# Influence of Frost Growth and Migration in Cryogenic Heat Exchanger on Air Refrigerator
## 2 Frost Formation in Heat Exchanger
### The Cryogenic Heat Exchanger
#### 2.3.1 Heat Transfer Characteristics
The convective heat transfer coefficient and pressure loss in the heat exchanger can be calculated as [6]:  
\[h=\frac{c_{p}\mu}{\rm Pr^{2/3}}\frac{1}{D_{h}}j\rm Re=j/Pr^{2/3}c_{p}G \tag{3}\]  
\[\Delta p=4f\frac{l}{D_{h}}\frac{G^{2}}{2\rho} \tag{4}\]  
where \(\rm Pr=c_{p}\mu/\lambda\), \(\Delta p\) is the pressure loss (Pa). The friction factor \(f\) and heat transfer factor \(j\) used for the plate-fin heat exchanger could be calculated as in [6; 11]. The heat exchanger was of two-fluid and there were 8 sets inside it. Thus, only a cold fluid and a hot fluid and the parting plate between them were calculated to speed up the calculation. The finite volume method was adopted to disperse the energy equations. The fluid and plate areas were divided with uniform grids. The grids of the heat exchanger in the simulation are presented in Figure 3. As shown in Figure 3, the heat transfer inside the plates consists of the heat convection between the parting plate and fluid and the heat conduction inside the plates. The heat transfer could be calculated through the non-steady equation of heat conduction, as shown in Equation (5). In the differential equation, the heat convection could be presented as a source term. \[\frac{\partial T}{\partial\tau}=\frac{\lambda}{\rho c}\frac{\partial^{2}T}{ \partial^{2}x}+\frac{\stackrel{{\bullet}}{{\Phi}}}{\rho c} \tag{5}\]  
Figure 3: The grids and heat transfer inside the plate. Integrating Equation (5) in the time interval \(\Delta\tau\) and the volume shown in Figure 5, the expression of integral form is as below:  
\[\int_{x}^{x+\Delta x}\int_{\tau}^{\tau+\Delta x}\frac{\partial T}{\partial\tau}d \tau dx=\int_{\tau}^{\tau+\Delta\tau}\int_{x}^{x+\Delta x}\frac{\partial^{2}T}{ \partial^{2}x}dxd\tau+\int_{\tau}^{\tau+\Delta x}\int_{x}^{x+\Delta x}\dot{ \Phi}_{i}dxd\tau \tag{6}\]  
For integrating the diffusion term, the first derivative was changed explicitly with time. The heat convection between the plate and fluid was presented as the heat source of the control volume. Using the lumped parameter method, the following equation is obtained [6]:  
\[\frac{\rho_{w}c_{p,w}\left(\frac{T_{w}^{j+1}-T_{w,i}^{j}}{\Delta\tau}\right)}{ \Delta\tau}=\frac{\lambda_{w}\left(T_{w,i+1}^{j}-2T_{w,i}^{j}+T_{w,i-1}^{j} \right)}{\Delta x^{2}}+\frac{1}{\
u_{w}}\left[h_{h,i}^{j}b_{h,i}(T_{h,i}^{j}- T_{w,i}^{j})+h_{c,i}^{j}b_{c,i}(T_{c,i}^{j}-T_{w,i}^{j})\right] \tag{7}\]  
where \(V_{w}\) is a single control volume (m\({}^{3}\)), \(h_{h,i}\) is the coefficient of convective heat transfer in hot side (W m\({}^{-2}\) K\({}^{-1}\)), \(b_{h,i}\) is the area of convective heat transfer in hot side (m\({}^{2}\)), \(h_{c,i}\) is the coefficient of convective heat transfer in cold side (W m\({}^{-2}\) K\({}^{-1}\)), and \(b_{c,i}\) is the area of convective heat transfer in cold side (m\({}^{2}\)). As shown in Equation (7), the diffusion term is discretized into a central difference scheme. Also, it has a 2-order intercept precision. The heat transfer of the fluid is the transient heat convection. The transient equation of convection-diffusion heat transfer is:  
Figure 2:0  
The energy transfer of the control volume in fluid usually contains the convection term cause by flow, the diffusion term caused by conduction, and the source term. The heat transfers of the hot fluid in the moment of \(j\) and \(j+1\) are shown in Figure 2:1 Compared with the one in [6], the energy flow conditions in fluid area were improved with both the blue and cyan bars to reflect the convection flow more clearly. The heat coming in and out of the control volume was transferred through the convective flow. The heat transited between the fluid and the wall could be recognized as a source item. According to the physical properties of air, the Peclet Number is great under low velocity. Therefore, the effect of diffusion could be ignored and the first term on the right side of Equation (9) could be omitted.