Ind. Eng. Chem. Res. 1987, 26, 2274-2286
2274
Morari, M. Chem. Eng. Sci. 1983, 38, 1881-1891. Ogunnaike, B. A. “A Statistical Appreciation of Dynamic Matrix Control”, American Control Conference, San Francisco, 1983, pp 1126-1 131. Reid, J. G.; Chafin, D. A.; Silverthorn, J. T. “Output Predictive Algorithmic Control: Precision Tracking with Application to Terrain Following”, Joint Automatic Control Conference, San Francisco, 1980, paper FA-9F. Reid, J. G.; Chafin, D. E.; Silverthorn, J. T. J. Guidance Control 1981, 4 , 502-509. Reid, J. G.; Mehra, R. K.; Kirkwood, E. “Robustness Properties of Output Predictive Deadbeat Control: SISO Case“, IEEE Conference on Decision and Control, Fort Lauderdale, 1979, pp 307-314.
Richalet, J. “General Principles of Scenario Predictive Control Techniques”, Joint Automatic Control Conference, San Francisco, 1980, paper FA9-A. Richalet, J. A.; Rault, A.; Testud, 3. D.; Papon, J. Automatica 1978, 14, 413-428. Rouhani, R.; Mehra, R. K. Automatica 1982, 18, 401-414. Yeo, Y. K. “Adaptive and Non-adaptive Bilinear Model Predictive Control”, Ph.D. Dissertation, Auburn University, Auburn, AL, 1986. Zames, G. IEEE Trans. Autom. Control 1981, AC-26, 301-320. Received for review October 15, 1985 Revised manuscript received July 13, 1987 Accepted August 19, 1987
A Modified UNIFAC Group-Contribution Model for Prediction of Phase Equilibria and Heats of Mixing Bent L. Larsen DECHEMA, Frankfurt (M), BRD
Peter Rasmussen and Aage Fredenslund* Instituttet for Kemiteknik, The Technical University of Denmark, DK-2800Lyngby, Denmark
The Modified UNIFAC model for predicting activity coefficients presented in this work is based on the well-known UNIFAC model. Two changes are introduced in Modified UNIFAC: (1)the group-interaction parameters have been made temperature-dependent and (2) the combinatorial term is slightly modified. Group-interaction parameters have been determined for 21 different main groups. It is shown that Modified UNIFAC gives somewhat better predictions of vapor-liquid equilibria than does UNIFAC, while the predictions of excess enthalpies are much improved. Hence, Modified UNIFAC has a better built-in temperature dependence than UNIFAC. Modified UNIFAC gives in general predictions of liquid-liquid equilibria (LLE) of the same quality as the original UNIFAC with LLE-based parameters. A Modified UNIQUAC model corresponding to the Modified UNIFAC model is also presented. The UNIFAC group-contribution method is widely applied to the prediction of liquid-phase activity coefficients in nonelectrolyte, nonpolymeric mixtures at low to moderate pressures and at temperatures between 275 and 425 K. The parameters needed for the use of UNIFAC are group volumes (Ilk),group surface areas (Qk), and groupinteraction parameters (amnand an,,,). Extensive tables with revised and updated values for 44 commonly needed groups are published in collaboration between University of Dortmund (BRD) and Technical University of Denmark (DK);see Gmehling et al. (1982),Macedo et al. (1983),and Tiegs et al. (1987). The uses and shortcomings of UNIFAC are reviewed by Fredenslund and Rasmussen (1985). One of the shortcomings of UNIFAC is that the built-in temperature dependence is not good enough for simultaneous prediction of vapor-liquid equilibria (VLE) and excess enthalpies (HE). This paper presents a Modified UNIFAC model which permits simultaneous representation and prediction of VLE and HE through the introduction of temperaturedependent group-interaction parameters. In addition, the combinatorial term is modified according to the ideas of Kikic et al. (1980).
The Modified UNIQUAC and UNIFAC Models As in the UNIQUAC (Abrams and Prausnitz, 1975) and UNIFAC (Fredenslund et al., 1977) models, the excess Gibbs function is calculated as a sum of a combinatorial and a residual contribution: 0888-5885/87/2626-2274$01.50/0
gE = gcE + g,E The Combinatorial Term. It is assumed that for mixtures containing alkanes only, the residual excess Gibbs function is zero. Thus, mixtures containing different alkanes are described by the combinatorial term only. We must then seek an expression for the combinatorial term, which can describe as well as possible the activity coefficients of alkanes in mixtures with other alkanes. Combinatorial expressions can be obtained from statistical mechanical arguments. The Flory-Huggins combinatorial can be written as
while the Staverman-Guggenheim combinatorial, which is used in UNIQUAC and UNIFAC, is
(3) The molecular volume fraction (ai) and surface area fractions (ei) are defined in terms of the segment numbers (r) and contact number parameters [(z/2)q] as
0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2275 Table I. Deviations between Experimental a n d Calculated Pressures for Alkane Systems dev, % of exptl
E
iD
U
P system hexane-decane hexane-undecane hexane-dodecane hexane-hexadecane 2-methylpentane-hexadecane 3-methylpentane-hexadecane 2,3-dimethylbutanehexadecane 2,2-dimethylbutanehexadecane
rep 1 2 3 4 5 5 5
temp, K 308.15 308.15 308.15 293.15 293.15 293.15 293.15
S-Gb 1.50 1.97 3.33 9.03 6.38 6.87 5.45
F-Hc (2/3) 0.29 0.24 0.53 1.99 1.36 2.36 1.07
5
293.15
6.26
2.01
"References for experimental data: 1, Marsh et al. (1980a); 2, Marsh et al. (1980b); 3, Ott et al. (1981); 4, McGlashan and Williamson (1961); 5, Fernandez-Garcia et al. (1968). *S-G : Staverman-Guggenheim. F-H = Flory-Huggins.
Lc 0
1 .o
c t ,
.-W .-U v-
0.8
4-
0
u
0.6
h .-+
>
W + .-
0.4
0
10
20
30
40
50
.-c
The r and ( z / 2 ) q values are assumed to be proportional to the van der Waals volumes and surface areas. The last term in eq 3 may be viewed as a Staverman-Guggenheim correction to the Flory-Huggins combinatorial. The effect from the Staverman-Guggenheim correction is often quite small. As pointed out by Sayegh and Vera (1980), it may, however, in some cases give large corrections leading to negative values of the combinatorial excess entropy, which is not realistic. As a consequence, the Staverman-Guggenheim correction has been dropped in Modified UNIQUAC and UNIFAC. The combinatorial expression used in Modified UNIFAC then becomes (5)
with modified volume fractions following Kikic et al. (1980):
x;ri2I3
60 n
4c -
Carbon number o f second alkane Figure 1. Calculated and experimental infinite dilution activity coefficients for hexane in normal alkanes: (-) Staverman-Guggen3
heim; (- - -) Flory-Huggins with modified volume parameter (exponent 2/3); (A,0,@) experimental ?- values (GLC).
parameters [(z/2)q] as shown in eq 4 and the local area fractions (8ji) are calculated from
m
The Boltzmann factors ( r j i )are obtained from the temperature-dependent interaction parameters (aji)as r j i = exp(-aji/T) (10) Three coefficients are used to describe the temperature dependence of the interaction parameters,
( :
)
aji = ajCl+ aji,,(T- To)+ aji,3 T In - + T - To
(11)
Table I shows a comparison of the Staverman-Guggenheim combinatorial with the Flory-Huggins combinatorial with r2I3. The two different combinatorials are used to predict activity coefficients in binary alkane mixtures using the group volume parameters (Rk) given in Table VI. The numbers given in the last two columns are the mean absolute deviations between experimental and calculated pressures, in percent of experimental pressures. The modified (r2I3)Flory-Huggins combinatorial is seen to be much better in describing VLE of alkane mixtures than the Staverman-Guggenheim combinatorial, especially when the difference in size of the components is large. The same conclusion may be drawn from Figure 1. In terms of gE, the Modified UNIFAC combinatorial is given by
The Residual Term. The residual contribution is z
g?/RT = ZxiiqiIn
(eii/&)
(8)
1
in Modified UNIQUAC where the surface area fractions (4) are calculated from the molecular contact number
where Tois an arbitrary reference temperature, here 298.15 K. The temperature dependence shown in eq 11was chosen from considering aji as related to an interaction Gibbs function, Agji = ajik (12) In this case, the enthalpy of interaction is Ahji = (aji,l - aji,zTo + aji,S(T- T0))lz
and the interaction heat capacity is Acji = aji,3k
(13) (14)
where lz is the Boltzmann constant. It may be shown by means of eq 8 that at the reference temperature, gE depends only on the fmt interaction Gibbs parameter coefficient, while the excess enthalpy (hE)depends on the first and second coefficients. The excess heat capacity (CPE)depends on all three coefficients. The molecular volume parameters ( r ) are calculated as in UNIQUAC, while the contact number parameters [(z/2)q] are calculated as q for UNIQUAC, Le., r = Vw/15.17 (Vw in cm3/mol) (15) z iiq = Aw/(2.5 X lo9)
(Aw in cm2/mol)
(16)
2276 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 Table 11. Structural Parameters for Modified UNIQUAC component r (z/2)9 cyclohexane 4.0464 3.240 hexane 4.4998 3.856 methanol 1.0000 1.000 Interaction Parameter Coefficients for the Table 111. Cyclohexane-Hexane-Methanol System
i
i
k
cyclohexane
1 2
hexane
1 2
methanol
1 2 3
hexane 74.95 -0.2497 1.577 0.0 0.0 0.0 -0.6989 -0.4015 -0.1112
cyclohexane 0.0 0.0 0.0 -59.15 0.1083 -1.640 54.13 -0.4864 -0.6890
methanol 1160.0 0.4499 -1.157 1382.0 0.0191 25.20 0.0 0.0 0.0
where V , is the van der Waals volume and Aw is the van der Waals area of a molecule (Abrams and Prausnitz, 1975). Differentiation of gE leads to the following activity coefficient expressions for Modified UNIQUAC: In y1 = In : y
+ In 7,'
In ylc = In ( w , / x l ) i- 1 - (w,/x,) z
In 7: = -s,(-[ln 2
+ 1-
(c0J7Jl)1 Co
J
J
j7,j/(COm7mI)) m
(17)
(18) (19)
Equations 17 and 18 also apply for the Modified UNIFAC model, while eq 19 is replaced by In y; = Cvkl(lnr k - In k
rk')
(20)
where v k l is the number of groups of type k in molecule i, r k is the activity coefficient of group k at mixture composition, and rk1is the activity coefficient of group k at a group composition corresponding to pure component i. r k and r k L are given by
Table IV. Representation with Modified UNIQUAC of Binary Systems within Cyclohexane-Hexane-Methanol" rmsd HE, temp or pressure rmsd .Y rmsd P, % J/mol rep Cyclohexane-Hexane 1 VLE 343.15 K 0.0049 0.35 2 760.0 mmHg 0.0036 0.40 3 760.0 mmHg 0.0030 0.23 HE 298.15K 4 0.88 4 318.15 K 0.82 VLE
HE VLE
HE
Cyclohexane-Methanol 2.21 318.15K 0.0131 328.15 K 0.0104 2.08 0.0100 1.45 760.0 mmHg 328.15 K 0.0063 0.90 308.15-320.15 K 308.15 K 313.15 K 318.15 K 323.15 K 333.15 K 348.15 K 298.15K 303.15 K 306.85 K 313.15 K 318.15 K 323.15 K
5 5 5 6 7
38.81
Hexane-Methanol 1.74 1.60 1.43 1.23 0.88 0.68
8 8 8
16.76 16.58 25.63 20.21 23.47 28.70
8 8 8 9 9 9 9 9 9
a rmsd = root mean square deviation between experimental and calculated data. *1, Susarev and Chen (1963); 2, Ridgway and Butler (1967); 3, Myers (1957); 4, Marsh (1973); 5, Madhavan and Murti (1966); 6, Morachevski and Komarova (1957); 7, Belousov and Morachevski (1970); 8, Wolff and Hoeppel (1968); 9, Savini et al. (1965).
250
I
A 0
. z. 7
200
150 100
50
In Modified UNIFAC,
0 0.0
=
exp(-(lmk/T)
0.4
0.6
0.8
1.0
Figure 2. Excess enthalpy for cyclohexane-hexane a t 298.15 K (0) and at 318.5 K (A). Experimental data: Marsh (1973).
Xmn m i Q m Tmk
0.2
(24)
The structural parameters [ R k and ( z / 2 ) Q k ] for the groups are generally obtained from eq 15 and 16, and the temperature-dependent group-interaction parameters (amk) are described by eq 11. The total number of groups ( k ) in the mixture is nk.
Correlation of VLE and LLE Using Modified UNIQUAC The Modified UNIQUAC model, as given by eq 17-19, may be used to correlate simultaneouslyVLE and HE data. In this case, the parameters a;ih of eq 11are fitted to VLE and HE data for binary systems
To illustrate the capability of Modified UNIQUAC, we have chosen the ternary system methanol-hexane-cyclohexane and the binaries included in it. The structural parameters for the components are shown in Table 11, and the aijh parameters obtained from binary VLE and HE data are given in Table 111. The results of the correlation are shown in Table IV. The ability of the Modified UNIQUAC model to represent both negative and positive excess heat capacities is illustrated in Figures 2 and 3, showing the fit to experimental excess enthalpy data at different temperatures for hexane-methanol and cyclohexane-hexane. It may be noted that the original UNIQUAC model with temperature-independent parameters cannot describe the
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2277 1 .o
1000 J
0
z
Y1
\
-3
800
0.8
600
0.6
400
0.4
200
0.2
0
0.0 0.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
1.0
xl
xl Figure 3. Excess enthalpy for hexane-methanol a t 298.15 K ( O ) , a t 313.15 K (A),and a t 323.15 K (X). Experimental data: Savini et al. (1965).
m a
110
1
I
Y
100 .M
Table V. Predictions of Vapor-Liquid Equilibria in the Ternary System Cyclohexane-Hexane-Methanol (Goral et al., 1983) temp, K rmsd P, % 293.15 1.37 313.15 0.74
negative excess heat capacities. Representation of experimental VLE data is illustrated for cyclohexane-methanol in Figure 4. The calculated curves are seen to represent the data from the two different sources well. Although it is difficult to see from the figures, there is a calculated phase split a t 328.15 K. This is erroneous, as the experimental upper critical solution temperature is 318 K. The predicted and experimental mutual solubilities are compared in Figure 5, and it is seen that Modified UNIQUAC predicts the upper critical solution temperature 40 K too high. From the above results and others (Larsen, 1986),it can be concluded that the Modified UNIQUAC model is well suited for simultaneous correlation of binary VLE and excess enthalpy data. Predictions of ternary vapor-liquid equilibria with the parameters in Tables 11and I11 have been compared with experimental data. The results are summarized in Table V. It is seen that the ternary systems are represented quite accurately.
