method of measuring .the temperature, the different mass velocities used, and the different approach. Though higher values of U are indicated from iour data than those obtained by Fritz and Ferron, the trend is more or less similar. The data of Fritz and Ferron were obtained for one fluidizing material (aluminasilica) only and for much loiver mass rate of flow of gas. The present data included a variety and sizes of glass beads. Acknowledgment
The authors are grateful to the National Research Council of Canada and the Ontario Research Foundation for financial assistance. Nomenclature
At
CP, C,,
D, DPj
D, GI kj,
k,
Lelptl. L,
= = = = = = = =
T,, T , T,, Bv.
= = = = = =
TI, Tp U
= =
W,
= = =
41 42
TV,
Xi
e Pa
= =
surface area of entire solid bed, sq. ft. specific heat of fluid, B.t.u.,/(lb. mass)(” F.) specific heat of solid, B.t.u./(lb. mass)(’ F.) particle diameter particle diameter of ith weight fraction of solid tube diameter, ft. fluid mass .velocity, lb./hr. sq. ft. thermal conductivity of fluid and solid, respectively, B.t.u./(hr.)(sq. ft.)(’ F./ft.) expanded bed height, ft. bed settled height, ft. rate of heat loss from fluidized bed, B.t.u./hr. rate of heat loss due to reaction, B.t.u./hr. temperature of fluid and solid, respectively, ” F. mean temperature of room? jacket, reaction hardware, arid insulation, ’ F. inlet and cutlet temperatures, ’ F. space-averaged over-all heat transfer coefficient, B.t.u./(hr.)(sq.ft.) F. fluid flow I ate, Ib./hr. weight of solid fluidized, lb. Tveight fraction of particle passing through ith opening of screen time, minutes particle density, Ib./cu. ft.
literature Cited
Feng, L., Ph.D. thesis, University of Ottawa, 1966. Ferron, J. R., Ph.D. thesis, University of TYisconsin, 1961. Frantz. J. F., Cheni. Eng. Pragr. 57, 35-42 (1961). Fritz, J. C., Ph.D. thesis, University of \Yisconsin, 1956. Heerden, C. Van, Nobel, A. P. P., Krevelen, D. 1%’.Van, Chem. Eng. Sa. 1 (2), 51-66 (1951). Heertjes, P. M., McKibbins, S. TV., Chem. Eng. Sa. 5 , 161 (1956). Johnstone, H. F., Pigford, R. L., Chapin, J. H., Trans. A m . Inst. Chem. Eners. 37. 95 11941). Kettenringr K. N:, Mander’sfield, E. L., Smith, J. M., Chem. Eng. Progr. 46, 139 (1950). Koble, R. A,, Ademino, J. H., Bartkus, E. P., Corrigan, T. E., Ckem. Eng. 58, 174 (1951). Leva. M.. “Fluidization.” McGraw-Hill. New York. 1959. Leva; M.; Tt’eintraub, Grumrner, M., Chem. Eng. Progr. 45, 563-72 (1949). Richardson, 3. F., Ayers, P., Trans. Znst. Chem. Engrs. 37, 314-21 (1959). Rozenthal, E. O., Teplo-i Massoobmen V Protses. Ispareniza, Akad. n h u k S S S R , Energet, Znst. 1958, p. 87. Shakhova, N. A., Rychkov, A. I., Tr. Moskoc. Znst. Khim. Mashinostroeniva 12. 119 119.37). Toome;, R. ’D., Johnstone, H. F., Chem. Eng. Progr. Symp. Ser. 5 , 49, 51-63 (1953). TValton, J. S., Olson, R. S., Levenspiel, O., Znd. Eng. Chem. 44, 1474-80 (1952). TYamsley, \Y. TV., Johanson, L. N., Chem. Eng. Progr. 50, 347 (1954). Zenz, F’, A , , Othmer, D. F., “Fluidization and Fluid Particle System,” pp. 433-7, Reinhold, New York, 1960. RECEIVED for review October 17, 1966 RESUBMITTED hTovember 27, 1967 ACCEPTEDApril 15, 1968 Material supplementary to this article has been deposited as Document No. 9977 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, TVashington, D. C . .4 copy may be secured by citing the document number and by remitting $1.25 for photoprints of $1.25 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
PREDICTION OF VAPOR COMPOSITIONS IN VAPOR=LIQUID SYSTEMS I V A R S N E R E T N I E K S
Department of (,’hemica1 Engmeering, Royal Institute of Technology, Stockholm 70, Sweden
A method of calculating vapor-liquid equilibria in multicomponent mixtures is described.
