LKP (Lee-Kesler-Plöcker)

The LKP model is a sort of hybrid between corresponding states and multiparameter EOS, simple EOS are developed for a reference fluid, and a simple fluid, and the acentric factor of the mixture is used to weight the two.

The reduced residual Helmholtz energy for the mixture is evaluated from

$${\alpha }^{\mathrm{r}}=(1-\frac{{\omega }_{\mathrm{m}\mathrm{i}\mathrm{x}}}{{\omega }_{\mathrm{r}\mathrm{e}\mathrm{f}}}){\alpha }_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}^{\mathrm{r}}+\frac{{\omega }_{\mathrm{m}\mathrm{i}\mathrm{x}}}{{\omega }_{\mathrm{r}\mathrm{e}\mathrm{f}}}{\alpha }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{r}}$$

where the contributions are each of the form

$${\alpha }_X^{\mathrm{r}}(\tau ,\delta )=B\left(\frac{\delta }{Z_c}\right)+\frac{C}{2}{\left(\frac{\delta }{Z_c}\right)}^2+\frac{D}{5}{\left(\frac{\delta }{Z_c}\right)}^5-\frac{c_4{\tau }^3}{2\gamma }(\gamma {\left(\frac{\delta }{Z_c}\right)}^2+\beta +1)\exp (-\gamma {\left(\frac{\delta }{Z_c}\right)}^2)+\frac{c_4{\tau }^3}{2\gamma }(\beta +1)$$

where $X$ is one of simple or reference (abbreviation: ref) with the matching sets of coefficients taken from this table:

var	simple	reference
$b_1$	0.1181193	0.2026579
$b_2$	0.265728	0.331511
$b_3$	0.154790	0.276550e-1
$b_4$	0.303230e-1	0.203488
$c_1$	0.236744e-1	0.313385e-1
$c_2$	0.186984e-1	0.503618e-1
$c_3$	0	0.169010e-1
$c_4$	0.427240e-1	0.41577e-1
$d_1$	0.155428e-4	0.487360e-4
$d_2$	0.623689e-4	0.740336e-5
$\beta$	0.653920	1.226
$\gamma$	0.601670e-1	0.03754
$\omega$	0.0	0.3978

The terms in the contributions are given by:

$$B=b_1-b_2\tau -b_3{\tau }^2-b_4{\tau }^3$$

$$C=c_1-c_2\tau +c_3{\tau }^3$$

$$D=d_1+d_2\tau$$

For density, the reduced density $\delta$ is defined by

$$\delta =\frac{\rho }{{\rho }_{\mathrm{r}\mathrm{e}\mathrm{d}}}=v_{\mathrm{c},\mathrm{m}\mathrm{i}\mathrm{x}}\rho$$

in which the reducing density is the reciprocal of the pseudo-critical volume obtained from

$$v_{\mathrm{c},\mathrm{m}\mathrm{i}\mathrm{x}}=\sum _{i=1}^{N-1}\sum _{j=i+1}^N{x}_i{x}_j{v}_{ij}$$

$$v_{c,ij}=\frac{1}{8}(v_{c,i}^{1/3}+v_{c,j}^{1/3}{)}^3$$

and the critical volumes are estimated from

$$v_{c,i}=(0.2905-0.085{\omega }_i)\frac{R{T}_{c,i}}{p_{c,i}}$$

For temperature, the reciprocal reduced density is defined by

$$\tau =\frac{T_{\mathrm{c},\mathrm{m}\mathrm{i}\mathrm{x}}}{T}$$

with

$$T_{\mathrm{c},\mathrm{m}\mathrm{i}\mathrm{x}}=\frac{1}{v_{c,mix}^{\eta }}\sum _{i=1}^{N-1}\sum _{j=i+1}^N{x}_i{x}_j{v}_{c,ij}^{\eta }{T}_{c,ij}$$

with $\eta =0.25$ and

$$T_{c,ij}=k_{ij}\sqrt{T_{c,i}{T}_{c,j}}$$

Note: the default interaction parameter $k_{ij}$ is therefore 1, rather than 0 in the case of SAFT and cubic models.

Finally the parameter $Z_c$ is defined by

$$Z_c=0.2905-0.085{\omega }_{\mathrm{m}\mathrm{i}\mathrm{x}}$$

with the mixture acentric factor defined by

$${\omega }_{\mathrm{m}\mathrm{i}\mathrm{x}}=\sum _i{x}_i{\omega }_i$$

import teqp, numpy as np
spec = {
    "Tcrit / K": [190.564, 126.192],
    "pcrit / Pa": [4.5992e6, 3.3958e6],
    "acentric": [0.011, 0.037],
    "R / J/mol/K": 8.3144598,
    "kmat": [[1.0, 0.977],[0.977, 1.0]]
}
model = teqp.make_model({'kind': 'LKP', 'model': spec}, validate=True)

# A little sanity check, with the check value from TREND
expected = -0.18568096994998817
diff = abs(model.get_Ar00(300, 8000.1, np.array([0.8, 0.2])) - expected)
assert(diff < 1e-13)