Introduction
The Peng-Robinson equation of state (Ding-Yu Peng and Donald B. Robinson, "A New Two-Constant Equation of State", Ind. Eng. Chem. Fundam., Vol. 15, No. 1, pp. 59-64,1976) is widely used for calculating the thermodynamic properties of both pure fluids and fluid mixtures. Although not as accurate as the equations of state used in the EES pure fluid property data base, the Peng-Robinson equation offers algebraic simplicity and generality since it requires minimal data. Only the critical temperature and pressure and the acentric factor are needed to determine the properties for a pure fluid. Binary interaction parameters are needed in addition to determine the properties of a non-ideal mixture. The Peng-Robinson equation of state is claimed to provide reasonably accurate estimates of liquid as well as vapor phase densities. As a result, the Peng-Robinson equation can be used to determine the vapor pressure of pure fluids. With known binary interaction parameters, the Peng-Robinson equation can be used to do multi-component liquid-vapor phase equilibrium calculations. See the Thermodynamics book by Klein and Nellis for more information.
The Peng-Robinson equation is:
where
P is the pressure
T is the absolute temperature
v is specific volume
a(T) is a fluid-specific constant that depends on temperature
b is a fluid-specific constant.
The Peng-Robinson equation can be written in terms of the dimensionless compressibility factor as:
where
Procedures AB_PR and AB_MIX_PR determine the values of these parameters for pure fluids and mixtures, respectively.
Note the equation for Z is cubic. In general, the solution of this equation can have one or three real roots. Only one root exists if the fluid is above the critical point. When two phases coexist, two of the three possible roots correspond to saturated liquid and vapor, respectively, while the third root is a numerical aberration and has no physical meaning. Functions Z_G_PR and Z_L_PR have been developed to return the roots of the equation. If there is only one root, both functions will return the same value, but if there are two phases, Z_L_PR will return the compressibility for the liquid phase (the smallest non-zero root) and Z_G_PR will return the compressibility for the vapor phase (the largest root). Note that for temperatures below the critical temperature, the compressibility factors returned by Z_L_PR and Z_G_PR may differ, but this alone does not indicate a two-phase system.
The Peng-Robinson equation of state can be used to determine the enthalpy and entropy of a fluid or fluid mixture provided that the specific heat of the pure fluid(s) in the ideal gas state is known as a function of temperature. If the fluid obeyed the ideal gas law, then that is all of the information that would be needed to calculate the enthalpy and enthalpy relative to a selected reference state. When the fluid does not obey the ideal law, additional terms must be included in the enthalpy and entropy functions. These terms are most conveniently presented in terms of the enthalpy and entropy departures which are provided in functions H_DEP_PR and S_DEP_PR. When calculating the departure functions for a mixture, it is necessary to numerically evaluate da/dT. Function DSADT_MIX_PR provides this service.
A criterion of phase equilibrium for a pure fluid is that the fugacity of the liquid must be equal to the fugacity of the vapor. The ratio of the pure component fugacity to pressure is called the fugacity coefficient. The fugacity coefficient for a pure fluid is returned using the Peng-Robinson equation of state in function PHI_PR.
The fugacity concept can be extended to mixture by defining the partial fugacity coefficient as the ratio of the partial fugacity to the product of the mole fraction and pressure for the fluid component. At equilibrium, the partial fugacity of each component in the liquid phase must equal its partial fugacity in the vapor phase. Function PHI_I_PR returns the partial fugacity of one component in one phase of a mixture.
The solution to a phase equilibrium problem, though simple in concept, is mathematically difficult. The solution is necessarily numerical and iterative and a poor guess will result in an attempt to take the log of a negative value. Consequently, good guess values for the unknown quantities are required for a successful calculation.
The functions and procedure composing the Peng_Robinson library are listed below. Click on the name for more information.
AB_MIX_PR(N, X[1..N], A[1..N], B[1..N], k_12, k_13, .. k_23,.. : A, B)
DADT_MIX_PR(N, T, Tc[1..N], Pc[1..N], w[1..N], X[1..N], k[1..N,1..N])