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Packed Bed Reactor
Introduction
The packed bed reactor is used in heterogeneous catalytic processes and is one of the most common reactors in chemical industry. Its basic design is a column filled with porous catalyst particles, and these particles can be contained within a supporting structure, such as tubes or channels, or they can be packed in one single compartment in the reactor. In this example, volatile organic compounds (VOC) and CO are oxidized in a catalytic converter. Propylene is used as a representative for hydrocarbons present in the feed stream, which could, for example, be exhaust gas from a combustion process.
The structure of the packed catalyst particles makes the modeling of mass and energy transport in the reactor a challenge. The difficulty lies in the description of the porous structure, which gives transport of different orders of magnitudes within and between the particles. In most cases the structure in between particles is described as macroporous and the particle radius can be of the order of magnitude of 1 mm. When a pressure difference is applied across the bed, convection arises in the macropores. The pores inside the catalyst particles form the microstructure of the bed. The pore radii in the particles is often between 1 and 10 micrometers.
Figure 1: An example of the macro dimension (bed height) and the microdimension (pellet radial position) in a packed bed reactor.
This model presents a multigeometry approach to study microscale and macroscale mass balances in packed beds and other heterogeneous reactors with bimodal pore distribution. The example provides the mass and reaction distributions along the reactor and within each catalyst pellet along the reactor length. The Transport of Diluted Species and Chemistry interfaces are used and enable evaluation of the catalyst load utilization, optimal pellet size, or inlet temperature.
Model Definition
The following chemical reactions describe oxidation of carbon monoxide and an organic volatile by-product such as propylene in an automobile catalytic converter:
(1)
(2)
For these heterogeneous catalytic reactions the rates (SI unit: mol/(m3·s)) are given by (Ref. 2):
The rate and adsorption constants are defined by Arrhenius expressions which are available in the Chemistry interface. The values of the frequency factors and activation energies (SI unit: J/mol) are taken from the literature (Ref. 3) and listed in Table 1,
Table 1: Arrhenius parameters.
Rate/Adsorption Constant
Frequency factor
Activation energy
(Joule/mol)
k1
7.07·1013
1.09·105
k2
1.47·1015
1.26·105
KCO
8.1
-3400
KC3H6
257.9
1588
The chemical reactions in Equation 1 and Equation 2 describe a process limited by the second-order rate expression at high temperatures when the surface coverage is low. At low temperatures the adsorption reactions are slower and limit the reaction rate due to a shortage of free catalytic sites.
Macroscale Equations
The pressure drop along the reactor is described by the Ergun equation and is solved with a Coefficient Form PDE:
(3)
In Equation 3, P is the pressure (SI unit: Pa), εb the porosity, Dp the particle diameter (SI unit: m), μ denotes the gas viscosity (SI unit: Pa·s), ρ the gas density (SI unit: kg/m3), and x the reactor length coordinate (SI unit: m). u is the reactor flow velocity (SI unit: m/s) that depends on the pressure and velocity of the feed (SI unit: m/s), as described by Equation 4:
(4)
This relation applies since the flux across the reactor is set as constant, the system is assumed to be isothermal, and the fluid (air) is considered to be an ideal gas.
The mass transport on the macroscale level, along the reactor, takes place through convection and diffusion. This is easily accounted for using the Transport of Diluted Species interface in which the following equation (stationary) is solved:
(5)
The reaction rate R source term (SI unit: mol/(m3·s)) in Equation 5 depends on the transport inside the catalyst particles. The molar flux at the outer surface of the particles multiplied by the available outer surface area of the particles per unit volume gives the proper source term:
(6)
In Equation 6, εb is the bed porosity, N denotes the flux vector inside the porous particle (SI unit: mol/(m2·s)) and n is the outward unit vector normal to the particle surface. The equation is only valid at the particle surface, where the independent radius variable r (introduced below) equals the particle radius, rp. Furthermore, Ap denotes the pellet surface to volume ratio (SI unit: m2/m3). This property is related to the pellet radius as:
The boundary conditions for the macroscale mass balances are as follows:
At the inlet, the reactant concentrations are known:
cCO = cCOin
cC3H6 = cC3H6in
cO2 = cO2in
cCO2 = 0
cH2O = 0
At the outlet, the Outflow condition states that convective mass transport dominates the species transport across the boundary.
Microscale Equations
To properly calculate R, mass balances are required for the catalyst pellet interior, that is, the microscale. In the catalyst pores, mass transport can be assumed to take place by diffusion only. If the pellets are selected with spherical and dimensionless coordinates the following equation applies:
(7)
Here, Dcp is the diffusion coefficient in the particle, cp is the species concentration in the particle, and Rp is the reaction rate for the heterogeneous reaction in the particle. r (SI unit: m) is the independent variable for the position along the radius of the particle. Equation 7 is modeled with a Transport of Diluted Species interface with corrections for spherical coordinates and dimensionless particle radius. The boundary conditions are symmetry at the center of the particle and concentration at the surface. The latter is described by cp = εc where c and cp represent the species concentrations in the bulk and in the particle, respectively, and ε is the catalyst porosity. The concentration at the surface of the particle is equal to the concentration outside the particle compensated to account for the part of the particle volume that is occupied by solid catalyst support.
The concentration distribution in the particle gives the molar flux at every point along the reactor. This implies that the source term (Equation 6) in the macroscale mass balance becomes:
This type of problem exists for many chemical reaction engineering applications and is often solved by using an analytical approximation of the solution to the microscale mass balance. However, such an approach cannot be used for complicated reaction mechanisms involving several reaction species. The approach shown here is general and can be used for very complex reaction mechanisms involving a large number of species.
Macroscale and Microscale implementation
A complication in solving the derived system of equations is that the macroscale and microscale balances are defined in different coordinate systems. In this example, the macroscale part (along the reactor length) is set up in 1D and the microscale (along the pellet radius) in 2D. In the latter part, the x direction represents the surface concentration of the pellet at each part along the reactor and the y direction the concentration distribution along the dimensionless pellet radius.
To achieve the coupling of the fluxes and concentrations between the two scales of the system, general extrusion features are used.
Results and Discussion
Figure 2 shows the concentration of reacting species as a function of position in the reactor. The levels of carbon monoxide and propylene are significantly reduced.
Figure 2: The concentration of reactants and products along the reactor length. The pellet radius rp is 2.5 mm.
The concentration can also be evaluated within each pellet, revealing whether or not the catalyst comes to efficient use. Figure 3 is an example of this.
Figure 3: The concentration of in-pellet CO as a function of reactor position. The scaled pellet radius is given on the y-axis and the reactor position along the x-axis.
From Figure 3 it is clear that the CO concentration is low at the center of the pellet at all reactor positions. This means that reactor performance is limited by diffusion within the pellets and that active catalyst material in the center of the pellet is not used. Reducing the pellet diameter could potentially fix this situation and simulation results (see Figure 4) confirm this.
With a pellet radius of 2.5 mm, the reactor is limited by the in-pellet diffusion, as is indicated by the slope of the solid lines. The effect of this is that the active sites in the interior of the pellets are not used to their full potential. If the pellet size is reduced (represented by the lines with a triangular marker) the lines level out faster, suggesting larger reaction limited regions.
Figure 4: The concentration of in-pellet C3H6 at different reactor positions (5, 15, and 25 cm). Solid lines represent a pellet radius of 2.5 mm and lines with triangular markers represent a pellet radius of 1.8 mm.
References
1. J.M. Coulson and J.F. Richardson, Chemical Engineering, vol. 2, 4th ed., Pergamon Press, 1990.
2. S.H. Oh, J.C. Cavendish, and L.L. Hegedus. “Mathematical modeling of catalytic converter lightoff: Single-pellet studies”, AIChE Journal, vol. 26, no. 6, pp. 935–943, 1980.
3. J.B. Rawlings and J.G. Ekerdt, Chemical Reactor Analysis and Design Fundamentals, Nob Hill Publishing, Madison, 2002.
Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Porous_Catalysts/packed_bed_reactor
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, Start by adding the necessary 1D physics interfaces: Transport of Diluted Species and Coefficient Form PDE.
2
click  1D.
3
In the Select Physics tree, select Chemical Species Transport > Transport of Diluted Species (tds).
4
Click Add.
5
In the Number of species text field, type 5.
6
In the Concentrations (mol/m³) table, enter the following settings:
C3H6
CO
CO2
H2O
O2
7
In the Select Physics tree, select Mathematics > PDE Interfaces > Coefficient Form PDE (c).
8
Click Add.
9
In the Field name (1) text field, type P.
10
In the Dependent variables (1) table, enter the following settings:
P
11
Click  Study.
12
In the Select Study tree, select General Studies > Stationary.
13
Click  Done.
Geometry 1
Interval 1 (i1)
1
In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and choose Interval.
2
In the Settings window for Interval, locate the Interval section.
