edit field, type step1(pH(x)-iep_1). 9 In the z c2 edit field, type step1(pH(x)-iep_2). 10 In the z c3 edit field, type step1(pH(x)-iep_3). 11 In the z c4 edit field, type step1(pH(x)-iep_4). 12 In the z c5 edit field, type step1(pH(x)-iep_5). 0 x v_lam y Solved with COMSOL Multiphysics 4.4 10 | I S O E L E C T R I C S E P A R A T I O N 13 In the z c6 edit field, type step1(pH(x)-iep_6). This creates a step change in the charge from -1 to +1 at the isoelectric point. Inflow 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 Select Boundary 4 only. 3 In the Inflow settings window, locate the Concentration section. 4 In the c 0,c1 through c 0,c6 edit field, type c_in. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 3 only. ME S H 1 1 In the Model Builder window, under Component 1 (comp1) click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Element size list, choose Extremely fine. You need a fine mesh to resolve the model properly. To reduce artificial oscillations, refine the mesh further at the inlet by following the steps below. Free Triangular 1 Right-click Component 1 (comp1)>Mesh 1 and choose Edit Physics-Induced Sequence. Size 1 1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Point. 4 Select Points 3 and 4 only. 5 Locate the Element Size section. From the Predefined list, choose Extremely fine. 6 Click the Custom button. 7 Locate the Element Size Parameters section. Select the Maximum element size check box. 8 In the associated edit field, type 0.01. Solved with COMSOL Multiphysics 4.4 11 | I S O E L E C T R I C S E P A R A T I O N 9 Click the Build All button. S T U D Y 1 Step 1: Stationary 1 On the Study toolbar, click Study Steps and choose Stationary>Stationary. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Step 2: Stationary 2 1 In the Model Builder window, under Study 1 click Step 2: Stationary 2. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Physics Solve for Discretization Transport of Diluted Species × physics Physics Solve for Discretization Electrostatics × physics Solved with COMSOL Multiphysics 4.4 12 | I S O E L E C T R I C S E P A R A T I O N 4 Click to expand the Study extensions section. Locate the Study Extensions section. Select the Adaptive mesh refinement check box. This will automatically refine the mesh where concentration gradients are sharp. 5 On the Home toolbar, click Compute. R E S U L T S Electric Potential (es) To reproduce the plot in Figure 3, add a surface arrow plot of the electric field to the first default plot showing the potential. 1 In the Model Builder window, right-click Electric Potential (es) and choose Arrow Surface. 2 In the Arrow Surface settings window, locate the Arrow Positioning section. 3 In the Points edit field, type 5. 4 In the Points edit field, type 10. 5 Locate the Coloring and Style section. From the Color list, choose Black. 6 On the 2D plot group toolbar, click Plot. 7 Click the Zoom Extents button on the Graphics toolbar. Concentration (chds) Modify the second default plot to show the total concentration. Compare the result with that in Figure 4. 1 In the Model Builder window, expand the Results>Concentration (chds) node, then click Surface 1. 2 In the Surface settings window, locate the Expression section. 3 Click Total protein concentration (c_tot) in the upper-right corner of the section. On the 2D plot group toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. Create a 1D plot at the outlet to see if the different proteins are well resolved. 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 3. This is the solution for the automatically refined mesh. Solved with COMSOL Multiphysics 4.4 13 | I S O E L E C T R I C S E P A R A T I O N 4 On the 1D plot group toolbar, click Line Graph. 5 Select Boundary 3 only. 6 In the Line Graph settings window, locate the y-Axis Data section. 7 In the Expression edit field, type c_tot. 8 Locate the x-axis data section. Click x-coordinate (x) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Solved with COMSOL Multiphysics 4.4 14 | I S O E L E C T R I C S E P A R A T I O N Solved with COMSOL Multiphysics 4.4 1 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Che mi c al V apor De pos i t i on of GaAs Introduction This example illustrates the modeling of a reactor for chemical vapor deposition (CVD). CVD is an important process for the electronics industry in which a thin film is grown on a substrate by allowing molecules and molecular fragments to adsorb and react on a surface. Combining detailed chemical reaction kinetics with transport models of a CVD reactor allows for realistic modeling of the deposition process. Such simulations in turn minimize the large number of expensive and time-consuming trial runs typically required for a reactor design. In the CVD process described here, triethyl-gallium (Ga(C 2 H 5 ) 3 ) first decomposes into a gas phase. The reaction products, along with arsine (AsH 3 ), then adsorb and react on a substrate to form GaAs layers. The CVD system is modeled using momentum, energy, and mass balances including a detailed description of the gas phase and adsorption kinetics (Ref. 1). The model highlights the usability of the Reaction Engineering interface together with its capability to generate space-dependent models for simulation of complex reaction/transport systems. In the Reaction Engineering interface you can easily study the transient behavior of different sets of reactions in a perfectly mixed system. On the basis of this analysis, you can then choose a reaction set, couple it seamlessly with a space-dependent model, and investigate how the chemistry behaves in a detailed reactor geometry with surface reactions present. Note: This model requires the Chemical Reaction Engineering Module. and either of the Heat Transfer Module or the CFD module. Model Definition You can describe the gas phase decomposition of Ga(C 2 H 5 ) 3 with three irreversible reactions: (1) Ga(C 2 H 5 ) 3 C 2 H 5 + k 1 Ga(C 2 H 5 ) 2 Solved with COMSOL Multiphysics 4.4 2 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S (2) (3) A number of radical reactions also take place in the gas phase: (4) (5) (6) (7) (8) (9) The growth of GaAs at the surface is governed by the adsorption of gas phase species and the subsequent reaction of the surface-bonded molecular fragments. Equation 10 to Equation 14 give the surface reactions involving the Ga and As species. S A and S G represent surface sites, corresponding to dangling bonds of As or Ga atoms, respectively. (10) (11) (12) (13) Ga(C 2 H 5 ) 2 Ga(C 2 H 5 )H C 2 H 4 + k 2 Ga(C 2 H 5 )H GaH 2 C 2 H 4 + k 3 + H 2 C 2 H 6 H + f r C 2 H 5 k 4 k 4 + 2CH 3 C 2 H 5 H k 5 + H 2 CH 4 + CH 3 H k 6 C 2 H 6 2CH 3 k 7 C 2 H 4 + C 2 H 5 k 8 H 2H 2 + H 2 2H k 9 Ga(C 2 H 5 ) 3 k 10 + 2C 2 H 5 + S A GaC 2 H 5 * Ga(C 2 H 5 )H + S A Ga(C 2 H 5 )H* f r k 11 k 11 GaH 2 k 12 Ga* + 2H + S A Ga* Ga(C 2 H 5 )H* k 13 + H C 2 H 5 + Solved with COMSOL Multiphysics 4.4 3 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S (14) The surface reactions of carbon and hydrogen fragments are given by: (15) (16) (17) (18) (19) (20) Finally, surface reactions leading to GaAs growth are given by: (21) (22) The reaction rates (mol/(m 3 ·s)) corresponding to the chemistry just described involve the mass action law . Here, and denote the forward and reverse rate constants, respectively. The concentration of species i is denoted c i (mol/m 3 ). The stoichiometric coefficients are denoted v ij , and are defined as negative for reactants and positive for products. The temperature dependence of the reaction rates is included through Arrhenius expressions for the rate constants: AsH 3 k 14 As* + 3H + S G + S A C 2 H 5 C 2 H 5A * f r k 15 k 15 + S G C 2 H 5 C 2 H 5G * f r k 16 k 16 k 17 + C 2 H 5A * C 2 H 4 H A * k 18 + C 2 H 5G * C 2 H 4 H G * k 19 + S A H A * H k 20 + S G H G * H k 21 GaAs + C 2 H 5 GaC 2 H 5 * As* + + S A + S G k 22 GaAs Ga* As* + + S A + S G r j k j f c i v – j k j r c i v ij i prod e [ – i react e [ = k j f k j r Solved with COMSOL Multiphysics 4.4 4 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S . In this equation, A denotes the frequency factor, T the temperature (K), n the temperature exponent, E the activation energy (J/mol), and R g the ideal gas constant, 8.314 J/(mol·K). The frequency factor is expressed in the units , where o is the order of the reaction. Adsorption kinetics can also affect the overall rates. The rate of adsorption of species i at the substrate surface is where c i is the species concentration (mol/m 3 ), M i denotes the molecular weight (kg/m 3 ), x s is the fraction of available surface sites, and E gives the activation energy (J/mol). k AT n E R g T ----------- – \ . | | exp = (m 3 /mol) o 1 – /s r i c i R g T 2tM i --------------x s E R g T ----------- – \ . | | exp = Solved with COMSOL Multiphysics 4.4 5 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S In this model, the chemical species occurring in the reactions just outlined have the following labels: any label starting with SA indicates a surface species adsorbed at an arsenic site, while SG denotes a surface species adsorbed at a gallium site. Figure 22-1: Species labels used in the model. The analysis process follows these steps: First, take the chemistry just outlined and enter it into the Reaction Engineering interface. Next study the material balances of different reaction networks—while assuming they are working in a perfectly mixed batch reactor—to quickly get an overview of the reaction kinetics. Then use the Reaction Engineering interface’s predefined expressions for species transport properties. With this preprocessing completed, generate the space dependent model to investigate the effects of the CVD reactor geometry on the reacting system. You can also study the effects of the adsorption and surface reactions. Figure 1 shows the CVD reactor as drawn in COMSOL Multiphysics. The reactor is 40 cm long and 10 cm high. Located in the center is the substrate, 5 cm across and Ga(C 2 H 5 ) 3 GaC 2 H 5 * Ga(C 2 H 5 )H Ga(C 2 H 5 )H* GaH 2 Ga* AsH 3 As* H 2 C 2 H 6 H C 2 H 5 Ga(C 2 H 5 ) 2 C 2 H 4 CH 3 CH 4 H* C 2 H 5 * GaAs gaet3 gaet2 gahet gah2 SAgahet SAgaet SAga ash3 SGas gaas ethane et ethene methane met h2 h SAet, SGet SAh, SGh Solved with COMSOL Multiphysics 4.4 6 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S tilted 10° with respect to the vertical position. Gas enters the reactor at the inlet with a velocity of 0.4 m/s and at a pressure of 4000 Pa. Figure 1: The modeling domain consists of the CVD reactor and the substrate surface. You solve the 2D space-dependent problem using three interfaces: • Transport of diluted species • Heat Transfer in Fluids • Laminar Flow As such, the gas in the reactor domain is described by momentum, heat, and mass-balance equations. The substrate wall is modeled with the feature Interior Wall in the Laminar flow interface. Results and Discussion As noted, the first step in the modeling process is to enter the complete set of gas phase reactions, Equation 1 to Equation 9, into the Reaction Engineering interface for analysis. Figure 2 shows the species concentrations as functions of time in a perfectly wall wall outlet inlet substrate Solved with COMSOL Multiphysics 4.4 7 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S mixed batch reactor kept at 900 K. Figure 2: The complete set of gas phase reactions including decomposition reactions of gallium species as well as radical reactions. The chemistry occurs in a perfectly mixed batch reactor held at 900K. Radical species are not shown in the graph. As a test, omit the radical reactions given by Equation 4 to Equation 9 from the set of gas phase reactions. Once again analyze the kinetics of the reactions describing gallium species decomposition, Equation 1 to Equation 3, at 900 K. The results appear in Figure 3. Solved with COMSOL Multiphysics 4.4 8 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Figure 3: A reduced set of gas phase reactions including only the decomposition reactions of gallium species. Reactions occur in a perfectly mixed system held at 900 K. Reducing the gas phase reaction set does not affect the reactions of the gallium species. However, excluding the radical reactions has a considerable influence on the carbon-species distribution. For the reduced reaction set, ethene and ethyl radicals are the main carbon products; for the full reaction set the main products are ethene and methane. The various species have different characteristics with respect to surface adsorption and reaction. Furthermore, the net concentration of carbon species is higher for the full reaction set. Both these factors can significantly influence the growth of surface layers. For a first study of geometrical effects on the reacting system, you can bring the reduced reaction model into the actual geometry of the CVD reactor and then solve the space-dependent problem. Solved with COMSOL Multiphysics 4.4 9 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Figure 4: The gas phase velocity in the reactor domain. Figure 5: The temperature distribution in the reactor domain. Figure 4 shows the fluid velocity and Figure 5 the temperature distribution in the reactor domain. The gas mixture enters the reactor with a velocity of 0.4 m/s and a Solved with COMSOL Multiphysics 4.4 10 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S temperature of 300 K with the substrate held at a constant temperature of 900 K. Notice the large effect that the heating plate has on the temperature and the expansion this causes in the fluid. This effect is seen in the average velocity, which increases downstream after the position of the substrate. Figure 6: Concentration distribution of triethyl-gallium in the reactor domain. Solved with COMSOL Multiphysics 4.4 11 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Figure 7: Concentration profiles of triethyl-gallium (blue line) and gallium hydride (green line) along the reactor centerline. In Figure 6 shows the concentration distribution of the triethyl-gallium species in the reactor domain, while Figure 7 displays the concentration profile along the reactor centerline for triethyl-gallium together with that of the final product gallium hydride. Triethyl-gallium is stable at the inlet temperature (300 K) and then rapidly decomposes near the hot substrate. Figure 8 shows the concentration profile of arsine along the reactor centerline. This species does not decompose in the gas phase. The decrease in concentration at the Solved with COMSOL Multiphysics 4.4 12 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S substrate surface (at the 0 length coordinate) is due to the adsorption of arsine at the surface. Figure 8: Concentration profile of arsine along the reactor centerline. Arsine is adsorbed at the substrate surface, which is located at the center of the length scale. Figure 9 and Figure 10 depict a few of the transport properties set up in the Reaction Engineering interface and coupled to the physics interfaces of the space-dependent model. Figure 9 shows the diffusivities of triethyl-gallium (bottom) and arsine (top). Figure 10 shows the thermal conductivity of the hydrogen carrier gas. All variables are plotted as functions of temperature. Solved with COMSOL Multiphysics 4.4 13 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Figure 9: The diffusivities of triethyl-gallium (bottom) and arsine (top) as functions of temperature. Figure 10: The thermal conductivity of the hydrogen carrier gas. Solved with COMSOL Multiphysics 4.4 14 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S In summary, this model illustrates how to solve an applied multiphysics problem using the Chemical Reaction Engineering Module. Set up the chemical kinetics in the Reaction Engineering interface by entering the chemical reaction formulas. It is then easy to study different models by effortlessly activating/deactivating reactions. Then, making use of a predefined library of expressions, set up the transport properties of the reacting system before generating the space-dependent model. Finally, add the effect of the CVD reactor’s geometry and of surface reactions to the model. Reference 1. N.K. Ingle, C. Theodoropoulos, T.J. Mountziaris, R.M. Wexler, and F.T.J. Smith, J. Crystal Growth, vol. 167, p. 543, 1996. Model Library path: Chemical_Reaction_Engineering_Module/Surface_Reac tions_and_Deposition_Processes/gaas_cvd Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Time Dependent. 6 Click the Done button. R E A C T I O N E N G I N E E R I N G 1 In the Model Builder window, under Component 1 click Reaction Engineering. 2 In the Reaction Engineering settings window, locate the Reactor Settings section. 3 Select the Calculate thermodynamic properties check box. Solved with COMSOL Multiphysics 4.4 15 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 4 Select the Calculate transport properties check box. 5 Locate the General section. In the T edit field, type 900[K]. Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type gaet3=>gaet2+et. 4 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 5 In the A f edit field, type 1e15. 6 In the E f edit field, type 195e3. Reaction 2 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type gaet2=>gahet+ethene. 4 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 5 In the A f edit field, type 1e18. 6 In the E f edit field, type 195e3. Reaction 3 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type gahet=>gah2+ethene. 4 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 5 In the A f edit field, type 1e18. 6 In the E f edit field, type 195e3. Reaction 4 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type et+h2<=>ethane+h. 4 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 5 In the A f edit field, type 3.1e-6. 6 In the n f edit field, type 3.6. 7 In the E f edit field, type 35.6e3. Solved with COMSOL Multiphysics 4.4 16 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 8 In the A r edit field, type 5.5e-4. 9 In the n r edit field, type 3.5. 10 In the E r edit field, type 21.8e3. Reaction 5 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type et+h=>2met. 4 Locate the Rate Constants section. In the k f edit field, type 3.6e7. Reaction 6 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type met+h2=>methane+h. 4 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 5 In the A f edit field, type 2.9e-4. 6 In the n f edit field, type 3.12. 7 In the E f edit field, type 36.4e3. Reaction 7 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type 2met=>ethane. 4 Locate the Rate Constants section. In the k f edit field, type 2e6. Reaction 8 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type ethene+h=>et. 4 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 5 In the A f edit field, type 1.5e2. 6 In the n f edit field, type 1.49. 7 In the E f edit field, type 41.8e3. Reaction 9 1 On the Physics toolbar, click Global and choose Reaction. Solved with COMSOL Multiphysics 4.4 17 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type 2h+h2=>2h2. 4 Locate the Rate Constants section. In the k f edit field, type 1e6. Verify that 11 species are added automatically. For each species, specify the molecular weight and the initial concentration. Species: gaet3 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gaet3. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 156.7e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 6.415e-3. Species: gaet2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gaet2. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 127.7e-3. Species: et 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: et. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 29e-3. Species: gahet 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gahet. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 99.7e-3. Species: ethene 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: ethene. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 28e-3. Solved with COMSOL Multiphysics 4.4 18 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Species: gah2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gah2. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 71.7e-3. Species: h2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: h2. 2 In the Species settings window, locate the Species Formula section. 3 From the list, choose Solvent. Note that this step automatically selects the Lock concentration/activity check box. This includes the influence of hydrogen concentration as a constant in the rate expressions. 4 Locate the General Parameters section. In the M w edit field, type 2e-3. 5 Locate the General Expressions section. In the c 0 edit field, type 1.44. Species: ethane 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: ethane. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 30e-3. Species: h 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: h. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 1e-3. Species: met 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: met. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 15e-3. Species: methane 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: methane. Solved with COMSOL Multiphysics 4.4 19 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 16e-3. S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S Concentration (re) Delete the expression for c_gaet2, c_et, c_gahet, c_h2, c_h, and c_met. 4 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Cycle. 5 Click to expand the Legends section. From the Legends list, choose Manual. 6 In the table, enter the following settings: 7 Click the x-Axis Log Scale button on the Graphics toolbar. 8 Click the y-Axis Log Scale button on the Graphics toolbar. 9 In the Model Builder window, click Concentration (re). 10 In the 1D Plot Group settings window, click to expand the Axis section. 11 Select the Manual axis limits check box. 12 In the y minimum edit field, type 1e-5. 13 In the y maximum edit field, type 1e-1. 14 Click to expand the Legend section. From the Position list, choose Lower right. 15 On the 1D plot group toolbar, click Plot. 16 Click to expand the Title section. From the Title type list, choose None. 17 On the 1D plot group toolbar, click Plot. To reduce the model before simulating the process in a 2-dimensional model, study whether it is possible to remove the non-gallium species and reactions and yet obtain approximately the same results. To do so, modify the existing reaction model by first Legends gaet3 ethene gah2 ethane methane Solved with COMSOL Multiphysics 4.4 20 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S deactivating reactions of non-gallium species. Then re-solve the material balances and compare the results with the full reaction model. In order not to lose the previous solution, which is to be used for comparison, copy the solution. S T U D Y 1 In the Model Builder window, expand the Study 1 node. Solver 1 In the Model Builder window, expand the Study 1>Solver Configurations node. Copy 2 1 Right-click Solver 1 and choose Solution>Copy. 2 In the Model Builder window, under Study 1>Solver Configurations right-click Copy 2 and choose Rename. 3 Go to the Rename Solver dialog box and type complete_set in the New name edit field. 4 Click OK. R E S U L T S Concentration (re) 1 In the 1D Plot Group settings window, locate the Data section. 2 From the Data set list, choose Solution 2. R E A C T I O N E N G I N E E R I N G 4: et+h2<=>ethane+h In the Model Builder window, under Component 1>Reaction Engineering, multi-select Reactions 4-9, and right-click and choose Disable. S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S Concentration (re) 1 1 In the Model Builder window, expand the Concentration (re) 1 node, then click Global 1. 2 In the Global settings window, locate the Coloring and Style section. Solved with COMSOL Multiphysics 4.4 21 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 3 Find the Line markers subsection. From the Marker list, choose Cycle. 4 Locate the Legends section. From the Legends list, choose Manual. 5 In the table, enter the following settings: 6 Click the x-Axis Log Scale button on the Graphics toolbar. 7 Click the y-Axis Log Scale button on the Graphics toolbar. 8 In the Model Builder window, click Concentration (re) 1. 9 In the 1D Plot Group settings window, locate the Axis section. 10 Select the Manual axis limits check box. 11 In the y minimum edit field, type 1e-5. 12 In the y maximum edit field, type 1e-1. 13 Locate the Legend section. From the Position list, choose Lower right. 14 Locate the Title section. From the Title type list, choose None. 15 On the 1D plot group toolbar, click Plot. A comparison of Figure 2 and Figure 3 reveals that the gallium-related reactions remain approximately the same (see the Results and Discussion section for a deeper analysis and discussion). This means that you can go on to set up a dimensional CVD model based on the reduced model instead of the one comprising all species. In this CVD model, you add the surface adsorption/desorption kinetics to the reactions of the gas phase. Furthermore, before coupling the reduced model to the reactor geometry, the exercise makes use of predefined expressions for species transport properties in the Reaction Engineering interface. Briefly summarized, the step-by-step instructions below implement the following procedure: First remove the non-gallium reactions from the reaction engineering interface. Then add the necessary data for describing the thermodynamics and transport parameters necessary for the dimensional model. Finally generate the space dependent model of the reactor subdomain. Legends gaet3 gaet2 et gahet ethene gah2 Solved with COMSOL Multiphysics 4.4 22 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Now proceed to create the surface reactions that occur at the substrate. You first modify the reaction model and then couple it to the substrate boundary of the 2D model. The reaction engineering model describes the source terms necessary for defining the mass balance boundary conditions on the substrate surface. In this case you must first set up the surface temperature, Tsurf, as a model parameter. R E A C T I O N E N G I N E E R I N G Species: h2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: h2. 2 In the Species settings window, locate the Species Formula section. 3 From the list, choose None. Surface Reaction 1 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type gaet3=>SAgaet+2et. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the e f edit field, type 20.9e3. 6 Locate the Kinetics Expressions section. In the q edit field, type ksf_1*c_gaet3*((R_const*T)/(2*pi*M_gaet3))^0.5*x_A_surf. Surface Reaction 2 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type gahet=>SAgahet. 4 Locate the Kinetics Expressions section. In the q edit field, type ksf_2*c_gahet*((R_const*T)/(2*pi*M_gahet))^0.5*x_A_surf. Surface Reaction 3 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SAgahet=>gahet. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. Solved with COMSOL Multiphysics 4.4 23 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 5 In the a f edit field, type 5e8. 6 In the e f edit field, type 146e3. Surface Reaction 4 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type gah2=>SAga+2h. 4 Locate the Kinetics Expressions section. In the q edit field, type ksf_4*c_gah2*((R_const*T)/(2*pi*M_gah2))^0.5*x_A_surf. Surface Reaction 5 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SAgahet=>SAga+et+h. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 5e11. 6 In the e f edit field, type 134e3. Surface Reaction 6 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type ash3=>SGas+3h. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the e f edit field, type 20.9e3. 6 Locate the Kinetics Expressions section. In the q edit field, type ksf_6*c_ash3*((R_const*T)/(2*pi*M_ash3))^0.5*x_G_surf. Surface Reaction 7 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type et=>SAet. 4 Locate the Kinetics Expressions section. In the q edit field, type ksf_7*c_et*((R_const*T)/(2*pi*M_et))^0.5*x_A_surf. Solved with COMSOL Multiphysics 4.4 24 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Surface Reaction 8 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SAet=>et. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 7.9e11. 6 In the e f edit field, type 151e3. Surface Reaction 9 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type et=>SGet. 4 Locate the Kinetics Expressions section. In the q edit field, type ksf_9*c_et*((R_const*T)/(2*pi*M_et))^0.5*x_G_surf. Surface Reaction 10 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SGet=>et. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 7.9e11. 6 In the e f edit field, type 151e3. Surface Reaction 11 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SAet=>ethene+SAh. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 2.5e10. 6 In the e f edit field, type 134e3. Surface Reaction 12 1 On the Physics toolbar, click Global and choose Surface Reaction. Solved with COMSOL Multiphysics 4.4 25 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SGet=>ethene+SGh. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 2.5e10. 6 In the e f edit field, type 134e3. Surface Reaction 13 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SAh=>h. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 7.2e5. 6 In the e f edit field, type 67.4e3. Surface Reaction 14 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SGh=>h. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 7.2e5. 6 In the e f edit field, type 67.4e3. Surface Reaction 15 1 On the Physics toolbar, click Global and choose Surface Reaction. 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SAgaet+SGas=>gaas+et. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 1.1e6. 6 In the e f edit field, type 4.18e3. Surface Reaction 16 1 On the Physics toolbar, click Global and choose Surface Reaction. Solved with COMSOL Multiphysics 4.4 26 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 2 In the Surface Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type SAga+SGas=>gaas. 4 Locate the Arrhenius Parameters section. Select the Use Arrhenius expressions check box. 5 In the a f edit field, type 1.1e6. 6 In the e f edit field, type 4.18e3. Species: gaet3 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gaet3. 2 In the Species settings window, click to expand the Species transport parameters section. 3 Locate the Species Transport Parameters section. In the o edit field, type 6.245[angstrom]. 4 In the c/k b edit field, type 478.6. Species: gaet2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gaet2. 2 In the Species settings window, click to expand the Species transport parameters section. 3 Locate the Species Transport Parameters section. In the o edit field, type 5.487[angstrom]. 4 In the c/k b edit field, type 516.7. Species: et 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: et. 2 In the Species settings window, click to expand the Species transport parameters section. 3 Locate the Species Transport Parameters section. In the o edit field, type 4.291[angstrom]. 4 In the c/k b edit field, type 184.5. Species: gahet 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gahet. Solved with COMSOL Multiphysics 4.4 27 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 2 In the Species settings window, click to expand the Species transport parameters section. 3 Locate the Species Transport Parameters section. In the o edit field, type 4.720[angstrom]. 4 In the c/k b edit field, type 554.8. Species: ethene 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: ethene. 2 In the Species settings window, click to expand the Species transport parameters section. 3 Locate the Species Transport Parameters section. In the o edit field, type 4.163[angstrom]. 4 In the c/k b edit field, type 224.7. Species: gah2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gah2. 2 In the Species settings window, click to expand the Species transport parameters section. 3 Locate the Species Transport Parameters section. In the o edit field, type 3.664[angstrom]. 4 In the c/k b edit field, type 345.3. Species: h2 1 In the Model Builder window, under Component 1>Reaction Engineering right-click Species: h2 and choose Enable. 2 In the Species settings window, locate the Species Formula section. 3 From the list, choose Solvent. 4 Locate the General Expressions section. In the c 0 edit field, type 1.44. 5 Click to expand the Species thermodynamic parameters section. Locate the Species Thermodynamic Parameters section. In the T lo edit field, type 200. 6 In the T hi edit field, type 3000. 7 In the a low,k table, enter the following settings: 2.3433 7.9805e-3 Solved with COMSOL Multiphysics 4.4 28 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 8 In the a hi,k table, enter the following settings: 9 Click to expand the Species transport parameters section. Locate the Species Transport Parameters section. In the o edit field, type 2.827[angstrom]. 10 In the c/k b edit field, type 59.7. Species: ash3 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: ash3. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 77.9e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 0.16. 5 Click to expand the Species transport parameters section. Locate the Species Transport Parameters section. In the o edit field, type 4.145[angstrom]. 6 In the c/k b edit field, type 259.8. Species: SAgaet 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: SAgaet. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 98.7e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 1.4e-9. 5 Select the Lock concentration/activity check box. -1.9481e-5 2.0172e-8 -7.3761e-12 -9.1793e2 6.8301e-1 3.3373 -4.9402e-5 4.9946e-7 -1.7957e-10 2.0026e-14 -9.5016e2 -3.2050 Solved with COMSOL Multiphysics 4.4 29 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Species: SAgahet 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: SAgahet. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 99.7e-3. 4 Locate the General Expressions section. Select the Lock concentration/activity check box. Species: SAga 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: SAga. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 69.7e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 7.2e-6. 5 Select the Lock concentration/activity check box. Species: SGas 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: SGas. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 74.9e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 7.3e-6. 5 Select the Lock concentration/activity check box. Species: SAet 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: SAet. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 29e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 2.9e-8. 5 Select the Lock concentration/activity check box. Species: SGet 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: SGet. 2 In the Species settings window, locate the General Parameters section. Solved with COMSOL Multiphysics 4.4 30 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 3 In the M w edit field, type 29e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 1.7e-8. 5 Select the Lock concentration/activity check box. Species: SAh 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: SAh. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 1e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 6.8e-8. 5 Select the Lock concentration/activity check box. Species: SGh 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: SGh. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 1e-3. 4 Locate the General Expressions section. In the c 0 edit field, type 4e-8. 5 Select the Lock concentration/activity check box. Species: gaas 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: gaas. 2 In the Species settings window, locate the General Parameters section. 3 In the M w edit field, type 144.6e-3. 4 Locate the General Expressions section. Select the Lock concentration/activity check box. Species: h 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: h. 2 In the Species settings window, click to expand the Species transport parameters section. 3 Locate the Species Transport Parameters section. In the o edit field, type 1.825[angstrom]. 4 In the c/k b edit field, type 2.31. Solved with COMSOL Multiphysics 4.4 31 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Generate Space-Dependent Model 1 1 In the Model Builder window, right-click Reaction Engineering and choose Generate Space-Dependent Model. 2 In the Generate Space-Dependent Model settings window, locate the Geometry Settings section. 3 From the Geometry to use list, choose 2D: New. 4 Locate the Physics Interfaces section. From the Energy balance list, choose Heat Transfer in Fluids: New. 5 From the Momentum balance list, choose Laminar Flow: New. 6 Locate the Space-Dependent Model Generation section. Click the Create/Refresh button. G E O ME T R Y 1 In the Model Builder window, expand the Component 2 node. Rectangle 1 1 Right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 0.4. 4 In the Height edit field, type 0.1. 5 Locate the Position section. From the Base list, choose Center. 6 Click the Build Selected button. Polygon 1 1 In the Model Builder window, right-click Geometry 1 and choose Polygon. 2 In the Polygon settings window, locate the Coordinates section. 3 In the x edit field, type 0 0. 4 In the y edit field, type -0.025 0.025. 5 Click the Build Selected button. Rotate 1 1 On the Geometry toolbar, click Rotate. 2 Select the object pol1 only. 3 In the Rotate settings window, locate the Rotation Angle section. 4 In the Rotation edit field, type -10. 5 Click the Build Selected button. Solved with COMSOL Multiphysics 4.4 32 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Form Union In the Model Builder window, under Component 2>Geometry 1 right-click Form Union and choose Build Selected. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file gaas_cvd_parameters.txt. 5 In the Model Builder window’s toolbar, click the Show button and select Advanced Physics Options in the menu. TR A N S P O R T O F D I L U T E D S P E C I E S 1 1 In the Model Builder window, under Component 2 click Transport of Diluted Species 1. 2 In the Transport of Diluted Species settings window, click to expand the Advanced settings section. 3 Locate the Advanced Settings section. From the Convective term list, choose Conservative form. Flux Discontinuity 1 In the Model Builder window, expand the Transport of Diluted Species 1 node, then click Flux Discontinuity. 2 Select Boundary 4 only. Inflow 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 Select Boundary 1 only. 3 In the Inflow settings window, locate the Concentration section. 4 In the c 0,cgaet3 edit field, type c_gaet3_in. 5 In the c 0,cash3 edit field, type c_ash3_in. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 5 only. Solved with COMSOL Multiphysics 4.4 33 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S H E A T TR A N S F E R I N F L U I D S 1 Heat Transfer in Fluids 1 1 In the Model Builder window, expand the Component 2>Heat Transfer in Fluids 1 node, then click Heat Transfer in Fluids 1. 2 In the Heat Transfer in Fluids settings window, locate the Model Inputs section. 3 Clear the Reference pressure check box. Initial Values 1 1 In the Model Builder window, under Component 2>Heat Transfer in Fluids 1 click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the T edit field, type T_in. Temperature 1 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 Select Boundaries 1–3 only. 3 In the Temperature settings window, locate the Temperature section. 4 In the T 0 edit field, type T_in. Temperature 2 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 Select Boundary 4 only. 3 In the Temperature settings window, locate the Temperature section. 4 In the T 0 edit field, type T_surf. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 5 only. L A MI N A R F L OW 1 Initial Values 1 1 In the Model Builder window, expand the Component 2>Laminar Flow 1 node, then click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the p edit field, type p_0. Solved with COMSOL Multiphysics 4.4 34 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S Inlet 1 1 In the Model Builder window, under Component 2>Laminar Flow 1 click Inlet 1. 