# Model for Cryogenic Flashing LNG Leak
## 2 Materials and Methods  
The proposed model tracks the different stages of an LNG leak. Figure 1 shows that the leak starts at the breach, undergoes de-pressurization in the expansion zone, undergoes air mixing in the entrainment zone, and ends at the dispersion zone [18]. In the entrainment zone, LNG that does not vaporize will fall onto the ground to form a rainout. The leak model consists of 9 equations to quantify the leak scenario. Equations (1) and (2) are for the breach orifice zone. Equations (3)-(7) cover the expansion zone leak development. Equations (8) and (9) quantify the leak parameter at the entrainment zone. The mass flow rate across the orifice for subcooled stagnation condition is provided, as in [19]:  
\[\text{Q}=\text{A}_{1}*\text{C}_{\text{d}}\ \forall 2\times[\text{P}_{0}-\text{P}_{ \text{V}}(\text{T}_{0})]\times\rho_{0} \tag{1}\]  
Q is the mass flow rate. A\({}_{1}\) is the orifice cross-sectional area. C\({}_{\text{d}}\) is the discharge coefficient set as 0.62, which is typical for fluid flow across a sharp orifice. P\({}_{0}\) is the stagnation pressure. P\({}_{\text{V}}(\text{T}_{0})\) is vapour pressure at stagnation temperature T\({}_{0}\). \(\rho_{0}\) is the LNG liquid density at stagnation pressure. The leak velocity, V\({}_{1}\), at the orifice is calculated by:  
\[\text{V}_{1}=\text{Q}/(\text{A}_{1}\times\rho_{0}) \tag{2}\]  
In the expansion zone, the stagnation pressure will be depressurized to atmospheric pressure and will undergo volume expansion. If the stagnation temperature is higher than the LNG ambient boiling temperature, a portion of the liquid will be vaporized to reduce the remaining liquid temperature to boiling temperature. The depressurization process occurs rapidly, and heat from nearby air will not be absorbed by the cold LNG. Hence, both the liquid and gas LNG will be at the same temperature (LNG ambient boiling temperature). To calculate the two-phase composition from the depressurization, HEM was used to determine the vapour mass fraction at the end of the expansion zone [18]. There are three assumptions made in HEM. 1. Both the liquid and gas phase temperature will be LNG ambient boiling temperature at the end of the expansion zone;  
Figure 1: Schematic representation of the zones in jet leak. The heat exchange from vaporization is much greater than heat gain from surrounding so the warming effect is ignored;  
The vapour mass fraction is much lower than 1. The vapour mass fraction at the end of the expansion zone is calculated as follows:  
\[X=C_{\rm pl}\times(T_{0}-T_{1})/L_{\rm v} \tag{3}\]  
\(X\) is the vapour mass fraction. \(C_{\rm pl}\) is LNG liquid heat capacity at \(P_{0}\) and \(T_{0}\) conditions. \(T_{1}\) is LNG ambient boiling temperature set at 112 K. \(L_{\rm V}\) is LNG latent heat of vaporization. The two-phase leak density at the end of the expansion zone is given as follows:  
\[\rho_{2}=(1-X)\times\rho_{\rm L}+X\times\rho_{\rm g} \tag{4}\]  
\(\rho_{2}\) is the leak density at end of the expansion zone. \(\rho_{\rm L}\) and \(\rho_{\rm g}\) are the respective LNG liquid and gas densities at boiling point temperature in atmospheric pressure. Based on the conservation of mass and momentum, the condition at the beginning and end of the expansion zone can be expressed, respectively, as follows:  
\[\rho_{1}*V_{1}*A_{1}=\rho_{2}\times V_{2}\times A_{2} \tag{5}\]  
\[\rho_{1}\times{V_{1}}^{2}\times A_{1}+A_{1}\times P_{1}=\rho_{2}\times{V_{2}}^ {2}\times A_{2}+A_{2}\times P_{2} \tag{6}\]  
\(\rho_{1}\) is the LNG density at the start of the expansion zone. \(V_{1}\) is the leak velocity calculated from Equation (2). \(A_{1}\) is the orifice cross-sectional area. \(\rho_{2}\) is the density of LNG calculated from Equation (4). \(V_{2}\) is the depressurized velocity at the end of the expansion zone. \(A_{2}\) is the expanded cross-sectional leak area at the end of the expansion zone. \(P_{2}\) is atmospheric pressure. Combing Equations (5) and (6) will form quadratic Equation (7).