Theory

Equation of state (EOS) of a real fluid can be expressed in a general form as

pv = ZRT,

where p is the pressure, v is the specific volume, T is the temperature, Z is the compressibility factor, and R = R_0 / M is the specific gas constant (R_0 is the universal gas constant and M is the molar mass of the fluid). Various EOS differ based on how they determine Z.

The density of the fluid can be easily obtained from the specific volume as \rho=M/v. The speed of sound, on the other hand requires knowledge of the heat capacities, and isothermal compressibility of the fluid. Tee expression for speed of sound of a real fluid is

c^2 = \frac{1}{\rho} \frac{c_p}{c_v} \frac{1}{\chi_T},

where c_p and c_v are the isobaric and isochoric specific heat capacities, and chi_t is the isothermal compressibility.

Ideal Gas EOS

The ideal gas (IG) EOS is obtained by assuming Z=1. This EOS is quite satisfactory at lower pressures. However, at higher pressures it is unreliable as it does not take into account the finite volume of the fluid molecules or the interactions between them.

For an ideal gas, the heat capacities can be expressed as polynomials of temperature. In the present work, the isobaric specific heat capacity is assumed to be modelled by the following polynomial.

c_p = a + bT + cT^2 + dT^3.

The coefficients (a, b, c, d) for various fluids can be found in the tables provided by ThermoNet (reproduced here in Ideal gas polynomials for various fluids). The ideal gas isochoric specific heat capacity can then be obtained by the relation c_p -c_v = R.

The isothermal compressibility is of an ideal gas is given by

\chi_T = -\frac{1}{v} \left. \frac{\partial v}{\partial p} \right|_T = \frac{1}{p}

Cubic EOS

The cubic EOS are an improvement on the ideal gas model in the sense that they try to model the finite volume of the fluid molecules as well as the interactions between them. The present work considers three different cubic EOS – Redlich-Kwong (RK) [redlich1949], Soave-Redlich-Kwong (SRK) [soave1972], and Peng-Robinson(PR) [peng1976]. A general expression for cubic EOS is given by

p = \frac{RT}{v-b+c} - \frac{a(T)}{v^2+dv+e},

where (a(T), b, c, d, e) are the coefficients of the EOS. Substituting the general EOS pv = ZRT into the above equation results in a cubic equation in terms of Z given by

Z^3 + A Z^2 + B Z + C = 0.

The coefficients A, B, C are expressed as

A(T, p) = \frac{-RT - bp + cp + dp}{RT} \\ B(T, p) = \frac{-RTdp + a(T)p - bdp^2 + cdp^2 + ep^2}{R^2 T^2} \\ C(T, p) = \frac{-RTep^2 - a(T)bp^2 + a(T)cp^2 - bep^3 + cep^3}{R^3 T^3}.

For the cubic EOS considered here, these coefficients are as follows.

  • Redlich-Kwong (RK):

    a(T) = \frac{0.4278 R^2 T_c^2}{p_c} \sqrt{\frac{T_c}{T}} \\ b = d = \frac{0.08664 R T_c}{p_c} \\ c = e = 0.

  • Soave-Redlich-Kwong (SRK)

    a(T) = \frac{0.4278 R^2 T_c^2}{p_c} \left[1 + \kappa \left(1 - \sqrt{\frac{T}{T_c}} \right) \right]^2 \\ \kappa = 0.48508 + 1.55171 \omega - 0.15613 \omega^2 \\ b = d = \frac{0.08664 R T_c}{p_c} \\ c = e = 0.

  • Peng-Robinson (PR)

    a(T) = \frac{0.45724 R^2 T_c^2}{p_c} \left[1 + \kappa \left(1 - \sqrt{\frac{T}{T_c}} \right) \right]^2 \\ \kappa = 0.37464 + 1.54226 \omega - 0.26992 \omega^2 \\ b = \frac{0.07780 R T_c}{p_c} \\ c = 0 \qquad d = 2b \qquad e = -b^2.

Here, (T_c, p_c) are the critical temperature and pressure of the fluid, and \omega is its acentric factor.

The heat capacities can be obtained from the enthalpy and internal energy of the fluid. The isobaric specific heat capacity c_p is given by c_p = \left. \frac{\partial H}{\partial T} \right|_p, where H is the specific enthalpy. Similarly, the isochoric specific heat capacity c_v is given by c_v = \left. \frac{\partial U}{\partial T} \right|_v, where U is the specific internal energy. The expressions for H and U can be found in [assael1996], and are not reproduced here for the sake of brevity. After incorporating these expressions, the heat capacities can be written as

c_p = c_p^0 % + R T Z_p' % + R (Z-1) % + T \frac{\log(h)}{\Delta} \frac{\mathrm d^2 a}{\mathrm d T^2} % + \frac{h g}{\Delta} (T \frac{\mathrm d a}{\mathrm d T} - a) % \\ c_v = c_v^0 % + \frac{T}{\Delta} \frac{\mathrm d^2 a}{\mathrm d T^2} \log(h),

where Z=pv/RT, and c_p^0 and c_v^0 are the ideal gas specific heat capacities; and

\Delta = \sqrt{d^2 - 4 e} \qquad Z_p' = \left. \frac{\partial Z}{\partial T} \right|_p \\ h = \frac{\Delta + 2 R T Z/p + d}{-\Delta + 2 R T Z/p + d} \qquad g = \frac{-4 \Delta R p (T Z_p' + Z)}{(2 R T Z - p(\Delta - d))^2}.

Lastly, the isothermal compressibility of the fluid is obtained using the following relation.

\chi_T = -\frac{1}{v} \left. \frac{\partial v}{\partial p} \right|_T = \frac{1}{p} - \frac{1}{Z} \left. \frac{\partial Z}{\partial p} \right|_T.

Lee-Kesler EOS

The Lee-Kesler (LK) EOS [lee1975] is based on the correlation developed by Pitzer and co. according to which Z of a fluid can be written as

Z(T_r, p_r) = Z_0(T_r, p_r) + \omega Z_1 (T_r, p_r),

where Z_0 is the compressibility factor of a simple fluid whose molecules are spherical, and Z_1 is the deviation of compressibility factor. \omega is the acentric factor of the fluid – a measure of the non-spherical nature of the molecules. Also, T_r=T/T_c and p_r=p/p_c are the reduced temperature and pressure. The deviation Z_1 is expressed as a linear function of Z_0 and the compressibility factor Z_2 of a heavy non-spherical reference fluid (with acentric factor \omega_2) in the following manner.

Z_1(T_r, p_r) = \frac{Z_2(T_r, p_r) - Z_0(T_r, p_r)}{\omega_2}.

Z_0 and Z_2 are obtained by solving the following nonlinear equation.

Z_{0,2} = \frac{p_r v_r}{T_r} = 1 + \frac{B(T_r)}{v_r} + \frac{C(T_r)}{v_r^2} + \frac{D(T_r)}{v_r^5} + \frac{c_4}{T_r^3 v_r^2} \left( \beta + \frac{\gamma}{v_r^2} \right) \exp{\left(\frac{-\gamma}{v_r^2}\right)},

where v_r is called the reduced volume, and

B(T_r) = b_1 - \frac{b_2}{T_r} - \frac{b_3}{T_r^2} - \frac{b_4}{T_r^3} \\ C(T_r) = c_1 - \frac{c_2}{T_r} + {c_3}{T_r^3} \\ D(T_r) = d_1 + \frac{d_2}{T_r}.

The constant b_i, c_i, d_i, \beta and \gamma are different for the simple and reference fluids, and are listed in the table below.

Constant Simple fluid Reference fluid Constant Simple fluid Reference fluid
b_1 0.1181193 0.2026579 b_2 0.265728 0.331511
b_3 0.154790 0.027655 b_4 0.030323 0.203488
c_1 0.0236744 0.0313385 c_2 0.0186984 0.0503618
c_3 0.0 0.016901 c_4 0.042724 0.041577
d_1 \times 10^4 0.155488$ 0.48736 d_2 \times 10^4 0.623689$ 0.0740336
\beta 0.65392 1.226 \gamma 0.060167 0.03754

The departure (difference between real and ideal fluid) in c_p and c_v for the simple and reference fluids is given by [assael1996]

\frac{\left. c_v^r \right|_{0, 2}}{R} = 2 \frac{(b_3 + 3 b_4/T_r)}{T_r v_r} - 3 \frac{c_3}{T_r^3 v_r^2} - 6 E \\ \frac{\left. c_p^r \right|_{0,2}}{R} = \frac{c_v^r|_i}{R} - 1 - T_r \left. \left( \left. \frac{\partial p_r}{\partial T_r} \right|_{v_r} \right)^2 \right/ \left( \left. \frac{\partial p_r}{\partial v_r} \right|_{T_r} \right),

where

E = \frac{c_4}{2 T_r^3 \gamma} \left( \beta + 1 - \left(\beta + 1 + \frac{\gamma}{v_r^2}\right) \exp\left(\frac{-\gamma}{v_r^2}\right) \right).

Then, the real gas heat capacities are given by

c_p = c_p^0 + c_p^r|_0 + \omega \frac{(c_p^r|_2 - c_p^r|_0)}{\omega_2} \\ c_v = c_v^0 + c_v^r|_0 + \omega \frac{(c_v^r|_2 - c_v^r|_0)}{\omega_2},

where c_p^0 and c_v^0 are the ideal gas heat capacities.

The isothermal compressibility of the simple and reference fluids is given by

\chi_T|_{0,2} = -\frac{1}{v} \left. \frac{\partial v}{\partial p} \right|_T = \frac{1}{p} - \frac{1}{Z_{0,2}} \left. \frac{\partial Z_{0,2}}{\partial p} \right|_T.

The isothermal compressibility of the real fluid can then be obtained by

\chi_T = \chi_T|_0 + \omega \frac{(\chi_T|_2 - \chi_T|_0)}{\omega_2}.

References

[redlich1949]Redlich O, Kwong JNS. On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions. Chemical Reviews 1949;44:233–44. doi:10.1021/cr60137a013.
[soave1972]Soave G. Equilibrium constants from a modified Redlich-Kwong equation of state. Chemical Engineering Science 1972;27:1197–203. doi:10.1016/0009-2509(72)80096-4.
[peng1976]Peng D-Y, Robinson DB. A New Two-Constant Equation of State. Industrial & Engineering Chemistry Fundamentals 1976;15:59–64. doi:10.1021/i160057a011.
[assael1996](1, 2) Assael MJ, Trusler JPM, Tsolakis TF. Thermophysical Properties of Fluids: An Introduction to Their Prediction. World Scientific; 1996.
[lee1975]Lee BI, Kesler MG. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J 1975;21:510–27. doi:10.1002/aic.690210313.