Ind. Eng. Chem. Fundam. 1983, 22,
364
Table 111. Kihara Parameters for Hydrate-Gas Interactions
.. methane propane (I
K
.. 3.2363 3.3049
0.30017 0.67964
153.22 200.92
= Boltzmann constant.
t 5
c'H'
?Q 20
30 40 50 MI 70 80 MOLE%INGASPHASE
90
'"
Figure 5. Hydrate point conditions for propane-hydrogen sulfide mixtures at -3 O C . The solid points are the data of Plateeuw and Van der Waals. The line corresponds to the calculations of the computer model.
Only the data for the mixture containing 4% H2S were used to determine Amiand Bmi. The same empirical expression is used to represent the Langmuir constant of propane a t temperatures below 60 O F . The values of the constants Amiand Bmiare given in Table I1 for each size cavity of structure I1 hydrates. The Langmuir constants for methane at all temperatures and for propane above 60 O F were evaluated using the Kihara potential function r I 2ai U(r)= m;
V(r)= 4ei
[(- ( ui
r - 2ai
)12-
ui
r - 2ai
)"I
;
r
> 2ai (4)
The Kihara potential parameters (ai, ui, and ci) for methane and propane are given in Table 111. Predictions using the computer model are shown in Figure 4, along with the data. The predictions are good for all three gas compositions, with a maximum temperature deviation from the data of approximately 2 "C.
364-366
There is one data set available in the oDen literature for structure I1 hydrates containing H2S (PGtteeuw and Van der Waals, 1959). These data are displayed in Figure 5. In that figure, the hydrate formation pressure is plotted vs. the C3H8-H2Sgas-phase composition in mole percent. The data are shown as points, and the model predictions as a solid line. Again, agreement between the predictions and data is good. Consequently, predictions of hydrate formation conditions in sour systems using the computer model appear to be accurate within 2 "C, just as in sweet systems. Conclusion Using a computerized data collection device, we have examined the hydrate formation/ decomposition behavior in the system H2S-methane-propane by the pressure change technique. The experiment revealed the expected hysteresis behavior in pressure vs. temperature space near the hydrate point. We observe a repeatable hydrate decomposition point, which can be located with an accuracy of better than 0.1 "C and 0.1 psia. The data on the H2Smethane-propane system were used to adjust parameters in a computer model so as to fit the behavior of sour gas systems which form structure I1 hydrates. As seen in Figure 4, the model agrees quite well with the data. Registry No. H2S, 7783-06-4; CHI, 74-82-8; C3Hs, 74-98-6.
Literature Cited Hammerschmidt, E. G. Ind. Eng. Chem. 1934, 26, 851. Kobayashi, R.; Katz. D. L. Trans. AIME 1949, 126, 66. Lin, C. J.; Hopke, S. W. AIChE Symp. Ser. No. 140, 1974, 70, 37. Marshall, D. R.; Saito, S.; Kobayashi, R. AIChEJ. 1964, 10(2),202. Parrish, W. R.; Prausnitz, J. M. Ind. Eng. Chem. Process D e s . Dev. 1972, 1 1 , 26. Piatteeuw, J. C.; Van der Waals, J. H. Recl. Trav. Chem. 1959, 78, 126. Schroeter, J. P.; Kobayashi, R. "ComputerizedControl, Data Acquisition, and Storage in the Determination of Hydrate Formation Conditions",Presented at 72nd AIChE Meeting, San Francisco, Nov 1979. Van der Waals, J. H.; Platteeuw, J. C. A&. Chem. fhys. 1959, 2 , 1. Verma, V. W.D. Thesis, University of Michigan, Ann Arbor, MI, 1974.
Received for reuiew March 1, 1982 Revised manuscript received May 12, 1983 Accepted July 20, 1983 The authors acknowledge the support of the Exxon Production Research Company over the duration of these studies.
Equation Suitable for Estimation of Ternary Liquid-Liquid Equilibria with Binary Wilson Parameters MRsuyasu Hlranuma Tomakomai Technical College, 443, Nishikloka, Tomakomai, Mkkaido, 059- 12, Japan
A four-parameter equation is proposed for systems having unusual behavior or for partially miscible systems. An extension of the equation appks to mutkomponent solutions. I f one binary pair of compounds in a ternary mixture is completely miscible, Wilson's parameters already collected from binary data may be used for the ternary system. The proposed method yields almost the Same size of immiscibility regions as those observed for the ternary liquid mixtures tested.
