# Numerical and Experimental Investigations of a Micromixer with Chicane Mixing Geometry
## Materials and Methods
### Numerical Simulation  
In order to achieve high accuracy of the simulation and due to a limited performance capacity of the computing system, only the Y-mixer and the first 4 meanders have been computed. The simulation was performed using the CFD software ANSYS CFX, version 14.0, utilizing a finite volume method and a coupled solver.  
The mathematical model consists of a steady-state Navier-Stokes equation for incompressible flow of Newtonian fluids and the diffusion-convection transport equation.  
\[\begin{split}\rho\overset{\rightarrow}{\mathrm{v}}\cdot\
abla \overset{\rightarrow}{\mathrm{v}}&=-\
abla\rho+\eta\
abla^{2} \overset{\rightarrow}{\mathrm{v}}\\ \
abla\overset{\rightarrow}{\mathrm{v}}&=0\\ \
abla\cdot(D\
abla C)-\
abla\cdot(\overset{\rightarrow}{ \mathrm{v}}C)&=0\end{split} \tag{1}\]  
The fluid is assumed to be isothermal and the process is steady-state.  
The walls of the micromixer have no-slip boundary conditions. Special consideration should be given to inlet boundary conditions. It was shown by [PERSON] et al. [29], that the inlet velocity profile has a significant influence on the mixing process. For this reason, two 4.5 mm long channels were added before the Y-mixer. Moreover, fully developed velocity profiles in a round channel were applied on both inlets. At the outlet, the zero static pressure was used as the outlet boundary condition. An additional section with a straight channel was added after the last meander element in order to avoid the backflow effect.  
Because strongly diluted solutions were used in the experiments, the properties of the two working fluids were determined as related parameters of water at a temperature T = 20 \({}^{\circ}\)C. The density, viscosity and diffusion coefficient were set as \(\rho\) = 998 kg/m\({}^{3}\), \(\mu\) = 10\({}^{-3}\) Pa\(\cdot\)s and D = 2 \(\times\) 10\({}^{-9}\) m\({}^{2}\)/s.  
For the determination of flow regimes, the Reynolds number was calculated as:  
\[\mathrm{Re}=\
u L\rho/\mu \tag{2}\]  
where L = 1 mm.  
A mixing index was used to estimate the grade of mixing. It is derived from the degree of segregation, proposed by [PERSON] [30]. The mixing index I\({}_{\mathrm{M}}\) represents the ratio of species homogeneity and varies from 0 (for a completely segregated species) to 1 (for a completely mixed species):  
\[\mathrm{I}_{\mathrm{M}}=1-(\mathrm{I}_{\mathrm{S}})^{1/2}=1-(\sigma^{2}/\sigma ^{2}_{\mathrm{max}})^{1/2} \tag{3}\]  
where \(\sigma\)--is the variance of concentration, \(\sigma_{\mathrm{max}}\)--the maximal variance of concentration, I\({}_{\mathrm{S}}\)--the degree of segregation. These parameters can be estimated in this way:  
Figure 3: Image post processing.  
\[\sigma=\frac{1}{\text{A}}\int\limits_{\text{A}}\left(\text{C}-\overline{\text{C}} \right)^{2}\text{dA and }\sigma_{\text{max}}=\frac{1}{\text{A}}\int\limits_{\text{A}}\text{C}(1- \text{C})\text{dA} \tag{4}\]  
where C--the concentration and A--the cross-sectional area.  
The discretization in CFD applications is a very important procedure [31]. The so-called numerical diffusion plays a crucial role in the simulation of the mixing process. To decrease numerical induced diffusion, the second-order discretization scheme is used. According to high values of gradients, the double precision for the calculations has been used, therefore increasing accuracy. Convergence criteria were defined as followed:  
* Residuals RMS are less than 1 \(\times\) 10\({}^{-6}\);
* Domain imbalances are less than 0.1%.  
Average mixing index in outlet section reaches a steady solution (variation is less than 1% in comparison with the previous iteration).  
Moreover, the grid independence analysis was carried out and the results are shown in Figure 4. A grid convergence study in terms of the grid convergence index (GCI) is presented in Appendix A.  
According to the results, the mesh with 23.2 Mio tetrahedral elements was chosen with an average element size of 20 \(\upmu\)m.