2 Select Boundary 1 only. 3 In the Inlet settings window, locate the Velocity section. 4 In the U 0 edit field, type u_in. Outlet 1 1 In the Model Builder window, under Component 2>Laminar Flow 1 click Outlet 1. 2 Select Boundary 5 only. 3 In the Outlet settings window, locate the Pressure Conditions section. 4 In the p 0 edit field, type p_0. 5 Select the Normal flow check box. Interior Wall 1 1 On the Physics toolbar, click Boundaries and choose Interior Wall. 2 Select Boundary 4 only. ME S H 1 1 In the Model Builder window, under Component 2 click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Element size list, choose Finer. 4 Click the Build All button. S T U D Y 2 On the Home toolbar, click Compute. R E S U L T S Velocity (spf1) To create Figure 4, follow these steps: 1 In the 2D Plot Group settings window, click to expand the Color legend section. 2 Locate the Color Legend section. From the Position list, choose Bottom. 3 On the 2D plot group toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. Temperature (ht1) To reproduce Figure 5, do the following Solved with COMSOL Multiphysics 4.4 35 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 1 In the Model Builder window, under Results click Temperature (ht1). 2 In the 2D Plot Group settings window, locate the Color Legend section. 3 From the Position list, choose Bottom. 4 Click the Zoom Extents button on the Graphics toolbar. Concentration (chds1) You can reproduce Figure 6 as follows: 1 In the Model Builder window, under Results click Concentration (chds1). 2 In the 2D Plot Group settings window, locate the Color Legend section. 3 From the Position list, choose Bottom. 4 Click the Zoom Extents button on the Graphics toolbar. Data Sets 1 On the Results toolbar, click Cut Line 2D. 2 In the Cut Line 2D settings window, locate the Line Data section. 3 In row Point 1, set x to -0.2. 4 In row Point 2, set x to 0.2. 5 Click the Plot button. In order to produce the remaining figures, illustrating various results along the reactor centerline, use the CutLine2D data set. 1D Plot Group 8 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Line 2D 1. 4 On the 1D plot group toolbar, click Line Graph. 5 In the Line Graph settings window, locate the x-axis data section. 6 Click x-coordinate (x) in the upper-right corner of the section. Click to expand the Legends section. Select the Show legends check box. 7 From the Legends list, choose Manual. 8 In the table, enter the following settings: 9 On the 1D plot group toolbar, click Line Graph. Legends gaet3 Solved with COMSOL Multiphysics 4.4 36 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 10 In the Line Graph settings window, locate the y-axis data section. 11 Click Concentration (cgah2) in the upper-right corner of the section. Locate the x-axis data section. Click x-coordinate (x) in the upper-right corner of the section. Locate the Legends section. Select the Show legends check box. 12 From the Legends list, choose Manual. 13 In the table, enter the following settings: 14 In the Model Builder window, click 1D Plot Group 8. 15 In the 1D Plot Group settings window, locate the Title section. 16 From the Title type list, choose None. 17 On the 1D plot group toolbar, click Plot. 1D Plot Group 9 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Line 2D 1. 4 On the 1D plot group toolbar, click Line Graph. 5 In the Line Graph settings window, locate the y-axis data section. 6 Click Concentration (cash3) in the upper-right corner of the section. Locate the x-axis data section. Click x-coordinate (x) in the upper-right corner of the section. In the Model Builder window, click 1D Plot Group 9. 7 In the 1D Plot Group settings window, locate the Title section. 8 From the Title type list, choose None. 9 Click to expand the Grid section. Select the Manual spacing check box. 10 In the x spacing edit field, type 0.05. 11 In the y spacing edit field, type 0.01. 12 On the 1D plot group toolbar, click Plot. The Reaction Engineering interface calculates the diffusivities, the thermal conductivity, and other fluid properties, including their temperature dependence. Next, plot the diffusivities along the reactor centerline for two of the species as functions of the temperature. Legends gah2 Solved with COMSOL Multiphysics 4.4 37 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 1D Plot Group 10 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Line 2D 1. 4 On the 1D plot group toolbar, click Line Graph. 5 In the Line Graph settings window, locate the y-axis data section. 6 Click Average diffusion coefficient (chds.Dav_cgaet3) in the upper-right corner of the section. Locate the x-axis data section. Click Temperature (T) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. 7 Locate the Legends section. Select the Show legends check box. 8 On the 1D plot group toolbar, click Plot. 9 From the Legends list, choose Manual. 10 In the table, enter the following settings: 11 On the 1D plot group toolbar, click Line Graph. 12 In the Line Graph settings window, locate the y-axis data section. 13 Click Average diffusion coefficient (chds.Dav_cash3) in the upper-right corner of the section. Locate the x-axis data section. Click Temperature (T) in the upper-right corner of the section. Locate the Legends section. Select the Show legends check box. 14 From the Legends list, choose Manual. 15 On the 1D plot group toolbar, click Plot. 16 In the table, enter the following settings: 17 In the Model Builder window, click 1D Plot Group 10. 18 In the 1D Plot Group settings window, locate the Title section. 19 From the Title type list, choose None. 20 Click to expand the Grid section. Select the Manual spacing check box. 21 In the x spacing edit field, type 100. 22 In the y spacing edit field, type 1e-3. Legends gaet3 Legends ash3 Solved with COMSOL Multiphysics 4.4 38 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S 23 Locate the Legend section. From the Position list, choose Upper left. 24 On the 1D plot group toolbar, click Plot. Finally, plot the thermal conductivity of the hydrogen carrier gas, displayed in the right panel of Figure 8: 1D Plot Group 11 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Line 2D 1. 4 On the 1D plot group toolbar, click Line Graph. 5 In the Line Graph settings window, locate the y-axis data section. 6 Click Mean effective thermal conductivity (ht.kmean) in the upper-right corner of the section. Locate the x-axis data section. Click Temperature (T) in the upper-right corner of the section. In the Model Builder window, click 1D Plot Group 11. 7 In the 1D Plot Group settings window, locate the Title section. 8 From the Title type list, choose None. 9 On the 1D plot group toolbar, click Plot. Solved with COMSOL Multiphysics 4.4 1 | L A M I N A R S T A T I C M I X E R L a mi na r S t a t i c Mi x e r Introduction In static mixers, also called motionless or in-line mixers, a fluid is pumped through a pipe containing stationary blades. This mixing technique is particularly well suited for laminar flow mixing because it generates only small pressure losses in this flow regime. This example studies the flow in a twisted-blade static mixer. It evaluates the mixing performance by calculating the concentration’s standard deviation. Model Definition This model studies the mixing of one species dissolved in water at room temperature. The geometry consists of a tube with three twisted blades of alternating rotations (Figure 1). Figure 1: Depiction of a laminar static mixer containing three blades with alternating rotations. The tube’s radius, R, is 6 mm; the length is 14R, and the length of each blade is 3R. The inlet flow is laminar and fully developed with an average velocity of 1 cm/s. At the outlet, the model specifies a constant reference pressure of 0 Pa. The equations for the momentum transport are the stationary Navier-Stokes equations in 3D: Solved with COMSOL Multiphysics 4.4 2 | L A M I N A R S T A T I C M I X E R (1) Here q denotes the dynamic viscosity (kg/(m·s)), u is the velocity (m/s), µ represents the fluid density (kg/m 3 ), and p denotes the pressure (Pa). The fluid’s properties are not affected by the change in concentration of the dissolved species. The model studies the mixing performance by assuming a discontinuous concentration profile at the mixer’s inlet. The inlet concentration is defined as (2) with the line x = 0 separating the two inlet sides. Diffusion and convection contribute to the mass flux, and the resulting mass transport equation is: (3) Here D denotes the diffusion coefficient (m 2 /s), and c is the concentration (mol/m 3 ). At the outlet, the mass transport is mainly driven by convection. That is, the transport by diffusion is neglected in the normal direction of the pipe’s cross section. Because the convective term leads to instabilities in the solution, you need a fine mesh to obtain a stable solution for the concentration field. The low Reynolds numbers, in the mixer implies that the Navier-Stokes equations do not require a particularly dense mesh. You can therefore first solve the Navier-Stokes equations on a coarse mesh and then map the solution onto a finer mesh. In the last solution step you use this mapped velocity field in the convective mass-transport term. Results Figure 2 shows a slice plot of the concentration in the mixer. The slice at the bottom shows the lighter and darker halves of the fluid with and without the dissolved species, respectively. As the fluid flows upward through the system, the two solutions are mixed and an almost constant concentration is obtained at the outlet. µ u V · ( )u V pI – q Vu Vu ( ) T + ( ) + | | · = V u · 0 = c inlet c 0 x 0 < 0 x 0 > ¹ ´ ¦ = V D c cu) + V – ( · 0 = Solved with COMSOL Multiphysics 4.4 3 | L A M I N A R S T A T I C M I X E R Figure 2: Slice plot of the concentration at different distances from the inlet. Solved with COMSOL Multiphysics 4.4 4 | L A M I N A R S T A T I C M I X E R Figure 3 shows the flow field responsible for the mixing. The streamlines clearly reveal the twisting motion in the fluid that is induced by the mixer blades. Figure 3: Slice plots of the velocity magnitude field inside the mixer. The streamlines show the flow direction. You can also visualize the mixing through a series of cross-section plots. Figure 4 contains such a series of plots showing the concentration in the mixer’s cross section Solved with COMSOL Multiphysics 4.4 5 | L A M I N A R S T A T I C M I X E R along the direction of the flow. The results show that most of the mixing takes place where the blades change rotational direction (the three middle figures). Figure 4: Cross-sectional plots of the concentration at different distances from the inlet. The nine plots shows the concentration at z =- 2 mm to z = 30 mm in steps of 4 mm. References 1. R. Perry and D. Green, Perry’s Chemical Engineering Handbook, 7th ed., McGraw-Hill, 1997. 2. J.M. Coulson and J.F. Richardson, Chemical Engineering, vol. 1, 4th ed., Pergamon Press, 1990. Model Library path: Chemical_Reaction_Engineering_Module/Mixing/ laminar_static_mixer Solved with COMSOL Multiphysics 4.4 6 | L A M I N A R S T A T I C M I X E R Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 3D button. 2 In the Select physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf). 3 Click the Add button. 4 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 5 Click the Add button. 6 Click the Study button. 7 In the tree, select Preset Studies for Selected Physics>Stationary. 8 Click the Done button. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Step 1 (step1) 1 On the Home toolbar, click Functions and choose Global>Step. 2 In the Step settings window, locate the Parameters section. 3 In the To edit field, type 5. 4 Click to expand the Smoothing section. In the Size of transition zone edit field, type 3e-4. Name Expression Value Description ra 3[mm] 0.003000 m Tube radius u_mean 10[mm/s] 0.01000 m/s Mean inlet velocity c0 5[mol/m^3] 5.000 mol/m³ Inlet concentration D 5e-8[m^2/s] 5.000E-8 m²/s Diffusion coefficient Solved with COMSOL Multiphysics 4.4 7 | L A M I N A R S T A T I C M I X E R G E O ME T R Y 1 Create the geometry. To simplify this step, insert a prepared geometry sequence: 1 On the Geometry toolbar, click Insert Sequence. 2 Browse to the model’s Model Library folder and double-click the file laminar_static_mixer.mph. Then click Build all on the Geometry toolbar. 3 Click the Zoom Extents button on the Graphics toolbar. MA T E R I A L S On the Home toolbar, click Add Material. A D D MA T E R I A L 1 Go to the Add Material window. 2 In the tree, select Built-In>Water, liquid. 3 In the Add material window, click Add to Component. MA T E R I A L S Water, liquid (mat1) The first material you add applies to all domains by default, so you do not need to change any settings. L A MI N A R F L OW ( S P F ) Initial Values 1 1 In the Model Builder window, expand the Component 1 (comp1)>Laminar Flow (spf) node, then click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 Specify the u vector as Inlet 1 1 On the Physics toolbar, click Boundaries and choose Inlet. 2 Select Boundary 20 only. 3 In the Inlet settings window, locate the Velocity section. 0 x 0 y u_mean z Solved with COMSOL Multiphysics 4.4 8 | L A M I N A R S T A T I C M I X E R 4 In the U 0 edit field, type 2*(1-(x^2+y^2)/ra^2)*u_mean. This gives a parabolic inlet velocity profile appropriate for fully developed laminar flow with mean velocity u_mean. Outlet 1 1 On the Physics toolbar, click Boundaries and choose Outlet. 2 Select Boundary 23 only. TR A N S P O R T O F D I L U T E D S P E C I E S ( C H D S ) Convection and Diffusion 1 1 In the Model Builder window, expand the Component 1 (comp1)>Transport of Diluted Species (chds) node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Diffusion section. 3 In the D c edit field, type D. 4 Locate the Model Inputs section. From the u list, choose Velocity field (spf/fp1). Inflow 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 In the Inflow settings window, locate the Concentration section. 3 In the c 0,c edit field, type step1(x[1/mm]). 4 Select Boundary 20 only. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 23 only. ME S H 1 1 In the Model Builder window, under Component 1 (comp1) click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Element size list, choose Extra coarse. 4 Click the Build All button. C O MP O N E N T 1 ( C O MP 1 ) On the Mesh toolbar, click Add Mesh. Solved with COMSOL Multiphysics 4.4 9 | L A M I N A R S T A T I C M I X E R ME S H 2 Size 1 In the Model Builder window, under Component 1 (comp1)>Meshes right-click Mesh 2 and choose Free Tetrahedral. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type 0.7. 5 In the Minimum element size edit field, type 0.35. 6 Click the Build All button. S T U D Y 1 Step 1: Stationary 1 On the Study toolbar, click Study Steps and choose Stationary>Stationary. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Step 2: Stationary 2 1 In the Model Builder window, under Study 1 click Step 2: Stationary 2. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: 4 On the Home toolbar, click Compute. R E S U L T S Velocity (spf) To reproduce plot in Figure 2 that visualizes the velocity field, follow these steps. Physics Solve for Discretization Transport of Diluted Species × physics Physics Solve for Discretization Laminar Flow × physics Solved with COMSOL Multiphysics 4.4 10 | L A M I N A R S T A T I C M I XE R Concentration (chds) 1 In the Model Builder window, expand the Results>Concentration (chds) node, then click Slice 1. 2 In the Slice settings window, locate the Plane Data section. 3 From the Plane list, choose xy-planes. 4 In the Planes edit field, type 8. 5 On the 3D plot group toolbar, click Plot. Velocity (spf) 1 In the Model Builder window, expand the Results>Velocity (spf) node, then click Slice 1. 2 In the Slice settings window, locate the Plane Data section. 3 From the Plane list, choose xy-planes. 4 In the Planes edit field, type 8. 5 Locate the Expression section. From the Unit list, choose mm/s. 6 In the Model Builder window, right-click Velocity (spf) and choose Streamline. 7 In the Streamline settings window, locate the Streamline Positioning section. 8 From the Positioning list, choose Magnitude controlled. 9 In the Min distance edit field, type 0.025. 10 In the Max distance edit field, type 0.1. 11 Locate the Coloring and Style section. From the Line type list, choose Tube. 12 In the Tube radius expression edit field, type 0.05. 13 From the Color list, choose Yellow. 14 Select the Radius scale factor check box. 15 In the associated edit field, type 2. 16 On the 3D plot group toolbar, click Plot. Finally, reproduce the series of cross-sectional concentration plots for different z-coordinates shown in Figure 4 with the following steps. Data Sets 1 On the Results toolbar, click Cut Plane. 2 In the Cut Plane settings window, locate the Plane Data section. 3 From the Plane list, choose xy-planes. 4 In the z-coordinate edit field, type -2. Solved with COMSOL Multiphysics 4.4 11 | L A M I N A R S T A T I C M I X E R 2D Plot Group 5 1 On the Home toolbar, click Add Plot Group and choose 2D Plot Group. 2 In the 2D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Plane 1. 4 Right-click Results>2D Plot Group 5 and choose Surface. 5 In the Surface settings window, locate the Expression section. 6 Click Concentration (c) in the upper-right corner of the section. Locate the Coloring and Style section. Clear the Color legend check box. 7 On the 2D plot group toolbar, click Plot. 8 Click the Zoom Extents button on the Graphics toolbar. Data Sets 1 In the Model Builder window, under Results>Data Sets click Cut Plane 1. 2 In the Cut Plane settings window, locate the Plane Data section. 3 In the z-coordinate edit field, type 2. 2D Plot Group 5 Repeat these steps for z-coordinate 6, 10, 14, 18, 22, 26, and 30 to reproduce the remaining plots in Figure 4. Data Sets 1 In the Model Builder window, under Results>Data Sets click Cut Plane 1. 2 In the Cut Plane settings window, locate the Plane Data section. 3 In the z-coordinate edit field, type 6. Data Sets 1 In the Model Builder window, under Results>Data Sets click Cut Plane 1. 2 In the Cut Plane settings window, locate the Plane Data section. 3 In the z-coordinate edit field, type 10. Data Sets 1 In the Model Builder window, under Results>Data Sets click Cut Plane 1. 2 In the Cut Plane settings window, locate the Plane Data section. 3 In the z-coordinate edit field, type 14. Data Sets 1 In the Model Builder window, under Results>Data Sets click Cut Plane 1. Solved with COMSOL Multiphysics 4.4 12 | L A M I N A R S T A T I C M I XE R 2 In the Cut Plane settings window, locate the Plane Data section. 3 In the z-coordinate edit field, type 18. Data Sets 1 In the Model Builder window, under Results>Data Sets click Cut Plane 1. 2 In the Cut Plane settings window, locate the Plane Data section. 3 In the z-coordinate edit field, type 22. Data Sets 1 In the Model Builder window, under Results>Data Sets click Cut Plane 1. 2 In the Cut Plane settings window, locate the Plane Data section. 3 In the z-coordinate edit field, type 26. Data Sets 1 In the Model Builder window, under Results>Data Sets click Cut Plane 1. 2 In the Cut Plane settings window, locate the Plane Data section. 3 In the z-coordinate edit field, type 30. Solved with COMSOL Multiphysics 4.4 1 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L Ma x we l l - S t e f a n Di f f us i on i n a F ue l C e l l Uni t C e l l Introduction In concentrated gases and liquids, where the concentrations of all species are of the same order of magnitude, there is no obvious solvent-solute relationship. Fick’s law for diffusion accounts only for 1-way solute-solvent interactions whereas the Maxwell-Stefan equations account for all interactions of species in a solution. In a system with three components, three pair-wise interactions are present, while for a system of four components there are six such interactions. These interactions are described as Fick-analogous Maxwell-Stefan diffusion coefficients, D ij . This example models the steady-state mass transport in the cross section of a proton exchange membrane fuel cell cathode. It models the mass transport in the 3-component gas mixture by using a Concentrated Species physics interface including a Maxwell-Stefan diffusion model. The cross section includes the channel and current collector in the bipolar plate, at the upper boundary, while the active layer defines the lower boundary. The purpose of this model is to show how to consider Maxwell-Stefan diffusion in mass transport. Model Definition Figure 1 describes the computational domain. The insulating boundary at the top of the domain is the current-collector boundary corresponding to the position of the bipolar plate. The vertical boundaries are symmetry boundaries, while the lower boundary, denoted as reactive boundary, represents the position of the active layer. Figure 1: Depiction of the modeling domain with descriptions of the boundary conditions. Air saturated with steam Insulation Reactive boundary Insulation Solved with COMSOL Multiphysics 4.4 2 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L The model equations are defined by a simple mass transport taking the divergence of the mass flux through diffusion and convection. This yields the following expression for species i: (1) where M denotes the total molar mass of the mixture (kg/mol), M j gives the molar mass of species j (kg/mol), and e j is the mass fraction of species j. M can also be expressed in terms of the mass fractions, e j . The symmetric multicomponent Fick diffusivities strongly depend on the composition and are given by these expressions (see Ref. 2 and Ref. 3): (2) (3) where x j is the mol fraction of species j (which can be expressed in terms of the mass fractions e j ), and D ij are the Maxwell-Stefan diffusivities (m 2 /s). Additional entries of the symmetric diffusivities are constructed by permutation of the indices; that is, D 12 = D 21 . The Maxwell-Stefan diffusivities can be described with an empirical equation (Ref. 4) based on the kinetic gas theory: (4) where k is a constant with the value 3.16·10 ÷8 Pa·m 2 /s, T is the temperature expressed in kelvin, p denotes the pressure (Pa), v i equals the molar diffusion volume of species i expressed in m 3 /mol, and M i is the molar mass of species i expressed in kg/mol. The V µe i D ˜ ij M M j ------- Ve j e j VM M ---------- + \ . | | x j e j – ( ) Vp p -------- + ¹ ) ´ ` ¦ ¹ e i µu + j 1 = N ¿ – · 0 = D ˜ 11 e 2 e 3 + ( ) 2 x 1 D 23 --------------------------- e 2 2 x 2 D 13 ---------------- e 3 2 x 3 D 12 ---------------- + + x 1 D 12 D 13 -------------------- x 2 D 12 D 23 -------------------- x 3 D 13 D 23 -------------------- + + --------------------------------------------------------------------------- = D ˜ 12 e 1 e 2 e 3 + ( ) x 1 D 23 -------------------------------- e 2 e 1 e 3 + ( ) x 2 D 13 -------------------------------- e 3 2 x 3 D 12 ---------------- – + x 1 D 12 D 13 -------------------- x 2 D 12 D 23 -------------------- x 3 D 13 D 23 -------------------- + + ---------------------------------------------------------------------------------------------- – = D ij k T 1.75 p v i 1 3 v j 1 3 + ( ) 2 ----------------------------------------- 1 M i ------- 1 M j ------- + 1 2 = Solved with COMSOL Multiphysics 4.4 3 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L molar diffusion volumes are given in Table 1 (Ref. 1). At the reactive boundary (the electrode), the flux of oxygen is (5) where n j represents the mass flux of j, and i c is the reaction current given by the Tafel expression: . (6) Here S a denotes the specific surface area (m 2 /m 3 ), and o is the thickness of the active layer (m). In the Tafel equation, F denotes Faraday’s number (C/mol), R is the gas constant (J/(mol·K)), T represents the temperature (K), i 0 denotes the exchange current density (A/m 2 ), and q is the overpotential (V). The subscript zero in the mass fraction for oxygen represents the reference state. Similarly, the flux of water is (7) where t H2O is the transport number for water (that is, the number of water molecules dragged with each proton migrating through the membrane). At the reactive boundary there is no flux of nitrogen gas because it does not take part in the reactions. This boundary condition results in zero total flux of nitrogen in the entire domain at steady state. Included this fact in the model by specifying the gas velocity according to (8) TABLE 1: DIFFUSION VOLUMES SPECIES DIFFUSION VOLUME O 2 16.6·10 -6 m 3 /mol H 2 O 12.7·10 -6 m 3 /mol N 2 17.9·10 -6 m 3 /mol n O 2 n · i c 4F ------- – \ . | | M O 2 = i c S a oi 0 4F --------------- – \ . | | e O2 e O2 0 ------------ 0,5Fq RT ---------------- \ . | | exp = n H2O n · i c F ---- 1 2 --- t H 2 O + \ . | | M H 2 O = u 1 e N 2 µ ------------- – \ . | | V µe N 2 D N 2 j , M M j ------- Ve j e j VM M ---------- + \ . | | x j e j – ( ) Vp p -------- + ¹ ) ´ ` ¦ ¹ j 1 = N ¿ \ . | | | | – · = Solved with COMSOL Multiphysics 4.4 4 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L At the inlet, a fixed composition is applied. This model uses a composition of air saturated with steam at 80 °C as according to the following table (Ref. 1). Results Figure 2 shows the mass fraction of oxygen at 0.8 V cathode overvoltage (representing a short circuit of the fuel cell). Figure 2: Mass fraction of oxygen at 0.8 V overvoltage of the cathode (an almost short circuit of the fuel cell). The oxygen concentration in the active layer is close to zero in the positions far away from the inlet. The large variation in concentration has a direct influence on the value of the diffusion coefficients for oxygen and the other involved species, nitrogen and water. TABLE 2: INPUT DATA FOR THE MASS FRACTIONS AT THE INLET SPECIES MASS FRACTION O 2 0.1447 H 2 O 0.3789 N 2 0.4764 Solved with COMSOL Multiphysics 4.4 5 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L To illustrate the composition dependence in the diffusivities, you can plot the Multicomponent Fick diffusivity (the water diffusivity); see Figure 3. Figure 3: Variation of multicomponent Fick diffusivity of water, , within the gas mixture at 0.8 V overvoltage. The water diffusivity decreases with decreasing oxygen concentration. This is probably due to the decreased interaction among oxygen molecules. Figure 4 shows the convective gas velocity induced by the drag and production of water at the reactive boundary. The induced gas velocity causes significant convective fluxes out from the reactive boundary, significantly reducing the concentration of oxygen at the boundary. This causes a poorer performance of the cell. In addition, there is a significant D ˜ 22 D ˜ 22 Solved with COMSOL Multiphysics 4.4 6 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L velocity peak at the corner of the inlet. This is caused by the close to spherical diffusion situation at that point. Figure 4: Velocity field in the fuel cell cathode gas compartment at 0.8 V cathode overvoltage. This example also investigates cathode performance in terms of a polarization curve (Figure 5). At high cathode overvoltages, the total electrode current levels off. This is caused by mass transport limitations, which is important to consider when designing or operating fuel cells. Solved with COMSOL Multiphysics 4.4 7 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L Figure 5: Polarization curve for the cathode. References 1. R. Perry, D. Green, Perry’s Chemical Engineering Handbook, 7th ed., McGraw-Hill, 1997. 2. C.F. Curtiss and R.B. Bird, “Multicomponent Diffusion,” Ind. Eng. Chem. Res., pp. 2515–2522, vol. 38, 1999. 3. R. Bird, W. Stewart, and E. Lightfoot, Transport Phenomena, 2nd ed., John Wiley & Sons, 2002. 4. J.A. Wesselingh and R. Krishna, Mass Transfer in Multicomponent Mixtures, Delft University Press, 2000. Model Library path: Chemical_Reaction_Engineering_Module/Mass_Transport/ maxwell_stefan_diffusion Solved with COMSOL Multiphysics 4.4 8 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Chemical Species Transport>Transport of Concentrated Species (chcs). 3 Click the Add button. 4 In the Number of species edit field, type 3. 5 In the Mass fractions table, enter the following settings: 6 Click the Study button. 7 In the tree, select Preset Studies>Stationary. 8 Click the Done button. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: wO2 wH2O wN2 Name Expression Value Description k 3.16e-8[Pa*m^2 /s] 3.160E-8 W/m MS diffusivity prefactor v_N2 17.9e-6 1.790E-5 Molar diffusion volume, N2 v_O2 16.6e-6 1.660E-5 Molar diffusion volume, O2 v_H2O 12.7e-6 1.270E-5 Molar diffusion volume, H2O Solved with COMSOL Multiphysics 4.4 9 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L M_O2 32[g/mol] 0.03200 kg/mol Molar diffusion volume, H2O M_H2O 18[g/mol] 0.01800 kg/mol Molar mass, H2O M_N2 28[g/mol] 0.02800 kg/mol Molar mass N2 w_O20 0.1447 0.1447 Inlet mass fraction, O2 w_H2O0 0.3789 0.3789 Inlet mass fraction, H2O T0 353[K] 353.0 K Temperature p0 101[kPa] 1.010E5 Pa Pressure D_O2_N2 k*(T0[1/ K])^1.75/ (p0*(v_O2^(1/ 3)+v_N2^(1/ 3))^2)*(1[kg/ mol]/ M_O2+1[kg/ mol]/M_N2)^0.5 2.757E-5 m²/s MS diffusivity, O2-N2 component D_O2_H2O k*(T0[1/ K])^1.75/ (p0*(v_O2^(1/ 3)+v_H2O^(1/ 3))^2)*(1[kg/ mol]/ M_O2+1[kg/ mol]/ M_H2O)^0.5 3.513E-5 m²/s MS diffusivity, O2-H2O component D_H2O_N2 k*(T0[1/ K])^1.75/ (p0*(v_H2O^(1/ 3)+v_N2^(1/ 3))^2)*(1[kg/ mol]/ M_H2O+1[kg/ mol]/M_N2)^0.5 3.508E-5 m²/s MS diffusivity, H2O-N2 eta 0.1[V] 0.1000 V Overpotential S_a 1e7[1/m] 1.000E7 1/m Specific surface area d 10[um] 1.000E-5 m Active-layer thickness i_0c 1[A/m^2] 1.000 A/m² Exchange current density t_H2O 3 3.000 Drag number, H2O Name Expression Value Description Solved with COMSOL Multiphysics 4.4 10 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L D E F I N I T I O N S Variables 1 1 In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Here F_const and R_const refer to Faraday's constant and the gas constant, respectively. G E O ME T R Y 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose mm. Rectangle 1 (r1) 1 Right-click Component 1 (comp1)>Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 1. 4 In the Height edit field, type 0.2. 5 Click the Build Selected button. Point 1 (pt1) 1 In the Model Builder window, right-click Geometry 1 and choose Point. 2 In the Point settings window, locate the Point section. 3 In the x edit field, type 0.5. 4 In the y edit field, type 0.2. 5 Click the Build All Objects button. 6 Click the Zoom Extents button on the Graphics toolbar. Name Expression Unit Description i_c -S_a*d*i_0c*exp(0.5 *F_const*eta/ (R_const*T0))*(abs( wO2)/w_O20) A/m² Reaction current Solved with COMSOL Multiphysics 4.4 11 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S ( C H C S ) 1 In the Model Builder window, under Component 1 (comp1) click Transport of Concentrated Species (chcs). 2 In the Transport of Concentrated Species settings window, locate the Transport Mechanisms section. 3 From the Diffusion model list, choose Maxwell-Stefan. 4 Locate the Species section. From the From mass constraint list, choose wN2. Convection and Diffusion 1 1 In the Model Builder window, expand the Transport of Concentrated Species (chcs) node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Density section. 3 In the M wO2 edit field, type M_O2. 4 In the M wH2O edit field, type M_H2O. 5 In the M wN2 edit field, type M_N2. 6 Locate the Diffusion section. In the D ik table, enter the following settings: 7 Specify the u vector as 8 In the p A edit field, type p0. 9 In the T edit field, type T0. Initial Values 1 1 In the Model Builder window, under Component 1 (comp1)>Transport of Concentrated Species (chcs) click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the w 0,wO2 edit field, type w_O20. 4 In the w 0,wH2O edit field, type w_H2O0. Symmetry 1 1 On the Physics toolbar, click Boundaries and choose Symmetry. 1 D_O2_H2O D_O2_N2 D_O2_H2O 1 D_H2O_N2 D_O2_N2 D_H2O_N2 1 -chcs.jx_wN2/(wN2*chcs.rho) x -chcs.jy_wN2/(wN2*chcs.rho) y Solved with COMSOL Multiphysics 4.4 12 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L 2 Select Boundaries 1, 3, and 5 only. Mass Fraction 1 1 On the Physics toolbar, click Boundaries and choose Mass Fraction. 2 Select Boundary 4 only. 3 In the Mass Fraction settings window, locate the Mass Fraction section. 4 Select the Species wO2 check box. 5 In the e 0,wO2 edit field, type w_O20. 6 Select the Species wH2O check box. 7 In the e 0,wH2O edit field, type w_H2O0. Flux 1 1 On the Physics toolbar, click Boundaries and choose Flux. 2 Select Boundary 2 only. 3 In the Flux settings window, locate the Inward Flux section. 4 Select the Species wO2 check box. 5 In the N 0,wO2 edit field, type i_c*M_O2/(4*F_const). 6 Select the Species wH2O check box. 7 In the N 0,wH2O edit field, type -i_c*(1/2+t_H2O)*M_H2O/F_const. ME S H 1 Free Triangular 1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Free Triangular. Size 1 1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Point. 4 Select Point 3 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type 5[um]. 8 In the Model Builder window, right-click Mesh 1 and choose Build All. Solved with COMSOL Multiphysics 4.4 13 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L D E F I N I T I O N S Integration 1 (intop1) 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 2 only. Variables 1 1 In the Model Builder window, under Component 1 (comp1)>Definitions click Variables 1. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: S T U D Y 1 Step 1: Stationary 1 In the Model Builder window, under Study 1 click Step 1: Stationary. 2 In the Stationary settings window, click to expand the Study extensions section. 3 Locate the Study Extensions section. Select the Auxiliary sweep check box. 4 Click Add. 5 In the table, enter the following settings: 6 From the Run continuation for list, choose eta. 7 On the Home toolbar, click Compute. R E S U L T S Mass Fraction (chcs) 1 In the Model Builder window, expand the Mass Fraction (chcs) node, then click Surface 1. 2 In the Surface settings window, locate the Expression section. 3 Select the Description check box. 4 In the associated edit field, type Mass fraction, O2. Name Expression Unit Description I_tot intop1(i_c) A/m Total current Auxiliary parameter Parameter value list eta range(0.05,0.05,0.8) Solved with COMSOL Multiphysics 4.4 14 | M A X WE L L - S T E F A N D I F F U S I O N I N A F U E L C E L L U N I T C E L L 5 On the 2D plot group toolbar, click Plot. 6 Click the Zoom Extents button on the Graphics toolbar. 2D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 2D Plot Group. 2 In the Model Builder window, under Results right-click 2D Plot Group 2 and choose Surface. 3 In the Surface settings window, locate the Expression section. 4 Click Multicomponent Fick diffusivities, 22 component (chcs.DE22) in the upper-right corner of the section. On the 2D plot group toolbar, click Plot. 2D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 2D Plot Group. 2 In the Model Builder window, under Results right-click 2D Plot Group 3 and choose Surface. 3 In the Surface settings window, locate the Expression section. 4 In the Expression edit field, type sqrt(comp1.chcs.u^2+comp1.chcs.v^2). 5 Select the Description check box. 6 In the associated edit field, type Velocity. 7 In the Model Builder window, right-click 2D Plot Group 3 and choose Streamline. 8 In the Streamline settings window, locate the Expression section. 9 Click Velocity field (chcs.u,chcs.v) in the upper-right corner of the section. Locate the Streamline Positioning section. From the Positioning list, choose Magnitude controlled. 10 Locate the Coloring and Style section. From the Color list, choose White. 11 On the 2D plot group toolbar, click Plot. 1D Plot Group 4 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Point Graph. 3 Select Point 1 only. 4 In the Point Graph settings window, locate the y-axis data section. 5 Click Total current (I_tot) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Solved with COMSOL Multiphysics 4.4 1 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R Hy dr ode al ky l at i on i n a Me mbr ane Re a c t or Introduction At high temperatures and pressures, and in the presence of hydrogen, toluene can be demethylated to produce benzene. Furthermore, benzene can react reversibly to produce biphenyl. The following example illustrates the simulation of the hydrodealkylation process, carried out in a membrane reactor. This reactor arrangement allows for continuous addition of hydrogen to the process, increasing the selectivity for the desired benzene product. The example shows how you can easily modify the predefined Plug flow reactor type in the Chemical Reaction Engineering Module to set up a membrane reactor model. You will also learn how to access and include external thermodynamic and physical property calculations in your simulation, using the Thermodynamics feature. This feature can connect to free to use and non-proprietary databases that follows rules and interfaces that allow CAPE (Computer-Aided Process Engineering) applications or components to interoperate. Model Definition Two important reactions occur in the thermal hydrodealkylation (HDA) of toluene. The main reaction involves toluene reacting with hydrogen to produce benzene and methane: (1) The dealkylation reaction rate is first order in the toluene concentration and half order in the hydrogen concentration: (2) At the same time, biphenyl is reversibly formed from benzene: k 1 f CH 4 + CH 3 + H 2 r 1 k 1 c C7H8 c H2 = Solved with COMSOL Multiphysics 4.4 2 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R (3) The rate of the coupling reaction follows the mass action law: (4) In the above rate expressions, the rate constants follow Arrhenius type behavior: (5) The values of the frequency factors and activation energies (J/mol) are taken from the literature (Ref. 1 and Ref. 2) and are presented in Table 1. The chemical reactions given in Equation 1 and Equation 3 suggest that maintaining high concentrations of hydrogen would be beneficial to ensure a high benzene yield. Such process conditions can be achieved using a membrane reactor. As illustrated schematically in Figure 1, hydrogen can be supplied continuously across the porous membrane. Figure 1: Hydrogen is continuously supplied to the reactor through a porous membrane. The species mass balance for hydrogen in the membrane reactor is given by: (6) where F is the molar flow rate (mol/s) in the reactor, V is the reactor volume (m 3 ), R is the species rate expression (mol/(m 3 ·s)), and f is the molar flow rate per unit volume TABLE 1: ARRHENIUS PARAMETERS FREQUENCY FACTOR ACTIVATION ENERGY Forward reaction 1 5.67e9 228.2e3 Forward reaction 2 1e8 167.5e3 Reverse reaction 2 1e8 149.8e3 k 2 r k 2 f 2 + H 2 r 2 k 2 f c C6H6 2 k 2 r c H2 c C12H10 – = k Ae E R g T ---------- – = C 6 H 5 CH 3 + H 2 H 2 C 6 H 6 + CH 4 C 12 H 10 dF H2 dV -------------- R H2 f H2 + = Solved with COMSOL Multiphysics 4.4 3 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R (mol/(m 3 ·s)) across the membrane. The velocity of the hydrogen gas across the porous membrane can be described by Darcy’s Law: (7) where K (m 3 /(N·s)) is a proportionality constant, p shell (Pa) is the gas pressure on the shell side of the membrane, and p reactor (Pa) the pressure on the reactor side. The molar flow rate per unit volume across the membrane then becomes: (8) Above, a is the membrane surface area per unit volume (m 2 /m 3 ), and c shell is the concentration of hydrogen (mol/m 3 ) on the shell side. Except for hydrogen, the other chemical species in the reactor do not pass through the membrane and their material balances thus follow the standard plug flow equations: (9) The adiabatic energy balance for the reactor is given by: (10) In Equation 10, C p,i represents the species molar heat capacity (J/(mol·K)), and Q denotes the heat due to chemical reaction (J/(m 3 ·s)): (11) where H j is the heat produced by reaction j, calculated from: (12) In Equation 12 h i represents the species molar enthalpy (J/mol) and v ij the stoichiometric coefficients. The last term in the energy balance accounts for the energy transfer associated with the flow of hydrogen across the membrane: (13) u K p shell p reactor – ( ) = f H2 uac shell = dF i dV --------- R i = F i C p i , dT dV -------- i ¿ Q Q mem + = Q H j r j j ¿ – = H j v ij h j i ¿ = Q mem f H2 h H2 = Solved with COMSOL Multiphysics 4.4 4 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R The Chemical Reaction Engineering Module automatically sets up and solves Equation 9 and Equation 10 when you select the predefined Plug flow reactor type. To adjust the default model to account for hydrogen entering the reactor through the membrane, the flow term f H2 has to be specified and included into the H2 material balance. Solving the energy balance, Equation 10, requires the input of molar heat capacities C p,i (J/(mol·K)), and the molar enthalpies, h i (J/mol), of the reacting species. In this example, these thermodynamic properties will be calculated by an external compliant property package, accessed by means of the Thermodynamics feature. A more detailed description follows below. Results and Discussion In a first simulation, the reactor is assumed to be a standard tubular reactor, that is, without hydrogen entering through the reactor circumference. The reactor is fed with equal molar flows (10 mol/s) of hydrogen and toluene. At the inlet the reactant gas is held at 1200 K and 2 atmospheres. Figure 2: Molar flow rates (mol/s) as function of reactor volume (m 3 ) for a tubular reactor design. Solved with COMSOL Multiphysics 4.4 5 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R A second model simulates the membrane reactor, with a continuous supply of hydrogen through the membrane. Figure 3 shows the corresponding concentration distributions. Figure 3: Molar flow rates (mol/s) as function of reactor volume (m 3 ) for a membrane reactor design. Clearly, the membrane reactor produces benzene with greater selectivity. This example has summarized how to model a membrane reactor by modifying the predefined reactor equations for the Plug flow reactor type. Furthermore, calculations of thermodynamic properties in the model are performed by an external property pack, TEA, by means of the Thermodynamics feature. References 1. K.C. Hou and H.B. Palmer, “The Kinetics of Thermal Decomposition of Benzene in a Flow System,” The Journal of Physical Chemistry, vol. 69, p. 863, 1965. 