# Influence of Frost Growth and Migration in Cryogenic Heat Exchanger on Air Refrigerator
## 2 Frost Formation in Heat Exchanger
### The Cryogenic Heat Exchanger  
The structure sketch of the plate-fin heat exchanger is shown in Figure 2. The main design parameters of the heat exchanger are listed in Table 1. The feature of the plate-fin heat exchanger is the fin with secondary heat transfer. As the basic element of plate-fin heat exchanger, the fin increases  
Figure 1: Test rig and \(T\)-\(s\) diagram of the air refrigerator: (**a**) Test rig; (**b**) \(T\)-\(s\) diagram and thermodynamic process. the heat transfer area and makes the structure compact, resulting in the large specific surface area. The heat transfer efficiency is improved and the thermal resistance is reduced. Also, the strength of the heat exchanger is increased. As the refrigerator runs, new air enters the system continuously. Although most of the water is removed by the precooling system, some of the water vapor still enters the system. In the operation process, the temperature at cold end of the heat exchanger might be below \(-100\)\({}^{\circ}\)C or even lower for a long time. The frosting will occur when the wet air flows through the middle of the heat exchanger. When the humidity of inlet air is low, a thin layer of frost is formed along the wall in the heat exchanger channel, as shown in Figure 2. Currently, the convection heat transfer coefficient decreases and thermal resistance increases. The performance of heat exchanger is greatly affected, and its outlet temperature arises. When the humidity is high, the thick frost layer will accumulate in the middle of the channel, which further reduces the heat transfer coefficient and increases the flow resistance. More seriously, the flow channel of heat exchanger will be blocked and the system will be unable to operate. Moreover, the pressure air at the hot side will carry frost particles into the expander. Because of high rotating speed, the blades of expansion wheel will be damaged by frost particles  
### The Cryogenic Heat Exchanger  
#### 2.3.1 Heat Transfer Characteristics  
In the plate-fin heat exchanger, the energy is transferred mainly through the heat conduction inside the fin and heat convection between the fin and fluid. The fin efficiency \(\eta_{f}\) should be considered when the temperature difference of the primary heat transfer surface is applied in calculation. The fin efficiency \(\eta_{f}\) and surface efficiency \(\eta_{s}\) can be obtained as:  
\[\eta_{f}=\frac{\tanh(ml)}{ml} \tag{1}\] \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Parameters** & \(\mathbf{H_{f}}\) & \(\mathbf{s_{f}}\) & \(\mathbf{\delta_{f}}\) & \(\mathbf{x_{f}}\) & \(\mathbf{y_{f}}\) & \(\mathbf{\delta_{W}}\) & \(\mathbf{a_{f}}\) \\ \hline Value/mm & 9.5 & 1.2 & 0.2 & 1.0 & 9.3 & 1.2 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the cryogenic heat exchanger [6]. Figure 2: The structure sketch of the plate-fin heat exchanger. where \(m\) is the fin parameter, \(l\) is the conduction distance. \[\eta_{\rm s}=1-\frac{A_{2}}{A_{S}}(1-\eta_{f}) \tag{2}\]  
According to the structure and heat transfer features of the plate-fin heat exchanger, some assumptions were introduced for solving the problem easily [6; 11]. There was a one-dimensional flow, and the distributions of fluid were even among channels. The heat from friction and the pressure drop gradient in the equation of momentum were negligible. These can be validated by the three-dimensional CFD method [10]. With the assumptions, conservation laws of the mass and momentum were satisfied, and only the energy conservation equation needed to be resolved [10; 11]. With numerical methods of flow and heat transfer, a transient simulation method of the plate-fin heat exchanger was newly proposed [6]. In the model, influences of the cold loss, axial conduction, and thermal capacity were all considered. The methods of control volume integral and finite difference were applied to disperse the partial differential equations.