docum

doc/page/usage/eos/cubics/index.md

@@ -2,13 +2,73 @@
 title: Cubics
 ---
 
-** Table of contents**
-[TOC]
+All our Cubic Equations of State are implemented based on the generic Cubic
+Equation:
+
+\[
+    P = \frac{RT}{V-b} - \frac{a_c\alpha(T_r)}{(V + \delta_1 b)(V - \delta_2 b)}
+\]
 
 ## SoaveRedlichKwong
+[[SoaveRedlichKwong]]
+
+Using the critical constants setup the parameters to use the 
+SoaveRedlichKwong Equation of State
+
+- \[\alpha(T_r) = (1 + k (1 - \sqrt{T_r}))^2\]
+- \[k = 0.48 + 1.574 \omega - 0.175 \omega^2 \]
+- \[a_c = 0.427480  R^2 * T_c^2/P_c\]
+- \[b = 0.086640  R T_c/P_c\]
+- \[\delta_1 = 1\]
+- \[\delta_2 = 0\]
+
+There is also the optional posibility to include the k_{ij} and l_{ij}
+matrices. Using by default Classic Van der Waals mixing rules. For more information
+about mixing rules look at [Mixing Rules](mixing.md)
 
 ## PengRobinson76
+[[PengRobinson76]]
+
+- \[\alpha(T_r) = (1 + k (1 - \sqrt{T_r}))^2\]
+- \[k = 0.37464 + 1.54226 * \omega - 0.26993 \omega^2 \]
+- \[a_c = 0.45723553  R^2 T_c^2 / P_c\]
+- \[b = 0.07779607r  R T_c/P_c\]
+- \[\delta_1 = 1 + \sqrt{2}\]
+- \[\delta_2 = 1 - \sqrt{2}\]
 
 ## PengRobinson78
+[[PengRobinson78]]
+
+- \[\alpha(T_r) = (1 + k (1 - \sqrt{T_r}))^2\]
+- \[k = 0.37464 + 1.54226 \omega - 0.26992 \omega^2  \text{ where } \omega <=0.491\]
+- \[k = 0.37464 + 1.48503 \omega - 0.16442 \omega^2  + 0.016666 \omega^3 \text{ where } \omega > 0.491\]
+- \[a_c = 0.45723553  R^2 T_c^2 / P_c\]
+- \[b = 0.07779607r  R T_c/P_c\]
+- \[\delta_1 = 1 + \sqrt{2}\]
+- \[\delta_2 = 1 - \sqrt{2}\]
+
+## RKPR
+[[RKPR]]
+
+The RKPR EoS extends the classical formulation of Cubic Equations of State by
+freeing the parameter \(\delta_1\) and setting 
+\(\delta_2 = \frac{1+\delta_1}{1-\delta_1}\).
+This extra degree provides extra ways of implementing the equation in
+comparison of other Cubic EoS (like PR and SRK) which are limited to
+definition of their critical constants.
+
+Besides that extra parameter, the RKRR includes another \(\alpha\)
+function:
+
+\[
+ \alpha(T_r) = \left(\frac{3}{2+T_r}\right)^k
+\]
+
+In this implementation we take the simplest form which correlates
+the extra parameter to the critical compressibility factor \(Z_c\) and
+the \(k\) parameter of the \(\alpha\) function to \(Z_c\) and \(\omega\):
+
+\[\delta_1 = d_1 + d_2 (d_3 - Z_c)^d_4 + d_5 (d_3 - Z_c) ^ d_6\]
+\[k = (A_1  Z_c + A_0)\omega^2 + (B_1 Z_c + B_0)\omega + (C_1 Z_c + C_0)\]
 
-## RKPR
+It is also possible to include the parameters as optional arguments.

doc/page/usage/eos/cubics/mixing.md

@@ -62,8 +62,21 @@ subroutine my_aij_implementation(self, ai, daidt, daidt2, aij, daijdt, daijdt2)
 end subroutine
 ```
 
-### Constant \(k_{ij}\)
-
 ## \(G^E\) Models Mixing Rules
+It is possible to mix the attractive parameter of Cubic Equations with an 
+excess Gibbs-based model.
+This can be useful for cases of polar molecules and/or systems that have been 
+fitted to \(G^E\) models.
+
+### Michelsen's Modified Huron-Vidal Mixing Rules
+This mixing rule is based on the aproximate zero-pressure limit 
+of a cubic equation of state. At the aproximate zero-pressure limit the
+attractive parameter can be expressed as:
+
+\[
+\frac{D}{RTB}(n, T) = \sum_i n_i \frac{a_i(T)}{b_i} + \frac{1}{q}
+\left(\frac{G^E(n, T)}{RT} + \sum_i n_i \ln \frac{B}{nb_i} \right)
+\]
 
-### Michelsen's Modified Huron-Vidal Mixing Rules
+Where \(q\) is a weak function of temperature. In the case of `MHV`
+and simplicity it is considered that depends on the model used.

doc/page/usage/newmodels/autodiff.md

@@ -91,6 +91,7 @@ contains
 
         integer :: i, j
 
+        ! Associate allows us to keep the later expressions simple.
         associate(&
             pc => self%composition%pc, ac => self%ac, b => self%b, k => self%k,&
             kij => self%kij, lij => self%lij, tc => self%compostion%tc &