solved with comsol multiphysics 4.2 porous reactor · pdf filethe model couples the free fluid...

Solved with COMSOL Multiphysics 4.2

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Po r ou s R e a c t o r w i t h I n j e c t i o n Ne ed l e

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 Cin 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:

(1)A B C+

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D O M A I N E Q U A T I O N S

The stationary Navier-Stokes equations describe the fluid flow in the free-flow regions:

(2)

In the porous bed, use the Brinkman equations:

(3)

In the above equations, denotes the viscosity of the fluid (Ns/m2), p is the porosity (dimensionless), u the velocity (m/s), the density (kg/m3), p the pressure (Pa), and the permeability (m2).

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

(4)

In this equation, ci denotes the concentration (mol/m3), Di the diffusivity (m2 /s), and Ri the reaction rate for species i (mol/(m3·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

(5)

where k is the reaction rate constant.

B O U N D A R Y C O N D I T I O N S

A constant velocity profile is assumed at the inlet boundaries:

(6)

The boundary condition for the Navier-Stokes equations at the outlet reads

– u u T+ pI+ u u–=

u 0=

p-----– u u T+ pI+

k---u–=

u 0=

Di ci ciu+– Ri=

RA kc– AcB=

RB kc– AcB=

RC kcAcB=

u uin=

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(7)

where t is any tangential vector to the boundary.

In the mass transport, the concentrations at the inlet are fixed:

(8)

At the outlet, assume that convection dominates the mass transport:

(9)

This implies that the gradient of ci 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:

(10)

Results

Figure 2 shows the modulus of the velocity vector. The flow is almost homogeneous in the porous part of the reactor.

t u 0=

p 0=

ci ci0 inlet=

n Di ci ciu+– n ciu=

n Di ci ciu+– 0=



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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.



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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.



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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

M O D E L W I Z A R D

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:

7 Click Next.

8 In the Studies tree, select Preset Studies>Stationary.

c_A

c_B

c_C



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9 Click Finish.

G E O M E T R Y 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 11 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 10.

5 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 21 In the Model Builder window, right-click Geometry 1 and choose Cylinder.


3 Locate the Size and Shape section. In the Radius edit field, type 1.


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.




3 Locate the Size and Shape section. In the Radius edit field, type 0.4.







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3 Locate the Size and Shape section. In the Radius edit field, type 0.3.




Block 11 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 10.

4 In the Depth edit field, type 1.


6 Locate the Position section. In the y edit field, type -1.

7 In the z edit field, type -1.


Compose 11 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.



Operations>Compose.

2 Select the objects cyl4 and co1 only.


4 Locate the Compose section. In the Set formula edit field, type co1+cyl4.

5 Clear the Keep interior boundaries check box.




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Operations>Compose.

2 Select the objects cyl2 and co2 only.


4 Locate the Compose section. In the Set formula edit field, type co2+cyl2.



Operations>Compose.

2 Select the objects blk1 and co3 only.


4 Locate the Compose section. In the Set formula edit field, type co3-blk1.


Form Union1 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.



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M A T E R I A L S

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.

R E A C T I N G F L O W, D I L U T E D S P E C I E S

Transport Properties 11 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 Di edit field, type 1e-6[m^2/s].

5 In the Di edit field, type 1e-6[m^2/s].

6 In the Di edit field, type 1e-6[m^2/s].

Porous Matrix Properties 11 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

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 0.3.

5 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 11 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

the domain setting Species Transport>Reactions.


3 Go to the Settings window for Reactions.

4 Locate the Reactions section. In the RcA edit field, type -k*c_A*c_B.

5 In the RcB edit field, type -k*c_A*c_B.

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6 In the RcC edit field, type k*c_A*c_B.

Inlet 11 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

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 U0 edit field, type 2.5[cm/s].

Inlet 21 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

the boundary condition Fluid Flow>Inlet.


3 Go to the Settings window for Inlet.

4 Locate the Velocity section. In the U0 edit field, type 2.5[cm/s].

Outlet 11 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

the boundary condition Fluid Flow>Outlet.


Symmetry 11 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

the boundary condition Fluid Flow>Symmetry.

2 Select Boundaries 2, 12, and 16 only.

Inflow 11 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

the boundary condition Species Transport>Inflow.


3 Go to the Settings window for Inflow.

4 Locate the Concentration section. In the c0,cB edit field, type 1[mol/m^3].

Inflow 21 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

the boundary condition Species Transport>Inflow.


3 Go to the Settings window for Inflow.



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4 Locate the Concentration section. In the c0,cA edit field, type 7[mol/m^3].

Outflow 21 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

the boundary condition Species Transport>Outflow.


Symmetry 21 In the Model Builder window, right-click Reacting Flow, Diluted Species and choose

the boundary condition Species Transport>Symmetry.

2 Select Boundaries 2, 12, and 16 only.

D E F I N I T I O N S

Variables 11 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.


5 Locate the Variables section. In the Variables table, enter the following settings:

G L O B A L D E F I N I T I O N S

Parameters1 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:

S T U D Y 1

1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Build All.

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

NAME EXPRESSION DESCRIPTION

T_iso 300[K] Temperature



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2 Right-click Study 1 and choose Compute.

R E S U L T S

3D Plot Group 31 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 41 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).


7 Locate the Levels section. In the Total levels edit field, type 20.

8 Click the Plot button.

3D Plot Group 51 In the Model Builder window, right-click Results and choose 3D Plot Group.

2 Right-click Results>3D Plot Group 5 and choose 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 20.

7 Click the Plot button.



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solved with comsol multiphysics 4.2 porous reactor · pdf filethe model couples the free fluid...

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