A METHOD FOR PHASE EQUILIBRIUM
CALCULATIONS BASED ON GENERALIZED BENED I CT-W EBB-RU BI N CONSTANTS THOMAS G. KAUFMANN Esso Research and Engineering Co., Florham Park, N . J .
A method based on the generalized Benedict-Webb-Rubin equation of state was used to develop the BWR constants for cis-2-butene, 1 -pentenet and 1,3-butadiene. This development consisted of three steps. First, the coefficients of the reduced BWR equation of state were determined based on 1-butene and methylacetylene, respectively. Second, with the aid of the reduced equations of state, the BWR constants for cis-2-butene, 1 -pentene, and 1,3-butadiene were established. Finally, in conjunction with the new constants, a set of temperature-dependent C i s was developed for each substance. This last step assured the accurate prediction of the vapor and liquid fugacities up to the critical point. The new BWR constants were evaluated by calculating the equilibrium constants of these components in mixtures. Predicted values were in excellent agreement with experimental results, indicating the applicability of these constants to phase equilibrium calculations.
Benedict-Webb-Rubin equation of state is one of the best procedures for predicting the thermodynamic properties of complex hydrocarbon systems, and has also been used for certain nonhydrocarbons. I t contains eight constants which are usually determined by fitting the equation to purecomponent P-V-T data. T h e evaluation of these constants is a tedious task requiring extensive experimental data not always available. I n view of the lack of data for many nonparaffinic hydrocarbons and the (difficultiesinvolved in obtaining them for heavier substances, a generalized correlation among the BWR constants would greatly extend the applicability of this method. There have been several attempts to develop correlations for the constants of the BWR equation of state. Such correlations have been established for normal (Canjar et a/., 1955), 2-methyl (Griskey and Canjar, 1963), 3-methyl, 2,2-dimethyl, and 2,3dimethyl (Beyer and Griskey, 1964) paraffins, based on simple structural characteristics of the various homologs and on the critical temperature. T h e validity of the constants obtained by these methods was demonstrated by reproducing the compressibility data and the critical properties for the various paraffins. I n all cases, good agreement was observed between experimental and predicted quantities. However, the authors state (Beyer and Griskey, 1964; Griskey and Canjar, 1963) that the developed constants hold only for the critical and superheated vapors and not for the two-phase region. Therefore, the constants cannot be used for predicting equilibrium K values or other properties within the vapor-liquid dome. This significantly restricts thleir usefulness, since from a practical point of view the K value is the thermodynamic property most often used. Lin and Naphtali (1963) pointed out that the adequacy of a n equation of state merely to reproduce the volumetric behavior of a substance does not assure its reliability in predicting derived properties, such as fugacities, enthalpies, and entropies. T h e derivatives and integrals of the equation must also be sufficiently accurate. T o assure this quality, at least one of the derived properties should also be included in testing a :new set of BWR constants. HE
T h e Benedict-Webb-Rubin equation of state, expressing pressure as a function of temperature and molar density, is of the form (Benedict et al., 1942):
P = RTd $? (B,RT
- A , - Co/T2)d2+ (bRT - a)d3 +
aad6
+ cd3/T2
(1 f yd2) exp(--y#)
(1)
T h e objectives of this study were to develop the BWR constants for cis-2-butene, 1-pentene, and 1,3-butadiene, and then to apply them to vapor-liquid equilibrium calculations. T h e available information on the P-V-T behavior of these substances is relatively meager and, therefore, a generalized procedure was sought to obtain the coefficients. I n correlating the constants of normal paraffins, Canjar et al. (1955) found that none of the light paraffins, methane, ethane, and propane, could be included in the correlation for predicting the coefficients of the heavier hydrocarbons. This was attributed to the fact that a correlation which does not contain “shape factors” is valid only over the range where additional methyl groups do not seriously alter the shape of the molecule-i.e., when a carbon chain is long enough so that the addition of another methyl group does not have any significant effect. Starting with a n approach similar to that developed for paraffins was not possible in this investigation, since of the homologs of cis-2-butene and 1-pentene probably only 1butene could be used, and no BWR coefficients exist for any of the homologs of 1,3-butadiene. Consequently, a different approach was necessary. T h e main advantage of the method developed in the present study is that it results in a set of constants which are reliable for accurate phase equilibrium calculations. Reduced Forms of Benedict-Webb-Rubin Equation of State
