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

\[\alpha^{\rm r} = \left(1-\frac{\omega_{\rm mix}}{\omega_{\rm ref}}\right)\alpha^{\rm r}_{\rm simple} + \frac{\omega_{\rm mix}}{\omega_{\rm ref}}\alpha^{\rm r}_{\rm ref}\]

where the contributions are each of the form

\[\alpha^{\rm r}_{X}(\tau, \delta) = B\left(\frac{\delta}{Z_c}\right) + \frac{C}{2}\left(\frac{\delta}{Z_c}\right)^2 + \frac{D}{5}\left(\frac{\delta}{Z_c}\right)^5 - \frac{c_4\tau^3}{2\gamma}\left(\gamma\left(\frac{\delta}{Z_c}\right)^2+\beta+1\right)\exp\left(-\gamma\left(\frac{\delta}{Z_c}\right)^2\right) + \frac{c_4\tau^3}{2\gamma}(\beta+1)\]

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

\[\delta = \frac{\rho}{\rho_{\rm red}} = v_{\rm c,mix}\rho\]

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

\[v_{\rm c, mix} = \sum_{i=1}^{N-1}\sum_{j=i+1}^N x_ix_jv_{ij}\]

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

and the critical volumes are estimated from

\[v_{c,i} = (0.2905-0.085\omega_i)\frac{RT_{c,i}}{p_{c,i}}\]

For temperature, the reciprocal reduced density is defined by

\[\tau = \frac{T_{\rm c,mix}}{T}\]

with

\[T_{\rm c,mix} = \frac{1}{v_{c,mix}^{\eta}}\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}x_ix_jv_{c,ij}^{\eta}T_{c,ij}\]

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

\[Z_c = 0.2905-0.085\omega_{\rm mix}\]

with the mixture acentric factor defined by

\[\omega_{\rm mix} = \sum_i x_i\omega_i\]

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)