Models.chem.Reacting Pillars

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Solved with COMSOL Multiphysics 4.4 1 | SURFACE REACTIONS IN A BIOSENSOR Surface Reactions in a Biosensor This example illustrates how to use the Surface Reactions physics interface, the Transport of Diluted Species physics interface, and the Laminar Flow physics interface to set up a model of a biosensor application. Combining these physics interfaces makes it straightforward to couple surface reactions to mass transport in a fluid stream. Introduction A flow cell in a biosensor contains an array of micropillars. The curved side of the pillars are coated with an active material that allows for the selective adsorption of analyte species in the sample stream. The adsorbed species produce a signal that is dependent upon the local concentration at the pillar surfaces. This example investigates the surface concentration distribution in the cell while an analyte pulse is transported through it. It also studies the effect of a quenching surface reaction where adsorbed species are converted into an inactive state. Model Definition GEOMETRY The flow cell contains seven rows containing four pillars each. The curved surfaces of the pillars are the only active surfaces for adsorbing the analyte molecules. The flow cell has two planes of symmetry that allow for reduction of the modeling domain to one fourth of the full geometry.

Transcript of Models.chem.Reacting Pillars

Page 1: Models.chem.Reacting Pillars

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S u r f a c e R e a c t i o n s i n a B i o s e n s o r

This example illustrates how to use the Surface Reactions physics interface, the Transport of Diluted Species physics interface, and the Laminar Flow physics interface to set up a model of a biosensor application. Combining these physics interfaces makes it straightforward to couple surface reactions to mass transport in a fluid stream.

Introduction

A flow cell in a biosensor contains an array of micropillars. The curved side of the pillars are coated with an active material that allows for the selective adsorption of analyte species in the sample stream. The adsorbed species produce a signal that is dependent upon the local concentration at the pillar surfaces. This example investigates the surface concentration distribution in the cell while an analyte pulse is transported through it. It also studies the effect of a quenching surface reaction where adsorbed species are converted into an inactive state.

Model Definition

G E O M E T R Y

The flow cell contains seven rows containing four pillars each. The curved surfaces of the pillars are the only active surfaces for adsorbing the analyte molecules. The flow cell has two planes of symmetry that allow for reduction of the modeling domain to one fourth of the full geometry.

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Outlet

Inlet

Active surface

Figure 1: The flow cell holds seven rows of pillars with four pillars in each row. The curved pillar surface is the only surface that allows the adsorption of analyte molecules. The modeling geometry can be reduced to one fourth of the full geometry due to mirror symmetry.

S U R F A C E R E A C T I O N S

Analyte molecules (P) can adsorb and desorb from surface sites (S) on the micropillar surfaces according to

(1)

The adsorbed analyte (PS) can transform into a quenched state (QS) that does not contribute to the sensor signal. The quenching reaction is reversible:

(2)

The rate of adsorption is

(3)

where cp is the concentration of P in the stream. The desorption rate is linear in the concentration of surface adsorbed species, cPS:

+P S PSkads

kdes

PS QSk1

k2

rads kadscP=

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(4)

The rate of the reversible quenching reaction is given by

(5)

M A S S TR A N S P O R T I N T H E A N A L Y T E S T R E A M

The equations in the Transport of Diluted Species interface describe the transport of the species, P, in the analyte stream according to

(6)

Here DP denotes the diffusion coefficient (m2/s), cP the species concentration (mol/m3), and u the velocity vector (m/s).

The sample pulse that enters the sensor array is described by a Gaussian distribution at flow cell inlet with a maximum concentration of 80 mol/m3.

At the outlet, the Outflow condition is used:

(7)

The adsorption and desorption of analyte at the active pillar surfaces give rise to a net flux at the corresponding boundaries:

(8)

The mass flux due to desorption is dependent upon local concentration of adsorbed surface species and is hence coupled to the equations in the Surface Reactions interface, described next.

