levels of 52, 30, 15, and 9%. These results are given in the following table with the resulting starch graft cost per pound : AN-starch grafts
Annual operating costs, $1000’s AS-starch graft cost, %/lb
52%AN
30%AN
15%AN
9%AN
1600
1450
1350
1300
0 32
0 29
0 27
0 26
literature Cited
Cleland, It. L., Stockmayer, W. H., J . Poiymer S a . , 17, 473 (1955).
Fanta, G. F., Burr, R. C., Russell, C. R., Rist, C. E., J . A p p l . Polymer Scz., 10,929 (1966). Reyes, Zoila, unpublished work. Reyes, Zoila, Rist, C. E., Russell, C. It., J . Polymer S a . , Part A-1, 4, 1031-43 (1966). Schouteden, F. L. M., Xakromol. Chem., 2 4 , 2 5 (1957). RECEIIEDfor review 3Iarch 6, 1972 ACCLPTEDOctober 2, 1972 Presented at the Division of Carbohydrate Chemistry, 161st Meeting, rlCS, Los Angeles, Calif., Xarch 1971.
Effective local Compositions in Phase Equilibrium Correlations Jose M. Marina’ and Dimitrios P. Tassios Newark College of Engineering, nTewark, N . J . 0’7102
The ambiguity in choosing the proper value of cy in the NRTL equation is eliminated by replacing alpha by - 1 . For miscible systems, no loss in accuracy is observed when correlating binary vapor-liquid equilibrium (VLE) or predicting ternary behavior. For binary immiscible systems, better accuracies are obtained.
F o r a quantitative treatment of separation techniques, such as distillation and extraction, we must’ rely heavily on the phase equilibria relationships. There arises, therefore, the necessity of a functional expression between activity coefficient, temperature, and composition. Throughout the years several such expressions have been proposed. Among these, the equations of Van Laar (1913), lllargules (1895), and Wohl (1946) are probably the best known. More recently Wilson (1964), by introducing the concept of local mole fractions, developed a n expression which provides a very good represent,ation of miscible systems but fails to describe immiscible behavior. Later Renon and Prausnitz (1968) based on the concept of local mole fractions and Scott’s (1956) two-liquid theory, developed the nonrandom two-liquid ( S R T L ) equation, which describes with good accuracy miscible and immiscible systems. Both of these expressions seem to represent the data with a greater degree of accuracy than the previously proposed expressions. Since the Wilson equation contains only two parameters per binary system, it lends itself to the study of equilibrium from a pair of data points, such as in the case of azeotropic data. The XRTL equation, on the other hand, contains three parameters, thus requiring a minimum of three data points for their evaluation. Renon had some success in trying to overcome this problem by proposing rules for evaluation of the parameter CY from qualitative coilaiderations on the nature of the system and its components. However, a t times, these rules are ambiguous and difficult to apply. Our purpose in this work is to fill the gap between these two equatioiis, that is, to develop a true two-constant expres-
sion for miscible and immiscible systems. Such an expression would combine bhe advantages of both t’heWilson and S R T L expressions thus allowing not only prediction of vapor-liquid equilibrium from azeotropic data but also prediction of vaporliquid equilibrium (T’LE) from mutual solubility data and vice versa. The expression presented here is the result of an extensive study of the parameter a in the S R T L equation, indicating that the substitution a = - 1, not only yields the same degree of accuracy previously obtained for miscible systems, but improves data prediction for immiscible systems. NRTL Equation
Renon and Prausnitz (1968) modified Wilson’s equation for local mole fractions by introducing the constant CY to account for the nonrandom~iessof liquid solutions:
where S,j
=
local mole fraction of component i around a central molecule of j
gij
=
residual Gibbs energies
X1 and S2= mole fractioiis of components 1 and 2 LIaking use of this expression arid Scott’s (1956) two-liquid theory, they dereloped the N R T L equation, a three-parameter equation capable of describing VLE of miscible and immiscible systems with remarkable accuracy:
Present address, Merck & Co., Rahway, K.J. To whom correspondence should be addressed. Ind. Eng. Chem. Process Des. Develop., Vol. 1 2 , No. 1 , 1 9 7 3
67
Table 1. Comparison of l o c a l M o l e Fraction Equations LEMF
Acetone-chloroform a t 50°C Acetone-methanol a t 50°C Acetone-methyl acetate a t
50°C Acetone-toluene a t 45°C Acetonitrile-toluene a t 45OC Carbon tetrachloridebenzene a t 760 mm Hg Chloroform-methanol a t
50" C
-23
-16
-I2
-c3
-34
3
04
09
a
IC:
Figure 2. Standard deviation vs. alpha 0 Methanol-benzene at 35OC Carbon tetrachloride-nitromethane at 45'C Methanol-heptane at 7 6 0 rnm Hg 0 Ethanol-benzene a t 7 6 0 mm Hg
I
I n the above expression the differences ( g i j - gjj) are fitted to the data and the third constant, a , may eit'her be fitted to t'he data or evaluated a priori from the nature of the system.
