Constitutive Equations

Mixture State:	Pm = $\rho_{m}^{}$Tm$\left(\vphantom{ X_n R_n+\left( 1-X_n \right) R_g }\right.$XnRn + $\left(\vphantom{ 1-X_n }\right.$1 - Xn$\left.\vphantom{ 1-X_n }\right)$Rg$\left.\vphantom{ X_n R_n+\left( 1-X_n \right) R_g }\right)$
Liquid State:	$\rho_{l}^{}$ = $\rho_{l}^{}$$\left(\vphantom{ P_l,T_l }\right.$Pl, Tl$\left.\vphantom{ P_l,T_l }\right)$
Solid State:	$\rho_{s}^{}$ = $\rho_{s}^{}$$\left(\vphantom{ T_s }\right.$Ts$\left.\vphantom{ T_s }\right)$
Volume Fraction Sum:	$\sum_{{x=m,l,s}}^{}$    $\alpha_{x}^{}$ = 1
Capillary Pressure Relation:	Pm - Pl = $\mathcal {L}$$\left(\vphantom{ \Delta P_{cap} \left( \alpha_m \right) }\right.$$\Delta$Pcap$\left(\vphantom{ \alpha_m }\right.$$\alpha_{m}^{}$$\left.\vphantom{ \alpha_m }\right)$$\left.\vphantom{ \Delta P_{cap} \left( \alpha_m \right) }\right)$

Grand total of 15 equations, 15 unknowns at each node.

System variables are:
$\rho_{m}^{}$ , $\rho_{l}^{}$ , $\rho_{s}^{}$ , $X_n$ , $\alpha_{m}^{}$ ,$\alpha_{l}^{}$ , $\alpha_{s}^{}$ , $P_m$ , $P_l$ , $V_m$ , $V_l$ , $T_m$ , $T_l$ , $T_s$ and $T_w$ .

The interphase transfer terms ( $\textcolor{magenta}{\Gamma_{xy}}$ , $\textcolor{magenta}{Q_{xy}}$ , $\textcolor{magenta}{Q_{xy}^\Gamma}$ ) are all functions of $\rho_{m}^{}$, X_n, T_m, T_l, T_s , and T_w and are defined implicitly in the Radial Model.