# Influence of Frost Growth and Migration in Cryogenic Heat Exchanger on Air Refrigerator
## 3 Frost Model of the Cryogenic Heat Exchange  
### Heat and Mass Transfer During Frosting  
Based on computational fluid dynamics and numerical heat transfer, the semi-empirical quasi-steady state frosting model was adopted for calculating frost growth. The heat transfer coefficient is calculated firstly and the corresponding heat transfer number is obtained. Then the mass transfer number could be calculated according to the heat transfer number and analogy formula. The total frosting amount is gained and the average thickness of the frosting layer at different times could be obtained according to variations of the frost layer density. In order to simplify the simulation, the following assumptions are made:  
1. Frosting process in the heat exchanger is quasi-steady. 2. Frosting occurs in the hot side of the heat exchanger, the frost layer is filled with the entire fin spacing, and the frost is evenly distributed in the same control volume. 3. The density and thermal conductivity of frost layer are characterized by average values. 4. The heat transfer in the frost layer is one-dimensional heat conduction along the direction of thickness. In the process of frosting, the frost layer can grow continuously as the water vapor in the air moves towards the cold surface and condenses under the action of transferring pressure. \[\dot{m}_{fr}=M_{w}h_{m}(C_{Ah}-C_{As}) \tag{14}\]  
For the water vapor on the surface of frost layer is rare, it could be ignored. With the ideal gas state equation, the mass transfer equation could be expressed as:  
\[\dot{m}_{fr}=0.018h_{m}\frac{p_{a}}{RT_{a}} \tag{15}\]  
Then the key to calculate mass transfer is to determine the coefficient \(h_{m}\). The determination of \(h_{m}\) is complicated, which is related to the fluid properties, flow rate, and the geometrical shape and roughness of the wall surface. When the fluid flows through the object surface, there is mass and heat exchange between the fluid and the surface. According to the analogy theory of heat and mass transfer, the mass transfer coefficient can be calculated by analogy from the heat transfer coefficient. The well-known [PERSON] analogy theory was introduced. The Lewis relation was based on the analogy of heat and mass transfer [31]:  
\[\frac{h_{m}}{h}=\frac{\text{Le}^{-\frac{2}{3}}}{\rho c_{p}}=\frac{(a/D)^{-\frac{ 2}{3}}}{\rho c_{p}} \tag{16}\]  
where \(\rho\) is the density of wet air (kg m\({}^{-3}\)), \(c_{p}\) is the specific heat capacity of wet air (J kg\({}^{-1}\) K\({}^{-1}\)). According to Equations (15) and (16), \(\dot{m}_{fr}\) can be expressed as:  
\[\dot{m}_{fr}=0.018h\frac{(a/D)^{-\frac{2}{3}}}{\rho c_{p}}\frac{p_{a}}{RT_{a}} \tag{17}\]  
When the water vapor concentration in wet air is expressed in terms of moisture content, the above equation is as follows:  
\[\dot{m}_{fr}=0.018h\frac{(a/D)^{-\frac{2}{3}}}{\rho c_{p}}\frac{p_{h}d_{a}}{(0.622+d_{a})RT_{a}} \tag{18}\]  
The heat exchange between the frost and wet air is composed of two parts: one is the sensible heat, which is the convection heat transfer between the frost layer and the wet air. The other is the latent heat, which is released by the condensation of water from mass transfer. Therefore, the total heat exchange amount is:  
\[q_{a}=h_{h}(T_{h}-T_{s})+\dot{m}_{fr}\gamma_{fr} \tag{19}\]  
The diffusion heat transfer coefficient \(h_{s,i}\) is introduced to calculate the latent heat flow:  
\[q_{a}=(h_{h,i}+h_{s,i})(T_{h,i}-T_{s,i})=(h_{h,i}+h_{s,i})\Delta T_{s,i} \tag{20}\]  
\[h_{s,i}=\frac{\dot{m}_{fr}\gamma_{fr}}{T_{h,i}-T_{s,i}} \tag{21}\]  
For frosting areas, the control equation is discretized by the control volume balance method. The control volume balance method directly applies the conservation law to the control volume. It's a deformation and supplement of the control volume integral method. \[