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Protein Adsorption
This application simulates an ion-exchange column for protein adsorption. The fluid phase contains four components: two proteins, one solvent, and one salt. The adsorption/desorption kinetics is described by two equilibrium reactions where proteins displace ions adsorbed at the surface and vice versa.
The example highlights how reactions at chemical equilibrium can be studied in a 0D reactor system in Reaction Engineering. In addition, it also shows how the kinetics from the 0D setup is exported to a 3D model where the reacting surface in the column can be studied in detail. The 3D model incorporates mass transport through diffusion and convection, and the reactions at the surface of the ion-exchange mass with Transport of Diluted Species, Laminar Flow, and Surface Reactions interfaces.
Introduction Protein Ion-Exchange
The binding of proteins to ion exchangers can be described within the steric mass action approximation (SMA). This approach assumes that the adsorption of a protein can be considered as an exchange reaction of the protein with a given number of adsorbed ions, the process is schematically depicted in Figure 1.
Figure 1: Proteins A and B displacing salt ions S at an ion-exchanger surface.
The equilibrium describing the adsorption/desorption reactions is
(1)
Here, S denotes the salt ion, P stands for either protein A and B. Once P is adsorbed, P(ads), salt ions are displaced, reducing the concentration of adsorbed salt ions, S(ads).
Model Definition
The system is modeled both in 0D and in 3D. The former model setup is adequate to investigate the kinetics of the equilibrium reactions within the column. The latter makes it possible to study the surface of the ion-exchange beads that make up the porous structure of the ion-exchange mass. In Figure 2, the two model approaches are presented.
Figure 2: Ion-exchange column and model geometries. The 0D model approximates the entire column, while the 3D geometry is a detailed representation of a section at the top of the column.
The 0D model utilizes the reactor type CSTR, constant volume in the Reaction Engineering interface. Ideal conditions are assumed within the reactor, meaning that well-mixed conditions with no concentration gradients apply. The reaction kinetics are described by Equation 1 for proteins A and B are entered directly into the interface as equilibrium surface reactions. The equilibrium constants for reaction 1, adsorption of A, is K1eq = 2 and for reaction 2, adsorption of B, is K2eq = 5. These are entered as well, relating the concentrations as followed
(2)
(3)
In order to compute the concentrations, both protein surface concentrations need to be set as dependent in the Equilibrium Species Vector section.
The proteins enter the reactor with a Feed Inlet with concentrations that vary in accordance to a 10 s Gaussian pulse with a maximum of 0.05 mol/m3. The outlet flow rate regulates so that the volume of the reactor is constant. Initially, no protein is available in the column and the ion-exchange mass is set to be completely adsorbed with salt, that is, the initial site density, γ0, is equal to the initial surface concentration of S, S(ads).
For the 3D model, as shown in Figure 3, only one quarter of the top section of the column is simulated due to symmetry. The proteins enter the top with a constant concentration of 0.05 mol/m3. In the column initially, the pores (the bulk) is filled with solvent and the ion-exchange beads are saturated with adsorbed salt.
Figure 3: 3D model geometry.
The reaction kinetics are taken directly from the 0D model with the Generate Space-Dependent Model feature and are collected in a Chemistry interface that also supplies computed diffusion coefficients and fluid density for the 3D setup.
The mass balances describing the species mass transport in the pores of the column are set up with the Transport of Diluted Species interface with diffusion and convection accounted for
(4)
On the right hand side of Equation 4 no reaction source is present for the bulk. However, the surface of the ion-exchange beads produces a reaction source that needs to be coupled to this equation. This is simply done with a Surface Reaction interface. Such an interface is automatically generated with the use of the Generate Space-Dependent Model feature if surface reactions are present.
The 3D model also computes the convective velocity, u, with a Laminar Flow interface. The velocity is based on the assumption that the velocity, U, within the reactor is 0.1 mm/s and that the inlet is open to the surroundings, that is, exposed to the atmospheric pressure.
Study settings
The 3D problem is solved in a two-step study. First, the Laminar Flow interface is first solved with a Stationary study step. Second, the rest of the interfaces (Chemistry, Transport of Diluted Species, and Surface Reactions) are solved with a Time Dependent study step. A Fully Coupled Direct solver is required in the second step to obtain stable computations.