n-Heptane-toluene a t 760 mm Hg Methanol-benzene a t 55°C Methanol-carbon tetrachloride a t 55'C Methanol-*heptane a t 760 mm Hg Methanol-toluene a t 760 m m Hg Methylethylketone-toluene at 760 mm Hg Nitroethane-toluene a t
45oc I-Propanol-ethyl acetate a t 60°C 1-Propanol-water a t 60°C 2-Propanol-ethyl acetate at 60°C
Standard deviation NRTL Wilson
0.0282 0.0323
0.0277 0.0329
0.0278 0.0327
0.0314 0,0024 0.0109
0,0314 0.0003 0.0221
0,0312 0.0011 0.0150
0.0040
0.0040
0.0040
0.0224
0.0210
0.0293
0.0355 0.0073
0.0357 0.0073
0.0357 0.0076
0.0593
0.0469
0.0777
0.0380
0,0426
0,0177
0.0134
0.0150
0,0148
0.0478
0,0481
0.0478
0,0054
0.0081
0.0063
0,0241 0.0544
0.0235 0.0589
0.0237 0,0588
0.0249
0.0245
0.0247
Value of Alpha
According to Renon, binary mixtures can be classified into seven different types from considerations on the polarity and self-association characteristics of the pure components and the value of the excess free energy of the mixture. Each of these misture types is associated with a value of a in the range 0.2-0.47, therefore the value of a follows from the type of binary mixture under consideration. When using these rules, however, one finds situations in which the same are ambiguous or lack specificity. Esamples of these are the binaries riitroethane-carbon tetrachloride and nitromethane-carbon tetrachloride. .Ilthough these systems are quite similar, one finds that the recommended values of a are 0.3 and 0.47, respectively. -4nother case in point is the partially miscible system water-ethy; acetate for which Renon and Prausnitz (1969) recommend a = 0.4, even though a value of a = 0.2 was originally recommended for these type of systems (Renon, 1966). Therefore, i t was decided to study the effect of the value of a on the accuracy of the correlation. For this purpose, 18 systems were chosen for study and a plot of a vs. the standard deviation of calculated activity coefficients was made. For each system, a was assigned values bebween -2.0 and SO.7, and the corresponding activity coefficients and standard deviation were calculated: Std dev =
[Xote that Figures 1 and 3-6 plus Table V have been deposited nith the -4CS llicrofilm Depository Service. See instructions a t end of paper ] Results representative of the 18 systems studied are presented in Figures 1-3 Figure 4 I' a n extended study of the 68
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1, 1973
system methanol-benzene covering a range of a values from - 10 to + l o . Data sources are presented in Table V. Although for some systems the equation is rather insensitive to the value of a! for others two minima are observed-a sharp one in the range of 0.2 to 0.5, and a gradually approached one in the vicinity of CY = - 1.Extension of these results to a number of other systems has shown that a value of a = - 1 consistently yields accuracies comparable to those obtained with a set a t the values recommended by Renon. Furthermore the new value of a predicts phase immiscibility with better accuracy than the recommended value of 0.2. This suggests that a true two-parameter expression may be obtained by substituting a by - 1. A total of 55 binary and 11 ternary systems have been studied. I n all cases the single value of a = - 1 yields excellent results. Local Effective Mole Fraction (LEMF) Equation