M A S S TR A N S P O R T A N D R E A C T I O N S O N T H E A C T I V E S U R F A C E S

Transport of adsorbed species occurs in the tangential direction along the surface. The Surface Reactions interface models the tangential flux in the along the surface, the surface molar flux, Nt,i (mol/(m·s)), according to

where Ds,i (m2/s) is the surface diffusion coefficient for species i.

The governing equation for the surface concentrations is written as

rdes kdescPS=

rquench k1cPS– k2cQS+=

∂cP∂t--------- ∇+ DP∇– cP( )⋅ u ∇cP⋅+ 0=

n D∇– c( )⋅ 0=

Np rads– rdes+=

Nt i, Ds i, ∇tcs i,–=

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where Rs,i (mol/(m2·s)) is the sum of all sources due to surface reactions and adsorption/desorption phenomena.

In this example, surface diffusion is ignored. Using the reactions described by Equation 3 through Equation 5, the balances equation for the surface species P and Q become:

The rate of adsorption depends on the concentration of the P species in the analyte stream and is therefore coupled to the equations in the Surface Reactions interface to those provided by the Transport of Diluted Species interface.

F L U I D F L O W

The flow in the flow cell is laminar and given by the Navier-Stokes equations:

(9)

where ρ denotes density (kg/m3), u represents the velocity (m/s), η denotes viscosity (kg/(m· s)), and p equals the pressure in the tubes (Pa).

The calculated flow field serves as input to the Transport of Diluted Species interface, to describe the convective mass transport.

The boundary conditions are

(10)

At the outlet, viscous stresses are ignored and the pressure is set to 1 atmosphere.

t∂∂ cs i, ∇t Ds i, ∇tcs i,–( )⋅+ Rs i,=

dcs P,dt-------------- rads rdes– rquench–=

dcs Q,dt-------------- rquench=

ρu ∇⋅ u ∇ pI– η ∇u ∇u( )T+( ) 2η 3⁄( ) ∇ u⋅( )I–+[ ]⋅=

∇ ρu( )⋅ 0=

u n⋅ v0=

u 0 =

p pref =

inletwalls

outlet

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Results and Discussion

Figure 2 shows the magnitude of the laminar velocity field in the flow cell.

Figure 2: The velocity magnitude of the laminar flow field in the biosensor flow cell

Figure 3 through Figure 6 show the concentration of the species, P, in the stream as well the relative coverage of surface adsorbed species, PS, as the analyte pulse passes through the flow cell.

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Figure 3: Concentration distribution in the analyte stream and surface coverage of adsorbed species at t = 35 s.

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Figure 4: Concentration distribution in the analyte stream and surface coverage of adsorbed species at t = 45 s.

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Figure 5: Concentration distribution in the analyte stream and surface coverage of adsorbed species at t = 55 s.

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Figure 6: Concentration distribution in the analyte stream and surface coverage of adsorbed species at t = 75 s.

The velocity distribution of the flow field will cause pillars near the wall to reach their maximum adsorption level at a later time compared to pillars in the center of the stream. Pillars near the wall will also take longer to release adsorbed analyte. The position of a pillar in a row also has an effect on the maximum adsorption level, and the time at which it is reached. This effect is highlighted in Figure 7. These geometrical effects will cause the sensor signal to become relatively diffuse.

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Figure 7: Average fractional surface coverage of adsorbed analyte PS.

Figure 8 shows the relative surface concentration of the quenched surface species QS for the same pillar positions. The geometrical effect is once again evident yet more importantly, the plot shows that a relatively long time is required to purge the reactive sites in preparation for a new analysis.

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Figure 8: Average fractional surface coverage of adsorbed quenched species QS.

Notes About the COMSOL Implementation

In this example, there is a one-way coupling between the stationary flow field and the mass transport equations. This means that the equations for the laminar flow need only be solved once, and that the results can be used for calculating the transient mass transport problem. This is accomplished by using two study steps when setting up and solving the model. The first step solves for the stationary flow field. The solution is then used in the second step where the transient mass balance equations are solved.

Model Library path: Chemical_Reaction_Engineering_Module/Surface_Reactions_and_Deposition_Processes/reacting_pillars

Modeling Instructions

From the File menu, choose New.

