Cubic Mixing Rules

All CubicEoS in yaeos include a CubicMixRule derived type, which handles how the $D$ and $B$ parameters in the CubicEoS are calculated.

Quadratic Mixing Rules (QMR)

Quadratic mixing rules are the ussually most used mixing rules for cubic equations of state.

$$nB = \sum_i \sum_j n_i n_j \frac{b_i + b_j}{2} (1 - l_{ij})$$

$$D = \sum_i \sum_j n_i n_j a_{ij}$$

QMR are handled by the QMR derived type. Which can be used like:

use yaeos, only: pr, QMR

type(QMR) :: mixrule
real(pr) :: kij(2, 2), lij(2, 2)

kij(1, :) = [0.0, 0.1]
kij(2, :) = [0.1, 0.0]

lij(1, :) = [0.0, 0.01]
lij(2, :) = [0.01, 0.0]

mixrule = QMR(k=kij, l=lij)

By default the $a_{ij}$ matrix will be calculated with a constant $k_{ij}$ value (as shown below). But this functionality can be modified by replacing the get_aij procedure

use yaeos, only: pr, QMR

type(QMR) :: mixrule
real(pr) :: kij(2, 2), lij(2, 2)

kij(1, :) = [0.0, 0.1]
kij(2, :) = [0.1, 0.0]

lij(1, :) = [0.0, 0.01]
lij(2, :) = [0.01, 0.0]

mixrule = QMR(k=kij, l=lij)
mixrule%get_aij => my_aij_implementation

subroutine my_aij_implementation(self, ai, daidt, daidt2, aij, daijdt, daijdt2)
    class(QMR) :: self
    real(pr), intent(in) :: ai(:), daidt(:), daidt2(:)
    real(pr), intent(out) :: aij(:, :), daijdt(:, .), daijdt2(:, :)
    ! Implementation here
end subroutine

$G^E$ 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:

$$\frac{D}{RTB}(n, T) = \sum_i n_i \frac{a_i(T)}{b_i} + \frac{1}{q} \left(\frac{G^E(n, T)}{RT} + \sum_i n_i \ln \frac{B}{nb_i} \right)$$

Where $q$ is a weak function of temperature. In the case of MHV and simplicity it is considered that depends on the model used.