Soave Equation

In an effort to improve the equation of state representation of the effect of temperature, the Soave equation (Soave, 1972) replaces the a/T° term in the Redlich-Kwong equation with a more general temperature-dependent parameter, a(T) ... 

A revival of interest in cubic equations in the sixties and seventies led to the introduction of two equations which have found great acceptance among engineers the Soave-Redlich-Kwong (SRK) equation and the Peng-Robinson (PR) equation. The SRK equation (Soave 1972) has the form... 

Given the estimate of the reactor effluent in Example 4.2 for fraction of methane in the purge of 0.4, calculate the.actual separation in the phase split assuming a temperature in the phase separator of 40°C. Phase equilibrium for this mixture can be represented by the Soave-Redlich-Kwong equation of state. Many computer programs are available commercially to carry out such calculations. 

TABLE 4.3 Vapor-Liquid Phase Split Using the Soave-Redlich-Kwong Equation of State... 

Hydrocarbon mixtures are most often modeled by the equations of state of Soave, Peng Robinson, or Lee and Kesler. 

In 1972, Soave published a method of calculating fugacities based on a modification of the Redlich and Kwong equation of state which completely changed the customary habits and became the industry standard. In spite of numerous attempts to improve it, the original method is the most widespread. For hydrocarbon mixtures, its accuracy is remarkable. For a mixture, the equation of state is ... 

In practice, however, it is recommended to adjust the coefficient m, in order to obtain either the experimental vapor pressure curve or the normal boiling point. The function f T ) proposed by Soave can be improved if accurate experimental values for vapor pressure are available or if it is desired that the Soave equation produce values estimated by another correlation. 

At low temperatures, using the original function/(T ) could lead to greater error. In Tables 4.11 and 4.12, the results obtained by the Soave method are compared with fitted curves published by the DIPPR for hexane and hexadecane. Note that the differences are less than 5% between the normal boiling point and the critical point but that they are greater at low temperature. The original form of the Soave equation should be used with caution when the vapor pressure of the components is less than 0.1 bar. In these conditions, it leads to underestimating the values for equilibrium coefficients for these components. 

Soave, G. (1972), Equilibrium constants from a modified Redlich-Kwong equation of state . Chem. Eng. Sci., Vol. 27, p. 1197. 

In each of these expressions, ie, the Soave-Redhch-Kwong, 9gj j (eq. 34), Peng-Robinson, 9pj (eq. 35), and Harmens, 9 (eq. 36), parameter 9, different for each equation, depends on temperature. Numerical values for b and 9(7) are deterrnined for a given substance by subjecting the equation of state to the critical derivative constraints of equation 20 and by requiring the equation to reproduce values of the vapor—Hquid saturation pressure,... 

Vapor densities for pure compounds can also be predicted by cubic equations of state. For hydrocarbons, relatively accurate Redlich-Kwong-type equations such as the Soave and Peng-Robinson equations are often used. Both require only T, and (0 as inputs. For organic compounds, the Lee-Erbar-EdmisteF" equation (which requires the same input parameters) has been used with errors essentially equivalent to those determined for the Lydersen method. While analytical equations of state are not often used when only densities are required, values from equations of state are used as inputs to equation of state formulations for thermal and equilibrium properties. 

An afternate method with approximately the same accuracy as the Rackett method is the COSTALD metnod of Hanldnson and Thomson.The critical temperature, a characteristic volume near the critical volume, and an acentric factor optimized for vapor pressure prediction by the Soave equation of state are required input parameters. The method is detailed in the Technical Data Book ... 

Example Many equations of state involve solving ciihic equations for the compressibility factor Z. For example, the Redlich-Kwong-Soave equation of state requires solving... 

The Soave/Redhch/Kwoug (SRK) and the Peug/Robiusou (PR) equations of state, both expressed by Eqs. (4-230) and (4-231), were developed specifically for X T.E calculations. The fugacity coefficients imphcit in these equations are given by Eq. (4-232). When combined... 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

The existing equations of state (i.e., Benedict-Webb-Rubin (BWR), Soave-Redlich-Kwang, and Peng-Robinson) have some practical limitations. The equations of state developed by the University of Illinois... 

The fugacity coefficient can be calculated from other equations of state such as the van der Waals, Redlick-Kwong, Peng-Robinson, and Soave,d but the calculation is complicated, since these equations are cubic in volume, and therefore they cannot be solved explicitly for Vm, as is needed to apply equation (6.12). Klotz and Rosenburg4 have shown a way to get around this problem by eliminating p from equation (6.12) and integrating over volume, but the process is not easy. For the van der Waals equation, they end up with the relationship... 

Equations of state that are cubic in volume are often employed, since they, at least qualitatively, reproduce the dependence of the compressibility factor on p and T. Four commonly used cubic equations of state are the van der Waals, Redlich-Kwong, Soave, and Peng-Robinson. All four can be expressed in a reduced form that eliminates the constants a and b. However, the reduced equations for the last two still include the acentric factor u> that is specific for the substance. In writing the reduced equations, coefficients can be combined to simplify the expression. For example, the reduced form of the Redlich-Kwong equation is... 

Figure A3.3 compares the experimental (corresponding states) results with the predictions from the van der Waals. modified Berthelot, Dieterici, and Redlich-Kwong equations of state.b The comparison is not so direct for the Soave and Peng-Robinson equations of state, since the reduced equation still includes to, the acentric factor. Figure A3.4 compares the corresponding states line, with the prediction from the Soave equation, using four different values of to. The acentric factors chosen are those for H (o> = —0.218), CH4 (to = 0.011),...

Figure A3.4 Comparison of the experimental r (dashed lines) with the r values calculated from the Soave equation of state (solid lines). Values for the acentric factor are (a) oj = -0.218 (the value for EC), (b) a, = 0.011 (the value for CH4), (c) lU = 0,250 (the value for NEC), and (d) = 0.344 (the value for ECO).

For both the Soave and Peng-Robinson equations, the fit is best for uj — 0. The Soave equation, which essentially reduces to the Redlich-Kwong equation when ui — 0, does a better job of predicting than does the Peng-Robinson equation. The acentric factors become important when phase changes occur, and it is likely that the Soave and Peng-Robinson equations would prove to be more useful when 77 < 1. 

Soave (1972) modified the Redlich-Kwong equation to extend its usefulness to the critical region, and for use with liquids. 

The Peng-Robinson equation is related to the Redlich-Kwong-Soave equation of state and was developed to overcome the instability in the Redlich-Kwong-Soave equation near the critical point Peng and Robinson (1970). 

The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, Thermodynamics. It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vapor-liquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. 

The need for methods of accurately describing the thermodynamic behavior of natural and synthetic gas systems has been well established. Of the numerous equations of state available, three--the Soave-Redlich-Kwong (SRK) (19), the Peng-Robinson (PR) (18) and the Starling version of the Benedict-Webb-Rubin (BWRS) (13, 20)--have satisfied this need for many hydrocarbon systems. These equations can be readily extended to describe the behavior of synthetic gas systems. At least two of the equations (SRK and PR) have been further extended to describe the thermodynamic properties of water-light hydrocarbon systems. 

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