# A Thermodynamic Model for Cryogenic Liquid Hydrogen Fuel Tanks
## 2 Thermodynamic Model  
A cryogenic fuel tank consists of a tank structure, insulation, and fuel. External heat is transferred to the inside of the tank via the tank structure and insulation. The heat flow rate depends on the shape of the tank, the thickness of the tank structure, and the insulation [20; 21]. In addition, the insulation performance of the tank varies depending on the materials used in the structure and the insulation, because the thermal conductivity of materials varies. Cryogenic fuel exists as a mixture of liquid and gaseous phases inside the tank. The liquid fuel undergoes a phase change into a gas by the heat inflow from the outside of the tank. The gas vaporized due to the heat inflow increases the temperature and the pressure inside the tank. The tank pressure is reduced when the fuel is supplied to a fuel consumption device, such as fuel cell, or discharged to the outside, acting as pressure relief. Therefore, a thermodynamic model for a cryogenic fuel tank needs to be developed to consider the amount of heat transfer through the structure, the filling ratio of liquid fuel, the phase change, and the fuel consumption rate. A thermodynamic phenomenon occurring in the cryogenic fuel tank can be expressed in a schematic diagram, as shown in Figure 1. Several assumptions were made to develop the thermodynamic model. First, it was assumed that the heat introduced from the outside of the tank is transferred to the liquid and gas regions separately [29; 30]. This can be expressed as Equations (1) and (2). \[q_{heat\_liq}=h_{wall\_liq}A_{liq}\left(T_{amb}-T_{liq}\right) \tag{1}\]  
\[q_{heat\_gas}=h_{wall\_gas}A_{gas}\left(T_{amb}-T_{gas}\right) \tag{2}\]  
\[h_{wall\_liq}=\frac{1}{1/h_{amb}+L_{wall\_wall}+1/h_{liq}} \tag{3}\]  
\[h_{wall\_gas}=\frac{1}{1/h_{amb}+L_{wall\_wall}+1/h_{gas}} \tag{4}\]  
Figure 1: Schematic illustration of the thermodynamic model for cryogenic tanks. In applying the overall heat transfer coefficient between the outside and inside of the tank, convection by ambient air was not considered, in order to ignore the influence of the convective environment outside the tank. In addition, it was assumed that sufficient convection occurs within gaseous and liquid hydrogen [31] through boil-off and fuel emission flow. Therefore, Equations (3) and (4) are simplified to Equation (5), in which only the terms due to heat conduction are considered. \[h_{wall,liq}=h_{wall,gas}=\frac{k_{wall}}{L_{wall}} \tag{5}\]  
Second, the heat introduced into the liquid region is used to vaporize the liquid. The mass corresponding to the vaporization energy of the liquid is transferred to the gas region. The vaporized mass flow is calculated as in Equation (6). \[\dot{m}_{boil}=\frac{q_{heat,liq}}{h_{fg}} \tag{6}\]  
The differential mass in the liquid and gas regions by the vaporized mass flow is expressed as Equations (7) and (8). \[dm_{liq}=-\dot{m}_{boil}dt \tag{7}\]  
\[dm_{gas}=\dot{m}_{boil}dt \tag{8}\]  
Lastly, the heat introduced into the gas region only changes the temperature and pressure of the gas. The condensation caused by the rise in gas pressure was not considered. Hence, the mass only moves from the liquid region to the gas region. In the case of using fuel, the fuel may be supplied in a gaseous or liquid form or both. When discharge of fuel from the tank to the outside occurs, the fuel consumption term is added to Equations (7) and (8), and expressed as Equations (9) and (10). \[q_{heat\_liq}=h_{wall\_liq}A_{liq}\left(T_{amb}-T_{liq}\right) \tag{1}\]0  
\[q_{heat\_liq}=h_{wall\_liq}A_{liq}\left(T_{amb}-T_{liq}\right) \tag{1}\]1  
The differential heat energy of the gas region consists of the heat energy transferred from the outside, the heat added by vaporized hydrogen from liquid hydrogen, and the heat lost from being used as fuel. It is expressed as Equation (11).