3144
Ind. Eng. Chem. Res. 1999, 38, 3144-3161
Solubility of Hydrocarbons and CO2 Mixtures in Water under High Pressure Aleksande1 r Dhima,†,‡ Jean-Charles de Hemptinne,*,† and Jacques Jose§ Institut Franc¸ ais du Pe´ trole, 1 a` 4 avenue de Bois Pre´ au, Rueil-Malmaison Cedex, 92852 France, and Universite´ Claude BernardsLyon 1, 43, Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
New solubility data for methane, carbon dioxide, and their binary mixtures in pure water were obtained at 344.25 K and from 10 to 100 MPa. A thermodynamic model that combines a cubic equation of state (Peng-Robinson with kij given in the literature) with the Henry’s law approach has been elaborated. It has been shown that the (P,T) functional form of Henry’s constant given by the Krichevsky-Kasarnovsky equation is applicable. It involves two important parameters: partial molar volume at infinite dilution and Henry’s constant at the vapor pressure of water. Neither of these two parameters is a function of pressure or composition. The Henry’s constant correlations of Li and Nghiem and Harvey, and the partial molar volume correlation of Lyckman et al. have been validated using a great number of solubility data. 1. Introduction Solubilities of hydrocarbons and natural-gas components such as CO2 and H2S in water are of great interest to petroleum engineers. Especially under high-pressure/ high-temperature conditions, hydrocarbon-water mutual solubilities become important. Unfortunately, the experimental data and therefore the theoretical knowledge still present some gaps. In a previous work,1 we have published some solubility data of methane, ethane, n-butane, and the binary mixtures of methane and ethane or n-butane in pure water at 71 °C (160 °F) and at high pressures. These data have been analyzed on the basis of Henry’s law, which seems to be a very simple and powerful approach. Contradicting some ambiguous literature data,2,3 we concluded that, using this approach, the ternary solubility data can be predicted by the binary ones. This paper gives, for the same conditions, some new data on the solubility of carbon dioxide and carbon dioxide + methane mixtures in water. The analysis of these data allows us to reach the same conclusion: the binary solubility data, as well as the ternary ones, can be predicted using the Henry’s law approach. Henry’s law is expressed in its most general form by4,5 aq faq j) ) γH i (P,T,x i (Pref,T,x)‚xi ‚H° i,w(Pref,T)
(∫
exp
)
j aq j) P v i (P,T,x dP (1) Pref RT
with
lim γH i ) 1 xif0
aq aq aq where xj ) xj(xaq 1 , ..., xi , ..., xn-1, xw ) is the vector of the aq aqueous phase composition, xi is the mole fraction of a
* Corresponding author. E-mail: J-charles.de-hemptinne@ ifp.fr. Fax: 33 (1) 47 52 70 25. Phone: 33 (1) 47 52 71 28. † Institut Franc ¸ ais du Pe´trole. ‡ Present address: ENSM de Paris, 35, rue Saint-Honore ´, 77305 Fontainebleau Cedex, France. § Universite ´ Claude BernardsLyon 1.
j) is its given solute i in the aqueous phase, faq i (P,T,x aqueous-phase fugacity under the system pressure and temperature conditions, and H°i,w(Pref,T) is its corresponding Henry’s constant defined at the reference pressure, Pref. The use of eq 1 to calculate the solute aqueous-phase fugacity requires the knowledge of three parameters: j); (1) the aqueous phase activity coefficient γH i (Pref,T,x (2) Henry’s law constant of component i in water H°i,w (Pref,T); (3) the aqueous partial molar volume of component i vj aq i . The use of eq 1 for a phase equilibrium calculation requires, in addition, an equation of state (EOS) for evaluating the hydrocarbon (vapor or liquid)-phase fugacity of each component in the presence of water. This work has two objectives. First, we want to complete the solubility data published in our previous paper.1 Second, we will determine all three parameters j),H°i,w(Pref,T),vj aq mentioned above (γH i (Pref,T,x i ) for several important hydrocarbons and CO2, using a large number of solubility data. 2. Experimental Data and Discussion of the Results In this work we complete our previously published solubility data1 with solubility measurements of pure carbon dioxide and its binary mixtures with methane in pure water at 71 °C (160 °F). The high-pressure apparatus (given in Figure 1) and the experimental procedure are described in our previous work.1 In contrast to the previously published method, no inert gas is used during the flash (see Figure 1). The residual solubility of gases in the trapped water at atmospheric pressure is also measured by chromatography using a technique described by Benoit and Monot6 and was nearly 800 ppm mole fraction in the case of pure carbon dioxide and between 150 and 500 ppm mole fraction for the methane-carbon dioxide mixtures. The results presented in Table 1 take into account these corrections. It should be noted that, since the solubility of carbon dioxide in water is very high, the composition of the gaseous mixtures has changed with equilibrium pres-
10.1021/ie980768g CCC: $18.00 © 1999 American Chemical Society Published on Web 07/16/1999
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3145
Figure 1. Schematic diagram of experimental apparatus. Table 1. Corrected Solubility of Carbon Dioxide and Methane-Carbon Dioxide Mixtures in Pure Water at 71 °C (160 °F) aqueous mole fraction of CO2 C1 + CO2
total pressure (MPa)
mole fraction of pure CO2 in aqueous phase
total pressure (MPa)
equilibrium methane mole fraction in gaseous mixturea
C1
10 12.5 15 20 50
0.0166 0.0182 0.0197 0.0213 0.0256
75
0.0285
100
0.0317
10 10 20 20 50 50 75 75 100
0.5670 0.8045 0.5700 0.8102 0.5850 0.8205 0.5870 0.8260 0.5940
0.000 776 0.001 100 0.001 310 0.001 820 0.002 434 0.003 190 0.003 027 0.003 893 0.003 610
a
0.008 346 0.003 555 0.011 300 0.005 390 0.012 670 0.006 265 0.014 347 0.007 105 0.015 071
0.009 122 0.004 655 0.012 610 0.007 210 0.015 104 0.009 455 0.017 374 0.010 998 0.018 681
On dry basis.
