Chapter 2: Numerical MethodsPrepared by: Lim Ying Pei FKK, UiTM Shah Alam PART I: ITERATIVE CONVERGENCE METHODS Introduction Digital simulation is a powerful tool for solving the equations describing chemical engineering systems. One of the most common problems in digital simulation is the solution of simultaneous nonlinear algebraic equations. If these equations contain transcendental functions, analytical solutions are impossible. Therefore, an iterative trial-and error procedure of some sort must be devised. If there is only one unknown, a value for the solution is guessed. It is plugged into the equation or equations to see if it satisfies them. If not, a new guess is made and the whole process is repeated until the iteration converges (we hope) to the right value. the series of new guesses will oscillate around the correct solution with ever-increasing deviations. There are a host of techniques. . This is one kind of numerical instability. We will discuss only a few of the simplest and most useful methods. Some methods that converge very rapidly for some equations will diverge for other equations.. Unfortunately there is no best method for all equations.e. i.Introduction The key problem is to find a method for making the new guess that converges rapidly to the correct answer. Method 1: Newton-Raphson Newton-Raphson method is using the slope of the function curve to extrapolate to the correct value. . . a spherical tank is used to store liquid raw material. Given that the radius of the tank is 3 m. Take initial depth to be 3 m (ho) and state the calculated values correct to 4 decimal places and with tolerance error less than 1x10-4 . and R is the tank radius (m). h is the depth of the water in the tank (m). The capacity of the liquid in the tank is described by the following equation: V h 2 3R h 3 where V is the volume (m3). determine the depth of the tank when the volume is 30 m3.Example: In a small food industry. Using the NR method. Method 2: False Position (Secant Method) . .B: False Position Combination of Newton-Raphson and interval halving. Needs two guess values to start the iteration. Considered stable than Secant iterative method. . April 2010 .Example: Refer to handout Question 5 Final Exam. liquid Y is pumped into the tank at 160 kg/hr and pure orange juice is being added at 30 kg/hr. Final Exam. Given that the density of liquid Y vapor = 0. Concentration of the outlet solution is the same as that within the tank. The resulting orange juice solution (pure orange juice + Y liquid) is leaving the tank at 120 kg/hr. At the start of operation (t = 0).796 kg/m3. liquid Y is evaporating from the tank at a velocity of 1m/hr. . Because of faulty design work. Assume that the rates and of input and output of the tank remain constant at the start of the operation.Tutorial (Q4. Oct 2010) A well-stirred cylindrical tank with radius 2 m contains 100 kg of liquid Y. 6t 3 15 marks b) By using False Position method. Start the initial guess value as 0. determine the time when the mass fraction of orange juice in the tank is 0.1. State your calculation values to four decimal places and tolerance less than 10-4 16 marks .5 hr and 1 hr for the time.a) Prove that the mathematical equation to model the mass fraction of orange juice (w) in the tank as a function of time (t) is given by 1 1 w 1 6 1 0. Method 3: Wegstein You need two sets of guess values and the other two values will be calculated from given function. . Example Refer to handout Question 6 Final Exam. Oct 2007 . PART II: NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS . 1: EULER ALGORITHM . k3=2.2 min starting at t = 0min to t = 0.0 min-1 and k2= 0 min-1. CBO = 0 mol/l and Cco = 0 mol/l a) Derive the component equation (rate of formation) for each component.6min by using explicit Euler method. Δt = 0. (State your calculation value to 4 decimal places) .0 min-1 and k4= 3 min-1 with initial concentration of CAO = 1 mol/l. k1=1.Consider a batch reactor which has a combination of consecutive and reversible chemical reactions as the follow : K1 K3 A B C K2 K4 Given that the kinetic rate constants and the initial concentrations of the three components are as follows. b) Simulate the concentration profile of this reaction system with step size. 2000 0 0 0.ANSWER t dCA/dt CA dCB/dt CB dCC/dt CC 0 -1.2800 0.3440 0.4 -0.0000 1.3200 0.6400 0.0000 0.3200 