PR Equation of State

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... 

Example 5 Calculation of the SR Method Use the SR method with the PR equation of state for K values and enthalpy departures. The oil was taken as n-dodecane. To compute stage temperatures and interstage vapor and hquid flow rates and compositions for ahsorher-column specifications shown in Fig. 13-52. Note that a secondary ahsorher oil is used in addition to the main ahsorher oil and that heat is withdrawn from the seventh theoretical stage. 

The fugacity coefficient < >2 is calculated by using a thermodynamic model. In this work the SRK (Soave) and Peng Robinson (PR) Equations of State were considered. 

Figure 2 also shows the density (p) vs temperature phase diagrams of ( () -(, I Ii -N and CO -C Hi -() systems calculated from PR equation of state.The system is single phase outside the envelopes, and separates into two phases in the envelopes. The results in Figure 2 indicate that the differ-rence of p T curves of the two systems is not considerable. The main reasons are that the physical properties of N2 and O2 are similar and their concentrations in the corresponding solutions are low (0.1 mol%). This suggests that N2 can be used to replace O2 for the phase behavior measurements. 

Figure 16. The solubility of benzaldehyde and benzyl alcohol in CO2 at 318.2 K and different pressures calculated from PR equation of state Benzaldehyde, O Benzyl alcohol.

The van der Waals, SRK, and PR equations of state have only two parameters per pure fluid (a and b) that need to be estimated for solvents and polymers. The Sako et al. equation of state has three parameters. It can be shown that the Sako et al. repulsive term is essentially equivalent to the chain-free volume expression presented previously (Equation 16.54). 

Various EoS/G models have been proposed over the last several years for polymers. These models combine the SRK, the PR equation of state, or the Sako et al. cubic equation of state with FV activity coefficient models such as UNIFAC-FV, Entropic-FV, EH and the ASOG. 

Limited comparisons of SAFT with other models have been reported. In a recent review, Kang et al." compared SAFT to the PR equation of state. Similar deviations are observed for the VLB of a few polymer-solvent systems investigated. 

Peng-Robinson (PR) Equation of State The Peng-Robinson equation of state is given by... 

This extension is just a first step, however, because it will not be a good approximation at extremely high pressmes. The Redlich-Kwong equation of state is a modification of van der Waal s equation of state, and then was modified further by Soave to give the Soave-Redlich-Kwong (SRK) equation of state, which is a common one in process simulators. Another variation of Redlich-Kwong equation of state is Peng-Robinson (PR) equation of state. 

Find the molar volume of methanol gas at 100 atm and 300 °C using Peng-Robinson (PR) equation of state. Compare its molar volume when you are using Soave-Redlich-Kwong (SRK) equation of state. 

Consider a mixture of 25 percent ammonia, and the rest nitrogen and hydrogen in a 1 3 ratio. The gas is at 270 atm and 550 K. Use Peng-Robinson (PR) equation of state to compute the specific volume of this mixture. 

Figure 5.36 Experimental [3] and calculated vapor pressures for selected solvents using the PR equation of state and the Twu-of-function.

In Figure 5.37 the experimental and calculated enthalpies of vaporization using the SRK and the volume translated PR equation of state for 11 different compounds in a wide temperature range up to the critical temperature are shown. It can be... 

Calculate the liquid density of cyclohexane at the normal boiling point Tb = 353.85 K, P = 1 atm) with the help of the PR equation of state. 

For the calculation of the liquid density at the normal boiling point for cyclohexane first the parameters a and b of the PR equation of state have to be determined from the critical data and the acentric factor using Eqs. (2.167)-(2 169). 

In the next step, the molar liquid volume has to be determined for which the right-hand side of the PR equation of state gives a value of 1 atm. This can be done iteratively or by solving the cubic equation. 

Figure 5.38 Experimental and calculated liquid densities using the PR equation of state for six different solvents in the temperature range Tr = 0.5-0.8.