2. S.E. Shull and A.N. Hixson, I&EC Process Design and Development, vol. 5, p. 147, 1966. 3. http://www.colan.org Solved with COMSOL Multiphysics 4.4 6 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R 4. http://www.cocosimulator.org Model Library path: Chemical_Reaction_Engineering_Module/ Tubular_Reactors/membrane_hda Adding a Thermodynamics Package for Physical and Thermodynamic Property Calculations In this model, an external third-party package for chemical process simulation software (Ref. 3) is used to supply physical and thermodynamic property calculations to the COMSOL model. TEA is a thermodynamics property package that handles the physical and thermodynamic property calculations for the simulation environment COCO. The property data bank contains of over 190 commonly used chemicals, and the package exhibits more than 100 property calculation routines with their analytical or numerical derivatives. COCO is maintained by AmsterCHEM and is free to download from the Internet (Ref. 4). To use the external physical and thermodynamic property calculations in COMSOL models, you need to go through the following steps: 1 Download and install COCO, which includes the TEA property package manager. The software is available from www.cocosimulator.org/index_download.html. 2 Create and configure a property package template that handles physical and thermodynamics calculations needed for your model. If you have already created a package template earlier, or if an adequate property package already exists in the installation, this step is not needed. 3 Create a functions using the Thermodynamics feature, choosing relevant species and properties from the property package template. 4 Use the functions directly in the physics interfaces of COMSOL to evaluate a property as a function of model variables such as temperature, pressure, and composition. Note. In this example, the function enthalpyF will be used. The difference between enthalpy and enthalpyF is that the latter is guaranteed to contain the enthalpy of Solved with COMSOL Multiphysics 4.4 7 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R formation. This is useful if you want to account for the heat of reaction in a chemical reaction. Downloading and Installing a Property Package Using TEA This section describes how to set up and configure a property package for thermodynamic calculations using the TEA property package manager (Ref. 4). The property package will include the reacting species hydrogen (H2), methane (CH4), benzene (C6H6), toluene (C6H5CH3), and biphenyl (C12H10), and the functions EnthalpyF and heatCapacityCp will be used for calculating enthalpy and heat capacity. If not already on your system, download the COCO simulation environment. COCO is maintained by AmsterCHEM and is available free of charge on the Internet (Ref. 4). Download and install the package from www.cocosimulator.org/ index_download.html. An equivalent property package can be set up using other property pack managers as well. For more information about providers, see The Thermodynamics Feature. 1 Start the application ConfigureTEA, by selecting Start menu>All Programs>COCO>ConfigureTEA (or similar, depending on your operating system version). 2 Click the Create template button. 3 In the Property pack definition page, enter membrane_hda in the Name edit field. 4 Type an optional description in the Description field, for example Hydrodealkylation chemistry. 5 From the Model set menu, select Soave Redlich Kwong. 6 Click the Add button on the right. 7 Type hydrogen in the Filter by edit field. Solved with COMSOL Multiphysics 4.4 8 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R 8 Select Hydrogen in the Compound selection list by clicking the formula CH4 in the Formula column, then click OK. 9 In the same way, add the compounds Methane, Benzene, Toluene, and Biphenyl. 10 Click OK and Done. Solved with COMSOL Multiphysics 4.4 9 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R Adding the Thermodynamics Package to the COMSOL Model Right-click Global Definitions under the top node in the Model Builder window, then select Functions->Thermodynamics and select Thermodynamics Package. Solved with COMSOL Multiphysics 4.4 10 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R Right-click the Thermodynamics package feature to see the available thermodynamic property functions classes. Choose Single-Phase Property: Figure 4: Select the property calculation class to set up by right-clicking the Thermodynamics Package feature. For more information about the wizard, see The Thermodynamics Feature in the Chemical Reaction Engineering Module User’s guide. In a first step, the Wizard displays available property packages and property package managers available on your system. You can browse the compounds as well as physical property and Solved with COMSOL Multiphysics 4.4 11 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R thermodynamic calculations supported by a property packages before selecting it by clicking the blue arrow button. Figure 5: The Thermodynamics Wizard lets you browse the contents of the available property packages on your system. In the next step selections are made from the available property calculations. If applicable, you can also set if you want the property calculations returned on a mole or mass basis. For instance, you could get the property heatCapacityCp returned either with the unit J/(mol·K) or with J/(kg·K). Mole basis is the default and is the proper setting for thermodynamic properties in the Chemical Reaction Engineering Module. Figure 6 shows a selection of available single-phase property calculations. Solved with COMSOL Multiphysics 4.4 12 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R Figure 6: Property selections for Single-Phase Property calculations in the Thermodynamics wizard. Clicking the Next button reveals the compound selection page of the Wizard. It is possible to select one or more entries from the Available compounds list. In the case of single-phase and two-phase properties, a single compound selection leads to the generation of pure compound property functions. For multiple compound selections the software assumes that mixture property functions are to be generated. Figure 7: Compound selections for Single-Phase Property calculations in the Thermodynamics wizard. Solved with COMSOL Multiphysics 4.4 13 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R For single-phase property calculations, the final step of the Wizard lets you to select the fluid phase and also displays a summary of the functions that will be created as you click the Finish button. In addition to property calculations, CAPE-OPEN compliant package providers can also often perform calculations of the property derivatives with respect to the dependent variables. When the Include derivatives check box is selected the appropriate derivative calculations are automatically set up. Figure 8: The property functions that will be created by the wizard are summarized in the Function Overview dialog. Clicking the Finish button exits the Wizard and generates the corresponding property function features in the Model Builder. In feature settings window you can review the Function Information, change the Function Name, as well as plot the property function over specified intervals. Note that the Thermodynamics Wizard can be started not only from the Thermodynamics node but also from any Property Package node in the Model Builder. In such a case the Wizard starts with the Select Properties dialog. It is still possible to change the property package selection by clicking the Back button. Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. Solved with COMSOL Multiphysics 4.4 14 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Stationary Plug Flow. 6 Click the Done button. Access the Thermodynamics Package feature to define the external thermodynamic property calculations required for the model. Following the steps below you will create single phase property functions for pure gas phase species. G L O B A L D E F I N I T I O N S In the Model Builder window, right-click Global Definitions and choose Thermodynamics Package. T H E R MO D Y N A MI C S 1 In the Model Builder window, under Global Definitions>Thermodynamics right-click Thermodynamics Package and choose Single-Phase Property. In the first step of the Thermodynamics wizard you select the property package that will perform the thermodynamic calculations. 2 In the Property Package Wizard settings window, locate the Property Packages section. 3 In the tree, select TEA (CAPE-OPEN 1.1)>membrane_hda. 4 Click Next. 5 Find the Available properties subsection. In the Available properties list, first choose enthalpyF and add it to the lower list by clicking the + button, then add heatCapacityCp in the same way. 6 Click Add Selected. 7 Click Next. 8 Find the Available compounds subsection. In the list, select Hydrogen. 9 Click Add Selected. Selecting a single compound generates functions for calculating pure species properties. Multiple compound selections at this point lead to mixture property calculations. Solved with COMSOL Multiphysics 4.4 15 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R 10 Click Next, accept Vapor as the phase by clicking Next again, and the click Finish. Proceed in the same manner to add functions for pure component thermodynamic calculations for methane, benzene, toluene, and biphenyl. Renaming the functions To make it easier to call the functions you can rename them to shorter names. 1 In the Model Builder window, under Global Definitions>Thermodynamics>Thermodynamics Package>Property Package 1 click Single-Phase Property 1. 2 In the Single-Phase Property settings window, locate the Function Name section. 3 In the Function name edit field, type h_H2, for the example of enthalpy of hydrogen. Do the same and rename the heat capacity function for hydrogen to cp_H2. Note that the order of the functions presented may be vary depending on your platform. Make sure to match the right short name, for example h_CH4 to the right function. Rename all functions according to this list: h_H2 (enthalpy, Hydrogen) cp_H2 (heat capacity, Hydrogen) h_CH4 (enthalpy Methane) cp_CH4 (heat capacity, Methane) h_C6H6 (enthalpy Benzene) cp_C6H6 (heat capacity, Benzene) h_C6H5CH3 (enthalpy, Toluene) cp_C6H5CH3 (heat capacity, Toluene) h_C12H10 (enthalpy Biphenyl) cp_C12H10 (heat capacity, Biphenyl) G L O B A L D E F I N I T I O N S Now, define some model parameters and variables. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file membrane_hda_parameters.txt. Solved with COMSOL Multiphysics 4.4 16 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R D E F I N I T I O N S Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 2 Click Load from File. 3 Browse to the model’s Model Library folder and double-click the file membrane_hda_variables.txt. Note that some expressions use variables defined in the Reaction Engineering feature. These variables need to be specified with the scope of the Reaction Engineering node. For example, re.T specifies the temperature variable defined by the Reaction Engineering node with the identifier re. With all property functions and model specific expressions in place, move on to select the reactor model and defined the reaction kinetics. R E A C T I O N E N G I N E E R I N G 1 In the Model Builder window, under Component 1 click Reaction Engineering. 2 In the Reaction Engineering settings window, locate the Reactor Settings section. 3 From the Reactor type list, choose Plug flow. 4 Locate the General section. In the p edit field, type p_reactor. 5 Locate the Reactor Settings section. Select the Calculate thermodynamic properties check box. Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type C6H5CH3+H2=>C6H6+CH4. 4 In the Reaction settings window, locate the Rate Constants section. 5 Select the Use Arrhenius expressions check box. 6 In the A f edit field, type 5.67e9. 7 In the E f edit field, type 228.2e3. 8 Locate the Reaction Rate section. From the Reaction rate list, choose User Defined. 9 In the r edit field, type kf_1*c_C6H5CH3*c_H2^0.5. Change the default kinetics expression by modifying the reaction order for hydrogen. Solved with COMSOL Multiphysics 4.4 17 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R Reaction 2 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type 2C6H6<=>C12H10+H2. 4 In the Reaction settings window, locate the Rate Constants section. 5 Select the Use Arrhenius expressions check box. 6 In the A f edit field, type 1e8. 7 In the E f edit field, type 167.5e3. 8 In the A r edit field, type 1e8. 9 In the E r edit field, type 149.8e3. Following the steps below, type in settings for each Species. The Plug flow reactor model requires you to input the Inlet molar flow. You find this edit field in the Species Feed Stream section. To solve the reactor energy balance the software also prompts you for the heat capacity and enthalpy for each of the species. Provide the Thermodynamics functions directly in the appropriate edit fields in the Thermodynamic Expressions sections. Species: C6H5CH3 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: C6H5CH3. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type 10. 4 Click to expand the Species thermodynamic expressions section. Overwrite the predefined thermodynamic expressions with the Thermodynamics functions. 5 Locate the Species Thermodynamic Expressions section. In the h edit field, type h_C6H5CH3(re.T,p_reactor). 6 In the C p edit field, type cp_C6H5CH3(re.T,p_reactor). Species: H2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: H2. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type 10. Solved with COMSOL Multiphysics 4.4 18 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R 4 Click to expand the Species thermodynamic expressions section. Locate the Species Thermodynamic Expressions section. In the h edit field, type h_H2(re.T,p_reactor). 5 In the C p edit field, type cp_H2(re.T,p_reactor). Species: C6H6 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: C6H6. 2 In the Species settings window, click to expand the Species thermodynamic expressions section. 3 Locate the Species Thermodynamic Expressions section. In the h edit field, type h_C6H6(re.T,p_reactor). 4 In the C p edit field, type cp_C6H6(re.T,p_reactor). Species: CH4 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: CH4. 2 In the Species settings window, click to expand the Species thermodynamic expressions section. 3 Locate the Species Thermodynamic Expressions section. In the h edit field, type h_CH4(re.T,p_reactor). 4 In the C p edit field, type cp_CH4(re.T,p_reactor). Species: C12H10 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: C12H10. 2 In the Species settings window, click to expand the Species thermodynamic expressions section. 3 Locate the Species Thermodynamic Expressions section. In the h edit field, type h_C12H10(re.T,p_reactor). 4 In the C p edit field, type cp_C12H10(re.T,p_reactor). Energy Balance 1 In the Model Builder window, right-click Reaction Engineering and choose Energy Balance. 2 In the Energy Balance settings window, locate the Energy Balance section. Solved with COMSOL Multiphysics 4.4 19 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R 3 In the T 0 edit field, type T_inlet. You have now set up a model for a nonisothermal tubular reactor. First solve this model and review the results, then move on to study the related membrane reactor model. In the present example the total reactor volume is 1m^3, so default solver settings can be used. S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S Flow Rate (re) The default plot shows the molar flow rates of all species as a function of the reactor volume. S T U D Y 1 Store a copy of the solution for the tubular reactor model. This way you can readily access the results for comparison with the membrane reactor model. 1 In the Model Builder window, expand the Study 1 node. Solver 1 In the Model Builder window, expand the Study 1>Solver Configurations node. Copy 2 1 Right-click Solver 1 and choose Solution>Copy. 2 In the Model Builder window, under Study 1>Solver Configurations right-click Copy 2 and choose Rename. 3 Go to the Rename Solver dialog box and type tubular reactor in the New name edit field. 4 Click OK. You will now modify the existing Reaction Engineering node to create a membrane reactor model. R E A C T I O N E N G I N E E R I N G Species: H2 1 In the Species settings window, locate the General Expressions section. 2 From the Rate expression list, choose User Defined. Solved with COMSOL Multiphysics 4.4 20 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R 3 In the R edit field, type -r_1+r_2+f_H2. The f_H2 term corresponds to the flow of hydrogen across the membrane. Energy Balance 1 In the Model Builder window, under Component 1>Reaction Engineering click Energy Balance. 2 In the Energy Balance settings window, locate the Energy Balance section. 3 In the Q ext edit field, type Q_mem. This accounts for the heat transferred into the reactor due to the flow across the membrane. S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S Flow Rate (re) Plot Group 1 shows the results for the current problem solved. This means that the membrane reactor results overwrite the tubular reactor results. Create a new Plot Group where you recall and plot the results from the tubular reactor model. 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 On the 1D plot group toolbar, click Global. 5 On the 1D plot group toolbar, click Plot. 6 In the Model Builder window, under Results>1D Plot Group 3 click Global 1. 7 In the Global settings window, click to expand the Legends section. 8 Find the Include subsection. Select the Expression check box. 9 On the 1D plot group toolbar, click Plot. Solved with COMSOL Multiphysics 4.4 1 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R Opt i mi z a t i on of a C a t a l y t i c Mi c r or e a c t or Introduction In this example model, a solution is pumped through a catalytic bed, where a reactant undergoes chemical reaction as it gets in contact with the catalyst. The purpose of the model is to maximize the total reaction rate for a given total pressure difference across the bed by finding an optimal catalyst distribution. The distribution of the porous catalyst determines the total reaction rate in the bed. A large amount of catalyst results in a low flow rate through the bed while less catalyst gives a high flow rate but low conversion of the reactant. This model is based on Ref. 1. Note: This model requires the Optimization Module. Model Definition The model geometry is depicted in Figure 1. The reactor consists of an inlet channel, a fixed catalytic bed, and an outlet channel. Solved with COMSOL Multiphysics 4.4 2 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R Figure 1: Model geometry. The optimal catalyst distribution should maximize the average reaction rate, which is expressed as the integral of the local reaction rate, r (mol/(m 3 ·s)), over the domain, O. This is equivalent to minimizing the negative of this average reaction rate: Assuming a first-order catalytic reaction with respect to the reactant species, the local reaction rate is determined by (1) where c denotes the volume fraction of solid catalyst, c refers to the concentration (mol/m 3 ), and k a is the rate constant (1/s). The mass transport is described by the convection and diffusion equation where u denotes the velocity vector (m/s) and D is the diffusion coefficient (m 2 /s). The Navier-Stokes equations describe the fluid flow: (2) Inlet Reacting domain Outlet Symmetry boundary min 1 vol(O) ------------------- – r O d O } ¹ ) ´ ` ¦ ¹ r k a 1 c – ( )c = V DVc – ( ) · r u Vc · – = µ u V · ( )u V – p V + µ Vu Vu ( ) T + ( ) o c ( )u – · = V u · 0 = Solved with COMSOL Multiphysics 4.4 3 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R The coefficient o(c) depends on the distribution of the porous catalyst as (3) where Da is the Darcy number; L is the length scale (m); and q is a dimensionless parameter, the interpretation of which is discussed in the next section. From Equation 3, the direct conclusion is that when c equals 1, o equals zero and Equation 2 reduces to the ordinary Navier-Stokes equations. In this case the reaction rate is zero; see Equation 1. To summarize, the optimization problem is (4) where and physical boundary conditions apply. C O NV E X O P T I MI Z A T I O N P R O B L E MS One of the most important characteristics of an optimization problem is whether or not the problem is convex. This section therefore briefly describes this property. For a more general discussion of the subject, see for example Ref. 2. A set C is said to be convex if for any two members x, y of C, the following relation holds: that is, the straight line between x and y is fully contained in C. A convex function is a mapping f from a convex set C such that for every two members x, y of C (5) o c ( ) µ Da L 2 · ------------------- q 1 c – ( ) q c + -------------------- · = min c 1 vol(O) ------------------ – k a 1 c – ( )c ( ) O d O } ¹ ) ´ ` ¦ ¹ µ u V · ( )u V – p V + µ Vu Vu ( ) T + ( ) o c ( )u – · = V u · 0 = V DVc – ( ) · r u Vc · – = 0 c 1 s s tx 1 t – ( )y C for every t 0 1 , | | e e + f tx 1 t – ( )y + ( ) t f x ( ) 1 t – ( )f y ( ) for every t 0 1 , | | e + s Solved with COMSOL Multiphysics 4.4 4 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R An optimization problem is said to be convex if the following conditions are met: • the design domain is convex • the objective and constraints are convex functions The importance of convexity follows simply from the result that if x* is a local minimum to a convex optimization problem, then x* is also a global minimum. This is easily proven by simply assuming that there is a y such that f(y) < f(x*), and then using Equation 5. This particular optimization problem is nonlinear, because a change in c implies a change in the concentration, c. Because of this implicit dependence, it is very difficult to determine whether or not the objective is convex. There is therefore no guarantee that the optimal solution you obtain is globally optimal or unique. In the best of cases, running the optimization will give a good local optimum. The parameter q can be used to smoothen the interfaces between the catalyst and the open channel. To see the effect of this parameter, rewrite Equation 3 as o c ( ) µ Da L 2 · ------------------- 1 c – 1 c q --- + ------------- · = Solved with COMSOL Multiphysics 4.4 5 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R It follows that when q approaches infinity, o is the (inverse) porosity. On the other hand, lowering the value of q decreases the magnitude of o. Figure 2: q(1 ÷ c)/(q ÷ c) plotted as a function of c for different values of q. Figure 2 shows q(1 ÷ c)/(q ÷ c) plotted as a function of c for different values of q. This plot shows that lowering the value of q, increases the convexity of the force coefficient. For a low q value, an increase in c around 0.5, imposes a small increase of the force coefficient, while for a higher value of q, a change in c imposes an almost equal change for the whole range. Therefore, for a lower q value, the solution is not sharp at the interfaces. On the other hand, for small values of c, the force term decreases rapidly when q is small, and thus affects the flow field to a much wider extent. In the limit when q approaches infinity, o as a function of c is a straight line. Results and Discussion Figure 3 shows the velocity field in the empty channel, this is the starting point for the optimization. Solved with COMSOL Multiphysics 4.4 6 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R Figure 3: Velocity field in the open channel. Figure 4: Distribution of the porous catalyst seen in black and open channel in white. Solved with COMSOL Multiphysics 4.4 7 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R Figure 4 shows the distribution of the porous catalyst in black and the open channels in white. This result shows that, optimally, the supply of the reactant should be distributed over a large area of the reactor. Note also that the amount of open channel volume is significant. Figure 5 shows the concentration distribution in the reactor. This plot shows how the porous catalyst is fed with the reactant through the open channels. The plot naturally resembles that of Figure 4. Figure 5: Concentration distribution in the reactor after optimization. Let where n flow refers to the normal to the boundary cO i in the flow direction (that is, pointing in to the domain at the inlet and out from the domain at the outlet). Then F i is a measurement of the flow of the species with concentration c through the boundary cO i per unit length in the transverse dimension. The conversion, X, of the reactant is defined as F i n flow DVc – cu + ( ) · s d cO i } = Solved with COMSOL Multiphysics 4.4 8 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R In this case, the conversion of the reactant is around 50%. Figure 6 shows the velocity field in the reactor. The porous catalyst slows down the flow significantly compared to Figure 3. Figure 6: Velocity field in the reactor after optimization. References 1. F. Okkels and H. Bruus, “Scaling Behavior of Optimally Structured Catalytic Microfluidic Reactors,” Phys. Rev. E, vol. 75, pp. 016301 1–4, 2007. 2. S.G. Nash and A. Sofer, Linear and Nonlinear Programming, McGraw-Hill, 1995. Model Library path: Chemical_Reaction_Engineering_Module/ Optimization/microreactor_optimization X F in F out – F in -------------------------- = Solved with COMSOL Multiphysics 4.4 9 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf). 3 Click the Add button. 4 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 5 Click the Add button. 6 In the Select physics tree, select Mathematics>Optimization and Sensitivity>Optimization (opt). 7 Click the Add button. 8 Click the Study button. 9 In the tree, select Preset Studies for Selected Physics>Stationary. 10 Click the Done button. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File and browse to the model’s model library folder and double-click the file microreactor_optimization_parameters.txt. G E O ME T R Y 1 Next, create the geometry. The reactor consists of three domains: the inlet channel, the reacting domain, and the outlet channel (see Figure 1). 1 In the Model Builder window, under Component 1 click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose mm. Solved with COMSOL Multiphysics 4.4 10 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R Rectangle 1 1 Right-click Component 1>Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 2*L. 4 In the Height edit field, type L. Rectangle 2 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 6*L. 4 In the Height edit field, type 3*L. 5 Locate the Position section. In the x edit field, type 2*L. Rectangle 3 1 Right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 2*L. 4 In the Height edit field, type L. 5 Locate the Position section. In the x edit field, type 8*L. 6 Click the Build Selected button. 7 Click the Zoom Extents button on the Graphics toolbar. The geometry should now look like that in Figure 1. D E F I N I T I O N S Define integration couplings to use for calculating the conversion of the reactant. Integration 1 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 1 only. Integration 2 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. Solved with COMSOL Multiphysics 4.4 11 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R 4 Select Boundary 12 only. Variables 1 1 In the Model Builder window, right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File and browse to the model’s model library folder and double-click the file microreactor_optimization_variables1.txt. In the resulting list, chds.tfluxx_c is the COMSOL Multiphysics variable for the x-component of the total flux. Variables 2 1 Right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domain 2 only. 5 Locate the Variables section. 6 Click Load from File and browse to the model’s model library folder and double-click the file microreactor_optimization_variables2.txt. MA T E R I A L S On the Home toolbar, click Add Material. A D D MA T E R I A L 1 Go to the Add Material window. 2 In the tree, select Built-In>Water, liquid. 3 In the Add material window, click Add to Component. MA T E R I A L S L A MI N A R F L OW Volume Force 1 1 On the Physics toolbar, click Domains and choose Volume Force. 2 Select Domain 2 only. 3 In the Volume Force settings window, locate the Volume Force section. Solved with COMSOL Multiphysics 4.4 12 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R 4 Specify the F vector as Inlet 1 1 On the Physics toolbar, click Boundaries and choose Inlet. 2 Select Boundary 1 only. 3 In the Inlet settings window, locate the Boundary Condition section. 4 From the Boundary condition list, choose Pressure, no viscous stress. 5 Locate the Pressure, No Viscous Stress section. In the p 0 edit field, type delta_p. Symmetry 1 1 On the Physics toolbar, click Boundaries and choose Symmetry. 2 Select Boundaries 2, 5, and 9 only. Outlet 1 1 On the Physics toolbar, click Boundaries and choose Outlet. 2 Select Boundary 12 only. TR A N S P O R T O F D I L U T E D S P E C I E S Convection and Diffusion 1 1 In the Model Builder window, expand the Component 1>Transport of Diluted Species node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 From the u list, choose Velocity field (spf/fp1). 4 Locate the Diffusion section. In the D c edit field, type D. Reactions 1 1 On the Physics toolbar, click Domains and choose Reactions. 2 Select Domain 2 only. 3 In the Reactions settings window, locate the Reactions section. 4 In the R c edit field, type -phi. Concentration 1 1 On the Physics toolbar, click Boundaries and choose Concentration. 2 In the Concentration settings window, locate the Concentration section. -alpha*u x -alpha*v y Solved with COMSOL Multiphysics 4.4 13 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R 3 Select the Species c check box. 4 In the c 0,c edit field, type c_in. 5 Select Boundary 1 only. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 12 only. This completes the setup of the physics. Now set up the optimization problem. O P T I MI Z A T I O N Define the control variable epsilon, select its shape, and constrain its values to the interval [0, 1]. Control Variable Field 1 1 In the Model Builder window, under Component 1 click Optimization. 2 On the Physics toolbar, click Domains and choose Control Variable Field. 3 Select Domain 2 only. Because the porous catalyst is only used in the reacting domain you can deactivate the inlet and outlet channels. 4 In the Control Variable Field settings window, locate the Control Variable section. 5 In the Control variable name edit field, type epsilon. 6 In the Initial value edit field, type 1. 7 Locate the Discretization section. From the Element order list, choose Linear. Control Variable Bounds 1 1 In the Model Builder window, right-click Control Variable Field 1 and choose Control Variable Bounds. 2 In the Control Variable Bounds settings window, locate the Bounds section. 3 In the Upper bound edit field, type 1. Pointwise Inequality Constraint 1 1 On the Physics toolbar, click Domains and choose Pointwise Inequality Constraint. 2 Select Domain 2 only. 3 In the Pointwise Inequality Constraint settings window, locate the Constraint section. 4 In the Constraint expression edit field, type epsilon. Solved with COMSOL Multiphysics 4.4 14 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R 5 Locate the Bounds section. In the Upper bound edit field, type 1. Next, define the objective function. Integral Objective 1 1 On the Physics toolbar, click Domains and choose Integral Objective. 2 Select Domain 2 only. 3 In the Integral Objective settings window, locate the Objective section. 4 In the Objective expression edit field, type -phi/vol. This example requires a fine mesh, both to solve the physics problem and to resolve the topology optimization problem. ME S H 1 1 In the Model Builder window, under Component 1 click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Element size list, choose Finer. 4 Click the Build All button. S T U D Y 1 Although you can choose to solve the optimization problem directly, it can be useful to check that the solution for the PDE problem looks sound before starting the optimization. 1 On the Home toolbar, click Compute. R E S U L T S Velocity (spf) The first default plot (see Figure 3) shows the velocity field in the reactor. Now solve the optimization problem. S T U D Y 1 Step 1: Stationary 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Stationary. 2 In the Stationary settings window, click to expand the Results while solving section. 3 Locate the Results While Solving section. Select the Plot check box. This setting gives a plot of the evolving velocity distribution in the Graphics window. Solved with COMSOL Multiphysics 4.4 15 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R Optimization 1 On the Study toolbar, click Optimization. 2 In the Optimization settings window, locate the Optimization Solver section. 3 From the Method list, choose SNOPT. Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Optimization Solver 1 node, then click Stationary 1. 3 In the Stationary settings window, locate the General section. 4 In the Relative tolerance edit field, type 1e-6. 5 On the Home toolbar, click Compute. R E S U L T S Velocity (spf) The velocity field in the reactor after optimization should resemble that in Figure 6. Concentration (chds) The third default plot shows the concentration distribution in the reactor after optimization (Figure 5). To reproduce the plot in Figure 4, modify the default plot with the following steps. 2D Plot Group 4 1 In the Model Builder window, under Results click 2D Plot Group 4. 2 In the 2D Plot Group settings window, click to expand the Title section. 3 From the Title type list, choose Manual. 4 In the Title text area, type Distribution of porous catalyst. 5 In the Model Builder window, expand the 2D Plot Group 4 node, then click Surface 1. 6 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Optimization>Control variable epsilon (epsilon). 7 Locate the Coloring and Style section. From the Color table list, choose GrayScale. 8 Clear the Color legend check box. 9 On the 2D plot group toolbar, click Plot. 10 Click the Zoom Extents button on the Graphics toolbar. Solved with COMSOL Multiphysics 4.4 16 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R 11 Right-click Results>2D Plot Group 4>Surface 1 and choose Rename. 12 Go to the Rename Surface dialog box and type Porous Catalyst in the New name edit field. 13 Click OK. Derived Values To display the result for the conversion rate, continue as follows. 1 On the Results toolbar, click Global Evaluation. 2 In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Definitions>Conversion of reactant (X). 3 Click the Evaluate button. T A B L E The value appears in the Table window below the Graphics window. Solved with COMSOL Multiphysics 4.4 1 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R NOx Re duc t i on i n a Monol i t hi c Re a c t or Introduction This example illustrates the modeling of selective NO reduction, that occurs as flue gases pass through the channels of a monolithic reactor in the exhaust system of a motored vehicle. The simulations are aimed at finding the optimal dosing of NH 3 , the reactant that serves as reducing agent in the process. Figure 1: Catalytic converters reduce the NOx levels in the exhaust gases emitted by combustion engines. It is central that NH 3 is present in amounts that do not limit the NO reduction reaction. At the same time, excess NH 3 at the outlet of the reactor will lead to undesirable waste. The situation is complicated by the fact that NH 3 can be depleted by oxidation, a reaction that is parallel to the NO reduction, and that the rates of the two competing reactions are affected by temperature as well as composition. In this model you apply three different analyses to address the issue of proper NH 3 dosing: 1 Kinetic analysis—Study selectivity aspects of the kinetics by modeling initial reaction rates as function of temperature and relative reactant amounts. This analysis gives information about the conditions that are preferred to attain the desired selectivity. Solved with COMSOL Multiphysics 4.4 2 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 2 Process modeling—Model a reactive monolith channel as a non-isothermal plug flow reactor. This model reveals necessary minimal dosing level of NH 3 , based on the working conditions of the catalytic converter. 