According to Renon, a was introduced into Wilson's expression for local mole fractions as a measure of the nonrandomness of the solution, and as such its lower value should be zero, corresponding to the case of a n ideal solution. Furthermore from comparison with Guggenheim's quasichemical t'heory (1952), Renon concluded that CY is the equivalent' of 1/Z, where 2 is the coordinat:on numbe: Therefore, the expected value of a is of the order of 0.1 to 0.3 From these observations it is concluded that a negative value for CY would be inconsistent with its original meaning. A new interpretation, based on the concept of local effective mole fractions (LEAIF), has been presented elseiThere (Marina, 1971). The new value of a when applied to Renon's (1968) expression for multicomponents systems yields:
Table II. Prediction of Ternary Vapor-liquid Equilibrium Average absolute errar System
Data points
Acetone-methanol-chloroform a t 50’ C Alcetone-metlianol-n-ater a t 100°C Acetone-methyl acetate-methanol a t 50°C Chloroform-methanol-ethyl acetate a t 760 m m H g Ethanol-benzene-n-heptane a t 400 m m Hg Ethanol-ethyl acetate-water a t 760 m m H g Methaiiol-carbon tetrachloride-benzene a t 55OC 1Iethanol-ethanol-water a t 760 m m H g Methanol-n-heptane-toluene a t 760 m m H g Methyl acetate-chloroform-benzene a t 760 m m H g llethylethylketone-n-heptane-toluenea t 760 m m H g
30 46 35 69 50 14 8 19 8 91 36
of predicted vapor compositions LEMF equation
0 064 0 016 0 006 0 034 0 034 0 010 0 006 0 020 0 012 0 007 0 023
0 083 0 024 0 009 0 039 0 019 0 017 0 005 0 015 0 009 0 003 0 019
NRTL equation
0 045 0 011 0 013 0 020 0 017 0 018 0 002 0 030 0 004 0 008 0 009
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
056 019 006 018 022 009 006 021 012 013 020
067 025 010 019 010 018 005 016 008 012 020
0 0 0 0 0 0 0 0 0 0 0
052 012 012 011 014 006 002 032 004 011 008
Table Ill. Prediction of M u t u a l Solubilities from Vapor-liquid Equilibrium Data
Data System
Ethyl acetate-water n-Butanol-water Isobutanol-water Water-aniline Propylene oxide-n ater
Experimental VLE data Compn Temp. range, range,
XI
points
3 12 9 4 13
Xi’
O C
0 94-0 976 74 1-75 8 0 015-0 961 93 0-111 5 0 010-0 938 90 9-99 0 0 025-0 200 152-105 0 046-0 925 50 8-34 9
Experimental mutual solubilities
Predicted mutual solubilities LEMF NRTL ____
0 0 0 0 0
Xi’
Xi”
05 02 025 26 13
where
0 0 0 0 0
0 0 0 0 0
68 55 65 98 55
hT
=
35 02 025 27 135
Xi‘
XI’’
0 0 0 0 0
72 61 69 98
80
0 0 0 0 0
014 021 021 372 166
Temp,
Xi”
0 0 0 0 0
768 364 402 985 625
O C
70 90 90 100 36 3
number of experimental points
Ternary Systems
.ind for the activity coefficient we obtain 111
yz =
k=l
k = l
The performance of the L E l I F and K R T L equations in predicting multicomponent VLE is compared in this study covering 11 ternary miscible systems. Equation 5 mas emp1o)ed with the binary constants A t , obtained from binary data. Vapor phase compositions were calculated by employiiig the method of P r a u w t z et al. (1967). Table I1 presents the average abqolute error of calculated vapor phase compositions for the S R T L and L E N F equationq. The results are very good for both equations which provide practically the same accuracy. Data sources for these systems are given in Table T’. Partially Miscible Binary Systems
Where the parameters A,, are evaluated from data on the binaries making u p the multicomponent system. Binary Systems
An extensive comparison between the K R T L and L E X F equations covering 55 binary systems revealed that they both yield the same accuracy in correlating binary activity coefficients (Marina, 1971). Table I presents a representatlve sample and iiicludes the Kilson equation for comparison. All three equationb provide practically the same accuracy. Activity coefficients were calculated according to the method of Prausnitz et a]. (1967). Data sources are given in Table V. The standard deviation is defined hereafter in terms of relative errors:
T o test the performance of the L E X F equation in this area several systems were studied. Our purpose was first to verify if the equation would predict immiscibility from VLE data in the miscible region and conversely to predict T‘LE in the miscible region from mutual solubility data. A% cornparisoil with the S R T L equation m s also sought. The constants for the S R T L and L E N F equations were calculated from the arailable VLE points for these systems and then were employed in calculating free energies of mising as a function of composition. The results for the system isobutanol-water are presented in Figure 5 and for studied in Table 111.Data sources are given in Table T’. It can be seen that the LEAIF equation yields better resul Turning to prediction of \-LE from mutual s o h we note that the constants for the L E N F and K R T L equations were obtained from the requirements:
Ind. Eng. Chem. Process Des. Develop., Vol. 1 2 , No. 1, 1973
69
Table IV. Prediction of VLE from Mutual Solubility Data a = 0.2 X
1-Butanol-water
Y1 EXP
Y1 Calcd
CY
= -1 Y1 Calcd
0.015 0.020 0.025 0.423 0,448 0.494 0.504 0.695 0,725 0.743 0.930 0,961
37 14 30 69 25 26 1 50 1 41 1 30 1 29 1 07 1 05 1 03 0 95 0 94
20.61 18.23 16.20 0.98 0.96 0.95 0.95 0.96 0.97 0.97 0.99 0.99
27 55 24 56 21 95 1 27 1 22 1 16 1 15 1 03 1 02 1 02 1 00 1 00
1 00 0 99 1 00 1 68 1 75 1 85 1 87 2 47 2 57 2 63 3 73 4 65
Ethyl acetate-water
0,940 0,958 0.976
1.00 1.00 1.00
Isobutanol-water
0,010 0.015 0,021 0.371 0,525 0.693 0.759 0,906 0.938 0.118 0.166 0.625 0.686 0,756 0.792 0.831 0.847 0.894 0.925
1 01 1 00 1 00 43 8 38 2 33 1 1 78 1 25 1 14 1 08 1 03 1 04 6 67 4 31 1 37 1 26 1 16 1 12 1 07 1 06 1 03 1 01 4 410 4 739 4 490 3 885
1 01 1 00 1 00 32 9 29 3 26 6 1 39 1 09 1 04 1 02 1 00 1 00 7 053 5 284 1 386 1 273 1 170 1 127 1 087 1 072 1 037 1 019 3 422 3 507 3 477 3 208
6 91 8 33 8 30 1 09 1 04 1 09 1 60 2 28 2 75 3 04 4 65 5 17 1 05 1 25 2 67 2 76 3 56 4 12 5 47 5 47 7 44 7 78 1 164 1 192 1 134 0 910
Propylene oxide-water
Water-aniline
0.025 0.059 0.093 0.200
26.8 23.72 21.00 1.14 0.98 0.97 0.98 1.00 1.00 6.052 4.735 1.258 1,168 1.094 1.067 1.043 1.034 1.016 1.008 3.434 3.323 3,239 3.006
Typical results are presented in Figure 6 and Table IV. As may be expected, largest deviations from experimental values occur a t both extremes of the concentration range where experimental errors are large. However, irrespective of this, the L E M F equation shows better agreement with the data than the X R T L equation. Discussion and Conclusions
Renon has recommended optimum values for alpha in the N R T L equation, for several classes of binary systems, based on qualitative considerations about the system and its components. Therefore a n ambiguity is encountered sometimes in choosing the proper value of alpha. For example, while for the system nitrometliane-carbon tetrachloride a value of CY = 0.47 is recommended, a value of CY = 0.30 is recommended for the system nitroethane-carbon tetrachloride. Furthermore, as shown in Figures 1 and 2 a small change in alpha, in the positive region, can resuit in very high errors. On the other hand, the lack of steepness of the curves in the negative region of 01, as shown in Figures 1 through 4, suggests that a value of CY = -1 satisfies all systems. This conclusion is further documeiited with the results presented in Table I. 70 Ind.