Study 1 — Space Independent
Figure 4 shows how the concentrations of the reacting species change with the time. Initially, only adsorbed salt species are present in the column. The concentrations of proteins A and B are seen to change with the Gaussian concentration pulse feed inlet. A stronger adsorption affinity of protein B compared to protein A is readily observed. Note also how the concentration of bulk salt species S increases as the proteins adsorb at the surface. Toward the end of the pulse, most proteins have been adsorbed in the column and the bulk salt species have exited the system.
Figure 4: Concentrations of the reacting species as functions of time (s).
Study 2 — Space Dependent
The 3D model is solved for 30 s and a selection of the results are displayed in this section. Figure 5, Figure 6, Figure 7, and Figure 8 show the behavior of protein B after 5 s and 30 s.
From Figure 6, in which the bulk concentration is shown, indicates that the beads at the center of the column are less accessible for adsorption or that the protein is more rapidly adsorbed at the center, both phenomena lowering the bulk concentration there.
Figure 5: Bulk concentration of B at 5 s.
Figure 6: Bulk concentration of B at 30 s.
To get a proper understanding of the ion-exchange beads, a comparison of the adsorbed B surface concentration at the two times is also made with the results in Figure 7 and Figure 8. The lower surface concentration of B in the center suggests that less B is adsorbed there and that the porous structure obstructs the species transport. Note that the bead surface cannot adsorb more than the (initial) site density of the ion-exchange material, γ0 (SI unit: mol/m2).
Figure 7: Surface concentration of B at 5 s.
Figure 8: Surface concentration of B at 30 s.
In Figure 9, the velocity field is displayed. It shows that the porous structure causes a quite distorted velocity field. The exception is at the walls where the flow is less obstructed due to the relatively large gap between beads and wall.
Figure 9: Velocity field in the pores of the column section.
Application Library path: Chemical_Reaction_Engineering_Module/Mixing_and_Separation/protein_adsorption
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  0D.
2
In the Select Physics tree, select Chemical Species Transport > Reaction Engineering (re).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Global Definitions
Import the model parameters from a file from the application’s Application Library folder.
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 protein_adsorption_parameters.txt.
Gaussian Pulse 1 (gp1)
Reactants are injected to the system in a pulse. Choose a Gaussian Pulse function to describe the injection.
1
In the Home toolbar, click  Functions and choose Global > Gaussian Pulse.
2
In the Settings window for Gaussian Pulse, locate the Parameters section.
3
In the Location text field, type 5.
4
In the Standard deviation text field, type 1.5.
Use the Gaussian Pulse function to set up a time-dependent pulse variable with an amplitude of 1.
Definitions
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
pulse11
3.76*gp1(t/1[s])
 
Pulse with amplitude 1
Reaction Engineering (re)
Select a constant volume CSTR to model the ion-exchange column in 0D.
1
In the Model Builder window, under Component 1 (comp1) click Reaction Engineering (re).
2
In the Settings window for Reaction Engineering, locate the Reactor section.
3
From the Reactor type list, choose CSTR, constant volume.
4
Locate the Mixture Properties section. From the Phase list, choose Liquid.
Reaction 1
Set up the equilibrium reaction for adsorption/desorption of protein A.
1
In the Reaction Engineering toolbar, click  Reaction.
2
In the Settings window for Reaction, locate the Reaction Formula section.
3
In the Formula text field, type S(ads)+A=S+A(ads).
4
Click Apply.
5
Locate the Equilibrium Settings section. In the Keq0j text field, type Keq01.
Enter densities and molar masses in the bulk species nodes for Reaction 1.
Surface species: S(ads)
COMSOL will automatically assign the S to be sulfur. However, in this case is S a descriptive variable for the salt ion.
1
In the Model Builder window, click Surface species: S(ads).
2
In the Settings window for Species, locate the Chemical Formula section.
3
Clear the Enable formula checkbox.
4
In the M text field, type MS.
Species: A
1
In the Model Builder window, click Species: A.
2
In the Settings window for Species, locate the Chemical Formula section.
3
In the M text field, type MA.
4
In the ρ text field, type rho_p.
Species: S
1
In the Model Builder window, click Species: S.
2
In the Settings window for Species, locate the Chemical Formula section.
3
Clear the Enable formula checkbox.
4
In the M text field, type MS.
5
In the ρ text field, type rho_S.