3 Detailed process modeling—Set up and solve a 3D model of the full reactor monolith including mass transport, heat transfer, and fluid flow. The model reveals the full space-dependency of the problem and lets you confirm or further adjust the dosing levels. Kinetic Analysis C H E MI C A L R E A C T I O N S Two parallel reactions occur in the V 2 O 5 /TiO 2 washcoat of the monolithic reactor. The desired reaction is NO reduction by ammonia: (1) However, ammonia can at the same time undergo oxidation: (2) The heterogeneous catalytic conversion of NO to N 2 is described by an Eley-Rideal mechanism. A key reaction step involves the reaction of gas-phase NO with surface-adsorbed NH 3 . The following rate equation (mol/(m 3 ·s)) has been suggested in Ref. 1 for Equation 1: (3) where (4) and (5) For Equation 2, the reaction rate (mol/(m 3 ·s)) is given by (6) 4NO 4N 2 O 2 + 6H 2 O + 4NH 3 + + 3O 2 2N 2 6H 2 O + 4NH 3 r 1 k 1 c NO ac NH3 1 ac NH3 + --------------------------- = k 1 A 1 E 1 R g T ----------- – \ . | | exp = a A 0 E 0 R g T ----------- – \ . | | exp = r 2 k 2 c NH3 = Solved with COMSOL Multiphysics 4.4 3 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R where (7) R E S U L T S The competing chemical reactions raise issue of optimal dosing of NH 3 to handle the reduction process. Stoichiometry suggests a 1:1 ratio of NH 3 to NO as a lower limit. It is likely that a stoichiometric excess of NH 3 will be necessary but, at the same time, it is desirable to avoid unnecessarily high levels of NH 3 in the gas stream leaving the catalytic converter. Analyzing the kinetics can help you identify conditions favoring the desirable reduction reaction. A first simulation looks at the initial reaction rates of the reduction and oxidation reactions as function of temperature and relative amounts of reactants. Figure 2 shows initial rates for the reduction reaction (Equation 1). The curves represent a set of NH 3 :NO ratios ranging from 1 to 2. The concentration of NO in the exhaust gas entering the catalytic converter is known to be 0.0411 mol/m 3 . Figure 2: Initial reaction rates of the NO reduction reaction (r 1 ) as a function of temperature. The NH 3 :NO ratio ranges from 1 to 2. The rate of NO reduction goes through a maximum and falls off at higher temperatures. Higher concentrations NH 3 in the gas phase will lead to increased levels k 2 A 2 E 2 R g T ----------- – \ . | | exp = Increasing NH 3 :NO ratio Solved with COMSOL Multiphysics 4.4 4 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R of surface-adsorbed NH 3 , in turn favoring the conversion of gas-phase NO to N 2 . This explains the shifts of the rate maximum towards higher temperatures as the NH 3 :NO ratio increases. The decrease in the NO reduction rate at the highest temperatures is explained by the desorption rate of NH 3 from the catalyst surface becoming faster than the reaction of adsorbed NH 3 with gas-phase NO. According to Equation 6, the oxidation rate increases with temperature and NH 3 concentration. A plot of selectivity parameter, S, giving the ratio of reaction rates according to (8) will indicate the concentration and temperature effects on the reaction preference. A value greater than one means that NO reduction is favored, while a value of less than one means NH3 oxidation is the preferred reaction pathway. Clearly the selectivity for NO reduction drops both with temperature and increasing NH 3 :NO ratio. Figure 3: Selectivity parameter (r 1 /r 2 ) as a function of temperature. The NH 3 :NO ratio ranges from 1 to 2. The kinetic investigation suggests that preferred working conditions involve moderate temperatures and relatively low ratios of NH 3 :NO. S r 1 r 2 ----- = Increasing NH 3 :NO ratio Solved with COMSOL Multiphysics 4.4 5 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R Process Modeling—Single Channel Model To find the minimal level of NH 3 required to reduce the NO present in the exhaust gas requires a reactor model accounting for changing reactant concentrations and system temperature. From a mass transfer point of view, channels of the reactor monolith can be considered to be uncoupled to one another. Therefore, it is reasonable to perform initial simulations where a single reactive channel, modeled by non-isothermal plug flow equations, represents the monolith reactor. Set up and solve this model using the Reaction Engineering interface. MO D E L E Q U A T I O N S Assuming steady state, the mass balance equation for a plug flow reactor is given by: (9) where F i is the species molar flow rate (mol/s), V represents the reactor volume (m 3 ), and is R i the species net reaction rate (mol/(m 3 ·s)). The molar flow rate is related to the species concentrations, c i (mol/m 3 ), through the volumetric flow rate, v (m 3 /s): (10) where the volumetric flow rate is given by the average flow velocity, u (m/s), multiplied by the reactor cross-section A (m 2 ): (11) The energy balance for the ideal reacting gas is: (12) F d i V d --------- R i = F i vc i = v uA = F i C p i , dT dV -------- i ¿ Q ext Q + = Solved with COMSOL Multiphysics 4.4 6 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R where C p,i is the species molar heat capacity (J/(mol·K)), and Q ext is the heat added to the system per unit volume (J/(m 3 ·s)). Q denotes the heat due to chemical reaction (J/(m 3 ·s)) (13) where H j the heat of reaction (J/mol), and r j the reaction rate (mol/(m 3 ·s)). R E S U L T S For an exhaust gas containing 41.1 mmol/m 3 of NO, with a temperature of 523 K, passing through the channel at 0.3 m/s, the model suggests a NH 3 :NO ratio of at least 1.3 to guarantee that NH 3 is available as a reductive agent throughout the entire length of the reactive channel. Figure 4: Molar flow rate of NO as function of channel volume. Q H j r j j ¿ – = Solved with COMSOL Multiphysics 4.4 7 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R A plot of the selectivity parameter confirms that the NO reduction chemistry will be favored throughout the monolith reactor at NH 3 :NO ratios 1.3 and 1.4. NO oxidation starts to dominate near the channel outlet for NH 3 :NO = 1.5. Figure 5: Selectivity parameter (r 1 /r 2 ) as a function of channel volume. The NH 3 :NO ratio ranges from 1.3 to 1.5. Solved with COMSOL Multiphysics 4.4 8 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R Settling at NH 3 :NO ratio of 1.35 at the inlet generates the following flow rate and temperature profiles in the single channel model. Figure 6: Molar Flow rates (mol/s) of NO and NH 3 as function of channel volume. The conversions of NO and NH 3 of 98.7% and 97.2%, respectively. Figure 7: Temperature of the reacting gas as function of channel volume. Solved with COMSOL Multiphysics 4.4 9 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R Results show a moderate temperature increase for the given dosing level, which is desirable from a selectivity point of view. A plot of the selectivity parameter, below, confirms that NO reduction is favored in the entire channel (2.3 < S < 3.4). Figure 8: Selectivity parameter (r 1 /r 2 ) as function of channel volume. Detailed Process Modeling—3D Reactor Model It is clear from the kinetic analysis as well as from the single channel model that temperature will play a central role in affecting the optimal dosing of NH 3 . Because the temperature distribution is sure to vary from channel to channel in a full reactor monolith, a full 3D reactor model is called for. MO D E L G E O ME T R Y The monolithic reactor has a modular structure made up of reactive channel blocks and supporting solid walls. The reactor is 0.36 m long with a 0.1 m radius. Each reactive Solved with COMSOL Multiphysics 4.4 10 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R channel has a cross-sectional area of 12.6 mm 2 , and the void fraction of a channel block is 0.75. Figure 9: NO reduction chemistry takes place in the channel blocks. Supporting walls hold together the full reactor geometry. Symmetry reduces the modeling domain to one eighth of the full reactor geometry. MO D E L E Q U A T I O N S The present example takes a pseudo-homogeneous approach to model the hundreds of channels present in the monolith reactor. No mass is exchanged between channels, so each channel is described by 1D mass-transport equations. Furthermore, assume fully developed laminar flow in the channels, such that the average flow field is proportional to the pressure difference across the reactor. The fluid flow transports mass and energy only in the channel direction. The energy equation describes the temperature of the reacting gas in the channels, as well as the conductive heat transfer in the monolith structure and the supporting walls. Because the temperature affects not only reaction kinetics but also the density and viscosity of the reacting gas, the energy equation is what really connects the channels in the reactor structure turning this into a 3D model. Mass Transport The mass balances describing transport and reaction in the monolith channels are given by diffusion-convection equations at steady state: Channel block Inlet Outlet Supporting wall Reactive channel Solved with COMSOL Multiphysics 4.4 11 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R (14) Here D i denotes the diffusion coefficient (m 2 /s), c i is the species concentration (mol/ m 3 ), and u equals the velocity vector (m/s). The term R i (mol/(m 3 ·s)) corresponds to the species’ rate expression, which is a function of the reaction rates, Equation 3 and Equation 6, and the reaction stoichiometry. Mass transport is only allowed in the direction of the channels, corresponding to direction of the x-axis in the 3D geometry used in this example. For the diffusive transport, this is accomplished by setting the y- and z-components of the diffusivity matrix to zero. The pressure-driven flow in the monolith is also defined in the direction of the x-axis, hereby restricting the convective mass transport to the channel direction as well. Each monolith channel thus behaves like a 1D plug flow model with included diffusion. These separate channel models are connected through the heat transfer equations for the reactor monolith. Species concentrations are defined at the reactor inlet boundaries: (15) At the outlet, use the Outflow condition: (16) Fluid Flow The flow of reacting gas through the monolith is modeled using a Darcy’s Law interface with the governing equations: (17) (18) The monolith block is treated as a porous matrix with the effective permeability k (m 2 ). The density, µ (kg/m 3 ), and viscosity, µ (Pa·s), of the gas are assumed to be well represented by the temperature-dependent properties of air, as only relatively small concentrations of NO and NH 3 are present. Pressure conditions are set at the reactor inlet and outlet boundaries. Heat Transfer A single temperature equation describing the heat transfer in the porous monolith reactor can be written as: V D i V – c i ( ) · u Vc i · + R i = c c in = n DV – c ( ) · 0 = V µu ( ) · 0 = u k µ ---Vp – = Solved with COMSOL Multiphysics 4.4 12 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R (19) where µ L (kg/m 3 ) is the fluid density, C pL (J/(kg·K)) is the fluid heat capacity, (µC p ) eq (J/(m 3 ·K)) is the equivalent volumetric heat capacity, and k eq (W/(m·K)) is the equivalent thermal conductivity. Furthermore, u (m/s) is the fluid velocity field, which in this model is calculated in the Darcy’s Law interface. Q (W/m 3 ) is the heat source, which is due to exothermic chemical reactions: (20) Above, H 1 and H 2 (J/(mol·K)) are the heats of reaction. In the stationary case this implies (21) The equivalent conductivity of the solid-fluid system, k eq , is related to the conductivity of the solid, k p , and to the conductivity of the fluid, k L , by (22) Here O p denotes the solid material’s volume fraction, which is related to the volume fraction of the liquid O L (or porosity) by: (23) The Heat Transfer interface sets up Equation 21 for a fluid domain. For the supporting walls in the reactor, only heat transfer by conduction applies: (24) where k s (W/(m·K)) is the thermal conductivity for the solid walls. The temperature is specified at the reactor inlet boundaries: (25) At the outlet, use the Outflow condition: (26) For the reactor walls the heat flux through the boundaries is given by (27) µC p ( ) eq cT ct ------- µ L C pL u VT · + V k eq VT ( ) · Q + = Q Q 1 Q 2 + r 1 H 1 – r 2 H 2 – = = µ L C pL u VT · V k eq VT ( ) · Q + = k eq O p k p O L k L + = O L O p + 1 = V – k s VT ( ) · 0 = T T 0 = n kVT ( ) · 0 = q m n · h T T – amb ( ) = Solved with COMSOL Multiphysics 4.4 13 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R where h (W/(m 2 ·K)) denotes the heat transfer coefficient, and T amb (K) equals the ambient temperature. As mentioned, the temperature affects not only reaction kinetics but also the density and viscosity of the reacting gas. In this way the heat transfer equation connects the channels in the reactor structure. T H E R MO D Y N A MI C A N D TR A N S P O R T P R O P E R T I E S Accurate thermodynamic data is required as input to energy balance equations, both in the plug flow model (Equation 12) and the 3D monolith model (Equation 19). The Reaction Engineering Module uses the following set of polynomials as default expressions describing species thermodynamic properties: (28) (29) (30) Here, C p,i denotes the species’ heat capacity (J/(mol·K)), T the temperature (K), and R g the ideal gas constant, 8.314 (J/(mol·K)). Further, h i is the species’ molar enthalpy (J/mol), and s i represents its molar entropy (J/(mol·K)). A set of seven coefficients per species are taken as input for the polynomials above. The coefficients a 1 through a 5 relate to the species heat capacity, the coefficient a 6 is associated with the species enthalpy of formation (at 0 K), and the coefficient a 7 comes from the species entropy of formation (at 0 K). The equation form outlined by Equation 28 through Equation 30 is referred to as CHEMKIN or NASA format (Ref. 2). Database resources list the needed coefficients for different temperature intervals (Ref. 3). In the this example files with thermodynamic coefficients will be read into the software such that the predefined thermodynamic property expressions can be used directly. In addition to thermodynamic properties, the model equations also require transport properties to accurately describe the space dependent reactor model. For instance, the mass transport (Equation 14) needs species specific diffusion coefficients as input. For reacting gas mixtures, the Reaction Engineering feature makes use of kinetic gas theory to set up expressions for transport properties such as diffusivities, viscosity, and C p i , R g a 1 a 2 T a 3 T 2 a 4 T 3 a 5 T 4 + + + + ( ) = h i R g a 1 T a 2 2 ------T 2 a 3 3 ------T 3 a 4 4 ------T 4 a 5 5 ------T 5 a 6 + + + + + \ . | | = s i R g a 1 T ln a 2 T a 3 2 ------T 2 a 4 3 ------T 3 a 5 4 ------T 4 a 7 + + + + + \ . | | = Solved with COMSOL Multiphysics 4.4 14 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R thermal conductivity as functions of temperature, pressure, and composition. In this example, the diffusivities (m 2 /s) are calculated using the formula (31) where O D is a collision integral . (32) To evaluate Equation 31 you need to specify the characteristic length and energy minimum of the Lennard-Jones interaction potential, that is o (10 ÷10 m) and c/k b (K), respectively. The species dipole moment, µ D (Debye), can also be provided. Each species in the reacting gas has a characteristic set of these constants, and you find their values in the literature, in databases, or from experiments. The parameters o, c/k b , and µ D can either be entered manually in the Reaction Engineering interface, or you can import text files containing the data, as is done in this example. The data parameters used for this model have been compiled from Ref. 4 and are summarized below: R E S U L T S Figure 10 shows the conversion of NO in the monolith channel blocks. The average conversion at the outlet is 97.5 %. This is somewhat lower than the 98.7 % conversion predicted by the single channel model. The isosurfaces in the plot show how a SPECIES o [Å] c/k b [K] µ D [D] H2O 809.1 2.640 1.8 N2 71.4 3.798 0.0 NH3 558.3 2.900 1.5 NO 116.7 3.492 0.2 O2 106.7 3.467 0.0 D 2.695 10 3 – T 3 M A M B + ( ) 2 10 3 M · A M B ( ) ( ) po A o B O D ------------------------------------------------------------------------------------------- · · = O D f T o c k b ------ µ , , , \ . | | = Solved with COMSOL Multiphysics 4.4 15 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R channel’s performance depends on its position in the reactor, clearly pointing to the 3D nature of the problem. Figure 10: Isosurfaces showing the conversion of NO in the reactor. The individual channels, although they do not exchange mass, are connected through the temperature distribution in the reactor. The temperature affects both the flow velocity of the reacting gas as well as the reaction rates. Cross sections of the reactor Solved with COMSOL Multiphysics 4.4 16 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R temperature are shown in Figure 11. Figure 11: Temperature distribution in cross sections of the reactor. The exothermic reactions increase the temperature in the central parts of the reactor, while the temperature is decreased through heat loss to the surroundings. The maximum temperature calculated for the 3D reactor is 541.5 K, which is higher than for the single channel model, where T max = 533.9 K. The effect of the relatively high thermal conductivity of the supporting walls is clearly visible. A seen from the initial kinetic analysis, elevated temperatures have a detrimental effect on the selectivity, leading to ammonia getting oxidized (Equation 2) rather than be consumed in the NO reducing reaction (Equation 1). A plot of r 1 /r 2 on the symmetry surface of the monolith is shown in Figure 12. The fact that r 1 /r 2 is greater than 1 Solved with COMSOL Multiphysics 4.4 17 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R signals that the selectivity favors the desired reducing reaction. Figure 12: The fact that r 1 /r 2 is greater than 1 signals that the selectivity favors the desired NO reducing reaction. The selectivity plot once again reveals the space-dependent nature of the problem. Channels in the relatively cold region near the reactor outer surface display high selectivity throughout, whereas channels in the region close to the center see selectivity falling off comparatively fast. Compared to the single channel model, the 3D reactor shows notably lower values of the selectivity parameter near the center of the outlet. Nevertheless, NO reduction is still favored throughout. Summary The model uses the Chemical Reaction Engineering Module to perform three different analyses concerning the reduction of NO in a monolithic reactor: 1 Kinetic analysis—to explore the system of competing reactions and learn what conditions that promote selectivity towards NO reduction. 2 Process modeling—to explore the coupled mass and energy balance equations in a single channel model, resulting in a first estimate of the NH 3 dosing level. Solved with COMSOL Multiphysics 4.4 18 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 3 Detailed process modeling—testing the reactor operating conditions in a full 3D reactor representation, noting the space-dependent effect due to coupling between monolith channels. A reactant ratio NH 3 :NO of approximately 1.35 is found to be close to optimal under the investigated conditions. This ratio leads to minimal waste of ammonia without limiting the NO reduction chemistry. It also favors the selectivity for NO reduction while at the same time restricting the possible heat evolved through the chemical reactions. This in turn helps control the temperature in the reactor, again favoring the NO reduction. The sequential approach used in this example—going from kinetic analysis, to process modeling, to detailed process modeling—has several advantages. Starting with fast simulations using models that are easy to set up allows you to identify and narrow down the process condition envelope before you move to more advanced and computationally demanding models. The sequential modeling approach also lets you identify when and how effects such as temperature dependency and space dependency come into play. This deepened system understanding leads to efficient model setup and solution strategies. Going from perfectly mixed conditions to full space-dependency in 3D also puts you in the position to decide what level of detail is needed for the particular system. References 1. G. Schaub, D. Unruh, J. Wang, and T. Turek, “Kinetic analysis of selective catalytic NOx reduction (SCR) in a catalytic filter,” Chemical Engineering and Processing, vol. 42, p. 365, 2003. 2. S. Gordon and B.J. McBride, Computer Program for Calculation of Complex Chemical Equilibrium Compositions, Rocket Performance, Incident and Reflected Shocks, and Chapman-Jouquet Detonations, NASA-SP-273, 1971. 3. This example uses data from the GRI-Mech 3.0. http://www.me.berkeley.edu/ gri-mech/ 4. B.E. Poling, J.M. Prausnitz, and J.P. O’Connell, The Properties of Gases and Liquids, fifth ed., McGraw-Hill, 2001. Solved with COMSOL Multiphysics 4.4 19 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R Model Library path: Chemical_Reaction_Engineering_Module/ Heterogeneous_Catalysis/monolith_3d Modeling Instructions The step-by-step instructions below guide you through the process of setting up and solving two models simulating the catalytic reduction of NO in a monolith reactor. First, a plug flow reactor model is used to simulate the non-isothermal chemistry taking place in a single monolith channel. This model outputs the results presented in Figure 6 through Figure 8. The second model accounts for the full 3D monolith reactor, coupling mass transport to heat transfer and fluid flow. Figure 10 through Figure 12 show sample results from this model. Note: The step-by-step instructions to set up the model generating the results shown in Figure 2 through Figure 5 are presented in the documentation for the model monolith_kinetics.mph, available in the Chemical Reaction Engineering Module Model Library. This model is also used in the Introduction to the Chemical Reaction Engineering Module From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Stationary Plug Flow. 6 Click the Done button. Solved with COMSOL Multiphysics 4.4 20 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R C O MP O N E N T 1 1 In the Model Builder window, right-click Component 1 and choose Rename. 2 Go to the Rename Component dialog box and type Channel Model in the New name edit field. 3 Click OK. G L O B A L D E F I N I T I O N S Start by reading in a set of global parameters defining the process conditions for the monolith reactor, including the dimensions or the reactive channels, the flow rate of the reacting gas, and the temperature conditions. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file monolith_3d_parameters.txt. R E A C T I O N E N G I N E E R I N G Now define the chemical reactions. First, type in the reaction formula for NO reduction. The Reaction Engineering feature will automatically interpret the reaction formula and suggest a reaction rates based on the mass-action law. Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type 4NO+4NH3+O2=>4N2+6H2O. In this example, replace the automatically generated reaction rate expression with the rate expression known from the literature. 4 Locate the Reaction Rate section. From the Reaction rate list, choose User Defined. 5 In the r edit field, type k1*c_NO*a*c_NH3/(1+a*c_NH3). Reaction 2 1 On the Physics toolbar, click Global and choose Reaction. In the same fashion define the reaction for NH3 oxidation. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type 4NH3+3O2=>2N2+6H2O. Solved with COMSOL Multiphysics 4.4 21 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 4 Locate the Reaction Rate section. From the Reaction rate list, choose User Defined. 5 In the r edit field, type k2*c_NH3. D E F I N I T I O N S Read in expressions defining the rate constants and the selectivity parameter S. Variables 1 1 In the Model Builder window, under Channel Model right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file monolith_3d_variables.txt. R E A C T I O N E N G I N E E R I N G After setting up the reaction kinetics, move on to define the reactor model where the chemistry takes place. Selecting the Reaction Engineering node you can choose one of the predefined Reactor types. In this case, select a nonisothermal plug-flow reactor to represent a reactive channel in the monolith. 1 In the Model Builder window, under Channel Model click Reaction Engineering. 2 In the Reaction Engineering settings window, locate the Reactor Settings section. 3 From the Reactor type list, choose Plug flow. The Reaction Engineering feature can set up predefined expressions for species transport and thermodynamic properties. Thermodynamic property expressions follow the NASA polynomial format while transport property expressions are based of the kinetic gas theory. Input files on the CHEMKIN format can be read into the software supplying all necessary input parameters for the expressions. The thermodynamic expressions enter the energy balance of the Plug flow reactor. The transport expressions, for instance describing species diffusivity coefficients, will be used in the 3D monolith model set up later on. 4 Select the Calculate thermodynamic properties check box. 5 Select the Calculate transport properties check box. 6 Locate the General section. In the T edit field, type T_in. 7 Locate the Mass Balance section. In the v edit field, type v_av*A. Solved with COMSOL Multiphysics 4.4 22 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 8 Click to expand the CHEMKIN section. In the Thermo input file edit field, type monolith_3d_thermo.txt. 9 Click the Import button. 10 In the Transport input file edit field, type monolith_3d_transport.txt. 11 Click the Import button. At this point you have set up the reaction kinetics and chosen a plug flow reactor to model NO reduction in a monolith channel. In the next part of the set up, define the species molar flow rates at the reactor inlet. Species: NO 1 In the Model Builder window, under Channel Model>Reaction Engineering click Species: NO. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type 1.55e-7[mol/s]. Species: NH3 1 In the Model Builder window, under Channel Model>Reaction Engineering click Species: NH3. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type 2.1e-7[mol/s]. Species: O2 1 In the Model Builder window, under Channel Model>Reaction Engineering click Species: O2. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type 2.71e-6[mol/s]. Species: N2 1 In the Model Builder window, under Channel Model>Reaction Engineering click Species: N2. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type 6.86e-5[mol/s]. Species: H2O 1 In the Model Builder window, under Channel Model>Reaction Engineering click Species: H2O. 2 In the Species settings window, click to expand the Species feed stream section. Solved with COMSOL Multiphysics 4.4 23 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 3 Locate the Species Feed Stream section. In the F 0 edit field, type 7.34e-6[mol/s]. Species: N2 1 In the Model Builder window, under Channel Model>Reaction Engineering click Species: N2. 2 In the Species settings window, locate the Species Formula section. 3 From the list, choose Solvent. N2 is the dominant species in the reacting mixture and does not take part in the chemical reactions. Assigning this species as a Solvent removes the associated mass balance. Also, automatically generated model expressions assume that the physical properties of the reacting mixture are the same as the properties of N2. Energy Balance 1 In the Model Builder window, right-click Reaction Engineering and choose Energy Balance. Finalize the set up of the nonisothermal plug flow reactor by adding an Energy Balance feature. Note that all thermodynamic properties that go into the energy balance are automatically set up by the software. You need only supply in the temperature at the inlet and heat transferred to the reactor surroundings. 2 In the Energy Balance settings window, locate the Energy Balance section. 3 In the Q ext edit field, type (T_amb-T)*UA. 4 In the T 0 edit field, type T_in. S T U D Y 1 Step 1: Stationary Plug Flow 1 In the Model Builder window, under Study 1 click Step 1: Stationary Plug Flow. 2 In the Stationary Plug Flow settings window, locate the Study Settings section. 3 In the Volumes edit field, type 0 0.36*A. 4 Select the Relative tolerance check box. 5 In the associated edit field, type 1e-5. Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solver 1 node, then click Plug Flow Solver 1. 3 In the Plug Flow Solver settings window, click to expand the Absolute tolerance section. Solved with COMSOL Multiphysics 4.4 24 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 4 Locate the Absolute Tolerance section. In the Tolerance edit field, type 1e-6. 5 On the Home toolbar, click Compute. R E S U L T S Flow Rate (re) 1 In the Model Builder window, expand the Flow Rate (re) node, then click Global 1. 2 In the Global settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Reaction Engineering>Molar flow rate (comp1.re.F_NO). 3 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Reaction Engineering>Molar flow rate (comp1.re.F_NH3). 4 On the 1D plot group toolbar, click Plot. Conversion is close to 99% for NO and 97% for NH3. Temperature (re) 1 On the 1D plot group toolbar, click Plot. The temperature goes through a maximum at 534 K in the first half of the channel as the chemical reactions expel heat. 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Global. 3 In the Global settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Definitions>Selectivity parameter (comp1.S). 4 On the 1D plot group toolbar, click Plot. The selectivity for the NO reduction reaction falls off in the second half of the reactor even though the decreasing temperature should favor this reaction. The reason for this is that relatively low ratio of NO to NH3. R E A C T I O N E N G I N E E R I N G In the next phase of the example you prepare to set up a 3D model of the monolithic reactor, including mass transport and reaction, heat transfer, and fluid flow. The Generate Space-Dependent Model feature creates a active link between the plug flow channel model and the full 3D monolith model. It allows you to transfer reaction Solved with COMSOL Multiphysics 4.4 25 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R kinetics, thermodynamics, and transport properties set up in the Reaction Engineering feature to the physics interfaces describing space and time-dependent systems. Generate Space-Dependent Model 1 1 In the Model Builder window, under Channel Model right-click Reaction Engineering and choose Generate Space-Dependent Model. 2 In the Generate Space-Dependent Model settings window, locate the Physics Interfaces section. 3 From the Energy balance list, choose Heat Transfer in Fluids: New. 4 Select the Create inflow and outflow features check box. 5 Locate the Space-Dependent Model Generation section. Select the Enable space-dependent physics interfaces check box. C O MP O N E N T 2 1 In the Model Builder window, right-click Component 2 and choose Rename. 2 Go to the Rename Component dialog box and type 3D Model in the New name edit field. 3 Click OK. 3 D MO D E L The Generate Space-Dependent Model has set up the Transport of Diluted species interface and the Heat Transfer in Fluids interface. You are free to include additional physics interfaces through the Model Wizard. Here, add a Darcy's Law interface in order to model the flow through the porous channel blocks. 1 On the Home toolbar, click Add Physics. A D D P HY S I C S 1 Go to the Add Physics window. 2 In the Add physics tree, select Fluid Flow>Porous Media and Subsurface Flow>Darcy's Law (dl). 3 In the Add physics window, click Add to Component. G E O ME T R Y 1 Import a file with the reactor geometry. Symmetry reduces the modeling domain to one eighth of the full monolith. Solved with COMSOL Multiphysics 4.4 26 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R Import 1 1 On the Home toolbar, click Import. 2 In the Import settings window, locate the Import section. 3 Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file monolith_3d.mphbin. 5 Click the Import button. D E F I N I T I O N S A central part of the model set up consists of assigning features in the Model Builder to subdomains and boundaries of the model geometry. In this work it is often efficient to create selection groups that can be referenced by name. Explicit 1 1 On the Definitions toolbar, click Explicit. 2 Select Domain 1 only. 3 Right-click 3D Model>Definitions>Explicit 1 and choose Rename. 4 Go to the Rename Explicit dialog box and type supporting walls in the New name edit field. 5 Click OK. Explicit 2 1 On the Definitions toolbar, click Explicit. 2 Select Domains 2–6 only. 3 Right-click 3D Model>Definitions>Explicit 2 and choose Rename. 4 Go to the Rename Explicit dialog box and type channel blocks in the New name edit field. 5 Click OK. Explicit 3 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Click the Zoom Extents button on the Graphics toolbar. 5 Select Boundaries 4, 9, 13, 19, and 23 only. 6 Right-click 3D Model>Definitions>Explicit 3 and choose Rename. Solved with COMSOL Multiphysics 4.4 27 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 7 Go to the Rename Explicit dialog box and type inlet in the New name edit field. 8 Click OK. Explicit 4 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 30–34 only. 5 Right-click 3D Model>Definitions>Explicit 4 and choose Rename. 6 Go to the Rename Explicit dialog box and type outlet in the New name edit field. 7 Click OK. Explicit 5 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 2, 3, 6, 8, 15, and 18 only. 5 Right-click 3D Model>Definitions>Explicit 5 and choose Rename. 6 Go to the Rename Explicit dialog box and type symmetry in the New name edit field. 7 Click OK. Explicit 6 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 1 only. 5 Right-click 3D Model>Definitions>Explicit 6 and choose Rename. 6 Go to the Rename Explicit dialog box and type inlet walls in the New name edit field. 7 Click OK. Explicit 7 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. Solved with COMSOL Multiphysics 4.4 28 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 4 Select Boundary 29 only. 5 Right-click 3D Model>Definitions>Explicit 7 and choose Rename. 6 Go to the Rename Explicit dialog box and type outlet walls in the New name edit field. 7 Click OK. Explicit 8 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 27 only. 5 Right-click 3D Model>Definitions>Explicit 8 and choose Rename. 6 Go to the Rename Explicit dialog box and type reactor surface in the New name edit field. 7 Click OK. Union 1 1 On the Definitions toolbar, click Union. 2 In the Union settings window, locate the Geometric Entity Level section. 3 From the Level list, choose Boundary. 4 Locate the Input Entities section. Under Selections to add, click Add. 5 Go to the Add dialog box. 6 In the Selections to add list, choose inlet and inlet walls. 7 Click the OK button. 8 Right-click 3D Model>Definitions>Union 1 and choose Rename. 9 Go to the Rename Union dialog box and type inlet end in the New name edit field. 10 Click OK. MA T E R I A L S Move on to specify material properties for the model. Ready to use materials can be selected from available libraries. You can also define your own materials. 1 On the Home toolbar, click Add Material. A D D MA T E R I A L 1 Go to the Add Material window. Solved with COMSOL Multiphysics 4.4 29 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 2 In the tree, select Liquids and Gases>Gases>Nitrogen. 3 In the Add material window, click Add to Component. MA T E R I A L S Nitrogen Assigning a material to selections in the geometry makes the physical properties of the material available to the physics interfaces. 1 In the Model Builder window, under 3D Model>Materials click Nitrogen. 2 In the Material settings window, locate the Geometric Entity Selection section. 3 From the Selection list, choose channel blocks. Next, create a user-defined material and associate it with the supporting walls. Material 2 1 In the Model Builder window, right-click Materials and choose New Material. 2 In the Material settings window, locate the Geometric Entity Selection section. 3 From the Selection list, choose supporting walls. 4 In the Model Builder window, expand the Material 2 node, then click Basic. 5 In the Property Group settings window, locate the Output Properties and Model Inputs section. 6 Find the Model inputs subsection. In the tree, select Output Properties>Density. 7 Click Add. 8 In the table, enter the following settings: 9 In the tree, select Output Properties>Heat Capacity at Constant Pressure. 10 Click Add. 11 In the table, enter the following settings: 12 In the tree, select Output Properties>Thermal Conductivity. 13 Click Add. Property Variable Expression Unit Size Density rho 2970[kg/m^3] kg/m³ 1x1 Property Variable Expression Unit Size Heat capacity at constant pressure Cp 975[J/kg/K] J/(kg·K) 1x1 Solved with COMSOL Multiphysics 4.4 30 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 14 In the table, enter the following settings: 15 In the Model Builder window, right-click Material 2 and choose Rename. 16 Go to the Rename Material dialog box and type Walls in the New name edit field. 17 Click OK. In the next stage of the modeling process you will set up the physics interfaces describing the mass transport, heat transfer, and fluid flow in the monolithic reactor. TR A N S P O R T O F D I L U T E D S P E C I E S 1 1 In the Transport of Diluted Species settings window, locate the Domain Selection section. 2 From the Selection list, choose channel blocks. Convection and Diffusion 1 Couple the mass transport to the fluid flow by selecting the Darcy's Law velocity field as Model Input. 1 In the Model Builder window, expand the Transport of Diluted Species 1 node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 From the u list, choose Darcy's velocity field (dl/dlm1). The mass transport model for the monolith channels assume that there is only diffusive mass transport in the axial direction of the reactor, here, along the x-axis. This can be accomplished by specifying the diffusivity only in the first element of the diagonal diffusion matrix. Note also the variables predefined in the Diffusion coefficient edit fields, corresponding to diffusivity expressions, are set up by the Generate Space-Dependent Model feature. 4 Locate the Diffusion section. From the symmetry property list, choose Diagonal. 5 In the D cNO table, enter the following settings: Property Variable Expression Unit Size Thermal conductivity k; kii = k, kij = 0 35[W/m/K] W/(m·K) 3x3 root.comp1.re.D_NO 0 0 0 0 0 0 0 0 Solved with COMSOL Multiphysics 4.4 31 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 6 From the symmetry property list, choose Diagonal. 7 In the D cNH3 table, enter the following settings: 8 From the symmetry property list, choose Diagonal. 9 In the D cO2 table, enter the following settings: 10 From the symmetry property list, choose Diagonal. 11 In the D cH2O table, enter the following settings: Features defining reaction rates and inlet concentrations have also been set up during the model generation procedure. Definitions correspond to the reactor conditions specified for the plug-flow channel model. All you have to do is to assign the Reactions and Inlet features with the proper domains and boundaries of the 3D reactor. Inflow 1 1 In the Model Builder window, under 3D Model>Transport of Diluted Species 1 click Inflow 1. 2 In the Inflow settings window, locate the Boundary Selection section. 3 From the Selection list, choose inlet. Outflow 1 1 In the Model Builder window, under 3D Model>Transport of Diluted Species 1 click Outflow 1. 2 In the Outflow settings window, locate the Boundary Selection section. 3 From the Selection list, choose outlet. root.comp1.re.D_NH3 0 0 0 0 0 0 0 0 root.comp1.re.D_O2 0 0 0 0 0 0 0 0 root.comp1.re.D_H2O 0 0 0 0 0 0 0 0 Solved with COMSOL Multiphysics 4.4 32 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R H E A T TR A N S F E R I N F L U I D S 1 Next, set up the Heat Transfer interface. Start by defining the conductive heat transfer in the supporting solid walls. Note that physical properties of the walls are taken from the material called Walls, associated with that domain. Heat Transfer in Solids 1 1 On the Physics toolbar, click Domains and choose Heat Transfer in Solids. 2 In the Heat Transfer in Solids settings window, locate the Domain Selection section. 3 From the Selection list, choose supporting walls. Next specify the Heat transfer in Fluids feature, accounting for convective and conductive heat transfer in the channel blocks. Heat Transfer in Fluids 1 1 In the Model Builder window, click Heat Transfer in Fluids 1. 2 In the Heat Transfer in Fluids settings window, locate the Model Inputs section. 