No experimental information of multicomponent mixtures is necessary. The data needed can b e obtained from the binaries. Two models were used to calculate the activity coefficients: the Wilson equation and the Black equation. The Black equation has been extended in order to make it possible to extrapolate to other temperatures. Parameters in the equations are calculated by a nonlinear least-squares fit of observed data. The efficiency of the method when using the Black and Wilson equations in predicting vapor-liquid equilibria has been compared. The efficiency in predicting equilibria at other temperatures, some more than 40” C. from those where datlo was obtained, has also been investigated. The Wilson equation has been more accurate in nearly all cases studied.
separation processes, reliable data on equicompositions are needed for good design. For binary systems data are usually available, but at some other temperature. I t is therefore convenient to have a mathematical model \vhich can predict compositions in one phase a t pressures, temperatures, and compositions in the other phase, using data a t other pressures, temperatures, and compositions. N VAPOR-LIQUID
I librium
For multicomponent systems, data are scarce and much experimental work is needed to obtain them. Methods whereby data for a multicomponent mixture can be calculated from binary data only are therefore of great value. As data for different binaries have seldom been determined at the same temperature, the mcthod has to predict the temperature dependence as well. VOL. 7
NO. 3
JULY
1968
335
Thermodynamic Relations
T h e composition in the vapor phase, y,, is related to the composition in the liquid phase, x,, by
Another way of calculating activity coefficients in multicomponent mixtures has been developed by Black (195810, 1959, 1963). Log y i is given by log
~i
=
[CaiFRj22~t+ 0.5 3
(ai?
Methods for calculating Bi, y t , and P,O are given below. T h e equations have been chosen with regard to both simplicity and accuracy. None of the equations used need other experimental data than what is readily available in the literature -i.e., critical constants, liquid density, boiling point, or heat of vaporization, besides vapor-liquid equilibrium data, Equation 2 gives the imperfection pressure coefficient, O,,
+
%t2
XjxkRjzRkz X Ik+
- ajk”,j)I/YC
XjRjd2 3
+ E,,
(7)
where
Est =
C R
[(X,- X d Z (
cirxr)I r
+
P
Equations 7 and 8 are modified van Laar equations. As 7, for distillation systems changes little with pressure, the integral in Equation 2 can be written (3)
A t j is the logarithm of the activity coefficient of component i at infinite dilution in liquid j .
mean value of 8, in the pressure range Pi0 to P. T h e fugacity coefficients, pt, can be calculated by an equation proposed by Redlich and Kwong (1949).
Parameters at? are not independent of temperature, but it has been shown (Hala et al., 1958) that the temperature dependence of A t j is
( P - Pro) 8 t , m
Vi,,is the
In p, = ( z
- 1) B2 - In B
(z
- BP) - A’ B
(7-2) 2Ai
Parameters A and B are dependent only on temperature, critical properties, and vapor phase compositions. Equation 4 applies best to nonpolar substances. Errors introduced in p t for polar substances are generally small and are to a large extent eliminated in calculating 8,. For most practical purposes Equation 4 is sufficiently accurate and has been used in this study. For very accurate determinations other equations must be used. Black (1958a) has made a modified van der Waals equation which can be used with polar substances and systems with “special chemical effects,” but is rather complicated and needs many experimentally determined constants. Two methods for determining activity coefficients in the liquid phase were used, the Wilson and Black equations. Wilson (1964) gives a two-parameter equation which has been tried (Eckertet al., 1965; Orye and Prausnitz, 1965) with good results for a variety of components including polar substances. For a multicomponent mixture the activity coefficient can be calculated by
‘ I t j is defined by -~ v, - X e
[Xij
A,, =
-
= ___
1 -
2.303R T
+ Konstrj - c t j
or
TU,? = aJ’(1
+ TKi,) - Tcij
(134
where a,?’, K i j , and c t j are independent of temperature. K i j is the temperature-dependence parameter. For convenience Equation 13a is lvritten
Tal? = aij2‘(1
+ TKtj’)
(13b)
Kij’ is also independent of temperature. T o use the Black equation for other temperatures, five parameters are needed for each pair of binaries: ai?’, a,:‘, cpl, Ktj’, and Kjt’. T h e Kij”s can be determined simultaneously with the composition parameters or from two sets of determinations a t different temperatures from vapor-liquid equilibrium data only. For simultaneous determination isothermal data will not do. A thorough review and comparison of equations for pure component vapor pressure were done recently by Miller (1 964). T h e recommended equation is
Xii]
-
can be considered independent of temperature. k This makes it possible to predict activities at other temperatures than those used in obtaining the parameters. 336
Uij2
kT
vi [“j
where LtjO is the molar heat of solution of component i in infinite dilution in liquid j . LrjO can for our purpose be considered independent of temperature. T h e temperature dependence of a,? can then be written
l&EC PROCESS DESIGN A N D DEVELOPMENT
log Pr =
G
-- [I - T,2 + k(l - Tr)a] Tr
(14)
G can be determined from data on the normal boiling point and critical pressure or critical temperature and heat of vaporization, whereas k must be determined from data on the vapor pressure.