Surface species: A(ads)
1
In the Model Builder window, click Surface species: A(ads).
2
In the Settings window for Species, locate the Chemical Formula section.
3
In the M text field, type MA.
Reaction 2
1
In the Reaction Engineering toolbar, click  Reaction.
In similar fashion, set up the equilibrium reaction for adsorption/desorption of protein B.
2
In the Settings window for Reaction, locate the Reaction Formula section.
3
In the Formula text field, type S(ads)+B=S+B(ads).
4
Click Apply.
5
Locate the Equilibrium Settings section. In the Keq0j text field, type Keq02.
As before, enter densities and molar masses in the bulk species nodes, but now for Reaction 2.
Species: B
Similar to S, B does not represent boron in this case, but is instead a descriptive variable of protein B.
1
In the Model Builder window, click Species: B.
2
In the Settings window for Species, locate the Chemical Formula section.
3
Clear the Enable formula checkbox.
4
In the M text field, type MB.
5
In the ρ text field, type rho_p.
Surface species: B(ads)
1
In the Model Builder window, click Surface species: B(ads).
2
In the Settings window for Species, locate the Chemical Formula section.
3
Clear the Enable formula checkbox.
4
In the M text field, type MB.
Replace the default dependent variables and replace these with A(ads) and B(ads). This omits any cyclic dependence of the reaction rates.
5
In the Model Builder window, click Reaction Engineering (re).
6
In the Settings window for Reaction Engineering, locate the Equilibrium Species section.
7
In the User defined equilibrium species (comma separated) text field, type A(ads):B(ads).
8
Locate the Reactor section. Find the Mass balance subsection. From the Volumetric rate list, choose User defined.
Set vp=0, to neglect the volumetric production rate.
9
In the vp text field, type 0.
Select the reactive surface area available in the reactor. This regulates, when multiplied with the site density, how much the ion-exchange column can adsorb.
10
Find the Surface reaction area subsection. In the Ar text field, type Arsurf.
Initial Values 1
In the Initial Values feature, set all species except adsorbed S (S(ads)) to zero. Also, set the initial density of surface sites for the ion-exchange mass. Note that the latter property sets the upper limit of how many moles can be adsorbed. Assume that each species takes up one reactive site.
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, click to expand the Surface Species Initial Values section.
3
In the Γs text field, type G0.
4
In the table, enter the following settings:
Species
Surface concentration (mol/m^2)
Site occupancy number (1)
S(ads)
CS0surf
1
Feed Inlet 1
Last, in the OD model, add the Feed Inlet feature containing bulk species A and B. Add the concentration with the Gaussian pulse in the feed stream.
1
In the Reaction Engineering toolbar, click  Feed Inlet.
2
In the Settings window for Feed Inlet, locate the Feed Inlet Properties section.
3
In the vf text field, type vfp.
4
Locate the Feed Inlet Concentration section. In the table, enter the following settings:
Species
Concentration (mol/m^3)
A
CAmax_inlet*pulse11
B
CBmax_inlet*pulse11
Solve the 0D model for 14 s.
Study 1 - Space Independent
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1 - Space Independent in the Label text field.
Step 1: Time Dependent
1
In the Model Builder window, under Study 1 - Space Independent click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,0.1,14).
4
In the Home toolbar, click  Compute.
Results
Concentrations 0D model
Create Figure 4 in which all species concentrations are displayed.
1
In the Settings window for 1D Plot Group, type Concentrations 0D model in the Label text field.
2
Click to expand the Title section. From the Title type list, choose None.
3
Locate the Plot Settings section. Select the x-axis label checkbox.
4
Select the y-axis label checkbox. In the associated text field, type Concentration (mol/m<sup>3</sup>) | Surface concentration (mol/m<sup>2</sup>).
5
Locate the Legend section. From the Layout list, choose Outside graph axis area.
Global 1
1
In the Model Builder window, expand the Concentrations 0D model node, then click Global 1.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Reaction Engineering > re.c_A - Concentration - mol/m³.
3
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Reaction Engineering > re.c_B - Concentration - mol/m³.
4
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Reaction Engineering > re.c_S - Concentration - mol/m³.
5
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Reaction Engineering > re.csurf_A_surf - Surface concentration - mol/m².
6
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Reaction Engineering > re.csurf_B_surf - Surface concentration - mol/m².
7
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Reaction Engineering > re.csurf_S_surf - Surface concentration - mol/m².
8
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
re.csurf_A_surf*1e4
mol/m^2
Surface concentration per square dm
re.csurf_B_surf*1e4
mol/m^2
Surface concentration per square dm
re.csurf_S_surf*1e4
mol/m^2
Surface concentration per square dm
9
Click to expand the Coloring and Style section. From the Width list, choose 2.
10
Click to expand the Legends section. From the Legends list, choose Manual.
11
In the table, enter the following settings:
Legends
A
B
S
Surface A
Surface B
Surface S
12
In the Concentrations 0D model toolbar, click  Plot.
Continue setting up a space-dependent model to investigate the top section of the ion-exchange column. Start by adding a solvent to the system. The solvent in these types of systems is often water.
Reaction Engineering (re)
Species 1
1
In the Reaction Engineering toolbar, click  Species.
2
In the Settings window for Species, locate the Name section.
3
In the text field, type H2O.
4
Locate the Type section. From the list, choose Solvent.
5
Locate the Chemical Formula section. In the M text field, type MH2O.
6
In the ρ text field, type rho_H2O.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Volumetric Species Initial Values section.
3
In the table, enter the following settings:
Species
Concentration (mol/m^3)
H2O
CH2O
Specify the transport properties of the solvent. These will transferred to the Chemistry interface in the space-dependent model below.
4
In the Model Builder window, click Reaction Engineering (re).
5
In the Settings window for Reaction Engineering, click to expand the Calculate Transport Properties section.
6
Select the Calculate mixture properties checkbox.
7
From the Dynamic viscosity list, choose User defined.
8
In the μ text field, type muH2O.
Use the Generate Space-dependent Model feature. Select to create a 3D geometry together with a Transport of Diluted Species and a Laminar Flow interface and a Time Dependent study type.
Generate Space-Dependent Model 1
1
In the Reaction Engineering toolbar, click  Generate Space-Dependent Model.
2
In the Settings window for Generate Space-Dependent Model, locate the Study Type section.
3
From the Study type list, choose Time dependent.
4
Locate the Physics Interfaces section. Find the Fluid flow subsection. From the list, choose Laminar Flow: New.
5
Locate the Space-Dependent Model Generation section. Click Create/Refresh.
Geometry 1(3D)
Insert the 3D geometry sequence file protein_adsorption_geom_sequence.mph from the application’s Application Library folder.
1
In the Model Builder window, expand the Component 2 (comp2) node, then click Geometry 1(3D).
2
In the Geometry toolbar, click Insert Sequence and choose Insert Sequence.
3
Browse to the model’s Application Libraries folder and double-click the file protein_adsorption_geom_sequence.mph.
4
In the Geometry toolbar, click  Build All.
5
Click the  Zoom Extents button in the Graphics toolbar.
Transport of Diluted Species (tds)
Start with the Transport of Diluted Species interface.
Fluid 1
The Laminar Flow interface is Coupled to the velocity field.
A and B enter the column at the top. Select Danckwerts (Flux) for a more stable computation.
Despite the fact that only solvent makes up the fluid within the column initially, adding initial trace concentrations of all dependent species gives a more stable solution to the equilibrium reactions.
Initial Values 1
1
In the Model Builder window, expand the Component 2 (comp2) > Transport of Diluted Species (tds) node, then click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the cA text field, type CAmax_inlet*0.01.
4
In the cB text field, type CBmax_inlet*0.01.
5
In the cS text field, type 1e-7.
Inflow 1
1
In the Model Builder window, click Inflow 1.
2
In the Settings window for Inflow, locate the Boundary Selection section.
3
From the Selection list, choose Inlet.
4
Locate the Concentration section. In the c0,cA text field, type CAmax_inlet.
5
In the c0,cB text field, type CBmax_inlet.
6
Locate the Boundary Condition Type section. From the list, choose Flux (Danckwerts).
Outflow 1
Set an Outflow condition at the bottom of the geometry. This means that the transport along the z direction (the height) of the reactor is dominated by convection.
1
In the Model Builder window, click Outflow 1.
2
In the Settings window for Outflow, locate the Boundary Selection section.
3
From the Selection list, choose Outlet column section.
Surface Equilibrium Reaction 1
The surface of the beads is the boundary where reactions take place.