3
In the table, enter the following settings:
Coordinates (m)
0
0.3
4
Click  Build Selected.
Root
Add the model parameters from a text file.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file packed_bed_reactor_parameters.txt.
Variables 1
1
In the Home toolbar, click  Variables and choose Global Variables.
Import also the global variables from a text file.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file packed_bed_reactor_variables_global.txt.
Follow these steps to set up mass transport equations for the packed bed. Start with the Transport of Diluted Species.
Transport of Diluted Species (tds)
Fluid 1
1
In the Model Builder window, under Component 1 (comp1) > Transport of Diluted Species (tds) click Fluid 1.
2
In the Settings window for Fluid, locate the Convection section.
3
Specify the u vector as
u
x
4
Locate the Diffusion section. In the DC3H6 text field, type D_C3H6.
5
In the DCO text field, type D_CO.
6
In the DCO2 text field, type D_CO2.
7
In the DH2O text field, type D_H2O.
8
In the DO2 text field, type D_O2.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the C3H6 text field, type 1e-6.
4
In the CO text field, type 1e-6.
5
In the CO2 text field, type 1e-6.
6
In the H2O text field, type 1e-6.
7
In the O2 text field, type 1e-6.
Reactions 1
1
In the Physics toolbar, click  Domains and choose Reactions.
2
Select Domain 1 only.
3
In the Settings window for Reactions, locate the Reaction Rates section.
4
In the RC3H6 text field, type -Ap*C3H6flux*(1-por_b).
5
In the RCO text field, type -Ap*COflux*(1-por_b).
6
In the RCO2 text field, type -Ap*CO2flux*(1-por_b).
7
In the RH2O text field, type -Ap*H2Oflux*(1-por_b).
8
In the RO2 text field, type -Ap*O2flux*(1-por_b).
You will define the flux variables later on, after having set up the pellet model. Note that the volume fraction of the catalytic particles in the reactor is (1-por_b).
Inflow 1
1
In the Physics toolbar, click  Boundaries and choose Inflow.
2
Select Boundary 1 only.
3
In the Settings window for Inflow, locate the Concentration section.
4
In the c0,C3H6 text field, type x_C3H6_feed*Ctot_feed.
5
In the c0,CO text field, type x_CO_feed*Ctot_feed.
6
In the c0,O2 text field, type x_O2_feed*Ctot_feed.
7
Locate the Boundary Condition Type section. From the list, choose Flux (Danckwerts).
Outflow 1
1
In the Physics toolbar, click  Boundaries and choose Outflow.
2
Select Boundary 2 only.
Coefficient Form PDE: Ergun Equation
Follow the steps below to specify the Ergun equation in the Coefficient Form PDE.
1
In the Model Builder window, under Component 1 (comp1) click Coefficient Form PDE (c).
2
In the Settings window for Coefficient Form PDE, type Coefficient Form PDE: Ergun Equation in the Label text field.
3
Locate the Units section. Click  Select Dependent Variable Quantity.
4
In the Physical Quantity dialog, type pressure in the text field.
5
In the tree, select General > Pressure (Pa).
6
Click OK.
7
In the Settings window for Coefficient Form PDE, locate the Units section.
8
In the Source term quantity table, enter the following settings:
Source term quantity
Unit
Custom unit
N/m^3
Coefficient Form PDE 1
1
In the Model Builder window, under Component 1 (comp1) > Coefficient Form PDE: Ergun Equation (c) click Coefficient Form PDE 1.
2
In the Settings window for Coefficient Form PDE, locate the Diffusion Coefficient section.
3
In the c text field, type 0.
4
Locate the Source Term section. In the f text field, type Px+(150*mu*u/(2*Rr)^2*(1-por_b)^2/por_b^3+1.75*rho_feed*u^2/(2*Rr)*(1-por_b)/por_b^3).
5
Locate the Damping or Mass Coefficient section. In the da text field, type 0.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the P text field, type P_feed.
Dirichlet Boundary Condition 1
1
In the Physics toolbar, click  Boundaries and choose Dirichlet Boundary Condition.
2
Select Boundary 1 only.
3
In the Settings window for Dirichlet Boundary Condition, locate the Dirichlet Boundary Condition section.
4
In the r text field, type P_feed.
The source terms in the reactor mass balances depend on the surface fluxes from the catalyst pellets. Therefore, set up a separate model component calculating the mass transport and reaction in the pellets.The model geometry is in 2D. The x-coordinate represents reactor length and the y-coordinate the pellet radius.