Figure 5.42 Experimental and calculated VLE data for the system acetone (1)-water (2) using the PR equation of state with classical mixing rules k = —0.2428) (a) and the Soave-Redlich-Kwong equation of state with g -mixing rules (NRTL, Agi2 = 257.9 cal/mol, Ag2i = 1069 cal/mol, cri2 = 0.2) (b) at 308, 323 and 333 K.

In this section interfacial tensions are computed with the gradient theoiy of van der Waals in which the Peng Robinson (PR) equation of state [9] has been incorporated. The combination of these two models originally was presented by Carey et al. [8]. 

However, it must be stated that the fitting procedure of the influence parameter cancels out most errors that arise from inaccuracies incorporated in the PR equation. Another aspect of this approach is that by choosing the PR model, the critical temperature and pressure are always found to be equal to the experimental ones. This is due to the fact that the critical temperature and pressure are incorporated in the volume and energy parameters (b and a, respectively) of the PR equation of state, which assures that the critical temperature and pressure are always correct, and that the interfacial tension at the critical temperature is always equal to zero, as it should. 

During the fitting of the influence parameter it was noticed that the interfacial tension shows a very large relative error at temperatures near the critical point. The reason for this behaviour is that the PR equation of state is a classical equation of state, and it is to be expected that such an equation will lead to classical interfacial tensions. A fact that has been observed already by several other authors [15,16]. For the same reason the relative errors were computed only for reduced temperatures (T/Tc) in the region between 0.3 and 0.85. 

To improve the description of the interfacial tensions by the gradient theory and the PR equation of state a temperature dependent influence parameter c = c(T) was used. As a first attempt we use a simple linear temperature dependence. In this approach the influence parameter is written as ... 

Also for the parameters that are required in the APACT we refer to these papers and [11]. One important difference between the PR equation of state and the APACT is that the critical temperature of the APACT, which is not an input parameter, is overestimated. This should not be a surprise since the APACT is also an analytic equation of state, in which no special actions are taken to include the proper behaviour in the vicinity of critical points. Another difference is that the liquid densities are much better described by the APACT then by the PR equation of state. 

Some of the computed interfacial tensions have been included in Figures 2a and b (solid lines). In the first figure the results of the calculations for octane are presented. These results do not show a large deviation from the experiments, an effect that has been observed for the PR equation of state as well. This is not very surprising since both the PR equation and APACT are able to describe the thermodynamic properties of the alkanes accurately. 

In order to be able to model the interfacial tensions with a higher accuracy, again a temperature dependence is introduced for the influence parameter. Therefore, the influence parameter was modelled with a linear relationship, which has shown to give very accurate results for the PR equation of state. Fitting yields the results included in Table 4. 

Having included the extra parameter in the model for the influence parameter, the calculated interfacial tensions appear to be much more in accordance with the experimental tensions, which is shown in Figures 2a and b. Comparison of the computed interfacial tensions of the APACT model with those obtained from the PR equation of state shows that the APACT yields similar results. However, the results obtained with the APACT at near critical states appear to be not as accurate as was found for the PR equation of state. This is due to the fact that the APACT theoiy predicts a critical point while the PR equation of state is fixed in that point. Other differences between the two equations of state are almost completely absorbed by the fitting of the calculated interfacial tensions to experimental data. 

In binaiy mixtures of caibon dioxide and the normal hydrocarbons heavier than C7, coexisting liquid-vapour, liquid-liquid, and liquid-liquid-vapour phase splits have been observed above 273.15 K. Pressure-composition diagrams predicted by the PR equation of state for carbon dioxide/decane mixtures at two different temperatures are shown in Figures 5. At 260 K, binaiy mixtures of carbon dioxide and decane separate into a liquid and a vapour phase at low pressures. As the pressure is increasexl, a value is... 

Figure 3. The pressure composition sections of the binary system carbon dioxide/decane at (a) 260.0 K, with liquid-liquid phase split and (b) 319.3 K. Both figures have been obtained from the PR equation of state. Open ircles are experiments [22].

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