## 2 Numerical Methods
### Dispersion Model for Leaked Liquid Hydrogen
#### 2.2.2 Liquid Pool Model  
The FLACS code employs the two-dimensional shallow water equation to model the behavior of the pool. The underlying concept of this theory is that properties within the pool, such as temperature and velocity, are uniform along its depth, making them a function exclusively of the horizontal coordinates (x, y). The shallow water equations for the conservation of mass and momentum are written as follows:  
\[\rm{\frac{\partial h}{\partial t}+\frac{\partial hu_{i}}{\partial X_{i}}=\frac {\dot{m}_{L}-\dot{m}_{V}}{\rho_{l}}} \tag{18}\]  
\[\rm{\frac{\partial hu_{i}}{\partial t}+u_{i}\frac{\partial hu_{i}}{\partial X_ {j}}=F_{g,i}+F_{\tau,j}} \tag{19}\]  
where h, t, u, \(\dot{m}_{L}\), \(\dot{m}_{V}\) and \(\rho_{l}\) represent pool height, time, liquid spill rate into the pool, evaporation rate, and density of the liquid, respectively. \(\rm{F_{g,i}}\) and \(\rm{F_{\tau,j}}\) in the right-hand side of the momentum Equation (19) are the gravity term and shear stress between the pool and the substrate under the pool. These terms are written as follows:  
\[\rm{F_{g,i}=g\Delta\frac{\partial(h+z)}{\partial X_{i}}} \tag{20}\]  
\[\rm{F_{\tau,j}=\frac{1}{8}f_{t}u_{i}|\dot{\overrightarrow{u}}|} \tag{21}\]where \(\Delta=1\) for the pools spreading on the solid surfaces and \(\Delta=(1-\rho_{1}/\rho_{\rm w})\) for pools on water, \(\rm f_{t}\) is the friction coefficient between pool and substrate, and \(\rm z\) is the ground elevation.  
The vaporization rate from the pool, denoted as \(\rm\dot{m}_{V}\) is influenced by several factors: heat transfer from the substrate, solar radiation, and convective heat transfer from the air above the pool. The components of heat transfer are integrated into the energy equation, which is resolved using specific enthalpy in the following:  
\[\frac{\partial\theta}{\partial t}+\rm u_{i}\frac{\partial\theta}{\partial x_{i }}=\frac{\rm\dot{m}_{L}}{h}(\theta_{L}-\theta)+(\rm\dot{q}_{c}+\rm\dot{q}_{rad }+\rm\dot{q}_{g}+\rm\dot{q}_{evap}) \tag{22}\]  
where the first term on the right hand side of Equation (22) is due to the leak; h, \(\rm\theta\), \(\rm\dot{q}_{c}\), \(\rm\dot{q}_{rad}\), \(\rm\dot{q}_{g}\) and \(\rm\dot{q}_{evap}\) are pool height, specific enthalpy, convective heat transfer between pool and air, radiative heat transfer from the surroundings and sun, heat transfer to the pool from the substrate, and heat loss due to the evaporation, respectively.  
For cryogenic spills such as hydrogen, liquid nitrogen and LNG, the heat transfer from the substrate is often the dominant source of heat input to the pool. In the FLACS pool model, heat transfer from a solid substrate is calculated as a function of position and time according to the semi-infinite solid heat transfer theory, assuming perfect contact between pool and substrate:  
\[\rm q_{g}(x,y,t)=\frac{\lambda_{s}\Big{(}T_{s}^{0}-T_{I}\Big{)}}{\sqrt{\pi \alpha_{s}(t-t_{w}(x,y))}} \tag{23}\]  
where \(\lambda_{s}\) and \(\alpha_{s}\), represent the thermal conductivity, thermal diffusivity of the substrate; \(T_{s}^{0}\), \(T_{I}\) and \(t_{w}\) are the initial temperature of the substrate, the temperature of the pool and the time at which the substrate (at position x, y) is first wetted by the liquid; the details have been published [25].