1
In the Model Builder window, click Surface Equilibrium Reaction 1.
2
In the Settings window for Surface Equilibrium Reaction, locate the Boundary Selection section.
3
From the Selection list, choose Array 1.
Surface Equilibrium Reaction 2
1
In the Model Builder window, click Surface Equilibrium Reaction 2.
2
In the Settings window for Surface Equilibrium Reaction, locate the Boundary Selection section.
3
From the Selection list, choose Array 1.
Surface Reactions (sr)
In the surface reactions interface make sure that the reactions take place on the surface of the beads.
1
In the Model Builder window, under Component 2 (comp2) click Surface Reactions (sr).
2
In the Settings window for Surface Reactions, locate the Boundary Selection section.
3
From the Selection list, choose Array 1.
Reactions 1
1
In the Model Builder window, expand the Surface Reactions (sr) node, then click Reactions 1.
2
In the Settings window for Reactions, locate the Boundary Selection section.
3
From the Selection list, choose Array 1.
For the same reason as in the Transport of Diluted Species interface, add initial trace concentrations of species A and B.
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 csA text field, type CS0surf*0.01.
4
In the csB text field, type CS0surf*0.01.
Laminar Flow (spf)
At the inlet, or the top of the column, atmospheric pressure is applied since the column is open to the surroundings.
Inlet 1
1
In the Model Builder window, expand the Laminar Flow (spf) node, then click Inlet 1.
2
In the Settings window for Inlet, locate the Boundary Selection section.
3
From the Selection list, choose Inlet.
4
Locate the Boundary Condition section. From the list, choose Pressure.
The average velocity through the xy-plane of the geometry is assumed to be constant everywhere along the column height. Set a constant velocity within the column, in this case, at the bottom of the 3D model geometry.
Outlet 1
1
In the Model Builder window, click Outlet 1.
2
In the Settings window for Outlet, locate the Boundary Selection section.
3
From the Selection list, choose Outlet column section.
4
Locate the Boundary Condition section. From the list, choose Velocity.
5
Locate the Velocity section. In the U0 text field, type U_column.
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
In the Settings window for Symmetry, locate the Boundary Selection section.
3
From the Selection list, choose Symmetry.
Wall 1
1
In the Model Builder window, click Wall 1.
2
In the Settings window for Wall, locate the Boundary Selection section.
3
Click  Create Selection.
4
In the Create Selection dialog, type Walls in the Selection name text field.
5
Click OK.
Mesh 1
Select a mesh that resolves the surface of the beads well.
Free Triangular 1
1
In the Mesh toolbar, click  More Generators and choose Free Triangular.
2
In the Settings window for Free Triangular, locate the Boundary Selection section.
3
From the Selection list, choose Array 1.
Size 1
1
Right-click Free Triangular 1 and choose 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.
5
Select the Maximum element size checkbox. In the associated text field, type 9E-2.
Free Tetrahedral 1
1
In the Mesh toolbar, click  Free Tetrahedral.
2
In the Settings window for Free Tetrahedral, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose All domains.
Size 1
1
Right-click Free Tetrahedral 1 and choose 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.
5
Select the Maximum element size checkbox. In the associated text field, type 5E-1.
This model requires that a two-step study node is used. The first step solves the stationary solution of the Laminar Flow interface and the second the time-dependent solution for the rest of the interfaces.
Study 2 - Space Dependent
1
In the Model Builder window, click Study 2.
2
In the Settings window for Study, type Study 2 - Space Dependent in the Label text field.
Stationary - Laminar Flow
1
In the Study toolbar, click  Stationary.
2
In the Settings window for Stationary, type Stationary - Laminar Flow in the Label text field.
3
Locate the Physics and Variables Selection section. In the Solve for column of the table, under Component 2 (comp2), clear the checkboxes for Chemistry (chem), Transport of Diluted Species (tds), and Surface Reactions (sr).
Time Dependent - the Rest of the Interfaces
1
In the Model Builder window, click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, type Time Dependent - the Rest of the Interfaces in the Label text field.
3
Locate the Study Settings section. In the Output times text field, type range(0,1,30).
4
From the Tolerance list, choose User controlled.
5
In the Relative tolerance text field, type 2e-4.
6
Locate the Physics and Variables Selection section. In the Solve for column of the table, under Component 2 (comp2), clear the checkbox for Laminar Flow (spf).
