Peng-Robinson equations

This equation (Peng and Robinson, 1976) was developed with the goal of overcoming some of the deficiencies of the Soave equation, namely its inaccuracy in the critical region and in predicting liquid densities. The equation is similar to the Soave equation in that it is cubic in the volume, expresses its parameters in terms of the critical temperature, critical pressure, and acentric factor, and is based on correlating pure-component vapor pressure data. The equation is written as 

Mixtures are handled by calculating mixture parameters, using the same mixing rules used with the Soave equation (Equation 1.15). 

The Peng-Robinson equation is widely used for the same applications as the Soave equation. Although it may be more accurate than the Soave equation in the critical region and for calculating liquid densities, it is not generally recommended for the latter since better methods for predicting liquid densities are available. 

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Corresponding states have been used in other equations. For example, the Peng-Robinson equation is a modified RedHch-Kwong equation formulated to better correlate vapor—Hquid equiHbrium (VLE) vapor pressure data. This equation, however, is not useful in reduced form because it is specifically designed to calculate accurate pressure data. Reduced equations generally presuppose knowledge of the 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. 

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 methanol(l)/acetone(2) system serves as a specific example in conjunction with the Peng/Robinson equation of state. At a base temperature To of 323.15 K (50°C), both XT E data (Van Ness and Abbott, Jnt. DATA Ser, Ser A, Sel. Data Mixtures, 1978, p. 67 [1978]) and excess enthalpy data (Morris, et al., J. Chem. Eng. Data, 20, pp. 403-405 [1975]) are available. From the former. 

A variety of equations-of-state have been applied to supercritical fluids, ranging from simple cubic equations like the Peng-Robinson equation-of-state to the Statistical Associating Fluid Theoiy. All are able to model nonpolar systems fairly successfully, but most are increasingly chaUenged as the polarity of the components increases. The key is to calculate the solute-fluid molecular interaction parameter from the pure-component properties. Often the standard approach (i.e. corresponding states based on critical properties) is of limited accuracy due to the vastly different critical temperatures of the solutes (if known) and the solvents other properties of the solute... 

In some earlier work the shift reaction was assumed always at equilibrium. Fiigacities were calculated with the SRK and Peng-Robinson equations of state, and correlations were made of the equilibrium constants. 

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 A3.5 Comparison of the experimental r (dashed lines) with the r values calculated from the Peng-Robinson equation of state (solid lines). Values for the acentric factor are (a) = —0.218 (the value for HU), (b) = 0.011 (the value for CH4),...

NH3 (a = 0.250), and H20 (u> = 0.344). Thus, results for a wide range of acentric factors are compared. In Figure A3.5, we make the same comparisons with the Peng-Robinson equation. 

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. 

The value of v is important both in equation 7 and for accurate calculation of concentrations in other equations. For simplicity and accuracy, the Peng-Robinson equation of state has been used to calculate v for the model O). This equation expresses the P-V-T relationship as follows ... 

Volumetric equations of state (EoS) are employed for the calculation offluid phase equilibrium and thermo-physical properties required in the design of processes involving non-ideal fluid mixtures in the oil, gas and chemical industries. Mathematically, a volumetric EoS expresses the relationship among pressure, volume, temperature, and composition for a fluid mixture. The next equation gives the Peng-Robinson equation of state, which is perhaps the most widely used EoS in industrial practice (Peng and Robinson, 1976). 

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). 

In process design calculations, cubic equations of state are most commonly used. The most popular of these cubic equations is the Peng-Robinson equation of state given by3 ... 

Example 4.1 Using the Peng-Robinson equation of state ... 

The value of (8Z/dT)P can be obtained from an equation of state, such as the Peng-Robinson equation of state, and the integral in Equation 4.78 evaluated3. The enthalpy departure for the Peng-Robinson equation of state is given by3 ... 

Example 6.5 Repeat the calculations from Example 6.4 taking into account vapor-phase nonideality. Fugacity coefficients can be calculated from the Peng-Robinson Equation of State (see Poling, Prausnitz and O Connell6 and Chapter 4). 

Now calculate the molar enthalpies of the vapor and liquid streams. Enthalpies were calculated here from ideal gas enthalpy data corrected using the Peng-Robinson Equation of State (see Chapter 4) ... 

Example 11.1 Each component for the mixture of alkanes in Table 11.2 is to be separated into relatively pure products. Table 11.2 shows normal boiling points and relative volatilities to indicate the order of volatility and the relative difficulty of the separations. The relative volatilities have been calculated on the basis of the feed composition to the sequence, assuming a pressure of 6 barg using the Peng-Robinson Equation of State with interaction parameters set to zero (see Chapter 4). Different pressures can, in practice, be used for different columns in the sequence and if a single set of relative volatilities is to be used, the pressure at which the relative volatilities are calculated needs, as much as possible, to be chosen to represent the overall system. 

Example 11.2 Using the Underwood Equations, determine the best distillation sequence, in terms of overall vapor load, to separate the mixture of alkanes in Table 11.2 into relatively pure products. The recoveries are to be assumed to be 100%. Assume the ratio of actual to minimum reflux ratio to be 1.1 and all columns are fed with a saturated liquid. Neglect pressure drop across each column. Relative volatilities can be calculated from the Peng-Robinson Equation of State with interaction parameters assumed to be zero (see Chapter 4). Determine the rank order of the distillation sequences on the basis of total vapor load for ... 

This calculation assumed the gas to be ideal. For comparison, the calculation can be based on the Peng-Robinson Equation of State (see Chapter 4). A number of commercial physical property software packages allow the prediction of gas density and y for a mixture of hydrogen and methane using the Peng-Robinson Equation of State. Using this, the gas density at normal conditions is 0.1651 kg-rn 3. At 40°C and 81 bar, the density is 11.2101 kg-rn 3. Thus, suction volume of gas... 

Table 13.10 Phase separation calculated using the Peng-Robinson Equation of State.

Penetration tests, 21 743 Penetration theory, in absorption, 1 46 Peng-Robinson equation of state, 24 656, 665, 685... 

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