3 From the p list, choose Pressure (dl/dlm1). 4 From the u list, choose Darcy's velocity field (dl/dlm1). 5 Locate the Heat Conduction, Fluid section. From the k list, choose User defined. From the list, choose Diagonal. Specifying the diagonal thermal conductivity elements allows you to represent anisotropic conductive heat transfer in the channel blocks. 6 In the k table, enter the following settings: 7 Locate the Thermodynamics, Fluid section. From the µ list, choose From material. 8 From the C p list, choose From material. 9 From the ¸ list, choose From material. Associate the Heat Source due to the chemical the exothermic chemistry with the channel blocks. Note that the feature and the expressions describing the heat source are generated by the Generate Space-Dependent Model feature, linking the 3D monolith model to the plug-flow channel model. 0.13 0 0 0 0.25 0 0 0 0.25 Solved with COMSOL Multiphysics 4.4 33 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R Heat Source 1 1 In the Model Builder window, under 3D Model>Heat Transfer in Fluids 1 click Heat Source 1. 2 In the Heat Source settings window, locate the Domain Selection section. 3 From the Selection list, choose channel blocks. Complete the set up of the Heat Transfer interface by assigning the following boundary conditions. Temperature 1 1 In the Model Builder window, under 3D Model>Heat Transfer in Fluids 1 click Temperature 1. 2 In the Temperature settings window, locate the Boundary Selection section. 3 From the Selection list, choose inlet. Outflow 1 1 In the Model Builder window, under 3D Model>Heat Transfer in Fluids 1 click Outflow 1. 2 In the Outflow settings window, locate the Boundary Selection section. 3 From the Selection list, choose outlet. Temperature 1a 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 In the Temperature settings window, locate the Boundary Selection section. 3 From the Selection list, choose inlet walls. 4 Locate the Temperature section. In the T 0 edit field, type T_in. Heat Flux 1 1 On the Physics toolbar, click Boundaries and choose Heat Flux. 2 In the Heat Flux settings window, locate the Boundary Selection section. 3 From the Selection list, choose reactor surface. 4 Locate the Heat Flux section. Click the Inward heat flux button. 5 In the h edit field, type 10. 6 In the T ext edit field, type T_amb. Heat Flux 2 1 On the Physics toolbar, click Boundaries and choose Heat Flux. 2 In the Heat Flux settings window, locate the Boundary Selection section. Solved with COMSOL Multiphysics 4.4 34 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 3 From the Selection list, choose outlet walls. 4 Locate the Heat Flux section. Click the Inward heat flux button. 5 In the h edit field, type 1. 6 In the T ext edit field, type T_amb. Symmetry 1 1 On the Physics toolbar, click Boundaries and choose Symmetry. 2 In the Symmetry settings window, locate the Boundary Selection section. 3 From the Selection list, choose symmetry. Follow these steps to set up the Darcy's Law interface describing the fluid flow. D A R C Y ' S L A W 1 In the Model Builder window, under 3D Model click Darcy's Law. 2 In the Darcy's Law settings window, locate the Domain Selection section. 3 From the Selection list, choose channel blocks. Fluid and Matrix Properties 1 1 In the Model Builder window, expand the Darcy's Law node, then click Fluid and Matrix Properties 1. 2 In the Fluid and Matrix Properties settings window, locate the Model Inputs section. 3 From the T list, choose Temperature (ht). 4 Locate the Matrix Properties section. From the k list, choose User defined. In the associated edit field, type 4.27e-8. 5 From the c p list, choose User defined. In the associated edit field, type 0.75. Pressure 1 1 On the Physics toolbar, click Boundaries and choose Pressure. 2 In the Pressure settings window, locate the Boundary Selection section. 3 From the Selection list, choose inlet. 4 Locate the Pressure section. In the p 0 edit field, type 70. Pressure 2 1 On the Physics toolbar, click Boundaries and choose Pressure. 2 In the Pressure settings window, locate the Boundary Selection section. Solved with COMSOL Multiphysics 4.4 35 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 3 From the Selection list, choose outlet. This completes the set up of the model equations describing the reacting flow and heat transfer in the monolith. Before solving the problem numerically the geometry needs to be discretized with a mesh. First create a free triangular mesh at the reactor inlet and then complete the mesh by sweeping in the axial direction of the reactor. ME S H 1 Free Triangular 1 1 In the Model Builder window, under 3D Model right-click Mesh 1 and choose Free Triangular. 2 In the Free Triangular settings window, locate the Boundary Selection section. 3 From the Selection list, choose inlet end. Size 1 1 Right-click 3D Model>Mesh 1>Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Selection list, choose inlet walls. 4 Locate the Element Size section. Click the Custom button. 5 Locate the Element Size Parameters section. Select the Maximum element size check box. 6 In the associated edit field, type 0.0022. 7 Select the Resolution of narrow regions check box. 8 In the associated edit field, type 0.85. 9 Click the Build All button. Swept 1 In the Model Builder window, right-click Mesh 1 and choose Swept. Distribution 1 1 In the Model Builder window, under 3D Model>Mesh 1 right-click Swept 1 and choose Distribution. 2 In the Distribution settings window, locate the Distribution section. 3 In the Number of elements edit field, type 20. 4 Click the Build All button. Solved with COMSOL Multiphysics 4.4 36 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R S T U D Y 2 It is now time to solve the monolith model. Solver 2 1 On the Study toolbar, click Show Default Solver. Changing from an iterative to a direct solver leads to faster solution times at the expense of increased memory requirements. Changing the defaults solver settings in this way makes sense for the medium-sized 3D model you are working with here. 2 In the Model Builder window, expand the Study 2>Solver Configurations>Solver 2>Stationary Solver 1 node. 3 Right-click Direct and choose Enable. 4 On the Home toolbar, click Compute. R E S U L T S 3D Plot Group 10 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the Model Builder window, under Results right-click 3D Plot Group 10 and choose Isosurface. 3 In the Isosurface settings window, locate the Expression section. 4 In the Expression edit field, type (cNO_0-cNO)/cNO_0. 5 Locate the Levels section. In the Total levels edit field, type 20. 6 On the 3D plot group toolbar, click Plot. The average conversions of NO and NH3 at the outlet are 97.5% and 95.5%, respectively. 3D Plot Group 11 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the Model Builder window, under Results right-click 3D Plot Group 11 and choose Slice. 3 In the Slice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Heat Transfer in Fluids 1>Temperature>Temperature (T). 4 Locate the Plane Data section. In the Planes edit field, type 10. Solved with COMSOL Multiphysics 4.4 37 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R 5 On the 3D plot group toolbar, click Plot. The following steps generate plot of the selectivity parameter S, defined on the mirror plane cutting the channel blocks in half. Data Sets 1 On the Results toolbar, click More Data Sets and choose Surface. 2 Select Boundaries 6 and 15 only. 2D Plot Group 12 1 On the Home toolbar, click Add Plot Group and choose 2D Plot Group. 2 In the Model Builder window, under Results right-click 2D Plot Group 12 and choose Surface. 3 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Definitions>Selectivity parameter (comp1.S). 4 Right-click Results>2D Plot Group 12>Surface 1 and choose Height Expression. 5 On the 2D plot group toolbar, click Plot. 6 Click the Go to Default 3D View button on the Graphics toolbar. Solved with COMSOL Multiphysics 4.4 38 | N O X R E D U C T I O N I N A M O N O L I T H I C R E A C T O R Solved with COMSOL Multiphysics 4.4 1 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S Anal y s i s of NO Re ac t i on Ki ne t i c s Introduction The present example takes a closer look at the selective reduction of NO by ammonia that occurs as flue gases pass through the channels of a monolithic reactor in the exhaust system of a car. It is central that NH 3 is present in amounts that do not limit the NO reduction reaction. At, the same time, excess NH 3 at the outlet of the reactor will lead to undesirable waste. The situation is complicated by the fact that ammonia can be depleted by oxidation, a reaction that is parallel to the NO reduction, and that the rates of the two competing reactions are affected by temperature as well as composition. An analysis of the systems reaction kinetics is performed in an effort to find the optimal NH 3 dosing levels. Model Definition C H E MI C A L R E A C T I O N S NO reduction by ammonia can be summarized by the following reaction: (1) However, ammonia can at the same time undergo oxidation: (2) The heterogeneous catalytic conversion of NO to N 2 is described by an Eley-Rideal mechanism. A key reaction step involves the reaction of gas-phase NO with surface-adsorbed NH 3 . The following rate equation (mol/(m 3 ·s)) has been suggested in Ref. 1for Equation 1: (3) where (4) 4NO 4N 2 O 2 + 6H 2 O + 4NH 3 + + 3O 2 2N 2 6H 2 O + 4NH 3 r 1 k 1 c NO ac NH3 1 ac NH3 + --------------------------- = k 1 A 1 E 1 R g T ----------- – \ . | | exp = Solved with COMSOL Multiphysics 4.4 2 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S and (5) For Equation 2, the reaction rate (mol/(m 3 ·s)) is given by (6) where (7) The competing chemical reactions raise issue of optimal dosing of NH 3 to handle the reduction process. Stoichiometry suggests a 1:1 ratio of NH 3 to NO as a lower limit. It is likely that a stoichiometric excess of NH 3 will be necessary but, at the same time, we want to avoid unnecessarily high levels of NH 3 in the gas stream leaving the catalytic converter. Results and Discussion I N I T I A L R E A C T I O N R A T E S Analyzing the kinetics can help us narrow down a proper NH 3 :NO ratio. A first study looks at the initial reaction rates of the reduction and oxidation reactions as function of temperature and relative amounts of reactants. Figure 1 shows initial rates for the reduction reaction (Equation 1). The curves represent a set of NH 3 :NO ratios ranging a A 0 E 0 R g T ----------- – \ . | | exp = r 2 k 2 c NH3 = k 2 A 2 E 2 R g T ----------- – \ . | | exp = Solved with COMSOL Multiphysics 4.4 3 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S from 1 to 2. The concentration of NO in the exhaust gas entering the catalytic converter is known to be 0.0411 mol/m 3 . Figure 1: Initial reaction rates of the NO reduction reaction (r 1 ) as a function of temperature. The NH 3 :NO ratio ranges from 1 to 2. The rate of NO reduction goes through a maximum and falls off at higher temperatures. The maximum shifts towards higher temperatures as the ratio NH 3 :NO increases. This behavior is explained by the desorption rate of NH 3 from the catalyst surface becoming faster than the reaction of adsorbed NH 3 with gas-phase NO. According to Equation 6, the oxidation rate increases with temperature and NH 3 concentration. A plot of selectivity parameter, S, giving the ratio of reaction rates according to (8) S r 1 r 2 ----- = Solved with COMSOL Multiphysics 4.4 4 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S will indicate the concentration and temperature effects on the reaction preference. A value greater than one means that NO reduction is favored. Clearly the selectivity for NO reduction drops both with temperature and increasing NH 3 :NO ratio. Figure 2: Selectivity parameter (r 1 /r 2 ) as a function of temperature. The NH 3 :NO ratio ranges from 1 to 2. The kinetic analysis suggests that preferred working conditions involve moderate temperatures and relatively low ratios of NH 3 :NO. N O N I S O T H E R MA L C H A N N E L MO D E L To find the minimal level of NH 3 required to reduce the NO present in the exhaust gas requires a reactor model accounting for changing reactant concentrations and system temperature. For this purpose we use a nonisothermal plug flow reactor serves to simulate the behavior of a reactive channel. For an exhaust gas containing 41.1 mmol/m 3 of NO, with a temperature of 523 K, passing through the monolith at 0.3 m/s, the model suggests a NH 3 :NO ratio of at Increasing NH 3 :NO ratio Solved with COMSOL Multiphysics 4.4 5 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S least 1.3 to guarantee that NH 3 is available as a reductive agent throughout the entire length of the reactive channel. Figure 3: Molar flow rate of NO as function of channel volume. Solved with COMSOL Multiphysics 4.4 6 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S A plot of the selectivity parameter confirms that the NO reduction chemistry will be favored in the monolith reactor at NH 3 :NO ratios ranging from 1.3 to 1.5. Figure 4: Selectivity parameter (r 1 /r 2 ) as a function of channel volume. The NH 3 :NO ratio ranges from 1.3 to 1.5. It is clear from the kinetic analysis as well as from the nonisothermal channel model that temperature will play a central role in affecting the optimal dosing of NH 3 . As the temperature distribution is sure to vary from channel to channel in a full reactor monolith, a full 3D reactor model is called for. Based on the initial single channel simulations a NH 3 :NO ratio between 1.3 and 1.4 appears appropriate for the 3D model. Note: For details on the full 3D model of the reactor monolith, see the example monolith_3d.mph in the Chemical Reaction Engineering Module’s model library. Model Library path: Chemical_Reaction_Engineering_Module/ Heterogeneous_Catalysis/monolith_kinetics Solved with COMSOL Multiphysics 4.4 7 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S Reference 1. G. Schaub, D. Unruh, J. Wang, and T. Turek, “Kinetic analysis of selective catalytic NOx reduction (SCR) in a catalytic filter,” Chemical Engineering and Processing, vol. 42, p. 365, 2003. Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Stationary Plug Flow. 6 Click the Done button. G L O B A L D E F I N I T I O N S Start by reading in a set of global parameters defining the process conditions for the monolith reactor, including the dimensions or the reactive channels, the flow rate of the reacting gas, and the temperature conditions. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file monolith_kinetics_parameters.txt. R E A C T I O N E N G I N E E R I N G Now define the chemical reactions. First, type in the reaction formula for NO reduction. The Reaction Engineering feature will automatically interpret the reaction formula and suggest a reaction rates based on the mass-action law. Solved with COMSOL Multiphysics 4.4 8 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type 4NO+4NH3+O2=>4N2+6H2O. 1: 4NO+4NH3+O2=>4N2+6H2O In this example, replace the automatically generated reaction rate expression with the rate expression known from the literature. 1 In the Model Builder window, under Component 1>Reaction Engineering click 1: 4NO+4NH3+O2=>4N2+6H2O. 2 In the Reaction settings window, locate the Reaction Rate section. 3 From the Reaction rate list, choose User Defined. 4 In the r edit field, type kf_1*c_NOm*a*c_NH3m/(1+a*c_NH3m). 5 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. Enter the Arrhenius parameters for the temperature dependent rate constant. 6 In the A f edit field, type 1e6. 7 In the E f edit field, type 60e3. Reaction 2 1 On the Physics toolbar, click Global and choose Reaction. In the same fashion define the reaction for NH3 oxidation. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type 4NH3+3O2=>2N2+6H2O. 4 Locate the Reaction Rate section. From the Reaction rate list, choose User Defined. 5 In the r edit field, type kf_2*c_NH3m. 6 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 7 In the A f edit field, type 6.8e7. 8 In the E f edit field, type 85e3. D E F I N I T I O N S Read in expressions defining the rate constant a and selectivity parameter S. Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. Solved with COMSOL Multiphysics 4.4 9 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file monolith_kinetics_variables.txt. The following steps allow you to study the initial reaction rates as functions of the temperature. First, use the independent variable t (time) to set up a temperature ramp between 500 K and 750 K. Next, lock the concentrations of reactant NO and NH3. R E A C T I O N E N G I N E E R I N G 1 In the Model Builder window, under Component 1 click Reaction Engineering. 2 In the Reaction Engineering settings window, locate the Reactor Settings section. 3 From the Reactor type list, choose Plug flow. 4 Locate the General section. In the T edit field, type 500+250*t. 5 Locate the Mass Balance section. In the v edit field, type v_av*A. Species: NO 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: NO. 2 In the Species settings window, locate the General Expressions section. 3 Select the Lock concentration/activity check box. 4 Click to expand the Species feed stream section. Locate the Species Feed Stream section. In the F 0 edit field, type F_NO_in. Species: NH3 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: NH3. 2 In the Species settings window, locate the General Expressions section. 3 Select the Lock concentration/activity check box. 4 Click to expand the Species feed stream section. Locate the Species Feed Stream section. In the F 0 edit field, type F_NH3_in. Species: O2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: O2. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type F_O2_in. Solved with COMSOL Multiphysics 4.4 10 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S Species: N2 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: N2. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type F_N2_in. Species: H2O 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: H2O. 2 In the Species settings window, click to expand the Species feed stream section. 3 Locate the Species Feed Stream section. In the F 0 edit field, type F_H2O_in. S T U D Y 1 Now add a Parametric Sweep feature to solve the model for a set of NH3:NO ratios. Parametric Sweep 1 On the Study toolbar, click Extension Steps and choose Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: 5 Click Range. 6 Go to the Range dialog box. 7 In the Start edit field, type 1. 8 In the Stop edit field, type 2. 9 In the Step edit field, type 0.2. 10 Click the Replace button. Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solver 1 node, then click Plug Flow Solver 1. 3 In the Plug Flow Solver settings window, click to expand the Absolute tolerance section. 4 Locate the Absolute Tolerance section. In the Tolerance edit field, type 1E-6. Parameter names Parameter value list X0 Solved with COMSOL Multiphysics 4.4 11 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S Step 1: Stationary Plug Flow 1 In the Model Builder window, under Study 1 click Step 1: Stationary Plug Flow. 2 In the Stationary Plug Flow settings window, locate the Study Settings section. 3 Select the Relative tolerance check box. 4 In the associated edit field, type 1e-5. 5 On the Home toolbar, click Compute. R E S U L T S Flow Rate (re) Create the plots shown in Figure 1 and Figure 2 by following the steps below. 1 In the 1D Plot Group settings window, click to expand the Legend section. 2 From the Position list, choose Upper left. 3 In the Model Builder window, expand the Flow Rate (re) node, then click Global 1. 4 In the Global settings window, locate the y-axis data section. 5 Click Reaction rate (comp1.re.r_1) in the upper-right corner of the section. Locate the x-Axis Data section. From the Parameter list, choose Expression. 6 In the Expression edit field, type comp1.re.T. 7 Click to expand the Legends section. Find the Include subsection. Clear the Expression check box. 8 On the 1D plot group toolbar, click Plot. 9 In the Model Builder window, click Flow Rate (re). 10 In the 1D Plot Group settings window, locate the Legend section. 11 From the Position list, choose Upper right. 12 In the Model Builder window, under Results>Flow Rate (re) click Global 1. 13 In the Global settings window, locate the y-axis data section. 14 Click Reaction rate (comp1.re.r_2) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. 15 Click Selectivity parameter (comp1.S) in the upper-right corner of the y-axis data section. On the 1D plot group toolbar, click Plot. It is now time to model a nonisothermal monolith channel. To do this, read in thermodynamic property data for the reacting species and set up an Energy Balance feature. Solved with COMSOL Multiphysics 4.4 12 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S R E A C T I O N E N G I N E E R I N G Species: NO 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: NO. 2 In the Species settings window, locate the General Expressions section. 3 Clear the Lock concentration/activity check box. Species: NH3 1 In the Model Builder window, under Component 1>Reaction Engineering click Species: NH3. 2 In the Species settings window, locate the General Expressions section. 3 Clear the Lock concentration/activity check box. 4 In the Model Builder window, click Reaction Engineering. 5 In the Reaction Engineering settings window, locate the Reactor Settings section. 6 Select the Calculate thermodynamic properties check box. 7 Locate the General section. In the T edit field, type T_in. 8 Click to expand the CHEMKIN section. Click the Browse button. 9 Browse to the model’s Model Library folder and double-click the file monolith_kinetics_thermo.txt. 10 Click the Import button. Energy Balance 1 Right-click Reaction Engineering and choose Energy Balance. 2 In the Energy Balance settings window, locate the Energy Balance section. 3 In the Q ext edit field, type (T_amb-T)*UA. 4 In the T 0 edit field, type T_in. S T U D Y 1 Step 1: Stationary Plug Flow 1 In the Model Builder window, under Study 1 click Step 1: Stationary Plug Flow. 2 In the Stationary Plug Flow settings window, locate the Study Settings section. 3 In the Volumes edit field, type 0 0.36*A. 4 On the Home toolbar, click Compute. Solved with COMSOL Multiphysics 4.4 13 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S R E S U L T S Flow Rate (re) Follow the instructions below to create Figure 3 and Figure 4. 1 In the Model Builder window, under Results>Flow Rate (re) click Global 1. 2 In the Global settings window, locate the y-axis data section. 3 Click Molar flow rate (comp1.re.F_NH3) in the upper-right corner of the section. Locate the x-Axis Data section. From the Parameter list, choose Volume. 4 On the 1D plot group toolbar, click Plot. 5 Locate the y-axis data section. Click Temperature (comp1.re.T) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. S T U D Y 1 Parametric Sweep 1 In the Model Builder window, under Study 1 click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 In the table, enter the following settings: 4 On the Home toolbar, click Compute. R E S U L T S Flow Rate (re) 1 In the Global settings window, locate the y-axis data section. 2 Click Selectivity parameter (comp1.S) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Parameter names Parameter value list X0 range(1.3,0.1,1.5) Solved with COMSOL Multiphysics 4.4 14 | A N A L Y S I S O F N O R E A C T I O N K I N E T I C S Solved with COMSOL Multiphysics 4.4 1 | M O N O L I T H R E A C T O R C H A N N E L Monol i t h Re a c t or C ha nne l This is a template MPH-file that you can use as the starting point to build a full 3D monolith reactor model including mass transfer and chemical reactions, heat transfer, and fluid flow. For a description of that model, including detailed step-by-step instructions showing how to build it, see the book Introduction to the Chemical Reaction Engineering Module. Model Library path: Chemical_Reaction_Engineering_Module/ Heterogeneous_Catalysis/monolith_plugflow Solved with COMSOL Multiphysics 4.4 2 | M O N O L I T H R E A C T O R C H A N N E L Solved with COMSOL Multiphysics 4.4 1 | N O N I S O T H E R M A L H I R E A C T O R Noni s ot he r ma l HI Re a c t or Introduction In the case of a perfectly mixed nonisothermal system, both the time-dependent material and energy balances must be set up. There are no spatial concentration gradients because the system is perfectly mixed, so the Reaction Engineering interface can be used to create a model without evaluating the material-transport properties. This exercise continues the previous example, modeling the formation of HI described by the reaction The energy balance for the Batch reactor is by default defined according to the equation In the above equation V r denotes the system volume (m 3 ), c i is the species concentration (mol/m 3 ), C p,i is the species molar heat capacity (J/(mol·K)), T is the temperature (K), and p the pressure (Pa). On the right hand side, w s represents the shaft work (J/s), Q is the heat due to chemical reaction (J/s), and Q ext denotes heat added to the system (J/s). The heat of reaction is given as: For this scenario, the modeling procedure in the Reaction Engineering interface is: 1 Start COMSOL Multiphysics. 2 In the Model Wizard, select 0D on the Select Space Dimension page and then add a Reaction Engineering interface on the Select Physics page. 3 Select the type of fluid (gas) and activate the energy balance. 4 Set a new reaction and type in the expression for the reaction just described. 5 Set the initial concentrations for H 2 , I 2 , and HI. 6 Set the thermodynamic properties of H 2 , I 2 , and HI. 7 Set the time interval for the reactor simulation. H 2 g ( ) I 2 g ( ) + 2HI g ( ) = V r c i C p i , dT dt -------- i ¿ w s Q Q + ext V r dp dt ------- + + = Q V r H j r j j ¿ – = Solved with COMSOL Multiphysics 4.4 2 | N O N I S O T H E R M A L H I R E A C T O R 8 Compute the solution. 9 Plot the solution. At this stage can be included the polynomials that describe the heat capacity, C p,i , for each species i as input data to the model. In addition, the enthalpy of formation and the entropy of formation at 0 K for each of the three species is needed. These are usually tabulated in NASA polynomial files for thermodynamic properties of pure gases. The following list summarizes the input data for the model: • Frequency factor for the forward reaction A f = 8.87·10 7 m 3 /(mol · s) • Frequency factor for the reverse reaction A r = 3.00·10 7 m 3 /(mol · s) • Activation energy for the forward reaction E f = 167·10 3 J/mol • Activation energy for the reverse reaction E r = 184·10 3 J/mol • Initial concentrations = 5.8 mol/m 3 • Initial system temperature = 700 K The polynomials for C p (J/(mol · K)) are arranged according to the expression and their coefficients take on the following values: Note that h 0,0K and s 0,0K are fictitious enthalpies and entropies calculated as The equation uses the C p polynomials in the integral expression despite the fact that these might not be valid all the way down to 0 K. This inclusion does not introduce an error, because contributions outside the interval are cancelled out in every integration in the nonisothermal system (when going from T 1 to T 2 ) as long as the system temperature is within the correct interval. To calculate the entropy, follow a procedure C p R g ------- a 1 a 2 T a 3 T 2 a 4 T 3 a 5 T 4 + + + + = TABLE 1: COEFFICIENT VALUES FOR VARYING TEMPERATURE a 1 a 2 a 3 a 4 a 5 h 0,0K /R g s 0,0K /R g H 2 2.883 3.681·10 -3 -7.720·10 -6 6.920·10 -9 -2.130·10 -12 -9.671·10 2 -1.034 I 2 3.508 6.303·10 -3 -1.461·10 -5 1.470·10 -8 -5.310·10 -12 6.287·10 3 1.002·10 1 HI 3.648 -1.392·10 -3 3.890·10 -6 -3.260·10 -9 1.100·10 -12 2.131·10 3 4.334 h 0 0K , h 0 298K , C p T d 298 0 } + = Solved with COMSOL Multiphysics 4.4 3 | N O N I S O T H E R M A L H I R E A C T O R for s 0,0K analogous to the one described earlier for h 0,0K . This is also a standard way of calculating s 0,0K and h 0,0K in NASA polynomials for thermodynamic data. Results and Discussion The Figure 1 shows the concentrations of the reactants and products as functions of time. The system reaches steady state after approximately 2000 s, which is considerably faster compared to the isothermal case discussed previously. Figure 1: Reactant and product concentrations (mol/m 3 ) as functions of time (s). Figure 2 Shows the reactor temperature increasing from the initial temperature of 700 K to a final temperature of 809 K. Solved with COMSOL Multiphysics 4.4 4 | N O N I S O T H E R M A L H I R E A C T O R Figure 2: Reactor temperature (K) as function of time (s). Solved with COMSOL Multiphysics 4.4 5 | N O N I S O T H E R M A L H I R E A C T O R Below, Figure 3 shows the relationship between the forward and reverse reaction rates as well as the equilibrium expression throughout the experiment. At steady state the expressions coincide. Figure 3: The relationship between the forward and reverse reaction rates as well as the equilibrium expression as function of time (s). Model Library path: Chemical_Reaction_Engineering_Module/ Batch_Reactors/nonisothermal_hi_reactor Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 0D button. Solved with COMSOL Multiphysics 4.4 6 | N O N I S O T H E R M A L H I R E A C T O R 2 In the Select physics tree, select Chemical Species Transport>Reaction Engineering (re). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Time Dependent. 6 Click the Done button. R E A C T I O N E N G I N E E R I N G ( R E ) 1 In the Model Builder window, under Component 1 (comp1) click Reaction Engineering (re). 2 In the Reaction Engineering settings window, locate the Reactor Settings section. 3 Select the Calculate thermodynamic properties check box. Reaction 1 1 On the Physics toolbar, click Global and choose Reaction. 2 In the Reaction settings window, locate the Reaction Formula section. 3 In the Formula edit field, type H2+I2<=>2HI. 4 Locate the Rate Constants section. Select the Use Arrhenius expressions check box. 5 In the A f edit field, type 8.87e7. 6 In the E f edit field, type 167e3. 7 In the A r edit field, type 3e7. 8 In the E r edit field, type 184e3. Species: H2 1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Species: H2. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 5.8. 4 Click to expand the Species thermodynamic parameters section. Locate the Species Thermodynamic Parameters section. In the a low,k table, enter the following settings: 2.883 3.681e-3 -7.720e-6 6.920e-9 -2.130e-12 Solved with COMSOL Multiphysics 4.4 7 | N O N I S O T H E R M A L H I R E A C T O R Species: I2 1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Species: I2. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 5.8. 4 Click to expand the Species thermodynamic parameters section. Locate the Species Thermodynamic Parameters section. In the a low,k table, enter the following settings: Species: HI 1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Species: HI. 2 In the Species settings window, locate the General Expressions section. 3 In the c 0 edit field, type 5.8. 4 Click to expand the Species thermodynamic parameters section. Locate the Species Thermodynamic Parameters section. In the a low,k table, enter the following settings: Energy Balance 1 In the Model Builder window, right-click Reaction Engineering (re) and choose Energy Balance. -9.671e2 -1.034 3.508 6.303e-3 -1.461e-5 1.470e-8 -5.310e-12 6.287e3 1.002e1 3.648 -1.392e-3 3.890e-6 -3.260e-9 1.100e-12 2.131e3 4.334 Solved with COMSOL Multiphysics 4.4 8 | N O N I S O T H E R M A L H I R E A C T O R 2 In the Energy Balance settings window, locate the Energy Balance section. 3 In the T 0 edit field, type 700. S T U D Y 1 Step 1: Time Dependent 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type 6.3e3. 4 On the Home toolbar, click Compute. R E S U L T S 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Global. 3 In the Global settings window, locate the y-Axis Data section. 4 In the table, enter the following settings: 5 On the 1D plot group toolbar, click Plot. Expression Unit Description comp1.re.kf_1/ comp1.re.kr_1 1 comp1.re.c_HI^2/ (comp1.re.c_I2*comp1.re. c_H2) 1 (comp1.re.T-700[K])/10 K Solved with COMSOL Multiphysics 4.4 1 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R Opt i ma l C ool i ng of a T ubul a r Re a c t or Introduction Maximizing product yield is a main task in chemical reaction engineering. This can be especially challenging if the desired product, once formed, can be consumed by further reactions. This example investigates such a series reaction as it occurs in a tubular reactor. You will start by setting up the tightly coupled mass and energy balance equations describing the reactor applying predefined physics interfaces of the Chemical Reaction Engineering Module. In a second step, you add an Optimization Study node to calculate the temperature conditions the reactor that maximize the production of the intermediary product. Note: This model requires the Optimization Module. Model Definition Two consecutive reactions take place in a tubular reactor. A heat exchanger jacket, run in co-current mode, is used to control the reaction rates and hence the product distribution in the reactor. Figure 1: Reactions occur in a tubular reactor equipped with a heat exchanger jacket, run in co-current mode. Temperature control in the reactor involves a delicate balance, where on the one hand, energy has to be supplied to the system to achieve acceptable reaction rates. On the Cooling stream Reacting stream Solved with COMSOL Multiphysics 4.4 2 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R other hand, the energy transfer to the reacting stream must be limited so that the desired intermediate product is not consumed by further reaction. The situation is further complicated by the fact that the temperature of the reacting stream is not only affected by the heat transfer from the heat exchanger jacket, but also by the endothermic nature of the reactions. The idea for this challenge in reactor optimization is taken from a literature example (Ref. 1), although the present reactor model is considerably more detailed. The model is set up in 1D, coupling mass and energy balances in the reactor tube with an energy balance for the heat exchanger jacket. Streams in both the tube and jacket are treated as plug flows. C H E MI S T R Y Two consecutive reaction occur in water, where the desired product is species B: (1) (2) The following rate equations apply: (3) (4) where the rate constants are temperature dependent according to the Arrhenius relation: (5) The kinetic parameters are summarized in the table below: MA S S T R A N S P O R T The mass transport is modeled by the convection-diffusion equation at steady-state: J AJ [1/S] EJ [J/MOL] 1 1.6e8 75e3 2 1e15 125e3 A B k 1 B C k 2 r 1 k 1 c A = r 2 k 2 c B = k j A j E j R g T ----------- – \ . | | exp = Solved with COMSOL Multiphysics 4.4 3 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R (6) In this equation, c i denotes the concentration (mol/m 3 ) and D i is the diffusivity (m 2 /s). R i is the rate expression for species i (mol/(m 3 ·s)). The velocity u (m/s) of the fluid in the reactor is represented by a constant profile: (7) At the inlet, the concentration of the reactant A is 700 mol/m 3 . At the outlet, it is assumed that convective mass transport is dominant: (8) E N E R G Y T R A N S P O R T - R E A C T O R The energy transport in the reactor is described by: (9) Above, k is the thermal conductivity (W/(m·K)) and T the temperature of the reacting stream (K). µ is the density (kg/m 3 ) and C p the heat capacity (J/(kg·K)). The reacting species are diluted in water, and hence, the physical properties of the reacting mixture are assumed to be those of water. The heat source due to reaction, Q rxn (W/m 3 ), is calculated from the reaction rates and the enthalpies of reaction: (10) Both of the reactions are endothermic, with AH 1 = 200 kJ/mol and AH 2 = 100 kJ/ mol. Furthermore, the heat transferred from the reactor to the cooling jacket is given by: (11) Here, U is the overall heat transfer coefficient (J/(K·m 2 ·s)), and A represents the heat exchange area per unit volume (m 2 /m 3 ). The temperature of the reacting fluid at the inlet is 400 K. At the outlet, it is assumed that convective heat transport is dominant: (12) V D i V – c i ( ) · u Vc i · + R i = u 0.0042 m/s = V D i V – c i ( ) · 0 = V k – VT ( ) · µC p u VT · + Q rxn Q j + = Q rxn AH j r j – j 1 2 – = ¿ = Q j UA T T j – ( ) – = V k – VT ( ) · 0 = Solved with COMSOL Multiphysics 4.4 4 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R E N E R G Y T R A N S P O R T - C O O L I N G J A C K E T Water serves as the cooling medium in the jacket, and the energy transport is given by (13) The cooling stream is assumed to have plug flow character, and hence a constant velocity profile: (14) The optimal temperature of the cooling fluid at the inlet is to be found such that the maximum concentration of species B is achieved at the outlet. Results and Discussion In a first simulation, the inlet temperatures of the jacket stream and the reacting stream are set to be equal, at 400 K. In a second simulation, an optimization calculation is performed to find the inlet temperature of the jacket stream that maximizes the concentration of the desired intermediary product (B) at the reactor outlet. Comparisons between the two cases follow below. Figure 2 shows the concentration of reacting species as a function of the reactor length when the inlet temperature of the jacket stream is 400 K. Figure 3 shows concentration curves for the optimal inlet temperature of the jacket stream, found to be 336 K. Clearly, when the inlet temperature is 400 K the conversion of reactant A is high, but at the same time, the selectivity for the desired product B is unfavorable. Under the optimized conditions, the concentration of B at the reactor outlet is 355 mol/m 3 , to be compared to a concentration of 159 mol/m 3 when the inlet temperature is 400 K. V k – VT j ( ) · µC p u j VT · + Q j – = u j 0.001 m/s = Solved with COMSOL Multiphysics 4.4 5 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R Figure 2: Species concentrations (blue c_A, green c_B, red c_C) as function of reactor position when the inlet temperature of the cooling fluid is 400 K. Solved with COMSOL Multiphysics 4.4 6 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R Figure 3: Species concentrations (blue c_A, green c_B, red c_C) as function of reactor position when the inlet temperature of the cooling fluid is 336 K. Plots of reacting stream and jacket stream temperatures are shown in Figure 4 and Figure 5. The jacket stream heats up the reacting stream when its inlet temperature is kept at 400 K. Figure 4: Temperature distribution for the reacting stream (blue) and jacket stream (green) when the inlet temperature of the jacket stream is 400 K. In contrast, the jacket stream cools the reacting stream when its inlet temperature is 336 K. Solved with COMSOL Multiphysics 4.4 7 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R . Figure 5: Temperature distribution for the reacting stream (blue) and jacket stream (green) when the inlet temperature of the jacket stream is 336 K. The reaction rates are illustrated in Figure 6 and Figure 7. When the inlet temperature of the jacket stream is 400 K, the rate at which B is consumed (r 2 ) dominates over the production rate (r 1 ) from a point approximately 0.65 m down the reactor. This effect Solved with COMSOL Multiphysics 4.4 8 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R is due to heat being transferred from the jacket stream, counteracting the cooling effect of the endothermal reactions. Figure 6: Rate of the production r 1 (blue) and rate consumption r 2 (green) of species B when the inlet temperature of the cooling fluid is 400 K. Solved with COMSOL Multiphysics 4.4 9 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R At an inlet temperature of 336 K, the combined effect of cooling by the jacket stream and energy consumption due to reaction work together to quench the system, resulting in increased concentrations levels of B at the outlet. Figure 7: Rate of the production r 1 (blue) and rate consumption r 2 (green) of species B when the inlet temperature of the cooling fluid is 336 K. Reference 1. T.F. Edgar and D.M. Himmelblau, Optimization of Chemical Processes, McGraw-Hill, 1988. Model Library path: Chemical_Reaction_Engineering_Module/ Optimization/optimal_cooling Modeling Instructions From the File menu, choose New. Solved with COMSOL Multiphysics 4.4 10 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 1D button. 