# Influence of Frost Growth and Migration in Cryogenic Heat Exchanger on Air Refrigerator
## 2 Frost Formation in Heat Exchanger
### The Cryogenic Heat Exchanger
#### 2.3.2 Property Characteristics of Frost Layer  
The study of frost formation mainly consists of two aspects: one is to analyze the formation and growth of the whole frost layer from a macro perspective. Another is to analyze the nucleation and growth of single ice crystals from a microscopic perspective. From a macro perspective, variations of the frost density, thermal conductivity, and thickness are studied. Because the research mainly focuses the overall effect of frosting on the system, the macro analysis was conducted in the present study. In order to solve the heat and mass transfer of the frost layer, it is necessary to determine the relevant thermal parameters of the frost layer in combination with boundary conditions. The structure of frost layer is extremely complex, and the properties of the frost layer change with the time, boundary conditions, and locations of the frosting. In the present study, the experimental correlations were adopted for predict the distribution and variation of the frost property. The density and thermal conductivity are two important physical properties of frost layer. Frost is a soft and loose pore structure composed of fine ice crystals. The average density of frost layer is generally considered in calculation. Based on experimental data, [PERSON] et al. [25] put forward a widely used experimental correlation of frost density, in which the density was considered as a function of the temperature and wind speed on the layer surface. In this paper, the formula was applied to calculate the density of the frost layer. Many experimental correlations of the frost density were related to the cold surface temperature of above \(-25\,^{\circ}\mathrm{C}\), which might be enough to calculated the frosting condition in evaporators. Whereas, in a cryogenic heat exchanger, the wall temperature is much low., When the temperature of the cold surface is low, the frost will grow rapidly and the physical properties stabilized soon [26, 27]. Then the density of the frost layer under cryogenic condition was given according to [28] and is expressed as below:  
\[ \begin{array}{ll}\rho_{fr}=340|T_{w}|^{-0.455}+u_{0}&-25\,^{\circ}\mathrm{C} <T_{w}<-5\,^{\circ}\mathrm{C}\\ \rho_{fr}=180&T_{w}<-25\,^{\circ}\mathrm{C}\end{array}  \tag{12}\]  
where \(\rho_{fr}\) is the density of the frost layer (kg m\({}^{-3}\)), \(T_{w}\) is the local temperature of the cold surface (\({}^{\circ}\mathrm{C}\)), \(u_{0}\) is the velocity of mainstream (m s\({}^{-1}\)). The heat transfer within the frost layer includes the thermal conduction in ice crystal, convection heat transfer of wet air, latent heat transfer caused by the phase change of water vapor, etc. The thermal conductivity of frost layer refers to the effective thermal conductivity including the above heat transfer forms. The structure of frost layer directly determines the proportion of various heat transfer forms and has obvious effects on the thermal conductivity. The simplest parameter reflecting the structure of frost layer is its average density. The average density is influenced by both the frost quality and the thickness of the frost layer in growth direction. Therefore, the thermal conductivity can be considered as a function of the average density. Many experiments were conducted to find out the relationship between the two parameters. The formula proposed by [PERSON] et al. [29] was used with its accuracy and adaptability, as shown below:  
\[\lambda_{fr}=0.001202\rho_{fr}{}^{0.963} \tag{13}\]  
where \(\lambda_{fr}\) is the thermal conductivity of the frost layer (W m\({}^{-1}\) K\({}^{-1}\)). The frosting process of wet air on cold surface is extremely complex, which is accompanied by phase transitions of gas-liquid, liquid-solid, gas-solid and moving boundary problems. Using the numerical heat transfer and computational fluid dynamics to illustrate the frosting process is one of the main research methods [30]. In the method, the semiempirical correlations of the frost properties are essential, which affects the accuracy of the simulation.