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Model

This year our modeling was mainly to simulate the mass transfer and fluid concentration in different channels of microfluidic chips. Because the principle calculation in the microchannel was quite complicated, we follow the calculation of the equivalent circuit: We used resistor analog flow resistance, using current analog flow, and using voltage analog fluid pressure. The liquid inlet can be regarded as a power source, and we designed the liquid outlet as grounding. Then we calculated the concentration gradient of microfluidic chip with Kirchhoff Voltage Law and Kirchhoff’s Current Law. As shown in the Figure 1:

1. Fluid mass transfer modeling

In the microchannel, the Reynolds number of the fluid is small (Re<1), so the liquid to be diluted and the dilution liquid exchange solute mainly through laminar diffusion in the channel. The resistance in the microchannel is mainly the liquid viscous force. Under the influence of viscous force, the liquid flow is stable, and under this circumstance we can use the fluid convection mass transfer law to calculate the fluid concentration in the mixing channel.

At the micro-nano scale, the diffusion between multiple substances is

(2.1)

c represents the material concentration, v stands for the flow velocity vector, and D is the diffusion coefficient.

If the velocity of the fluid is set as u, then the hydraulic radius is:

(2.2)

H represents the height of the chip, w represents the width of the chip.

Simultaneous equations 2.1 and 2.2:

(2.3)

In the above equation, Sc = μ / ρd is the Schmidt number, and Re = ρuL / μ is the Reynolds number.

According to Fick formula, the dimension of diffusion coefficient, which is represented by D, can be obtained:

(2.4)
(2.5)

From the mass transfer equation, we can obtain the mixing situation of liquid, by which the flow rate and concentration of liquid can be calculated:

(2.6)

c represents the concentration, D is the liquid diffusion coefficient, and μ is the fluid velocity.

2. Microchannel and internal fluid modeling

When the fluid flows through the microchannel, the state of the fluid is different due to the difference of the microchannels. For example, the narrower the microchannel, the greater the fluid resistance, and the longer the microchannel, the greater the fluid resistance. The resistance of the microchannel to the fluid can be determined by Poiseuille's law:

(2.7)

Let R = 8ηL / (πr4), R is the flow resistance, then Q =δp/R.

In the above equation, q is liquid flow, r is channel radius, and η is viscosity.

3. Microchannel equivalent circuit modeling

According to the relationship between mixed liquid concentration and flow rate in laminar diffusion:

(2.8)

C is the liquid concentration and Q is the liquid flow rate. By this formula, we can calculate the flow rate at the mixed outlet of each layer. If the outlet flow is Q0, according to Kirchhoff's current law, the flow in the channel can be calculated iteratively:

(2.9)

The flow at the exit is equal to the higher flow.

According to Kirchhoff Voltage Law, the flow resistance of each channel is obtained:

(2.10)

After that, the formula for calculating the relevant parameters of the chip channel can be derived, then we can obtain the liquid concentration in the channel

References

  • [1]Wang H , Chen C H , Xiang Z , et al. A convection-driven long-range linear gradient generator with dynamic control[J]. Lab Chip, 2015, 15(6):1445-1450.
  • [2]Yang Yu B , Elbuken C , Ren C L , et al. Image processing and classification algorithm for yeast cell morphology in a microfluidic chip[J]. Journal of Biomedical Optics, 2011, 16(6):066008.