Add Component
In the Model Builder window, right-click the root node and choose Add Component > 2D.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
Select a Chemistry interface handling the reactions and a Transport of Diluted Species interface for the mass transport within the pellets.
3
In the tree, select Chemical Species Transport > Chemistry (chem).
4
Click the Add to Component 2 button in the window toolbar.
5
In the tree, select Chemical Species Transport > Transport of Diluted Species (tds).
6
Click to expand the Dependent Variables section. In the Number of species text field, type 5.
7
In the Concentrations (mol/m³) table, enter the following settings:
C3H6p
COp
CO2p
H2Op
O2p
8
Click the Add to Component 2 button in the window toolbar.
9
In the Home toolbar, click  Add Physics to close the Add Physics window.
Geometry 2
Square 1 (sq1)
1
In the Geometry toolbar, click  Square.
2
In the Settings window for Square, click  Build Selected.
Chemistry (chem)
Start setting up the Chemistry interface for the two reactions present in the system.
Reaction 1
1
In the Physics toolbar, click  Domains and choose Reaction.
2
In the Settings window for Reaction, locate the Reaction Formula section.
3
In the Formula text field, type CO+1/2O2=>CO2.
4
Click Apply.
5
Locate the Reaction Rate section. In the rj text field, type chem.kf_1*chem.c_CO*chem.c_O2/(1+KCO*chem.c_CO+KC3H6*chem.c_C3H6)^2.
6
Locate the Reaction Orders section. Find the Volumetric overall reaction order subsection. In the Forward text field, type 2.
7
Locate the Rate Constants section. Select the Use Arrhenius expressions checkbox.
8
In the Af text field, type A1.
9
In the Ef text field, type E1.
Reaction 2
1
In the Physics toolbar, click  Domains and choose Reaction.
2
In the Settings window for Reaction, locate the Reaction Formula section.
3
In the Formula text field, type C3H6+9/2O2=>3H2O+3CO2.
4
Click Apply.
5
Locate the Reaction Orders section. Find the Volumetric overall reaction order subsection. In the Forward text field, type 2.
6
Locate the Reaction Rate section. In the rj text field, type chem.kf_2*chem.c_C3H6*chem.c_O2/(1+KCO*chem.c_CO+KC3H6*chem.c_C3H6)^2.
7
Locate the Rate Constants section. Select the Use Arrhenius expressions checkbox.
8
In the Af text field, type A2.
9
In the Ef text field, type E2.
Species 1
1
In the Physics toolbar, click  Domains and choose Species.
2
In the Settings window for Species, locate the Name section.
3
In the text field, type N2.
4
Locate the Type section. From the list, choose Solvent.
5
In the Model Builder window, collapse the Chemistry (chem) node.
Note that the molar masses are computed from the chemical formula.
6
In the Model Builder window, click Chemistry (chem).
7
In the Settings window for Chemistry, locate the Model Input section.
8
From the T list, choose User defined. In the associated text field, type Tr.
9
From the p list, choose User defined. In the associated text field, type P_feed.
10
Locate the Species Matching section. Find the Bulk species subsection. From the Species solved for list, choose Transport of Diluted Species 2.
11
In the table, enter the following settings:
Species
Type
Molar concentration
Value (mol/m^3)
C3H6
Variable
C3H6p
Solved for
CO
Variable
COp
Solved for
CO2
Variable
CO2p
Solved for
H2O
Variable
H2Op
Solved for
N2
Solvent
User defined
Cinert_feed
O2
Variable
O2p
Solved for
Continue with the Transport of Diluted Species interface to set the mass transport conditions.
Transport of Diluted Species 2 (tds2)
1
In the Model Builder window, under Component 2 (comp2) click Transport of Diluted Species 2 (tds2).
2
In the Settings window for Transport of Diluted Species, locate the Transport Mechanisms section.
3
Clear the Convection checkbox.
Fluid 1
Use the full diffusivity matrix to specify that diffusion occurs only in the radial pellet direction corresponding to the direction of the y-axis in the 2D geometry.
1
In the Model Builder window, under Component 2 (comp2) > Transport of Diluted Species 2 (tds2) click Fluid 1.
2
In the Settings window for Fluid, locate the Diffusion section.
3
From the list, choose Diagonal.
4
Specify the DC3H6p matrix as
0
0
0
(D_C3H6/rp^2)*y^2
5
From the list, choose Diagonal.
6
Specify the DCOp matrix as
0
0
0
(D_CO/rp^2)*y^2
7
From the list, choose Diagonal.
