CAIRO UNIVERSITY FACULTY OF ENGINEERING CHEMICAL ENGINEERING DEPARTMENTModeling of Newtonian and Non-Newtonian Fluid Flow in a Capillary Tube Viscometer using COMSOL Transport Phenomena I Under supervision of: Dr. Ahmed Sherif Prepared by: Ahmed Osama Elnabawy Aly Magdy Aly Esraa Elmitainy Mennah Mohamed Reda Zein 1/1/2011 ..................................................................................................................................................................................................................... 3................ 2 COMSOL Results ............................................................................................................ Problem Statement ............................................................................................................................... 1 Solving Procedure...................................Table of Contents 1.... 5 Conclusion ...... 4................ 9 List of Figures............................... 10 i ................... 2........................ e. Center line Wal l r Figure 1: A Simple Illustrative Diagram of the Capillary Tube Viscometer Case 1 . Problem Statement We consider a capillary viscometer. Of special interest is to measure the pressure drop phenomenon which affects any fluid that passes inside a capillary tube. and hence the difference between them is the pressure drop needed. how Figure 1. is well known In case of non-Newtonian fluid. a case of falling film on a wall. Hagen-Poiseuille equation is applicable) and its viscosity as well. This problem (i. where and the length of which . the capillary viscometer) is a little bit more sophisticated in geometry and represents also an application that has been long sought of. This problem is the development of a simpler case that we have thought of. The treatment covered by the next few pages shows how the pressure could be found at the entrance and the exit of the capillary tube. The case in hand makes use of the Hagen-Poiseuille equation to check the consistency of the COMSOL model because we’ll apply our model on water which is a Newtonian fluid (i. It is required to generate the velocity profile across the flow of the fluid in the changes with as shown in capillary.1. that is. one needs to use more general equation than Hagen-Poiseuille in order to calculate the pressure drop . the radius of capillary of which is is .e. which is a Newtonian fluid. and applying it to water. in this window it was checked that the model used is the power law according to the following relation: ( Where: : is the shear stress at the pipe wall. dimensionless. the model space dimension was defined as Axial Symmetry (2D) to be able to model half the capillary tube around an axis of symmetry. Also as water is a fully defined fluid it allows the user to compare the pressure drop from the model to the pressure drop resulting from the HagenPoiseuille equation. Where m=µ = viscosity of fluid for Newtonian fluids n: flow behavior index. The laminar model is logical since the capillary tube is very thin so there is no chance of turbulence within the system. Pa-sn. As the tube is very small drawing it directly will not give accurate dimensions because the COMSOL scale is in meters and the model is in millimeters. s-n (in a direction perpendicular to the shear stress) According to the equation above.2. The next step was to define the Sub-domain settings. Pa m: flow consistency index. Solving Procedure During this model it is assumed that the flowing fluid is water. The values of n and m entered were: n=1 and m=1e-3 Pa-s as shown in Figure 2. That was why the model was defined through specifying an object in the Draw option where the width was specified as 5e-4 m and the length as 2e-2 m. Where n= 1 for Newtonian fluids : Shear rate at the pipe wall. which is the fluid mostly used to calibrate viscometers before being used. also the non Newtonian model is assumed to be able to use this program later for any type of fluid (Newtonian or non Newtonian) without needing to specify the whole model again.5 mm radius ( ) and 20 mm in length ( ). The capillary viscometer is considered as a very small tube with the dimensions of 0. 2 ) . The Momentum transfer model used is a steady state analysis for a non Newtonian fluid flowing in a laminar way. Therefore. 2is the outlet with a pressure of 0 Pa (used as a reference so the inlet pressure will be the pressure drop).Figure 2: Power Law Parameters Definition The last step before solving was to define the boundaries of the tube.02 m and =2 s to be satisfy the human reaction in measuring time) and 4 is a wall with no slip conditions. The model shape just before solving is shown in Figure 3.01 m/s in the negative z-direction (calculated through the equation where =0. 3 is the inlet with a z-velocity of 0. Boundary 1 is the axis of symmetry. 3 . Figure 3: Capillary Tube Viscometer Case before Solving The model is now ready to solve after refining the mesh to contain 960 elements. The results obtained are displayed in the next chapter. 4 . COMSOL Results The first result obtained is the surface velocity profile over the whole model.3. with the maximum velocity at =0 and a zero velocity at =5e-4 as shown in Figure 5 Figure 5: General Velocity Profile 5 . as expected. Figure 4: Surface Velocity Profile The general velocity profile at a point in the middle of the tube shows a parabolic shape. which shows a maximum velocity at the axis of symmetry and zero velocity at the wall of the tube as shown in Figure 4. In which the z-inlet and outlet were given values inside the domain but very near the end. the values used were as follows z-inlet = 0.The pressure change over the whole body can be seen in the surface plot from Figure 6 Figure 6: Surface Pressure Plot To get specific plots at certain cross sections the Cross-section plot parameters from the post processing tab was used. Figure 7: Cross-Section Plot Parameters to Get Pressure Profile at Inlet 6 .outlet=1e-6. Therefore.019995 and the z. An example of the data entry is shown in Figure 7 where the pressure profile at the inlet. Figure 9: Outlet Pressure Profile 7 . Figure 8: Inlet and Outlet Velocity Profiles The outlet pressure profile shows a constant pressure of a very small value (3e-4 Pa) which is near enough to the zero value specified and the profile shape is as seen in Figure 9.At the inlet and outlet a velocity profile was obtained for both on the same plot as shown in Figure 8 which shows a constant inlet velocity and a parabolic outlet velocity as expected. Figure 10: Inlet Pressure Profile This pressure drop is then compared to the one obtained from Hagen.The inlet pressure profile is displayed in Figure 10 and the pressure drop can be obtained from it directly (as the outlet pressure is set to zero) by getting the average of the profile.Poiseuille equation as discussed in the next chapter. 8 . Conclusion After getting ΔP using COMSOL. : Average velocity =0.s. µ: viscosity of water =0. This proves the accuracy of this model for calibrating the viscometer and checking the pressure drop of different non Newtonian fluids.001 Pa.02 m.01 m/s. : Length of the tube= 0. 9 .0005 m. We can investigate the accuracy of the result by comparing it with that of Hagan-Poiseuille equation. Pa Which is approximately equals to that found by COMSOL with an error 2.4. : Radius of the tube =0.8%. ....................... 1 Figure 2: Power Law Parameters Definition ......................... 6 Figure 8: Inlet and Outlet Velocity Profiles .................................................................................................... 5 Figure 5: General Velocity Profile ................................................ 3 Figure 3: Capillary Tube Viscometer Case before Solving ................... 6 Figure 7: Cross-Section Plot Parameters to Get Pressure Profile at Inlet ............... 7 Figure 9: Outlet Pressure Profile................................................................................................................................... 7 Figure 10: Inlet Pressure Profile ............................................. 4 Figure 4: Surface Velocity Profile .............................................................................................................................................................................................................................................................. 8 10 ............................................................................................................ 5 Figure 6: Surface Pressure Plot ..... List of Figures Figure 1: A Simple Illustrative Diagram of the Capillary Tube Viscometer Case ...................