Mass Transfer in a Stirred Batch Reactor

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Milton Reynolds
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1 Mass Transfer in a Stirred Batch Reactor In many processes, efficient reactor usage goes hand in hand with efficient mixing. The ability to accurately examine the effects of impeller placement, speed, or shape can lead to increased conversion and space velocity as well as improved safety. One of many example where this can be important is in enzyme chemistry. Enzymes are used as catalysts not only for industrial processes such as manufacturing and waste treatment, but also in the production of pharmaceuticals and biotechnical research. One problem when working with enzymes is their sensitivity to fluid properties and mechanical stress. Because a change in temperature or ph can lead to geometric changes of the molecules, efficient stirring is needed but without stressing the molecules too much. This makes agitation interesting to study. An example of this can be found in ethanol production. The added enzymes mix with a cellulose solution under agitation and break it down into glucose. This is currently one of the key reactions in the effort to make ethanol a cheap transportation fuel. This example simulates a stirred research reactor using the Rotational Machinery feature of the Chemical Engineering Module. Using this application mode, this study first sets up a rotating geometry that represents the effect of an impeller on the fluid. It then solves a momentum balance to fully describe the stirred fluid. It also solves a material balance to describe how the concentration distribution is affected by the stirring. The features of the Chemical Engineering Module used in this model are mainly: The Rotating Machinery feature that, with just a few mouse clicks, allows rotation of one geometry while keeping neighboring geometries fixed Assemblies that connect geometries User-defined variables that can be used both for modeling and plotting; in this example the concentration of enzymes is integrated in an effort to quantify the mixing COMSOL Multiphysics transient solver to examine transient problems MASS TRANSFER IN A STIRRED BATCH REACTOR 33

2 On Rotating Machinery The Rotating Machinery application mode is designed to make modeling rotating objects in fixed surroundings as easy and fast as possible. This makes modeling equipment such as pumps, stirred tanks, rotors, mixers, and fans much more accessible and straightforward. To implement this functionality, you must divide the geometry into two parts: one that rotates and one that does not. The predefined application mode then takes care of the formulation of the momentum balance in the rotating coordinate system and linking it to the momentum balance in the fixed coordinate system. The boundary conditions for the affected equations are available as groups in their respective application modes. The rotational speed must also be defined for time-dependent problems. On Assemblies This model uses COMSOL Multiphysics assembly functionality. This advanced feature lets the software represent the geometry in a fashion different than normally done. In this way users can connect physics from different geometries to each other, geometries that are not part of the same geometry object. This allows for meshing two sides of the same boundary differently, and it also open the possibility for special boundary conditions. The connections between the geometries are called identity pairs. This model uses that feature to let the geometry on one side of a boundary have a rotating coordinate system while the other has a fixed coordinate system. Problem Definition A cellulose and water suspension is stirred in a batch reactor when a solution of enzymes is added though a pipe. Orifices in the pipe are opened for 1 sec and the enzyme solution flows in. The volume of fluid entering the reactor is small compared to the reactor volume. The enzymes are also very sensitive to fluid conditions and mechanical stress. Thus a quantification of the mixing rate as a function of time is desirable. This example calculates the concentration of enzymes, and it then determines the mixing efficiency for the proposed reactor configuration. 34 CHAPTER : COMSOL MULTIPHYSICS: CHEMICAL ENGINEERING MODULE MINICOURSE

Model Definition This model demonstrates how to model a stirred reactor with a momentum and mass transport balance. The reactor is represented by a 2D geometry as seen in Figure 2.

The reduction from 3D to 2D also saves considerable computer resources and time. Figure 2: 2D representation of the tank along with a 3D representation.