Table 1.
Calculated Parameters and Standard Errors of Estimate in y for Black and Wilson Equations
Kzi ‘, K.-I
a1z2/,
System
O
Hex-MCP Hex-benz Eth-hex MCP-benz Eth-MCP Eth-benz Meth-benz Meth-CClc CCl4-benz
K.
25.88 68.22 359.85 50.20 374.28 407.69 276.91 508.10 25.56 A12
SY,
%
O
-
BLACKEQUATION -6.57 4.94 x 2.35 1.05 x 57.06 -4.63 x 3.43 x -6.61 12.23 -5.63 x -8.00 x -21 . I 5 11.6G 3.63 x -1.06 x 108.67 -9.33 7.52 x
5.03 59.41 290.13 42.94 266.25 193.54 170.10 278.97 IO. 10 x21
All,
4.67 x 1.64 x -1.77 x 3.28 x -1.09 x 2.65 x 9.71 X -2.20 x -7.82 x
10-3 10-4 10-5 10-4
10-5 10-4 10-4 10-3 10-4
- xzz,
System
Cal./MoCe
Cal./Mole WILSON EQUATION 0.14 327.54 323.66 124.82 324.42 285.87 351.53
Hex-MCP Hex-benz Eth-hex MCP-benz Eth-MCP Eth-benz Meth-benz
60.41 5.32 2212.58 126.25 2352.42 1329.87 1616.95
Meth-CCl4
2248.53
277.66
3.49
CClrbenz
287.65
- 188.97
0.36
10-5 10-4
1.52 3.00 4.08 0.79 5.18 3.78 1.60 4.46 0.40
10-4
10-4 10-4 10-4 10-4 10-4 Ref.
1.75 2.99 2.87 1.11 3.88 5.27 2.25
Myers and Braun (1957) Chu et al. (1956) Sinor and Weber (1960) Chu et al. (1956) Sinor and Weber (1960) Chu et al. (1956) Chu et al. (1956j, Scatchard et al. (1940) Chu et al. (1956), Scatchard and Tichnor (1952). Chu et al. (1956). Scatchard et al. (1940) ’
”
Determination of Partrmeters
I n the Wilson as well as the Black equation the parameters A,,, at?, and cu have i o be determined with experimental d a t a from binary mixtures. Data need not be isobaric or isothermal. As the equations are not linear with respect to the parameters, these cannot be determined by simple regression analysis. A general minimum seeking method is used instead. I t operates o n an error function of the type shown below. 2N
E =
(Yk,obsd
k=1
- YR,oslcd)’
(15)
By minimizing the error function, E, which is a function of the parameters, the “best” value of the parameters is obtained. T h e minimum seeking method used by the author operates on one parameter at a time by calculating E at three points by variation of one pammeter, the others being held constant. A parabola is fitted tlirough these points and the minimum of the parabola calculated. This is repeated until the improvement is negligible. This procedure is then repeated for the parameters in a circular sequence. After a few cycles, when the direction of the minimum valley is known, the search is continued along the valley. When no improvement can be obtained in this direction, the search starts again with one parameter a t a time. T h e search ends when no improvement can be made in any direction. T h e error functions are constructed as below. EBiack
=
Ewileon
zi: T 1% == Z(ln
’Yobad
- T 1%
Yobsd
- In Yoalcd)’
Yesled)’
(16)
(17)
T h e error function has been constructed so that the “best” fit of the y’s will be obtained-in the sense that the relative errors in y are smallest-for use in multicomponent equilibrium calculations. If the parameters are going to be used only for binary equilibrium calculations, the error function, E, should be made in a different way-e.g.,
Table II.
