40
3 Computational Analysis of Flow and Mixing in Micromixers
differencing scheme is likely to introduce considerable discretization error in the
analysis of complex flows. However, higher-order upwind (second- and third-order
accurate) schemes tend to reduce numerical diffusion. Kim and coworkers [2, 10,
14, 23, 24] conducted flow and mixing analyses in different micromixers. They used
a high resolution scheme to discretize the advection terms in the governing equations. The scheme reduced the numerical discretization errors using an automatic
correction algorithm [12].
The linearized algebraic system of equations resulting from discretization can
be solved using direct and iterative methods. However, the use of a direct method
is usually not economical, and the iterative methods are more widely used. Multigrid accelerated incomplete lower–upper (ILU) factorization procedure was used in
many studies. Hong et al. [1] used conjugate gradient with preconditioning solver,
and convergence limit for mass fraction set to 10
–6 for mixing study in a mixer
with modified Tesla structure. Glatzel et al. [25] performed a comparative evaluation
of commercial CFD packages for microfluidic applications. Four commercial finite
volume codes, CFD-ACE + ®, ANSYS-CFX®, ANSYS-Fluent® and Flow-3D®
have been tested to study the flow and mixing in a SAR micromixer and a rotating
microchannel. For the numerical solution of the advection–diffusion equation, all
the tested codes showed good predictions qualitatively, but suffered from numerical diffusion. However, any quantitative assessment of them by comparing with
experimental data was not performed.
3.2 Lagrangian Approach to Mixing
As explained earlier, the numerical solution of the advection–diffusion equation
for concentration field can be affected by discretization errors through numerical
diffusion. To avoid this problem, Lagrangian particle tracking has been used by
many researchers to study mixing in different micromixers [26–32].
In the Lagrangian formulation, a large number of particles are introduced at the
inlet of the micromixer, and tracked through the flow field for the continuous phase
obtained from the solution of governing Eqs. (3.1) and (3.2). The effect of particles
on the flow field is negligible (one-way coupling). Movement of a massless particle
inside the flow is determined by integrating the vector equation of motion for the
particle:
d x p
dt
= V p
(3.4)
For the massless particles, the particle velocity is equal to the velocity of the
continuous phase. Hence, the trajectory of each particle can be obtained using the
particle velocity V p = V, where V p and V are velocities of the particle and continuous
phase, respectively. The new particle location along the trajectory is calculated using
Eq. (3.4).
3 Computational Analysis of Flow and Mixing in Micromixers
differencing scheme is likely to introduce considerable discretization error in the
analysis of complex flows. However, higher-order upwind (second- and third-order
accurate) schemes tend to reduce numerical diffusion. Kim and coworkers [2, 10,
14, 23, 24] conducted flow and mixing analyses in different micromixers. They used
a high resolution scheme to discretize the advection terms in the governing equations. The scheme reduced the numerical discretization errors using an automatic
correction algorithm [12].
The linearized algebraic system of equations resulting from discretization can
be solved using direct and iterative methods. However, the use of a direct method
is usually not economical, and the iterative methods are more widely used. Multigrid accelerated incomplete lower–upper (ILU) factorization procedure was used in
many studies. Hong et al. [1] used conjugate gradient with preconditioning solver,
and convergence limit for mass fraction set to 10
–6 for mixing study in a mixer
with modified Tesla structure. Glatzel et al. [25] performed a comparative evaluation
of commercial CFD packages for microfluidic applications. Four commercial finite
volume codes, CFD-ACE + ®, ANSYS-CFX®, ANSYS-Fluent® and Flow-3D®
have been tested to study the flow and mixing in a SAR micromixer and a rotating
microchannel. For the numerical solution of the advection–diffusion equation, all
the tested codes showed good predictions qualitatively, but suffered from numerical diffusion. However, any quantitative assessment of them by comparing with
experimental data was not performed.
3.2 Lagrangian Approach to Mixing
As explained earlier, the numerical solution of the advection–diffusion equation
for concentration field can be affected by discretization errors through numerical
diffusion. To avoid this problem, Lagrangian particle tracking has been used by
many researchers to study mixing in different micromixers [26–32].
In the Lagrangian formulation, a large number of particles are introduced at the
inlet of the micromixer, and tracked through the flow field for the continuous phase
obtained from the solution of governing Eqs. (3.1) and (3.2). The effect of particles
on the flow field is negligible (one-way coupling). Movement of a massless particle
inside the flow is determined by integrating the vector equation of motion for the
particle:
d x p
dt
= V p
(3.4)
For the massless particles, the particle velocity is equal to the velocity of the
continuous phase. Hence, the trajectory of each particle can be obtained using the
particle velocity V p = V, where V p and V are velocities of the particle and continuous
phase, respectively. The new particle location along the trajectory is calculated using
Eq. (3.4).