PSRK: agroup contribution equation of state based on unifac, Essays (university) of Thermodynamics

Download PSRK: agroup contribution equation of state based on unifac and more Essays (university) Thermodynamics in PDF only on Docsity! Fluid Phase Equilibria, 70 (1991) 251-265 251 Elsevier Science Publishers B.V., Amsterdam PSRK: A Group Contribution Equation of State Based on UIVIFAC T. Holderbaum and J . Gmehling University of Oldenburg, Chair of Industrial Chemistry, PO Box 2503, D2900 Oldenburg (FRG) Keywords: group contribution equation of state, PSRK, vapor liquid equilibria, UNIFAC ABSTRACT A group contribution equation of state called PSRK (Predictive Soave-Redlich-Kwong) which is based on the Soave-Redlich-Kwong equation (Soave, 1972) has been developed . It uses the UNIFAC method to calculate the mixture parameter a and includes all already exist- ing UNIFAC parameters. This concept makes use of recent developments by Michelsen (1990b) and has the main advantage, that vapor-liquid-equilibria (VLE) can be predicted for a large number of systems without introducing new model parameters that must be fitted to experimental VLE-data. The PSRK equation of state can be used for VLE-predictions over a much larger temperature and pressure range than the UNIFAC -y-c-approach and is easily extended to mixtures containing supercritical compounds . Additional PSRK parameters, which allow the calculation of gas/gas and gas/alkane phase equilibria, are given in this pa- per. In addition to those mixtures covered by UNIFAC, phase equilibrium calculations may also include gases like CH4, C2H6, C3H6, C4H10, C02, N2, H2 and CO. THE PSRK EQUATION OF STATE The PSRK equation is based on the modified Soave-Redlich-Kwong (Soave, 1972) equa- tion of state, which yields good results for vapor-liquid-equilibria (VLE) of nonpolar or slightly polar mixtures : RT a P = -- (1)v - b v(v + b) 0378-3812/91/$03 .50 © 1991 Elsevier Science Publishers B .V. All rights reserved 252 Two modifications are necessary to obtain equation of state for redicting vapor-liquid- equilibria of polar as well as nonpolar mix odification concerns the tempera- ture dependence of the pure component parameter a, which was originally expressed by Soave in terms of the acentric factor w: f(T) = [ 1 + ct(1-T°-5) f(T) = [ 1 + ci(1-T°•5) j2 This temperature dependence yields sufficiently accurate vapor pressure data for nonpolar substances, but improvements are still necessary for polar components . Therefore, the ex- pression proposed by Mathias and Copeman (1983) is used in the PSRK equation : 0.5) j2 (5) The use of the three adjustable parameters especially improves the description of the pure component vapor pressures for polar components . This is of course important when a reli- able prediction of the real behavior of polar mixtures is required . Table 1 summarizes the required PSRK pure component parameters for some selected substances . The second modification concerns the mixing rule for the parameter a . Recent develop- ments of Heidemann and Kokal (1990) and Michelsen (1990a,b) lead to simple, density in- dependent mixing rules, which link the mixture parameter a to the excess Gibbs energy 96F at zero pressure. The pressure dependence of gE is small at low pressures an group contribution method like UNIFAC or ASOG can be used to calculate mixing rules involving the excess Gibbs energy at infinite pressure (Huron an Tochigi et al. 1990) a recalculation of existing parameter tables is not necessary . Michelsen proposed a mixing rule based on the zero pressure reference state and a first- and second-order approximation . The first one includes an extrapolation scheme and is therefore called "extrapolation method" (Dahl and Michelsen, 1990) . The latter approxima- tions are called "modified Huron-Vidal" mixing rules (MHV1, MHV2) . The simplest first-or- der approximation is used in the PSRK equation : goE a • RT a=b[-+zxi-'+-zxi A1 bi Al (6) 2 2Tai ai = 0.42748 . f(T) P R (2) c,i f(T) = [1 + cl(1-T0.1) j2 (3) c1 = 0.48 + 1 .574w - 0.176w2 (4) of the chosen go- del. The PSRK equation and the model proposed b Dahl and Michelsen yield similar results. However, one great advantage of the PSRK equation is evident in Table 2: Ethane- (and also propane- and butane-) systems can be treated without introducing new model parameters . These gases are divided into structure groups in the same way as higher alkanes. Taking into account that ethane-, propane- and butane-systems have never been used to develop the UNIFAC parameter tables, the obtained results are surprisingly good . Figure 1 shows additional VLE predictions for light hydrocarbon systems . Ethane/Toluene Xi, Yi FIGURE 1 VLE predictions with the PSRK eq a a:Ethane/Acetone, 298 K 50 b :Ethane/Diethyl ether, 298 K 255 256 ESTIMATION OF NEW PSRK PARAMETERS The main advantage of equations of state in comparison with y-c-approaches is their abili- ty to calculate phase equilibria of systems containing supercritical components . Therefore, the UNIFAC interaction parameter table was extended as shown in Figure 2 . Six gases (CO2, CH4, N2, H2S, H2, CO) are included and parameters for alkane/gas and gas/gas phase equi- libria predictions are now available . The missing van der Waals volumes rk and surface areas qk are summarized in Table 3 and were estimated using Bondi's method (1968) . In some cases, slight changes were found useful to improve VLE predictions . FIGURE 2 PSRK interaction parameter matrix TABLE 3 Van der Waals properties for the PSRK equation of state Nearly all gas/alkane interaction parameters could be estimated using a large