Step 2: Stationary - Laminar Flow
In the Model Builder window, under Study 2 - Space Dependent right-click Step 2: Stationary - Laminar Flow and choose Move Up.
Solution 2 (sol2)
1
In the Study toolbar, click  Show Default Solver.
Replace the Segregated solver with a Fully Coupled one since this fits the quick shifts in concentration originating from the equilibrium reactions in the system. Also apply a Direct solver to further improve the stability of the computations.
2
In the Model Builder window, expand the Solution 2 (sol2) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Absolute Tolerance section.
4
From the Tolerance method list, choose Manual.
5
In the Absolute tolerance text field, type 1e-4.
6
Right-click Study 2 - Space Dependent > Solver Configurations > Solution 2 (sol2) > Time-Dependent Solver 1 and choose Fully Coupled.
7
In the Settings window for Fully Coupled, locate the General section.
8
From the Linear solver list, choose Direct.
9
In the Model Builder window, click Study 2 - Space Dependent.
10
In the Settings window for Study, locate the Study Settings section.
11
Clear the Generate default plots checkbox.
12
In the Study toolbar, click  Compute.
Results
Set up two Sector 3D datasets to rotate the 3D geometry according to its symmetry.
Sector 3D 1
1
In the Results toolbar, click  More Datasets and choose Sector 3D.
2
In the Settings window for Sector 3D, locate the Symmetry section.
3
In the Number of sectors text field, type 4.
4
Click  Plot.
Sector 3D 2
1
Right-click Sector 3D 1 and choose Duplicate.
2
In the Settings window for Sector 3D, locate the Symmetry section.
3
From the Sectors to include list, choose Manual.
4
In the Number of sectors to include text field, type 3.
5
In the Start sector text field, type 3.
6
Click  Plot.
The following step creates Figures 3-6.
Bulk concentration B, 5 s
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Bulk concentration B, 5 s in the Label text field.
3
Locate the Data section. From the Time (s) list, choose 5.
Surface 1
1
Right-click Bulk concentration B, 5 s and choose Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Sector 3D 1.
4
From the Time (s) list, choose 5.
5
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 2 (comp2) > Transport of Diluted Species > Species cB > cB - Molar concentration, cB - mol/m³.
6
Click the  Show Grid button in the Graphics toolbar.
7
In the Bulk concentration B, 5 s toolbar, click  Plot.
8
Click the  Zoom Extents button in the Graphics toolbar.
Bulk concentration B, 30 s
1
In the Model Builder window, right-click Bulk concentration B, 5 s and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Bulk concentration B, 30 s in the Label text field.
3
Locate the Data section. From the Time (s) list, choose 30.
Surface 1
1
In the Model Builder window, expand the Bulk concentration B, 30 s node, then click Surface 1.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Sector 3D 2.
4
From the Time (s) list, choose 30.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
In the Bulk concentration B, 30 s toolbar, click  Plot.
Surface concentration B, 5 s
1
In the Model Builder window, right-click Bulk concentration B, 30 s and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Surface concentration B, 5 s in the Label text field.
3
Locate the Data section. From the Time (s) list, choose 5.
Surface 1
1
In the Model Builder window, expand the Surface concentration B, 5 s node, then click Surface 1.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Sector 3D 1.
4
From the Time (s) list, choose 5.
5
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 2 (comp2) > Surface Reactions > Surface species cs_B > cs_B - Surface concentration - mol/m².
6
Click the  Transparency button in the Graphics toolbar.
7
Click the  Zoom Extents button in the Graphics toolbar.
8
In the Surface concentration B, 5 s toolbar, click  Plot.
Surface concentration B, 30 s
1
In the Model Builder window, right-click Surface concentration B, 5 s and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Surface concentration B, 30 s in the Label text field.
3
Locate the Data section. From the Time (s) list, choose 30.
Surface 1
1
In the Model Builder window, expand the Surface concentration B, 30 s node, then click Surface 1.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Sector 3D 2.
4
From the Time (s) list, choose 30.
5
Click the  Transparency button in the Graphics toolbar.
6
Click the  Zoom Extents button in the Graphics toolbar.
7
In the Surface concentration B, 30 s toolbar, click  Plot.
Velocity field
Last, create Figure 7 to visualize the velocity field within the 3D geometry.