Determination of Modified UNIFAC Group-Interaction Parameters For the estimation of group-interaction parameters, extensive use has been made of computerized data banks with VLE, LLE, and HEdata. The VLE data bank has been established by the University of Dortmund, BRD, and the LLE and the HE data banks have been established in collaboration between the University of Dortmund and the Technical University of Denmark. Most of the information in these banks is also available in book form (VLE, Gmehling and Onken (1977);LLE, Christensen et al. Sarensen and Arlt (1979),and HE, (1984)). The objective function, which is minimized by the parameter estimation program (Skjold-Jrargensen, 1983),is a sum of squared deviations between experimental and calculated properties. The properties included in this s u m of squares (SSQ)are the directly measured data; i.e., for
a
90 80 70 60 50 40 0.0
0.2
0.4
0.6
0.8
1.0 X1’ y1
Figure 4. VLE for cyclohexane-methanol at 328.15 K. Experimental data: Madhavan and Murti (1966). 360 K
340
320
300
280
260
I
1
I
I
0
20
40
60
I
80
100 x1
Figure 5. Mutual solubility for cyclohexane-methanol. Experimental data: Ssrensen and Arlt (1979).
binary VLE data it is the temperature (T), the pressure (P),the liquid mole fraction of component 1 (xJ, and the vapor mole fraction of component 1 (yl). The mole fractions for component 2 are not included, as they are obtained directly from the mole fractions of component 1. For binary excess enthalpy data, the measured properties are T, xl, and hE. The deviations are normalized relative
2278 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 Table VI. Modified UNIFAC GrouDs" ~~
main group no. name 1 CHZ
alkanes
olefins
2
c=c
aromatic hydrocarbons
3
ACH
alcohols water ketones
4 5 6 7
OH CH3OH H20 CHzCO
aldehydes esters
8 9
CHO
ethers
10
CHZO
amines
11 12
NH2 CHzNH
13
CHzN
pyridines
14 15
ANHZ pyridine
nitriles
16
CHzCN
acids chlorinated hydrocarbons
17 18
COOH CCI
19
CClZ
20
CC13
21
CCl,
ccoo
subgroup no. name 1 CHn 2 CH, 3 CH 4 c 5 CHZ.---CH 6 CH=CH 7 CH2=C 8 CH=C c=c 9 10 ACH 11 AC 12 OH 13 CH3OH 14 H20 15 CH3C0 16 CHZCO 17 CHO 18 CH3CO0 19 CHzCOO 20 CH30 21 CHzO 22 CHO 23 FCHZO 24 NHZ 25 CH3NH 26 CHzNH 27 CHNH 28 CH3N 29 CHzN 30 ANHz 31 CSH5N 32 CbHdN 33 CSHSN 34 CH3CN 35 CHZCN 36 COOH 37 CHzCl 38 CHCl 39 cc1 40 CH2C12 41 CHC12 42 CClz 43 CHC13 44 CC13 cc1, 45
structural parameters R (2/2)Q 0.9011 0.848 0.6744 0.540 0.228 0.4469 0.000 0.2195 1.176 1.3454 0.867 1.1168 0.985 1.1173 0.676 0.8887 0.485 0.6606 0.400 0.5313 0.120 0.3652 1.200 1.oOoo 1.000 1.0000 1.400 0.9200 1.488 1.6724 1.488 1.4457 0.948 0.9980 1.728 1.9031 1.420 1.6764 0.900 1.1450 0.780 0.9183 0.650 0.6908 1.100 0.9183 1.150 0.6948 1.050 1.4337 0.936 1.2070 0.624 0.9795 0.940 1.1865 0.632 0.9597 1.400 0.6948 2.113 2.9993 1.833 2.8332 1.553 2.6670 1.724 1.8701 1.416 1.6434 1.224 1.3013 1.264 1.4654 0.952 1.2380 0.724 1.0060 1.988 2.2564 1.684 2.0606 1.448 1.8016 2.410 2.8700 2.184 2.6401 2.910 3.3900
subgroup assignment examples 2,2,4-trimethylpente: 5(1), 1(2), 1(3), l(4)
isoprene: 1(1), 1(5), l(7)
toluene: 1(1), 5(10), l(11) ethanol: 1(1), 1(2), l(12) methanol: l(13) water: l(14) 2-butanone: 1(1), 1(2), l(15) acetaldehyde: 1(1), l(17) acetic acid ethyl ester: 1(1), 1(2), l(18) methyl ethyl ether: 1(1), 1(2), l(20) tetrahydrofuran: 3(2), l(23) isopropylamine: 2(1), 1(3), l(24) diethylamine: 2(1), 1(2), l(26) triethylamine: 3(1), 2(2), l(29) aniline: 5(10), 1(11), l(30) 2-methylpyridine: 1(1), l(32) acetonitrile: l(34) propionitrile: 1(1), l(35) acetic acid: 1(1), l(36) chloroethane: 1(1), l(37) 1,l-dichloroethane: 1(l), l ( 4 1) 1,l,I-trichloroethane: 1(1), l(44) tetrachloromethane: l(45)
"Main group and subgroup definitions in Modified UNIFAC and structural parameters for subgroups. In examples of group assignments for the molecules, the number in parentheses indicates the identification number of the subgroup and the number in front of the parentheses indicates the number present of the specific subgroup; e.g., 5(1) means that five methyl groups (no. 1)are present in the molecule.
to a preset standard deviation (a), and the resulting expression for the objective function is then n m
In eq 25, X is a measured quantity, m is summed over the number of measured variables in a data point, and n is summed over the total number of data points. Usually the following standard deviations were used in the initial calculation of parameters: uy = 0.005, u p = 1.0 mmHg, and UH = 5.0 J/mol. The optimization algorithm applied in the parameter estimation program is the Levenberg-Marquard algorithm (see, e.g., Fletcher (1980)). Up to six coefficients can totally be determined for the two temperature-dependent interaction parameters characterizing a binary group combination. The actual number of optimized coefficients depends on the amount of experimental information available. In general, if only VLE information was used in the parameter estimation, the interaction parameters were assumed to be independent
of temperature, i.e., only the first coefficient was optimized for each interaction. If excess enthalpy data at one temperature were included, the second coefficients could also be established, and if excess enthalpy information at different temperatures were available, the third coefficients could also be determined. Although it is possible to determine all coefficients from VLE information at three different temperatures, they will because the obin general give poor predictions of HE, tained temperature variation of GE is highly influenced by inaccuracies in the experimental VLE data. Predictions of VLE with parameters, which have been obtained from HEdata alone, will also in general give poor results. The resulting group volume and surface area constants and group-interaction parameters for 21 different main groups are given in Tables VI and VII. It should be noted that the quantity ( z / 2 ) Q in Table VI (see eq 4) for all practical purposes is completely equivalent to the quantity Q in the original UNIFAC model.