2 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 3 Click the Add button. 4 In the Number of species edit field, type 3. 5 In the Concentrations table, enter the following settings: 6 In the Select physics tree, select Heat Transfer>Heat Transfer in Fluids (ht). 7 Click the Add button. 8 In the Select physics tree, select Heat Transfer>Heat Transfer in Fluids (ht). 9 Click the Add button. 10 In the Temperature edit field, type Tj. 11 Click the Study button. 12 In the tree, select Preset Studies for Selected Physics>Stationary. 13 Click the Done button. G E O ME T R Y 1 Interval 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Interval. 2 In the Interval settings window, locate the Interval section. 3 In the Right endpoint edit field, type 2. 4 Click the Build Selected button. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. c_A c_B c_C Solved with COMSOL Multiphysics 4.4 11 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file optimal_cooling_parameters.txt. Variables 1 1 In the Model Builder window, right-click Global Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file optimal_cooling_variables.txt. D E F I N I T I O N S Integration 1 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 2 only. Variables 2 1 In the Model Builder window, right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: MA T E R I A L S On the Home toolbar, click Add Material. A D D MA T E R I A L 1 Go to the Add Material window. 2 In the tree, select Liquids and Gases>Liquids>Water. 3 In the Add material window, click Add to Component. Name Expression Unit Description c_B_out intop1(c_B) mol/m³ Outlet concentration Solved with COMSOL Multiphysics 4.4 12 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R TR A N S P O R T O F D I L U T E D S P E C I E S Convection and Diffusion 1 1 In the Model Builder window, expand the Component 1>Transport of Diluted Species node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 Specify the u vector as 4 Locate the Diffusion section. In the D cA edit field, type D. 5 In the D cB edit field, type D. 6 In the D cC edit field, type D. Inflow 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 Select Boundary 1 only. 3 In the Inflow settings window, locate the Concentration section. 4 In the c 0,cA edit field, type 700. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 2 only. Reactions 1 1 On the Physics toolbar, click Domains and choose Reactions. 2 Select Domain 1 only. 3 In the Reactions settings window, locate the Reactions section. 4 In the R cA edit field, type -r1. 5 In the R cB edit field, type r1-r2. 6 In the R cC edit field, type r2. H E A T TR A N S F E R I N F L U I D S Heat Transfer in Fluids 1 1 In the Model Builder window, expand the Component 1>Heat Transfer in Fluids node, then click Heat Transfer in Fluids 1. 2 In the Heat Transfer in Fluids settings window, locate the Model Inputs section. u x Solved with COMSOL Multiphysics 4.4 13 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R 3 Specify the u vector as Temperature 1 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 Select Boundary 1 only. 3 In the Temperature settings window, locate the Temperature section. 4 In the T 0 edit field, type T_in. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 2 only. Heat Source 1 1 On the Physics toolbar, click Domains and choose Heat Source. 2 Select Domain 1 only. 3 In the Heat Source settings window, locate the Heat Source section. 4 In the Q edit field, type -UA*(T-Tj)+Q_rxn. H E A T TR A N S F E R I N F L U I D S 2 Heat Transfer in Fluids 1 1 In the Model Builder window, expand the Component 1>Heat Transfer in Fluids 2 node, then click Heat Transfer in Fluids 1. 2 In the Heat Transfer in Fluids settings window, locate the Model Inputs section. 3 Specify the u vector as Temperature 1 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 Select Boundary 1 only. 3 In the Temperature settings window, locate the Temperature section. 4 In the T 0 edit field, type Tj_in. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. u x uj x Solved with COMSOL Multiphysics 4.4 14 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R 2 Select Boundary 2 only. Heat Source 1 1 On the Physics toolbar, click Domains and choose Heat Source. 2 Select Domain 1 only. 3 In the Heat Source settings window, locate the Heat Source section. 4 In the Q edit field, type UA*(T-Tj). ME S H 1 Size 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Edge. 2 In the Size settings window, locate the Element Size section. 3 From the Predefined list, choose Extra fine. 4 Click the Build All button. 5 Click the Zoom Extents button on the Graphics toolbar. The mesh should look as below. Solved with COMSOL Multiphysics 4.4 15 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R S T U D Y 1 On the Home toolbar, click Compute. Go through the steps below to save a copy of the solution where the coolant temperature is 400 K at the inlet. Copy 2 1 In the Model Builder window, under Study 1>Solver Configurations right-click Solver 1 and choose Solution>Copy. 2 Right-click Copy 2 and choose Rename. 3 Go to the Rename Solver dialog box and type Tj_in 400K in the New name edit field. 4 Click OK. R E S U L T S Concentration (chds) 1 In the Model Builder window, expand the Results>Concentration (chds) node. 2 Right-click Line Graph 1 and choose Duplicate. 3 In the Line Graph settings window, locate the y-axis data section. 4 Click Concentration (c_B) in the upper-right corner of the section. Click to expand the Title section. From the Title type list, choose None. 5 Right-click Results>Concentration (chds)>Line Graph 2 and choose Duplicate. 6 In the Line Graph settings window, locate the y-axis data section. 7 Click Concentration (c_C) in the upper-right corner of the section. On the 1D plot group toolbar, click Plot. Temperature (ht) 1 In the Model Builder window, under Results>Temperature (ht) right-click Line Graph 1 and choose Duplicate. 2 In the Line Graph settings window, locate the y-axis data section. 3 Click Temperature (Tj) in the upper-right corner of the section. Locate the Title section. From the Title type list, choose None. 4 On the 1D plot group toolbar, click Plot. Temperature (ht2) 1 In the Model Builder window, expand the Results>Temperature (ht2) node, then click Line Graph 1. 2 In the Line Graph settings window, locate the y-axis data section. Solved with COMSOL Multiphysics 4.4 16 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R 3 Click Reaction rate (r1) in the upper-right corner of the section. Right-click Results>Temperature (ht2)>Line Graph 1 and choose Duplicate. 4 In the Line Graph settings window, locate the y-axis data section. 5 Click Reaction rate (r2) in the upper-right corner of the section. Locate the Title section. From the Title type list, choose None. 6 On the 1D plot group toolbar, click Plot. S T U D Y 1 Now, solve the optimization problem. Optimization 1 On the Study toolbar, click Optimization. 2 In the Optimization settings window, locate the Optimization Solver section. 3 From the Method list, choose BOBYQA. 4 In the table, enter the following settings: 5 From the Type list, choose Maximization. 6 Locate the Control Variables and Parameters section. Click Add. 7 In the table, enter the following settings: 8 On the Home toolbar, click Compute. R E S U L T S The plots you set up earlier now show the optimized solution. Temperature (ht2) Scroll down the table to find the resulting values of the inlet temperature. Expression Description comp1.c_B_out Outlet concentration Parameter names Initial value Lower bound Upper bound Tj_in 400[K] Solved with COMSOL Multiphysics 4.4 1 | P A C K E D B E D R E A C T O R Pa c ke d Be d Re a c t or Introduction The packed bed reactor is one of the most common reactors in the chemical industry, for use in heterogeneous catalytic processes. In essence, the reactor consists of a container filled with catalyst particles. These particles can be contained within a supporting structure, like tubes or channels, or they can be packed in one single compartment in the reactor. The structure that is formatted by the packed catalyst particles makes the modeling of mass and energy transport in the reactor a challenging task. The difficulty lies in the description of the porous structure, which gives transport of different orders of magnitudes within the particles and between the particles. In most cases, the structure in between particles is described as macroporous and the particle radius can be of the order of magnitude of 1 mm. When a pressure difference is applied across the bed, convection arises in the macropores. The pores inside the catalyst particles form the microstructure of the bed. The pore radii in the particles is often between one and ten micrometer. This model presents a simple and fast route to studying microscale and macroscale mass balances in packed beds and other heterogeneous reactors with bimodal pore distribution. Using a multigeometry approach, the model provides the mass and reaction distributions along the reactor and within each catalyst pellet along the reactor length. This makes it possible to evaluate the utilization of catalyst load, optimal pellet size, or inlet temperature. In this example, volatile organic compounds (VOC) and CO are oxidized in a catalytic converter. Propylene is used as representative for hydrocarbons present in the feed stream, which could be, for example, exhaust gas from a combustion process. Model Definition The following chemical reactions describe the catalytic conversion carbon monoxide and by-product hydrocarbon: (1) CO 1 2 ---O 2 + CO 2 ÷ Solved with COMSOL Multiphysics 4.4 2 | P A C K E D B E D R E A C T O R (2) For these heterogeneous catalytic reactions the rates (mol/(m 3 ·s)) are given by (Ref. 1): (3) (4) The rate and adsorption constants are given by Arrhenius expressions: (5) The values of the frequency factors and activation energies (J/mol) are taken from the literature (Ref. 2) and presented in Table 1, The chemical reactions given in Equation 1 and Equation 2 describe a process that is limited by the second-order rate expression at high temperatures when the surface coverage is low. At low temperatures the adsorption reactions will be slower and limit the reaction rate due to a shortage of free catalytic sites. The pressure drop in the reactor is described by the Ergun equation (6) where P is the pressure (kPa), u the porosity, D p the particle diameter (m), q denotes the gas viscosity (Pa·s)), µ the gas density (kg/m 3 ), and x the reactor length (m). u is the reactor flow velocity (m/s) that depends on the pressure drop according to TABLE 1: ARRHENIUS PARAMETERS RATE/ADSORPTION CONSTANT FREQUENCY FACTOR (JOULE/MOL) ACTIVATION ENERGY (JOULE/MOL) k 1 7.07·10 13 1.09·10 5 k 2 1.47·10 15 1.26·10 5 K CO 8.1 -3400 K C3H6 257.9 1588 C 3 H 6 9 2 ---O 2 + 3H 2 O 3CO 2 + ÷ r 1 k 1 c CO c O2 1 K CO c + CO K C3H6 c C3H6 + ( ) 2 ------------------------------------------------------------------------------------- = r 2 k 2 c C3H6 c O2 1 K CO c + CO K C3H6 c C3H6 + ( ) 2 ------------------------------------------------------------------------------------- = k A E R g T ----------- – \ . | | exp = dP dx -------- µu µg c D p ----------------- – \ . | | 1 u – ( ) u 3 ------------------ 150 1 u – ( )q D p --------------------------------- 1,75µu + = Solved with COMSOL Multiphysics 4.4 3 | P A C K E D B E D R E A C T O R (7) where u feed is the inlet velocity and c the total concentration (mol/m 3 ). The mass transport in the reactor is given by the convection and diffusion equation (8) where D is the diffusion coefficient (m 2 /s) and R is a source term (mol/(m 3 ·s)). Equation 8 needs to be solved for all species participating in Equation 1 and Equation 2. Each equation also needs proper boundary conditions. At the inlet the reactant concentration is known: (9) (10) (11) (12) (13) At the outlet, the Outflow condition states that convective mass transport will dominate the species transport across the boundary: (14) At first glance, these equations appear straight forward, especially if the velocity vector is given by an analytical expression, which is the case for plug flow in a packed bed reactor. However, the source term, R, depends on the transport inside the catalyst particles. The molar flux at the outer surface of the particles multiplied by the available outer surface area of the particles per unit volume gives the proper source term: (15) In this equation, N denotes the flux vector inside the porous particle (mol/(m 2 ·s)) and n is the outward unit vector normal to the particle surface. Equation 15 is only valid at the particle surface, where the independent radius variable r (introduced below) u u feed c c feed ----------- P feed P ------------ · · = V DV – c ( ) · R u c V · – = c CO c COin = c C3H6 c C3H6in = c O2 c O2in = c CO2 0 = c H2O 0 = n DV – c ( ) · 0 = R A p N n · ( ) = at r r p = Solved with COMSOL Multiphysics 4.4 4 | P A C K E D B E D R E A C T O R equals the particle radius, r p . Furthermore, A p denotes the pellet surface to volume ratio (m 2 /m 3 ). This property is related to the pellet radius as (16) To properly calculate R, a mass balance is required for the catalyst pellet interior. Such a mass balance can be expressed as (17) Here, D pc is the effective diffusion coefficient in the particle, c p is the species concentration in the particle, and R p is the reaction rate for the heterogeneous reaction in the particle. In the catalyst pores, transport takes palace by diffusion only. Equation 17 gives mass balance for each species for the catalyst pellet. Since the pellets are spherical, it is convenient to express these mass balances in spherical coordinates. Equation 17 then becomes (18) where r (m) is the independent variable for the position along the radius of the particle. The equation is scaled with the particle radius to get a geometry with a lower aspect ratio. This gives the final formulation (19) The diffusion-reaction equation, combined with the boundary conditions for the particle, give the concentration distribution in the particle. The boundary conditions are (20) and (21) A p r p 3 ----- = V D pc Vc p – ( ) · R p + 0 = for x 2 y 2 + r p 2 < V r 2 D pc Vc p – ( ) · r 2 R p + 0 = for 0 r r p < < V r r p ----- \ . | | 2 D cp Vc p – \ . | | · r r p ----- \ . | | 2 R p + 0 = for 0 r 1 < < D pc Vc p – n · 0 = at r 0 = c p cc = at r 1 = Solved with COMSOL Multiphysics 4.4 5 | P A C K E D B E D R E A C T O R where c denotes the porosity of the particle, and c and c p represent the species concentrations in the bulk and in the particle, respectively. The concentration at the surface of the particle is equal to the concentration outside the particle compensated to account for the part of the particle volume that is occupied by solid catalyst support. The concentration distribution in the particle gives the molar flux at every point in the particles. This implies that the source term for the catalyst bed is given by the solution of the microscale mass balance: (22) A complication in solving the derived system of equations is that the macroscale and microscale balances are defined in different coordinate systems. This problem is general for many chemical reaction engineering applications and is often solved by using an analytical approximation of the solution to the microscale balance. However, such an approach cannot be used for complicated reaction mechanisms involving several reaction species. The approach exemplified here is general and can be used for very complex reaction mechanisms involving a large number of species. R A p D pc V – c p n · ( ) = Solved with COMSOL Multiphysics 4.4 6 | P A C K E D B E D R E A C T O R Results and Discussion Figure 1 shows the concentration of reacting species as function of position in the reactor. The levels of carbon monoxide and propylene are significantly reduced. Figure 1: The concentration of reactants and products along the reactor length. The pellet radius r p is 2.5 mm. The concentration can also be evaluated within each pellet, revealing whether or not the catalyst comes to efficient use. Figure 2 shows an example of this. Solved with COMSOL Multiphysics 4.4 7 | P A C K E D B E D R E A C T O R Figure 2: The concentration of in-pellet CO as function of reactor position. The scaled pellet radius is given on the y-axis and the reactor position along the x-axis. From Figure 2 it is clear that the CO concentration is low at the center of the pellet at all reactor positions. This means that reactor performance is limited by diffusion within the pellets and that active catalyst material in the center of the pellet will never come to use. Reducing the pellet diameter could potentially remedy this situation, and simulation results shown in Figure 3 confirm this notion. Scaled reactor length Scaled pellet radius Solved with COMSOL Multiphysics 4.4 8 | P A C K E D B E D R E A C T O R Figure 3: The concentration of in-pellet C 3 H 6 at three different reactor positions (5, 15, and 25 cm). Solid lines represents a pellet radius of 2.5 mm and lines with triangular markers represents a pellet radius of 1.8 mm. With a pellet radius of 2.5 mm, the reactor is limited by the in-pellet diffusion, as is indicated by the slope of the solid lines. The effect of this is that the active sites in the interior of the pellets are not used to their full potential. If the pellet size is reduced (represented by the lines with a triangular marker), the lines level out faster, suggesting larger reaction limited regions. The effect on C 3 H 6 conversion in the simulated case is an improvement from 94 % to 97 %. References 1. S.H. Oh, J.C. Cavendish, and L.L. Hegedus. “Mathematical modeling of catalytic converter lightoff: Single-pellet studies,” AIChE Journal, vol. 26, no. 6, pp. 935– 943, 1980. 2. J.B. Rawlings and J.G. Ekerdt, Chemical Reactor Analysis and Design Fundamentals, Nob Hill Publishing, Madison, 2002. Solved with COMSOL Multiphysics 4.4 9 | P A C K E D B E D R E A C T O R Model Library path: Chemical_Reaction_Engineering_Module/ Packed_Bed_Reactors/packed_bed_reactor Modeling Instructions Start by selecting physics user interfaces for the reactor mass transport and pressure equations. From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 1D button. 2 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 3 Click the Add button. 4 In the Number of species edit field, type 5. 5 In the Concentrations table, enter the following settings: 6 In the Select physics tree, select Mathematics>PDE Interfaces>Coefficient Form PDE (c). 7 Click the Add button. 8 In the Field name edit field, type P. 9 In the Dependent variables table, enter the following settings: 10 Click the Study button. 11 In the tree, select Preset Studies for Selected Physics>Stationary. C3H6 CO CO2 H2O O2 P Solved with COMSOL Multiphysics 4.4 10 | P A C K E D B E D R E A C T O R 12 Click the Done button. G E O ME T R Y 1 Interval 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Interval. 2 In the Interval settings window, locate the Interval section. 3 In the Right endpoint edit field, type 0.3. 4 Click the Build Selected button. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file packed_bed_reactor_parameters.txt. Variables 1 1 In the Model Builder window, right-click Global Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file packed_bed_reactor_variables_global.txt. TR A N S P O R T O F D I L U T E D S P E C I E S Follow these steps to set up mass transport equations for the packed bed. Convection and Diffusion 1 1 In the Model Builder window, expand the Component 1>Transport of Diluted Species node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 Specify the u vector as 4 Locate the Diffusion section. In the D C3H6 edit field, type D_C3H6. u x Solved with COMSOL Multiphysics 4.4 11 | P A C K E D B E D R E A C T O R 5 In the D CO edit field, type D_CO. 6 In the D CO2 edit field, type D_CO2. 7 In the D H2O edit field, type D_H2O. 8 In the D O2 edit field, type D_O2. Initial Values 1 1 In the Model Builder window, under Component 1>Transport of Diluted Species click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the C3H6 edit field, type 1e-6. 4 In the CO edit field, type 1e-6. 5 In the CO2 edit field, type 1e-6. 6 In the H2O edit field, type 1e-6. 7 In the O2 edit field, type 1e-6. Reactions 1 1 On the Physics toolbar, click Domains and choose Reactions. 2 Select Domain 1 only. 3 In the Reactions settings window, locate the Reactions section. 4 In the R C3H6 edit field, type -Ap*C3H6flux. 5 In the R CO edit field, type -Ap*COflux. 6 In the R CO2 edit field, type -Ap*CO2flux. 7 In the R H2O edit field, type -Ap*H2Oflux. 8 In the R O2 edit field, type -Ap*O2flux. You will define the flux variables later on, after having set up the pellet model. Inflow 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 Select Boundary 1 only. 3 In the Inflow settings window, locate the Concentration section. 4 In the c 0,C3H6 edit field, type x_C3H6_feed*Ctot_feed. 5 In the c 0,CO edit field, type x_CO_feed*Ctot_feed. 6 In the c 0,O2 edit field, type x_O2_feed*Ctot_feed. Solved with COMSOL Multiphysics 4.4 12 | P A C K E D B E D R E A C T O R Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 2 only. C O E F F I C I E N T F O R M P D E Follow the steps below to specify the Ergun equation for the pressure drop in the reactor. 1 In the Model Builder window, under Component 1 click Coefficient Form PDE. 2 In the Coefficient Form PDE settings window, locate the Units section. 3 Find the Source term quantity subsection. From the list, choose Pressure (Pa). 4 In the Unit edit field, type N/m^3. Coefficient Form PDE 1 1 In the Model Builder window, expand the Coefficient Form PDE node, then click Coefficient Form PDE 1. 2 In the Coefficient Form PDE settings window, locate the Diffusion Coefficient section. 3 In the c edit field, type 0. 4 Locate the Source Term section. In the f edit field, type Px+beta*(F/F0)*(P_feed/ P). 5 Locate the Damping or Mass Coefficient section. In the d a edit field, type 0. Initial Values 1 1 In the Model Builder window, under Component 1>Coefficient Form PDE click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the P edit field, type P_feed. Dirichlet Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Dirichlet Boundary Condition. 2 Select Boundary 1 only. 3 In the Dirichlet Boundary Condition settings window, locate the Dirichlet Boundary Condition section. 4 In the r edit field, type P_feed. Solved with COMSOL Multiphysics 4.4 13 | P A C K E D B E D R E A C T O R R O O T The source terms in the reactor mass balances depend on the surface fluxes from the catalyst pellets. Therefore, set up a separate model calculating the mass transport and reaction in the pellets. On the Home toolbar, click Add Component and choose 2D. C O MP O N E N T 2 On the Home toolbar, click Add Physics. A D D P HY S I C S 1 Go to the Add Physics window. 2 In the Add physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 3 Click to expand the Dependent variables section. Locate the Dependent Variables section. In the Number of species edit field, type 5. 4 In the Concentrations table, enter the following settings: 5 In the Add physics window, click Add to Component. G E O ME T R Y 2 Square 1 1 In the Model Builder window, under Component 2 right-click Geometry 2 and choose Square. 2 Right-click Square 1 and choose Build Selected. TR A N S P O R T O F D I L U T E D S P E C I E S 2 Convection and Diffusion 1 Use the full diffusivity matrix to specify that diffusion occurs only in the radial pellet direction, corresponding to the direction of the y-axis in the 2D geometry. C3H6p COp CO2p H2Op O2p Solved with COMSOL Multiphysics 4.4 14 | P A C K E D B E D R E A C T O R 1 In the Model Builder window, expand the Component 2>Transport of Diluted Species 2 node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Diffusion section. 3 From the symmetry property list, choose Anisotropic. 4 In the D C3H6p table, enter the following settings: 5 From the symmetry property list, choose Anisotropic. 6 In the D COp table, enter the following settings: 7 From the symmetry property list, choose Anisotropic. 8 In the D CO2p table, enter the following settings: 9 From the symmetry property list, choose Anisotropic. 10 In the D H2Op table, enter the following settings: 11 From the symmetry property list, choose Anisotropic. 12 In the D O2p table, enter the following settings: Initial Values 1 1 In the Model Builder window, under Component 2>Transport of Diluted Species 2 click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the C3H6p edit field, type 1e-6. 4 In the COp edit field, type 1e-6. 0 0 0 (D_C3H6/rp^2)*y^2 0 0 0 (D_CO/rp^2)*y^2 0 0 0 (D_CO2/rp^2)*y^2 0 0 0 (D_H2O/rp^2)*y^2 0 0 0 (D_O2/rp^2)*y^2 Solved with COMSOL Multiphysics 4.4 15 | P A C K E D B E D R E A C T O R 5 In the CO2p edit field, type 1e-6. 6 In the H2Op edit field, type 1e-6. 7 In the O2p edit field, type 1e-6. Reactions 1 1 On the Physics toolbar, click Domains and choose Reactions. 2 In the Reactions settings window, locate the Domain Selection section. 3 From the Selection list, choose All domains. 4 Locate the Reactions section. In the R C3H6p edit field, type -y^2*R2. 5 In the R COp edit field, type -y^2*R1. 6 In the R CO2p edit field, type y^2*(R1+3*R2). 7 In the R H2Op edit field, type y^2*3*R2. 8 In the R O2p edit field, type -y^2*(0.5*R1+4.5*R2). Concentration 1 1 On the Physics toolbar, click Boundaries and choose Concentration. Species concentrations calculated by the reactor model serve as boundary conditions for the pellet model. You define the variables in a later step. 2 Select Boundary 3 only. 3 In the Concentration settings window, locate the Concentration section. 4 Select the Species C3H6p check box. 5 In the c 0,C3H6p edit field, type C3H6bulk. 6 Select the Species COp check box. 7 In the c 0,COp edit field, type CObulk. 8 Select the Species CO2p check box. 9 In the c 0,CO2p edit field, type CO2bulk. 10 Select the Species H2Op check box. 11 In the c 0,H2Op edit field, type H2Obulk. 12 Select the Species O2p check box. 13 In the c 0,O2p edit field, type O2bulk. 14 In the Model Builder window’s toolbar, click the Show button and select Advanced Physics Options in the menu. 15 Click to expand the Constraint settings section. Locate the Constraint Settings section. From the Apply reaction terms on list, choose Individual dependent variables. Solved with COMSOL Multiphysics 4.4 16 | P A C K E D B E D R E A C T O R D E F I N I T I O N S Complete the model setup by coupling the reactor and pellet models. First use a general extrusion component coupling to make the reactor species concentrations available in the pellet model. General Extrusion 1 1 On the Definitions toolbar, click Component Couplings and choose General Extrusion. 2 In the General Extrusion settings window, locate the Source Selection section. 3 From the Selection list, choose All domains. 4 Locate the Destination Map section. In the x-expression edit field, type x*0.30. 5 Locate the Source section. Select the Use source map check box. General Extrusion 2 Next, set up another general extrusion coupling in the pellet model for use when calculating the species flux at the pellet boundary. 1 On the Definitions toolbar, click Component Couplings and choose General Extrusion. 2 In the General Extrusion settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 3 only. 5 Locate the Destination Map section. In the x-expression edit field, type x/0.30. 6 Clear the y-expression edit field. 7 Locate the Source section. Select the Use source map check box. 8 Clear the y-expression edit field. Variables 2 Read in the variable file that defines the species fluxes at the pellet boundary. 1 In the Model Builder window, right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file packed_bed_reactor_variables_1d.txt. Note that these variables use the extrusion coupling operator that you just defined. Variables 3 Now, read in the variable file that defines the bulk species concentrations. Solved with COMSOL Multiphysics 4.4 17 | P A C K E D B E D R E A C T O R 1 Right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 3 only. 5 Locate the Variables section. Click Load from File. 6 Browse to the model’s Model Library folder and double-click the file packed_bed_reactor_variables_2d.txt. As you can see, these variables use the extrusion coupling operator you defined in the reactor model. ME S H 1 Size 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Edge. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type 0.0025. 5 Click the Build All button. ME S H 2 Mapped 1 In the Model Builder window, under Component 2 right-click Mesh 2 and choose Mapped. Distribution 1 1 In the Model Builder window, under Component 2>Mesh 2 right-click Mapped 1 and choose Distribution. 2 Select Boundaries 1 and 4 only. 3 In the Distribution settings window, locate the Distribution section. 4 From the Distribution properties list, choose Predefined distribution type. 5 In the Number of elements edit field, type 30. 6 In the Element ratio edit field, type 0.2. Distribution 2 1 Right-click Mapped 1 and choose Distribution. 2 Select Boundary 3 only. Solved with COMSOL Multiphysics 4.4 18 | P A C K E D B E D R E A C T O R 3 In the Distribution settings window, locate the Distribution section. 4 In the Number of elements edit field, type 120. 5 Click the Build All button. S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S Concentration (chds) Fist adapt the default plot to show the species concentrations in the reactor as functions of the position. 1 In the 1D Plot Group settings window, click to expand the Title section. 2 From the Title type list, choose Manual. 3 In the Title text area, type Concentration along reactor length. 4 Locate the Plot Settings section. Select the x-axis label check box. 5 In the associated edit field, type Reactor length (m). 6 In the Model Builder window, expand the Concentration (chds) node, then click Line Graph 1. 7 In the Line Graph settings window, click to expand the Legends section. 8 Select the Show legends check box. 9 From the Legends list, choose Manual. 10 In the table, enter the following settings: 11 On the 1D plot group toolbar, click Plot. 12 Right-click Results>Concentration (chds)>Line Graph 1 and choose Duplicate. 13 In the Line Graph settings window, locate the y-Axis Data section. 14 In the Expression edit field, type CO. 15 Locate the Legends section. In the table, enter the following settings: Legends C3H6 Legends CO Solved with COMSOL Multiphysics 4.4 19 | P A C K E D B E D R E A C T O R 16 Right-click Results>Concentration (chds)>Line Graph 2 and choose Duplicate. 17 In the Line Graph settings window, locate the y-Axis Data section. 18 In the Expression edit field, type CO2. 19 Locate the Legends section. In the table, enter the following settings: 20 Right-click Results>Concentration (chds)>Line Graph 3 and choose Duplicate. 21 In the Line Graph settings window, locate the y-Axis Data section. 22 In the Expression edit field, type O2. 23 Locate the Legends section. In the table, enter the following settings: 24 On the 1D plot group toolbar, click Plot. Concentration (chds2) Now plot the concentration of CO in the pellet of the catalytic bed. 1 In the Model Builder window, expand the Results>Concentration (chds2) node, then click Surface 1. 2 In the Surface settings window, locate the Expression section. 3 In the Expression edit field, type COp. 4 Right-click Results>Concentration (chds2)>Surface 1 and choose Height Expression. S T U D Y 1 To investigate if better pellet utilization can be achieved, reduce the pellet size. This affects the packing of the catalyst and reduces the diffusive length within the pellets. Step 1: Stationary 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Stationary. 2 In the Stationary settings window, click to expand the Study extensions section. 3 Locate the Study Extensions section. Select the Auxiliary sweep check box. 4 Click Add. Legends CO2 Legends O2 Solved with COMSOL Multiphysics 4.4 20 | P A C K E D B E D R E A C T O R 5 In the table, enter the following settings: 6 On the Home toolbar, click Compute. R E S U L T S Data Sets Create plots of the propene concentration distribution in the pellets, evaluated at reactor positions 5 cm, 15 cm, and 25 cm. 1 On the Results toolbar, click Cut Line 2D. 2 In the Cut Line 2D settings window, locate the Line Data section. 3 In row Point 1, set x to 5/30. 4 In row Point 2, set x to 5/30. 5 In row Point 2, set y to 1. 6 Select the Additional parallel lines check box. 7 In the Distances edit field, type -10/30 -20/30. 8 Click the Plot button. 1D Plot Group 4 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Line Graph. 3 In the Line Graph settings window, locate the Data section. 4 From the Data set list, choose Cut Line 2D 1. 5 From the Parameter selection (rp) list, choose From list. 6 In the Parameter values (rp) list, choose 0.0025 and 0.0018. 7 Locate the y-Axis Data section. In the Expression edit field, type C3H6p. 8 Locate the Legends section. Select the Show legends check box. 9 From the Legends list, choose Manual. 10 In the table, enter the following settings: Auxiliary parameter Parameter value list rp 0.0025 0.0018 Legends 5 cm Solved with COMSOL Multiphysics 4.4 21 | P A C K E D B E D R E A C T O R 11 Right-click Results>1D Plot Group 4>Line Graph 1 and choose Duplicate. 12 In the Line Graph settings window, locate the Data section. 13 In the Parameter values (rp) list, select 0.0018. 14 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Marker list, choose Triangle. 15 In the Model Builder window, click 1D Plot Group 4. 16 In the 1D Plot Group settings window, click to expand the Legend section. 17 From the Position list, choose Upper left. 18 Locate the Title section. From the Title type list, choose Manual. 19 In the Title text area, type Concentration, C3H6p (mol/m³). 20 On the 1D plot group toolbar, click Plot. 15 cm 25 cm Legends Solved with COMSOL Multiphysics 4.4 22 | P A C K E D B E D R E A C T O R Solved with COMSOL Multiphysics 4.4 1 | P L A T E R E A C T O R Pl a t e Re a c t or Introduction Plate reactors running under continuous conditions have emerged as candidates to replace batch reactors, primarily in fine chemicals and pharmaceuticals production. One of the advantages of the plate reactor design is that it allows for efficient temperature control of the reacting fluid. For instance, this means that the heat released from strongly exothermic reactions can be readily dissipated and more concentrated reaction mixtures can be run through the system. Plate reactors show promise to provide more energy-efficient production in a smaller package. The model presented here shows you how to set up and solve the coupled flow, mass, and energy transport equations describing the reacting flow in a plate reactor. Model Definition A plate reactor is similar to a heat exchanger in design, where reactor plates and cooling/heating plates are stacked on top of one another. Figure 1 shows the winding Solved with COMSOL Multiphysics 4.4 2 | P L A T E R E A C T O R interior of a reactor plate treated in the present model. Reactants enter the system through two inlet streams. Two heat exchange zones affect the outer boundaries. Figure 1: 3D geometry of a reactor plate. Two inlet streams are indicated as are the two heat exchange zones. C H E MI S T R Y Two exothermic chemical reactions take place. The first reaction generates the desired product D. In the second reaction the desired product proceeds to react with B to generate the unwanted product U. The reaction rates (mol/(m 3 ·s)) are given by: where rate constants are temperature dependent according to the Arrhenius equation: inlet 1 heat exchange A + B inlet 2 B region 1 heat exchange region 2 + A B D k 1 + D B U k 2 r 1 kc A c B = r 2 kc D c B = Solved with COMSOL Multiphysics 4.4 3 | P L A T E R E A C T O R (1) Both reactions are exothermic, and the rate of energy expelled is given by: (2) The Arrhenius parameters and heat of reaction are given below: The higher activation energy of reaction 2 makes the reaction rate more temperature sensitive compared to reaction 1. As both reactions are exothermic there is a risk that elevated temperatures will make the second reaction dominant, producing the unwanted product U. From this point of view, it is important to dissipate the heat of the reaction in such a way that the temperature allows for reaction 1 to proceed at a reasonable rate while reaction 2 is inhibited. In the present model, the second half of the reactor exchanges heat with a cooling medium that is at lower temperature compared to the first half. MO ME N T U M TR A N S P O R T In this model the fluid flow is described by the Navier-Stokes equations at steady state: (3) Here, q denotes the dynamic viscosity (Ns/m 2 ), u the velocity (m/s), µ the density of the fluid (kg/m 3 ), p the pressure (Pa), and F is a body force term (N/m 3 ). Apart from the domain equations you also need to select proper boundary conditions. At the inlets you specify a velocity vector normal to the boundary: (4) At the outlet boundary you specify a pressure: (5) Finally, at the reactor walls, a no-slip boundary condition is applied: REACTION FREQUENCY FACTOR ACTIVATION ENERGY HEAT OF REACTION 1 1(m 3 /mol/s) 40·10 3 (J/mol) -1.1·10 5 (J/mol) 2 1000(m 3 /mol/s) 60·10 3 (J/mol) -1·10 6 (J/mol) k A E R g T ----------- – \ . | | | exp = Q j r j H j = µ u V · ( )u V p – I q Vu Vu ( ) T + ( ) 2q 3 ------- k dv – \ . | | V u · ( )I – + · F + = V µu ( ) · 0 = u n · u 0 = p p 0 = Solved with COMSOL Multiphysics 4.4 4 | P L A T E R E A C T O R (6) E N E R G Y TR A N S P O R T The energy balance equation applied to the reactor domain considers heat transfer through convection and conduction: (7) In Equation 7, C p denotes the specific heat capacity (J/(kg·K)), k is the thermal conductivity (W/(m·K)), and Q is a sink or source term (W/m 3 ). At the inlets you set a temperature boundary condition: (8) At the outlet you set a Outflow boundary condition. This assumes that all energy passing through this boundary does so by means of convective transport. Equivalently this means that the heat flux due to conduction across the boundary is zero (9) so that the resulting equation for the total heat flux becomes (10) Finally, set two Heat flux conditions on the reactor walls, describing the two different heat exchange zones: (11) MA S S TR A N S P O R T The mass transfer in the reactor domain is given by the convection and diffusion equation: (12) where D i denotes its diffusion coefficient (m 2 /s), and R i denotes the reaction term (mol/(m 3 ·s)). For the boundary conditions, specify the concentrations at the inlets: u 0 = µC p t c cT V k T) µC p u VT · + V – ( · + Q = T T 0 = q cond n · k T V – n · 0 = = q n · µC p Tu n · = k T V – n · h T x T – ( ) = t c cc i V D i c i c i u + V – ( ) · + R i = Solved with COMSOL Multiphysics 4.4 5 | P L A T E R E A C T O R (13) At the outlet, specify that the mass flow through the boundary is convection dominated. This assumes that any mass flux due to diffusion across this boundary is zero (14) so that (15) Finally, at the reactor walls, assume that no mass is transported across the boundaries, that is, an insulation boundary condition: (16) c i c i 0 , = n D i c i V – ( ) · 0 = N i n c i u n · = · N i n 0 = · Solved with COMSOL Multiphysics 4.4 6 | P L A T E R E A C T O R Results Figure 2 shows the streamlines of the fluid flow in the reactor plate. The color scale indicates the concentration of reactant A. Figure 2: Streamlines of the fluid flow with the concentration of reactant A indicated by the color scale. The isosurfaces for the concentration of reactant B are shown in Figure 3. The chemical reactions clearly consume the reactant along the entire reactor volume. The Solved with COMSOL Multiphysics 4.4 7 | P L A T E R E A C T O R injection stream at the second inlet port mixes with the main stream, in effect making the distribution of B non-uniform in the second part of the reactor. Figure 3: The concentration of reactant B (mol/m 3 ) across the reactor volume. Solved with COMSOL Multiphysics 4.4 8 | P L A T E R E A C T O R Figure 4 shows the temperature distribution, represented by horizontal and vertical cut planes. Figure 4: Temperature distribution in the reactor plate. Heat expelled by the reaction dominate the temperature distribution in the first half of the reactor. In the second half, the reactions are quenched by the increased cooling. Model Library path: Chemical_Reaction_Engineering_Module/ Homogeneous_Reactions_and_Catalysis/plate_reactor Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. Solved with COMSOL Multiphysics 4.4 9 | P L A T E R E A C T O R MO D E L WI Z A R D 1 In the Model Wizard window, click the 3D button. 