Unconditionally Energy Stable Linear Schemes for the Diffuse Interface Model with Peng–Robinson Equation of State

Abstract

In this paper, we investigate numerical solution of the diffuse interface model with Peng–Robinson equation of state, that describes real states of hydrocarbon fluids in the petroleum industry. Due to the strong nonlinearity of the source terms in this model, how to design appropriate time discretizations to preserve the energy dissipation law of the system at the discrete level is a major challenge. Based on the “Invariant Energy Quadratization” approach and the penalty formulation, we develop efficient first and second order time stepping schemes for solving the single-component two-phase fluid problem. In both schemes the resulted temporal semi-discretizations lead to linear systems with symmetric positive definite spatial operators at each time step. We rigorously prove their unconditional energy stabilities in the time discrete sense. Various numerical simulations in 2D and 3D spaces are also presented to validate accuracy and stability of the proposed linear schemes and to investigate physical reliability of the target model by comparisons with laboratory data.

Appendix

All the following parameters are classical definitions, and can be found in the references [15, 16, 18, 22] and the references cited therein. The universal gas constant R has a value of approximately $8.31432\,\text {JK}^{-1}\text {mol}^{-1}$, and the (temperature-dependent) energy parameter $a=a(T)$ and the co-volume parameter b in the Peng–Robinson equation of state are defined as

$$\begin{aligned} a(T)=\sum \limits _{i=1}^M\sum \limits _{j=1}^M (1-k_{ij}) y_iy_j\sqrt{a_i(T)a_j(T)},\qquad b = \sum \limits _{i=1}^My_ib_i, \end{aligned}$$

with $\displaystyle y_i=\frac{n_i}{n}$ being the mole fraction of component i. The binary interaction coefficient $0\le k_{ij}\le 1$ is assumed to be a constant for a fixed species pair and usually computed from experimental correlation. The Peng–Robinson parameters for the pure-substance component i, $a_i$ and $b_i$, are calculated from the critical properties of the specie

$$\begin{aligned} a_i(T)&=\;0.45724\frac{R^2T^2_{c_i}}{P_{c_i}}\left( 1+m_i\left( 1 -\sqrt{\frac{T}{T_{c_i}}}\right) \right) ^2,\\ b_i&=\,0.07780\frac{RT_{c_i}}{P_{c_i}}, \end{aligned}$$

where $T_{c_i}$ and $P_{c_i}$ represent the critical temperature and pressure of the pure substance component i respectively, which are intrinsic properties of the specie and available for most substances encountered in engineering applications. The parameter $m_i$ for modeling the influence of temperature on $a_i$ is experimentally correlated to the acentric parameter of the specie $\omega _i$ by

$$\begin{aligned} {\left\{ \begin{array}{ll} m_i= 0.37464 +1.54226\omega _i -0.26992\omega _i^2, &{}\omega _i\le 0.49,\\ 0.379642+1.485030\omega _i-0.164423\omega _i^2+0.016666\omega _i^3, &{}\omega _i>0.49, \end{array}\right. } \end{aligned}$$

with

$$\begin{aligned} \omega _i=\frac{3}{7}\left( \frac{\log _{10}\left( \frac{P_{c_i}}{14.695~\text {PSI}}\right) }{\frac{T_{c_i}}{T_{b_i}}-1}\right) -1=\frac{3}{7}\left( \frac{\log _{10}\left( \frac{P_{c_i}}{1~\text {atm}}\right) }{\frac{T_{c_i}}{T_{b_i}}-1}\right) -1, \end{aligned}$$

where $T_{b_i}$ represents the normal boiling point of the pure substance i, “PSI” is “pounds per square inch”, and “atm” refers to the standard atmosphere pressure (equal to 101325 Pa).

The dependence of the influence parameter $c_{ij}$ on the molar concentrations is practice very weak, thus it is common to assume that $c_{ij}=c_{ij}(T)$ is just a temperature-dependent parameter, which often can be obtained by adopting the modified geometric mean

$$\begin{aligned} c_{ij}(T)=(1-\beta _{ij})\sqrt{c_i(T)c_j(T)}. \end{aligned}$$

Note $\beta _{ij}$ is the binary interaction coefficient for the influence parameter, usually required to be included between 0 and 1 and $\beta _{ij}=\beta _{ji}$ to maintain the stability of the interfaces, and $c_i$ is the influence parameter of the pure substance component i, computed by

$$\begin{aligned} c_i = a_ib_i^{\frac{2}{3}}\left( m^c_{1,i}\left( 1-\frac{T}{T_{c_i}}\right) +m^c_{2,i}\right) \end{aligned}$$

with $m^c_{1,i}$ and $m^c_{2,i}$ being the coefficients correlated merely with the acentric factor $\omega _i$ by

$$\begin{aligned} m^c_{1,i}=-\frac{10^{-16}}{1.2326+1.3757\omega _i}, \qquad m^c_{2,i}=\frac{10^{-16}}{0.9051+1.5410\omega _i}. \end{aligned}$$