Predictions with Modified UNIFAC Compounds presently described by Modified UNIFAC
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2279 parameters include hydrocarbons (alkanes, olefins, aromates), alcohols, water, ketones and aldehydes, ethers, esters, carboxylic acids, amines, pyridines, nitriles, and chlorinated hydrocarbons. Hydrocarbons. The hydrocarbon main groups are CH2 (alkanes), C=C (olefins), and ACH (aromates). The special ACCHz (alkyl aromate) main group in UNIFAC has been avoided, and alkyl aromates are described by the CH2 and ACH main groups. Alcohols. Methanol is described by the structural group CH,OH, and other alcohols are described by the OH structural group. The structural parameters R and ( z / 2 ) Qfor methanol were determined by fitting interaction parameters to binary VLE data of methanol with cyclohexane (CH,), benzene (ACH), and tetrachloromethane (CC14),using different values of R and Q for methanol. It was found that R = 1.0 and ( z / 2 ) Q = 1.0 gave the best result. Ketones and Aldehydes. The ketones are described by one main group with two structural subgroups, the CH3C0 and the CH2C0 subgroups. It was from correlations of experimental data found that increasing the ( z / 2 ) Q value for CH2C0 from 1.18 (the value of Q for the group in UNIFAC) to 1.488 (the same as for CH,CO) gave a better overall representation of ketone systems. Aldehydes are described by one main group. Experimental data with aldehydes are scarce in the literature, and the available data are often of poor quality. Therefore, a number of the aldehyde group interaction parameters have not been established at all. The aldehyde interaction parameters for Modified UNIFAC have been established by Schmelzer (1984). Ethers. The ethers are a large group of compounds, which are described by a single main group in Modified UNIFAC. The same main group is used to describe acyclic ethers (e.g.,diethyl ether) as well as cyclic ethers (dioxanes, tetrahydrofurans, epoxides). As in UNIFAC, four subgroups are used for description of ethers, i.e., CH30, CH20,CHO, and a special subgroup FCHzO for the ether group in tetrahydrofuran. It was found that by varying the values of ( z / 2 ) Q for the subgroups, some improvement could be obtained in the representation of experimental data using only one ether main group. Esters. It was attempted to describe the different esters by a single COO group. It was, however, found that this group could not give results that were as good as the CH2CO0 group, used in UNIFAC, even when the ( z / 2 ) Q parameter was optimized. The COOCH2 group was also investigated as a possibility but was not found to give as good a representation of the different ester systems as CH2CO0. The CH2CO0 main group has therefore been retained in Modified UNIFAC. Two ester subgroups, CH3CO0 and CH2CO0,describe most of the commonly encountered esters. The most important exceptions are formates, acrylates, and benzoates. In UNIFAC, one special main group is used for formates, and a special COO main group is used for other esters, which cannot be represented by the above two subgroups. Amines and Pyridines. Three main groups are used for aliphatic amines and one for aromatic amines. Primary aliphatic amines are described by the NH2 group, but for the secondary and tertiary amines, it was found necessary to include a neighboring methyl or methylene group into the group definition. The main groups describing secondary and tertiary amines are CH2NH and CH2N. These groups are also used in the case of N-substituted anilines, while for the description of aniline and ring-substituted
anilines the ANH2 main group is used. The pyridines are described by a single main group. Nitriles. As in UNIFAC, it has been found necessary to include a CH2group in the main group definition. Often parameters for nitriles are based on experimental data for systems with acetonitrile, as these are far more common in the literature than other nitrile systems. Chlorinated Hydrocarbons. Four main groups are used for description of chlorinated aliphatic compounds. The group definitions are as in UNIFAC, i.e., CC1, CC12, CCl,, and CC14according to the number of chlorine atoms on the saturated carbon atom. Although a careful study may reveal that good results can be obtained with less main groups, if, e.g., the C1 group is used with different R and Q values in different compounds, this has not been attempted here. Carboxylic Acids. The carboxylic acids are described with the COOH main group. In UNIFAC, formic acid is included among other carboxylic acids in the same main group. It was, however, found from comparison with experimental information that predictions for formic acid with UNIFAC are generally of poor quality. As changing R and Q for HCOOH did not lead to improved results for Modified UNIFAC, HCOOH is not included within the COOH main group. Water. Many systems with water are difficult to describe with UNIFAC as well as with Modified UNIFAC, due to large nonidealities present. Some of the water interaction parameters were determined from liquid-liquid equilibrium data.
Representation of VLE and HEUsing Modified UNIFAC This section shows the results when Modified UNIFAC is used for the prediction of activity coefficients in VLE and for the prediction of HE. In all cases, the system pressures are low to moderate. Hence, for VLE, the gas phase is assumed to be ideal except for systems containing carboxylic acids. In this case, vapor nonidealities are taken into account by chemical theory as described in Fredenslund et al. (1977). The large amount of experimental VLE and HE data available in the data bank have also been used for a comparison with results calculated by UNIFAC and by Modified UNIFAC. The overall results of this extensive comparison are summarized in Table VI11 for VLE and Table IX for HE. In addition to the overall results, Table VI11 and Table IX show how different classes of systems are represented. The systems are classified as (1) “simple” binary, (2) “complex”binary, or (3) multicomponent, according to the presence of complex components and the total number of components. A complex compound is here a compound containing at least two different main groups others than the hydrocarbon main groups (CH2,C=C, ACH, or (for UNIFAC only) ACCH2). As an example, ethoxyethanol is considered as complex, but 1,2-diethoxyethane is not. No complex compounds can be present in simple binary systems, but a complex binary system will contain at least one complex compound. The distinction between simple binary systems, complex binary systems, and multicomponent systems is useful. Table VI11 shows that VLE for the simple binary systems and the multicomponent systems is well represented by both models, while the representation of complex binary systems is not quite as good. There is a general, but small, improvement from UNIFAC to Modified UNIFAC in the representation of VLE. The mean absolute deviations between experimental and calculated vapor-phase mole
2280 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 Table VII. Modified UNIFAC Interaction Parameters (a,.)
i i 1
CH,
2
C=C
3
ACH
4
OH
5
CHBOH
6
HzO
7
CH&O
8
CHO
9
CCOO
10
CHzO
11
NH,
12
CHzNH
13
CHzN
14
ANH,
1.5
pyridine
16
CH&N
17
COOH
18
CCI
19
CCl*
20
CC13
21
CC1,
1
0.0 0.0 0.0 -46.45 -0.1817 -0.4888 -1.447 -0.5638O -1.612 637.5 -5.832 -0.8703 16.25 -0.3005 0.6924 410.7 2.868 9.000 71.93 -0.7960 -2.916 313.5 -4.064 0.0 44.43 -0.9718 0.5518 369.9 -1.542 -3.228 346.5 1.595 0.0 149.5 1.336 0.0 -64.36 -0.1736 1.135 680.5 -5.470 0.0 -52.03 -0.5553 0.0 21.69 -1.226 0.0 171.5 -1.463 0.6759 -67.33 -0.6791 2.036 12.87 0.2650 0.0 -35.46 -0.1233 1.134 27.88 -0.1656 -0.6087
2 76.46 -0.1834 -0.3659 0.0 0.0 0.0 -0.2772" -0.7129" -0.3407 794.7 0.0 0.0 -6.808 0.0 0.0 564.4 0.0 0.0 -144.3 0.0 0.0 161.8 0.0 0.0 200.3 0.0 0.0 -17.23 -1.648 0.0 454.9 0.0 0.0 17.42 0.0 0.0 28.08 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -64.53 0.0 0.0 227.3 0.0 0.0 340.3 0.0 0.0 48.61 -1.484 0.0 38.70 0.1094 0.1165 -95.03 -0.6923 0.0
3 62.88 -0.2493 1.103 35.07 -0.8042" -0.3761 0.0 0.0 0.0 587.3 4.6787 9.000 10.97 -0.7308" 0.4966 736.7 1.965 0.0 92.19 0.6129 -8.963 125.4 -3.133 0.0 8.346 -0.5254 0.0 125.2 -1.093 0.5898 902.7 -5.763 0.0 188.9 0.9741" 8.732 -95.46 1.292 0.0 334.3 -1.655 0.0 -62.93 -0.1398 -0.9703 -3.280 -0.5814 -0.7119 62.32 0.0 0.0 -39.67 -1.457 -0.7822" 240.7 -0.1833 0.0 210.5 -0.2965 0.0 80.23 0.3871 1.830
"Value is actually times lo-'. Example: q5,, = -0.1261
4 972.8 0.2687 8.773 633.5 0.0 0.0 712.6 -1.459 9.000 0.0 0.0 0.0 66.34 -0.5845 0.0 -47.15 -0.4947 8.650 179.6 -1.285 -4.007 2553.0 0.0 0.0 266.9 -1.054 3.586 137.1 -1.115 -4.438 -173.7 1.642 0.0 -233.9 1.737 0.0 -287.6 0.3310 -1.907 170.3 0.0 0.0 28.72 -0.2570 9.000 291.1 -0.2758 0.0 -92.21 0.0 0.0 818.2 -4.270 -2.607 716.6 0.0 0.0 708.6 -2.613 7.771 918.5 -2.045 9.000 X
5 1318.0 -0.1261" 9.000 1155.0 0.0 0.0 979.8 -1.793 3.844 29.50 0.4043 0.0 0.0 0.0 0.0 265.5 3.540 8.421 263.3 -0.1553 1.768 -274.0 0.0 0.0
394.0 -0.5610 -0.1005 295.2 -0.2191 3.441 -297.4 0.8296 0.0 -440.5 -0.8291' -2.128 -440.2 0.3315 -2.960 170.1 1.738 0.0 -283.6 0.6258 0.0 367.1 -0.7673 0.0 757.2 1.502 0.0 892.3 -2.420 -1.140 947.5 -3.570 0.0 1029.0 -4.307 -8.902 1273.0 -1.618 9.000
6 1857.0 -3.322 -9.000 1049.0 -3.305 0.0 1055.0 -2.968 9.854 155.6 0.3761 -9.000 -75.41 -0.7570 -4.745 0.0 0.0 0.0 272.4 -1.842 0.3303 9999.0 9999.0 9999.0 245.0 -0.717" 2.754 183.1 -2.507 0.0 -244.5 0.2857 0.0 -342.4 2.640 13.09 -265.5 0.0 0.0 498.3 0.0 0.0 -58.20 1.231 1.509 233.6 -1.448 0.6395 86.44 0.9941 -12.74 862.1 -2.637 0.0
856.5 -2.549 0.0 837.8 0.0 0.0 1323.0 0.0 0.0
7 414.0 -0.5165 1.803 577.5 0.0 0.0 87.64 -0.4616 6.691 161.0 0.7501 9.000 -29.40 -0.7294 -1.670 40.20 1.668 -1.994 0.0 0.0 0.0 -53.04 -0.6270 0.0 43.65 0.1905 0.0 160.4 0.5484 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -330.7 -0.1526 0.0 -48.01 0.0 0.0 -249.4 -0.2123 0.0 -151.7 0.0 0.0 -93.46 0.6780 0.0 409.5 -0.3821 0.7678 245.2 -2.128 4.221 378.7 -0.4666 -2.138
8 721.5 -1.470 0.0 320.4 0.0 0.0 215.1 1.936 0.0 -325.2 0.0 0.0 177.2 0.0 0.0 9999.0 9999.0 9999.0 76.10 0.9203 0.0 0.0 0.0 0.0 241.7 0.0 0.0 -17.53 -0.71 22 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -15.23 1.532 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0
9 329.1 -0.1518 -1.824 -24.65 0.0 0.0 97.30 0.1902 -0.7515 169.1 0.1902 4.625 -49.46 -0.7764 0.4687 218.0 -0.4269 -6.092 -11.93 -0.4056" 0.0 -133.6 0.0 0.0 0.0 0.0 0.0 -129.4 -0.4136" 0.0 9999.0 9999.0 9999.0 -129.3 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -210.3 0.1672 0.0 -224.6 -0.7234 0.0 9999.0 9999.0 9999.0 74.00 1.064 0.0 180.0 0.5186 -0.7971 121.4 0.0 0.0
10 230.5 -1.328 -2.476 321.6 4.551 0.0 82.86 0.6106 -0.7392 227.0 1.364 3.324 -73.54 -1.237 -2.308 19.54 1.293 -8.850 -48.00 -0.5097 0.0 220.4 1.738 0.0 277.0 0.3255 0.0 0.0 0.0 0.0 9999.0 9999.0 9999.0 92.97 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -248.1 0.0 0.0 154.6 0.0 0.0 -58.45 1.549 0.0 127.5 2.681 0.0 97.70 0.6533 1.000
lo-'.
fractions (dev y) and the mean absolute deviations between experimental and calculated pressures, relative to the experimental pressure (dev P),have been used as measures for the quality of predictions. I t is noted that while the total mean deviations for UNIFAC are 0.022 for the vapor-phase composition and 5.6% for pressure, they are for Modified UNIFAC decreased to 0.018 for vapor-phase composition and 4.3% for pressure. The improvement may be ascribed to the improved temperature dependence.
Table IX shows both for UNIFAC and Modified UNIFAC the representation of excess enthalpy data. The mean absolute deviations between calculated and experimental excess enthalpy (dev H)have been used as a measure of the accuracy of predictions. It is noted that there is a large improvement from UNIFAC with a mean deviation of 323 J/mol to Modified UNIFAC with a mean deviation of 132 J/mol. The results are, however, not nearly as good as those obtained by simultaneous correlation of VLE and
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2281
i 12
13
14
248.0 -1.800 0.9722 223.9 0.0 0.0 29.25 -0.1847 -2.193 -199.9 -0.4746 0.0 -201.7 3.930 0.0 111.5 -3.302 9.347 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 312.5 0.0 0.0 13.40 -0.3964 0.0 0.0 0.0
217.7 -0.1509 1.117 54.21 0.0 0.0 88.03 -1.130 0.0 196.8 3.925 0.0 79.02 2.560 0.0 -15.80 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 0.0 0.0 0.0
580.8 -2.310 -16.05 9999.0 9999.0 9999.0 307.6 -0.9447 -8.767 -58.27 0.0 0.0 79.90 -2.152 0.0 -193.6 0.0 0.0 798.9 0.3800 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 0.0 0.0 0.0 9999.0 9999.0 9999.0 380.8 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 384.0 -1.957 0.0
11
420.7 -2.256 0.0 243.8 0.0 0.0 72.60 -0.4299 0.0 -176.5 -0.1073 -1.016 -182.9 1.257 0.0 -66.39 -1.053 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -129.3 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 337.4 0.0 0.0
0.0 0.0 0.0 9999.0 9999.0 9999.0 212.5 0.1049 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 168.9 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -131.7 0.0 0.0 -77.13 6.007 0.0 240.4 0.0 0.0
15 273.8 0.1763 0.0 9999.0 9999.0 9999.0 99.33 0.2329 1.530 311.8 2.405 9.000 491.8 -2.773 0.0 472.3 1.336 1.756 170.0 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 612.1 6.987 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 0.0 0.0 0.0 23.65 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 367.5 0.0 0.0 -37.94 -0.2759 0.0 323.6 -0.3216" 0.0
16
17
559.0 0.4537 0.0 294.4 0.0 0.0 198.3 0.9601 1.398 77.89 -0.4333 0.0 -6.177 -0.3781 0.0 338.4 1.900 -0.2065 387.5 0.4437" 0.0 9999.0 9999.0 9999.0 346.5 0.1126 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -187.1 0.0 0.0 64.57 0.0 0.0 0.0 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 126.0 0.7970 0.0 495.2 0.7697 1.477
664.1 1.317 -4.904 186.0 0.0 0.0 537.4 0.0 0.0 61.78 0.0 0.0 -321.2 -1.116 0.0 8.621 -1.709 6.413 230.0 0.0 0.0 9999.0 9999.0 9999.0 557.9 1.377 0.0 286.6 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 0.0 0.0 0.0 447.8 0.0 0.0 617.3 0.0 0.0 455.6 1.316 0.0 500.3 1.533 0.0
HEwith Modified UNIQUAC. The results indicate that the excess enthalpy is more difficult to represent with a group-contribution method than VLE. The enthalpy deviations (dev H) have for each data set been normalized. The values of dev H have thus been divided by the difference between the largest and smallest experimental value of HEin the data set (including the trivial HE= 0 for pure compounds). Table IX shows that the predictions in general do not give an accurate de-
18 264.3 0.2579 -0.4138 -135.3 0.0 0.0 49.15 1.656 -2.346 194.7 0.5463 0.0 -45.13 -0.1259 0.1048 527.0 1.416 0.0 194.4 -0.9735 0.0 227.3 -3.680 0.0 9999.0 9999.0 9999.0 156.2 0.0 0.0 571.8 0.0 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 113.7 0.0 0.0 0.0 0.0 0.0 108.1 0.0 0.0 145.1 0.0 0.0 35.10 -0.1413 0.0
19
20
101.2 -0.8471 0.0 -15.95 1.216 0.0 -195.1 -0.1411 0.0 230.7 -3.591 0.0 -88.72 -0.1812" -3.794 596.4 3.071 0.0 -285.7 -0.151 5" -1.678 9999.0 9999.0 9999.0 -142.8 -0.6688 -0.2909 -64.57 -0.6615 0.0 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -207.7 0.0 0.0 9999.0 9999.0 9999.0 -272.0 0.0 0.0 9999.0 9999.0 9999.0 -73.88 0.0 0.0 -95.08 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 141.9 0.2597 0.0
103.1 -0.1245 -1.818 -26.14 0.1241 0.0 -181.6 0.1135 0.0 112.8 1.955 8.077 -138.6 0.4331 -0.2909 674.5 0.0 0.0 -252.0 1.399 -6.332 9999.0 9999.0 9999.0 -222.5 0.2479 1.833 -241.8 -0.4315 -0.4 100 9999.0 9999.0 9999.0 9999.0 9999.0 9999.0 -403.9 -0.4903" 0.0 9999.0 9999.0 9999.0 -119.9 0.8788 0.0 -81.43 0.0 0.0 56.26 -1.041 0.0 -106.6 0.0 0.0 0.0 0.0 0.0 090 0.0 0.0 47.17 -0.5779" -0.1304
21 -12.65 0.4512" 0.3362 148.0 0.9241 0.0 -57.83 -0.3507 -1.284 415.3 1.391 9.600 -36.75 -0.1984" 0.8396 705.5 2.540 0.0 -49.29 0.4365 4.367 9999.0 9999.0 9999.0 46.32 0.0 0.0 188.7 -1.081 1.659 42.36 0.0 0.0 -14.72 0.0 0.0 -248.3 0.0 0.0 809.9 -4.505 0.0 -162.7 -0.2071" 0.0 -61.77 -0.4319 0.3858 148.3 -0.7082 0.0 42.60 0.0 0.0 -50.42 -0.4592 0.0 -24.81 -0.2760" 0.6462" 0.0 0.0 0.0
scription of the excess enthalpy curves. The difficulties in representation of complex systems, both with respect to VLE and HE, are not surprising. It is to be expected, because a simple group-contribution model like UNIFAC or Modified UNIFAC is not able to account for proximity effects, i.e., the influence on a group from nearby groups on the same molecule. These effects can influence the properties of a functional group significantly, especially when conjugation effects or intramo-
2282 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 Table VIII. Overall Representation of Vapor-Liquid Equilibrium Data by UNIFAC and Modified UNIFAC" systems data sets measd y measd P dev y dev P, % total UNIFAC 7749 84318 115326 0.0222 5.6 Mod. 6293 68944 93951 0.0184 4.3 UNIFAC simple binary UNIFAC 6960 66002 94930 0.0207 5.3 Mod. 5637 54028 77119 0.0170 4.1 UNIFAC complex binary UNIFAC 163 1626 2032 0.0349 14.1 Mod. 144 1386 1822 0.0260 10.1 UNIFAC multicomponent UNIFAC 626 16690 18364 0.0270 6.2 Mod. 512 13530 15010 0.0230 4.3 UNIFAC Dev y is the mean absolute deviation for vapor-phase composition and dev P is the mean absolute deviation for pressures relative to experimental values.
Table IX. Overall Representation of Excess Enthalpy Data by UNIFAC and Modified UNIFAC" dev H, norm systems data sets measd pts J/mol dev, % total 68.5 UNIFAC 3434 50717 322.6 33.8 2915 44285 132.2 Mod. UNIFAC simple binary 71.5 3333 46780 322.5 UNIFAC 35.2 2780 40453 127.6 Mod. UNIFAC complex binary 59.2 UNIFAC 5a 701 365.3 56 679 348.6 48.8 Mod. UNIFAC multicomponent IJNIFAC a2 3 236 314.1 25.8 Mod. UNIFAC 79 3 153 145.2 12.9 a The deviations are given as mean absolute deviations (dev H ) and as normalized deviations (norm dev), where dev H for each data set has been divided with the experimental range for the excess enthalpy (including the trivial HE = 0 for pure compounds).
lecular hydrogen bonds are present.
Representation of VLE and HEwithin Specific Systems From the results in Tables VI11 and IX, it is evident that
Modified UNIFAC both for VLE and excess enthalpy gives better predictions than UNIFAC. The numbers in these tables can, however, only be regarded as rough indications of the accuracies obtained for specific groups of systems, as there may be large variations hidden in the numbers. A more detailed picture may be obtained from Table X, which shows the representation of the different simple compound types described by Modified UNIFAC. From the tabulation it can be seen which compounds are generally well represented by the model and for which compounds one may expect difficulties. It may be observed that for VLE systems containing water, acids, or organic bases, the representation is relatively poor, while it for most other simple systems is good. Predictions of excess enthalpy seem especially to cause problems for systems containing ethers and amines. The results from Table X are given more detailed in Tables XI and XII. Here the mean representation of VLE and HEfor each system group is tabulated for Modified UNIFAC. Two different systems belong to the same system group, when they contain the same combination of component types from Table X (e.g., acetone-ethanol and 2-butanone-1-propanol are two systems within the same system group, i.e., the ketone-alcohol systems). Tables XI and XI1 may be used to get some indication of the representation within a specific system group. As an example, from Table XI it can be seen that the mean deviations in representation of VLE for the three alkane-dichlorinated hydrocarbon systems [CH2/CC12] found in the data bank are 0.008 in vapor-phase composition and 2.1 '% in relative pressure deviation. Similarly, the mean deviations for eight secondary amine-water systems in the VLE data bank are 0.022 for vapor-phase composition and 13.870 for relative pressure deviation. From Table XII, it can, as an example, be seen that for 31 different ether-aromatic hydrocarbon systems the mean deviation is 140 J/mol, and on the average the mean deviations are 102% of the measured experimental range for excess enthalpy within the data sets. The absolute error for the predictions is thus not very large, but the relative error is large. The opposite is the case for the nine secondary amine-alcohol systems. Here the mean absolute deviation is 315 J/mol, but relative to the experimental range of data within a data set this is only 13%. A number of system groups are only repesented by one or two systems in the data banks. It is for these system
Table X. Representation of Experimental Data with Modified IJNIFAC: Simple Binary Systems vapor-liquid equilibria excess enthalpy main group data sets dev y dev P, % data sets dev H, J/mol 1717 0.014 3.5 1239 114.0 1 CHZ 94.4 386 0.020 3.5 123 2 c=c 1557 0.013 3.2 817 72.4 3 ACH 4.3 684 125.3 1678 0.020 4 OH 507 0.019 4.6 150 111.9 5 CH,OH 6.7 218 203.3 773 0.025 6 H20 0.016 3.5 226 119.4 690 1 CHZCO 0.032 6.7 15 265.8 94 a CHO 3.8 122 71.0 9 ccoo 509 0.015 10 CHZO 508 0.020 4.6 298 278.3 145.7 11 NHZ 154 0.022 4.4 97 5.7 59 216.2 12 CHZNH 196 0.009 13 CHZN 126 0.024 8.4 113 210.7 14 ANH, 126 0.020 5.1 26 91.8 5.7 152 147.1 15 pyridines 196 0.019 16 CHZCN 119 0.019 2.6 68 187.7 138.0 415 0.022 6.6 45 1: COOH 3.6 71 77.0 18 cc1 191 0.014 247.8 19 cc1, 56 0.016 3.0 61 283.2 20 cc1, 206 0.011 3.0 174 2.6 284 63.3 21 cc1, 403 0.010 c
norm dev, % 30.7 80.1 36.0 21.2 15.6 21.3 26.6 23.1 8.3 50.6 10.1 20.0 69.4 11.4 39.0 23.2 14.2 16.2 32.7 60.5 49.4
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2283
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2284 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987
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Ind. Eng. Chem. Ree., Vol. 26, No. 11, 1987 2285 525
,
I
Work on the Modified UN'FAC model is continuing in collaboration with the University of Dortmund. Acknowledgment
425
We are grateful to Dr. J. Gmehling of the University of Dortmund for the use of the Dortmund Data Bank. We are grateful to DECHEMA and the Danish Council for Scientific and Technical Research for financial assistance.
0
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Nomenclature
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Unifac VLE Experimental Data
Figure 6. Mutual solubilities for 2-butanone(l)-water(2): (-) Modified UNIFAC, ( - - - ) UNIFAC VLE, experimental data.
(..e)
Aw = van der Waals surface area a, = interaction parameter for i-j interaction Ac, = excess heat capacity of i-j interaction E - Gibbs function, molar excess E= Gibbs function, molar, combinatorial = Gibbs function, molar, residual = Gibbs function, of interaction for 1-1 interaction excess enthalpy hE = molar excess enthalpy Ah = enthalpy of i-j interaction LLk = liquid-liquid equilibria P = pressure p a t = saturation pressure of component i Qk = surface area parameter, for group k q, = molecular surface area parameter, for component i R = universal gas constant Rk = volume parameter, for group k r, = molecular volume parameter, for component i T = temperature To = temperature, reference VLE = vapor-liquid equilibria Vw = van der Waals volume XnmexPtl= n, the variable for data point m, experimental XnmCalCd = n, the variable for data point m, calculated x , = mole fraction for component i (liquid) y l = mole fraction of component i (vapor phase) z = lattice coordination number Greek Symbols r k = residual group activity coefficient for group k ri = residual group activity coefficient, in pure component i yl = activity coefficient for component i yLc= activity coefficient, combinatorial part y: = activity coefficient, residual part Bt = surface area fraction, for i in mixture e,, = surface area fraction, local for j around i u = standard deviation for experimental measurement (or inverse weight factor) T,, = Boltzmann factor a, = volume fraction for component i in mixture w, = modified volume fraction of component i in mixture vk, = number of groups k present in molecule i
UNIFAC LLE, ( 0 )
groups uncertain how good the predictions will be for other systems within the same group. As an example, the ketone-nitrile interaction parameters are based on only one system (acetone-acetonitrile), and it is uncertain how well 4-methyl-2-pentanone-propionitrile is represented. On the other hand, there are many system groups which are represented by several systems, and in most cases the predictions are of reasonable quality, indicating that the group-contribution concept is a good approximation. Further notes and examples concerning predictions of VLE and HEare given by Larsen (1986). Prediction of LLE Magnussen et al. (1981) have published a separate UNIFAC group-interaction parameter table which may be used to predict LLE a t (or near) 25 "C. UNIFAC and Modified UNIFAC with parameters based on mainly VLE data may not be used to quantitatively predict LLE. An example of this is shown for the 2-butanone-water system in Figure 6. It may be noted that UNIFAC LLE performs well in this case near 25 OC, but the temperature-effect is poorly predicted. UNIFAC VLE has not even a correct qualitative picture of the two-phase LLE region. Modified UNIFAC is, due to the improved temperature dependence, able to give a better qualitative solubility curve. However, quantitatively the Modified UNIFAC LLE predictions are not satisfactory. More examples of LLE predictions with Modified UNIFAC are given by Larsen (1986), and in general Modified UNIFAC seems to give as good predictions as UNIFAC LLE. Conclusions The Modified UNIFAC model yields somewhat better predictions of VLE than does UNIFAC. Due to the introduction of temperature-dependent parameters, the predictions of HEare much improved. This means that accurate predictions of VLE may be obtained over a broader temperature range than is possible with UNIFAC. Gupte (1986) has applied Modified UNIFAC for predictions a t elevated temperatures and pressures with good results.
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Literature Cited Abrams, D. S.; Prausnitz, J. M. AZChE J. 1975, 21, 16. Belousov, W. P.; Morachevski, A. G. Temp1 Smech. Zh. Khim. (Leningrad) 1970, 1. Christensen, C.; Gmehling, J.; Rasmussen, P.; Weidlich, U. DECHEMA Chemistry Data Series; DECHEMA Frankfurt/M., 1984; Vol. 111, Parts 1 and 2. Fernandez-Garcia, J. G.; Guillamin, M.; Boissonnas, C. G. Helu. Chim. Acta 1968,51, 1733. Fletcher, R. Practical Methods of Optimization; Wiley: New York, 1980. Fredenslund, Aa.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria using UNZFAC; Elsevier: New York, 1977. Fredenslund, Aa.; Rasmussen, P. Fluid Phase Equilib. 1985,24,115. Gmehling, J.; Onken, U. DECHEMA Chemistry Data Series; DECHEMA: Frankfurt/M., 1977; Vol. I. Gmehling, J.; Rasmussen, P.; Fredenslund, Aa. Znd. Eng. Chem. Process Des. Deu. 1982,21, 118. Goral, M.; Orazc, P.; Warycha, S. Proceedings of the 5th Conference on Mixtures of Nonelectrolytes, Halle, 1983.
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Gupta, P. A.; Rasmussen, P.; Fredenslund, Aa. Ind. Eng. Chem. Fundam. 1986,25,636. Kikic, I.; Alessi, P.; Rasmussen, P.; Fredenslund, Aa. Can. J. Chem. Eng. 1980,58,253. Larsen, B. L. Ph.D. Thesis, Instituttet for Kemiteknik, The Technical University of Denmark, Denmark, 1986. Macedo, E. A.; Weidlich, U.; Gmehling, J.; Rasmussen, P. Ind. Eng. Chem. Process Des. Deu. 1983,22,676. Madhavan, S.; Murti, P. S. Chem. Eng. Sci. 1966,22,465. Magnussen, T.; Rasmussen, P.; Fredenslund, Aa. Ind. Eng. Chem. Process Des. Dev. 1981,20,331. Marsh, K. N., I.D.S ( A ) , 1973,1-4. Marsh, K. N.; Ott, J. B.; Costigan, M. J. J. Chem. Thermodyn. 1980a,12,343. Marsh, K. N.; Ott, J. B.; Richardo, A. E. J . Chem. Thermodyn. 1980b,12, 897. McGlashan, M. L.; Williamson, A. G. Trans. Faraday SOC.1961,57, 588. Morachewki, Komarova Vestn. Leningr. Univ., Fiz., Khim. 1957,12 (I), 118.
Myers, H. S. Pet. Refiner. 1957,36,175. Ott, J. B.;Marsh, K. N.; Stokes, R. H. J. Chem. Thermodyn. 1981, 13, 371. Ridgway, K.; Butler, P. A. J. Chem. Eng. Data 1967,12,509. Savini, C.G.; Wintherthaler, D. R.; Van Ness, H. C. J . Chem. Eng. Data 1965,10,171. Sayegh, S. G.; Vera, J. H. Chem. Eng. J . 1980,'19, 1. Schmelzer, J., Report, Instituttet for Kemiteknik, Technical University of Denmark, Denmark, 1984. Skjold-Jerrgeneen, S. Fluid Phase Equilib. 1983,13,273. Susarev, M. P.; Chen, S. T. Zh. Fiz. Khim. 1963,37,1739. Smensen, J. M.;Arlt, W. DECHEMA Chemistry Data Series; DECHEMA: Frankfurt/M., 1979; Vol. V, Parts 1-3. Tiegs, D.; Rasmussen, P.; Gmehling, J.; Fredenslund, Aa. Ind. Eng. Chem. Res. 1987,26,159. Wolff, H.; Hoeppel, H. E. Ber. Bunsen-Ges. Phys. Chem. 1968,72, 710.
Received for review May 12, 1986 Accepted June 23, 1987
Mass Transfer Accompanied by First-Order Chemical Reaction for Turbulent Duct Flow Owen T. Hanna,* Orville C. Sandall, and Creighton L. Wilson Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, California 93106
A large-Schmidt-number asymptotic approximation procedure is employed to derive an equation which represents mass transfer accompanied by a first-order chemical reaction for liquids in fully developed turbulent flow in a circular tube. However, since the concentration boundary layer is very thin, the results obtained should also apply to the parallel-plate and concentric-annulus geometries when proper scaling is employed. A modified Van Driest formula for the eddy diffusivity is used to model the turbulent transport. The asymptotic analysis leads to a simple nonlinear algebralc equation for the enhancement factor which involves only one dimensionless reaction rate parameter. The predictions of this equation agree within a maximum deviation of 1.7% with numerical solutions of the differential equation describing the process. A comparison is made with earlier studies of this problem. The objective of the research reported here is the development of an analytical relationship for the rate of mass transfer accompanied by homogeneous chemical reaction for turbulent flow in a circular duct. Several authors have previously considered the problem of mass transfer with homogeneous reaction in turbulent flow. Veith et al. (1963) used an eddy-diffusivity approach to model the turbulent transport and employed a finite difference numerical technique to solve the differential equation for mass transfer with first-order chemical reactions under fully developed conditions. These authors found that the enhancement factor, = k,/k$, could be considered to be essentially a function of only W J 2where , M is defined as
M = kRD/(k$)' (1) Marangozis et al. (1963) and Randhava and Wasan (1971) also utilized an eddy-diffusivity approach to theoretically describe mass transfer accompanied by chemical reaction in turbulent flow. Marangozis et al. derived an approximate analytical solution for the enhancement factor for the first-order reaction condition in the case where the eddy diffusivity is proportional to the distance from the wall. Randhava and Wasan obtained numerical results for developing concentration profiles by using a finite difference procedure. Randhava and Wasan considered cases of nonlinear kinetics but failed to account for the additional terms which, due to the nonlinear kinetics, arise 0888-5885/87/ 2626-2286$01.50/0
when carrying out the time averaging of the diffusion equation. Chung and Pang (1980) used a surface rejuvenation model to consider turbulent mass transfer accompanied by a first-order chemical reaction. These authors found that their numerical results for the enhancement factor were correlated by the parameter M . In the work reported here, we use an eddy-diffusivity model to describe the turbulent transport and consider the case of a first-order chemical reaction. The eddy-diffusivity expressed used is that proposed by Hanna et ai. (1981). This expression is the Van Driest (1955) model, modified in the region near the wall to give e+ proportional to (Y')~ as y+ 0 in accord with the large Sc mass-transfer results of Sandall and Hanna (1979). The Van Driest formula was modified so that the resulting expression agrees with the Notter and Sleicher (1971) equation for the eddy diffusivity in the limit as y+ 0. This modified Van Driest formula has been shown to give theoretical prediction8 in good agreement with experimental data for heat and mass transfer in turbulent pipe flow for the Pr or Sc range from 0.7 to 100000 (Sandall et al., 1980) and with experimental data for heat transfer across turbulent falling films of water (Sandall et al., 1984). The results presented here are obtained by integrating the basic differential equation describing the mass transfer utilizing a large Sc asymptotic procedure first given by
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0 1987 American Chemical Society
