# Water Flow Boiling in Micro/Mini Channels Using Volume of Fluid Model
## 2 Numerical Solution  
### Governing Equations  
The experimental setup in the reference [54] consisted of 20 parallel mini-channels with dimensions of 1 mm \(\times\) 1.5 mm \(\times\) 52 mm. The mini-channels were fabricated by electric discharge machining (EDM) on brass blocks. The top surface of the brass block was insulated by silicon, and the bottom surface was heated by cartridge heaters. The numerical study of 20 mini-channels would have been very expensive computationally. To simplify the numerical model, the domain was reduced to a single mini-channel by using periodic boundary conditions. As shown in Figure 1, the boundary conditions of the 3D model consisted of top-side adiabatic, constant heat flux at the bottom, and periodic walls at both sides. The VOF model is the most common method used to simulate FB. It is an inherently conservative method that captures the interface between two phases by assigning a color function \(\alpha\) to each phase. The value of \(\alpha\) varies from 0 to 1, where 0 means the cell is filled by one phase and 1 means the cell is filled by another phase. A piecewise linear interface capture (PLIC) scheme was used with the VOF method to capture the interface. The PLIC captures the interface as a straight line whose direction is given by a normal vector. The normal vector is oriented by interrogating the volume fraction of neighboring cells, and its direction is so selected that the volume fraction of cells is maintained [56]. The conservation equations are solved as follows [56]:  
Continuity:  
\[\frac{1}{\rho_{q}}\left[\frac{\partial}{\partial t}(\alpha_{q}\rho_{q})+\
abla \cdot(\alpha_{q}\rho_{q}\overset{\rightarrow}{v})=S_{q}+\sum_{p=1}^{n}(\dot{m }_{lp}-\dot{m}_{vl})\right] \tag{1}\]  
where primary phase volume is computed based on the following:  
\[\sum_{q=1}^{n}\alpha_{q}=1 \tag{2}\]  
Figure 1: Boundary conditions for the computational domain. where \(\alpha_{q}\) is the volume fraction of \(q\)th phase, \(S_{q}\) is a source term for each phase and is usually zero, \(m_{lv}\) is the mass transfer rate from phase \(l\) to phase \(v\) and \(m_{vl}\) is the mass transfer rate from phase \(v\) to phase \(l\). The Lee phase change model was selected to simulate evaporation. The phase change in the Lee model is directly proportional to the deviation from the saturation temperature, \(T_{sat}\). It is assumed that phase change occurs at constant pressure- the temperature of phases and interface is maintained at \(T_{sat}\). Using the Lee model [57], the mass transfer rates are given as follows:  
If \(T_{l}>T_{sat}\) (evaporation),  
\[m_{lv}=r_{i}\alpha_{l}\rho_{l}\frac{T_{l}-T_{sat}}{T_{sat}} \tag{3}\]  
If \(T_{l}<T_{sat}\) (condensation),  
\[m_{vl}=r_{i}\alpha_{v}\rho_{v}\frac{T_{sat}-T_{v}}{T_{sat}} \tag{4}\]  
The coefficient \(r_{i}\) is called the mass transfer intensity factor and has the unit of s\({}^{-1}\). There is a significant variation in the choice of \(r_{i}\) value. Researchers have used an extensive range of \(r_{i}\) values, ranging from 0.1 to \(1\times 10^{7}\) s\({}^{-1}\), to fine-tune the deviation of \(T_{sat}\), according to their experimental data. Momentum:  
\[\frac{\partial}{\partial t}(\overrightarrow{\vartheta})+\
abla\cdot( \rho\overrightarrow{v}\overrightarrow{v})=-\
abla p+\
abla\cdot[\mu( \
abla\overrightarrow{v}+\
abla\overrightarrow{v}^{T})]+\rho \overrightarrow{g}+\overrightarrow{F} \tag{5}\]  
where \(\rho\) is the mixture density, and \(\mu\) is the mixture viscosity given by the following:  
\[\rho=\alpha_{l}\rho_{l  
The overall HTC was worked out based on local wall temperatures and heat flux. To calculate the local parameters horizontal lines were drawn at various lengths along the channel, and quantities were line averaged as follows:  
\[T_{b}=\int_{0}^{l}T_{b}(x).dx \tag{14}\]  
\[T_{w}=\int_{0}^{l}T_{w}(x).dx \tag{15}\]  
\[q=\int_{0}^{l}\text{q}(x).dx \tag{16}\]  
\[h=\frac{q}{T_{w}-T_{b}} \tag{17}\]  
where \(q\) is wall heat flux, \(T_{w}\) is wall temperature and \(T_{b}\) is near wall temperature.