Introduction The original third Wilson parameter, c , is the same for all constituent binaries, but it must be changed for each multicomponent system. In a previous paper (Hiranuma,
19811, c was set as an adjustable parameter for component
i to remove the disadvantage. However, to correlate the VLE or the LLE data more precisely, it is convenient to use binary parameters because they can be selected in-
0196-4313/83/1022-0364$01.50/00 1983 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983 365
Table I. Ternary Partially Miscible Systems Tested system no.
system components benzene (1)-n-heptane (2)-acetonitrile (3) ethanol (1)-n-hexane (2)-acetonitrile (3) ethyl ether (1)-methanol (2)-cyclohexane (3) acetonitrile (1)-water (2)-ethyl acetate (3) acetone (1)-methyl acetate (2)-water (3) carbon tetrachloride (1)-methanol (2)-cyclohexane (3) 5-nonanone (1)-1-hexene (2)-Me,SO (3)
1 2 3 4
5 6 7
Table 11. The Necessary Binary Data Source ~___________________~~ system no. system components
~~
~~
~
4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
dependently of other constituent binary systems. The NRTL equation is the only equation whose third parameter is a binary one. The Wilson equation would be closer to the facts than the NRTL equation is, because the NRTL equation gives incorrect behavior for ideal mixtures when 7 i ’ -. This paper proposes a Wilson-like equation with additional binary parameters. Proposed New Equation The exact solution for a one-dimensional mixture yields Wilson’s equation (Hiranuma, 1974). A one-dimensional mixture does not show phase splitting. Unfortunately, there is no exact way to extend a one-dimensional model to a multi-dimensional one. Therefore, it is common in statistical thermodynamics that a mixture is assumed to contain ciNimolecules instead Ni molecules of component i and the number of the one-dimensional arrangement is derived from it. As a result (Hiranuma, 1974)
-
In a previous paper (Hiranuma, 1981), eq 1 was utilized with (cj’/ci? = 1.0 and ci # 1.0. The parameter ci is a third parameter which accounts a perturbation between the number of real molecular arrangement and that of the one-dimensional one. The value of cifor most substances was found to be nearly unity (Hiranuma, 1981). Now, suppose that ci = 1.0 and ( c { / c [ ) = aij # 1.0. Equation 1 is rewritten as gdRT = x1
+ a12A12x2
x1
+ a12x2
x2
+ a21A21x1
x2
source
45
Palmer and Smith (1972) Sugi and Katayama (1978) Sugi et al. (1976) Sugi and Katayama (1978) Venkataratnam e t al. (1957) Yasuda et al. (1975) Renon and Prausnitz (1968)
40 25 60 30 25 60
~
benzene-n-heptane acetonitrile-benzene ethanol-n- hexane acetonitrile-e thanol ethyl ether-methanol cyclohexane-ethyl ether acetonitrile-water ethyl acetate-acetonitrile acetone-methyl acetate acetone-water CC1,-methanol CCl,-cy clohexane 5-nonanone-Me,SO 5-nonanone- 1-hexene n-heptane-acetonitrile n-hexane-ace tonitrile methanol-cyclohexane water-ethyl acetate methyl acetate-water 1-hexene-Me,SO
1 2 3
temp, “C
+ a21x1
temp, “C
source
45 45 40 40 25 25 60 60 30 25 25 25 60 60 45 40 25 60 30 60
Palmer and Smith (1972) Palmer and Smith (1972j Sugi and Katayama (1978) Sugi and Katayama (1978) Arm and Bankay (1968) Arm et al. (1967) Sugi and Katayama (1978) Sugi and Katayama (1978) Bussei Jyosu (1970) Kagaku Binran (1966) Yasuda et al. (1975) Yasuda et al. (1975) Renon and Prausnitz (1968) Renon and Prausnitz (1968) Palmer and Smith (1972) Sugi and Katayama (1978) Sugi e t al. (1976) Sugi and Katayama (1978) Venkataratnam et al. (1957) Renon and Prausnitz (1968)
where aI1is independent of a,,. It makes aIl a binary parameter. It can be selected independently of other constituent binary systems.
Evaluation of the Proposed Method The following equation, which includes the newly defined parameters a,was derived and used.
/CIN In Yk = -In (C?x,akIAkI)- C , N ( ~ , ~ I ~ ~ , ~ + In (CINx,akI) + C?(~I~,~/C,N~ (3), a I l ) The four parameters A and a in eq 3 were determined from binary vapor-liquid equilibrium data. A nonlinear least-squares fitting method was used which minimizes the sum of squares of deviations in activity coefficients (7, = p,/PIox,)for all data points. It provided a good fit for the system chloroform-ethanol in which the activity coefficient exhibits rather unusual behavior since it has a maximum or minimum as in Figure 1. In this case, a’s would serve to account for composition dependence of A’s (if solved as an equality of Wilson’s equation (a’s = 1.0) with eq 3 (a’s # 1.0) a t any composition of liquid phase, A’s obtained in Wilson’s equation would depend on composition.) However, in most cases, the values of a’s obtained were close to unity and this method did not provide big improvement over Wilson’s equation. It means a one-dimensional model would be a good approximation even for a multi-dimensional one. Therefore, arl = 1.0 can be used except for a system with unusual behavior or a partially miscible system. Data for seven ternary partially miscible systems (type one: Sorensen, 1979) were chosen for the test along with all the necessary binary data as in Tables I and 11. When the constituent binary system was completely miscible, the A’s were determined with a’s = 1.0. Because Wilson’s parameters collected so far for many binary systems are available just as they are. Such treatment is possible because a’s are binary parameters.
~l~r
388
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983
2
3
1
NO. 0
/
I
/-----
/
--__ 3 NO 7 Figure 2. Experimental and estimated liquid-liquid equilibria. The number refers to Table I, which lists the components and references. The estimations are carried out using the parameters in Table 111, and Table I1 lists the binary system numbers, the components, and references: (-) observed; (- - -) the proposed method; (- - -) NRTL equation. 2
Figure 1. Activity coefficients for the system chloroform (1)ethanol (2) at 55 “C: (0) experimental data (Jakubicek et al., 1957); (- - -) Wilson; (-) the proposed method; AI2 = 1.576; AZ1= 0.1741; a12 0.1284; a21 = 1.1223. Table 111. Parameters for the Proposed Equation binary parameters system
no.
*
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1.390 0.4707 0.04598 0.4195 0.6132 0.7648 0.1394 0.8029 0.5970 0.3221 0.3372 0.6439 0.1525 0.5345
0.294 5 0.6009 0.2765 0.5242 0.2592 0.8701 0.3392 0.8014 1.326 0.2192 0.05208 1.282 0.4466 1.167
15 16 17 18 19 20
0.04600 0.06270 0.05731 0.2650 0.07263 0.06791
0.07890 0.07347 0.07389 0.01644 0.3623 0.03436
12
*
21
a11
a 21
1.0
1.0
Conclusions The proposed method yields almost the same size of immiscibility regions as those observed, and the estimation of the ternary tie lines from binary data was carried out with no problem. The Wilson parameters collected so far for many binary systems are available just as they are. Equation 3 can be utilized as a four-parameter equation in case of need. Nomenclature ci = parameter of component i in eq 1 = parameter of component i in eq 1 g = excess Gibbs energy of mixing Ni = total number of molecules i in mixture Pio = vapor pressure of pure component i, atm p i = vapor pressure of component i in mixture, atm R = gas constant T = absolute temperature, K xi = mole fraction of component i in liquid phase
ci
1.0 1.5 1.1 1.1
1.1 1.1 1.5 1.1
When the A parameters are determined from binary liquid phase solubility, the values of a’s must be set previously, unless the other data (for example, the experimental limiting activity coefficients except the solubility limits)are available. a’s = 1.1could correlate well the LLE data for systems no. 1 and 7. a’s would account for the perturbation between the one-dimensional arrangement and the real dimensional one. As a result, aij = 1.1 (i = a substance except a strongly hydrogen bonded molecule) is recommended. However, the common value of aij could not correlate LLE data for ternary aqueous mixtures and alcohol mixtures very well. It is supposed that the number of hydrogen bond formation of a substance would dominate the a parameter (Hiranuma, 1981). Then, aij = 1.5 (i = water) and aij = 1.0 (i = methanol or ethanol) were used as shown in Table 111. Figure 2 illustrates the agreement between the experimental data and the calculated values with the proposed ones for no. 6 and 7 . The agreement for no. 1-no. 5 is almost as good as the results of Hiranuma (1981). The results by the NRTL equation are also quoted from the literatures in Table 1.
Greek Letters aij = binary parameter in eq 2 and 3 y i = activity coefficient of component i in liquid phase ’ti, = Wilson-like parameter ri, = paramter in NRTL equation ( g i j - gjj)/RT Subscripts i, j , k , N = component
Literature Cited Arm, H.; Bankay. D.; Strub, K.; Waltl, M. &/v. Chim. Acta 1987, 50, 1013. Arm, H.; Bankay, D. Helv. Chim. Acta 1988, 51, 1243. Hiranuma, M. Ind. Eng. Chem. Fundem. t974, 13, 219; 1981, 20, 25. Jakubicek, J.; Fried, V.; Vahala, J. Chem. Listy 1957, 51, 1422. Kagaku Kogaku Kyokal “Bussel Jyosu“, 1970. 8 . 112. Nippon K8gaku Kal “Kagaku Binran, Kisohen 11”; Maruzen, 1966; p 596. Palmer, D. A,; Smlth, 8. 0. J . Chem. Eng. Data 1972, 17, 73. Renon, H.; Prausnk, J. M. I n d . Eng. Chem. Process D e s . D e v . 1988, 7 , 220. Ssrenson, J. M.; Magnussen, T.; Rasmussen, P.; Fredenslund, A. HUM Phase Equllib. 1979, 3 , 47. Sugi, H.; Nkta, T.; Kataymama, T. J . Chem. Eng. Jpn. 1978, 9 , 12. Sugi,ti.;Katayama, T. J . Chem. Eng. Jpn. 1978. 1 1 , 167. Yasuda, M.; Kawade, H.; Katayama, T. Kagaku Kogaku Ronbunshu 1975, 1 , 172. Venkataratnam, A,; Rao, R. J.; Rao, C. V. Chem. Eng. S d . 1957, 7, 104.
Received for review December 7, 1981 Revised manuscript received January 19, 1983 Accepted June 23, 1983
