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Model Library > Chemical Reaction Engineering Module > Mass Transport

Electroosmotic Flow in Porous Media

Introduction

This example treats the modeling of electroosmotic flow in porous media (Ref. 1, Ref. 2). The system consists of a

compartment of sintered porous material and two electrodes that generate an electric field. The cell combines pressure-

driven and electroosmotic flow.

The purpose of this model is to illustrate the modeling of electroosmosis and electrophoresis in porous media in COMSOL

Multiphysics.

Model Definition

Figure 1 shows the system’s geometry. It is made up of a domain of porous material containing the electrolyte and two

electrodes that generate a potential difference. The conductivity is very small, and the model assumes that the effects of

electrochemical reactions at the electrode surfaces are negligible.

Figure 1: The modeled domain consists of an electrolyte contained in a porous structure and two electrodes that generate an electric field.

In the first part of the model, you solve the continuity equations for the flow velocity and the current density at steady

state:

(1)

Here u denotes the velocity (m/s) and i represents the current-density vector (A/m2). The velocity includes two driving

forces—a pressure term and an electroosmotic term:

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7/11/2011http://127.0.0.1:1034/help/topic/com.comsol.help.models.chem.electroosmotic_flow/electr...

(2)

In this equation, εp denotes the porosity, a is the average radius of the pores (m), µ gives the fluid’s dynamic viscosity

(Pa·s), τ represents the tortuosity of the porous structure, εw is the fluid’s permittivity (F/m), p gives the pressure (Pa),

ζ is the zeta potential (V), and V equals the potential (V). The current density is given by

(3)

where κ denotes the conductivity (S/m).

At the solid walls, the normal velocity component vanishes:

(4)

At the inlet and the outlet, the pressure is fixed:

(5)

and

(6)

The boundary conditions for the current-density balance are insulating for all boundaries except the electrode surfaces,

where the potential is fixed:

(7)

In the second model stage, you use the steady-state velocity and potential fields in a transient simulation of the

concentration of a charged tracer species injected into the system, assuming that the tracer species does not influence

the conductivity or the set potential in the porous structure. The mass-transport equation for the tracer reads

(8)

where N is the flux vector given by the Nernst-Planck equation

(9)

In this equation, D denotes the tracer’s diffusivity (m2/s), c gives its concentration (mol/m

3), z represents the tracer’s

charge number, and F is Faraday’s constant (C/mol). The mobility, um (mol·m

2/(J·s)), is given by the Nernst-Einstein

equation

(10)

where Rg = 8.314 J/(mol·K) is the gas constant and T (K) is the temperature.

The boundary conditions for Equation 8, the mass-transport equation, are insulating except at the inlet and the outlet.

There you use the Flux condition to set the diffusive and convective contributions to the flux through the boundaries to

zero:



(11)

The initial concentration is given by a distribution that is uniform in the y direction and bell-shaped in the x direction:

(12)

Here ctop denotes the peak concentration, xm is the position of the peak along the x-axis, and σ equals the base width of

the peak.

Results

The upper plot in Figure 2 shows the flow distribution in the porous structure when only the electric field acts as a driving

force. In the lower plot, it is instead the pressure gradient that drives the flow. In both cases, the velocity is large around

the corners of the electrodes, where the field strength is large and the effect of the decrease in the geometry’s cross

section is most pronounced. A comparison of the maximum flow-velocity values shows that the pressure gradient is the

dominating driving force.



Figure 2: Velocity distributions in the cell with an applied electric field (top) and an applied pressure difference (bottom). The scale unit for

the magnitude surface plot is m/s.

Figure 3 shows the concentration field in the case where both pressure and electroosmotic forces are included. The figure

shows that the migration of the charged tracer also influences the transport rate. In this case, the tracer is transported

both by the movement of the flow, due to pressure and electroosmotic terms, and the electrophoretic effect on the

charged tracer itself. At t = 0, the tracer is introduced as a bell-shaped vertical distribution near the inlet. In the

subsequent simulation, this distribution is sheared and transported by diffusion, migration, and convection.



Figure 3: Concentration distribution in the domain at different times after injection. The effects of diffusion, convection, and migration

deform the initial bell-shaped pulse.

Figure 4 shows the cross sections of the pulse along a line at y = 2.5 mm for the times t = 0, 0.5 s, 1.0 s, 1.5 s, and t =

2.0 s. Diffusion smears out the pulse, while migration and convection mainly translate and shear it, depending on the

pressure and the electric field.

