Generalized Fundamental Equation of State for the Normal Alkanes $(\hbox {C}_{5}{-}\hbox {C}_{50})$

Abstract

Based on the extended three-parameter corresponding-states principle and the most reliable experimental data of $n$-alkanes, a generalized fundamental equation of state for technical calculations has been developed. This equation is in the form of the reduced Helmholtz free energy and takes the reduced density, reduced temperature, and acentric factor as variables. The proposed equation satisfies the critical conditions and Maxwell rule, shows correct behavior for the ideal curves and for the derivatives of the thermodynamic potentials, and allows the calculation of all thermodynamic properties including phase equilibrium of $n$-alkanes from $n$-pentane $(\hbox {C}_{5})$ to $n$-pentacontane $(\hbox {C}_{50})$ over a temperature range from the triple point to 700 K with pressures up to 100 MPa. The new equation differs from the previous generalized equations of other authors by a wider range of variation of the acentric factor $\omega =0.25$ to 1.8, and by more accurately predicting thermal properties.

Appendices

Appendix 1: Definitions of Common Thermodynamic Properties and their Relation to the reduced Helmholtz Energy

The functions used for calculating pressure $(p)$, compressibility factor $(Z)$, internal energy $(u)$, enthalpy $(h)$, entropy $(s)$, Gibbs energy $(g)$, isochoric heat capacity $(c_{v})$, isobaric heat capacity $(c_{p})$, Joule–Thomson coefficient ($\mu$), and the speed of sound $(w)$ from Eq. 1 are given below.

$$\begin{aligned} p&= \rho ^{2}\left( {\frac{{\partial a}}{{\partial \rho }}} \right) _{T} =\rho RT\left[ {1+\delta \;\left( {\frac{{\partial \alpha ^{{\text {r}}}}}{{\partial \delta }}}\right) _{\tau }}\right] \end{aligned}$$

(11)

$$\begin{aligned} Z&= \frac{p}{{\rho RT}}=1+\delta \,\left( {\frac{{\partial \alpha ^{{\text {r}}}}}{{\partial \delta }}}\right) _{\tau } \end{aligned}$$

(12)

$$\begin{aligned} \frac{u}{RT}&= \frac{a+Ts}{RT}=\tau \left[ {\left( {\frac{\partial \alpha ^{0}}{\partial \tau }}\right) _\delta +\left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \tau }}\right) _\delta }\right] \end{aligned}$$

(13)

$$\begin{aligned} \frac{h}{RT}&= \frac{u+pv}{RT}=\tau \left[ {\left( {\frac{\partial \alpha ^{0}}{\partial \tau }}\right) _\delta +\left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \tau }}\right) _\delta }\right] +\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau +1 \end{aligned}$$

(14)

$$\begin{aligned} \frac{s}{R}&= -\frac{1}{R}\left( {\frac{\partial a}{\partial T}}\right) _\rho =\tau \left[ {\left( {\frac{\partial \alpha ^{0}}{\partial \tau }}\right) _\delta +\left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \tau }}\right) _\delta }\right] -\alpha ^{0}-\alpha ^{\mathrm{r}} \end{aligned}$$

(15)

$$\begin{aligned} \frac{g}{RT}&= \frac{h-Ts}{RT}=1+\alpha ^{0}+\alpha ^{\mathrm{r}}+\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau \end{aligned}$$

(16)

$$\begin{aligned} \frac{c_v}{R}&= \frac{1}{R}\left( {\frac{\partial u}{\partial T}}\right) _\rho =-\tau ^{2}\;\left[ {\left( {\frac{\partial ^{2}\alpha ^{0}}{\partial \tau ^{2}}}\right) _\delta +\left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \tau ^{2}}} \right) _\delta }\right] \end{aligned}$$

(17)

$$\begin{aligned} \frac{c_p}{R}&= \frac{1}{R}\left( {\frac{\partial h}{\partial T}}\right) _p =\frac{c_v}{R}+\;\frac{\left[ {1+\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }} \right) _\tau -\delta \tau \left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta \partial \tau }}\right) } \right] ^{2}}{\left[ {1+2\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau +\delta ^{2}\left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta ^{2}}} \right) _\tau }\right] } \end{aligned}$$