2 In the Select physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf). 3 Click the Add button. 4 In the Select physics tree, select Heat Transfer>Heat Transfer in Fluids (ht). 5 Click the Add button. 6 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 7 Click the Add button. 8 In the Number of species edit field, type 4. 9 In the Concentrations table, enter the following settings: 10 Click the Study button. 11 In the tree, select Preset Studies for Selected Physics>Stationary. 12 Click the Done button. G E O ME T R Y 1 Import 1 1 On the Home toolbar, click Import. 2 In the Import settings window, locate the Import section. 3 Click the Browse button. 4 Browse to the model’s Model Library folder and double-click the file plate_reactor.mphbin. 5 Click the Import button. Form Union In the Model Builder window, under Component 1>Geometry 1 right-click Form Union and choose Build Selected. c_A c_B c_D c_U Solved with COMSOL Multiphysics 4.4 10 | P L A T E R E A C T O R D E F I N I T I O N S Explicit 1 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1>Definitions right-click Explicit 1 and choose Rename. 3 Go to the Rename Explicit dialog box and type Inlet 1 in the New name edit field. 4 Click OK. 5 In the Explicit settings window, locate the Input Entities section. 6 From the Geometric entity level list, choose Boundary. 7 Select Boundary 23 only. Explicit 2 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1>Definitions right-click Explicit 2 and choose Rename. 3 Go to the Rename Explicit dialog box and type Inlet 2 in the New name edit field. 4 Click OK. 5 In the Explicit settings window, locate the Input Entities section. 6 From the Geometric entity level list, choose Boundary. 7 Select Boundary 20 only. Explicit 3 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1>Definitions right-click Explicit 3 and choose Rename. 3 Go to the Rename Explicit dialog box and type Outlet in the New name edit field. 4 Click OK. 5 In the Explicit settings window, locate the Input Entities section. 6 From the Geometric entity level list, choose Boundary. 7 Select Boundary 2 only. Explicit 4 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1>Definitions right-click Explicit 4 and choose Rename. Solved with COMSOL Multiphysics 4.4 11 | P L A T E R E A C T O R 3 Go to the Rename Explicit dialog box and type Heat Exchanger 1 in the New name edit field. 4 Click OK. 5 In the Explicit settings window, locate the Input Entities section. 6 From the Geometric entity level list, choose Boundary. To make the selection below with the mouse in the graphics window: In the Graphics toolbar, click the button Go to ZX view once. The Z-axis indicator should point to the right. Then click the Select Box button and click and drag the mouse to select the right part, as shown in the figure below. 7 These boundaries should appear in the list: 8, 9, 11, 12, 14–18, 21, 22, 28, 30, 32– 34, 42–46, 48, 50, 52, 55, 56, 58, 60, 62, 66–68, 70, 72, 75, 76, 78, 80, 82, 86– 88, 90, 92, 95, 96, 98, 100, 102, 106–108, 110, 112, 115, 116, 118, 120, 122, 126–128, 130, 132, 135, 136, 138, 140, 142, 146–148, 150, 152, 155, 156, 158, 160, 162, 166–168, 170, 172, 175, 176, 178, 180, 182, 186–188, 190, 192, 195, 196, 198, 200, 202, 206–208, 210, 212, 215, 216, 218, 220, 222, 226–228, 230, 232, 244–247, 249, 250, 252, 255, 260–262, and 266–268 only. Explicit 5 1 On the Definitions toolbar, click Explicit. Solved with COMSOL Multiphysics 4.4 12 | P L A T E R E A C T O R 2 In the Model Builder window, under Component 1>Definitions right-click Explicit 5 and choose Rename. 3 Go to the Rename Explicit dialog box and type Heat Exchanger 2 in the New name edit field. 4 Click OK. 5 In the Explicit settings window, locate the Input Entities section. 6 From the Geometric entity level list, choose Boundary. To make the selection below with the mouse in the graphics window: In the Graphics toolbar, click the button Go to ZX view once. The Z-axis indicator should point to the right. Then click the Select Box button and click and drag the mouse to select the left part, as shown in the figure below. 7 These boundaries should be selected: 1, 3–7, 19, 24, 26, 27, 29, 36–40, 47, 49, 51, 53, 54, 57, 59, 61, 63–65, 69, 71, 73, 74, 77, 79, 81, 83–85, 89, 91, 93, 94, 97, 99, 101, 103–105, 109, 111, 113, 114, 117, 119, 121, 123–125, 129, 131, 133, 134, 137, 139, 141, 143–145, 149, 151, 153, 154, 157, 159, 161, 163–165, 169, 171, 173, 174, 177, 179, 181, 183–185, 189, 191, 193, 194, 197, 199, 201, 203– 205, 209, 211, 213, 214, 217, 219, 221, 223–225, 229, 231, 234–237, 239, 240, 242, 253, 257–259, and 263–265 only. Solved with COMSOL Multiphysics 4.4 13 | P L A T E R E A C T O R MA T E R I A L S Material 1 1 In the Model Builder window, under Component 1 right-click Materials and choose New Material. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: L A MI N A R F L OW Inlet 1 1 On the Physics toolbar, click Boundaries and choose Inlet. 2 In the Inlet settings window, locate the Boundary Selection section. 3 From the Selection list, choose Inlet 1. 4 Locate the Velocity section. In the U 0 edit field, type 2e-3. Inlet 2 1 On the Physics toolbar, click Boundaries and choose Inlet. 2 In the Inlet settings window, locate the Boundary Selection section. 3 From the Selection list, choose Inlet 2. 4 Locate the Velocity section. In the U 0 edit field, type 1e-3. Outlet 1 1 On the Physics toolbar, click Boundaries and choose Outlet. 2 In the Outlet settings window, locate the Boundary Selection section. 3 From the Selection list, choose Outlet. Activate normal flow to model that the channel continuous after the outlet. 4 Locate the Pressure Conditions section. Select the Normal flow check box. Property Name Value Unit Property group Density rho 1000[kg/m^3] kg/m³ Basic Dynamic viscosity mu 1e-3[Pa*s] Pa·s Basic Thermal conductivity k 0.65[W/m/K] W/(m·K) Basic Heat capacity at constant pressure Cp 4200[J/kg/K] J/(kg·K) Basic Ratio of specific heats gamma 1 1 Basic Solved with COMSOL Multiphysics 4.4 14 | P L A T E R E A C T O R H E A T TR A N S F E R I N F L U I D S Heat Transfer in Fluids 1 1 In the Model Builder window, expand the Component 1>Heat Transfer in Fluids node, then click Heat Transfer in Fluids 1. 2 In the Heat Transfer in Fluids settings window, locate the Model Inputs section. 3 From the u list, choose Velocity field (spf/fp1). Initial Values 1 1 In the Model Builder window, under Component 1>Heat Transfer in Fluids click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the T edit field, type T0. Heat Source 1 1 On the Physics toolbar, click Domains and choose Heat Source. 2 In the Heat Source settings window, locate the Heat Source section. 3 In the Q edit field, type Q_reac. 4 Locate the Domain Selection section. From the Selection list, choose All domains. Temperature 1 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 In the Temperature settings window, locate the Boundary Selection section. 3 From the Selection list, choose Inlet 1. 4 Locate the Temperature section. In the T 0 edit field, type T0. Temperature 2 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 In the Temperature settings window, locate the Boundary Selection section. 3 From the Selection list, choose Inlet 2. 4 Locate the Temperature section. In the T 0 edit field, type T0. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 In the Outflow settings window, locate the Boundary Selection section. 3 From the Selection list, choose Outlet. Solved with COMSOL Multiphysics 4.4 15 | P L A T E R E A C T O R Heat Flux 1 1 On the Physics toolbar, click Boundaries and choose Heat Flux. 2 In the Heat Flux settings window, locate the Boundary Selection section. 3 From the Selection list, choose Heat Exchanger 1. 4 Locate the Heat Flux section. In the q 0 edit field, type Q_exch1. Heat Flux 2 1 On the Physics toolbar, click Boundaries and choose Heat Flux. 2 In the Heat Flux settings window, locate the Boundary Selection section. 3 From the Selection list, choose Heat Exchanger 2. 4 Locate the Heat Flux section. In the q 0 edit field, type Q_exch2. TR A N S P O R T O F D I L U T E D S P E C I E S Convection and Diffusion 1 1 In the Model Builder window, expand the Component 1>Transport of Diluted Species node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Diffusion section. 3 In the D cA edit field, type D. 4 In the D cB edit field, type D. 5 In the D cD edit field, type D. 6 In the D cU edit field, type D. 7 Locate the Model Inputs section. From the u list, choose Velocity field (spf/fp1). Reactions 1 1 On the Physics toolbar, click Domains and choose Reactions. 2 In the Reactions settings window, locate the Domain Selection section. 3 From the Selection list, choose All domains. 4 Locate the Reactions section. In the R cA edit field, type -r_1. 5 In the R cB edit field, type -r_1-r_2. 6 In the R cD edit field, type r_1-r_2. 7 In the R cU edit field, type r_2. Inflow 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 In the Inflow settings window, locate the Boundary Selection section. Solved with COMSOL Multiphysics 4.4 16 | P L A T E R E A C T O R 3 From the Selection list, choose Inlet 1. 4 Locate the Concentration section. In the c 0,cA edit field, type 2e4. 5 In the c 0,cB edit field, type 2e4. Concentration 1 1 On the Physics toolbar, click Boundaries and choose Concentration. 2 In the Concentration settings window, locate the Boundary Selection section. 3 From the Selection list, choose Inlet 2. 4 Locate the Concentration section. Select the Species c_B check box. 5 In the c 0,cB edit field, type 2e4. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 In the Outflow settings window, locate the Boundary Selection section. 3 From the Selection list, choose Outlet. G L O B A L D E F I N I T I O N S Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: D E F I N I T I O N S Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. Name Expression Value Description T0 300[K] 300.00 K Initial temperature D 1e-7[m^2/s] 1.0000E-7 m²/s Diffusivity Solved with COMSOL Multiphysics 4.4 17 | P L A T E R E A C T O R 3 In the table, enter the following settings: ME S H 1 Free Triangular 1 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Free Triangular. 2 Select Boundaries 3, 7, 17, 27, 33, 37, 43, 235, and 245 only. Size 1 In the Model Builder window, under Component 1>Mesh 1 click Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type 1e-3. 5 In the Minimum element size edit field, type 5e-4. 6 In the Resolution of narrow regions edit field, type 0.2. 7 In the Model Builder window, right-click Mesh 1 and choose Swept. Name Expression Unit Description r_1 kf_1*c_A*c_B mol/(m³·s) Rate, reaction 1 r_2 kf_2*c_B*c_D mol/(m³·s) Rate, reaction 2 kf_1 Af_1*exp(-Ef_1/ (R_const*T0)) m³/(s·mol) Rate constant, reaction 1 kf_2 Af_2*exp(-Ef_2/ (R_const*T0)) m³/(s·mol) Rate constant, reaction 2 Af_1 1[m^3/(mol*s)] m³/(s·mol) Frequency factor Ef_1 40e3[J/mol] J/mol Activation energy Af_2 1000[m^3/(mol*s)] m³/(s·mol) Frequency factor Ef_2 60e3[J/mol] J/mol Activation energy Q_reac -r_1*H_1-r_2*H_2 W/m³ Heat source of reaction H_1 -1.1e5[J/mol] J/mol Enthalphy of reaction H_2 -1e6[J/mol] J/mol Enthalphy of reaction Q_exch1 (T0-T)*hx W/m² Heat exchange flux Q_exch2 (T0-20-T)*hx W/m² Heat exchange flux hx 1000[W/(m^2*K)] W/(m²·K) Heat transfer coefficient Solved with COMSOL Multiphysics 4.4 18 | P L A T E R E A C T O R 8 Right-click Mesh 1 and choose Build All. S T U D Y 1 On the Study toolbar, click Study Steps and choose Stationary>Stationary. Step 1: Stationary 1 On the Study toolbar, click Study Steps and choose Stationary>Stationary. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Step 2: Stationary 2 1 In the Model Builder window, under Study 1 click Step 2: Stationary 2. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Step 3: Stationary 3 1 In the Model Builder window, under Study 1 click Step 3: Stationary 3. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: 4 On the Home toolbar, click Compute. R E S U L T S 3D Plot Group 7 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. Physics Solve for Discretization Heat Transfer in Fluids × physics Transport of Diluted Species × physics Physics Solve for Discretization Laminar Flow × physics Heat Transfer in Fluids × physics Physics Solve for Discretization Laminar Flow × physics Transport of Diluted Species × physics Solved with COMSOL Multiphysics 4.4 19 | P L A T E R E A C T O R 2 In the Model Builder window, under Results right-click 3D Plot Group 7 and choose Streamline. 3 Select Boundary 23 only. 4 In the Streamline settings window, locate the Coloring and Style section. 5 From the Line type list, choose Tube. 6 In the Tube radius expression edit field, type 5e-4. 7 Click the Go to Default 3D View button on the Graphics toolbar. 8 Right-click Results>3D Plot Group 7>Streamline 1 and choose Color Expression. 9 In the Color Expression settings window, locate the Expression section. 10 Click Concentration (c_A) in the upper-right corner of the section. On the 3D plot group toolbar, click Plot. 3D Plot Group 8 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the Model Builder window, under Results right-click 3D Plot Group 8 and choose Slice. 3 In the Slice settings window, locate the Expression section. 4 Click Temperature (T) in the upper-right corner of the section. Locate the Plane Data section. From the Plane list, choose xy-planes. 5 In the Planes edit field, type 4. 6 In the Model Builder window, right-click 3D Plot Group 8 and choose Slice. 7 In the Slice settings window, locate the Expression section. 8 Click Temperature (T) in the upper-right corner of the section. Locate the Plane Data section. From the Plane list, choose zx-planes. 9 In the Planes edit field, type 1. As you can see, the two color legends are nearly aligned so a single legend is sufficient. 10 Locate the Coloring and Style section. Clear the Color legend check box. 11 On the 3D plot group toolbar, click Plot. 3D Plot Group 9 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the Model Builder window, under Results right-click 3D Plot Group 9 and choose Isosurface. Solved with COMSOL Multiphysics 4.4 20 | P L A T E R E A C T O R 3 In the Isosurface settings window, locate the Expression section. 4 Click Concentration (c_B) in the upper-right corner of the section. Locate the Levels section. In the Total levels edit field, type 20. 5 On the 3D plot group toolbar, click Plot. 6 Click the Zoom Extents button on the Graphics toolbar. Solved with COMSOL Multiphysics 4.4 1 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R S ur f a c e Re a c t i ons i n a Bi os e ns or This example illustrates how to use the Surface Reactions physics interface, the Transport of Diluted Species physics interface, and the Laminar Flow physics interface to set up a model of a biosensor application. Combining these physics interfaces makes it straightforward to couple surface reactions to mass transport in a fluid stream. Introduction A flow cell in a biosensor contains an array of micropillars. The curved side of the pillars are coated with an active material that allows for the selective adsorption of analyte species in the sample stream. The adsorbed species produce a signal that is dependent upon the local concentration at the pillar surfaces. This example investigates the surface concentration distribution in the cell while an analyte pulse is transported through it. It also studies the effect of a quenching surface reaction where adsorbed species are converted into an inactive state. Model Definition G E O ME T R Y The flow cell contains seven rows containing four pillars each. The curved surfaces of the pillars are the only active surfaces for adsorbing the analyte molecules. The flow cell has two planes of symmetry that allow for reduction of the modeling domain to one fourth of the full geometry. Solved with COMSOL Multiphysics 4.4 2 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Figure 1: The flow cell holds seven rows of pillars with four pillars in each row. The curved pillar surface is the only surface that allows the adsorption of analyte molecules. The modeling geometry can be reduced to one fourth of the full geometry due to mirror symmetry. S U R F A C E R E A C T I O N S Analyte molecules (P) can adsorb and desorb from surface sites (S) on the micropillar surfaces according to (1) The adsorbed analyte (PS) can transform into a quenched state (QS) that does not contribute to the sensor signal. The quenching reaction is reversible: (2) The rate of adsorption is (3) where c p is the concentration of P in the stream. The desorption rate is linear in the concentration of surface adsorbed species, c PS : Outlet Inlet Active surface + P S PS k ads k des PS QS k 1 k 2 r ads k ads c P = Solved with COMSOL Multiphysics 4.4 3 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R (4) The rate of the reversible quenching reaction is given by (5) MA S S TR A N S P O R T I N T H E A N A L Y T E S T R E A M The equations in the Transport of Diluted Species interface describe the transport of the species, P, in the analyte stream according to (6) Here D P denotes the diffusion coefficient (m 2 /s), c P the species concentration (mol/ m 3 ), and u the velocity vector (m/s). The sample pulse that enters the sensor array is described by a Gaussian distribution at flow cell inlet with a maximum concentration of 80 mol/m 3 . At the outlet, the Outflow condition is used: (7) The adsorption and desorption of analyte at the active pillar surfaces give rise to a net flux at the corresponding boundaries: (8) The mass flux due to desorption is dependent upon local concentration of adsorbed surface species and is hence coupled to the equations in the Surface Reactions interface, described next. MA S S TR A N S P O R T A N D R E A C T I O N S O N T H E A C T I V E S U R F A C E S Transport of adsorbed species occurs in the tangential direction along the surface. The Surface Reactions interface models the tangential flux in the along the surface, the surface molar flux, N t,i (mol/(m·s)), according to where D s,i (m 2 /s) is the surface diffusion coefficient for species i. The governing equation for the surface concentrations is written as r des k des c PS = r quench k 1 c PS – k 2 c QS + = cc P ct --------- V + D P V – c P ( ) · u Vc P · + 0 = n DV – c ( ) · 0 = N p r ads – r des + = N t i , D s i , V t c s i , – = Solved with COMSOL Multiphysics 4.4 4 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R where R s,i (mol/(m 2 ·s)) is the sum of all sources due to surface reactions and adsorption/desorption phenomena. In this example, surface diffusion is ignored. Using the reactions described by Equation 3 through Equation 5, the balances equation for the surface species P and Q become: The rate of adsorption depends on the concentration of the P species in the analyte stream and is therefore coupled to the equations in the Surface Reactions interface to those provided by the Transport of Diluted Species interface. F L U I D F L OW The flow in the flow cell is laminar and given by the Navier-Stokes equations: (9) where µ denotes density (kg/m 3 ), u represents the velocity (m/s), q denotes viscosity (kg/(m· s)), and p equals the pressure in the tubes (Pa). The calculated flow field serves as input to the Transport of Diluted Species interface, to describe the convective mass transport. The boundary conditions are (10) At the outlet, viscous stresses are ignored and the pressure is set to 1 atmosphere. t c c c s i , V t D s i , V t c s i , – ( ) · + R s i , = dc s P , dt -------------- r ads r des – r quench – = dc s Q , dt -------------- r quench = µu V · u V pI – q Vu Vu ( ) T + ( ) 2q 3 ( ) V u · ( )I – + | | · = V µu ( ) · 0 = u n · v 0 = u 0 = p p ref = inlet walls outlet Solved with COMSOL Multiphysics 4.4 5 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Results and Discussion Figure 2 shows the magnitude of the laminar velocity field in the flow cell. Figure 2: The velocity magnitude of the laminar flow field in the biosensor flow cell Figure 3 through Figure 6 show the concentration of the species, P, in the stream as well the relative coverage of surface adsorbed species, PS, as the analyte pulse passes through the flow cell. Solved with COMSOL Multiphysics 4.4 6 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Figure 3: Concentration distribution in the analyte stream and surface coverage of adsorbed species at t = 35 s. Solved with COMSOL Multiphysics 4.4 7 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Figure 4: Concentration distribution in the analyte stream and surface coverage of adsorbed species at t = 45 s. Solved with COMSOL Multiphysics 4.4 8 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Figure 5: Concentration distribution in the analyte stream and surface coverage of adsorbed species at t = 55 s. Solved with COMSOL Multiphysics 4.4 9 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Figure 6: Concentration distribution in the analyte stream and surface coverage of adsorbed species at t = 75 s. The velocity distribution of the flow field will cause pillars near the wall to reach their maximum adsorption level at a later time compared to pillars in the center of the stream. Pillars near the wall will also take longer to release adsorbed analyte. The position of a pillar in a row also has an effect on the maximum adsorption level, and the time at which it is reached. This effect is highlighted in Figure 7. These geometrical effects will cause the sensor signal to become relatively diffuse. Solved with COMSOL Multiphysics 4.4 10 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Figure 7: Average fractional surface coverage of adsorbed analyte PS. Figure 8 shows the relative surface concentration of the quenched surface species QS for the same pillar positions. The geometrical effect is once again evident yet more importantly, the plot shows that a relatively long time is required to purge the reactive sites in preparation for a new analysis. Solved with COMSOL Multiphysics 4.4 11 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Figure 8: Average fractional surface coverage of adsorbed quenched species QS. Notes About the COMSOL Implementation In this example, there is a one-way coupling between the stationary flow field and the mass transport equations. This means that the equations for the laminar flow need only be solved once, and that the results can be used for calculating the transient mass transport problem. This is accomplished by using two study steps when setting up and solving the model. The first step solves for the stationary flow field. The solution is then used in the second step where the transient mass balance equations are solved. Model Library path: Chemical_Reaction_Engineering_Module/ Surface_Reactions_and_Deposition_Processes/reacting_pillars Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. Solved with COMSOL Multiphysics 4.4 12 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R MO D E L WI Z A R D 1 In the Model Wizard window, click the 3D button. 2 In the Select physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf). 3 Click the Add button. 4 In the Select physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 5 Click the Add button. 6 In the Concentrations table, enter the following settings: 7 In the Select physics tree, select Chemical Species Transport>Surface Reactions (chsr). 8 Click the Add button. 9 In the Number of surface species edit field, type 2. 10 In the Surface (adsorbed) species concentrations table, enter the following settings: 11 Click the Study button. 12 In the tree, select Preset Studies for Selected Physics>Stationary. 13 Click the Done button. G E O ME T R Y 1 Start by creating the geometry. To simplify this step, insert a prepared geometry sequence from file. After insertion you can study each geometry step in the sequence. 1 On the Geometry toolbar, click Insert Sequence. 2 Browse to the model’s Model Library folder and double-click the file reacting_pillars_geom_sequence.mph. 3 On the Geometry toolbar, click Build all. 4 Click the Zoom Extents button on the Graphics toolbar. G L O B A L D E F I N I T I O N S Read in a text file with model parameters such as model rate constants and diffusion coefficients. c_P cs_P cs_Q Solved with COMSOL Multiphysics 4.4 13 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file reacting_pillars_parameters.txt. D E F I N I T I O N S Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file reacting_pillars_variables.txt. Gaussian Pulse 1 1 On the Home toolbar, click Functions and choose Global>Gaussian Pulse. 2 In the Gaussian Pulse settings window, locate the Parameters section. 3 In the Location edit field, type 20. 4 In the Standard deviation edit field, type 2. Note that you can plot the Gaussian function by clicking the Plot button on the Settings toolbar. Explicit 1 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Click the Wireframe Rendering button on the Graphics toolbar. You can select the surfaces boundary by boundary according to the list below. Alternatively, click the Go to YZ View button on the Graphics toolbar and then click Solved with COMSOL Multiphysics 4.4 14 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R the Select Box button. Enclose the concave pillar surfaces with the rubber band box and right-click the selection to confirm. 5 Select Boundaries 8, 9, 12, 13, 15, 16, 19, 20, 23, 26–28, 31, 32, 37, 38, 41, 42, 44, 45, 48, 49, 52, 55–57, 60, 61, 66, 67, 70, 71, 73, 74, 77, 78, 81, 84–86, 89, 90, 95, 96, 99, 100, 102, 103, 106, 107, 110, 113–115, 118, and 119 only. 6 Right-click Component 1>Definitions>Explicit 1 and choose Rename. 7 Go to the Rename Explicit dialog box and type Reacting surface in the New name edit field. 8 Click OK. Create a number of operators features that calculate the surface average. Average 1 1 On the Definitions toolbar, click Component Couplings and choose Average. 2 In the Average settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 23 and 28 only. 5 Locate the Operator Name section. In the Operator name edit field, type ave_center_1. Solved with COMSOL Multiphysics 4.4 15 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Average 2 1 On the Definitions toolbar, click Component Couplings and choose Average. 2 In the Average settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 110 and 115 only. 5 Locate the Operator Name section. In the Operator name edit field, type ave_center_4. Average 3 1 On the Definitions toolbar, click Component Couplings and choose Average. 2 In the Average settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 12, 13, 19, and 20 only. 5 Locate the Operator Name section. In the Operator name edit field, type ave_wall_1. Average 4 1 On the Definitions toolbar, click Component Couplings and choose Average. 2 In the Average settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 99, 100, 106, and 107 only. 5 Locate the Operator Name section. In the Operator name edit field, type ave_wall_4. MA T E R I A L S Select water as the material in the flow cell. 1 On the Home toolbar, click Add Material. A D D MA T E R I A L 1 Go to the Add Material window. 2 In the tree, select Liquids and Gases>Liquids>Water. 3 In the Add material window, click Add to Component. Solved with COMSOL Multiphysics 4.4 16 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R MA T E R I A L S Water Move on to set up the physics interfaces. L A MI N A R F L OW Inlet 1 1 On the Physics toolbar, click Boundaries and choose Inlet. 2 Select Boundary 1 only. 3 In the Inlet settings window, locate the Velocity section. 4 In the U 0 edit field, type u_in. Outlet 1 1 On the Physics toolbar, click Boundaries and choose Outlet. 2 Select Boundary 122 only. Symmetry 1 1 On the Physics toolbar, click Boundaries and choose Symmetry. 2 Select Boundaries 2, 4, 34, 63, 92, and 121 only. TR A N S P O R T O F D I L U T E D S P E C I E S Convection and Diffusion 1 1 In the Model Builder window, expand the Component 1>Transport of Diluted Species node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 From the u list, choose Velocity field (spf/fp1). 4 Locate the Diffusion section. In the D cP edit field, type D. Inflow 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 Select Boundary 1 only. 3 In the Inflow settings window, locate the Concentration section. 4 In the c 0,cP edit field, type c0. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 Select Boundary 122 only. Solved with COMSOL Multiphysics 4.4 17 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Symmetry 1 1 On the Physics toolbar, click Boundaries and choose Symmetry. 2 Select Boundaries 2, 4, 34, 63, 92, and 121 only. Flux 1 1 On the Physics toolbar, click Boundaries and choose Flux. 2 In the Flux settings window, locate the Boundary Selection section. 3 From the Selection list, choose Reacting surface. 4 Locate the Inward Flux section. Select the Species c_P check box. 5 In the N 0,cP edit field, type -r_ads+r_des. S U R F A C E R E A C T I O N S 1 In the Model Builder window, under Component 1 click Surface Reactions. 2 In the Surface Reactions settings window, locate the Boundary Selection section. 3 From the Selection list, choose Reacting surface. Reactions 1 1 On the Physics toolbar, click Boundaries and choose Reactions. 2 In the Reactions settings window, locate the Boundary Selection section. 3 From the Selection list, choose Reacting surface. 4 Locate the Reaction Rate for Surface Species section. In the R s,csP edit field, type r_ads-r_des-r_quench. 5 In the R s,csQ edit field, type r_quench. ME S H 1 Free Triangular 1 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Free Triangular. 2 In the Free Triangular settings window, locate the Boundary Selection section. 3 From the Selection list, choose Reacting surface. Size 1 1 Right-click Component 1>Mesh 1>Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. Solved with COMSOL Multiphysics 4.4 18 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R 4 Locate the Element Size Parameters section. Select the Maximum element size check box. 5 In the associated edit field, type 5e-5. Free Tetrahedral 1 In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral. Size 1 1 In the Model Builder window, under Component 1>Mesh 1 right-click Free Tetrahedral 1 and choose Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. Select the Maximum element size check box. 5 In the associated edit field, type 2e-4. 6 In the Model Builder window, right-click Mesh 1 and choose Build All. This creates a better overview of the model tree as you move to set up the solvers and work with results processing. Solve the problems using two study steps. The first step solves for the stationary flow field. In the second step, use the solution from the first step to solve the transient mass balance equations. S T U D Y 1 Step 1: Stationary 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Stationary. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Step 2: Time Dependent 1 On the Study toolbar, click Study Steps and choose Time Dependent>Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type range(0,1,100). Physics Solve for Discretization Transport of Diluted Species × physics Surface Reactions × physics Solved with COMSOL Multiphysics 4.4 19 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R 4 Locate the Physics and Variables Selection section. In the table, enter the following settings: Solver 1 1 On the Study toolbar, click Show Default Solver. The magnitude of the concentration in the analyte stream will be of many orders of magnitude higher compared to the concentrations of the surface species. Providing manual scales for the concentration variables will help convergence. 2 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Dependent Variables 2 node, then click Surface concentration (comp1.cs_Q). 3 In the Field settings window, locate the Scaling section. 4 From the Method list, choose Manual. 5 In the Scale edit field, type 1e-7. 6 In the Model Builder window, under Study 1>Solver Configurations>Solver 1>Dependent Variables 2 click Surface concentration (comp1.cs_P). 7 In the Field settings window, locate the Scaling section. 8 From the Method list, choose Manual. 9 In the Scale edit field, type 1e-7. 10 In the Model Builder window, under Study 1>Solver Configurations>Solver 1>Dependent Variables 2 click Concentration (comp1.c_P). 11 In the Field settings window, locate the Scaling section. 12 From the Method list, choose Manual. 13 In the Scale edit field, type 10. 14 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Time-Dependent Solver 1 node. 15 Right-click Study 1>Solver Configurations>Solver 1>Time-Dependent Solver 1 and choose Fully Coupled. 16 In the Fully Coupled settings window, click to expand the Method and termination section. 17 Locate the Method and Termination section. From the Jacobian update list, choose Once per time step. 18 On the Home toolbar, click Compute. Physics Solve for Discretization Laminar Flow × physics Solved with COMSOL Multiphysics 4.4 20 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R R E S U L T S Velocity (spf) Create Mirror 3D data sets to create plots for the full flow-cell geometry. Data Sets 1 On the Results toolbar, click More Data Sets and choose Mirror 3D. 2 In the Mirror 3D settings window, locate the Plane Data section. 3 From the Plane list, choose xy-planes. 4 In the z-coordinate edit field, type 5e-4. 5 On the Results toolbar, click More Data Sets and choose Mirror 3D. 6 In the Mirror 3D settings window, locate the Plane Data section. 7 From the Plane list, choose xz-planes. 8 In the y-coordinate edit field, type -3e-3. Velocity (spf) 1 In the Model Builder window, under Results click Velocity (spf). 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Mirror 3D 2. 4 In the Model Builder window, expand the Velocity (spf) node, then click Slice 1. 5 In the Slice settings window, locate the Plane Data section. 6 From the Plane list, choose xy-planes. 7 In the Planes edit field, type 1. 8 On the 3D plot group toolbar, click Plot. Next create the plots shown in Figure 3 through Figure 6, illustrating the analyte concentration and adsorbed species concentration as function of time. 3D Plot Group 6 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Mirror 3D 2. 4 From the Time (s) list, choose 35. 5 Right-click Results>3D Plot Group 6 and choose Slice. 6 In the Slice settings window, locate the Expression section. Solved with COMSOL Multiphysics 4.4 21 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R 7 Click Concentration (c_P) in the upper-right corner of the section. Locate the Plane Data section. From the Plane list, choose xy-planes. 8 In the Planes edit field, type 1. 9 Locate the Coloring and Style section. Clear the Color legend check box. 10 Right-click Results>3D Plot Group 6>Slice 1 and choose Deformation. 11 In the Deformation settings window, locate the Expression section. 12 In the z component edit field, type c_P. 13 On the 3D plot group toolbar, click Plot. 14 In the Model Builder window, click 3D Plot Group 6. 15 In the 3D Plot Group settings window, locate the Data section. 16 From the Time (s) list, choose 45. 17 On the 3D plot group toolbar, click Plot. 18 From the Time (s) list, choose 55. 19 On the 3D plot group toolbar, click Plot. 20 From the Time (s) list, choose 75. 21 On the 3D plot group toolbar, click Plot. 3D Plot Group 7 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Mirror 3D 2. 4 From the Time (s) list, choose 35. 5 Right-click Results>3D Plot Group 7 and choose Surface. 6 In the Surface settings window, locate the Expression section. 7 Click Surface coverage (chsr.theta_i_cs_P) in the upper-right corner of the section. Locate the Coloring and Style section. Clear the Color legend check box. 8 On the 3D plot group toolbar, click Plot. 9 In the Model Builder window, click 3D Plot Group 7. 10 In the 3D Plot Group settings window, locate the Data section. 11 From the Time (s) list, choose 45. 12 On the 3D plot group toolbar, click Plot. 13 From the Time (s) list, choose 55. 14 On the 3D plot group toolbar, click Plot. Solved with COMSOL Multiphysics 4.4 22 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R 15 From the Time (s) list, choose 75. 16 On the 3D plot group toolbar, click Plot. Finally create line plots showing the average coverage of surface species as a function of time and pillar position. Note how the Duplicate feature speeds up this procedure. 1D Plot Group 8 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, click to expand the Title section. 3 From the Title type list, choose Manual. 4 Locate the Plot Settings section. Select the y-axis label check box. 5 In the associated edit field, type Surface Fraction P. 6 On the 1D plot group toolbar, click Point Graph. 7 Select Point 10 only. 8 In the Point Graph settings window, locate the y-Axis Data section. 9 In the Expression edit field, type ave_center_1(chsr.theta_i_cs_P). 10 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Color list, choose Blue. 11 Click to expand the Legends section. Select the Show legends check box. 12 From the Legends list, choose Manual. 13 In the table, enter the following settings: 14 Right-click Results>1D Plot Group 8>Point Graph 1 and choose Duplicate. 15 In the Point Graph settings window, locate the y-Axis Data section. 16 In the Expression edit field, type ave_center_4(chsr.theta_i_cs_P). 17 Locate the Coloring and Style section. Find the Line markers subsection. From the Line list, choose Dash-dot. 18 Locate the Legends section. In the table, enter the following settings: 19 Right-click Point Graph 1 and choose Duplicate. Legends Center row, first pillar Legends Center row, last pillar Solved with COMSOL Multiphysics 4.4 23 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R 20 In the Point Graph settings window, locate the y-Axis Data section. 21 In the Expression edit field, type ave_wall_1(chsr.theta_i_cs_P). 22 Locate the Coloring and Style section. Find the Line style subsection. From the Color list, choose Red. 23 Locate the Legends section. In the table, enter the following settings: 24 In the Model Builder window, under Results>1D Plot Group 8 right-click Point Graph 2 and choose Duplicate. 25 In the Point Graph settings window, locate the y-Axis Data section. 26 In the Expression edit field, type ave_wall_4(chsr.theta_i_cs_P). 