Lee-Kesler-Plocker

The Lee-Kesler-Plocker package is a corresponding-states model: the mixture is represented by a simple and a reference fluid, each obeying a Benedict-Webb-Rubin equation, and its properties are interpolated with the mixture acentric factor. Its derivatives are the most involved of the packages, so every ingredient is written out below.

Mixture pseudo-critical rules

With the pair quantities

\[ v_{cjk}=\tfrac{1}{8}\!\left(V_{c,j}^{1/3}+V_{c,k}^{1/3}\right)^{3},\qquad t_{cjk}=\sqrt{T_{c,j}T_{c,k}}\;k_{jk} \]

the mixture critical properties are

\[ \begin{aligned}w_{m} & =\sum_{i}x_{i}w_{i},\qquad V_{cm}=\sum_{j}\sum_{k}x_{j}x_{k}v_{cjk},\qquad T_{cm}=V_{cm}^{-1/4}\sum_{j}\sum_{k}x_{j}x_{k}\,v_{cjk}^{1/4}\,t_{cjk}\\ P_{cm} & =(0.2905-0.085\,w_{m})\frac{RT_{cm}}{V_{cm}},\qquad z_{cm}=\frac{P_{cm}V_{cm}}{RT_{cm}} \end{aligned} \]

Hydrogen enters these rules with the effective (quantum-corrected) constants of Gunn, Chueh and Prausnitz, evaluated at the temperature of the calculation with the molar mass \(M\) (g/mol):

\[ T_{c}=\frac{43.6}{1+21.8/(MT)}\ \mathrm{K},\qquad P_{c}=\frac{20.5}{1+44.2/(MT)}\ \mathrm{atm},\qquad V_{c}=\frac{51.5}{1-9.91/(MT)}\ \mathrm{cm^{3}/mol},\qquad w=0 \]

The hydrogen \(k_{jk}\) of the table go with these constants. A hydrogen pair missing from the table takes

\[ k_{\mathrm{H_{2}},k}=0.2036+0.002246\,T_{c,k}+0.8364\,w_{k}+1.2111\,V_{c,k}^{1/3} \]

with \(T_{c,k}\) in K and \(V_{c,k}\) in m3/kmol, fitted to the eleven hydrogen pairs of the table and to hydrogen solubilities in n-dodecane, benzene, toluene, cyclohexane and methylcyclohexane. Table values and values set by the user are kept. Because the hydrogen constants depend on temperature, the temperature derivatives of the fugacity coefficients are taken by finite differences when hydrogen is present.

Other pairs missing from the table get an estimate when one compound is a hydrocarbon (elements C and H only, or a petroleum fraction) and the other a lighter hydrocarbon, nitrogen or carbon monoxide, provided \(T_{c2}/T_{c1}\geq1.3\) (component 2 the heavier):

\[ k_{12}=1+0.2802\,\ln\frac{T_{c2}}{T_{c1}}\,\ln\frac{V_{c2}}{V_{c1}}-0.0561\,\ln\frac{T_{c2}}{T_{c1}} \]

Carbon dioxide with a hydrocarbon of \(T_{c}\geq304.21\) K takes

\[ k_{\mathrm{CO_{2}},k}=0.9276+0.10719\,X+0.04505\,X^{2},\qquad X=\ln\frac{T_{c,k}}{304.21}\,\ln\frac{V_{c,k}}{0.094} \]

with \(V_{c,k}\) in m3/kmol, and hydrogen sulfide the carbon dioxide value less 0.027. Equation [eq:d-lkp-hckij] fits the 87 hydrocarbon pairs of the table with an rms deviation of 0.011, and the nitrogen row within 0.02; both correlations were fitted to the table and to parameters regressed from about 800 solubility points of methane, ethane, nitrogen, carbon monoxide, carbon dioxide and hydrogen sulfide in heavy alkanes, aromatics and naphthenes. All other pairs take 1. Table values and values set by the user are never replaced, and the parameter editor shows the estimates.

Compressibility and fugacity coefficient

Each fluid (simple \(s\) and reference \(h\) , with its own constant set) gives a compressibility from the reduced volume \(V_{r}=P_{c}V/(RT_{c})\) , and the mixture interpolates with \(w_{h}=0.3978\) :

\[ \begin{aligned}z=\frac{P_{r}V_{r}}{T_{r}} & =1+\frac{B}{V_{r}}+\frac{C}{V_{r}^{2}}+\frac{D}{V_{r}^{5}}+\frac{c_{4}}{T_{r}^{3}V_{r}^{2}}\!\left(\beta+\frac{\gamma}{V_{r}^{2}}\right)e^{-\gamma/V_{r}^{2}}\\ z_{m} & =z^{(s)}+\frac{w_{m}}{w_{h}}\!\left(z^{(h)}-z^{(s)}\right) \end{aligned} \]

with \(B=b_{1}-b_{2}/T_{r}-b_{3}/T_{r}^{2}-b_{4}/T_{r}^{3}\) , \(C=c_{1}-c_{2}/T_{r}+c_{3}/T_{r}^{3}\) , \(D=d_{1}+d_{2}/T_{r}\) (the two fluid constant sets are the standard Lee-Kesler table). The component fugacity coefficient is built from the composition derivatives of the mixing rules,

\[ \ln\varphi_{i}=\ln\varphi_{m}-\frac{\hat{H}}{T}\,S_{i}^{a}+\frac{z_{m}-1}{P_{cm}}\,S_{i}^{b}-\Big(\tfrac{\partial\ln\varphi_{m}}{\partial w_{m}}\Big)S_{i}^{c} \]

where \(\ln\varphi_{m}\) is the mixture (corresponding-states) fugacity coefficient, \(\hat{H}=H_{m}^{\mathrm{dep}}M_{m}/(RT_{cm})\) the reduced enthalpy departure, \(\partial\ln\varphi_{m}/\partial w_{m}=(\ln\varphi^{(h)}-\ln\varphi^{(s)})/w_{h}\) , and

\[ S_{i}^{a}=\sum_{j\neq i}x_{j}\,T_{cm}^{(ij)},\qquad S_{i}^{b}=\sum_{j\neq i}x_{j}\,P_{cm}^{(ij)},\qquad S_{i}^{c}=\sum_{j\neq i}x_{j}(w_{j}-w_{i}) \]

The pairwise mixing-rule derivatives are

\[ \begin{aligned}V_{cm}^{(ij)} & =2\sum_{l}x_{l}(v_{clj}-v_{cli}),\qquad Z_{cm}^{(ij)}=-0.085\,(w_{j}-w_{i})\\ T_{cm}^{(ij)} & =V_{cm}^{-1/4}\!\left[2\sum_{l}x_{l}\!\left(v_{clj}^{1/4}t_{clj}-v_{cli}^{1/4}t_{cli}\right)-\tfrac{1}{4}V_{cm}^{-3/4}V_{cm}^{(ij)}T_{cm}\right]\\ P_{cm}^{(ij)} & =P_{cm}\!\left(\frac{Z_{cm}^{(ij)}}{z_{cm}}+\frac{T_{cm}^{(ij)}}{T_{cm}}-\frac{V_{cm}^{(ij)}}{V_{cm}}\right) \end{aligned} \]

Temperature derivative

Since \(T_{r}=T/T_{cm}\) and \(P_{r}=P/P_{cm}\) is constant in \(T\) , each fluid needs the Benedict-Webb-Rubin derivatives at fixed \(P_{r}\) . Writing the residual \(\mathcal{R}=P_{r}V_{r}/T_{r}-1-B/V_{r}-C/V_{r}^{2}-D/V_{r}^{5}-\phi_{4}=0\) with \(\phi_{4}=(c_{4}/T_{r}^{3}V_{r}^{2})(\beta+\gamma/V_{r}^{2})e^{-\gamma/V_{r}^{2}}\) , implicit differentiation gives

\[ \begin{aligned}\frac{dV_{r}}{dT_{r}} & =-\frac{\partial\mathcal{R}/\partial T_{r}}{\partial\mathcal{R}/\partial V_{r}},\qquad\frac{dz}{dT_{r}}=\frac{P_{r}}{T_{r}}\frac{dV_{r}}{dT_{r}}-\frac{P_{r}V_{r}}{T_{r}^{2}}\\ \frac{\partial\mathcal{R}}{\partial V_{r}} & =\frac{P_{r}}{T_{r}}+\frac{B}{V_{r}^{2}}+\frac{2C}{V_{r}^{3}}+\frac{5D}{V_{r}^{6}}-\frac{\partial\phi_{4}}{\partial V_{r}}\\ \frac{\partial\mathcal{R}}{\partial T_{r}} & =-\frac{P_{r}V_{r}}{T_{r}^{2}}-\frac{B'}{V_{r}}-\frac{C'}{V_{r}^{2}}-\frac{D'}{V_{r}^{5}}+\frac{3\phi_{4}}{T_{r}} \end{aligned} \]

with \(B'=b_{2}/T_{r}^{2}+2b_{3}/T_{r}^{3}+3b_{4}/T_{r}^{4}\) , \(C'=c_{2}/T_{r}^{2}-3c_{3}/T_{r}^{4}\) , \(D'=-d_{2}/T_{r}^{2}\) . The single-fluid fugacity coefficient and its temperature derivative are

\[ \begin{aligned}\ln\varphi & =z-1-\ln z+\frac{B}{V_{r}}+\frac{C}{2V_{r}^{2}}+\frac{D}{5V_{r}^{5}}+E\\ \frac{d\ln\varphi}{dT_{r}} & =\frac{dz}{dT_{r}}\!\left(1-\frac{1}{z}\right)+\Big(\frac{B'}{V_{r}}-\frac{B}{V_{r}^{2}}\frac{dV_{r}}{dT_{r}}\Big)+\Big(\frac{C'}{2V_{r}^{2}}-\frac{C}{V_{r}^{3}}\frac{dV_{r}}{dT_{r}}\Big)\\ & \quad{}+\Big(\frac{D'}{5V_{r}^{5}}-\frac{D}{V_{r}^{6}}\frac{dV_{r}}{dT_{r}}\Big)+\frac{dE}{dT_{r}} \end{aligned} \]

where \(E=\dfrac{c_{4}}{2T_{r}^{3}\gamma}\big[\beta+1-(\beta+1+\gamma/V_{r}^{2})e^{-\gamma/V_{r}^{2}}\big]\) . The mixture derivatives interpolate the two fluids, \(dz_{m}/dT_{r}=dz^{(s)}/dT_{r}+(w_{m}/w_{h})(dz^{(h)}/dT_{r}-dz^{(s)}/dT_{r})\) and likewise for \(d\ln\varphi_{m}/dT_{r}\) , and the component derivative assembles as

\[ \frac{\partial\ln\varphi_{i}}{\partial T}=\frac{1}{T_{cm}}\frac{d\ln\varphi_{m}}{dT_{r}}+S_{i}^{a}\!\left(\frac{\hat{H}}{T^{2}}-\frac{1}{T}\frac{d\hat{H}}{dT}\right)+\frac{S_{i}^{b}}{P_{cm}T_{cm}}\frac{dz_{m}}{dT_{r}}-\frac{S_{i}^{c}}{T_{cm}}\frac{d}{dT_{r}}\!\Big(\tfrac{\partial\ln\varphi_{m}}{\partial w_{m}}\Big) \]

The reduced enthalpy-departure derivative is the only non-analytical ingredient and is taken by a tight central finite difference. The composition derivative differentiates the same fugacity expression through the pseudo-critical mixing rules; because those rules enter both directly and through the corresponding-states functions, a few mixture terms are likewise taken by finite differences, so the package uses a hybrid composition derivative.