Temp.,
Standard Error of Estimate in y for the System Methanol-Benzene
C. 100 Black JVilson No. of observations
sy
x
Table 111.
Temp., sy
x
58-74
O
O
55
0.75 0.71 7
40
2.43 1.50 5
2.09 2.13 13
Standard Error of Estimate in y for the System Carbon Tetrachloride-Benzene
C.
30
40
50
60
70
100
Black Wilson No. of observations
0.61 0.17 1
0.32 0.24 7
0.35 0.08 1
0.26 0.16 1
0.24 0.24 7
T h e errors can also be assigned weights, in order to make a better fit in the concentration ranges of greatest interest.
Results
I n Table I the parameters and the standard error of estimate sy of y are given. For four of the nine systems the Wilson equation gives a better fit and for one system the equations are equally good. T h e relative errors in y range from 0.40 to 5.18y0 for the Black equation and from 0.36 to 5.27Y0 for the Wilson equation. T h e parameters for the system methanol-benzene were evaluated a t 1 atm. and 58’ to 74’ C. Vapor compositions and pressure were predicted a t 55’ and 40’ C. T h e results are given in Table 11. T h e errors are larger the farther from the evaluation temperature the data. The azeotropic composition a t 40’ C. is in error by more than 1 mole %. T h e errors of prediction for the system carbon tetrachloridebenzene are given in Table 111. T h e errors for this system are much less than for the previous one. VOL. 7
NO. 3 J U L Y 1 9 6 8
337
Table IV. Standard Error of Estimate in for the System Carbon Tetrachloride (I)-Benzene(2)-);lethand(3)
(Data from Chu, 1956) Yl sy
x
100 Black Wilson
Y2
Ya
2.14 1.36 1.51 1.07 1.34 0.69
No. of observations Temp., O C.
Table V.
7 55
Yi
Yz
Ya
2.38 0.47 2.40 0.61 0.93 1.27 6 34.68
Standard Error of Estimate in y
Yl YZ Y3 Y4 System Hexane( 1)-Methylcyclopentane(2)-Ethanol(3)Benzene( 4)a sy x 100 0.56 2.61 0.75 Black 2.04 0.67 1.30 0.60 Wilson 1.05 sy
System Hexane( 1)-Methylcyclopentane(2)-Ben~ene(3)~ x 100 Black 0.72 0.51 0.70 LVilson 0.65 0.41 0.58
System Hexane( l)-Methylcyclopentane(2)-Ethanol(3)c sy x 100 Black 2.86 1.73 1.92 LVilson 1.27 1.11 1.29 System Methylcyclopentane(1)-Benzene(2)-Ethanold sy x 100 1.57 l-.08 Black 1.55 0.99 0.67 Wilson 0.89 Based on 37 observations. Data from Belknafi and Weber ( 7 9 6 1 ) , Kaes and Weber ( 1962), and Sinor and Weber (1960). Data from Belknap and Weber ( 1 9 6 1 ) . b Based on 54 observations. Based on 57 observations. Data from Kaes and Weber ( 1 9 6 2 ) . Data from Sinor and Weber ( 1960). d Based on 46 observations.