Eng. Chern. Process Des. Develop., Vol. 12, No. 1, 1973
= 0.2 YZ Calcd
01
YZ EXP
1.oo
1.00 1.01 1.60 1.62 1.65 1.66 1.62 1.60 1.59 1.44 1.41 5.74 5.91 6.08 1 .oo 1.00 1.00 1.60 1.81 1.81 1.79 1.68 1.66 1.048 1.093 2,431 2.799 3.311 3.620 3,996 4.163 4 , 705 5.107 1,000 1,000 1.002 1.013
01
= -1
Y2 Calod
r, o c
1 00 1 00 1 01 1 69 1 73 1 81 1 82 2 11 2 14 2 16 2 32 2 33
93.4 93.0 92.7 92.8 92.9 93.4 93.5 96.3 97.2 97.9 108.8 111.5
7 64 8 21 8 81 1 00 1 00 1 01 1 61 1 97 2 14 2 23 2 37 2 39 1 065 1 121 2 490 2 929 3 647 4 154 4 867 5 224 6 582 7 832 1 000 1 002 1 006 1 030
74.1 74.6 75.8 90.9 90.0 88.0 89.2 89.2 90.6 92.3 96.8 98.9 38.2 36.3 36.3 36.2 36.0 35.9 35.6 35.5 35.1 34.9 152 131 121 t05
Furthermore Figure 4 suggests that no other single value of CY can be found for all systems. The results presented in Table I1 suggest that the L E N F equation provides very good prediction of ternary VLE from binary data, a t least as good as the X R T L equation u l t h the Talue of 01 set according to the rules of Renon and Prausnitz I n this case the possibility that better ternary results could be obtained by fitting the value of CY to the binary data should be considered. Even though this seems attract11 e, a recent study (Larson and Tassios, 1972) has shonn this approach to give results not better than those obtained othernise. For biliary partiallt miscible s>stems, the L E N F equation seems to provide better results in the prediction of solubilities from VLE data (Table 111) as well as in the more important case of prediction of T’LE data from mutual solubility data (Table IV). .in attempt a t obtaining a semitheoretical explanation for the negative Talue of CY has been undertaken elsenhere (Marina, 1 9 i l ) . Here it suffices to say that if in Equation 1 for positile C Y , Sf,/iXll refers to the ratio of local mole fractions in the cell, then for a negative CY, X - 2 1 Silmubt refer to some other ratio nhich behaves just opposite to that of the
local mole fractions. One such ratio would be t h a t of the molecules in the cell with highest potential energy. Therefore the name, local effective mole fractions. Nomenclature = =
-
R T
X, X,,
2
= = = = =
difference between parameters, A,, = gzl - g j l residual Gibbs energy in the N R T L equation excess Gibbs energy gas constant absolute temperature overall mole fraction of component i local mole fraction of component i in a cell lattice coordination number
GREEKLETTERS CY
y
= =
nonrandomness parameter in the N R T L equation activity coefficient of component i
References
Benedict. 11..Johnson. C. 4..Rubin. L. C.. Trans. Amer. Znst. Chem. ’Ens.’, 41, 371 ’(1945): Delzenne, .4.O., J . Chem. Eng. Data, 3 , 224 (1958). Griswold, J., Andres, D., Arnett, E. F., Garland, F. M., Ind. Eng. Chew&.,32, 878 (1940). Griswold, J., Chu, P. L., Winsauer, W. O., i b i d . , 41, 2352 (1949). Griswold, J., Wong, S.Y., Chewa. Eng. Progr. Symp. Ser., 48, 1 (1952). Guggenheim, E. A., “Mxtures,” Clarendon Press, ($ford (1952). Hala, E., Wichterle, J., Polak, J., Boublik, J., Vapor-Liquid Equilibrium Data at Normal Pressures, Pergamon, Xew York, N.Y., 1968. Kincaid, J. F., Eyring, H., Stearn, A. E., Chem. Rev., 28, 301 (1941). Larson, C. D., Tasbios, I). P., Ind. Eng. Chem. Process Des. Develov.. 11. 35 11972). lIargulek, ’LI.,’ Sitz., Akad. W f s s . Wzen, Jlath. Saturwzss. Kl., 104, 1243 (1895). Marina, J . AI., Doctoral Dissertation, Xewark College of Engineering, Newark, N.J., 1971. Nurti, P. S., Van Winkle M., J . Chem. Eng. Data, 3 , 72 (1958). Kielsen, R. C., Weber, J. H., ibid.,4, 145 (1959).