N E W

1 In the New window, click the Model Wizard button.

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M O D E L W I Z A R D

1 In the Model Wizard window, click the 3D button.

2 In the Select physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf).

3 Click the Add button.

4 In the Select physics tree, select Chemical Species Transport>Transport of Diluted

Species (chds).

5 Click the Add button.

6 In the Concentrations table, enter the following settings:

7 In the Select physics tree, select Chemical Species Transport>Surface Reactions (chsr).

8 Click the Add button.

9 In the Number of surface species edit field, type 2.

10 In the Surface (adsorbed) species concentrations table, enter the following settings:

11 Click the Study button.

12 In the tree, select Preset Studies for Selected Physics>Stationary.

13 Click the Done button.

G E O M E T R Y 1

Start by creating the geometry. To simplify this step, insert a prepared geometry sequence from file. After insertion you can study each geometry step in the sequence.

1 On the Geometry toolbar, click Insert Sequence.

2 Browse to the model’s Model Library folder and double-click the file reacting_pillars_geom_sequence.mph.

3 On the Geometry toolbar, click Build all.

4 Click the Zoom Extents button on the Graphics toolbar.

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

Read in a text file with model parameters such as model rate constants and diffusion coefficients.

c_P

cs_P

cs_Q

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Parameters1 On the Home toolbar, click Parameters.

2 In the Parameters settings window, locate the Parameters section.

3 Click Load from File.

4 Browse to the model’s Model Library folder and double-click the file reacting_pillars_parameters.txt.

D E F I N I T I O N S

Variables 11 In the Model Builder window, under Component 1 right-click Definitions and choose

Variables.

2 In the Variables settings window, locate the Variables section.

3 Click Load from File.

4 Browse to the model’s Model Library folder and double-click the file reacting_pillars_variables.txt.

Gaussian Pulse 11 On the Home toolbar, click Functions and choose Global>Gaussian Pulse.

2 In the Gaussian Pulse settings window, locate the Parameters section.

3 In the Location edit field, type 20.

4 In the Standard deviation edit field, type 2.

Note that you can plot the Gaussian function by clicking the Plot button on the Settings toolbar.

Explicit 11 On the Definitions toolbar, click Explicit.

2 In the Explicit settings window, locate the Input Entities section.

3 From the Geometric entity level list, choose Boundary.

4 Click the Wireframe Rendering button on the Graphics toolbar.

You can select the surfaces boundary by boundary according to the list below. Alternatively, click the Go to YZ View button on the Graphics toolbar and then click

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the Select Box button. Enclose the concave pillar surfaces with the rubber band box and right-click the selection to confirm.

5 Select Boundaries 8, 9, 12, 13, 15, 16, 19, 20, 23, 26–28, 31, 32, 37, 38, 41, 42, 44, 45, 48, 49, 52, 55–57, 60, 61, 66, 67, 70, 71, 73, 74, 77, 78, 81, 84–86, 89, 90, 95, 96, 99, 100, 102, 103, 106, 107, 110, 113–115, 118, and 119 only.

6 Right-click Component 1>Definitions>Explicit 1 and choose Rename.

7 Go to the Rename Explicit dialog box and type Reacting surface in the New name edit field.

8 Click OK.

Create a number of operators features that calculate the surface average.

Average 11 On the Definitions toolbar, click Component Couplings and choose Average.

2 In the Average settings window, locate the Source Selection section.

3 From the Geometric entity level list, choose Boundary.

4 Select Boundaries 23 and 28 only.

5 Locate the Operator Name section. In the Operator name edit field, type ave_center_1.

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Average 21 On the Definitions toolbar, click Component Couplings and choose Average.

2 In the Average settings window, locate the Source Selection section.

3 From the Geometric entity level list, choose Boundary.

4 Select Boundaries 110 and 115 only.

5 Locate the Operator Name section. In the Operator name edit field, type ave_center_4.

Average 31 On the Definitions toolbar, click Component Couplings and choose Average.