system for which the coefficient were changed. The prediction of multicomponent compositions and pressures is, however, considerably improved. When the relation is satisfied, there no longer exists the need of numbering the components in the order proposed by Black. They can be numbered in any order. Conclusions
The results of prediction indicate that the equations used are of sufficient accuracy for most practical purposes. T h e larger standard errors of estimate for systems with a highly polar component-in Wilson equation predictions about 1% as compared to 0.5%-are partly due to vapor phase imperfections. If more accurate, and complicated, equations are used the accuracy can be further increased. The LYilson equation has in most cases been more accurate than the Black equation, and is also less complicated. The Black equation however, has the advantage of being able to handle systems which show partial miscibility and minima or maxima in the activity coefficients. Nomenclature
A,
Ai
A i j aij” ai,2,
Cij
= parameters in Black equation = parameters in Redlich, Kwong equation = parameters in Black equation
Esi E
= function: sum of squared errors
B, Bi
= correction term in Black equation
G Kij,
Kij’
Lip k
a
P PO P, R
Ri j s
‘The equilibrium compositions of the ternary system carbon tetrachloride (1)-benzene (2)-methanol (3) were calculated for t\+o temperatures, 55’ and 34.68’ C. (Table IV). I n nearly all cases the Wilson equation gave a more accurate prediction. The standard errors of estimate for the vapor composition of the quaternary system and for three of its ternary systems are given in Table V. T h e analytical errors are estimated to be 0.5 mole yo in the multicomponent systems. T h e W‘ilson equation has predicted compositions with an accuracy ranging from 0.41 to 1.30 mole % and the Black equation 0.51 to 2.86 mole %. T h e accuracy of prediction has been much better for the system hexane-methylcyclopentane-benzene, which is less nonideal than the others. For multicomponent equilibrium calculations with the Black equation the following relation should be satisfied :
T T, V
7
x
X, Y 2
338
I&EC PROCESS DESIGN AND DEVELOPMENT
= function: Equation 14 = temperature-dependence constants for a,,z
= constant = molar heat of solution of constituent
i in infinite dilution in liquid j = pressure = vapor pressure = reduced pressure = gas constant = a212/a,22 = standard error of estimate = temperature = reduced temperature = molar volume of pure component, volume = partial molar volume = liquid mole fraction = Zx, of class R components = mole fraction in gas phase = compressibility factor
GREEKLETTERS Y
e
A,, A%, cp
Po
activity coefficient imperfection pressure coefficient parameters in FVilson equation parameters in Wilson equation fugacity coefficient = fugacity coefficient in a standard state
= = A,, = = =
SUBSCRIPTS
i, j , k m M, R
S The experimentally determined a f t ’ s will not automatically satisfy the above relation, as for nearly ideal mixtures the atj2’s are small but R,ts differ much from 1, whereas for highly nonideal mixtures R,,’s are near to 1. T h e small a(j’s have been corrected so that the relation is satisfied in the temperature range considered. This introduces new errors, as it will not give the “best” fit to the binary
= parameter in Redlich, Kwong equation = logarithm of yi when xi 4 0
obsd calcd
= i, j , and kth component = mean value
= M , Rth class of components = special class having ith component = experimental values = calculated values
literature Cited
Belknap, R. C., Weber, J. €I., J. Chem. Eng. Data 6 , 485 (1961). Black, C., Znd. Eng. Chem. 50, 391 (1958a). Black, C., Znd. Eng. Chem. 50, 403 (1958b).
Black, C., Znd. Eng. Chem. 51, 211 (1959). Black, C., Derr, E. L., Papadopoulos, M. N., Znd. Eng. Chem. 55 ( 9 ) , 39 (1963). Chu, J . C., Wang, S. L., Sherman, L. L., Rajendra, P., “VaporLiquid Equilibrium Data,” J. LV. Edwards, Ann Arbor, Mich., 1956
Miller, D. G., Znd. Eng. Chem. 56 (3), 46 (1964). Myers, H. S., Braun, C. F., Petrol. Refiner 36 (3), 175 (1957). Orye, R. V., Prausnitz, J. M.,Znd. Eng. Chem. 57(5), 18(1965). Redlich, O., Kwong, J . N.S., Chem. Rev. 44, 233 (1949). Scatchard, G., Tichnor, L. B., J . A m . Chem. SOC. 74, 3724 (1952). Scatchard. G.. IVood. S. E.. Mochel. J. M.. J . A m . Chem. SOC.62, ..~ 712 (1940).’ Sinor, J. E., \Yeber, J. H., J . Chem. Eng. Data 5 , 243 (1960). Wilson, G. M., J . A m . Chem. SOC.86, 127 (1964). ~~~
Eckert, C. A,, Prausnitz, J. M., Orye, R. V., O’Connell, J. P., “Advances in Separation Techniques,” Preprint A.1.Ch.E.1.Chem.E. Meeting. June 1965. D. 58. Hala, E., Pick, J., Fried, V., Vifim, O., “Vapor-Liquid Equilibrium,” Pergamon Press, New York, 1958. Kaes, G. L., Weber, J. I, are expressed in terms of the binary diffusion coefficients, D , and composition as (Burchard and Toor, 1962) VOL. 7
NO. 3 J U L Y 1 9 6 8
339