binary VLE data base containing alkanes from ethane to decane . For example 107 data sets were used to fit H2/CH2 interaction parameters . The experimental data cover a temperature range from 93 K (H2/ethane) to 583 K (H2/n-decane). This range is remarkably larger than the one co ered by the UNIFAC 7-w-approach. Therefore, temperature dependent interaction parame- ters were introduced and the UNIFAC expression : a~ = exp - T eplaced by: The parameters bnm and cnm are zero for all UNIFAC ma pups with numbers up to 44 (see Figure 2) . One exception will be discussed later . Pressures are usually measured more accurately than vapor phase compositions . So only x,T,P data are used in the following objective function : nd nv ! F=E E fit =min i=1 j=1 nd = number of experimental data sets nv = number of experimental x,T,P-values fij = (Pij,cal - Pij,exp)/op Iii for isothermal data sets fij = (Tij cat - Tij exp)/oT ij for isobaric data sets op ij = assumed standard deviation of the pressure oT ij = assumed standard deviation of the temperature 257 (9) gas rk qk C02 1.300 0.482 CH4 1.129 1.124 N2 0.856 0.930 H2S 1.235 1.202 H2 0.416 0.571 Co 0.711 0.828 260 A w m A a 22 C02/Ethane 20 . 18 . 16 . 14 . 12 . 10 44. 8 . 6 . 4 . 32. 0.0 1000. 800. 600. A 0 .0 0 .5 X1, Y1 C02/n-Butane X1, YI FIGURE 4 PSRK results for CO2 systems 1 .0 0 .5 1 .0 56 . m A a 68 . C02/Ethane X1, Y1 X1, Y1 MULTICOMPONENT SYSTEMS The PSRK equation is based on the SRK equation of state and the UNIFAC method . It is well known, that both underlying models are able to predict multicomponent phase equilibria from binary data only . The most simple procedure to check this ability for the PSRK equa- tion is to generate multicomponent VLE-data at low pressures by the UNIFAC y-v-approach and to compare these data with PSRK predictions . This test shows clearly, that the PSRK equation of state is able to predict multicomponent VIE-data from binary information . Additional tests have been made using experimental data . Results for a 12-component re (Turek et al ., 1984) are given in Table 5 and Figure 5 . The deviations shown in Table 5 are based on a flash calculation at constant temperature and pressure . Turek et al. used a cubic equation of state with special modifications to improve the results for binary CO2 Sys - tems. Their approach requires the estimation of 66 binary parameters (if no simplifications ar oduced) from experimental VLE-data. The PSRK equation of state is able to predict these K-values by using the model parameters already given in Table 1, 3 and 4. Taking into account, that Turek et al. used an equation of state, that is directly based on binary VIE data and that they performed special modifications to improve their results, it is not surprising that they obtained smaller deviations . However, the pressure dependence shown in Figure 5 is predicted quite well with the PSRK equation of state . .0 -1 .0 -2 .0 -3 .0 -4.0 -5 .0 \ 9. M 10 1 1H 1B 2 I I k i 0. 50 . 100 . P [bar] 150 . 261 1 . Nitrogen 2. Methane 3. Carbon dioxide 4. Ethane 5 . Propane 6. Butane 7. Pentane 8. Hexane 9. Heptane 10. Octane 11 . Decane 12. Tetradecane FIGURE 5 Pressure dependence of K-values for a 12-component mixture (feed specified in Table 5) at 322 K 262 TABLE 5 K-value deviations for a 12-component mixture at 69 .29 bar and 322 K REVISION OF EXISTING PARAMETERS The UNIFAC interaction parameter matrix has been developed by using low pressure VLE-data. Taking into account that the PSRK equation can be used over a much larger tem- perature and pressure range, this data base is - strictly speaking - too small to be used in an equation of state . By using all available experimental data, improvements are e.g. possible for system- containing aromatic components and methanol . Aromatic compounds like benzene or naphthalene are built up by the UNIFAC structural groups AC- and AC-H . Therefore, only one parameter set characterizing the MeOH/ACH interaction is used to describe meth- anol/benzene- and methanol/naphthalene-systems. Table 6 summarizes the results obtained with original and revised parameters . Even at higher pressures the original parameters yield acceptable results for methanol/benzene VLE-data. However, large deviations are observed for the methanol/naphthalene system . This is not surprising, because these data have never been used to optimize the MeOH/ACH interaction parameters . Methanol is supercritical (TT = 512 K) at these conditions . The re- 'ch was obtained from a fit to the data shown in Table 6, yields good ets. Especially the naphthalene data are well represented over a large concentration range . Somewhat higher deviations occur only in the immediate vicinity of the critical point . Additional calculations were performed with binary systems containing methanol and sub- stituted aromatic components like toluene, 1-methyl naphthalene and 1,2,3,4-tetrahydro naphthalene. These calculations require three parameter sets for the MeOH/ACH, MeOH/ACCH2 and ACH/ACCH2 interaction. Remarkably better results were obtained with the revised parameters given in Table 6 without a further revision of MeOH/ACCH 2 or ACH/ACCH2 interaction parameters . component exp. Data zl Kl Turek et al . AK/K [%l PSRK AK/K [%J N2 0.00063 -19.7 CO2 0.20000 1.935 0.36 5.1 Methane 0.27551 3.545 3.68 -3.6 Ethane 0.02268 0.989 8.66 13.1 Propane 0.03009 0.426 -0.51 7.7 n-Butane 0.04572 0.181 -1.10 7.7 n-Pentane 0.03154 0.0794 -1.81 7.1 n-Hexane 0.02376 0.0352 -2.09 5.1 n-Heptane 0.03961 0.0157 -2.03 8.3 n-Octane 0.03970 0.0071 -2.41 17.5 n-Decane 0.24968 0.00195 -22.6 -0.5 n-Tetradecane 0.04108 0.0002 -61.0 -6.5