2 In the Average settings window, locate the Source Selection section.

3 From the Geometric entity level list, choose Boundary.

4 Select Boundaries 12, 13, 19, and 20 only.

5 Locate the Operator Name section. In the Operator name edit field, type ave_wall_1.

Average 41 On the Definitions toolbar, click Component Couplings and choose Average.

2 In the Average settings window, locate the Source Selection section.

3 From the Geometric entity level list, choose Boundary.

4 Select Boundaries 99, 100, 106, and 107 only.

5 Locate the Operator Name section. In the Operator name edit field, type ave_wall_4.

M A T E R I A L S

Select water as the material in the flow cell.

1 On the Home toolbar, click Add Material.

A D D M A T E R I A L

1 Go to the Add Material window.

2 In the tree, select Liquids and Gases>Liquids>Water.

3 In the Add material window, click Add to Component.

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M A T E R I A L S

WaterMove on to set up the physics interfaces.

L A M I N A R F L O W

Inlet 11 On the Physics toolbar, click Boundaries and choose Inlet.

2 Select Boundary 1 only.

3 In the Inlet settings window, locate the Velocity section.

4 In the U0 edit field, type u_in.

Outlet 11 On the Physics toolbar, click Boundaries and choose Outlet.

2 Select Boundary 122 only.

Symmetry 11 On the Physics toolbar, click Boundaries and choose Symmetry.

2 Select Boundaries 2, 4, 34, 63, 92, and 121 only.

TR A N S P O R T O F D I L U T E D S P E C I E S

Convection and Diffusion 11 In the Model Builder window, expand the Component 1>Transport of Diluted Species

node, then click Convection and Diffusion 1.

2 In the Convection and Diffusion settings window, locate the Model Inputs section.

3 From the u list, choose Velocity field (spf/fp1).

4 Locate the Diffusion section. In the DcP edit field, type D.

Inflow 11 On the Physics toolbar, click Boundaries and choose Inflow.

2 Select Boundary 1 only.

3 In the Inflow settings window, locate the Concentration section.

4 In the c0,cP edit field, type c0.

Outflow 11 On the Physics toolbar, click Boundaries and choose Outflow.

2 Select Boundary 122 only.

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Symmetry 11 On the Physics toolbar, click Boundaries and choose Symmetry.

2 Select Boundaries 2, 4, 34, 63, 92, and 121 only.

Flux 11 On the Physics toolbar, click Boundaries and choose Flux.

2 In the Flux settings window, locate the Boundary Selection section.

3 From the Selection list, choose Reacting surface.

4 Locate the Inward Flux section. Select the Species c_P check box.

5 In the N0,cP edit field, type -r_ads+r_des.

S U R F A C E R E A C T I O N S

1 In the Model Builder window, under Component 1 click Surface Reactions.

2 In the Surface Reactions settings window, locate the Boundary Selection section.

3 From the Selection list, choose Reacting surface.

Reactions 11 On the Physics toolbar, click Boundaries and choose Reactions.

2 In the Reactions settings window, locate the Boundary Selection section.

3 From the Selection list, choose Reacting surface.

4 Locate the Reaction Rate for Surface Species section. In the Rs,csP edit field, type r_ads-r_des-r_quench.

5 In the Rs,csQ edit field, type r_quench.

M E S H 1

Free Triangular 11 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Free

Triangular.

2 In the Free Triangular settings window, locate the Boundary Selection section.

3 From the Selection list, choose Reacting surface.

Size 11 Right-click Component 1>Mesh 1>Free Triangular 1 and choose Size.

2 In the Size settings window, locate the Element Size section.

3 Click the Custom button.

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4 Locate the Element Size Parameters section. Select the Maximum element size check box.

5 In the associated edit field, type 5e-5.

Free Tetrahedral 1In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral.

Size 11 In the Model Builder window, under Component 1>Mesh 1 right-click Free Tetrahedral

1 and choose Size.

2 In the Size settings window, locate the Element Size section.

3 Click the Custom button.

4 Locate the Element Size Parameters section. Select the Maximum element size check box.

5 In the associated edit field, type 2e-4.

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

This creates a better overview of the model tree as you move to set up the solvers and work with results processing.

