Cooling/Heating Mass Flow Rate at Micro/Nano Channels 
Written by Ehsan Roohi Golkhatmi 
We introduce a new technique named as “Iterative Technique” to implement both negative and positive wall heat flux rate values. The foundation of the iterative technique is based on the correction of the wall temperature during the DSMC simulation. We calculate wall temperature from the following relation
Eq. (1) is a local relation which is applied to every wall boundary cells in the domain. We should define wall temperature correction values ΔT_{w}(x) such that the iterative algorithm converges and the final wall temperature distribution becomes physical. Positive values of wall heat flux means that heat transfer from flow to the wall, i.e., cooling, and negative value showed that heat transfers from the wall to the flow, i.e., heating. Since total energy of reflected molecule is proportional to the wall temperature, higher wall temperature values result in more negative wall heat flux rates. Therefore, from the stability point of view, ΔT_{w}(x) must satisfy the following condition
According to Eq. (2), wall temperature correction for each boundary cell should be adjusted considering local heat flux rate of that cell. To apply this criterion, we introduce a nondimensional proportionality based on conditions given by Eq. (2)
The stability criteria given by Eq. (2) are well considered in the suggested proportionality, i.e., Eq. (3), for both positive and negative values of desired wall heat flux rates. With the substitution of Eq. (3) in Eq. (1), we have
where RF is a relaxation factor to control the convergence progress and avoid solution divergence. RF must be adjusted in a manner which ensures the best convergence behavior. Based on our numerical experiences, RF should be considered no greater than 0.03. The RF values greater than 0.03 may lead to divergence. However, it is observed in several simulation cases that RF could be considered up to 0.1 without divergence penalty. e_{0} is a nonzero and positive value which is negligible with respect to the incident energy fluxes. It is defined for adiabatic wall cases, i.e., . Equation (4) is called as “Iterative Equation” which implements a desired heat flux rate distribution by correcting the wall temperature. There must be an initial wall temperature distribution to start the iterative technique. We could use any desired initial wall temperature distribution, i.e., uniform distribution. Figure 1 shows the schematic algorithm of the iterative technique. During one wall sampling period ∆t, a specified wall temperature or classical DSMC simulation is performed and incident and reflected energy fluxes for all wall cells are sampled. At the end of sampling period ∆t, local wall temperature values T_{w}(x) are obtained by Eq. (4) and an updated wall temperature distribution is used for the DSMC simulation on the next sampling period. It should be noted that there is no additional complexity in the use of our iterative technique for flow with multiple species.
The effect of the heating and cooling processes on the mass flow rate of the channel is investigated in Fig. 2. The results are compared with those of Wang et al. [1], which correspond to the flow heating simulation. By comparison, it is well seen that there is a good agreement between current results with those of Ref. [1]. According to this figure, the flow heating by the walls decreases mass flow rate through the channel, and vice versa. The rate of mass flow rate increase is faster for more positive values of q_{des}. The increase in mass flow rate for this condition could be attributed to smaller decrease in the density field along the channel for cooling conditions, see Fig. 2. According to this observation, we could use normal heat flux parameter as a suitable tool to control and adjust desired mass flow rate in micro/nano devices.
Fig. 9. Dependence of the mass flow rate on the wall heat flux values and comparison of the current results with those of Wang et. al. [1].
Reference 1Q. W. Wang, C. L. Zhao, M. Zeng, and N. Y. E. Wu, Numerical Heat Transfer Part B 53, 174187 (2008).