0.0000 0.0000 0 0 0 0.8000 1.4000 0.8000 0.2 -1.0800 0.6400 0.1440 .4000 0.0000 1.6 -0.5120 0. However at tank 2. component A reacts with component C irreversibly to produce D as shown in equation 1. k1 A+C D ………(1) (a) Develop the dynamic model for volume hold up inside the tank 1 and tank 2. (4 marks) .Tutorial (April 2010) A mixing tank as shown in Figure 2 is used to mix component A and B. All liquid in the system have the same density and both tanks are perfectly mixed. The blending process outlet mixture from tank 1 flows to tank 2 where it was mixed with component C. Component A and B were recycled back to tank 1 and to retain component C and D in tank 2. a membrane was installed. (11 marks) . F1 F2 F5 C Ao C Bo C Co F6 C A2 F3 C A6 C A1 V2 C A3 CB 6 V1 CC 6 CB3 CD 6 F4 C A4 Membrane Tank 1 CB 4 Tank 2 Figure 2 A series of mixing tank (b) Develop the dynamic concentration models of component A in Tank 1 and Tank 2. Table 1 shows the initial parameters before valves failure.(c) Due to valves failure occurred during the process. F3 and F4.2 mole/ m3 V1 150 m3 13 marks . simulate the dynamic volume hold up and concentration of component A in tank 1 from a period of t = 0 to 3 min with a step change of 1 min. Table 1 Initial parameters for the mixing process Parameter Value F1 100 m3/min F2 25 m3/min F3 123 m3/min F4 35 m3/min CAo 5 mole/m3 CA1 3 mole/ m3 CA4 1. By using Euler method. there is no flow at streams. Your calculated values must be correct to four decimal places. 9375 0.01488 .8333 1 275 125 3.07577 2 400 125 3.8333 0.02841 3 525 125 3.ANSWER t (min) V1 (m3) dV1/dt CA1(mol/m3) dCA1/dt (m3/min) (mol/min) 0 150 125 3 0.9091 0. Assume the rates of input and output of the tank remain constant at the start of the operation and the contents of the tank are well mixed all times. .025 m3/hr. the level of the tank is kept constant with a total volume of 1 m3 of inlet solution.6 m3/hr and the resulting solution is leaving the tank at the same flow rate. a) Develop a total continuity equation for the well stirred tank b) Determine the time required for the tank to run dry if outlet pump is turned on. Because of faulty design work. c) Determine the salt concentration in the tank as a function of time for every hour until it reaches 4 hours using Euler method. At the start of the operation. water is evaporating from the tank at a rate of 0. Tutorial-Final Exam Oct 2007 Seawater with concentration of 8000 g/m3 is pumped into a well stirred tank at a rate of 0. 8813 4 8338.7071 3 8323.ANSWER t (hr) Cs (g/m3) dCs/dt 0 8000 200 1 8200 87.7873 .1795 2 8287.8860 14.1795 36.7679 5. Runge Kutta 4th Order . Y and Z to describe the above system. (19 marks) .2min with step size.1 (with 4 decimal places). Δt = 0. The reactions are as follow. (6 marks) b) Use Runge-Kutta 4th method to simulate the concentration profile of component X. Cyo = 0 mol/l and Czo = 0 mol/l X k1 Y k2 Z a) Derive the total continuity equation and component continuity equations for X.5 min-1 and k2=3min-1 with initial Concentration of Cxo = 3 mol/l.Tutorial-Final Exam. Given that the values of k1=1.Y and Z in the reactor at t = 0 min to t = 0. Oct 2007 First order consecutive reaction take place in a batch reactor. 2 1.2202 0.9133 0.72445 0.36223 0.59325 1.1865 .1 1.ANSWER t (min) CX (mol/L) CY (mol/L) CZ (mol/L) 0 3 0 0 0. 47390 0.18037 Z 0. ANSWER Time Compon K1 K2 K3 K4 (min) ent 0.2224 0.84843 Y 0.34875 0.28281 Z 0.2 X -0.71085 -0.9 0.44484 0.2870 0.04625 -1.1146 -0.8610 -0.36074 .23695 0.57399 0.56562 0.37153 0.35 -1.6975 0.74306 0.45 0.1 X -1.5411 Y 0.66726 -0. April 2007 The following reaction is taking place in an isothermal batch reactor K1 K2 A B C where B and C are the intermediate and final products. The rate constants. respectively. a)Derive the component continuity equation(s) for the reactor b)Given the step size as 0. CB = 0 and CC = 0. respectively and when t = 0s.m-3.Tutorial-Final Exam.5s. k1 and k2 are given as 3s-1 and 1s-1. solve the equations(s) using Runge-Kutta 4th order from t = 0s to t = 1s . CA = 1 mol. 2734 0.2266 1.0748 0.5 0.0 0.5 0.4401 0.ANSWER t (s) CA (mol/m3) CB(mol/m3) CC (mol/m3) 0 1 0 0 0.4851 . 2188 0.375 -1.5 0 1.3332 0.1025 -0.1875 0.375 0 0.2031 0.3282 B 1.0897 B 0.1301 -0.40475 C 0.25 0. ANSWER Time (s) Component K1 K2 K3 K4 0.2900 0.9376 C 0 0.31505 .5 A -1.2188 -0.0 A -0.4101 -0.6094 1.5 -0.1601 -0.