8
Specify the DCO2p matrix as
0
0
0
(D_CO2/rp^2)*y^2
9
From the list, choose Diagonal.
10
Specify the DH2Op matrix as
0
0
0
(D_H2O/rp^2)*y^2
11
From the list, choose Diagonal.
12
Specify the DO2p matrix as
0
0
0
(D_O2/rp^2)*y^2
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the C3H6p text field, type 1e-6.
4
In the COp text field, type 1e-6.
5
In the CO2p text field, type 1e-6.
6
In the H2Op text field, type 1e-6.
7
In the O2p text field, type 1e-6.
Reactions 1
1
In the Physics toolbar, click  Domains and choose Reactions.
Use the reaction rates calculated for each species in the Chemistry node.
2
In the Settings window for Reactions, locate the Domain Selection section.
3
From the Selection list, choose All domains.
4
Locate the Reaction Rates section. In the RC3H6p text field, type y^2/1[m^2]*chem.R_C3H6.
5
In the RCOp text field, type y^2/1[m^2]*chem.R_CO.
6
In the RCO2p text field, type y^2/1[m^2]*chem.R_CO2.
7
In the RH2Op text field, type y^2/1[m^2]*chem.R_H2O.
8
In the RO2p text field, type y^2/1[m^2]*chem.R_O2.
Concentration 1
1
In the Physics toolbar, click  Boundaries and choose Concentration.
Species concentrations calculated by the reactor model serve as boundary conditions for the pellet model. You define the variables in a later step.
2
Select Boundary 3 only.
3
In the Settings window for Concentration, locate the Concentration section.
4
Select the Species C3H6p checkbox.
5
In the c0,C3H6p text field, type C3H6bulk.
6
Select the Species COp checkbox.
7
In the c0,COp text field, type CObulk.
8
Select the Species CO2p checkbox.
9
In the c0,CO2p text field, type CO2bulk.
10
Select the Species H2Op checkbox.
11
In the c0,H2Op text field, type H2Obulk.
12
Select the Species O2p checkbox.
13
In the c0,O2p text field, type O2bulk.
14
Click the  Show More Options button in the Model Builder toolbar.
15
In the Show More Options dialog, in the tree, select the checkbox for the node Physics > Advanced Physics Options.
16
Click OK.
17
In the Settings window for Concentration, click to expand the Constraint Settings section.
18
From the Apply reaction terms on list, choose Individual dependent variables.
Complete the model setup by coupling the reactor and pellet models. First use a nonlocal general extrusion coupling to make the reactor species concentrations available in the pellet model. A correction of 0.30 is required since the 2D model uses the 1-by-1 dimension.
Definitions (comp1)
In the Model Builder window, under Component 1 (comp1) click Definitions.
General Extrusion 1 (genext1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose General Extrusion.
2
In the Settings window for General Extrusion, locate the Source Selection section.
3
From the Selection list, choose All domains.
4
Locate the Destination Map section. In the x-expression text field, type x*0.30.
5
Locate the Source section. Select the Use source map checkbox.
Next, set up another general extrusion coupling in the pellet model for use when calculating the species flux at the pellet boundary.
Definitions (comp2)
In the Model Builder window, under Component 2 (comp2) click Definitions.
General Extrusion 2 (genext2)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose General Extrusion.
2
In the Settings window for General Extrusion, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 3 only.
5
Locate the Destination Map section. In the x-expression text field, type x/0.30.
6
Clear the y-expression text field.
7
Locate the Source section. Select the Use source map checkbox.
8
Clear the yi-expression text field.
Definitions (comp1)
In the Model Builder window, under Component 1 (comp1) click Definitions.
Variables 2
1
In the Definitions toolbar, click  Local Variables.
Import a variable file that defines the species fluxes at the pellet boundary in the 1D model.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file packed_bed_reactor_variables_1d.txt.
Note that these variables use the nonlocal extrusion coupling that you just defined.
Definitions (comp2)
Now, read in the variable file that defines the bulk species concentrations in the 2D model.
1
In the Model Builder window, under Component 2 (comp2) click Definitions.
Variables 3
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 3 only.
5
Locate the Variables section. Click  Load from File.
6
Browse to the model’s Application Libraries folder and double-click the file packed_bed_reactor_variables_2d.txt.
As you can see, these variables use the nonlocal extrusion coupling you defined in the reactor model.
Mesh 1
Set up meshes for both the 1D and 2D geometries. A distributed mapped mesh is suitable in the 2D geometry to minimize the number of mesh elements solved for.