Solve the problems using two study steps. The first step solves for the stationary flow field. In the second step, use the solution from the first step to solve the transient mass balance equations.

S T U D Y 1

Step 1: Stationary1 In the Model Builder window, expand the Study 1 node, then click Step 1: Stationary.

2 In the Stationary settings window, locate the Physics and Variables Selection section.

3 In the table, enter the following settings:

Step 2: Time Dependent1 On the Study toolbar, click Study Steps and choose Time Dependent>Time Dependent.

2 In the Time Dependent settings window, locate the Study Settings section.

3 In the Times edit field, type range(0,1,100).

Physics Solve for Discretization

Transport of Diluted Species × physics

Surface Reactions × physics

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4 Locate the Physics and Variables Selection section. In the table, enter the following settings:

Solver 11 On the Study toolbar, click Show Default Solver.

The magnitude of the concentration in the analyte stream will be of many orders of magnitude higher compared to the concentrations of the surface species. Providing manual scales for the concentration variables will help convergence.

2 In the Model Builder window, expand the Study 1>Solver Configurations>Solver

1>Dependent Variables 2 node, then click Surface concentration (comp1.cs_Q).

3 In the Field settings window, locate the Scaling section.

4 From the Method list, choose Manual.

5 In the Scale edit field, type 1e-7.

6 In the Model Builder window, under Study 1>Solver Configurations>Solver

1>Dependent Variables 2 click Surface concentration (comp1.cs_P).

7 In the Field settings window, locate the Scaling section.

8 From the Method list, choose Manual.

9 In the Scale edit field, type 1e-7.

10 In the Model Builder window, under Study 1>Solver Configurations>Solver

1>Dependent Variables 2 click Concentration (comp1.c_P).

11 In the Field settings window, locate the Scaling section.

12 From the Method list, choose Manual.

13 In the Scale edit field, type 10.

14 In the Model Builder window, expand the Study 1>Solver Configurations>Solver

1>Time-Dependent Solver 1 node.

15 Right-click Study 1>Solver Configurations>Solver 1>Time-Dependent Solver 1 and choose Fully Coupled.

16 In the Fully Coupled settings window, click to expand the Method and termination section.

17 Locate the Method and Termination section. From the Jacobian update list, choose Once per time step.

18 On the Home toolbar, click Compute.

Physics Solve for Discretization

Laminar Flow × physics

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R E S U L T S

Velocity (spf)Create Mirror 3D data sets to create plots for the full flow-cell geometry.

Data Sets1 On the Results toolbar, click More Data Sets and choose Mirror 3D.

2 In the Mirror 3D settings window, locate the Plane Data section.

3 From the Plane list, choose xy-planes.

4 In the z-coordinate edit field, type 5e-4.

5 On the Results toolbar, click More Data Sets and choose Mirror 3D.

6 In the Mirror 3D settings window, locate the Plane Data section.

7 From the Plane list, choose xz-planes.

8 In the y-coordinate edit field, type -3e-3.

Velocity (spf)1 In the Model Builder window, under Results click Velocity (spf).

2 In the 3D Plot Group settings window, locate the Data section.

3 From the Data set list, choose Mirror 3D 2.

4 In the Model Builder window, expand the Velocity (spf) node, then click Slice 1.

5 In the Slice settings window, locate the Plane Data section.

6 From the Plane list, choose xy-planes.

7 In the Planes edit field, type 1.

8 On the 3D plot group toolbar, click Plot.

Next create the plots shown in Figure 3 through Figure 6, illustrating the analyte concentration and adsorbed species concentration as function of time.

3D Plot Group 61 On the Home toolbar, click Add Plot Group and choose 3D Plot Group.

2 In the 3D Plot Group settings window, locate the Data section.

3 From the Data set list, choose Mirror 3D 2.

4 From the Time (s) list, choose 35.

5 Right-click Results>3D Plot Group 6 and choose Slice.

6 In the Slice settings window, locate the Expression section.

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7 Click Concentration (c_P) in the upper-right corner of the section. Locate the Plane

Data section. From the Plane list, choose xy-planes.