3 Model Definition This model demonstrates how to model a stirred reactor with a momentum and mass transport balance. The reactor is represented by a 2D geometry as seen in Figure 2. A full 3D representation of this geometry (also in Figure 2) is not needed because the impeller goes from the top to the bottom of the reactor, so the reactor is invariant in the z-direction. The reduction from 3D to 2D also saves considerable computer resources and time. Figure 2: 2D representation of the tank along with a 3D representation. The model is solved in 2D since the tank can be assumed to be invariant in the z-direction. The batch reactor is filled with cellulose in a water suspension. Cellulose molecules are long chains of mono-saccharides, and the fluid is therefore quite viscous. Since the batch is full at t = 0 and the enzyme solution is only needed in a small volume it can be assumed that no inlet is affecting the flow. The flow is also well inside laminar limits. The behavior of the fluid is described by Navier-Stokes equations in the transient form: u T ρ η ( u + ( u) ) + ρu u + p = 0 t u = 0 where η denotes the dynamic viscosity (Pa s), u is the velocity vector (m/s), ρ gives the fluid s density (kg/m 3 ), and p is the pressure (Pa). This system is made up of three equations where the first two are the velocity in the x and y directions and the last one is the continuity equation that ensures the conservation of mass. When this system is solved, the flow velocity and the pressure are known in the entire domain. On the boundaries, No Slip conditions are imposed on the reactor walls: u = 0. MASS TRANSFER IN A STIRRED BATCH REACTOR 35

4 This is consistent with no inflow and outflow from the batch reactor, which is expected. Momentum is added to the fluid at the rotating impeller boundaries. The boundary conditions on the impeller represent No Slip conditions on the rotational domain, but to describe this state the model uses inflow/outflow conditions that match the effect of the rotating No Slip boundary. This way the momentum that causes the swirl is imposed on the system. The boundary conditions that COMSOL Multiphysics uses for clockwise rotation are: u = π n sin( π n t 30) ( X 30) + π n cos( π n t 30) ( Y 30) v = π n cos( ( π n t 30) ( X 30) π n sin( π n t 30) ( Y 30) ) where n is the revolution of the impeller (rpm), t is time (s), while X and Y are the independent coordinates in the fixed mesh (m). The pressure is set to 0 by the initial conditions and no further constrain is needed. In some cases it can be good to fix the pressure either in a point or in a boundary condition. This way some convergence issues can be avoided. This is not necessary in this model however. In many real conditions you can describe mass transport with Fick s law. This is certainly true in this case. The enzymes are injected into the cellulose mixture through a tube that goes all along the reactor wall in the z-direction. The volumes of enzymes is small compared to the batch because only a small volume is needed due to the catalytic nature of enzymes. This amount does not affect the flow. Fick s law is implemented in the Convection and Diffusion application mode. The mass transport equation is formulated in the transient form: c t + ( D c + cu) = 0 where c denotes the concentration of species i (mol/m 3 ), D denotes its diffusion coefficient (m 2 /s), and u denotes the velocity vector (m/s). The velocity in this equation is not unknown but is taken from the momentum balance just given. As boundary conditions, all the batch walls except the inlet have no-flux conditions, also called insulation/symmetry conditions: n A n = 0. At the impeller, the convection follows the flow so the impeller boundaries are set to convective flux: 36 CHAPTER : COMSOL MULTIPHYSICS: CHEMICAL ENGINEERING MODULE MINICOURSE

5 n ( D c) = 0 The inlet sends a 1-sec flow of enzymes into the tank, and the boundary is therefore set to flux. After that short time the flux drop to zero and then this boundary behaves just like the wall conditions. The boundary condition is formulated as n A n = 0.65 ( t < 1). Solving this equation gives the concentration of the enzyme in the entire domain at each solved time step. Results and Discussion The Figure 3 below displays the velocity field of the agitated fluid after 4 seconds. Arrows have also been added to further show the flow. It is clear from the length of the arrows that fluid moves slowly near the walls. Stagnant areas can also be found this way. Figure 3: The velocity field in the Batch after 4 seconds. How the distribution of enzymes vary with time can be seen in Figure 4, Figure 5 and Figure 6. After 4 seconds the enzymes are rather well distributed and the lowest MASS TRANSFER IN A STIRRED BATCH REACTOR 37

concentration in the batch is 0.221 mole/m 3. Note that the scales are different in the pictures, which can make the solution look less mixed than it really is.

6 concentration in the batch is mole/m 3. Note that the scales are different in the pictures, which can make the solution look less mixed than it really is. Figure 4: Concentration distribution after 0 and 1 seconds. Figure 5: Concentration distribution after 2 and 3 seconds. 38 CHAPTER : COMSOL MULTIPHYSICS: CHEMICAL ENGINEERING MODULE MINICOURSE

7 Figure 6: Concentration distribution after 4 seconds. Not that the full color scale is used which can make the enzymes less distributed than they are. Minimum concentration is moles/m 3 To quantify the mixing the difference of the actual concentration and the mean concentration is integrated over the subdomain. This can be expressed as: MixingValue = ( c c mean ) 2 dω MixingValue then gives an idea how the mixing varies with time, which is showed in Figure 7. This gives a relative value of how well the solution is mixing as well as how far from ideal the reactor is. MASS TRANSFER IN A STIRRED BATCH REACTOR 39

8 Figure 7: MixingValue as a function of time. MixingValue is the difference of the actual concentration and the average concentration integrated over the domain. The increase the first second is due to the inlet of enzymes. Modeling Using the Graphical User Interface 1 Double-click the COMSOL Multiphysics icon on the desktop to open the Model Navigator. 2 Go to the New page and set the Space dimension to 2D. 3 In the list of application modes select Chemical Engineering Module>Rotating Machinery>Rotating Navier Stokes>Transient analysis. 4 Click the Multiphysics button, then add the selected application mode by clicking the Add button. 5 In the list of application modes mark Chemical Engineering Module>Mass Balance>Convection and Diffusion>Transient analysis. 6 Click Add. 7 Click OK. 40 CHAPTER : COMSOL MULTIPHYSICS: CHEMICAL ENGINEERING MODULE MINICOURSE

9 GEOMETRY MODELING 1 Hold the Shift key and click the Ellipse/Circle (Centered) button on the Draw toolbar. 2 Set the Radius to 0.06 and click OK. This draws the tank walls. 3 To specify the rotating domain, again hold the Shift key and click the Ellipse/Circle (Centered) button on the Draw toolbar. 4 Set the Radius to 0.04 and click OK. 5 Click the Zoom Extents button on the Main toolbar. 6 You will now remove any overlap between the large and the small circle: Press Ctrl+A to select both objects. 7 Click the Union button. 8 To separate the circles into two non-overlapping geometries, click the Split Object button on the Draw toolbar. 9 To draw the impeller, hold the Shift key and click the Line button on the Draw toolbar. Then enter these coordinates: COORDINATES x y Set the Style drop-down menu to Closed Polyline (solid) and click OK. 11 To draw the inlet tube, select Options>Axis/Grid settings, then select the Grid page. 12 Clear the Auto check box and set both the X spacing and Y spacing to Click OK. 13 Draw the inlet by clicking the Line button on the Draw toolbar and, using the mouse, draw a line through the following coordinates:. POINT NO. X Y 1 5e e e e Right-click the end point in the polyline. All these points are on the grid, and you can see the shape of the pipe in Figure 8. MASS TRANSFER IN A STIRRED BATCH REACTOR 41

10 Figure 8: A cross section plot of the inlet pipe. 14 You will not model the interior of the tube or the impeller, so you can remove them from the geometry. Click the Create Composite Object button on the Draw toolbar and write CO3 CO1 in the Formula edit field. 15 Click the Apply button. 16 Now write CO2 CO4 in the Formula edit field and again click the Apply button. Doing so removes the impeller from the inner circle and the inlet from the outer circle. 17 Click OK. USING ASSEMBLIES 1 Select Draw>Use Assembly. This will turn on the assembly function. 2 Select Draw>Create Pairs to open the Create Pairs dialog box. 3 Select both geometries in the Object Selection list and click OK. You have now created an identity pair that connect the physics in the different geometries while still letting the coordinate systems differ. 42 CHAPTER : COMSOL MULTIPHYSICS: CHEMICAL ENGINEERING MODULE MINICOURSE

