Microchannel H-Cell

This example models an H-microcell for separation through diffusion. In Figure 1, it is shown how the cell puts two different laminar streams in contact for a controlled period of time. The contact surface is well defined, and by controlling the flow rate it is possible to control the amount of species transported from one stream to the other through diffusion. This example was originally formulated by Albert Witarsa under Professor Bruce Finlayson’s supervision at the University of Washington in Seattle. It was part of a graduate course in which the assignment consisted of using mathematical modeling to evaluate the potential of patents in the field of microfluidics.

The model utilizes the Laminar Flow and Transport of Diluted Species interfaces to fully capture the separation within the cell.

Figure 1: Diagram of the H-microcell (dimensions in meters).

Model Definition

The geometry of the microcell is shown in Figure 2. Due to symmetry, the model geometry can be simplified to half of it.

The design is aimed to eliminate upsets in the flow field when the two streams, A and B, are united. Thus, the cell enables mixing of A and B solely through diffusion. A system allowing convection would mix all species equally and lead to loss of control over the separation abilities.

Figure 2: Model geometry as given by Albert Witarsa’s and Professor Finlayson. To avoid any type of convective mixing, the design must smoothly let both streams come in contact with each other. Due to symmetry, it is sufficient to model half the geometry.

The simulations involve solving the fluid flow in the H-cell. According to the specifications, the flow rate at the inlet is roughly 0.1 mm/s. This implies a low Reynolds number, well inside the region of laminar flow:

(1)

Equation 1 gives a Reynolds number of 0.001 for a water solution and the channel dimensions given in Figure 1. This value is typical for microchannels. Additionally, this indicates that it is easy to get a numerical solution of the full momentum balance and continuity equations with a reasonable number of elements. The Laminar Flow interface can set up and solve the incompressible Navier-Stokes equations at steady state:

Here ρ denotes density (SI unit: kg/m3), u is the velocity (SI unit: m/s), μ denotes viscosity (SI unit: Pa·s), and p equals pressure (SI unit: Pa).

Separation in the H-cell involves species in relatively low concentrations compared to the solvent, in this case water. This means that the solute molecules interact only with water molecules, and it is safe to use Fick’s law to describe the diffusive transport in the cell. Use the Transport of Diluted Species interface to set up and solve the appropriate stationary mass-balance equation:

(2)

In this equation, D denotes the diffusion coefficient (SI unit: m2/s) and c represents the concentration (SI unit: mol/m3). In this model, you use the parametric solver to solve Equation 2 for three different values of D — 1·10−11 m2/s, 5·10−11 m2/s, and 1·10−10 m2/s — to simulate the mixing of different species.

You solve two versions of the model:

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In the first version, you assume that a change in concentration does not influence the fluid’s density and viscosity. This implies that it is possible to first solve for the fluid flow and then for the mass transport.

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In the second version, you include a correction term in the viscosity that depends quadratically on the concentration in Equation 1:

(3)

Here α is a constant of dimension (SI unit: m6/mol2). An influence of concentration on viscosity of this kind is usually observed in solutions of larger molecules. In this case the flow and mass transport equations have to be solved simultaneously.

Last, consider the boundary conditions. All boundaries are visualized in Figure 3.

Figure 3: Model domain boundaries.

For the Laminar Flow interface:

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At the inlets and outlets, Pressure conditions apply along with vanishing viscous stress. Setting the pressure at the outlets to zero, the pressure at the inlets represents the pressure drop over the cell. These inlet and outlet conditions comply with the H-cell being a part of a channel system of constant width, which justifies the assumption of developed flow.

•

At the walls, No slip conditions state that the velocity is zero.

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At the symmetry plane, using the Symmetry boundary condition sets the velocity component in the normal direction of the surface to zero.

For the Transport of Diluted Species:

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At the inlets, use the Concentration boundary condition to set concentration. At inlets A and B the concentration are 1 mol/m3 and 0 mol/m3, respectively.

•

At the outlets, apply the convective flux condition through the Outflow boundary condition, stating that the diffusive transport perpendicular to the boundary normal is negligible. This condition will thus eliminate concentration gradients in the flow direction.

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Model the symmetry plane and cell walls with the No Flux condition. This equation states that the flux of species perpendicular to the boundary equals zero.

Results and Discussion

Figure 4 shows the velocity field for the whole microcell. The flow is symmetric and is not influenced by the concentration field.

Figure 4: Flow velocity field.

Figure 5 shows the concentration distribution for the species with the highest simulated diffusivity.

Figure 5: Concentration distribution for a species with diffusivity 10−10 m2/s.

Because of the relatively large diffusion coefficient, the degree of mixing is almost perfect. The species with a diffusion coefficient ten times smaller shows a different result. The concentration distribution in Figure 6 shows that the diffusion coefficient for the species too low for achieving significant mixing of the streams.

Figure 6: Concentration distribution for a species with diffusivity 1·10−11m2/s.

The simulation clearly shows that the H-cell can separate lighter molecules from heavier ones. A cascade of H-cells can achieve a very high degree of separation.

In some cases, especially those involving solutions of macromolecules, the macromolecule concentration has a large influence on the liquid’s viscosity. In such situations, the model needs to be fully coupled and solve both interfaces simultaneously. Figure 7 shows the results of such a simulation. Here, the changes in viscosity have caused an asymmetry in the velocity. As a consequence of the modified flow field, the transport of molecules to outlet B is also different from the constant flow field case (Figure 8 versus Figure 5).

Figure 7: Velocity field. The viscosity varies with the concentration according to Equation 3 with α = 0.5 (m3/mol)2. The figure shows that the velocity field is affected by variations in concentration.

Figure 8: Concentration distribution for the species with diffusivity 1·10−10m2/s for the case where the fluid viscosity varies with concentration. Comparison with the plot in Figure 5 shows that fewer molecules of the species are transported to outlet B.

Chemical_Reaction_Engineering_Module/Mixing_and_Separation/microchannel_h_cell

Modeling Instructions

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