8 In the Planes edit field, type 1.

9 Locate the Coloring and Style section. Clear the Color legend check box.

10 Right-click Results>3D Plot Group 6>Slice 1 and choose Deformation.

11 In the Deformation settings window, locate the Expression section.

12 In the z component edit field, type c_P.

13 On the 3D plot group toolbar, click Plot.

14 In the Model Builder window, click 3D Plot Group 6.

15 In the 3D Plot Group settings window, locate the Data section.

16 From the Time (s) list, choose 45.

17 On the 3D plot group toolbar, click Plot.

18 From the Time (s) list, choose 55.

19 On the 3D plot group toolbar, click Plot.

20 From the Time (s) list, choose 75.

21 On the 3D plot group toolbar, click Plot.

3D Plot Group 71 On the Home toolbar, click Add Plot Group and choose 3D Plot Group.

2 In the 3D Plot Group settings window, locate the Data section.

3 From the Data set list, choose Mirror 3D 2.

4 From the Time (s) list, choose 35.

5 Right-click Results>3D Plot Group 7 and choose Surface.

6 In the Surface settings window, locate the Expression section.

7 Click Surface coverage (chsr.theta_i_cs_P) in the upper-right corner of the section. Locate the Coloring and Style section. Clear the Color legend check box.

8 On the 3D plot group toolbar, click Plot.

9 In the Model Builder window, click 3D Plot Group 7.

10 In the 3D Plot Group settings window, locate the Data section.

11 From the Time (s) list, choose 45.

12 On the 3D plot group toolbar, click Plot.

13 From the Time (s) list, choose 55.

14 On the 3D plot group toolbar, click Plot.

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15 From the Time (s) list, choose 75.

16 On the 3D plot group toolbar, click Plot.

Finally create line plots showing the average coverage of surface species as a function of time and pillar position. Note how the Duplicate feature speeds up this procedure.

1D Plot Group 81 On the Home toolbar, click Add Plot Group and choose 1D Plot Group.

2 In the 1D Plot Group settings window, click to expand the Title section.

3 From the Title type list, choose Manual.

4 Locate the Plot Settings section. Select the y-axis label check box.

5 In the associated edit field, type Surface Fraction P.

6 On the 1D plot group toolbar, click Point Graph.

7 Select Point 10 only.

8 In the Point Graph settings window, locate the y-Axis Data section.

9 In the Expression edit field, type ave_center_1(chsr.theta_i_cs_P).

10 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Color list, choose Blue.

11 Click to expand the Legends section. Select the Show legends check box.

12 From the Legends list, choose Manual.

13 In the table, enter the following settings:

14 Right-click Results>1D Plot Group 8>Point Graph 1 and choose Duplicate.

15 In the Point Graph settings window, locate the y-Axis Data section.

16 In the Expression edit field, type ave_center_4(chsr.theta_i_cs_P).

17 Locate the Coloring and Style section. Find the Line markers subsection. From the Line list, choose Dash-dot.

18 Locate the Legends section. In the table, enter the following settings:

19 Right-click Point Graph 1 and choose Duplicate.

Legends

Center row, first pillar

Legends

Center row, last pillar

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20 In the Point Graph settings window, locate the y-Axis Data section.

21 In the Expression edit field, type ave_wall_1(chsr.theta_i_cs_P).

22 Locate the Coloring and Style section. Find the Line style subsection. From the Color list, choose Red.

23 Locate the Legends section. In the table, enter the following settings:

24 In the Model Builder window, under Results>1D Plot Group 8 right-click Point Graph

2 and choose Duplicate.

25 In the Point Graph settings window, locate the y-Axis Data section.

26 In the Expression edit field, type ave_wall_4(chsr.theta_i_cs_P).

27 Locate the Coloring and Style section. Find the Line style subsection. From the Color list, choose Red.

28 Locate the Legends section. In the table, enter the following settings:

29 On the 1D plot group toolbar, click Plot.

Legends

Wall row, first pillar

Legends

Wall row, last pillar

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