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
Edge 1
In the Mesh toolbar, click  Edge.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type 0.0025.
5
Click  Build All.
Mesh 2
In the Model Builder window, under Component 2 (comp2) click Mesh 2.
Mapped 1
In the Mesh toolbar, click  Mapped.
Distribution 1
1
Right-click Mapped 1 and choose Distribution.
2
Select Boundaries 1 and 4 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 30.
6
In the Element ratio text field, type 0.2.
Distribution 2
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Boundary 3 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 120.
5
Click  Build All.
Study 1
In the Study toolbar, click  Compute.
Results
Concentrations, All Species (tds)
The first default plot shows the concentrations of all species in the reactor as functions of the position.
1
In the Settings window for 1D Plot Group, click to expand the Title section.
2
In the Title text area, type Concentration along reactor length.
3
Locate the Plot Settings section.
4
Select the x-axis label checkbox. In the associated text field, type Reactor length (m).
5
In the Concentrations, All Species (tds) toolbar, click  Plot.
Now modify the default plot of the concentration of CO within the pellets.
Height Expression 1
1
In the Model Builder window, expand the Results > Concentration, COp (tds2) node.
2
Right-click Surface 1 and choose Height Expression.
3
Click the  Zoom Extents button in the Graphics toolbar.
Study 1
To investigate if better pellet utilization can be achieved, reduce the pellet size. This affects the packing of the catalyst and reduces the diffusive length within the pellets.
Step 1: Stationary
1
In the Model Builder window, under Study 1 click Step 1: Stationary.
2
In the Settings window for Stationary, click to expand the Study Extensions section.
3
Select the Auxiliary sweep checkbox.
4
Click  Add.
5
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
rp (Particle radius)
0.0025 0.0018
m
6
In the Study toolbar, click  Compute.
Create plots of the propene concentration distribution within the pellets, evaluated at reactor positions 5 cm, 15 cm, and 25 cm.
Results
Cut Line 2D 1
1
In the Results toolbar, click  Cut Line 2D.
2
In the Settings window for Cut Line 2D, locate the Line Data section.
3
In row Point 1, set X to 5/30.
4
In row Point 2, set X to 5/30 and y to 1.
5
Select the Additional parallel lines checkbox.
6
In the Distances text field, type -10/30 -20/30.
7
Click  Plot.
Pellet size comparison
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Pellet size comparison in the Label text field.
For rp=2.5 mm
1
Right-click Pellet size comparison and choose Line Graph.
2
In the Settings window for Line Graph, type For rp=2.5 mm in the Label text field.
3
Locate the Data section. From the Dataset list, choose Cut Line 2D 1.
4
From the Parameter selection (rp) list, choose From list.
5
In the Parameter values (rp (m)) list, select 0.0025.
6
Locate the y-Axis Data section. In the Expression text field, type C3H6p.
7
Click to expand the Coloring and Style section. From the Width list, choose 2.
8
Click to expand the Legends section. Select the Show legends checkbox.
9
From the Legends list, choose Evaluated.
10
In the Legend text field, type eval(x*0.3,cm) cm, rp=eval(rp,mm) mm.
For rp=1.8 mm
1
Right-click For rp=2.5 mm and choose Duplicate.
2
In the Settings window for Line Graph, type For rp=1.8 mm in the Label text field.
3
Locate the Data section. In the Parameter values (rp (m)) list, select 0.0018.
4
Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Triangle.
5
From the Positioning list, choose Interpolated.
Pellet size comparison
1
In the Model Builder window, click Pellet size comparison.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Upper left.
4
Locate the Title section. From the Title type list, choose Manual.
5
In the Title text area, type Concentration, C3H6p (mol/m<sup>3</sup>).
6
In the Pellet size comparison toolbar, click  Plot.
The following steps fix the rest of the plot groups.
Pressure drop
1
In the Model Builder window, under Results click Coefficient Form PDE: Ergun Equation.
2
In the Settings window for 1D Plot Group, type Pressure drop in the Label text field.
3
Locate the Title section. From the Title type list, choose None.
4
Locate the Plot Settings section.
5
Select the y-axis label checkbox. In the associated text field, type Pressure (Pa).
6
In the Pressure drop toolbar, click  Plot.
Concentrations, All Species (tds)
1
In the Model Builder window, click Concentrations, All Species (tds).
2
In the Settings window for 1D Plot Group, locate the Data section.
3
From the Parameter selection (rp) list, choose First.
4
In the Concentrations, All Species (tds) toolbar, click  Plot.