(18)

$$\begin{aligned}&\mu R\rho =R\rho \left( {\frac{\partial T}{\partial p}}\right) _h \nonumber \\&=\frac{-\left( {\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau +\delta ^{2}\left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta ^{2}}}\right) _\tau +\delta \tau \left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta \partial \tau }}\right) }\right) }{\left( {1+\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau -\delta \tau \left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta \partial \tau }}\right) }\right) ^{2}-\tau ^{2}\left( {\left. {\left( {\frac{\partial ^{2}\alpha ^{0}}{\partial \tau ^{2}}}\right) _\delta +\left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \tau ^{2}}}\right) _\delta }\right) \left( {1+2\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau +\delta ^{2}\left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta ^{2}}}\right) _\tau }\right. }\right) }\nonumber \\ \end{aligned}$$

(19)

$$\begin{aligned} \frac{w^{2}M}{RT}&= \frac{M}{RT}\left( {\frac{\partial p}{\partial \rho }}\right) _s =1+2\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau +\delta ^{2}\left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta ^{2}}}\right) _\tau \nonumber \\&-\frac{\left[ {1+\delta \left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau -\delta \tau \left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta \partial \tau }}\right) }\right] ^{2}}{\tau ^{2}\left[ {\left( {\frac{\partial ^{2}\alpha ^{0}}{\partial \tau ^{2}}}\right) _\delta +\left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \tau ^{2}}} \right) _\delta }\right] } \end{aligned}$$

(20)

The fugacity coefficient and second and third virial coefficients are given in the following equations:

$$\begin{aligned} \varphi&= \exp \left[ {Z-1-\ln \left( Z\right) +\alpha ^{\mathrm{r}}}\right] \end{aligned}$$

(21)

$$\begin{aligned} B\left( T\right)&= \mathop {\lim }\limits _{\delta \rightarrow 0} \left[ {\frac{1}{\rho _c}\left( {\frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }}\right) _\tau }\right] \end{aligned}$$

(22)

$$\begin{aligned} C\left( T\right)&= \mathop {\lim }\limits _{\delta \rightarrow 0} \left[ {\frac{1}{\rho _c^{2}}\left( {\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta ^{2}}}\right) _\tau } \right] \end{aligned}$$

(23)

Other derived properties, given below, include the first derivative of pressure with respect to density at constant temperature $(\partial p/\partial \rho )_{T}$, the second derivative of pressure with respect to density at constant temperature $(\partial ^{2} p/\partial \rho ^{2})_{T}$, and the first derivative of pressure with respect to temperature at constant density $(\partial p/\partial T)_{\rho }$.

$$\begin{aligned} \left( {\frac{{\partial p}}{{\partial \rho }}}\right) _{T}&= RT\left[ {1+2\delta \;\left( {\frac{{\partial \alpha ^{{\text {r}}} }}{{\partial \delta }}}\right) _{\tau }\;+\delta ^{2}\;\left( {\frac{{\partial ^{2}\alpha ^{{\text {r}}}}}{{\partial \delta ^{2} }}}\right) _{\tau }}\right] \end{aligned}$$

(24)

$$\begin{aligned} \left( {\frac{{\partial ^{2}p}}{{\partial \rho ^{2}}}}\right) _{T}&= \frac{{RT}}{\rho }\left[ {2\delta \;\left( {\frac{{\partial \alpha ^{{\text {r}}}}}{{\partial \delta }}}\right) _{\tau }\;+4\delta ^{2}\;\left( {\frac{{\partial ^{2}\alpha ^{{\text {r}}} }}{{\partial \delta ^{2}}}}\right) _{\tau }+\delta ^{3}\;\left( {\frac{{\partial ^{3}\alpha ^{{\text {r}}}}}{{\partial \delta ^{3} }}}\right) _{\tau }}\right] \end{aligned}$$

(25)

$$\begin{aligned} \left( {\frac{{\partial p}}{{\partial T}}}\right) _{\rho }&= R\rho \left[ {1+\delta \;\left( {\frac{{\partial \alpha ^{{\text {r}}} }}{{\partial \delta }}}\right) _{\tau }\;-\delta \tau \;\left( {\frac{{\partial ^{2}\alpha ^{{\text {r}}}}}{{\partial \delta \partial \tau }}}\right) }\right] \end{aligned}$$

(26)

The derivatives of the residual Helmholtz energy are given in the following equations.

$$\begin{aligned} \delta \frac{\partial \alpha ^{\mathrm{r}}}{\partial \delta }&= \sum _{k=1}^6 {N_k \delta ^{d_k}\tau ^{t_k}d_k} +\sum _{k=7}^{14} {N_k \delta ^{d_k}\tau ^{t_k}\exp \left( {-\delta ^{l_k}}\right) \;\left[ {d_k -l_k \delta ^{l_k}}\right] } \end{aligned}$$

