Article pubs.acs.org/IECR
Predicting the Phase Equilibria, Critical Phenomena, and Mixing Enthalpies of Binary Aqueous Systems Containing Alkanes, Cycloalkanes, Aromatics, Alkenes, and Gases (N2, CO2, H2S, H2) with the PPR78 Equation of State Jun-Wei Qian, Romain Privat, and Jean-Noel̈ Jaubert* Université de Lorraine, Ecole Nationale Supérieure des Industries Chimiques, Laboratoire Réactions et Génie des Procédés (UMR CNRS 7274), 1 rue Grandville, 54000 Nancy, France ABSTRACT: The phase behavior of water/hydrocarbon mixtures in a wide range of concentrations, temperatures, and pressures is important in a variety of chemical engineering applications. For this reason, the physical understanding and mathematical modeling of these aqueous−organic mixtures constitute a challenging task, both for scientists and for applied engineers. In this work, mutual solubilities, critical loci, and mixing enthalpies of water + hydrocarbon, water + carbon dioxide, water + nitrogen, water + hydrogen sulfide, and water + hydrogen binary mixtures are predicted using the PPR78 cubic equation of state (EoS). The extremely nonideal behavior of these systems produces unusual and complex thermodynamic behavior. As an example, such mixtures often exhibit type III phase behavior in the classification scheme of Van Konynenburg and Scott and are characterized by a vapor−liquid critical line which first exhibits a temperature minimum and then extends to temperatures above the critical point of pure water. Such a behavior, called gas−gas equilibria of the second kind is a consequence of the large degree of immiscibility of the two components. The selected PPR78 model combines the Peng−Robinson cubic EoS and a groupcontribution method aimed at predicting the temperature-dependent binary interaction parameters, kij(T), involved in the Van der Waals one-fluid mixing rules. Although, it is acknowledged that cubic EoS with a constant kij are not suitable to predict phase equilibria of such highly nonideal systems, the addition of the H2O group to the PPR78 model makes it possible to conclude that the use of temperature-dependent binary interaction parameters not only results in qualitatively accurate predictions over wide pressure and temperature ranges but also leads to quantitatively reasonable predictions for many of the studied systems.
1. INTRODUCTION The study of phase equilibria for water containing binary systems is fundamental not only for practical purposes, but also because the strong nonideality of these systems, caused by the water hydrogen-bonding structure, produces unusual and complex thermodynamic behavior. From an industrial viewpoint, knowledge of phase equilibria in aqueous mixtures, from low to high temperatures and pressures, is vital for a large number of chemical engineering applications such as those related to refining, petrochemical processes, biofuels industry, and treatment of industrial liquid wastes. Let us indeed recall that although renewable energy sources are becoming increasingly important, fossil fuels continue to be the most important source of energy. The optimization of oil and gas processes (extraction, transportation, transformation, water injection to enhance the recovery of oil) passes through a better knowledge of the thermodynamic behavior of petroleum, which is, from a chemical point of view, mostly a mixture of water and hydrocarbons. Modeling of phase behavior, critical loci, and mixing enthalpies of water/hydrocarbon systems, which is the objective of this paper, is a difficult problem since such systems exhibit limited miscibility over a broad range of conditions. Moreover, the solubilities in the coexisting phases are strongly asymmetric: the solubility of the hydrocarbon in the water-rich phase is several orders of magnitude lower than that of water in the hydrocarbon-rich phase. It is also well-established that the hydrocarbon solubility presents a minimum value at relatively low © XXXX American Chemical Society
temperatures (the solubility of the hydrocarbon in water increases when the temperature is decreased), whereas the water solubility is a monotonic function of temperature. At room temperature, the phase equilibrium behavior of water + hydrocarbon mixtures may be treated with fair accuracy using the ideal simplification of absolute liquid-phase immiscibility. However, this is not the case when an accurate prediction of the solubility is needed or when we consider the equilibrium behavior in a wide range of temperatures and/or pressures, especially in the vicinity of the critical point. Over the last decades a number of approaches to the description of the mutual solubilities of hydrocarbons and water were attempted. Since, it is desirable to elaborate a model applicable over the entire temperature and pressure range, an equation of state is preferred over an activity coefficient model or a simple correlation.1 Such gE models and particularly UNIFAC and NRTL were however extensively used but with moderate success.2,3 Cubic equations of state developed originally for hydrocarbon mixtures were applied to aqueous systems containing hydrocarbons using various mixing rules. In general, cubic equations of state (EoSs) with a unique temperature-independent binary interaction parameter predict accurately the solubility of Received: August 3, 2013 Revised: September 23, 2013 Accepted: October 14, 2013
A
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a
Only the last line of this table, relative to the water group, was determined in this study. The 20 first lines of this table were determined in our previous papers.31−42
Table 1. Group Interaction Parameters: (Akl = Alk)/MPa and (Bkl = Blk)/MPaa
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B
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binary interaction parameter as a function of water content and temperature. Kabadi and Danner8 and Michel et al.9 proposed unconventional mixing rules by assigning a composition dependence not to a12 but, instead, to a11 (where index 1 denotes water) in the ordinary quadratic mixing rule of the attractive parameter (amixture) of a cubic EoS. Another approach is to use EoS that account specifically for association/hydrogen-bonding interactions through chemical equilibrium, lattice theory, or perturbation theory. The most used ones for aqueous systems were APACT,10,11 AEOS,12,13 MF-NLFHB,14,15 GCA-EoS,16 CPA,17−21 and SAFT-type11,19,20,22−25 EoS. Such equations are able to correlate the solubility of hydrocarbons in water with a much better accuracy than cubic EoS with a constant kij (the improvement is spectacular). They however fail (except Soft-SAFT23) to capture the solubility minima of n-alkanes in water. On the other hand such minima can be reproduced with cubic EoS via the so-called EoS-gE mixing rules.26 It is also well established that when the purecomponent parameters are classically fitted on experimental vapor pressures and liquid densities, EoS like CPA or SAFTtype models overpredict the pure-component critical properties. Such a drawback is a real concern since most of the aqueous systems containing hydrocarbons are characterized by a vapor-liquid critical line which starts from the critical point of pure water, exhibits a temperature minimum (the critical temperature of the mixtures diminishes of a few tens of kelvins with increasing pressure), and then extends to temperatures above the critical point of pure water. Such gas−gas equilibria of the second type can obviously not be reproduced in the correct domain if the EoS is not able to reproduce accurately the critical coordinates of pure water. As a consequence, the simultaneous correlation of phase equilibria at both low and high pressures (including the critical region) is few addressed by researchers working on SAFT-type EoS or CPA (this is however common with cubic EoS27). This failure was deeply ́ and discussed by Galindo et al.22 and by Aparicio-Martinez Hall20 who explained that a possible solution is to force-fitting the model parameters to the critical properties. The rescaling parameters however cause deterioration of saturated liquid density and vapor pressure accuracy. As explained by Klamt,28 COSMO-RS is able to predict the mutual solubilities of hydrocarbons and water with a good qualitative and even satisfactory quantitative agreement with the experimental data. Such a model can however not be used to correlate the critical loci of binary systems. In conclusion, the development of a predictive EoSin which the parameters could be, e.g., calculated by a group contribution methodable to accurately predict the mutual solubilities of hydrocarbons and water at low and high temperatures (including the critical locus) is still a challenge for thermodynamicists. In this study the PPR78 EoS was applied to the highly important class of water/hydrocarbons systems so as to predict mutual solubilities, critical loci, and mixing enthalpies. Such a model combines the widely used Peng−Robinson equation of state (EoS) with a group-contribution method aimed at estimating the temperature dependent binary interaction parameters (kij(T)). The PPR78 model may also be seen as the combination of the PR EoS and a Van Laar type activity coefficient (gE) model under infinite pressure. Such a model thus relies on the well-established Huron−Vidal mixing rules. More details can be found in our previous papers29−31 or in the next section of this publication. During the past decade,31−42
Table 2. List of the 57 Pure Components Used in This Study component
short name
methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-dodecane n-hexadecane n-eiscosane 2-methylpropane(isobutane) 2-methylbutane 2,2-dimethylpropane(neopentane) 2-methylpentane 3-methylpentane 2,2-dimethylbutane 2,4-dimethylpentane 2,2,4-trimethylpentane (isooctane) 2,2,5-trimethylhexane benzene methylbenzene (toluene) 1,3-dimethylbenzene (m-xylene) 1,2-dimethylbenzene (o-xylene) ethylbenzene 1,3,5-trimethylbenzene (mesitylene) 1,2,4-trimethylbenzene 1-metylethylbenzene (cumene) 1-methylnaphthalene phenanthrene cyclopropane cyclopentane methylcyclopentane cyclohexane methylcyclohexane cycloheptane ethylcyclohexane cyclooctane 1,2,3,4-tetrahydronaphthalene (tetralin) cis-decalin trans-decalin carbon dioxide nitrogen hydrogen sulfide water methyl mercaptan ethyl mercaptan propyl mercaptan hydrogen ethylene propene 1-butene 1,3-butadiene 1-hexene 1-octene
1 2 3 4 5 6 7 8 9 10 12 16 20 2m3 2m4 22m3 2m5 3m5 22m4 24m5 224m5 225m6 B mB 13mB 12mB eB 135mB 124mB iprB 1mBB Phe C3 C5 mC5 C6 mC6 C7 eC6 C8 tet cCC6 tCC6 CO2 N2 H2S H2O 1sh 2sh 3sh H2 a2 a3 1a4 13a4 1a6 1a8
water in hydrocarbon mixtures but fail to describe the solubility of hydrocarbons in water.4−6 To overcome this limitation, Daridon et al.7 developed an empirical formulation of the C
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Table 3. Binary Systems Database number of bubble points (T, p, x)
number of dew points (T, p, y)
number of binary critical points (Tcm, pcm, xc)
0.062000−0.999910 0.135000−0.999970
437 203 9 37 12 19 5 70 28 163 6 6 1 4 31 1 1 1 5 1 665 257
416 189 0 146 12 17 5 139 28 139 6 0 0 0 8 0 0 0 0 0 434 242
17 17 10 14 3 0 1 4 4 6 1 8 0 0 0 0 0 0 0 0 13 8
0.000100−0.999981
0.201465−0.938909
225
162
2
1.01−607.95
0.019230−0.999899
0.483000−0.971300
51
16
0
398.15−473.35 298.15−373.50 298.15−568.10 298.15−298.15 600.25−677.95 298.15−298.15 533.54−673.15 419.80−542.00 294.26−377.59 298.15−298.15 298.15−298.15 298.10−683.15
2.74−20.04 0.53−1.29 0.09−106.80 1.01−1.01 150.00−2225.00 1.01−1.01 17.23−174.50 3.61−10.55 1.17−42.74 1.01−1.01 1.01−1.01 0.32−1546.69
0.027850−0.999800 0.218700−0.999970 0.004300−0.999974 0.999991−0.999991 0.903150−0.983615 0.999993−0.999993 0.040000−0.996900 0.061000−0.072700 0.000072−0.002400 0.000040−0.000040 0.999991−0.999991 0.001130−0.999988
10 12 18 1 24 1 41 26 45 1 1 68
0 0 11 0 34 0 45 0 0 0 0 88
0 0 1 0 0 0 3 0 0 0 0 2
H2O−mC6 H2O−C7 H2O−eC6 H2O−C8 H2O−tet H2O−tCC6 H2O−cCC6 CO2−H2O
298.15−298.15 298.15−298.15 310.90−561.40 298.15−298.15 573.15−672.85 613.15−643.15 613.15−673.15 273.15−633.15
1.01−1.01 1.01−1.01 0.10−99.30 1.01−1.01 10.44−179.00 186.05−1551.62 180.00−1295.35 0.49−3600.00
0.999997−0.999997 0.999995−0.999995 0.000810−0.999998 0.999999−0.999999 0.026000−0.997000 0.994000−0.994000 0.925000−0.998000 0.000030−0.998100
0.251000−0.956000 0.722000−0.970100 0.733000−0.931000 0.055000−0.999300
1 1 19 1 55 4 4 1068
0 0 14 0 56 14 14 543
0 0 1 0 3 0 0 23
N2−H2O
257.44−659.00
3.40−2710.00
0.000023−0.480000
0.034000−1.000000
265
261
9
H2S−H2O
283.15−603.15
1.55−206.85
0.000320−0.987000
0.008500−0.996970
522
196
1
1sh−H2O 2sh−H2O 3sh−H2O H2−H2O
310.93−588.70 323.12−588.70 323.07−372.97 273.15−636.10
0.77−206.84 0.39−206.84 0.18−1.63 3.45−1013.25
0.002400−0.265000 0.000589−0.022700 0.000109−0.000567 0.0000−0.0164
0.138000−0.975700 0.122000−0.644000 0.1000−0.9994
19 13 11 146
16 6 0 82
0 0 0 0
a2−H2O
298.15−573.15
1.17−945.00
0.0000−0.0470
0.1600−0.9990
216
178
0
pressure range (bar)
x1 range (1st compound liquid mole fraction)
y1 range (1 compound gas mole fraction)
256.21−663.20 298.15−707.00 273.20−646.70 373.11−699.00 298.15−629.60 422.04−552.76 473.15−553.15 423.15−613.20 473.15−633.00 413.50−638.15 523.15−625.35 278.15−649.00 298.15−298.15 278.15−318.15 298.15−628.15 298.15−298.15 298.15−298.15 298.15−298.15 298.15−353.15 298.15−298.15 273.10−649.20 259.10−673.15
0.10−3300.00 1.04−3101.00 0.24−769.00 3.43−2467.00 0.08−324.00 6.55−88.60 20.10−88.26 4.90−303.00 18.63-219.40 2.68−294.20 41.38−181.42 1.01−1070.00 1.01−1.01 1.01−1.01 1.01−709.27 1.01−1.01 1.01−1.01 1.01−1.01 1.01−5.01 1.01−1.01 0.97−2500.00 0.63−3700.00
0.000001−0.999310 0.000002−0.999500 0.000010−0.115000 0.004249−0.998000 0.001000−0.999000 0.012000−0.999960 0.100000−0.575000 0.006700−0.999500 0.073000−0.997700 0.018000−0.999500 0.240000−0.923000 0.000010−0.999910 0.000012−0.000012 0.000006−0.000017 0.000003−0.055585 0.000003−0.000003 0.000004−0.000004 0.999999−0.999999 0.999999−1.000000 1.000000−1.000000 0.000026−0.295000 0.000008−0.340000
0.031000−0.999480 0.040000−0.999500 0.018000−0.115000 0.015000−0.977000 0.322000−0.971000 0.086000−0.736000 0.575000−0.840000 0.029000−0.989000 0.367000−0.985300 0.667500−0.999500 0.923000−0.995000 0.070000−0.200000
H2O−B
293.15−613.15
0.04−2976.68
H2O−mB
298.15−641.33
H2O−13mB H2O−12mB H2O−eB H2O−124mB H2O−135mB H2O−iprB H2O−1mBB H2O−Phe C3−H2O C5−H2O H2O−mC5 H2O−C6
binary system (1st compound− 2nd compound)
temperature range (K)
3−H2O 4−H2O 5−H2O 6−H2O H2O−7 H2O−8 H2O−9 H2O−10 H2O−12 H2O−16 H2O−20 2m3−H2O 2m4−H2O 22m3−H2O 2m5−H2O 3m5−H2O 22m4−H2O H2O−24m5 H2O−224m5 H2O−225m6 1−H2O 2−H2O
st
0.044049−0.124928
0.691000−0.778000 0.540666−0.910000 0.393000−0.960000
0.203940−0.994094
0.603000−0.755000
D
refs 47−55 56−62 63, 64 61, 65−68 63, 69, 70 71−73 70 66, 74−76 70, 77 78, 79 70 62, 80−82 81 81, 82 81, 83 81 81 81 81, 84 81 85−119 91, 92, 100, 104, 106, 113, 114, 120 −128 45, 68, 81, 83, 129−137 74, 81, 83, 129, 138 129 81, 139 71, 81, 140 81 141 81 71, 142 78 143 81 81 68, 81, 84, 135, 144 81 81 71, 140 81 142 145 145 95, 98, 122, 146 −181 48, 107, 113, 167, 182 −195 98, 185, 196 −201 202, 203 202, 203 203 113, 185, 204 −210 120, 167, 211 −216
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Table 3. continued binary system (1st compound− 2nd compound)
temperature range (K)
pressure range (bar)
x1 range (1st compound liquid mole fraction)
a3−H2O
294.26−623.15
1.50−2200.00
0.0000−0.9989
1a4−H2O 13a4−H2O 1a6−H2O H2O−1a8
279.15−417.15 280.15−377.59 366.48−496.26 366.48−549.82
1.83−68.26 1.32−19.01 3.18−53.78 1.24−92.60
0.0001−0.0010 0.0001−0.0011 0.0000−0.9898 0.0087−0.9999
y1 range (1 compound gas mole fraction) st
0.1830−0.9963 0.5278−0.9994 0.9980−0.9986 0.5647−0.7481 0.5870−0.7060 total number of points
number of bubble points (T, p, x)
number of dew points (T, p, y)
205
108
0
32 33 8 8 5117
42 6 4 5 3682
0 0 0 0 151
number of binary critical points (Tcm, pcm, xc)
refs 51, 214, 217 −220 80, 221−224 80, 225 71 71
Figure 1. Isothermal curves for the two binary systems CO2 (1) + H2O (2) and H2O (1) + n-decane (2) calculated with Peng−Robinson equation of state (EoS) with different kij values: (+) experimental bubble points, (∗) experimental dew points, (○) experimental critical points. Solid line: calculated curves. (a) System CO2 (1) + H2O (2) at T = 298.15 K with two different kij values: kij(1) = −0.119 (red) and kij(2) = 0.180 (green). (b) System H2O (1) + n-decane (2) at T = 583.15 K with two different kij values: kij(1) = 0.58 (red) and kij(2) = 0.30 (green).
twenty groups were defined: CH 3 , CH 2 , CH, C, CH 4 (methane), C2H6 (ethane), CHaro, Caro, Cfused aromatic rings, CH2,cyclic, CHcyclic ⇔ Ccyclic, CO2, N2, H2S, SH, H2, C2H4 (ethylene), CH2,alkenic ⇔ CHalkenic, Calkenic, and CHcycloalkenic ⇔ Ccycloalkenic. In this paper, the interactions between the group H2O and the twenty ones previously identified are determined. It therefore becomes possible to estimate, at any temperature, the kij value between two components i and j in any mixture containing paraffins, aromatics, naphthenes, CO2, N2, H2S, mercaptans, hydrogen, alkenes, and water. For people interested in predicting the phase behavior of mixtures containing fatty acid esters, the group interaction parameters determined by Jaubert et al.43,44 can also be used. They however need to be corrected following the theory developed by Jaubert and Privat.29,30 The fully predictive PPR78 model (no additional parameter besides the structure of the molecules is needed) can thus be used to not only predict isothermal or isobaric phase diagrams but also the mixing enthalpies and the critical loci of aqueous binary systems containing alkanes, cycloalkanes, aromatics, alkenes, and gases (N2, CO2, H2S, H2). It is thus possible to test the accuracy of the Huron−Vidal mixing rules on many aqueous systems and on a very wide range of concentrations, temperatures, and pressures. It thus will be possible to conclude on their accuracy for such complex systems.
ai(T ) RT − v − bi v(v + bi) + bi(v − bi)
(1)
⎧ R = 8.314472 J· mol−1· K−1 ⎪ ⎪ 3 3 ⎪ X = −1 + 6 2 + 8 − 6 2 − 8 ⎪ 3 ⎪ ≈ 0.253076587 ⎪ RTc, i ⎪ ⎪bi = Ωb P with c, i ⎪ ⎪ Ωb = X ≈ 0.0777960739 ⎪ X+3 ⎪ R2Tc, i 2 ⎪ ⎪ ai = Ωa P α(T ) with c, i ⎨ 8(5X + 1) ⎪ Ω = ≈ 0.457235529 and a ⎪ 49 − 37X ⎪ 2 ⎡ ⎛ ⎞⎤ ⎪ T ⎟⎥ ⎢ ⎜ ⎪ α(T ) = 1 + mi⎜1 − ⎢⎣ Tc, i ⎟⎠⎥⎦ ⎝ ⎪ ⎪ ⎪ if ωi ≤ 0.491 ⎪ 2 ⎪ mi = 0.37464 + 1.54226ωi − 0.26992ωi ⎪ ⎪ if ωi > 0.491 ⎪ mi = 0.379642 + 1.48503ωi − 0.164423ωi 2 ⎪ + 0.016666ωi 3 ⎩
(2)
P=
and
2. PPR78 MODEL In 1978, Peng and Robinson45 published an improved version of their well-known equation of state, referred as PR78 in this paper. For a pure component, the PR78 EoS is E
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factor of pure i. To apply this EoS to a mixture, mixing rules are necessary to calculate the values of a and b of the mixture. Classical Van der Waals one-fluid mixing rules are used in the PPR78 model:
where P is the pressure, R is the gas constant, T is the temperature, ai and bi are the EoS parameters of pure component i, v is the molar volume, Tc,i is the critical temperature, Pc,i is the critical pressure, and ωi is the acentric
Figure 2. continued F
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Figure 2. Study of the transition from type II to type III. Case of the naphthalene (1) + water (2) system: (+) pure component critical point, (▲) upper critical end point (UCEP), (▼) lower critical end point (LCEP), (○) starting and ending points of the azeotropic line (AEP = azeotropic end point, CAP = critical azeotropic point, DCEP = double critical end point). Dashed line: liquid−liquid−vapor equilibrium line (LLV). Dotted line: azeotropic line (Az). Thin ′ , or CL∞). continuous line: vapor pressure line (Psi (T) for component i). Bold continuous lines: liquid−liquid and liquid−vapor critical loci (CLLL, CLLV, CLLV (a and b) Type II global phase equilibrium diagrams (GPEDs) of the naphthalene (1) + water (2) system in the (P, T)- and (T, x1)-planes, as calculated by the PR EoS with k12 = 0.1. (c) Type II GPED of the naphthalene (1) + water (2) system in the (T, x1)-plane, as calculated by the PR EoS with k12 = 0.13. (d) GPED of the naphthalene (1) + water (2) system in the (T, x1)-plane, as calculated by the PR EoS with k12 = 0.139517. Coalescence of the CAP and of the AEP in a DCEP (border between types II and IV*). (e) Type IV* GPED of the naphthalene (1) + water (2) system in the (T, x1)-plane, as calculated by the PR EoS with k12 = 0.139520. (f) Schematic type IV* GPED in the (P, T)-plane. The phenomena are emphasized for a better understanding. (g) Type IV* GPED of the naphthalene (1) + water (2) system in the (P, T)-plane, as calculated by the PR EoS with k12 = 0.1397. The LCEP(LV) and the UCEP(LL) are about to merge in a tricritical point, and we are thus very close to the frontier between types IV* and III. (h) Type III GPED of the naphthalene (1) + water (2) system in the (P, T) plane, as calculated by the PR EoS with k12 = 0.145. N N ⎧ ⎪ a(T , z) = ∑ ∑ zizj ai(T )aj(T ) [1 − kij(T )] ⎪ ⎪ i=1 j=1 ⎨ N ⎪ ⎪b(z) = ∑ zibi ⎪ ⎩ i=1
parameter, whose choice is difficult even for the simplest systems, is the so-called binary interaction parameter (BIP) characterizing the molecular interactions between molecules i and j. Although the common practice is to fit kij to reproduce the vapor−liquid equilibrium data of the mixture under consideration, the predictive PPR78 model calculates the kij value, which is temperature-dependent, with a group contribution method using the following expression:31,32
(3)
where zi represents the mole fraction of component i and N is the number of components in the mixture. The kij(T)
kij(T ) =
1⎡ N N − 2 ⎢∑k =g 1 ∑l =g1 (αik − αjk)(αil − αjl)Akl ⎣
2
In eq 4, T is the temperature. The ai and bi values are given in eq 2. The Ng variable is the number of different groups defined by the group contribution method (for the time being, 21 groups are defined, and Ng = 21). The αik variable is the fraction of molecule i occupied by group k (occurrence of group k in molecule i divided by the total number of groups present in molecule i). The constant parameters, Akl = Alk and Bkl = Blk (where k and l are two different groups), were determined either in this study or in our previous papers31−42 (Akk = Bkk = 0). For the group added in this paper (group 21: H2O), 40 interactions (20 Akl and 20 Bkl values) between this new group and the 20 ones defined previously must be estimated. However, due to a lack of experimental data, it has been impossible to determine the interactions between group 21 (water) and groups 4, 19, and 20 (C, Calkenic, and CHcycloalkenic ⇔ Ccycloalkenic). These parameters were obtained by minimizing the deviations between
(
298.15 T/K
(Bkl / Akl − 1) ⎤
)
⎛ ⎥−⎜ ⎦ ⎝
ai(T ) bi
−
aj(T ) bj
⎞2 ⎟ ⎠
ai(T )aj(T ) bibj
(4)
calculated and experimental data from an extended database. The corresponding Akl and Bkl values (MPa) are summarized in Table 1. To be really exhaustive, let us recall that the PPR78 model may also be seen as the combination of the PR EoS and a Van Laar type activity coefficient (gE) model under infinite pressure. Indeed, as explained by Jaubert et al.,31 the well-established Huron−Vidal mixing rules p ⎧ a(T , x) a (T ) .E ⎪ = ∑ xi i − bi C EoS ⎪ b(x) i=1 ⎪ p ⎪ ⎨b(x) = ∑ xibi ⎪ i=1 ⎪ ⎪ 2 ln(1 + 2 ) for the PR EoS ⎪C EoS = ⎩ 2
G
(5)
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the most volatile component in the gas phase, and y2 is the mole fraction of the heaviest component (y2 = 1 − y1) at a fixed temperature and pressure. The xc1 variable is the critical mole fraction of the most volatile component, and xc2 is the critical mole fraction of the heaviest component at a fixed temperature. Pcm is the binary critical pressure at a fixed temperature. The minimization of the objection function is however not straightforward and care must be taken in order (i) to have not only the most accurate mutual solubility of hydrocarbon and water but also a fair balance between the correlation of vapor− liquid and liquid−liquid equilibria data and (ii) to have the correct type of phase diagram in the classification scheme of Van Konynenburg and Scott.226,227 These two difficulties are now addressed. 3.1. Which kij for Which Fluid-Phase Equilibrium? Binary aqueous systems containing hydrocarbons, all exhibit vapor−liquid, liquid−liquid, and liquid−liquid−vapor equilibria. We here want to graphically illustrate the well-known dilemma that at a given temperature, the kij suitable to describe the solubility of the hydrocarbon in water is different from the one required to correlate the solubility of the water in the hydrocarbon. By way of example, the phase diagram of the CO2 (1) + H2O (2) system was plotted at T/K = 298.15 in Figure 1a. At this temperature, both pure components are subcritical and a three-phase line is found since the experimental temperature of the three-phase upper critical end point (UCEP) is about 308 K. A negative kij value of −0.12 perfectly correlates the bubble curve, the dew curve and the solubility of CO2 in water in the liquid−liquid region. On the other hand, such a kij value fails to correlate the solubility of water in liquid CO2 which can only be captured with a positive kij value of +0.18. In order the PPR78 model could find the best compromise between these two situations, we made our best to maintain a similar number of data points to describe the solubility of water in the hydrocarbon and the solubility of hydrocarbon in H2O respectively. As shown in Figure 15a, for such a system, the PPR78 model predicts k12 = −0.098. Working with cubic EoS, another difficulty is the simultaneous correlation of the VL and LL regions when the temperature is above that of the three-phase UCEP (but below the critical temperature of the hydrocarbon). This instance is illustrated in Figure 1b for the H2O + n-decane system at 583.15 K. At this temperature, both pure components are subcritical. There is however no heteroazeotropic three-phase line since the temperature is above that of the UCEP. It is noticeable that a kij value of 0.58 perfectly correlates the VL domain, including the critical region but underestimates the water solubility in decane which can only be captured with a kij of 0.30 at the cost of a less accurate prediction of the VLE region. 3.2. Transition from Type II to Types IIIa and IIIb. It is extremely important that a predictive model detects the correct type of phase diagram in the classification scheme of Van Konynenburg and Scott226,227 revisited by Deiters and Kraska.228 As highlighted in the previous sections, aqueous systems containing hydrocarbons exhibit nearly exclusively type III phase behavior. Some exceptions229 can however be found as for instance the naphthalene + water system which displays a type II phase behavior. A correct prediction of the phase behavior type may be tricky because, as highlighted in this section, a small change of the kij may make switch the phase behavior from type III to type II. As a consequence, one needs to
are rigorously equivalent to the Van der Waals one-fluid mixing rules with temperature-dependent kij if a Van Laar type excess function p
p
. EVanLaar 1 ∑i = 1 ∑j = 1 xixjbibjEij(T ) = p ∑j = 1 bjxj C EoS 2
(6)
is used in eq 5. The mathematical relation between kij(T) [eq 3] and the interaction parameter of the Van Laar gE model [Eij(T) in eq 6] is kij(T ) =
Eij(T ) − (δi − δj)2 2δiδj
with δi =
ai bi
(7)
3. DATA BASE AND REDUCTION PROCEDURE Table 2 lists the 57 pure components involved in this study. The pure fluid physical properties (Tc, Pc, and ω) were obtained from two sources. Poling et al.46 was used for alkanes, cycloalkanes, aromatic compounds, CO2, N2, H2S, H2, and most of the mercaptans and alkenes. For the missing components (some mercaptans and some alkenes), the DIPPR database was employed. Table 3 details the sources of the binary experimental data used in our evaluations47−225 along with the temperature, pressure, and composition ranges for each binary system. Most of the data available in the open literature (5117 bubble points + 3682 dew points + 151 mixture critical points) were collected. Our database includes experimental data for 56 binary systems. The 34 parameters (17 Akl and 17 Bkl) determined in this study (see Table 1) were calculated by minimizing the following objective function: Fobj =
Fobj,bubble + Fobj,dew + Fobj,crit. comp + Fobj,crit. pressure nbubble + ndew + ncrit + ncrit (8)
nbubble ⎧ ⎛ |Δx| |Δx| ⎞⎟ ⎪F ⎜ = ∑ + 100 0.5 ⎜x ⎪ obj,bubble x 2,exp ⎟⎠ ⎝ 1,exp i=1 i ⎪ ⎪ with |Δx| = |x1,exp − x1,cal| = |x 2,exp − x 2,cal| ⎪ ndew ⎪ ⎞ ⎛ ⎪F ⎜ |Δy| + |Δy| ⎟ ∑ = 100 0.5 ⎜y ⎪ obj,dew y2,exp ⎟⎠ i=1 ⎝ 1,exp i ⎪ ⎪ |Δ | = | − | = | − y2,cal | y y y y with ⎪ 1,exp 1,cal 2,exp ⎨ ncrit ⎪ ⎛ |Δx | |Δxc| ⎞ c ⎪F ⎜ ⎟ ∑ = + 100 0.5 ⎜x ⎪ obj,crit. comp xc2,exp ⎟⎠ ⎝ c1,exp i=1 i ⎪ ⎪ with |Δxc| = |xc1,exp − xc1,cal| ⎪ = |xc2,exp − xc2,cal| ⎪ ⎪ ncrit ⎛ |P ⎞ ⎪ cm,exp − Pcm,cal| ⎟⎟ ⎪ Fobj,crit. pressure = 100 ∑ ⎜⎜ Pcm,exp ⎪ ⎠i i=1 ⎝ ⎩
where nbubble, ndew, and ncrit are the number of bubble points, dew points, and mixture critical points, respectively. The variable, x1, is the mole fraction of the most volatile component in the liquid phase, and x2 is the mole fraction of the heaviest component (x2 = 1 − x1) at a fixed temperature and pressure. Similarly, y1 is the mole fraction of H
dx.doi.org/10.1021/ie402541h | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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be extremely vigilant when fitting the group-interaction
(i) a continuous liquid−vapor critical line connecting the critical points of the two pure-components, (ii) a liquid−liquid critical line which starts at a three-phase UCEP and runs to infinite pressures, (iii) a liquid−liquid−vapor equilibrium (LLVE) line which starts at low temperature and goes up to the UCEP.
parameters. Systems exhibiting type II phase behavior227 are characterized in a (P, T) projection of the global phase equilibrium diagram (GPED) by the following:
Figure 3. continued I
dx.doi.org/10.1021/ie402541h | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Figure 3. Study of the transition from type IIIa to type IIIb. Case of the naphthalene (1) + water (2) system: (+) pure component critical point, (▲) upper critical end point (UCEP), (▼) lower critical end point (LCEP), (●) tricritical point (TCP), (■) double critical cusp (DCC), (○) double critic end point (DCEP). Dashed line: liquid−liquid−vapor equilibrium line (LLV). Bold continuous lines: stable critical locus (CL∞, CL′∞, CL2, CLLV). Dotted line: nonstable critical locus. Thin continuous line: vapor pressure line (Psi (T) for component i). (a and b) Global phase equilibrium diagrams (GPED) of the naphthalene (1) + water (2) system in the (P, T)- and (T, x1)-planes, as calculated by the PR EoS with k12 = 0.44. Border between types IIIa and IV*. (c) Type IV* GPED of the naphthalene (1) + water (2) system in the (T, x1)-plane, as calculated by the PR EoS with k12 = 0.441. (d) Type IV* GPED of the naphthalene (1) + water (2) system in the (T, x1)-plane, as calculated by the PR EoS with k12 = 0.4415. Coalescence of the critical cusps CC1 and CC2 in a DCC. (e) Type IV* GPED of the naphthalene (1) + water (2) system in the (T, x1)-plane, as calculated by the PR EoS with k12 = 0.447. (f) GPED of the naphthalene (1) + water (2) system in the (T, x1)-plane, as calculated by the PR EoS with k12 = 0.449. Coalescence of the UCEPLαV and of the LCEPLαV in a DCEP (border between types IV* and IIIb). (g and h) Type IIIb GPED of the naphthalene (1) + water (2) system in the (P, T)- and (T, x1)-planes, as calculated by the PR EoS with k12 = 0.5.
Systems exhibiting type III phase behavior227 are also characterized by three loci in a (P, T) projection: (i) a first critical line which starts at the critical point of the heaviest pure component and runs to infinite pressures, (ii) a second critical line which joins together the second pure component critical point and a three-phase UCEP, (iii) a LLVE line which starts at low temperature and goes up to the UCEP. In order to understand the relationships and the propinquity of type II and type III phase behaviors for water-containing systems, we propose to consider as an example, the naphthalene (1) + water (2) system and to study the transition from type II to type III. To do so, the Peng−Robinson EoS will be considered with a temperature-independent binary interaction parameter k12. To observe the transition, only the value of the k12 parameter will be increased in the course of this study. The first calculations were performed with kij = 0.1 (see Figure 2a and b) and the system exhibits a type II phase behavior. As classically observed with water-containing systems, a minimum is observed on the liquid−vapor critical curve (CLLV) and, as often in such a case, an azeotropic line also exists. By increasing the k12 parameter to 0.13, the GPED does not change significantly but the UCEP(LL) moves to higher temperature and higher pressure so that the three-phase line is now extremely close of the CLLV critical line. As a consequence, the azeotropic line which starts on the three-phase line (just below the UCEP(LL) on a so-called azeotropic end point230) and ends on the CLLV line (on a so-called critical azeotropic point) becomes extremely small. Figure 2c, which is an enlargement of the GPED, makes it possible to understand the relative locations of the different lines. By increasing again the kij, the temperature and the pressure of the UCEP(LL) still increase and the length of the azeotropic curve continues to decrease. In
other words, the azeotropic end point (AEP) and the critical azeotropic point (CAP) become extremely close. For k12 = 0.139517, the CAP (located on the critical line CLLV) and the AEP (located on the three-phase line) merge. We thus reach (see Figure 2d) the osculation point of a critical curve and a three-phase curve called a double critical end point (DCEP). In the present case, this coalescence point could be called a double critical azeotropic end point. The three-phase line and the liquid−vapor critical line (CLLV) are thus ready to split in two parts. We are thus at the border between types II and IV* (see Figure 2d). For a slightly higher k12 value, a type IV* phase diagram226 is observed with its three CEPs, its three critical curves, and its two three-phase lines (see Figure 2e in which k12 = 0.13952). The two liquid−vapor critical lines resulting from the splitting of CLLV are noted CLLV and CL′LV in Figure 2e−g. Let us recall that such type IV* phase diagrams which differ from type IV in the connectivity of the three critical lines, have not been found experimentally yet. The type IV* phase behavior, schematically drawn in Figure 2f, can only be observed on an extremely limited range (