sure. Another peculiarity of the CO2 is its high density. Under pressures higher than 800 bar the compressed pure supercritical carbon dioxide at 71 °C is heavier than the liquid water. In this case the flash is performed maintaining the equilibrium cell as presented in position 1 of Figure 1, while the equilibrium was attained in position 2. The fluids used in this work were all purchased from Air Liquide and have a degree of purity of g 99.995%, specified as N45. As in our previous work, the temperature measurement is accurate within 2 °C and the pressure is regulated and maintained constant during the flash with a precision of (1%. Subsequently, we estimate an overall experimental uncertainty of (2%. Figure 2 summarizes the binary solubility data obtained in this and previous work.1 A brief comparison with some other literature data shows good agreement.
It should be noted that the literature data of carbon dioxide dissolved in water correspond to 75 °C, as given by Wiebe and Gaddy,7 Gillespie and Wilson,8 and Sako et al.9). We observe that the Soreide and Whitson model10 describes very well the methane and ethane water solubility but slightly underestimates the solubility of n-butane in water compared to the solubility data reported by Reamer et al.11 and Sage and Lacey.12 The solubility of CO2 in water is well described at high pressure, but it is slightly underestimated at moderate pressure (up to 300 bar). Figure 3 compares our measurements on the aqueous solubility of methane-carbon dioxide mixtures with the Soreide and Whitson model.10 Again, the predicted values are slightly lower than the measured values, especially at moderate pressure.
3146 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999
Figure 2. Solubility of methane, ethane, butane, and carbon dioxide in pure water at 71 °C (160 °F).
Figure 3. Total solubility of methane-CO2 mixtures in water at 71 °C (160 °F).
3. Proposed Modeling Approach In this work, the fugacity coefficients of the hydrocarbon-phase components are described with a cubic equation of state, while those of hydrocarbons in the aqueous phase are calculated using a Henry’s law approach. 3.1. Hydrocarbon-Rich Phase. The equation that will be considered here is the Peng-Robinson13 equation of state: The temperature dependence of the attractive parameter for any component (i) other than water (w) is given
by the usual Soave relationship, with the parameters of Robinson and Peng.14 For water (w) we develop a specific R function of the form
Rw(Tr,w) ) 1 + R1(1 - xTr,w) + R2(1 - xTr,w)2 (2) Knowing the vapor pressure of water given by Saul and Wagner15 and on the basis of a very simple procedure given by Soave,16 we easily determine the value of the Peng-Robinson R function for each temperature. The
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3147
Figure 4. Aqueous activity coefficient ratio of methane and carbon dioxide at 71 °C (160 °F). Table 2. Binary Interaction Parameters Used in This Work methane methane ethane propane butane pentane decane toluene CO2 H2O
0.0079 0.0153 0.0218 0.0268 0.0431 0.0412 0.1300 0.5000
ethane
propane
butane
pentane
decane
toluene
CO2
0.0079
0.0153 0.0013
0.0013 0.0038 0.0063 0.0161 0.0146 0.1300 0.5000
0.0218 0.0038 0.0007
0.0007 0.0019 0.0086 0.0074 0.1268 0.5200
0.0268 0.0063 0.0019 0.0003
0.0003 0.0045 0.0036 0.1236 0.5400
0.0431 0.0161 0.0086 0.0045 0.0024
0.0024 0.0017 0.1204 0.5000
0.0412 0.0146 0.0074 0.0036 0.0017 0.0001
0.1300 0.1300 0.1268 0.1236 0.1204 0.1043
0.0001 0.1043 0.4540
0.2850
0.1896
Table 3. Critical Component Parameters and Acentric Factors Used in This Work methane ethane propane butane pentane decane toluene CO2 H2O
Tc (K)
Pc (bar)
Zc
ω
190.4 305.4 369.8 425.2 469.7 617.7 591.8 304.1 647.3
46.0 48.8 42.5 38.0 33.7 21.2 41.0 73.8 221.2
0.288 0.285 0.281 0.274 0.263 0.249 0.263 0.274 0.235
0.011 0.099 0.153 0.199 0.251 0.489 0.263 0.239 0.344
regression of R1 and R2 between 273 and 498 K yields
R1 ) 1.814 852 14 and
R2 ) 0.325 217 76 Using this procedure, the average error on the R parameter is 0.4%. The conventional mixing rule has been used for the parameters a and b of the equation of state. The use of binary interaction parameters coming from different sources is not recommended. Nevertheless, we could not find a single author providing all the parameters needed. We used the following references. The binary interaction parameters kij between all hydrocarbons from methane to decane are taken from Gao et al.17 The binary interaction parameters between hydrocarbons and water are taken from Davidson et al.18 for methane to butane and from Tsonopoulos and Heidman19 for
other hydrocarbons. The binary interaction parameters between carbon dioxide and hydrocarbons are also taken from Davidson et al.,18 while the CO2-water parameters are taken from Soreide and Whitson.10 The component critical parameters are from Reid et al.20 The binary interaction parameters used are summarized in Table 2, and the critical parameters in Table 3. 3.2. Aqueous Phase. The fugacity of water (w) in the aqueous phase will be calculated using the following equation proposed in our earlier work:1
[
sat sat sat j) = xaq faq w (P,T,x w ‚φw (Pw ,T)‚Pw exp
]
vlw (P - Psat w ) RT
(3)
Since the solubility of hydrocarbons in the aqueous phase is generally very low, the water activity coefficient in the aqueous phase is considered equal to unity. Furthermore, for temperatures far from the water critical temperature (we consider temperatures up to 200 °C), the saturated molar volume of the liquid water vlw is considered independent of pressure. The water fugacity coefficient at saturation sat φsat w (Pw ,T) has been calculated using the expression given by Li and Nghiem.21 The water vapor pressure, Psat w (T) and the saturated liquid water molar volume, vlw(T) are both calculated using the expressions given by Saul and Wagner.15 The fugacity of all hydrocarbons i dissolved in the aqueous phase will be calculated using the Henry’s law approach. This approach may be summarized as fol-
3148 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999
Figure 5. Relative behavior of ternary systems with methane in water of ethane, butane, and CO2 at 71 °C and for pressures from 10 to 100 MPa.
Figure 6. Behavior of some ternary systems compared to their respective binary systems.
lows:4 H faq j)‚xaq i (P,T,x) ) γi (Pref,T,x i ‚Hi,w(P,T) )
(∫
γH j)‚xaq i (Pref,T,x i ‚H° i,w(Pref,T) exp
)
vj aq j) i (P,T,x dP (1) Pref RT P
with
lim γH i ) 1 xif0
We will discuss each of the factors in turn.
The Henry’s constant, Hi,w(P,T) is always defined at infinite dilution of the solute in the solvent. When the system pressure reaches the solvent saturation pressure, both phases contain nothing but solvent molecules, and infinite dilution is thus automatically reached. It is therefore common to define the Henry’s constant at the solvent vapor pressure Psat w , considered as the reference pressure:
H°i,w(Psat w ,T)
≡
lim
sat (xaq i f0),(PfPw )
faq j) i (P,T,x xaq i
(4)
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3149
Figure 7. Ideal behavior of the aqueous phase at 71 °C (160 °F). Table 4. Observed and Calculated Partial Molar Volumes and Henry’s Constants at Saturated Water Vapor Pressure at 71 °C partial molar volume (cm3/mol)
ln Henry’s constant: H°(T) (MPa)
calcd value solute methane ethane n-butane CO2 a
our Lyckman Brelvi and measure- calcd value our 36 41 et al. ment Harvey42 measurement O’Connell 36 52 91 36
41 52 72 39
34 48 89 33
6178 6213 7980a 390
5930 6073 8002 374
Li and Nghiem.21
Table 5. References Providing Henry Constant Values of Light Hydrocarbons in Water aqueous solute methane, ethane, propane, butane, pentane, octane, carbon dioxide methane, ethane carbon dioxide for T < 353 K carbon dioxide for T > 353 K methane, ethane, carbon dioxide, hydrogen sulfide + gases methane, ethane, propane butane
independent of pressure, one obtains
ref Li and Nghiem21 Fernandez-Prini and Crovetto45 Crovetto46 Crovetto46 Harvey42 Carroll and Mather47 Carroll et al.48
However, the true molar fraction of solute in water may not be zero. A correction factor, or activity coefficient γH i , that is a function of the mole fraction must therefore be used. Although the activity coefficient can be taken as a function of pressure, it is most convenient to define it at the same reference pressure as that for the Henry’s constant, that is, the water vapor pressure. It is then only a function of temperature and mole fraction and can be considered as a correction on the Henry’s constant. The last factor of eq 1 provides the influence of pressure on the fugacity. When the partial molar volume of component i in the aqueous phase is considered
fhc i xaq i
)
ln(γH i H° i,w)
vj aq i (P - Psat + w ) RT
(5)
The well-known Krichevsky-Kasarnovsky [K-K] equation22 is written using vj aq j ∞i and γH i ) v i ) 1:
ln
vj ∞i ) H° + (P - Psat i,w w ) RT xaq i fhc i
(6)
Generally, it is admitted that at low temperatures the [K-K] equation works well for the hydrocarbons dissolved in water under high pressure (up to 1000 bar). The only paper presenting some doubts is presented by O’Sullivan and Smith.23 They conclude that the change of the aqueous partial molar volumes of methane and nitrogen with pressure is not negligible. Subsequently, the [K-K] equation cannot be applied to model the aqueous phase. This point is mentioned more recently by Lekvam and Bishnoi,24 without any analysis. We will further demonstrate that a constant γH i value can be used. However, for CO2 and at high temperatures some controversy remains concerning the use of the [K-K] equation: King25 and Weiss26 conclude that the [K-K] equation can be used to correlate the water solubility data of CO2 given by Wiebe and Gaddy7,27 while Parkinson and De Nevers,28 Gibbs and Van Ness,29 and Malinin30 conclude that the activity coefficient γH i cannot be neglected. This contradiction is analyzed by Carroll and Mather in a more recent paper31 that concludes that the [K-K] equation is valid below 100 °C, while at higher temperatures a γH i should be used. Using a detailed analysis of all available data, we investigate this question in a separate section.
3150 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 Table 6. Application of Eq 5 and Results Obtained on the Methane-Water System T (K)
H°γH (MPa)
3 vaq i (cm /mol)
σ (%)
Adev (%)
MaxDev (%)
N
P (MPa)
ref
323 298 311 344 378 411 444 298 325 376 398 311 344 423 473 523 473 523 423 473 523 427 479 423 478 298 313 338 298 323 283 288 293 298 283 303 323 343 274 283 286 324 344
5851 3864 4767 6271 6409 5748 4405 3889 5418 6167 5957 4798 6086 5092 3135 1853 3610 1935 5873 3445 2608 4394 2994 5139 3362 3889 4715 5748 4074 5490 2914 3267 3798 3757 1400 4458 2392 6148 2273 3039 3213 5391 5930
36 35 32 34 37 36 41
1.0 2.0 1.0 1.9 1.9 1.8 1.9 2.8 0.3 1.5 2.1 0.1 0.9 2.5 2.4 0.9 2.1 4.2 16.6 7.4 12.8 10.6 8.0 0.4 0.0 2.2 1.2 0.9 0.2 1.2 1.0 1.0 8.5 1.1
0.9 1.5 0.8 1.6 1.4 1.5 1.5 2.4 0.3 1.3 2.0 0.1 0.8 1.9 1.8 0.7 1.8 3.7 14.1 5.7 10.5 9.1 6.8 0.4 0.0 1.6 1.0 0.8 0.2 1.1 0.8 0.9 8.0 0.9
1.6 4.3 2.1 4.1 4.5 3.7 3.4 4.9 0.4 2.2 2.9 0.1 1.6 5.9 5.5 1.9 3.0 6.7 24.0 16.3 27.9 18.0 14.0 0.6 0.0 4.0 2.0 1.4 0.3 1.7 1.4 1.5 10.5 2.1
6 11 12 12 13 12 12 7 6 6 6 4 4 12 12 8 6 6 9 10 10 9 8 3 3 5 5 5 3 3 4 5 5 6 2 2 2 2 8 4 5 7 4
5-21 3-64 2-68 2-68 2-68 2-68 2-68 2-5 10-61 10-61 10-61 4-34 4-34 5-108 5-108 25-108 9-98 9-98 30-250 20-250 20-250 3-164 5-192 1-14 6-14 3-12 3-12 3-12 3-5 3-5 1-5 1-5 1-5 1-5 0.18-0.2 0.19-0.22 0.21-0.24 0.23-0.26 1-3 2-7 2-9 10-58 20-100
Michels et al.49 Culberson and McKetta50
a
34 40 46b 31 39 38 43 47 46 63 50 43 44 43 43 41 28 28 29 32
34 34
4.0 6.8 1.3 3.0 1.4
3.5 5.7 1.0 2.5 1.2
6.0 11.8 2.1 5.2 2.3
Duffy et al.51 a O’Sullivan and Smith23 Amirijafari and Campbell3 Sulltanov et al.52 Sulltanov et al.53 Sanchez54 Price55 Gillespie and Wilson8 Yarym-Agaev et al.56 Yokoyama et al.57 a Wang et al.58 a
Reichl59 a
Lekvam and Bishnoi24 a Gao et al.60 Dhima et al.1
Considered as low-pressure solubility data. b Unreasonable value.
Table 7. Application of Eq 5 and Results Obtained on the Ethane-Water System T (K)
H°γH (MPa)
3 vaq i (cm /mol)
σ (%)
Adev (%)
MaxDev (%)
N
P (MPa)
ref
311 344 378 411 444 473 523 573 344 344
4165 6303 7011 5958 4603 3037 2072 1144 6766 6073
48 49 50 59 61 61 51 57 38 48
2.9 2.6 3.9 3.0 2.1 19.0 15.0 15.0 4.4 0.9
2.8 2.0 3.3 2.4 1.5 16.4 12.7 13.4 4.0 0.8
4.6 6.3 7.5 5.5 4.4 32.7 26.3 22.7 6.2 1.2
10 11 12 12 13 8 8 8 3 4
4-65 2-66 2-68 2-67 5-68 20-350 20-350 20-350 3-27 20-100
Culberson and McKetta61
4. Can the Krichevsky-Kasarnovsky Equation Be Applied? An answer to this question requires a realistic analysis of all available experimental solubility data. However, the accuracy of this answer depends not only on the precision of the measurements but also on the way the hydrocarbon-phase fugacities are calculated. Generally, only the aqueous-phase composition is available. A flash calculation is therefore needed to estimate the hydrocarbon phase composition (in particular its water content) and hence the fugacity of each component. 4.1. Flash Calculation. The three-phase flash calculations are performed using a Rachford-Rice approach.
Danneil et al.62 Anthony and McKetta63 Dhima et al.1
Each flash calculation requires, in addition to pressure and temperature, the input of a global composition zj(z1, ...., zi, ..., zn-1, zw). When binary water-hydrocarbon mixtures are considered, the use of z1 ) z2 ) 0.5 is a aq aq ) xi,exp , a unique value of safe choice. By forcing xi,cal hc aq γiHi(P,T) ≡ fi (P,T)/xi is found for the hydrocarbon. When two or more hydrocarbons are used, two cases can be distinguished. Either the composition of the hydrocarbon mixtures is known before it is brought in contact with water, in which case the zj vector is composed of 50% water and 50% of this mixture, or the composition of the vapor in equilibrium with water is known. In the latter case, we assume that the solubility is small enough and does not change the ratio of their
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3151 Table 8. Application of Eq 5 and Results Obtained on the Propane-Water System T (K)
H°γH (MPa)
3 vaq i (cm /mol)
σ (%)
Adev (%)
MaxDev (%)
N
P (MPa)
285 311 329 361 370 383 400 422 273 344 289 311 344 378 411
1940 4990 6901 9013 8789 8810 8234 7195 1427 7326 1534 5428 9034 9357 8107
83 68 68 75 84 79 82 80
1.3 1.9 1.2 0.5 0.6 1.8 1.6 1.6 7.1 3.9 5.9 3.3 3.1 3.1 2.3
1.2 1.6 1.0 0.5 0.6 1.5 1.4 1.5 5.9 3.2 5.3 2.8 2.6 2.4 2.1
1.9 3.1 2.0 0.9 1.0 3.2 2.3 2.6 11.6 8.0 7.9 6.5 6.1 8.2 3.4
4 5 4 5 5 6 7 7 4 8 6 16 20 20 6
4-14 3-18 1-19 2-19 3-19 2-19 1-19 2-19 0.03-0.1 0.51-1.25 0.1-0.69 0.23-1.1 0.28-2.66 0.52-3.53 0.86-3.44
a
ref Kobayashi and Katz64
Umano et al.65 a Wehe and McKetta66 a Azarnoosh and McKetta67 a
Considered as low-pressure solubility data.
Table 9. Application of Eq 5 and Results Obtained on the Butane-Water System T (K)
H°γH (MPa)
3 vaq i (cm /mol)
σ (%)
Adev (%)
MaxDev (%)
N
P (MPa)
311 378 311 344 378 411 444 478 311 344 378 411 444 478 511 298 313 343 373 398 423 344
5754 8693 5195 8006 9014 9424 7139 5266 5315 7872 9157 9140 6901 5169 3536 4596 6016 8359 9793 9722 7836 8002
65 91 80 88 94 90 97 99 78 90 92 93 100 101 114
7.5 4.2 0.9 1.8 2.2 0.9 1.6 0.9 4.4 2.4 2.9 2.3 2.8 1.6 4.7 0.6 2.6 0.3 7.1 0.0 0.0 4.4
6.0 3.3 0.7 1.1 2.0 0.8 1.4 0.8 3.8 1.8 2.3 1.9 2.4 1.3 4.1 0.6 2.6 0.3 7.0 0.0 0.0 3.7
13.8 7.8 1.7 6.8 3.9 1.8 2.7 1.6 8.9 6.3 5.5 5.3 4.9 3.5 9.0 0.9 3.2 0.5 8.7 0.0 0.0 7.9
10 5 20 20 24 12 13 12 17 15 19 16 19 15 13 3 4 3 5 2 2 5
7-68 8-69 1-69 1-69 1-69 10-69 8-69 10-69 1-69 1-69 2-69 4-69 2-69 5-69 7-69 0.1-0.25 0.11-0.38 0.27-0.84 0.2-1.61 1.23-2.65 2.2-4.13 10-100
a
89
ref Brooks et al.68 Reamer et al.11
Sage and Lacey12
Carroll et al.48 a
Dhima et al.1
Considered as low-pressure solubility data.
compositions. The hydrocarbon part of the zj vector is then defined using the known hydrocarbon composition ratio in the vapor phase. Again, forcing the calculated aqueous mole fraction of each hydrocarbon to be equal to its experimental value allows the determination of a aq unique value of γiHi(P,T) ≡ fhc i (P,T)/xi for each hydrocarbon. 4.2. Analysis of Solubility Data. To evaluate whether the use of the [K-K] equation is justified, three arguments will be investigated: (i) the effect of mixture, (ii) the ln(f/x) versus P diagram, and (iii) comparison of j aq the values of γH i H° i,w and v i with literature data. 4.2.1. Effect of Mixture. Equation 1 indicates that at a given pressure and temperature, and for two different hydrocarbon compositions, the activity coefficients behave as follows
() γ(1) i
γ(2) i
)
[fhc,(1) (P,T,yj(1))]/x(1) i i [fhc,(2) (P,T,yj(2))]/x(2) i i
(7)
This approach has been proposed by Dhima et al.,1 where exponent (1) referred to the binary system and (2) to a ternary system. This ratio can also be interpreted as the ratio of the actual, measured solubility and its “theoretical” value. The ratio of hydrocarbon-phase fugacities and the aqueous mole fraction can be determined as explained
above, both for binary and ternary mixtures. Figure 4 thus shows the activity coefficient ratio for methane and carbon dioxide in the case of their corresponding binary and ternary systems. Two different ternary systems have been considered at four different pressures at 71 °C. Figure 5 gives this ratio as a function of aqueousphase mole fraction for all solutes considered in this work as well as in our previous work.1 We observe that for all our ternary systems (except the n-butane) this activity coefficient ratio (γ(1)/γ(2)) is close to unity within the experimental uncertainty. Hence, no interaction between solutes is observed in the aqueous phase. Furthermore, the activity coefficient is constant with composition. Since at infinite dilution its value must be 1, we conclude that it is 1 within the full concentration range of interest. The experimental uncertainty for the n-butane solubility is too large to draw any conclusion from the small activity coefficient ratio. This has been discussed in detail by Dhima et al.1 Figure 6 extends this analysis to the most important ternary systems for which data have been found for all temperatures. It can be observed that the evolution of the experimental values of the activity coefficient ratio with the solute aqueous-phase molar fraction is not very clear. Especially in the case of hydrocarbons such as n-butane
3152 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 Table 10. Application of Eq 5 and Results Obtained on the CO2-Water System T (K)
H°γH (MPa)
3 vaq i (cm /mol)
σ (%)
Adev (%)
MaxDev (%)
N
P (MPa)
285 291 298 304 308 313 323 348 373 374 393 323 373 423 473 383 423 473 523 533 543 289 298 303 304 348 366 394 373 393 413 433 453 473 323 348 353 393 433 471 348 421 288 293 298 323 313 323 333 278 303 308 333 338 344
114 135 163 188 206 228 275 393 492 476 510 285 498 583 518 502 556 478 416 391 375 131 163 185 200 412 531 561 496 558 617 609 549 528 260 415 427 526 535 535 239 408 127 147 165 261 224 293 304 77 148 161 328 355 374
32 32 32 32 32 32 32 32 32 37 40 30 31 32 35 34 36 37 26# 23# 10# 45# 39# 29 22# 23# 10# 25#
0.5 0.9 0.5 0.5 0.7 1.1 1.0 1.4 1.9 1.6 1.8 2.3 1.5 3.5 2.0 1.3 1.7 2.7 2.9 4.8 5.9 3.7 1.0 0.0 1.9 2.0 7.8 3.8 0.8 0.7 4.6 4.8 0.6 1.8 0.0 0.0 1.2 1.6 1.9 2.8 0.0 1.3 0.4 0.3 0.4 0.3 0.9 1.2 1.2 0.0 0.0 0.0 0.0 0.0 1.8
0.4 0.7 0.4 0.5 0.7 0.8 0.7 1.1 1.5 1.4 1.4 1.9 1.3 3.2 1.6 1.0 1.3 1.9 2.1 4.2 4.7 3.4 0.9 0.0 1.6 1.8 5.8 2.8 0.7 0.6 3.1 3.4 0.6 1.3 0.0 0.0 1.1 1.3 1.6 2.3 0.0 0.9 0.4 0.3 0.3 0.3 0.8 1.0 1.0 0.0 0.0 0.0 0.0 0.0 1.6
0.9 1.8 0.8 0.7 1.1 2.3 2.1 3.0 3.6 2.8 4.2 3.9 2.5 5.0 3.3 3.1 3.8 8.3 5.9 8.7 12.5 5.3 1.4 0.0 3.2 2.8 15.2 5.9 1.5 1.3 11.1 10.6 0.9 3.7 0.0 0.0 2.1 3.1 3.5 5.1 0.0 2.4 0.7 0.7 0.7 0.4 1.5 2.5 2.5 0.0 0.0 0.0 0.0 0.0 2.6
6 7 4 8 8 9 9 9 9 13 12 8 8 8 8 15 15 15 15 15 12 3 3 3 5 5 5 5 7 7 7 7 7 8 2 2 7 10 5 6 2 5 11 9 6 3 10 16 15 2 2 2 2 2 7
5.07-30.40 2.53-30.40 5.07-40.53 2.53-50.66 2.53-50.66 2.53-50.66 2.53-70.93 2.53-70.93 2.53-70.93 10.84-62.31 2.33-70.32 20-350 20-350 20-350 20-350 10-150 10-150 10-150 10-150 20-150 20-120 5.07-20.28 5.07-20.28 5.52-20.28 0.69-20.28 0.69-20.28 0.69-20.28 0.69-20.28 0.33-2.31 0.60-2.85 0.65-3.25 1.14-3.48 2.10-6.27 3.76-8.11 10-15 10-15 2.33-10.18 2.11-9.98 3.91-8.07 4.6-10.21 10-15 10-20 6.08-24.32 6.59-20.27 7.6-20.27 10.1-30.1 0.11-1.01 0.1-1.57 0.06-1.53 0.049-0.051 0.064-0.066 0.068-0.070 0.078-0.079 0.083-0.084 10-100
a
95b 29 29 31 36
33
ref Wiebe and Gaddy27
Wiebe and Gaddy7 Prutton and Savage69 To¨dheide and Franck70
Takenouchi and Kennedy71
Gillespie and Wilson8
Mu¨ller et al.72 a
D’Souza et al.73 Nighswander et al.74 a
Sako et al.9 King et al.35 Dohrn et al.75 Fischer et al.76 a Zheng et al.77 a
this work
b
Considered as low-pressure solubility data. Unreasonable data.
and heavier, this evolution becomes more irregular. This coincides with very low solubility values (lower than 100 ppm) where the experimental uncertainties are very large. For example in the case of n-pentane (solid triangle symbols) one observes that the experimental activity corresponding to the same amount of aqueous molar fraction may be greater or lower than unity. The butane data of McKetta and Katz2 also show very large activity coefficient ratios. Concerning the behavior of methane (open symbols) it can be observed that activity coefficient ratios are closer to unity. We believe that this distribution is probably the consequence of very large experimental uncertainties (Dhima et al.1). 4.2.2. ln(f/x) versus P Diagram. Equation 5 suggests that if ln(fi/xi) ) ln(γiHi(P,T)) is plotted as a function of (P - Psat w ), a straight line is obtained, on the condition that vj aq is a constant with pressure and i composition (and therefore equal to the aqueous partial molar volume at infinite dilution vj ∞i ), and γH i,w is i H° constant with composition (i.e.γH H° ) H° ). Figure 7 i i,w i,w presents the values of ln(fi/xi) as calculated using the above procedure, for all components investigated experimentally in this and previous work. Except for the case of n-butane, for which the experimental uncertainty is large, all data follow straight lines.
) 0 The intercepts of these lines with P - Psat w . The provide values for γHi H°i,w, and the slope vj aq i straight line is a necessary condition for the [K-K] equation to be valid, but is not sufficient. The last and perhaps most important argument is the comparison of j aq j ∞i obtained in this the values for γH i H° i,w and v i ) v work with other literature data. 4.2.3. Comparison of the Values of γHi H°i,w and aq v j i with Literature Data. The infinite dilution partial molar volume of hydrocarbons in water vj ∞i has been measured by a number of authors using techniques such as dilatometry (Krichevsky and Iliinskaya;32 Tiepel and Gubbins33) or densitometry (Masterton;34 King et al.35). Lyckman et al.36 developed a general correlation for vj ∞i based on Hildebrand theory and have shown good agreement between their correlation and experimental data of some gases in a certain number of organic solvents as well as in water given by Hildebrand and Scott,37 Horiuti,38 Schumm and Brown,39 and Walkley and Hildebrand.40 More recently, Brelvi and O’Connell41 developed another correlation. Although these correlations are not specific to aqueous solutes, as can be seen in Table 4, the agreement between the partial molar volumes obtained from our solubility data analysis and the results of two correlations is good.
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3153
Figure 8. Methane aqueous Henry’s constant.
Figure 9. Ethane aqueous Henry’s constant.
In addition, Table 4 compares the values of the o intercept with the y-axis, γH i Hi,w, to the values of Henry’s constant as correlated by Li and Nghiem21 or by Harvey.42 Other correlations, summarized in Table 5, give similar results. The same procedure can be extended to a much larger database. For this purpose, the flash procedure presented above has been adopted to analyze any isothermal data set at once. A single value for γH i H° i,w and for vj aq is obtained by minimizing the standard deviation, i
defined as
σ)
x ( 1
N
∑1
N
)
aq aq (xi,cal - xi,exp ) aq xi,exp
2
× 100
(8)
aq where N is the number of experimental points and xi,cal aq and xi,exp are respectively the calculated and experimental values of the solute aqueous phase molar fractions.
3154 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999
Figure 10. Propane aqueous Henry’s constant.
Figure 11. n-Butane aqueous Henry’s constant.
According to eq 5, the product γH i H° i,w is not a function of pressure but may be a function of the hydrocarbon content of water, which is higher at high pressure. Therefore, if we consider separately the highpressure and low-pressure solubility data, different values of γH i H° i,w would be obtained if the activity coefficient were not negligible. The results of this analysis are presented in Table 6-10, along with other statistical data. average deviation or mean absolute deviation (in %)
defined as
Adev )
1
N
∑1
N
aq aq (xi,cal - xi,exp ) aq xi,exp
× 100
(9)
and maximum absolute deviation (in %) defined as aq aq - xi,exp ) (xi,cal |N × 100 MaxDev ) Max| aq xi,exp
(10)
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3155
Figure 12. Carbon dioxide aqueous Henry’s constant.
Figure 13. Methane aqueous partial molar volume.
The references containing only low-pressure data (below 10 MPa) are identified with the symbol ‘a’. When an isotherm contains only low-pressure data, or only two points, the effect of pressure that is expressed by vj aq i is much more sensitive to the experimental uncertainties. In this case the values of volumes are not taken into consideration. Furthermore, when the volumes appeared unreasonable in view of the other data, they have been marked by a # (e.g. volumes corresponding to measurements of aqueous carbon dioxide solubility by Gillespie and Wilson8). We observe that, using the [K-K] equation (i.e.
j aq assuming constant γH i and v i ), the experimental soluaq , can be reproduced within a standard bility data, xi,exp deviation less than 5%, which is comparable to the experimental uncertainties. When only two data points are available on a given isotherm, the deviation is obviously zero. Table 10 shows that, even in the case of the carbon dioxide-water system, the [K-K] equation is applicable at least up to 200 °C, for pressures up to 100-120 MPa. 4.2.3.1. Henry’s Constants. Figures 8-12 present the results of the product of Henry’s constant with the
3156 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 Table 11. Henry’s Constant Values as Given by the Literature T (K)
H° methane
275 278 284 288 288 293 298 298 298 303 308 313 318 318 323 328
2329 2521 2938 3241 3243 3609 3969 3973 3970 4315 4648 4965 5238 5236 5487 5724
Rettich et al. 78
298 334 385 388 431 473 518
4133 6056 6392 6334 5058 3904 2609
Crovetto et al.79
277 286 315 334 364 384 432 457 461 466 484 513 538 542 554 573
2580 3430 5800 6260 6610 6310 4890 4050 3990 3120 3080 2580 2040 2240 1780 1130
Cramer80
298
3731
Tokunaga and Kawai81
274 283 286
2276 3042 3214
Lekvam and Bishnoi24
ref
T (K)
H° n-butane
283 298 303 313
1700 2700 4300 6300
T (K)
H° ethane
ref
T (K)
H° carbon dioxide
275 275 278 283 283 284 288 293 298 298 298 298 303 308 313 318 318 318 318 318 318 323 323
1396 1396 1559 1876 1876 1917 2225 2595 2974 2978 2977 2982 3370 3771 4176 4543 4545 4539 4550 4548 4542 4883 4887
Rettich et al. 78
278 278 303 303 308 308 333 333 338 338
87 88 183 184 205 207 329 328 354 355
Zheng et al.77
283 293 303 313
107 145 187 231
Tokunaga et al.84
288 293 298 308 313
123 144 165 211 236
King et al.35
Crovetto et al.82
124 143 166 189 212
Postigo and Katz85
2815 3245 3241 3121 4411 4345 4341 4674 4730 5191 5960 5954 6240 6593 6462 6713 6719 6534 6240 5223 4513 3102
288 293 298 303 308
295 301 301 301 314 315 315 318 322 330 336 342 349 351 352 364 369 396 404 429 442 474
303
181
Carvantes et al.86
303 313 323
187 234 284
Hagewiesche et al.87
303 323 353
192 288 443
Matous et al.88
623 632 643
198 169 139
Crovetto and Wood89
ref
T (K)
H° propane
Kazaryan and Ryabtsev83
311 328 344 361 378 394 411 428
4909 6646 7922 8845 9068 8837 8273 7590
activity coefficient as a function of temperature for all components. The values of the product [γH i H° i,w] obtained from the high-pressure solubility data (Tables 6-10) are compared to the values corresponding to lowpressure solubility data (Tables 6-10). Both these values are compared to the values of Henry’s constant given by the literature (Table 11), as well as to the values of Henry’s constant calculated using some existing literature correlations (Table 5). The values obtained from the low-pressure data and from the high-pressure data are very close, indicating that the activity coefficient has no effect. Furthermore, excellent agreement exists between the Henry’s constant values fitted here and the literature data or correlations presented. 4.2.3.2. Partial Molar Volume. The second condition necessary for evaluating the validity of the [K-K] equation is the independence of aqueous partial molar volume from pressure and composition. If this hypothesis is satisfied, the aqueous partial molar volume is equal to its value at infinite dilution. Figures 13-17 compare our fits for the aqueous-phase partial molar volume (Tables 6-10) with the values given in the literature (Table 12) that result from a direct measure-
ref
ref Kobayashi and Katz64
ment (when available) as well as with the correlations proposed by Lyckman et al.36 and Brelvi and O’Connell.41 Note that King et al.35 and Teng et al.43 measure the density of the water-rich phase dsolution (kg/m3) and do not give the values for the partial molar volumes. The following equation is used for converting their data to the apparent partial molar volume. aq
vj aq i ) 1000‚
[
]
1 - xi 18.015 18.015 44.10 + 1000‚ aq ‚ dsolution dsolution dwater x i
(11) Note that this equation only holds for CO2. A more general equation would have been
vj aqi )
Mi dsolution
+
[
]
1 - xaq 1 i 1 ‚Mw‚ dsolution dwater xaq i
(12)
One observes that the agreement between the values obtained using the above equation (Table 12) and the values resulting from our flash calculation given in Table 10 is surprising.
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3157
Figure 14. Ethane aqueous partial molar volume.
Figure 15. Propane aqueous partial molar volume.
Figure 16 shows, in addition, the saturated liquid molar volume as calculated by the Peng-Robinson equation. One observes that although at very low temperature its value is similar to that of the aqueous partial molar volume, at high temperatures no comparison can be made. From the above discussion we conclude that the [K-K] equation can be used for calculating waterhydrocarbon equilibria. If it is to be used for complex mixtures, however, the Henry constant and the partial molar volume at infinite dilution of each hydrocarbon component must be known. As we have seen, correla-
tions exist but they are limited to a few components, most often light hydrocarbons. Natural gases, however, also contain heavier components. It will be the purpose of another paper to propose a unified correlation for a great number of components. Conclusions New solubility data of carbon dioxide and its ternary mixtures with methane in water have been obtained at 71 °C and at pressures from 10 to 100 MPa. To get a thermodynamic model, these data as well as our previ-
3158 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999
Figure 16. n-Butane aqueous partial molar volume and its comparison to the saturated liquid volume.
Figure 17. Carbon dioxide aqueous partial molar volume.
ously published solubility data (Dhima et al.1) have been considered together with a large number of the published solubility data. From this analysis, we conclude the following: Our binary solubility data are in good agreement with the existing literature data. The ternary solubility data can be predicted by the binary ones within the experimental accuracy. As a direct consequence of this analysis, a very simple thermodynamic model can be proposed. The hydrocarbonrich phase is modeled using the Peng-Robinson equation of state where the binary interaction parameters
are taken from the literature. The aqueous phase is described by Henry’s law extrapolated by the Krichevsky-Kasarnovsky equation. The necessary parameters of this equation are the Henry’s constant at the water vapor pressure and the aqueous partial molar volume at infinite dilution. It is demonstrated that both of these parameters are only functions of temperature and are independent of pressure and composition. The hydrocarbon species dissolved in water do not interact. The values of Henry’s constant can be calculated using the literature expressions such as those given by Li and Nghiem21 or Harvey.42 These correlations are
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3159 Table 12. Values of the Directly Measured Apparent Partial Molar Volumes T (K)
methane
290 296 298 302 308 273 298 323 273 288 293 298 278 283 288 293
33 36 38 37 38 36 37 38 37
ethane 48 50 51 52
propane
carbon dioxide
benzene
64 67
ref
comments
Masterton34
densimetry
Krichevsky and Iliinskaya32
dilatometry
Tiepel and Gubbins35 King et al.43
dilatometry densimetry
Teng et al. (1997)
densimetry
83 83 84
68 32 33 33
53 29 30 30 32 32 32 32
given only for a limited number of solutes. Subsequently, a unique and general expression is desirable. The aqueous partial molar volumes at infinite dilution can be estimated by the correlation proposed by Lyckman et al.36
(2+) ) multiple hydrocarbons-water system - ) composition vector (xj and yj) - ) related to partial properties (vj aq i : aqueous solute partial molar volume)
Literature Cited Acknowledgment The authors are grateful to the ARTEP (Association de Recherche sur les Techniques d’Exploitation du Pe´trole) member societies (TOTAL, ELF Aquitaine, Institut Franc¸ ais du Pe´trole, Gaz de France) for financial support and to the thermodynamics group of Institut Franc¸ ais du Pe´trole, where this work was carried out. List of Symbols a, b ) cubic equation of state pure component parameters f ) fugacity H ) Henry’s law constant m ) mass M ) molecular weight P ) pressure R ) gas constant T ) absolute temperature V ) volume v ) molar volume s ) solubility (gas volume per unit mass of water) x ) hydrocarbon mole fraction in water-rich phase y ) hydrocarbon mole fraction in hydrocarbon phase Greel Letters ∆ ) change (of volume) γ ) activity coefficient Subscripts i, j ) hydrocarbon mixture component identification w ) water identification total ) total (mole fraction hydrocarbon in water-rich phase) g1, g2 ) volume of the gas meter before and after the flash a ) atmospheric or ambient (pressure) room ) room (temperature) ref ) reference (pressure) Superscripts aq ) aqueous phase H ) related to Henry’s constant l ) liquid ° ) reference (or standard) state H ) relative to Henry’s law (1), * ) pure hydrocarbon-water system
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Received for review December 7, 1998 Revised manuscript received May 26, 1999 Accepted May 28, 1999 IE980768G