Orye, R. V., Doctoral Dissertation, Univ. California, Berkeley, 1965. Perry, J. H., “Chemical Engineers Handbook,” 1IcGraw-Hill, New York, N.Y., 1963. Prausnitz, J. M.,“Molecular Thermodynamics of Fluid-Phase Equilibria,’] Prentice Hall, Englewood Cliffs, N.J., 1969. Prausnitz, J. LI.,Eckert, C. A., Orye, R. V., O’Connell, J. P., “Computer Calculations of Multicomponent Vapor-Liquid Equilibria,” Prentice Hall, Englewood Cliffs, K.J., 1967. Renon. H.. Doctoral Dissertation, Univ. California, Berkelev, 1966. Renon, H., Prausnitz, J. AI., A.1.Ch.E. J., 14, 135 (1968). Renon, H., Prausnitz, J . M., Ind. Eng. Chem. Process Des Develop., 8, 413 (1969). Scatchard, G., Ticknor, L. B., J . Amer. Chem. SOC.,74, 3724 (1952). Scott, R. L., J . Chem. Phys., 25, 193 (1956). Severns, W. H., Sesonske, A,, Perry, R. H., Pigford, K:. L., AIChE J., 1, 401 (1955). Steinhauser, H. H., White, R. R., Ind. Eng. Chem., 41, 2912 I.
119491 ~\ - -
I
Stockhardt, J. S., Hull, C. II.,ibid.,23, 1438 (1931). Tassios D. P., Preprint, A.1.Ch.E. National Meeting, Washington, b.C., 1969. Van Laar, J. J., Z. Physik Chem., 72, 723 (1910); 83, 599 (1913). Wickert. J. S . .Tamwlin. W.S.. Shank.’ R. L.. Chem. Ena. Prom.. Surn~.’Ser.-To. 2. 28. 92 119k2). Wilso< G. lI.,J . Arne;. Chem. Soc., 86, 127 (1964). Wohl, K., Trans. A.Z.Ch.E., 42, 215 (1946). RECEIVED for review hIarch 20, 1972 ACCEPTED July 12, 1972 Work sup orted by 1Ierck Sharp & Dohnie Research Laboratories, Ita!kvay, N. J. Figures 1 and 3-6 plus Table V will appear following these pages in the microfilm edition of this volume of the Journal. Table V gives the list of sources for data used in this presentation. Figures 1 and 3 and 4 show standard deviation vs. alpha. Figures 5 and 6 show predictions of liquidliquid equilibrium and vapor-liquid equilibrium from vapor-liquid equilibrium and mutual solubility data, respectively. Single copies of all 11 manuscript pages may be obtained from the Business Operat’ionsOffice, Books and Journalh Division, hmeriran Chemical Society, 1133 Sixteenth St., N.W., Washington, D. C. 20036. Refer to the following code number: PROC-7367. Itelnit by check or money order $4.00 for photocopy or $2.00 for microfiche.
Calculation of Solution Nonideality from Binary T-x Data Herbert E. Barrier' and Stanley B. Adler The JI. W . Kellogg Co., A Division of Pullman, Inc., 1300 Three Greenway Plaza, Houston, Tex. 77046
The calculation of equilibrium vapor compositions and liquid-phase activity coefficients from observed temperature-liquid composition ( T - x ) data is discussed. Reliable results can be obtained by numerically integrating the coexistence equation. A useful and simple procedure is also given for calculating the activity coefficient at infinite dilution from bubble point data. Validity of the calculations i s demonstrated for systems where experimental vapor compositions have been measured.
I n the customary method of measuring binary vapor-liquid equilibria, direct measurements of temperature, pressure, aiid composition of equilibrated vapor and liquid phases are made. address, Laboratory, Kennecott Copper Corp., 128 Spring St., Lexington, Mass. 02173. To whom correspondence should be sent.
Measurement of equilibrium vapor compositions, however, often is much more difficult and is subject t o greater uncertainties than the measurement of liquid compositions. It is therefore sometimes desirable t o find the vapor compositions by calculation from the properties of the liquid phase alone. Thus, numerous procedures for treating isothermal binary Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
71