Figure 4: Cross-section plot of the concentration along a horizontal line at y = 2.5 mm for the times t = 0, 0.5 s, 1.0 s, 1.5 s, and 2.0 s.

Despite its geometrical simplicity, this model includes the components needed to model complex-shaped domains.



References

1. M-S. Chun, “Electrokinetic Flow Velocity in Charged Slit-like Microfluidic Channels with Linearized Poisson-Boltzmann

Field,” Korean J. Chem. Eng., vol. 19, no. 5, pp. 729–734, 2002.

2. S. Yao, D. Huber, J. Mikkelsen, and J.G. Santiago, “A Large Flowrate Electroosmotic Pump with Micron Pores,” Proc.

IMECE, 2001 ASME International Mechanical Engineering Congress and Exposition, New York, November 2001.

Notes About the COMSOL Implementation

The COMSOL Multiphysics implementation is straightforward, and the only aspect to remember is to solve the steady-

state problem first and then the time-dependent problem. Do this by following these steps:

Model Library path: Chemical_Reaction_Engineering_Module/Mass_Transport/electroosmotic_flow

Modeling Instructions

MODEL WIZARD

GLOBAL DEFINITIONS

Parameters

For your convenience, a set of parameters are made available in a text file.

1 Solve the nonlinear steady-state problem for the flow and current distributions.

2 Store the solution.

3 Add a study for the transient simulation of the concentration field.

4 Solve the problem only for the concentration, using the stored solution for the pressure and potential fields.

1 Go to the Model Wizard window.

2 Click the 2D button.

3 Click Next.

4 In the Add physics tree, select Chemical Species Transport>Transport of Diluted Species (chds).

5 Click Add Selected.

6 In the Add physics tree, select Mathematics>PDE Interfaces>General Form PDE (g).

7 Click Add Selected.

8 Click Add Dependent Variable.

9 In the Dependent variables table, enter the following settings:

p

V

10 Click Next.

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

12 Click Finish.

1 In the Model Builder window, right-click Global Definitions and choose Parameters.

2 Go to the Settings window for Parameters.



In these expressions, the predefined physical constants epsilon0_const and R_const refer to the permittivity of

vacuum and the gas constant, respectively.

DEFINITIONS

Variables 1

In the above expressions, the explicit unit brackets implement the correct units for the pressure gradient and

potential gradient components defined using the unitless dependent variables of the General Form PDE interface.

GEOMETRY 1

Rectangle 1

Rectangle 2

Circle 1

Union 1

3 Locate the Parameters section. Click Load from File.

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

1 In the Model Builder window, right-click Model 1>Definitions and choose Variables.

2 Go to the Settings window for Variables.

3 Locate the Variables section. Click Load from File.

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

1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Rectangle.

2 Go to the Settings window for Rectangle.

3 Locate the Size section. In the Width edit field, type 8e-3.

4 In the Height edit field, type 3e-3.

5 Locate the Position section. In the x edit field, type -4e-3 .

6 Click the Build All button.

1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.

2 Go to the Settings window for Rectangle.

3 Locate the Size section. In the Width edit field, type 0.2e-3.

4 In the Height edit field, type 1.5e-3.

5 Locate the Position section. In the x edit field, type -2e-3.

6 Click the Build Selected button.

1 In the Model Builder window, right-click Geometry 1 and choose Circle.

2 Go to the Settings window for Circle.

3 Locate the Size and Shape section. In the Radius edit field, type 1e-4.

4 Locate the Position section. In the x edit field, type -1.9e-3.

5 In the y edit field, type 1.5e-3.


1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Union.

2 Select the objects r2 and c1 only.



Copy 1

Difference 1

TRANSPORT OF DILUTED SPECIES

Convection, Diffusion, and Migration

Initial Values 1

3 Go to the Settings window for Union.

4 Locate the Union section. Clear the Keep interior boundaries check box.


1 In the Model Builder window, right-click Geometry 1 and choose Transforms>Copy.

2 Select the object uni1 only.

3 Go to the Settings window for Copy.

4 Locate the Displacement section. In the x edit field, type 3.8e-3.


1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Difference.

2 Go to the Settings window for Difference.

3 Locate the Difference section. Under Objects to add, click Activate Selection.

4 Select the object r1 only to add it to the Objects to add list.

5 Under Objects to subtract, click Activate Selection.

6 Select the objects uni1 and copy1 only.

7 Click the Build All button.

1 In the Model Builder window, click Model 1>Transport of Diluted Species.

2 Go to the Settings window for Transport of Diluted Species.

3 Locate the Transport Mechanisms section. Select the Migration in electric field check box.

1 In the Model Builder window, expand the Transport of Diluted Species node, then click Convection, Diffusion,

and Migration.

2 Go to the Settings window for Convection, Diffusion, and Migration.

3 Locate the Diffusion section. In the Di edit field, type D.

4 Locate the Migration in Electric Field section. In the um,c edit field, type nu.

5 In the zc edit field, type z.

6 Locate the Model Inputs section. Specify the u vector as

u_flow x

v_flow y

7 In the V edit field, type V.

1 In the Model Builder window, click Initial Values 1.

2 Go to the Settings window for Initial Values.

3 Locate the Initial Values section. In the c edit field, type c_init.



Reactions 1

Flux 1

Symmetry 1

PDE

General Form PDE 1

Flux/Source 1

Dirichlet Boundary Condition 1


1 In the Model Builder window, right-click Transport of Diluted Species and choose Reactions.

2 Select Domain 1 only.

1 In the Model Builder window, right-click Transport of Diluted Species and choose Flux.

2 Select Boundaries 1 and 10 only.

3 Go to the Settings window for Flux.

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

5 In the N0,c edit field, type -chds.nmflux_c.

1 In the Model Builder window, right-click Transport of Diluted Species and choose Symmetry.

2 Go to the Settings window for Symmetry.

3 Locate the Boundary Selection section. From the Selection list, select All boundaries.

4 Select Boundaries 2–9 and 11–14 only.

1 In the Model Builder window, expand the Model 1>PDE node, then click General Form PDE 1.

2 Go to the Settings window for General Form PDE.

3 Locate the Conservative Flux section. Specify the first Γ vector as

u_flow x

v_flow y

4 Specify the 2nd Γ vector as

-kappa*Vx x

-kappa*Vy y

5 Locate the Source Term section. In the f edit-field array, type 0 on the first row.

6 In the f edit-field array, type 0 on the 2nd row.

1 In the Model Builder window, right-click PDE and choose Flux/Source.

2 Select Boundaries 2, 3, 6, and 9 only.

1 In the Model Builder window, right-click PDE and choose Dirichlet Boundary Condition.


3 Go to the Settings window for Dirichlet Boundary Condition.

4 Locate the Dirichlet Boundary Condition section. Clear the Prescribed value of p check box.





MESH 1

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

Size

STUDY 1

In Study 1, solve for the stationary pressure and potential fields only.

Step 1: Stationary

RESULTS

Visualize the stationary potential and flow fields in a combined surface, contour, and arrow plot.




4 Locate the Dirichlet Boundary Condition section. Clear the Prescribed value of p check box.

5 In the r2 edit field, type V1.


2 Select Boundary 1 only.


4 Locate the Dirichlet Boundary Condition section. Clear the Prescribed value of V check box.


2 Select Boundary 10 only.


4 Locate the Dirichlet Boundary Condition section. In the r1 edit field, type p1.

5 Clear the Prescribed value of V check box.

1 In the Model Builder window, click Size.

2 Go to the Settings window for Size.

3 Locate the Element Size section. Click the Custom button.

4 Locate the Element Size Parameters section. In the Maximum element size edit field, type 1e-4.

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

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

2 Go to the Settings window for Stationary.

3 Locate the Physics Selection section. In the associated table, enter the following settings:

PHYSICS INTERFACE USE

Transport of diluted species (chds) ×

4 In the Model Builder window, right-click Study 1 and choose Compute.



2D Plot Group 1

Add a separate study for the transient simulation of the concentration distribution.

MODEL WIZARD

STUDY 2

In this transient study, solve for the concentration field and use the solution from Study 1 for the flow and current

density distributions.

Step 1: Time Dependent

1 In the Model Builder window, expand the Results>2D Plot Group 1 node, then click Surface 1.

2 Go to the Settings window for Surface.

3 In the upper-right corner of the Expression section, click Replace Expression.

4 From the menu, choose PDE>Dependent variable V (V).

5 Click the Plot button.

6 Click the Zoom Extents button on the Graphics toolbar.

7 In the Model Builder window, right-click 2D Plot Group 1 and choose Contour.

8 Go to the Settings window for Contour.


10 From the menu, choose Dependent variable V (V).

11 Locate the Levels section. From the Entry method list, select Levels.

12 In the Levels edit field, type range(0,2.5,50).

13 Locate the Coloring and Style section. From the Color table list, select GrayScale.

14 Clear the Color legend check box.


16 In the Model Builder window, right-click 2D Plot Group 1 and choose Arrow Surface.

17 Go to the Settings window for Arrow Surface.

18 Locate the Expression section. In the x component edit field, type u_flow.

19 In the y component edit field, type v_flow.

20 Select the Description check box.

21 In the associated edit field, type Velocity field.

22 Locate the Coloring and Style section. In the Scale factor edit field, type 0.5.

23 From the Color list, select Black.


1 In the Model Builder window, right-click the root node and choose Add Study.

2 Go to the Model Wizard window.

3 In the Studies tree, select Preset Studies for Selected Physics>Time Dependent.

4 Click Finish.

1 In the Model Builder window, click Study 2>Step 1: Time Dependent.

2 Go to the Settings window for Time Dependent.



Solver 2

RESULTS

Concentration (chds)

The surface plot in 2D Plot Group 2 shows the concentration at t = 2 s. Under the Report branch in the Model Tree,

you can add an Animation feature to see how the distribution evolves with time. Alternatively, to reproduce the plot in

Figure 3, showing the concentration distribution at four different times, follow the steps below.

3 Locate the Study Settings section. In the Times edit field, type range(0,0.1,2).

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

PHYSICS INTERFACE USE

PDE (g) ×

5 Click to expand the Values of Dependent Variables section.

6 Select the Values of variables not solved for check box.

7 From the Method list, select Solution.

8 From the Study list, select Study 1, Stationary.

9 In the Model Builder window, right-click Study 2 and choose Show Default Solver.

10 Expand the Study 2>Solver Configurations node.

1 In the Model Builder window, expand the Study 2>Solver Configurations>Solver 2 node, then click Time-

Dependent Solver 1.

2 Go to the Settings window for Time-Dependent Solver.

3 Click to expand the Absolute Tolerance section.

4 In the Tolerance edit field, type 1e-8.

5 In the Model Builder window, right-click Study 2 and choose Compute.


2 In the Model Builder window, expand the Concentration (chds) node, then click Surface 1.



Proceed to study the effects of each of the two driving forces - the pressure gradient and the electric field- on the

velocity field by reproducing the plots in Figure 2. First visualize the velocity field that results from the electric field

alone.

2D Plot Group 3

Next, plot the velocity field that results from the pressure gradient.

2D Plot Group 4


4 Locate the Data section. From the Data set list, select Solution 2.

5 From the Time list, select 0.

6 Locate the Coloring and Style section. From the Color table list, select TrafficLight.


8 Locate the Data section. From the Time list, select 0.6.


10 From the Time list, select 1.2.


12 From the Time list, select 1.8.


1 In the Model Builder window, right-click Results and choose 2D Plot Group.

2 Go to the Settings window for 2D Plot Group.


4 Right-click Results>2D Plot Group 3 and choose Surface.



7 From the menu, choose Definitions>Flow-velocity electroosmosis term, magnitude (U_eo).

8 Locate the Expression section. From the Unit list, select mm/s.





13 Locate the Expression section. In the x component edit field, type u_eo.

14 In the y component edit field, type v_eo.


16 In the associated edit field, type Flow-velocity electroosmosis term.





4 Right-click Results>2D Plot Group 4 and choose Surface.



Finally, reproduce the cross-section plot of the concentration along the line y = 2.5 mm shown in Figure 4. First define a

Cut Line 2D Data Set feature.

Data Sets



7 From the menu, choose Definitions>Flow velocity pressure term, magnitude (U_p).

8 Locate the Expression section. From the Unit list, select mm/s.





13 Locate the Expression section. In the x component edit field, type u_p.

14 In the y component edit field, type v_p.


16 In the associated edit field, type Flow-velocity pressure term.

17 Locate the Coloring and Style section. In the Scale factor edit field, type 0.5.


1 In the Model Builder window, right-click Results>Data Sets and choose Cut Line 2D.

2 Go to the Settings window for Cut Line 2D.


4 Locate the Line Data section. In row Point 1, set x to -4e-3.

5 In row Point 2, set x to 4e-3.

6 In row Point 1, set y to 2.5e-3.

7 In row Point 2, set y to 2.5e-3.





1D Plot Group 5



3 Locate the Data section. From the Data set list, select Cut Line 2D 1.

4 From the Time selection list, select From list.

5 In the Time list, select 0, 0.5, 1, 1.5, and 2.

6 Right-click Results>1D Plot Group 5 and choose Line Graph.

7 Go to the Settings window for Line Graph.

8 In the upper-right corner of the x-Axis Data section, click Replace Expression.

9 From the menu, choose Geometry and Mesh>Coordinate>x-coordinate (x).

10 Locate the x-Axis Data section. From the Unit list, select mm.




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