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Velocity field in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Custom.
4
Find the Type and data subsection. Clear the Type checkbox.
5
Clear the Description checkbox.
6
Clear the Unit checkbox.
7
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
8
Locate the Color Legend section. Select the Show units checkbox.
Streamline 1
1
Right-click Velocity field and choose Streamline.
2
In the Settings window for Streamline, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 2 (comp2) > Laminar Flow > Velocity and pressure > u,v,w - Velocity field.
3
Locate the Streamline Positioning section. From the Positioning list, choose Magnitude controlled.
4
In the Minimum density level text field, type 9.
5
In the Maximum density level text field, type 10.5.
6
Locate the Coloring and Style section. Find the Point style subsection. From the Type list, choose Arrow.
7
Find the Line style subsection. From the Type list, choose Tube.
Color Expression 1
1
Right-click Streamline 1 and choose Color Expression.
2
In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 2 (comp2) > Laminar Flow > Velocity and pressure > spf.U - Velocity magnitude - m/s.
Surface 1
1
In the Model Builder window, right-click Velocity field and choose Surface.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Coloring list, choose Gradient.
4
From the Top color list, choose Custom.
5
Click Define custom colors.
6
Set the RGB values to 166, 214, and 208, respectively.
7
Click Add to custom colors.
8
Click Show color palette only or OK on the cross-platform desktop.
9
From the Bottom color list, choose Custom.
10
Click Define custom colors.
11
Set the RGB values to 255, 160, and 122, respectively.
12
Click Add to custom colors.
13
Click Show color palette only or OK on the cross-platform desktop.
Selection 1
1
Right-click Surface 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
From the Selection list, choose Walls.
4
In the Velocity field toolbar, click  Plot.
Appendix — Geometry Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Blank Model.
Add Component
In the Home toolbar, click  Add Component and choose 3D.
Geometry 1
1
In the Settings window for Geometry, locate the Units section.
2
From the Length unit list, choose mm.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 4.25.
4
In the Height text field, type 2.15.
Block 1 (blk1)
1
In the Geometry toolbar, click  Block.
2
In the Settings window for Block, locate the Size and Shape section.
3
In the Width text field, type 4.25.
4
In the Depth text field, type 4.25.
5
In the Height text field, type 2.15.
Intersection 1 (int1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Intersection.
2
Click in the Graphics window and then press Ctrl+A to select both objects.
Sphere 1 (sph1)
1
In the Geometry toolbar, click  Sphere.
2
In the Settings window for Sphere, locate the Size section.
3
In the Radius text field, type 0.5.
4
Locate the Position section. In the x text field, type 0.45.
5
In the y text field, type 0.45.
6
In the z text field, type 0.45.
Array 1 (arr1)
1
In the Geometry toolbar, click  Transforms and choose Array.
2
Select the object sph1 only.
3
In the Settings window for Array, locate the Size section.
4
In the x size text field, type 4.
5
In the y size text field, type 4.
6
In the z size text field, type 2.
7
Locate the Displacement section. In the x text field, type 0.95.
8
In the y text field, type 0.95.
9
In the z text field, type 0.95.
10
Locate the Selections of Resulting Entities section. Select the Resulting objects selection checkbox.
11
From the Show in physics list, choose Boundary selection.
12
Click  Build Selected.
Delete Entities 1 (del1)
1
In the Geometry toolbar, click  Delete.
2
Click the  Wireframe Rendering button in the Graphics toolbar.
3
Select the objects arr1(3,4,1), arr1(3,4,2), arr1(4,3,1), arr1(4,3,2), arr1(4,4,1), and arr1(4,4,2) only.
Difference 1 (dif1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object int1 only.
3
In the Settings window for Difference, locate the Difference section.
4
From the Objects to subtract list, choose Array 1.
Form Union (fin)
1
In the Model Builder window, click Form Union (fin).
2
In the Settings window for Form Union/Assembly, click  Build Selected.
Column wall
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, locate the Entities to Select section.
3
From the Geometric entity level list, choose Boundary.
4
On the object fin, select Boundary 37 only.
5
In the Label text field, type Column wall.
Inlet
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Inlet in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object fin, select Boundary 4 only.
Outlet column section
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Outlet column section in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object fin, select Boundary 3 only.
Symmetry
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Symmetry in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object fin, select Boundaries 1 and 2 only.