27 Locate the Coloring and Style section. Find the Line style subsection. From the Color list, choose Red. 28 Locate the Legends section. In the table, enter the following settings: 29 On the 1D plot group toolbar, click Plot. Legends Wall row, first pillar Legends Wall row, last pillar Solved with COMSOL Multiphysics 4.4 24 | S U R F A C E R E A C T I O N S I N A B I O S E N S O R Solved with COMSOL Multiphysics 4.4 1 | S T E F A N TU B E S t e f a n T ube Introduction This example, taken from Ref. 6, illustrates the use of the Maxwell-Stefan diffusion model available with the Transport of Concentrated Species physics interface. It models a multicomponent gas-phase diffusion problem in 1D. In this particular case, two of the three components experience a steady-state flux, whereas the third component has zero flux on one of the boundaries. The concentration profiles at steady state are modeled. Model Definition The Stefan tube, depicted in Figure 1, is a simple device used for measuring diffusion coefficients in binary vapors. Figure 1: Schematic diagram of a Stefan tube. At the bottom of the tube is a pool of mixture. The vapor that evaporates from this pool diffuses to the top of the tube, where a stream of air, flowing across the top of the tube, keeps the mole fraction of diffusing vapor there to be zero. The mole fraction of vapor above the liquid interface is at equilibrium. Because there is no horizontal flux inside the tube, you can analyze the problem using a 1D model. The system composition of acetone, methanol, and air has been extensively investigated, x Air Liquid mixture Gas/vapor phase Solved with COMSOL Multiphysics 4.4 2 | S T E F A N TU B E measuring both diffusion coefficients and composition at various positions within Stefan tubes. This makes it an ideal example for this model. As a comparison, one experiment measured the mole fraction at the liquid interface to be x Ac = 0.319 and x Me = 0.528 where the pressure, p, was 99.4 kPa and the temperature, T, was 328.5 K. The length of the diffusion path was 0.238 m. The respective Maxwell-Stefan diffusion coefficients, D ij , of the three binary pairs were calculated and are used in the model according to Table 1. To model this problem, use the Transport of Concentrated Species physics interface with the Maxwell-Stefan diffusion model. It solves for the fluxes in terms of mass fractions for two of the three components. The mass fraction, e, of the third is given by the two first. The three equations are: (1) (2) (3) where D is the diffusion coefficient (m 2 /s), p is the pressure (Pa), T is the temperature (K), u is the velocity (m/s), x and e are mole and mass fractions, respectively, and the mixture density, µ mix (kg/m 3 ), is a function of the average mixture mole fraction, M mix (kg/mol), according to Equation 5: (4) (5) TABLE 1: LABELS AND MAXWELL-STEFAN DIFFUSION COEFFICIENTS COMPONENT LABEL D ij VALUE Acetone 1 D 12 8.48·10 -6 m 2 /s Methanol 2 D 13 13.72·10 -6 m 2 /s Air 3 D 23 19.91·10 -6 m 2 /s V µe 1 D 1k x k V x k e k – ( ) p V ( ) p ( ) + ( ) | | k ¿ – · + D T T V ( ) T ( ) | R µu e 1 V · ( ) – = V µe 2 1 ( ) D 2k x k V x k e k – ( ) p V ( ) p ( ) + ( ) | | k ¿ – · + D T T V ( ) T ( ) | R µu e 2 V · ( ) – = e 3 1 e 1 – e 2 – = M mix x i M i i ¿ = µ mix p RT --------M mix = Solved with COMSOL Multiphysics 4.4 3 | S T E F A N TU B E In this case, there is no imposed fluid velocity. However, there will appear a fluid velocity due to the diffusive fluxes. At the top of the tube the mass fractions are fixed, with the fraction of air being unity. At the bottom (at the liquid interface), the fractions are also fixed according to the previously mentioned experimental conditions. The fact that there is no air flux at the interface results in the following relation for the convective velocity, at steady state: (6) where n diff,3 is the diffusive mass flux of air (kg/(m 2 ·s)). Results The steady-state mole fractions as a function of position are shown in Figure 2. For comparison, the results given from Ref. 6 are depicted in Figure 3. Figure 2: Steady-state mole fractions of: acetone, methanol, and air, in the Stefan tube according to the COMSOL Multiphysics model as solved using Maxwell-Stefan diffusivities. u n diff,3 e 3 µ -------------- = Solved with COMSOL Multiphysics 4.4 4 | S T E F A N TU B E Figure 3: Steady-state mole fractions of: acetone(-), methanol (- -), and air(- . -), in a Stefan tube according to experimental data (Ref. 6). We can see that the COMSOL Multiphysics model reproduces the results from Ref. 6 well, which means the Maxwell-Stefan equations describe the mass transport process in the system. The Maxwell-Stefan diffusion formulation includes the conservation of mass. In the absence of chemical reactions (source terms) and convective contributions, the Maxwell-Stefan formulation results in zero net mass flux. This is not the case for the Fickian diffusion formulation. A consequence of the conservation of mass is that if there is a net mass flux. In this example, it can be described by the convective term, which you can see in Figure 4. Solved with COMSOL Multiphysics 4.4 5 | S T E F A N TU B E Figure 4: Velocity of the gas mixture in the Stefan tube from the pool surface to the outlet. References 1. C.F. Curtiss and R.B. Bird, “Multicomponent diffusion,” Ind. Eng. Chem. Res., vol. 38, p. 2515, 1999. 2. R.B. Bird, W. Stewart, and E. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, 1960. 3. G.A.J. Jaumann, Wien. Akad. Sitzungsberichte (Math.-Naturw. Klasse), vol. 120, p. 385, 1911. 4. J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, USA, 1954. 5. E.N. Fuller, P.D. Schettler, and J.C. Giddings, Ind. Eng. Chem., vol. 58, p. 19, 1966. 6. R. Taylor and R. Krishna, Multicomponent Mass Transfer, John Wiley & Sons, NY, p. 21, 1993. Solved with COMSOL Multiphysics 4.4 6 | S T E F A N TU B E Model Library path: Chemical_Reaction_Engineering_Module/ Mass_Transport/stefan_tube Modeling Instructions From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 1D button. 2 In the Select physics tree, select Chemical Species Transport>Transport of Concentrated Species (chcs). 3 Click the Add button. 4 In the Review Physics section, click Add Mass Fraction. 5 In the Mass fractions table, enter the following settings: 6 Click the Study button. 7 In the tree, select Preset Studies>Stationary. 8 Click the Done button. G E O ME T R Y 1 Interval 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Interval. 2 In the Interval settings window, locate the Interval section. 3 In the Right endpoint edit field, type 0.238. 4 Click the Build All Objects button. w3 Solved with COMSOL Multiphysics 4.4 7 | S T E F A N TU B E G L O B A L D E F I N I T I O N S Next, add a set of model parameters by importing their definitions from a data text file provided with the Model Library. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Model Library folder and double-click the file stefan_tube_parameters.txt. TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S 1 In the Model Builder window, under Component 1 click Transport of Concentrated Species. 2 In the Transport of Concentrated Species settings window, locate the Transport Mechanisms section. 3 From the Diffusion model list, choose Maxwell-Stefan. 4 Locate the Species section. From the From mass constraint list, choose w3. 5 In the Model Builder window’s toolbar, click the Show button and select Discretization in the menu. 6 Click to expand the Discretization section. From the Mass fraction list, choose Quadratic. Convection and Diffusion 1 1 In the Model Builder window, expand the Transport of Concentrated Species node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 Specify the u vector as 4 In the T edit field, type T0. 5 In the p A edit field, type p0. 6 Locate the Density section. In the M w1 edit field, type M_ace. 7 In the M w2 edit field, type M_met. 8 In the M w3 edit field, type M_air. -chcs.dfluxx_w3/(w3*chcs.rho) x Solved with COMSOL Multiphysics 4.4 8 | S T E F A N TU B E 9 Locate the Diffusion section. In the D ik table, enter the following settings: Initial Values 1 1 In the Model Builder window, under Component 1>Transport of Concentrated Species click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the w 0,w1 edit field, type w_ace0. 4 In the w 0,w2 edit field, type w_met0. Mass Fraction 1 1 On the Physics toolbar, click Boundaries and choose Mass Fraction. 2 Select Boundary 1 only. 3 In the Mass Fraction settings window, locate the Mass Fraction section. 4 Select the Species w1 check box. 5 In the e 0,w1 edit field, type w_ace0. 6 Select the Species w2 check box. 7 In the e 0,w2 edit field, type w_met0. Mass Fraction 2 1 On the Physics toolbar, click Boundaries and choose Mass Fraction. 2 Select Boundary 2 only. 3 In the Mass Fraction settings window, locate the Mass Fraction section. 4 Select the Species w1 check box. 5 Select the Species w2 check box. ME S H 1 1 In the Model Builder window, under Component 1 click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Element size list, choose Extra fine. 4 Click the Build All button. 1 D12 D13 D12 1 D23 D13 D23 1 Solved with COMSOL Multiphysics 4.4 9 | S T E F A N TU B E S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S Mass Fraction (chcs) In order to reproduce the plot in Figure 2, do the following: 1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Line Graph. 3 Select Domain 1 only. 4 In the Line Graph settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Transport of Concentrated Species>Species w1>Mole fraction (chcs.x_w1). 5 Locate the x-Axis Data section. From the Parameter list, choose Expression. 6 In the Expression edit field, type x. 7 Click to expand the Title section. From the Title type list, choose None. 8 Click to expand the Legends section. Select the Show legends check box. 9 From the Legends list, choose Manual. 10 In the table, enter the following settings: 11 Right-click Results>1D Plot Group 2>Line Graph 1 and choose Duplicate. 12 In the Line Graph settings window, locate the y-Axis Data section. 13 In the Expression edit field, type chcs.x_w2. 14 Locate the Legends section. In the table, enter the following settings: 15 Right-click Results>1D Plot Group 2>Line Graph 2 and choose Duplicate. 16 In the Line Graph settings window, locate the y-Axis Data section. 17 In the Expression edit field, type chcs.x_w3. Legends Acetone Legends Methanol Solved with COMSOL Multiphysics 4.4 10 | S T E F A N TU B E 18 Locate the Legends section. In the table, enter the following settings: 19 In the Model Builder window, click 1D Plot Group 2. 20 In the 1D Plot Group settings window, click to expand the Legend section. 21 From the Position list, choose Upper left. To reproduce Figure 4, proceed as follows: 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D plot group toolbar, click Line Graph. 3 Select Domain 1 only. 4 In the Line Graph settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Transport of Concentrated Species>Velocity field>Velocity field, x component (chcs.u). 5 Locate the x-Axis Data section. From the Parameter list, choose Expression. 6 In the Expression edit field, type x. 7 On the 1D plot group toolbar, click Plot. Legends Air Solved with COMSOL Multiphysics 4.4 1 | T H E R M A L D E C O M P O S I T I O N T he r mal De c ompos i t i on Introduction In this tutorial model you will couple heat and mass transport equations to laminar flow in order to model exothermic reactions in parallel plate reactor. It exemplifies how COMSOL Multiphysics allows you to systematically set up and solve increasingly sophisticated models using predefined physics interfaces. Model Definition In this model you investigate the unimolecular decomposition of a chemical passing through a parallel plate reactor. A heat sensitive compound is present in a water solution. After entering the reactor, the liquid first experiences an expansion, due to a step in the bottom plate. Before exiting, the fluid also passes a heated cylinder. The full 3D representation of the reactor geometry is given in Figure 1. Figure 1: 3D geometry of a parallel plate reactor. The reacting fluid is heated as it passes the cylinder. The short inlet section of the reactor is considerably wider than it is high. With such a geometry, it is reasonable to assume that the laminar flow develops a parabolic velocity profile between the top and bottom plate. At the same time, the velocity between the side walls is expected to be close to constant (Ref. 1). As a consequence, you can reduce the modeling domain to 2D without dramatically reducing the validity of the inlet outlet cylinder heated top plate bottom plate Solved with COMSOL Multiphysics 4.4 2 | T H E R M A L D E C O M P O S I T I O N simulation (see Figure 2). Figure 2: Neglecting edge effects, the modeling geometry can be reduced to 2D. C H E MI S T R Y A heat sensitive chemical (A) undergoes thermal decomposition into fragments (F) according to the following unimolecular reaction: The reaction rate (mol/(m 3 ·s)) is given by: where rate constant k (s ÷1 ) is temperature dependent according to the Arrhenius equation: (1) In Equation 1, A is the frequency factor (1·10 10 1/s), E the activation energy (72·10 3 J/mol), R g the gas constant (8.314 J/(mol·K)), and T the temperature (K). In addition, the decomposition reaction is exothermic, and the rate of energy expelled is given by: where H is the heat of reaction (100 kJ/mol). The conversion of species A in the reactor is a function of the residence time, that is, dependent on the detailed fluid flow. Furthermore, the decomposition is influenced by the temperature distribution. A coupled system of transport equations thus describes the reactor. inlet outlet top plate heating cylinder bottom plate k A F rate kc A = k A E R g T ----------- – \ . | | | exp = Q rate H · = Solved with COMSOL Multiphysics 4.4 3 | T H E R M A L D E C O M P O S I T I O N MO ME N T U M TR A N S P O R T The equations governing momentum transport, the Navier-Stokes equations, solved by default in the single phase flow interfaces are the compressible formulation of the continuity: (2) and the momentum equations: (3) Here, µ denotes the dynamic viscosity (Ns/m 2 ), u the velocity (m/s), µ the density of the fluid (kg/m 3 ), p the pressure (Pa), and F is a body force term (N/m 3 ). In this particular model we will solve a stead-state problem and the first term in each equation will be canceled. Equation 2 governs the flow of a Newtonian fluid in the laminar flow regime. The flow type in this specific model will be determined in the section The Flow Regime. Apart from the domain equations you also need to select proper boundary conditions. At the inlet you specify a velocity vector normal to the boundary: (4) At the outlet boundary you specify a pressure: (5) Finally, at the surfaces of the reactor plates and the heating cylinder you set the velocity to zero, that is, a no-slip boundary condition: (6) By selecting the Laminar Flow physics interface you can easily associate the momentum balance (Equation 2) and boundary conditions (Equation 4 to Equation 6) with your modeling geometry. E N E R G Y TR A N S P O R T The energy balance equation applied to the reactor domain considers heat transfer through convection and conduction: (7) cµ ct ------ V + µu ( ) · 0 = µ cu ct ------- µ + u Vu · Vp – V µ Vu Vu ( ) T + ( ) 2 3 ---µ V u · ( )I – \ . | | F + · + = u n · u 0 = p p 0 = u 0 = V k T) µC p u V · ( )T + V – ( · Q = Solved with COMSOL Multiphysics 4.4 4 | T H E R M A L D E C O M P O S I T I O N In Equation 7, C p denotes the specific heat capacity (J/(kg·K)), k is the thermal conductivity (W/(m·K)), and Q is a sink or source term (W/m 3 ). At the inlet and at the surface of the heating cylinder you set a temperature boundary condition: (8) (9) At the outlet you set an Outflow boundary condition. This prescribes that all energy passing through this boundary does so by means of convective transport. Equivalently this means that the heat flux due to conduction across the boundary is zero (10) so that the resulting equation for the total heat flux becomes (11) This is a useful boundary condition, particularly in convection-dominated energy balances where the outlet temperature is unknown. Finally, assume that no energy is transported across the reactor plates, that is, a Thermal Insulation boundary condition: (12) Using the Heat Transfer physics interface, you can associate the energy balance (Equation 7) and boundary conditions (Equation 8 to Equation 12) with the modeling geometry. MA S S TR A N S P O R T The mass transfer in the reactor domain is given by the stationary convection and diffusion equation: (13) where D i denotes its diffusion coefficient (m 2 /s), and R i denotes the reaction term (mol/(m 3 ·s)). Equation 13 assumes that the species i is diluted in a solvent. For the boundary conditions, specify the concentration of A at the inlet T T 0 = T T cyl = q cond n · k T V – n · 0 = = q n · µC p Tu n · = q n · 0 = V D i c i V – ( ) u c V · i + · R i = Solved with COMSOL Multiphysics 4.4 5 | T H E R M A L D E C O M P O S I T I O N (14) At the outlet, specify that the mass flow through the boundary is convection dominated. This assumes that any mass flux due to diffusion across this boundary is zero (15) so that (16) Finally, at the surfaces of the reactor plates and the heating cylinder, assume that no mass is transported across the boundaries, that is, an insulation boundary condition: (17) By selecting the Diluted Species physics interface you can easily associate the mass balance (Equation 13) and boundary conditions (Equation 14 to Equation 17) with the modeling geometry. P R E P A R I N G F O R MO D E L I N G Before you can start modeling you need to gather the physical data that characterize your reacting flow. For instance, flow modeling requires you to supply the fluid density and viscosity. Mass transport requires knowledge of diffusivities and the reaction kinetics. Another part of the preparations involves selecting the appropriate physics interfaces and investigating the couplings between different transport equations. Transport Properties The term transport properties refers to the physical properties occurring in the transport equations (see the previous section). The momentum and heat transfer equations (Equation 2 and Equation 7) require fluid-specific transport properties: • Viscosity (q) • Density (µ) • Thermal conductivity (k) • Heat capacity (C p ) c i c i 0 , = n D i c i V – ( ) · 0 = N i n c i u n · = · N i n 0 = · Solved with COMSOL Multiphysics 4.4 6 | T H E R M A L D E C O M P O S I T I O N The mass transport equation (Equation 13) requires the species-specific: • Diffusivities (D i ) You need to supply appropriate values of the transport properties to the physics interfaces in order to ensure accurate simulation results. In the present example, water with the dissolved compound A enters the reactor at 300 K. Because water is the solvent, you can assume that its physical properties are representative for the entire fluid. The warmest part of the reactor is held at 325 K. Table 1 lists the transport properties of water as well as the diffusivity of A in water at 300 K and 325 K. When you build this model you will make use of the built-in materials databases of COMSOL Multiphysics, that automatically provides temperature-dependent properties. The Flow Regime The Reynolds number indicates whether a flow is in the laminar or turbulent regime: (18) As a rule of thumb, a Reynolds number between of 2000 and 2500 marks the transition from stable streamlines to stable turbulent flow. It is always good practice to evaluate the Reynolds number related to the specific flow conditions of the model, because its magnitude guides you to choose the appropriate flow model and corresponding physics interface. In the present example, you can evaluate the Reynolds number using values from Table 1 and setting the velocity to 5·10 ÷4 m/s and the characteristic length to 0.007 m: (19) TABLE 1: PHYSICAL PROPERTIES OF LIQUID WATER PROPERTY AT 300 K AT 325 K Density (kg/m 3 ) 997 987 Viscosity (Ns/m 2 ) 8.5·10 -4 5.3·10 -4 Thermal conductivity (W/(m·K)) 0.62 0.66 Heat capacity (J/(kg·K)) 4180 4182 Diffusivity (m 2 /s) 2.0·10 -9 2.0·10 -9 Re µud q ----------- = Re 997 5 10 4 – 0.007 · · · 8.5 10 4 – · ----------------------------------------------------- 4 = = Solved with COMSOL Multiphysics 4.4 7 | T H E R M A L D E C O M P O S I T I O N Calculating the Reynolds number at 325 K produces a near identical result. The Reynolds numbers are well within the limits of the laminar flow regime. Dilute or Concentrated Mixtures When modeling mass transport, it is advisable to discriminate between dilute and concentrated mixtures. For dilute mixtures, Fick’s Law is adequate to describe the diffusional transport. Furthermore, you can assume that the transport properties of the fluid are those of the solvent. For concentrated mixtures, on the other hand, other diffusion models, for example the Maxwell-Stefan model, may be required. Also, the transport properties of the fluid then depends of the mixture composition. In COMSOL Multiphysics, the Transport of Diluted Species interface is appropriate for dilute mixtures, while the Transport of Concentrated Species interface is recommended for concentrated mixtures. As a rule of thumb, you can consider concentrations of up to 10 mol% of a solute in a solvent as a dilute mixture. In the example at hand, the compound A is dissolved in water at concentration of 1000 mol/m 3 . As the concentration of pure water is 55,500 mol/m 3 , the molar fraction of A is approximately 2%. Because the mixture is dilute, it is appropriate to select the Diluted Species physics interface for mass transport and to select the transport properties of water as representative values for the mixture. Solving Coupled Models As noted previously, the chemistry occurring in the reactor depends both on the fluid flow and the temperature distribution in the reactor. More explicitly, the mass transport equation (20) depends on the velocity vector, u, which is solved for in the momentum transfer equation (Equation 3). Furthermore, the source term R i in Equation 20 is a function of the temperature, which in turn is the dependent variable of the energy transport equation (21) t c cc i V D i c i c i u + V – ( ) · + R i = µC p t c cT V k T V – ( ) · µC p u VT · + + Q = Solved with COMSOL Multiphysics 4.4 8 | T H E R M A L D E C O M P O S I T I O N When attempting to solve a coupled system of equations such as the one illustrated above, it is often a good idea analyze the couplings involved and the approach the solution in a stepwise fashion. In the current model, you will first neglect the heat of reaction, Q. This leads to a loose two-way coupling between the transport equations: • The momentum transport is weakly dependent of the energy and mass transport, through the material properties. • The energy transport depends only on the momentum transport • The mass transport depends on both the momentum transport and the energy transport This structure suggests that it is possible to solve the problem sequentially in the following order: First solve momentum transport + energy transport. Then add mass transport and investigate the difference. The last step leads to a fully coupled problem: • The momentum transport depends on the energy transport • The energy transport depends on both momentum and mass transport (added heat of reaction) • The mass transport depends on both the momentum transport and the energy transport In this case you must solve the equations describing all transport phenomena simultaneously. Solved with COMSOL Multiphysics 4.4 9 | T H E R M A L D E C O M P O S I T I O N Results and Discussion Figure 3 shows the velocity field in the reactor domain along with arrows indicating the velocity magnitude. Figure 3: Velocity field (m/s) in the reactor. The fluid cross-sectional area increases at the step and decreases at the cylinder, leading to a corresponding local reduction and increase in fluid velocity. Recirculation zones appear after the step and the cylinder. Solved with COMSOL Multiphysics 4.4 10 | T H E R M A L D E C O M P O S I T I O N The water solution enters the reactor at a temperature of 300 K and is heated as it passes the cylinder (325 K). Figure 4 shows the temperature distribution in the reactor domain at steady state. Figure 4: A water solution enters the reactor at 300 K and is heated by a cylinder kept at 325 K. Solved with COMSOL Multiphysics 4.4 11 | T H E R M A L D E C O M P O S I T I O N At the reactor inlet, the concentration of A is 1000 mol/m 3 . Figure 5 shows the concentration A as the compound undergoes decomposition. Figure 5: Concentration of the heat sensitive chemical (A) (mol/m 3 ) as function of position in the reactor. These plots make it possible to identify some general trends. It is clear that decomposition occurs mainly after the liquid has been heated by the cylinder. In the first half of the reactor, where the temperature is relatively low, however decomposition is still fairly advanced near the wall and after the step. This is due to the longer residence times in these areas. In the second part of the reactor, where heating takes place, regions with relatively high A concentration are visible. This also makes physical sense because the water velocity is relatively high. When the model accounts for the heat of reaction, the temperature distribution in the entire reactor is affected. As shown in Figure 6, the maximum fluid temperature now Solved with COMSOL Multiphysics 4.4 12 | T H E R M A L D E C O M P O S I T I O N exceeds the temperature of the heating cylinder. Furthermore, the water temperature is higher than 300 K also in the region between the inlet and the cylinder. Figure 6: Reactor temperature (K) when the heat of reaction is taken into account. Solved with COMSOL Multiphysics 4.4 13 | T H E R M A L D E C O M P O S I T I O N Figure 7 plots the rate of reaction as a function of the position in the reactor. Clearly, significant reaction now occurs in the first part of the reactor, before the heating cylinder. Figure 7: Significant decomposition of A occurs in the first half of the reactor. Reference 1. H. Schlichting, Boundary Layer Theory, 4th ed., McGraw Hill, p. 168, 1960. Model Library path: Chemical_Reaction_Engineering_Module/ Homogeneous_Reactions_and_Catalysis/thermal_decomposition Modeling Instructions Following the steps below you will create an increasingly detailed model of a parallel plate reactor. After modeling the isothermal flow in the reactor, you will set up and solve for the nonisothermal flow case. As a final modification you will investigate the fully coupled reacting flow. Solved with COMSOL Multiphysics 4.4 14 | T H E R M A L D E C O M P O S I T I O N Start by adding a Laminar Flow interface to the model. From the File menu, choose New. N E W 1 In the New window, click the Model Wizard button. MO D E L WI Z A R D 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Stationary. 6 Click the Done button. G L O B A L D E F I N I T I O N S Before creating the geometry, define some geometry parameters that will be easy to change later on. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: G E O ME T R Y 1 Rectangle 1 (r1) 1 In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and choose Rectangle. Name Expression Value Description H1 1[cm] 0.01000 m Reactor height W1 12[cm] 0.1200 m Reactor length H2 3[mm] 0.003000 m Step height W2 3[cm] 0.03000 m Step length R1 2[mm] 0.002000 m Cylinder radius xpos 6[cm] 0.06000 m Cylinder x-coordinate ypos 5[mm] 0.005000 m Cylinder y-coordinate Solved with COMSOL Multiphysics 4.4 15 | T H E R M A L D E C O M P O S I T I O N 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type W1. 4 In the Height edit field, type H1. Rectangle 2 (r2) 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type W2. 4 In the Height edit field, type H2. Circle 1 (c1) 1 Right-click Geometry 1 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type R1. 4 Locate the Position section. In the x edit field, type xpos. 5 In the y edit field, type ypos. Difference 1 (dif1) 1 On the Geometry toolbar, click Difference. 2 Select the object r1 only to add it to the Objects to add list. 3 In the Difference settings window, locate the Difference section. 4 Select the toggle button. 5 Select the objects r2 and c1 only. 6 Click the Build Selected button. MA T E R I A L S On the Home toolbar, click Add Material. A D D MA T E R I A L 1 Go to the Add Material window. 2 In the tree, select Liquids and Gases>Liquids>Water. 3 In the Add material window, click Add to Component. Solved with COMSOL Multiphysics 4.4 16 | T H E R M A L D E C O M P O S I T I O N MA T E R I A L S Water (mat1) By default, the first material you add applies on all domains so you can keep the Geometric Scope settings. D E F I N I T I O N S In preparation for defining boundary conditions, it is practical to define some named selections. Explicit 1 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1 (comp1)>Definitions right-click Explicit 1 and choose Rename. 3 Go to the Rename Explicit dialog box and type Inlet in the New name edit field. 4 Click OK. 5 In the Explicit settings window, locate the Input Entities section. 6 From the Geometric entity level list, choose Boundary. 7 Select Boundary 1 only. Explicit 2 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1 (comp1)>Definitions right-click Explicit 2 and choose Rename. 3 Go to the Rename Explicit dialog box and type Outlet in the New name edit field. 4 Click OK. 5 In the Explicit settings window, locate the Input Entities section. 6 From the Geometric entity level list, choose Boundary. 7 Select Boundary 6 only. Explicit 3 1 On the Definitions toolbar, click Explicit. 2 In the Model Builder window, under Component 1 (comp1)>Definitions right-click Explicit 3 and choose Rename. 3 Go to the Rename Explicit dialog box and type Heater in the New name edit field. 4 Click OK. 5 In the Explicit settings window, locate the Input Entities section. Solved with COMSOL Multiphysics 4.4 17 | T H E R M A L D E C O M P O S I T I O N 6 From the Geometric entity level list, choose Boundary. 7 Select Boundaries 7–10 only. Follow the instructions below to set up the Fluid Flow interface. The Fluid Properties are automatically taken from the Material assigned to the reactor domain, so all you need to do is to define inlet and outlet boundary conditions. L A MI N A R F L OW ( S P F ) Inlet 1 1 On the Physics toolbar, click Boundaries and choose Inlet. 2 In the Inlet settings window, locate the Boundary Selection section. 3 From the Selection list, choose Inlet. 4 Locate the Velocity section. In the U 0 edit field, type 5e-4. Outlet 1 1 On the Physics toolbar, click Boundaries and choose Outlet. 2 In the Outlet settings window, locate the Boundary Selection section. 3 From the Selection list, choose Outlet. 4 Locate the Pressure Conditions section. Select the Normal flow check box. This concludes the set up of the Fluid Flow interface. In the next step you will compute the solution. A mesh is created automatically. If you wish, you can inspect the mesh by clicking the Mesh node. S T U D Y 1 On the Home toolbar, click Compute. R E S U L T S Velocity (spf) Now create a copy of the solution to the isothermal flow calculation. You will use it at a later stage in the modeling process, as a starting guess to a nonisothermal flow situation. S T U D Y 1 In the Model Builder window, expand the Study 1 node. Solver 1 In the Model Builder window, expand the Study 1>Solver Configurations node. Solved with COMSOL Multiphysics 4.4 18 | T H E R M A L D E C O M P O S I T I O N Copy 2 1 Right-click Solver 1 and choose Solution>Copy. 2 In the Model Builder window, under Study 1>Solver Configurations right-click Copy 2 and choose Rename. 3 Go to the Rename Solver dialog box and type Isothermal flow in the New name edit field. 4 Click OK. R E S U L T S Velocity (spf) 1 In the Model Builder window, expand the Results>Velocity (spf) node. 2 Right-click Velocity (spf) and choose Rename. 3 Go to the Rename 2D Plot Group dialog box and type Flow field in the New name edit field. 4 Click OK. 5 In the 2D Plot Group settings window, locate the Data section. 6 From the Data set list, choose Solution 2. Flow field 1 Right-click Velocity (spf) and choose Arrow Surface. 2 On the 2D plot group toolbar, click Plot. 3 Click the Zoom Extents button on the Graphics toolbar. At this point, move on to include a Heat Transfer interface and extend the model to account for a nonisothermal flow situation. C O MP O N E N T 1 ( C O MP 1 ) On the Home toolbar, click Add Physics. A D D P HY S I C S 1 Go to the Add Physics window. 2 In the Add physics tree, select Heat Transfer>Heat Transfer in Fluids (ht). 3 In the Add physics window, click Add to Component. Solved with COMSOL Multiphysics 4.4 19 | T H E R M A L D E C O M P O S I T I O N H E A T TR A N S F E R I N F L U I D S ( H T ) Heat Transfer in Fluids 1 1 In the Model Builder window, expand the Heat Transfer in Fluids (ht) node, then click Heat Transfer in Fluids 1. 2 In the Heat Transfer in Fluids settings window, locate the Model Inputs section. 3 From the u list, choose Velocity field (spf/fp1). Selecting the velocity field from the Laminar Flow interface as model input automatically couples the flow field to the Heat Transfer interface. Temperature 1 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 In the Temperature settings window, locate the Boundary Selection section. 3 From the Selection list, choose Inlet. 4 Locate the Temperature section. In the T 0 edit field, type 300[K]. Temperature 2 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 In the Temperature settings window, locate the Boundary Selection section. 3 From the Selection list, choose Heater. 4 Locate the Temperature section. In the T 0 edit field, type 325[K]. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 In the Outflow settings window, locate the Boundary Selection section. 3 From the Selection list, choose Outlet. L A MI N A R F L OW ( S P F ) Fluid Properties 1 1 In the Model Builder window, under Component 1 (comp1)>Laminar Flow (spf) click Fluid Properties 1. 2 In the Fluid Properties settings window, locate the Model Inputs section. 3 From the T list, choose Temperature (ht). The Laminar Flow interface will now take the temperature field calculated by the Heat Transfer interface as input. The flow is affected through temperature-dependent fluid properties such as density and viscosity. Solved with COMSOL Multiphysics 4.4 20 | T H E R M A L D E C O M P O S I T I O N S T U D Y 1 Solver 1 1 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1 node, then click Dependent Variables 1. 2 In the Dependent Variables settings window, locate the General section. 3 From the Defined by study step list, choose User defined. 4 Locate the Initial Values of Variables Solved For section. From the Method list, choose Solution. 5 From the Solution list, choose Isothermal flow. With this selection the solver will take the solution to the isothermal flow case as starting guess for the flow field and pressure variables. 6 On the Home toolbar, click Compute. R E S U L T S S T U D Y 1 Copy 3 1 In the Model Builder window, under Study 1>Solver Configurations right-click Solver 1 and choose Solution>Copy. Again, create a copy of the current solution for use as starting guess to a nonisothermal reacting flow case. 2 In the Model Builder window, under Study 1>Solver Configurations right-click Copy 3 and choose Rename. 3 Go to the Rename Solver dialog box and type Nonisothermal flow in the New name edit field. 4 Click OK. R E S U L T S 2D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 2D Plot Group. 2 In the Model Builder window, under Results right-click 2D Plot Group 3 and choose Surface. 3 In the Surface settings window, locate the Expression section. Solved with COMSOL Multiphysics 4.4 21 | T H E R M A L D E C O M P O S I T I O N 4 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Heat Transfer in Fluids>Temperature>Temperature (T) and double click. 5 In the Model Builder window, right-click 2D Plot Group 3 and choose Rename. 6 Go to the Rename 2D Plot Group dialog box and type Temperature in the New name edit field. 7 Click OK. 8 Click the Zoom Extents button on the Graphics toolbar. 9 Click the Zoom In button on the Graphics toolbar. Note that you can select which results to plot from the Data set list. Solution 1 is the current solution. Solution 2 corresponds to the saved data set for the isothermal flow case and Solution 3 corresponds to the saved data set for the nonisothermal flow case. You can check this mapping by clicking the Solution subnode under the Data Set node. Now move on to extend the model to include mass transport and chemical reaction. A D D P HY S I C S 1 Go to the Add Physics window. 2 In the Add physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 3 Click to expand the Dependent variables section. Locate the Dependent Variables section. In the Concentrations table, enter the following settings: 4 In the Add physics window, click Add to Component. TR A N S P O R T O F D I L U T E D S P E C I E S ( C H D S ) Convection and Diffusion 1 1 In the Model Builder window, expand the Transport of Diluted Species (chds) node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Model Inputs section. 3 From the u list, choose Velocity field (spf/fp1). This couples the flow field to the mass transport. 4 Locate the Diffusion section. In the D cA edit field, type 2e-9. cA Solved with COMSOL Multiphysics 4.4 22 | T H E R M A L D E C O M P O S I T I O N Inflow 1 1 On the Physics toolbar, click Boundaries and choose Inflow. 2 In the Inflow settings window, locate the Boundary Selection section. 3 From the Selection list, choose Inlet. 4 Locate the Concentration section. In the c 0,cA edit field, type 1000. Outflow 1 1 On the Physics toolbar, click Boundaries and choose Outflow. 2 In the Outflow settings window, locate the Boundary Selection section. 3 From the Selection list, choose Outlet. The next step is to add some expressions for the reactive transport. D E F I N I T I O N S Variables 1a 1 In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: TR A N S P O R T O F D I L U T E D S P E C I E S ( C H D S ) Reactions 1 1 On the Physics toolbar, click Domains and choose Reactions. 2 Select Domain 1 only. 3 In the Reactions settings window, locate the Reactions section. 4 In the R cA edit field, type -rate. The chemical reactions generate heat. Take this into account by adding a Heat Source node to the Heat Transfer interface. Name Expression Unit Description E 72[kJ/mol] J/mol Activation energy H 100[kJ/mol] J/mol Heat of reaction A 1e10[1/s] 1/s Frequency factor k A*exp(-E/(8.314[J/ (mol*K)]*T)) 1/s Rate factor rate k*cA mol/(m³·s) Reaction rate Solved with COMSOL Multiphysics 4.4 23 | T H E R M A L D E C O M P O S I T I O N H E A T TR A N S F E R I N F L U I D S ( H T ) Heat Source 1 1 On the Physics toolbar, click Domains and choose Heat Source. 2 Select Domain 1 only. 3 In the Heat Source settings window, locate the Heat Source section. 4 In the Q edit field, type H*rate. S T U D Y 1 Solver 1 1 In the Model Builder window, under Study 1>Solver Configurations>Solver 1 click Dependent Variables 1. 2 In the Dependent Variables settings window, locate the Initial Values of Variables Solved For section. 3 From the Solution list, choose Nonisothermal flow. With this selection the solver will take the solution to the nonisothermal flow case as starting guess for the flow field, the pressure, and the temperature variables. 4 On the Home toolbar, click Compute. R E S U L T S Temperature 1 Click the Zoom Extents button on the Graphics toolbar. 2 Click the Zoom In button on the Graphics toolbar. Create a new plot groups and generate surface plots for the concentration and the reaction rate. 2D Plot Group 4 1 On the Home toolbar, click Add Plot Group and choose 2D Plot Group. 2 In the Model Builder window, under Results right-click 2D Plot Group 4 and choose Surface. 3 In the Surface settings window, locate the Expression section. 4 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Transport of Diluted Species>Species cA>Concentration (cA) and click Plot. 5 In the Model Builder window, right-click 2D Plot Group 4 and choose Rename. Solved with COMSOL Multiphysics 4.4 24 | T H E R M A L D E C O M P O S I T I O N 6 Go to the Rename 2D Plot Group dialog box and type Concentration in the New name edit field. 7 Click OK. 2D Plot Group 5 1 On the Home toolbar, click Add Plot Group and choose 2D Plot Group. 2 In the Model Builder window, under Results right-click 2D Plot Group 5 and choose Surface. 3 In the Surface settings window, locate the Expression section. 4 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Transport of Diluted Species>Species cA>Total rate expression (chds.R_cA), click Plot.