T h e reduced form of an equation of state is essentially a n analytical expression of the theorem of corresponding states or various modifications of this theory. T h e coefficients of the reduced BWR equation were evaluated by Joffe (1949), Opfell et al. (1956), and Su and Viswanath (1965). Opfell VOL. 7
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et al. determined the reduced coefficients based essentially on the two-parameter van der Waals equation (1873),
f(T7, p7, 2)
=
0
(2)
the three-parameter Meissner and Seferian equation (1951),
f ( T r , p7, zc,
2)
= 0
(3)
Development of Procedure
and the Pitzer (1954) relationship,
j(T7, Pr, W ,
2)
= 0
(4)
T h e best agreement between experimental and predicted compressibilities was found through Equation 4. T h e disadvantage of this method, however, is that it requires the evaluation of 15 constants. T h e other two methods yielded relatively poor agreement. Joffe (1949) and Su and Viswanath (1965) used Su’s (1937) modified theorem of corresponding states to evaluate the coefficients of the reduced BWR equation. I n view of the difficulties involved in measuring critical volumes, Su defined a pseudocritical volume term as Vc, = RTc/Pc,which results in
f ( T , , pr,
V7d
=
0
(5)
Combining Equations 1 and 5, the reduced BWR equation is then of the form
P, = T,d,
+ (B,’T7 - A,’
d,3 f a’a’dr6
+
- C,’/T7) d,2 + (b’T, c’dr8/T? ( 1 + r’d?) exp (-7d:)
a’) X
(6)
where
Bo’ A,‘ C,’ b’ a’
= BoPc/RTc = A,pc/RZT,2 = C,pc/R2Tc4
bp,2/R2T,2 = ap,2/R3T,3 6’ = cp,2/R3TC6 a’ = apC3/R3TC3 7 = 7p,2/R2Tc2 =
RT In f
(8)
fv”
I t was considered possible that a better agreement might be obtained between theoretical and predicted fugacity ratios ( f v o / f L o ) if, instead of the generalized constants, the coefficients of a structurally similar substance were used for estimating the coefficients of Equation 6. Canjar et al. (1955) established the BWR coefficients of 1-butene using the experimental data of Beattie and Marple (1950). T h e authors showed that these constants reproduce the P-V-T behavior of 1-butene with good accuracy. In the present study, the reliability of the constants was further confirmed by comparing the vapor-liquid equilibrium data of Sage and Lacey (1948) for the n-butane-1-butene system with those predicted by the equation of state. T h e conditions at which the data were tested were between 220’ and 280’ F. and 250 and 491 p.s.i.a. Excellent agreement was observed between experimental and predicted K values, the average deviation for 1-butene being less than 1%. T h e 1-butene constants were then converted to a reduced basis and used to calculate the BWR coefficients of n’s-2butene and 1-pentene (Table I). T h e values determined in this manner were then employed to calculate the fugacity ratios and are shown in Figure 1 for 1-pentene, together with those obtained through Su’s generalized constants. T h e fugacity of either of the phases, related to the fugacity a t unit pressure, was calculated from the following relation: =
RT In RTd
(7)
T h e constants of this equation have been established by Su and Viswanath (1965) from Su’s compressibility charts. Once these values have been determined, the BWR coefficients of any substance can be calculated by Equation 7. This approach was tried for cis-2-butene and 1-pentene using the generalized constants presented by Su and Viswanath (1965). However, when the resulting values were used to calculate the fugacities of saturated vapor and liquid at equilibrium, very large deviations were observed. T h e relation fLO =
inherently less accurate for a specific component, particularly when the derivatives of the equation of state are calculated. Furthermore, since the law of corresponding states is an approximation, further errors are introduced through its use. Therefore, the reduced equation of state approach had to be modified for application to phase equilibrium calculations.
must be satisfied by the equation of state for a pure component; otherwise, it cannot be expected to predict the phase behavior of the component in the mixture. T h e generalized constants are based on many types of substances and, therefore, are
+ 2 (BoRT - A , - C,/Tz) d +
T h e deviation of any of these fugacity ratios from unity directly reflects the error in the K value of the component when its concentration in a mixture approaches unity. Figure 1 indicates that errors up to a few hundred per cent exist when the generalized constants are used, and that these errors are reduced to a maximum of 20% when the constants based on 1-butene are employed. This clearly demonstrates the advantage of using a structurally similar substance. However, even a 20% deviation is large and not tolerable in many equilibrium calculations. Consequently, further modifications were necessary. Zudkevitch and Kaufmann (1966) showed that a very significant improvement can be achieved in the BWR equation of state by making the BWR coefficient, C,, a function of temperature. Particularly, this affects the prediction of equilibrium K values and enthalpies. Earlier it was pointed out that
Coefficients for the BWR Equation of State (English units). BO A0 c, x 10-0 b a 101,605 8.933752 1.858581 34,146.5 11.32932 115,822 9.865122 37,041.5 13.09672 1.953850 136,693 10.85090 2.049167 41,683.1 16.96739 42,113.1 3.80588 19,262.64 7.827279 1 .I12799 84,056.1 7.185653 1.529053 27,980.91 12.70685 T = temperature, R. d = molal density, lb. mofe/cu.ft. Table 1.
Substance I-Butene cis-2-Butene 1-Pentene Methylacetylene 1,3-Butadiene Z . P = pressure,p.s.i.a.
116
IhEC FUNDAMENTALS
c
x
10-0
53.80707 65.36291 88.81093 21,50043 47.96007
Ly
Y
3.74417 4 34468 5.011754 1.124815 2.918084
7.59454 8.38628 9.22429 3.19593 6.03404
4.2
0.6
I 0
I
I
I
I
I
I
I
1
I
I
I
I
I
I
10
20
30
40
50
60
70
80
90
100
I
I
I
I
I
T",F
Figure 1.
Comparison of predicted fugacity ratios for 1-pentene
frequently not known accurately or are not available and, therefore, have to be estimated through some correlation. These errors might also be offset through the temperaturedependent Go's. T h e BWR coefficients of 1,3-butadiene were also determined in this study. Since no constants are available for any of its homologs, both the 1-butene and methylacetylene constants were tried to obtain these values. T h e methylacetylene constants were established by Stark and Joffe (1964) based on the P-V-T data of Vohra et al. (1962). There are some minor errors in the original publication which were subsequently corrected (Stark, 1966). I n conjunction with these constants, a set of temperature-dependent C,'s has been developed in the present study, and its effect on the fugacity ratio of methylacetylene is shown in Figure 3. Without the use of C o ( T ) the fugacity ratios start to deviate significantly from unity around 120' F. ( T , = 0.8), which is the lowest temperature covered by the data of Vohra et al. (1962). T h e evaluation of the two sets of 1,3-butadiene coefficients showed that the values based on methylacetylene are superior to those obtained from 1-butene. T h e former are given in Table I. T h e coefficients of the C, polynomials are listed in Table 11.
when the equation of state is applied to either of the phases at equilibrium it should predict equal fugacities for both phases. T o satisfy the above criterion, the function C, = f ( T ) has to be determined from vapor pressure data (Rossini et al., 1953; Smith and Thodos, 1960; Vohra et al., 1962). T h e following procedure was utilized to establish a set of temperature-dependent C,s' for each of the components. Along the vapor pressure curve, a t each temperature, a value of C, was assumed and the fugacity of both phases calculated from Equation 9. This procedure was then repeated until at each temperature Equation 8 was satisfied. T h e resulting Go's were then expressed in the analytical form : 5
C,(T)
A5TTf-'
=
(10)
i-1
T h e values of the constants are presented in Table I1 and shown graphically in Figure 2. At the critical point C,(T) is equal to that established t h r o q h the generalized procedure. There are several advantages of determining the functionality C, = f ( T ) . First, this assures that in the limit the equation of state will predict the correct K value of the pure component. Second, some of the errors involved in estimating the BWR constants are canceled out by forcing the equation of state to predict fL' == fv' along the vapor pressure curve. T h e proposed procedure would be applied mainly when the data on the P-V-T behavior of a component are very limited. I n these cases, the critical properties of the substances are
Table 11.
Evaluation of Coefficients
T h e quality of the newly established coefficients was evaluated by comparing experimental and predicted vapor-liquid
Coefficients of C, Polynominals
+
+
+
Co X IO-'' = A I 4-AzT, AaT,' A4T,3 A5T,4 C, [ = ] (p.s.i.a.) (cu. ft.)' (" R.)2/(lb. mo1e)Z
Substance
cis-2-Butene 1-Pentene Methylacetylene 1,3-Butadiene
A1 -0,290680 - 3.58381 0.298698 0.308014
Az 7,32702 29.4598 1 .22342 2.92193
A3
- 12.6342 -62.1758 - 0.794138 - 3.15673
A4 9.74800 58.9453 -0.196283 1 .20545
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- 2.84040 -20.9321 - 0,251860 ...
FEBRUARY 1968
117
1.80
1.78
1.76
1.74
1.72
1.70
1.68
1.66
1.231 0.4
I
I
I
I
I
0.5
0.6
0.7
0.8
0.9
T,,
Figure 2.
REDUCED TEMPERATURE
C, vs. reduced temperature for cis-2-butene and 1 -pentene
equilibrium K values for mixtures containing these substances. T h e available data are not very extensive, although sufficient for the present purpose. T h e following systems were used: cis-2-buteneisobutene, cis-2-butene-l,3-butadiene, 1,3-butadiene-isobutane, 1,3-butadiene-isobutylene, and 1,3-butadiene-1-butene (Esso Research and Engineering Files, 1958; Union Oil Co. of California, 1944). The results are presented in Table 111. The comparison indicated excellent agreement
2-21
0 . 9 .5
11.64 1.0
between experimental and calculated values. T h e average deviation of the cis-2-butene and 1,3-butadiene K values is about 29r,, demonstrating the reliability of the newly established coefficients and the procedure used to obtain them. The BWR constants used in these calculations for isobutane and isobutylene were those published by Benedict et al. (1951). Only for isobutane were temperature-dependent C,)s employed.
I
I
I
I
I .7
.8
.6
1
I
I
.9
1.0
Tr
Figure 3. Effect of temperature-dependent Cds on the fugacity ratio of methylacetylene 118
I I E C FUNDAMENTALS
Table 111.
T, 'F. 28 30 32 155.4 155.8
Comparison of Experimental and Predicted K-Values by the BWR Equation of State 7,3-Butadiene x Y KP P, p.s.i.a. 1.02 0,648 0,292 0.814 0,708 1.09 14.7 0.352 0.839 1.02 0,490 0.565 1.13 0.510 0.435 14.7 1.02 0.575 0.870 14.7 0,350 0,425 1.18 0.650 0.882 1.04 120.0 0.491 0.441 0,509 0.559 1.06 1.02 0.557 120.0 0.443 0.896 0,524 1.08 0.476
0.649 0.706 0.768
18 22 26
14.7 14.7 14.7
0.366 0.420 0.665
0,240 0.380 0.528
14 16 18
14.7 14.7 14.7
0.515 0.648 0.753
7,3-Butadiene 0.823 0.427 0.559 0.849 0.672 0,880
143.1
120.0
0.559
0.542
1 .oo 1.02 1.03 1.03 0.99
1.03 1.03
0.634 0.480 0.335
Is0 butane 0.760 1.17 0.620 1.27 0,472 1.38
1.02 1.01 1.02
1.01 1.02 1.01
0,485 0.352 0.247
0.573 0.441 0,328
1.11 1.17 1.24
1.07 1.07 1.07
1.03
0.441
Isobutylene 0.458 1 .oo
1.03
1.01
0.940
KexplKprad
1-Butene
146.8
120.0
0.611
0.604
0,978
I n the absence of any available data on 1-pentene, no such comparison is possible for this substance. However, the results for cis-2-butene give confidence about the reliability of the I-pentene constants. Although the major objective of this work was to develop BWR coefficients that permit accurate phase equilibrium calculations for systems containing 1-pentene, cis-2-butene, and 1,3-butadiene in mixture with other hydrocarbons, the new coefficients were also tested for predicting the P-V-T behavior of one of these substances. Superheated vapor densities of 1,3-butadiene were measured by Scott et a / . (1945) over the temperature range 87.4' to 300.2' F. and in the pressure range 42.7 to 596.7 p.s.i.a. T h e comparison of experimental and calculated pressures indicated good agreement, the over-all average deviation being 0.50% with a maximum deviation of 1.1%. T h e results at the 300.2 isotherm (6' F. from the critical isotherm) are shown in Table IV. T h e comparison indicates that the coefficients established in this investigation can be used with confidence to predict the P-V-T behavior of these substances. When the coefficient C, is a function of temperature, the original BWR expression for calculating isothermal pressure effects on enthalpy, entropy, and heat capacity has to be modified to include explicitly the temperature dependence of this term.
Table IV. Comparison of Experimental and Predicted P-V-T Data for 1,3-Butacliene at the 300.2' F. Isotherm
T , a F. 300.22
300.16
(Critical remperature. u , cu. Ft./ Pexp, Lb. Mole P.S.I.A. 70.33 108.07 51.45 144.62 28.08 246.45 17.04 360.07 13.08 429.07 11.94 457.06 10.37 485.20 9.861 499.28 9.358 513.35 8.795 527.47 8.258 541.59 7.530 555.66 6.742 569.74
306" F.) Ppred,
P.S.I.A. 108.54 144.75 245.47 360.94 429.96 453.62 489.31 501.24 513.32 526.92 539.74 556.39 572.39
yo Dev. 0.43 0.09 -0.40 0.24 0.21 -0.75 0.85 0.39 0.01 -0.10
-0.34 -0.13 0.46
1.01
0.389
0.396
1.02
0.995
Conclusions
A simplified procedure has been developed for obtaining the constants of the Benedict-Webb-Rubin equation of state, which is particularly suitable to phase equilibrium calculations. T h e evaluation of the constants consists of three steps: T h e coefficients of the reduced BWR equation are determined based on a structurally similar substance. Su's modified theorem of corresponding states is used to obtain the reduced form of the equation of state. With the aid of the reduced coefficients, the constants for the components of interest are calculated. T h e function C, = f ( T ) is established in conjunction with the new coefficients, so that the relationfLo = fv" is satisfied. Using this method the BWR coefficients have been obtained for cis-2-butene, I-pentene, and 1,3-butadiene. T h e validity of the new constants was demonstrated by the good agreement between experimental and predicted K values.
Ac knowledgment
T h e author is grateful to the Esso Research and Engineering Co. for releasing this work for publication; to the Research Department of the Union Oil Co. of California for permitting the use of its vapor-liquid equilibrium data; and to R. H. Johnston for his helpful comments.
Nomenclature
A,, (I,Bo, b , C, G, C Y , y = constants of BWR equation of state Ao', a', Bo', b ' , Cot,G', C Y 'y' , = constants of reduced BWR d
f K P
P, P,
R T T, VCi
= =
= =
= = = = = =
z,
= =
w
=
I
equation of state molal density, lb. mole/cu. ft. fugacity, p.s.i.a. equilibrium constant, dimensionless pressure, p.s.i.a. critical pressure, p.s.i.a. reduced pressure, dimensionless universal gas constant temperature, ' R. reduced temperature, dimensionless pseudocritical volume, cu. ft./lb. mole compressibility factor, dimensionless critical compressibility factor, dimensionless acentric factor, dimensionless VOL. 7
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119
SUBSCRIPTS A N D SUPERSCRIPTS
exp = experimental L = liquid pred = predicted V = vapor O = pure component property
literature Cited
Beattie, J. A,, Marple, S. J., J . Am. Chem. Sac. 72,4143 (1950). Benedict, M., Webb, G. B., Rubin, L. C., Chem. Eng. Progr. 47, 413 (1951). Benedict, M., Webb, G. B., Rubin, L. C., J . Chem. Phys. 10, 747 (1942). Beyer, H. H., Griskey, R. G., A.Z.Ch.E. J . IO, 764 (1964). Canjar, L. N., Smith, R. F., Volianitis, E., Galluzzo, J. F., CabarCOS,M.,Znd. Eng. Chem. 47,1028 (1955). Esso Research and Engineering Files, unpublished data, 1958. Griskey, R. G., Canjar, L. N., A.1.Ch.E. J . 9,182 (1963). Joffe, J., Chem. Eng. Progr. 45, 160 (1949). Lin, M., Naphtali, L. M., A.Z.Ch.E. J. 9,580 (1963).
Meissner, H.P., Seferian, R., Chem. Eng. Progr. 47, 579 (1951). Opfell, J. B., Sage, B. H., Pitzer, K. S., Znd. Eng. Chem. 48, 2069 (1956). Pitzer, K. S., Lippmann, D. Z., API Project 50, University of California, Berkeley, 1954. Rossini, F. D., Pitzer, K. S., Arnett, R. L., Brown, R. M., Pimental, G. C., API Project 44, