Porous Reactor with Injection Needle

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Porous Reactor with Injection Needle Introduction This model treats the flow field and species distribution in an experimental reactor for studies of heterogeneous catalysis.

1 Porous Reactor with Injection Needle Introduction This model treats the flow field and species distribution in an experimental reactor for studies of heterogeneous catalysis. The model exemplifies the coupling of free and porous media flow in fixed bed reactors. The model was inspired by numerical experiments performed by Professor Finlayson s graduate students in chemical engineering at the University of Washington in Seattle. Model Definition The reactor consists of a tubular structure with an injection tube whose main axis is perpendicular to the reactor axis. The incoming species in the main and injection tubes react in a fixed porous catalyst bed. The model couples the free fluid and porous media flow through the Navier-Stokes equations and Brinkman s extension of Darcy s law. Because of symmetry, you need only model one half of the reactor; see Figure 1. Inlet species A Outlet A, B, C Production of C in porous bed Inlet species B Figure 1: The species A and B enter the reactor from the main and injection tubes, respectively, and react in a fixed porous catalyst bed to produce C. In the porous bed, a reaction takes place that consumes species A and B and produces C: A+ B C (1) 2011 COMSOL 1 POROUS REACTOR WITH INJECTION NEEDLE

2 DOMAIN EQUATIONS The stationary Navier-Stokes equations describe the fluid flow in the free-flow regions: u + u T + pi u = 0 = u u (2) In the porous bed, use the Brinkman equations: ---- u + u T + pi p = -- u k (3) u = 0 In the above equations, denotes the viscosity of the fluid (Ns/m 2 ), p is the porosity (dimensionless), u the velocity (m/s), the density (kg/m 3 ), p the pressure (Pa), and the permeability (m 2 ). Assume that the modeled species are present in low concentrations compared to the solvent gas. This means that you can use a Fickian approach for the diffusion term in the mass transport. Model the mass transport for the three species A, B, and C with the convection-diffusion equation D i c i + c i u = R i (4) In this equation, c i denotes the concentration (mol/m 3 ), D i the diffusivity (m 2 /s), and R i the reaction rate for species i (mol/(m 3 s)). Because the reaction takes place in the porous bed only, the reaction term is zero in the free-flow regions. The reaction rates are given by R A R B R C = = = kca c B kca c B kc A c B (5) where k is the reaction rate constant. BOUNDARY CONDITIONS A constant velocity profile is assumed at the inlet boundaries: u = u in (6) The boundary condition for the Navier-Stokes equations at the outlet reads 2 POROUS REACTOR WITH INJECTION NEEDLE 2011 COMSOL

3 t u = 0 p = 0 (7) where t is any tangential vector to the boundary. In the mass transport, the concentrations at the inlet are fixed: c i = c i0 inlet (8) At the outlet, assume that convection dominates the mass transport: n D i c i + c i u = n c i u (9) This implies that the gradient of c i in the direction perpendicular to the outlet boundary is negligible. This is a common assumption for tubular reactors with a high degree of transport by convection in the direction of the main reactor axis. The condition eliminates the need for specifying a concentration or a fixed value for the flux at the outlet boundary. At all other boundaries, insulating conditions apply: n D i c i + c i u = 0 (10) Results Figure 2 shows the modulus of the velocity vector. The flow is almost homogeneous in the porous part of the reactor COMSOL 3 POROUS REACTOR WITH INJECTION NEEDLE

4 Figure 2: Magnitude of the velocity field in the free and porous reactor domains. Figure 3 shows the pressure drop, which occurs mainly across the porous bed. Figure 3: The pressure drop across the reactor. 4 POROUS REACTOR WITH INJECTION NEEDLE 2011 COMSOL

5 Figure 4 and Figure 5 show the concentrations of species A and B, respectively. Figure 4: Iso-concentration surfaces for species A. Figure 5: Iso-concentration surfaces for species B COMSOL 5 POROUS REACTOR WITH INJECTION NEEDLE

6 Figure 4 shows that the concentration of the injected species A decays very rapidly with the distance from the injection point. This implies that the porous bed is not optimally used. Figure 5 shows the iso-concentration surfaces of species B, which is introduced in the main channel of the reactor. The reaction is not uniformly distributed in the catalytic bed and has its maximum close to the radial position of the injection pipe. The low concentration of B at the injection point is due to dilution by the solvent that carries species A. In summary, the model shows that the injection point is far too close to the porous bed. The reactants are not well mixed and only a fraction of the bed is utilized. A proper design should include a small static mixer after the injection point or a positioning of the injection point further upstream in order to obtain mixing through diffusion. The following path shows the location of the COMSOL Multiphysics model: Model Library path: Chemical_Reaction_Engineering_Module/ Packed_Bed_Reactors/porous_reactor Modeling Instructions MODEL WIZARD 1 Go to the Model Wizard window. 2 Click Next. 3 In the Add physics tree, select Chemical Species Transport>Reacting Flow, Diluted Species (rfds). 4 Click Add Selected. 5 In the Number of species edit field, type 3. 6 In the Concentrations table, enter the following settings: c_a c_b c_c 7 Click Next. 8 In the Studies tree, select Preset Studies>Stationary. 6 POROUS REACTOR WITH INJECTION NEEDLE 2011 COMSOL

7 9 Click Finish. GEOMETRY 1 1 In the Model Builder window, click Model 1>Geometry 1. 2 Go to the Settings window for Geometry. 3 Locate the Units section. From the Length unit list, select mm. Cylinder 1 1 Right-click Model 1>Geometry 1 and choose Cylinder. 2 Go to the Settings window for Cylinder. 3 Locate the Size and Shape section. In the Radius edit field, type 1. 4 In the Height edit field, type Locate the Axis section. In the x edit field, type 1. 6 In the z edit field, type 0. 7 Click the Build Selected button. Cylinder 2 1 In the Model Builder window, right-click Geometry 1 and choose Cylinder. 2 Go to the Settings window for Cylinder. 3 Locate the Size and Shape section. In the Radius edit field, type 1. 4 In the Height edit field, type 3. 5 Locate the Position section. In the x edit field, type 5. 6 Locate the Axis section. In the x edit field, type 1. 7 In the z edit field, type 0. 8 Click the Build Selected button. Cylinder 3 1 In the Model Builder window, right-click Geometry 1 and choose Cylinder. 2 Go to the Settings window for Cylinder. 3 Locate the Size and Shape section. In the Radius edit field, type In the Height edit field, type 2. 5 Locate the Position section. In the x edit field, type 3. 6 Click the Build Selected button. Cylinder 4 1 In the Model Builder window, right-click Geometry 1 and choose Cylinder COMSOL 7 POROUS REACTOR WITH INJECTION NEEDLE

8 2 Go to the Settings window for Cylinder. 3 Locate the Size and Shape section. In the Radius edit field, type In the Height edit field, type 2. 5 Locate the Position section. In the x edit field, type 3. 6 Click the Build Selected button. Block 1 1 In the Model Builder window, right-click Geometry 1 and choose Block. 2 Go to the Settings window for Block. 3 Locate the Size and Shape section. In the Width edit field, type In the Depth edit field, type 1. 5 In the Height edit field, type 3. 6 Locate the Position section. In the y edit field, type In the z edit field, type Click the Build Selected button. Compose 1 1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Compose. 2 Select the objects cyl1 and cyl3 only. 3 Go to the Settings window for Compose. 4 Locate the Compose section. In the Set formula edit field, type cyl1-cyl3. 5 Clear the Keep interior boundaries check box. 6 Click the Build Selected button. Compose 2 1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Compose. 2 Select the objects cyl4 and co1 only. 3 Go to the Settings window for Compose. 4 Locate the Compose section. In the Set formula edit field, type co1+cyl4. 5 Clear the Keep interior boundaries check box. 6 Click the Build Selected button. 8 POROUS REACTOR WITH INJECTION NEEDLE 2011 COMSOL

Compose 3 1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Compose. 2 Select the objects cyl2 and co2 only. 3 Go to the Settings window for Compose.

9 Compose 3 1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Compose. 2 Select the objects cyl2 and co2 only. 3 Go to the Settings window for Compose. 4 Locate the Compose section. In the Set formula edit field, type co2+cyl2. 5 Click the Build Selected button. Compose 4 1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Compose. 2 Select the objects blk1 and co3 only. 3 Go to the Settings window for Compose. 4 Locate the Compose section. In the Set formula edit field, type co3-blk1. 5 Click the Build Selected button. Form Union 1 In the Model Builder window, right-click Form Union and choose Build Selected. 2 Click the Go to Default 3D View button on the Graphics toolbar COMSOL 9 POROUS REACTOR WITH INJECTION NEEDLE

10 MATERIALS 1 In the Model Builder window, right-click Model 1>Materials and choose Open Material Browser. 2 Go to the Material Browser window. 3 Locate the Materials section. In the Materials tree, select Liquids and Gases>Gases>Nitrogen. 4 Right-click and choose Add Material to Model from the menu. REACTING FLOW, DILUTED SPECIES Transport Properties 1 1 In the Model Builder window, expand the Model 1>Reacting Flow, Diluted Species node, then click Transport Properties 1. 2 Go to the Settings window for Transport Properties. 3 Locate the Model Inputs section. In the T edit field, type T_iso. 4 Locate the Diffusion section. In the D i edit field, type 1e-6[m^2/s]. 5 In the D i edit field, type 1e-6[m^2/s]. 6 In the D i edit field, type 1e-6[m^2/s]. Porous Matrix Properties 1 Porous Matrix Properties. 2 Select Domain 2 only. 3 Go to the Settings window for Porous Matrix Properties. 4 Locate the Porous Matrix Properties section. From the p list, select User defined. In the associated edit field, type From the br list, select User defined. In the associated edit field, type 1e-9[m^2]. 6 From the Effective mass transport parameters list, select Bruggeman. Reactions 1 the domain setting Species Transport>Reactions. 2 Select Domain 2 only. 3 Go to the Settings window for Reactions. 4 Locate the Reactions section. In the R ca edit field, type -k*c_a*c_b. 5 In the R cb edit field, type -k*c_a*c_b. 10 POROUS REACTOR WITH INJECTION NEEDLE 2011 COMSOL

11 6 In the R cc edit field, type k*c_a*c_b. Inlet 1 the boundary condition Fluid Flow>Inlet. 2 Select Boundary 1 only. 3 Go to the Settings window for Inlet. 4 Locate the Velocity section. In the U 0 edit field, type 2.5[cm/s]. Inlet 2 the boundary condition Fluid Flow>Inlet. 2 Select Boundary 8 only. 3 Go to the Settings window for Inlet. 4 Locate the Velocity section. In the U 0 edit field, type 2.5[cm/s]. Outlet 1 the boundary condition Fluid Flow>Outlet. 2 Select Boundary 19 only. Symmetry 1 the boundary condition Fluid Flow>Symmetry. 2 Select Boundaries 2, 12, and 16 only. Inflow 1 the boundary condition Species Transport>Inflow. 2 Select Boundary 1 only. 3 Go to the Settings window for Inflow. 4 Locate the Concentration section. In the c 0,cB edit field, type 1[mol/m^3]. Inflow 2 the boundary condition Species Transport>Inflow. 2 Select Boundary 8 only. 3 Go to the Settings window for Inflow COMSOL 11 POROUS REACTOR WITH INJECTION NEEDLE

12 4 Locate the Concentration section. In the c 0,cA edit field, type 7[mol/m^3]. Outflow 2 the boundary condition Species Transport>Outflow. 2 Select Boundary 19 only. Symmetry 2 the boundary condition Species Transport>Symmetry. 2 Select Boundaries 2, 12, and 16 only. DEFINITIONS Variables 1 1 In the Model Builder window, right-click Model 1>Definitions and choose Variables. 2 Go to the Settings window for Variables. 3 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Domain. 4 Select Domain 2 only. 5 Locate the Variables section. In the Variables table, enter the following settings: NAME EXPRESSION DESCRIPTION k A*exp(-E/R_const/T_iso) Reaction rate A 1e6[m^3/mol/s] Frequency factor E 30e3[J/mol] Activation energy GLOBAL DEFINITIONS Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters. 2 Go to the Settings window for Parameters. 3 Locate the Parameters section. In the Parameters table, enter the following settings: NAME EXPRESSION DESCRIPTION T_iso 300[K] Temperature STUDY 1 1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Build All. 12 POROUS REACTOR WITH INJECTION NEEDLE 2011 COMSOL

13 2 Right-click Study 1 and choose Compute. RESULTS 3D Plot Group 3 1 In the Model Builder window, right-click Results and choose 3D Plot Group. 2 Right-click Results>3D Plot Group 3 and choose Surface. 3 Right-click Surface 1 and choose Plot. 3D Plot Group 4 1 Right-click Results and choose 3D Plot Group. 2 In the Model Builder window, right-click Results>3D Plot Group 4 and choose Isosurface. 3 Go to the Settings window for Isosurface. 4 In the upper-right corner of the Expression section, click Replace Expression. 5 From the menu, choose Concentration (c_a). 6 Go to the Settings window for Isosurface. 7 Locate the Levels section. In the Total levels edit field, type Click the Plot button. 3D Plot Group 5 1 In the Model Builder window, right-click Results and choose 3D Plot Group. 2 Right-click Results>3D Plot Group 5 and choose Isosurface. 3 Go to the Settings window for Isosurface. 4 In the upper-right corner of the Expression section, click Replace Expression. 5 From the menu, choose Concentration (c_b). 6 Locate the Levels section. In the Total levels edit field, type Click the Plot button COMSOL 13 POROUS REACTOR WITH INJECTION NEEDLE

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Solved with COMSOL Multiphysics 4.2

Solved with COMSOL Multiphysics 4.2 Laminar Static Mixer 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