# Thermal Performance of Cryogenic Micro-Pin Fin Coolers with Two-Phase Liquid Nitrogen Flows
## 2 Correlations
### Heat Transfer  
[PERSON] [10] used correlations, based on a Chen-type model, to evaluate heat transfer coefficients. [PERSON] [17] defined a heat transfer coefficient for two-phase state, \(h_{tp}\), for forced convective boiling in a tube as:  
\[h_{tp}=F\cdot h_{sp}+S\cdot h_{nb} \tag{1}\]  
where \(F\) denotes an enhancement factor, \(h_{sp}\) denotes a heat transfer coefficient for the single-phase state, \(S\) denotes a suppression factor and \(h_{nb}\) denotes a heat transfer coefficient for nucleate boiling.  
\(F\) is defined as the ratio of the two-phase Reynolds number to the liquid Reynolds number and \(S\) is defined as the ratio of the effective superheat to the total superheat of the wall. The decrease of \(S\) denotes the decrease of the effective superheat, and eventually induces the suppression of the nucleate boiling.  
[PERSON] [10] found that convective boiling is the dominant heat transfer mechanism, and thus the nucleate boiling term in Equation (1) is neglected.  
\(h_{tp}\) is re-expressed as:  
\[h_{tp}=Fh_{sp} \tag{2}\]  
For a case where \(Pr\approx 1\), \(F\) is defined as in [10]:  
\[F=\left(\phi_{l}^{2}\right)^{0.2475} \tag{3}\]For a case where \(Pr\
eq 1\), Equation (3) is multiplied by \(Pr^{0.333}\) by using the method adopted by [PERSON] et al. [18], as shown below:  
\[F=\left(\phi_{l}^{2}\right)^{0.2475}Pr^{0.333} \tag{4}\]  
where \(\phi_{l}\) is a two-phase frictional multiplier.  
Similar to [PERSON] and [PERSON] [19]'s method, an adjustment parameter \(\zeta\) is considered, and \(F\) is rewritten as:  
\[F=\zeta\left(\phi_{l}^{2}\right)^{0.2475}Pr^{0.333} \tag{5}\]  
Consequently, the two-phase heat transfer coefficient is written as:  
\[h_{tp}=\zeta\left(\phi_{l}^{2}\right)^{0.2475}Pr^{0.333}h_{sp} \tag{6}\]  
The value of the adjustment parameter, \(\zeta\), is 1.4. [20].  
A two-phase friction multiplier \(\phi_{l}\) is determined using the correlation proposed by [PERSON] and [PERSON] [21], as:  
\[\left(\phi_{l}\right)^{2}=\frac{\left(\Delta P_{f}\right)_{tp}}{\left(\Delta P _{f}\right)_{f}}=1+\frac{C}{X_{vv}}+\frac{1}{X_{pv}^{2}} \tag{7}\]  
where \(C=0.24\) is an empirically defined constant. \(X_{vv}\) is the Martinelli parameter [22], defined as:  
\[X_{vv}=\left[\frac{\left(\Delta P_{f}/\Delta Z\right)_{f}}{\left(\Delta P_{f} /\Delta Z\right)_{v}}\right]^{\frac{1}{2}} \tag{8}\]  
\[\left(\Delta P_{f}\right)_{f}=\frac{fN(G(1-x))^{2}}{2\rho_{f}} \tag{9}\]  
\[\left(\Delta P_{f}\right)_{v}=\frac{fN(Gx)^{2}}{2\rho_{v}} \tag{10}\]  
where \((\Delta P_{f}/\Delta Z)_{f}\) is the frictional pressure gradient assuming only liquid flows through the channel, \((\Delta P_{f}/\Delta Z)_{v}\) is the frictional pressure gradient assuming only vapor flows through the channel, \(f\) is the Blasius type friction factor, \(N\) is number of fin rows, \(G\) is the mass flux, \(x\) is the quality, \(\rho_{f}\) is the liquid density and \(\rho_{v}\) is the vapor density.  
\(f\) is defined as [23]:  
\[f=63.246(Re_{d})^{-0.7797} \tag{11}\]  
where \(Re_{d}\) is the Reynolds number (\(Re_{d}=Gd_{e}/\mu\)), \(d_{e}\) is the equivalent diameter of a square micro-pin fin and \(\mu\) is dynamic viscosity.  
In Equation (11), the correlation was developed to predict the experimental results within 5% [23].  
In Equation (6), the single-phase heat transfer coefficient term \(h_{sp}\) is defined as:  
\[h_{sp}=\frac{Nu\cdot k_{f}}{d_{e}} \tag{12}\]  
where \(Nu\) is the Nusselt number, \(k_{f}\) is the thermal conductivity of fluid and \(d_{e}\) is the equivalent diameter of a fin.  
To evaluate \(Nu\), the correlation by [PERSON] et al. [24] was utilized, defined as:  
\[Nu=0.76\left(\frac{S_{t}}{d_{e}}\right)^{0.16}\left(\frac{S_{t}}{d_{e}}\right)^ {0.2}\left(\frac{H_{fin}}{d_{e}}\right)^{-0.11}Re^{0.33} \tag{13}\]where \(S_{t}\), \(S_{l}\) and \(H_{fin}\) are transverse fin spacing, longitudinal fin spacing and fin height, respectively.