11 OPTIONS AND SETTINGS Open the Constants dialog box from the Options menu and enter the following names and settings; when done, click OK: PROPERTY EXPRESSION DESCRIPTION rpm 20 Revolution speed rho 920[kg/m^3] Density (kg/m^3) eta 1e-2[Pa*s] Viscosity (Pa*s) diff 1e-3[m^2/s] Diffusion (m^2/s) PHYSICS SETTINGS Select Moving Mesh (ale) (ALE) from the Model Tree view.that is the left-most window in the graphical user interface. You can also choose this from the Multiphysics menu. Subdomain Settings: Moving Mesh (ale) Application Mode 1 Select Subdomain Settings from the Physics menu. 2 Select the inner geometry with the impeller (subdomain number 2). 3 In the Group list choose Rotate_CW and note how the displacement expression changes. This sets the rotation of the inner circle to clockwise rotation with the speed rpm. 4 Now select the outer geometry (subdomain number 1) and choose Fixed in the Group drop down list. 5 Click OK. Subdomain Settings: Incompressible Navier Stokes Application Mode 1 From the Model Tree view select the Incompressible Navier Stokes application mode. 2 Select Subdomain settings from the Physics menu. 3 In the Subdomain selection list select 1 and 2. 4 Type rho in the Density edit field and eta in the Viscosity edit field. 5 Click OK. Boundary Conditions: Incompressible Navier Stokes Application Mode 1 Open the Boundary Settings dialog box from the Physics menu. 2 In the Boundary selection list select Do so either from the list or by clicking and dragging a box over the impeller. 3 From the Group list choose No_slip_CW. The boundary conditions are now set to No Slip conditions but in the rotational system. MASS TRANSFER IN A STIRRED BATCH REACTOR 43

12 4 Click OK. Subdomain Settings: Convection and Diffusion Application Mode Now configure the Convection and Diffusion application mode to the model. This application adds the concentration c as an unknown and the equation to solve for it. The new application mode, Convection and Diffusion (chcc), should be visible in the Model Tree on the left side of the screen. Select that application mode by clicking on it. 1 Select the application mode Convection and Diffusion (chcd) in the Model Tree on the left-hand side of the user interface. 2 Select Subdomain Settings from the Physics menu. 3 In the Subdomain selection list select 1 and 2. 4 Enter diff in the D isotropic edit field. 5 To use the flow field, enter u in the u edit field and v in the v edit field. 6 Click OK. Boundary Conditions: Convection and Diffusion Application Mode 1 Open the Boundary Settings dialog box from the Physics menu. 2 Select Boundary 2, which is the flat surface of the inlet pipe. 3 In the Boundary Condition list select Flux. 4 Enter 0.65*(t<1) in the N 0 edit field. 5 Mark Boundaries in the Boundary Selection list or drag a box over the impeller geometry. 6 Set the Boundary condition to Convective Flux. By default all the other boundaries are set to lsolation Symmetry, which is correct. 7 Click OK. COMPUTING THE SOLUTION Generate the mesh, set the solver settings, and solve for the flow field 1 Click the Initialize Mesh button on the Main toolbar. 2 Click the Solver Parameters button on the Main toolbar to open the Solver Parameters dialog box. 3 In the Times edit field enter 0:0.1:4. 4 Click OK. 5 Click the Solve button on the Main toolbar to calculate the flow filed. This takes approximately 4 minutes. 44 CHAPTER : COMSOL MULTIPHYSICS: CHEMICAL ENGINEERING MODULE MINICOURSE

13 POSTPROCESSING AND VISUALIZATION OF THE FLOW FIELD To create Figure 3 perform these steps: 1 From the Postprocessing menu open the Plot parameters dialog box. 2 On the General page select the plot types Surface and Arrow. 3 On the Surface page set the Surface data to Velocity field. 4 On the Arrow page set the Arrow data to Velocity field. 5 Click OK. To create the plots in Figure 4 to Figure 6 perform these steps: 1 On the General page select 0 from the Solution at time list. 2 On the Surface page set the Surface data to Concentration, c. 3 Click Apply. 4 Repeat this procedure for times 1, 2, 3, and 4. To create the plots in Figure 7perform these steps: 1 Select Options>Integration Coupling Variables>Subdomain Variables. 2 In the Subdomain selection list select 1 and 2. 3 In the Name edit field enter MixingValue. 4 In the Expression field enter (c-0.275)^2. This creates an integration of that quantified how far the concentration is from the average value Click OK. 6 Select Solve>Update Model to perform the integration. 7 Choose Postprocessing>Domain Plot Parameters. 8 Select the Point tab and click on any point. 9 Enter MixingValue the Expression edit field. 10 Click Apply. MASS TRANSFER IN A STIRRED BATCH REACTOR 45

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