## 2 Model Description
### Heat Transfer Model
#### 2.5.1 Fluid-Solid Heat Transfer  
The cryogenic fluid encounters a wide range of flow conditions during a cooling process. The resulting complex heat transfer regimes imply that the modelling of the convective heat transfer requires multiple correlations for each flow regime. Reference is made to the classical subdivision of two-phase flow regimes: single-phase gas flow, dispersed flow, annular flow, transition flow, bubbly flow, and single-phase liquid flow; see Figure 3.  
The popular Dittus-Boelter correlation [27; 28] was used in this model to calculate the heat transfer coefficient for _single-phase liquid or gas flows_:  
\[h_{\rm sp}=0.023\,\frac{k}{D_{\rm h}}\,Re^{0.8}\,Pr^{0.4} \tag{7}\]  
with \(Re=\frac{G\,D_{\rm h}}{\mu}\), \(Pr=\frac{\mu\,c_{\rm P}}{k}\), and \(G=\frac{\mu}{A}\). In the case of film boiling, the fluid can present two different flow regimes: _dispersed flow_ and _annular flow_. In the dispersed flow regime, the gas fraction is relatively high. The gas flows as a continuous phase in the pipe, and the liquid phase is present as dispersed droplets in the gas core. As the gas fraction reduces, a liquid film appears on the pipe walls, while the core of the pipe is still occupied by gas and liquid droplets. Several correlations are available in the literature to calculate the heat transfer coefficient in these regimes. For the liquid-deficient dispersed film boiling regime, we adopted the approach of [PERSON] et al. [29], modifying it to take into account the presence of an incondensable phase:  
\[h_{\rm disp}=0.023\,\frac{k_{\rm v}}{D_{\rm h}}\left\{Re_{\rm v}\left[\psi+ \frac{\rho_{\rm v}}{\rho_{1}}\left(1-\psi\right)\right]\right\}^{0.8}\,Pr_{ \rm v}^{0.4} \tag{8}\]  
In contrast with the original correlation, here, \(\psi\) is the overall gas fraction, expressed as \(\psi=\sum_{k=1}^{n_{\rm s}}\psi_{k}\,\beta_{k}\) for the generic case of a mixture. The gas fraction for the component \(k^{\rm th}\), \(\psi_{k}\), is \(\psi_{k}=1\) in the case of inert gas (He), while \(\psi_{k}=x_{k}=(h-h_{\rm sat})/h_{\rm lv}\) for the two-phase fluid (oxygen). For the annular film boiling, the Bromley-type relation [30; 31] modified by [PERSON] et al. [12] was applied:  
\[h_{\rm annular}=0.673\left\{\left[\frac{\sigma}{\left(\rho_{1}-\rho_{\rm v} \right)g}\right]^{0.5}2\pi\right\}^{-0.25}\left[\frac{g\,\rho_{\rm v}\left( \rho_{1}-\rho_{\rm v}\right)h_{\rm lv}^{\prime}\,k_{\rm g}^{3}}{\mu_{\rm v} \left(T-T_{\rm sat}\right)}\right]^{0.25}\left(\frac{u}{0.076}\right)^{0.5} \tag{9}\]  
Figure 3: Heat flux variation vs. temperature and overall gas fraction. where \(h^{\prime}_{\rm lv}\), the equivalent latent heat including the effect of vapour superheat, is expressed as \(h^{\prime}_{\rm lv}=h_{\rm lv}+0.4\left(T-T_{\rm sat}\right)c_{\rm p,v}\). Finally, at an even lower gas fraction, _bubbly flow_ and _transition flow_ occur: bubbles flow through a continuous liquid phase. For these two flow regimes, the heat flux is estimated by interpolating between the liquid single-phase and the critical and the annular heat flux as a function of the wall temperature. For the critical heat flux, the [PERSON] et al. correlation [20] was adopted to give encouraging results:  
\[\dot{q}_{CHF}=6.08\cdot 10^{-3}h_{\rm lv}\,G\left(\frac{\sigma\,\rho_{l}}{G^{2} \,D_{\rm h}}\right)^{-0.268} \tag{10}\]  
This correlation was derived from the Katto and Kurata relation and was obtained from experimental data of chill-down processes on LN\({}_{2}\), LAr and LO\({}_{2}\) conducted by [PERSON]. It returned a prediction of good accuracy, with a mean absolute error of 20%. As shown by Figures 3 and 4, the correlations for the heat transfer coefficient, \(h_{\rm fs}\), were selected according to the flow regimes. The flow regimes in two-phase conditions were determined by comparing the fluid temperature, \(T\), with the critical and minimum heat flux temperatures, \(T_{\rm CHF}\) and \(T_{\rm MHF}\), and as a function of the overall gas fraction, \(\psi\).