Peng-Robinson-Stryjek-Vera 2 (PRSV2 and PRSV2-VL)

PRSV2 keeps the Peng-Robinson assembly but makes the alpha parameter temperature-dependent:

\[ \begin{aligned}\kappa_{i} & =\kappa_{0,i}+\left[\kappa_{1,i}+\kappa_{2,i}\!\left(\kappa_{3,i}-T_{r,i}\right)\!\left(1-\sqrt{T_{r,i}}\right)\right]\\ & \quad{}\times\left(1+\sqrt{T_{r,i}}\right)\!\left(0.7-T_{r,i}\right) \end{aligned} \]

with \(T_{r,i}=T/T_{c,i}\) and \(\kappa_{0,i}=0.378893+1.4897153\,\omega_{i}-0.17131848\,\omega_{i}^{2}+0.0196554\,\omega_{i}^{3}\) . Because \(\kappa_{i}\) depends on temperature, the alpha derivative gains an extra term

\[ \frac{da_{i}}{dT}=\Omega_{a}\frac{R^{2}T_{c,i}^{2}}{P_{c,i}}\,2\sqrt{\alpha_{i}}\left[\frac{d\kappa_{i}}{dT}\!\left(1-\sqrt{T_{r,i}}\right)-\frac{\kappa_{i}}{2\sqrt{T\,T_{c,i}}}\right] \]

where \(d\kappa_{i}/dT=(1/T_{c,i})\,d\kappa_{i}/dT_{r,i}\) is obtained by differentiating the expression for \(\kappa_{i}\) above with respect to \(T_{r,i}\) . Everything else follows the Peng-Robinson assembly of the previous section. The composition derivative additionally differentiates the composition-dependent Stryjek-Vera mixing rule.

The PRSV2-VL variant replaces that mixing rule with the van Laar form

\[ a_{ij}=\sqrt{a_{i}a_{j}}\left(1-\frac{k_{ij}k_{ji}}{x_{i}k_{ij}+x_{j}k_{ji}}\right) \]

which carries two interaction matrices and depends on composition, but not on temperature. That single observation is what makes the temperature derivative tractable: the parenthesis is a constant with respect to \(T\) , so every \(a_{ij}\) and every sum built from it keeps its composition factor unchanged and differentiates through the pure-component \(a_{i}\) alone,

\[ \frac{\partial a_{ij}}{\partial T}=\frac{a_{ij}}{2}\left(\frac{1}{a_{i}}\frac{da_{i}}{dT}+\frac{1}{a_{j}}\frac{da_{j}}{dT}\right) \]

with \(da_{i}/dT\) from equation [eq:d-prsv2-dadt]. Only the temperature derivative is available in closed form for this variant; the composition derivative falls back to finite differences. Two guards keep the analytical result faithful to the function it differentiates: the routine declines to answer, and the caller reverts to finite differences, whenever the root-checking step has displaced the pressure, because that branch adds a term the derivative was not taken of; and it reproduces the same numerical guard the fugacity routine applies to its correction sum, so that the two can never end up differentiating different expressions.