Almajose-Dalida EOS | Mixing Rules

The vdW one-fluid mixing rule treats a mixture as if it were a single “pseudo-fluid.” The mixture parameters are calculated from the pure-component parameters using mole-fraction averages and pairwise interaction terms. Binary interaction parameters are often added to better match experimental mixture behavior. Because it is simple and efficient, this mixing rule is commonly used with cubic equations of state.

Binary assumption for van der Waals one-fluid

For a binary mixture, the mixing rule becomes:

In some mixtures, the interaction between two different components may be treated as asymmetric. This means the interaction parameter from component 1 to component 2 (k₁₂) does not have to be equal to the interaction parameter from component 2 to component 1 (k₂₁). Allowing k₁₂ ≠ k₂₁ provides additional flexibility when fitting experimental mixture data. In this case, the expression for the attractive mixture parameter becomes:

Van der Waals one-fluid mixing rule (vdW1F) derivatives for thermodynamic formulas

The following expressions give the derivatives of the van der Waals one-fluid mixing rule with respect to number of moles and temperature. The composition terms are written using z to keep the formulation general for either liquid or vapor mixtures in AD-EOS.

For a binary mixture, the derivative formulas reduce to the following expressions:

The Panagiotopoulos–Reid mixing rule extends the classical van der Waals one-fluid formulation by allowing asymmetric cross interactions between components. In this formulation the interaction parameter from component i toward component j does not have to equal the interaction from component j toward component i. This allows the model to represent systems where the unlike interactions are direction-dependent while still preserving the quadratic mixing structure used in cubic equations of state. 

The mixture attractive parameter is written as a double summation over all component pairs, but the cross interaction terms depend explicitly on composition. This additional dependence prevents the interaction terms from collapsing into a symmetric form and therefore maintains the asymmetric behavior of the model.

Binary assumption for Panagiotopoulos-Reid

For a binary mixture, the mixing rule becomes:

Panagiotopoulos-Reid mixing rule derivatives for thermodynamic formulas

The following expressions give the derivatives of the Panagiotopoulos–Reid mixing rule with respect to number of moles and temperature. The composition terms are written using z so the formulation remains valid for either liquid or vapor mixtures.

For a binary mixture, the derivative formulas reduce to the following expressions:

The Mathias–Klotz–Prausnitz mixing rule was developed to address the lack of composition invariance in asymmetric higher-order mixing rules used with cubic equations of state. In non-invariant formulations, the predicted thermodynamic properties of a mixture may change if a component is artificially divided into identical pseudo-components, which is physically unacceptable.

To resolve this, the MKP formulation restructures the attractive parameter as a hierarchical expansion in which nonlinear contributions are applied to composition-weighted sums, rather than to individual pairwise terms. This ensures that the resulting expression remains invariant with respect to composition splitting, while still allowing asymmetric interaction parameters between components.

The mixture attractive parameter is written as:

The two parameter and three-parameter MKP mixing rule can be written as:

The interaction parameters used in the MKP formulation follow specific symmetry relationships. The baseline parameter is symmetric, such that:

while the higher-order parameters are antisymmetric,

Binary assumption for Mathias-Klotz-Prausnitz

For a binary system, the MKP two-parameter and three-parameter mixing rules are expanded:

As such, the MKP two-parameter and three-parameter mixing rules applied to a binary system can be written as:

It is important to note that the values in the database are reported in terms of transformed parameters, Lij and Qij, which are related to lij and qij as:

Mathias-Klotz-Prausnitz mixing rule derivatives for thermodynamic formulas

The following expressions give the derivatives of the Mathias-Klotz-Prausnitz mixing rule with respect to number of moles and temperature. The composition terms are written using z so the formulation remains valid for either liquid or vapor mixtures. 

For a binary mixture applying the two-parameter MKP mixing rule, the derivative formulas reduce to the following expressions:

Likewise, a binary mixture applying the three-parameter MKP mixing rule results to the following expressions: