Wong–Sandler mixing rule

The Wong–Sandler mixing rule is a thermodynamic mixing rule used for vapor–liquid equilibrium and liquid-liquid equilibrium calculations.^[1]

Summary

The first boundary condition is

${\displaystyle b-{\frac {a}{RT}}=\sum _{i}\sum _{j}x_{i}x_{j}\left(b_{ij}-{\frac {a_{ij}}{RT}}\right)}$

which constrains the sum of a and b. The second equation is

${\displaystyle {\underline {A}}_{EOS}^{ex}(T,P\to \infty ,{\underline {x}})={\underline {A}}_{\gamma }^{ex}(T,P\to \infty ,{\underline {x}})}$

with the notable limit as ${\displaystyle P\to \infty }$ (and ${\displaystyle {\underline {V}}_{i}\to b,}$ ${\displaystyle {\underline {V}}_{mix}\to b}$) of

${\displaystyle {\underline {A}}_{EOS}^{ex}=C^{*}\left({\frac {a}{b}}-\sum x_{i}{\frac {a_{i}}{b_{i}}}\right).}$

The mixing rules become

${\displaystyle {\frac {a}{RT}}=Q{\frac {D}{1-D}},\quad b={\frac {Q}{1-D}}}$

${\displaystyle Q=\sum _{i}\sum _{j}x_{i}x_{j}\left(b_{ij}-{\frac {a_{ij}}{RT}}\right)}$

${\displaystyle D=\sum _{i}x_{i}{\frac {a_{i}}{b_{i}RT}}+{\frac {{\underline {G}}_{\gamma }^{ex}(T,P,{\underline {x}})}{C^{*}RT}}}$

The cross term still must be specified by a combining rule, either

${\displaystyle b_{ij}-{\frac {a_{ij}}{RT}}={\sqrt {\left(b_{ii}-{\frac {a_{ii}}{RT}}\right)\left(b_{jj}-{\frac {a_{jj}}{RT}}\right)}}(1-k_{ij})}$

or

${\displaystyle b_{ij}-{\frac {a_{ij}}{RT}}={\frac {1}{2}}(b_{ii}+b_{jj})-{\frac {\sqrt {a_{ii}a_{jj}}}{RT}}(1-k_{ij}).}$

See also

Vapor–liquid equilibrium

Equation of state

References

1. Wong, D. S. H. & Sandler, S. I. (1992). "A theoretically correct mixing rule for cubic equations of state". AIChE Journal. 38 (5): 671–680. doi:10.1002/aic.690380505.