(27)

$$\begin{aligned} \delta ^{2}\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \delta ^{2}}&= \sum _{k=1}^6 {N_k \delta ^{d_k}\tau ^{t_k}\left[ {d_k \left( {d_k -1}\right) }\right] }+\sum _{k=7}^{14} N_k \delta ^{d_k}\tau ^{t_k}\nonumber \\&\exp \left( {-\delta ^{l_k}}\right) \;\left[ {\left( {d_k -l_k \delta ^{l_k}}\right) \left( {d_k -1-l_k \delta ^{l_k}}\right) -l_k^{2}\delta ^{l_k}}\right] \qquad \end{aligned}$$

(28)

$$\begin{aligned} \delta ^{3}\frac{\partial ^{3}\alpha ^{\mathrm{r}}}{\partial \delta ^{3}}&= \sum _{k=1}^6 {N_k \delta ^{d_k}\tau ^{t_k}\left[ {d_k \left( {d_k -1}\right) \left( {d_k -2}\right) }\right] }+\sum _{k=7}^{14} N_k \delta ^{d_k}\tau ^{t_k}\exp \left( {-\delta ^{l_k}}\right) \nonumber \\&\left\{ d_k \left( {d_k -1}\right) \left( {d_k -2}\right) +l_k \delta ^{l_k}\left[ -2+6d_k -3d_k^2 -3d_k l_k \right. \right. \nonumber \\&\left. \left. +3l_k-l_k^2\right] +3l_k^2 \delta ^{2l_k}\left[ {d_k -1+l_k}\right] -l_k^3 \delta ^{3l_k}\right\} \end{aligned}$$

(29)

$$\begin{aligned} \tau \frac{\partial \alpha ^{\mathrm{r}}}{\partial \tau }&= \sum _{k=1}^6 {N_k \delta ^{d_k}\tau ^{t_k}t_k} +\sum _{k=7}^{14} {N_k \delta ^{d_k}\tau ^{t_k}\exp \left( {-\delta ^{l_k}} \right) \;t_k} \end{aligned}$$

(30)

$$\begin{aligned} \tau ^{2}\frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \tau ^{2}}\!&= \!\sum _{k=1}^6 {N_k \delta ^{d_k }\tau ^{t_k}\left[ {t_k \left( {t_k \!-\!1}\right) }\right] } \!+\!\sum _{k=7}^{14} {N_k \delta ^{d_k}\tau ^{t_k}\exp \left( {\!-\!\delta ^{l_k}}\right) \;\left[ {t_k \left( {t_k \!-\!1}\right) }\right] }\qquad \qquad \end{aligned}$$

(31)

$$\begin{aligned} \tau \delta \frac{\partial ^{2}\alpha ^{\mathrm{r}}}{\partial \tau \;\partial \delta }\!&= \!\sum _{k=1}^6 {N_k \delta ^{d_k}\tau ^{t_k}\left[ {d_k t_k}\right] } \!+\!\sum _{k=7}^{14} {N_k \delta ^{d_k}\tau ^{t_k}\exp \left( {\!-\!\delta ^{l_k}}\right) } \;\left[ {t_k \left( {d_k \!-\!l_k \delta ^{l_k}} \right) }\right] \end{aligned}$$

(32)

$$\begin{aligned} \delta \tau ^{2}\frac{\partial ^{3}\alpha ^{\mathrm{r}}}{\partial \delta \partial \tau ^{2}}&= \sum _{k=1}^6 {N_k \delta ^{d_k}\tau ^{t_k}\left[ {d_k t_k \left( {t_k -1}\right) }\right] }\nonumber \\&+\sum _{k=7}^{14} {N_k \delta ^{d_k}\tau ^{t_k}\exp \left( {-\delta ^{l_k}}\right) \;\left[ {t_k \left( {t_k -1}\right) \left( {d_k -l_k \delta ^{l_k}}\right) }\right] } \end{aligned}$$

(33)

Appendix 2: Comparison of Thermodynamic Properties of Selected $n$-alkanes $(\hbox {C}_{7},\,\hbox {C}_{8},\,\hbox {C}_{10},\,\hbox {C}_{12})$ calculated by Individual Equations of State with the Generalized Fundamental Equation of State, Eq. 3