Controlled Diffusion Micromixer 1
Introduction
This model treats an H-shaped microfluidic device for controlled mixing through diffusion. The device 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. The device concept is illustrated in Figure 1.Figure 1: Diagram of the device.
Model Definition
The geometry of the device is shown in Figure 2. The device geometry is split in two becau of symmetry. The design aims to maintain a laminar flow field when the two streams, A and B, are united and thus prevent uncontrolled convective mixing. The transport of species between streams A and B should take place only by diffusion in order that species with low diffusion coefficients stay in their respective streams.
1. This example was originally formulated by Albert Witarsa under Professor Bruce Finlayson’s supervi
sion at the University of Washington in Seattle. It was part of a graduate cour in which the assignment consisted of using mathematical modeling to evaluate the potential of patents in the field of microfluidics.
Diffusion perpendicular to
the flow transports species
from stream A to stream B Outlet A
Inlet A Stream A
Outlet B
Inlet B Stream B
Figure 2: Model geometry. 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, so the actual channel is twice as high in the z-direction.
The flow rate at the inlet is approximately 0.1 mm/s. The Reynolds number, which is important for characterizing the flow is given by:
where ρ is the fluid density (1000 kg/m 3), U is a characteristic velocity of the flow (0.1 mm/s), μ is the fluid viscosity (1 mPa ⋅s) and L is a characteristic dimension of the device (10 μm). When the Reynolds number is significantly less than 1, as in this example, the Creeping Flow interface can be ud. The convective term in the
Navier-Stokes equations can be dropped, leaving the incompressible Stokes equations:
where u is the local velocity (m/s) and p is the pressure (Pa).
Mixing in the device involves species at relatively low concentrations compared to the solvent, in this ca water. This means that the solute molecules interact only with Symmetry plane (blue)
Re ρUL μ
------------0,001==∇p I –μu ∇u ∇()T
+()+()⋅0
=∇u ⋅0
=
water molecules, and Fick’s law can be ud to describe the diffusive transport. The mass-balance equation for the solute is therefore:
where D is the diffusion coefficient of the solute (m 2/s) and c is its concentration (mol/m 3). Diffusive flows can be characterized by another dimensionless number: the Peclet number, which is given by:
In this model, the parametric solver is ud to solve Equation 1 for three different species, each with different values of D: 1×10-11 m 2/s, 5×10-11 m 2/s, and
1×10-10 m 2/s. The values of D correspond to Peclet numbers of 100, 20 and 10 respectively. Since the Peclet numbers are all greater than 1, implying a cell Peclet number significantly greater than 1, numerical stabilization is required when solving Fick’s equation. COMSOL automatically includes the stabilization by default, so no explicit ttings are required.
Two versions of the model are solved:
•In the first version, it is assumed that a change in solute concentration does not influence the fluid’s density and viscosity. This implies that it is possible to first solve the Navier-Stokes equations and then solve the mass balance equation.
•In the cond version, the viscosity depends quadratically on the concentration:
Here α is a constant of dimension m 6/(mol)2 and μ0 is the viscosity at zero
concentration. Such a relationship between concentration and viscosity is usually obrved in solutions of larger molecules.Results and Discussion
Figure 3 shows the velocity field for the ca where viscosity is concentration
independent. The flow is symmetric and is not influenced by the concentration field. Figure 4 shows the corresponding pressure distribution on the channel walls that
∇D c ∇–c u +()⋅–0
=Pe LU D
--------=μμ01αc 2
+()
=
results from the flow.
Figure 3: Flow velocity field.
Figure 4: Pressure distribution on the channel walls.
Figure 5: Concentration distribution for a species with diffusivity 1·10-11 m2/s.
Figure 6: Concentration distribution for a species with diffusivity 5·10-11 m2/s.
Figure 7: Concentration distribution for a species with diffusivity 1·10-10 m2/s.
coefficient.
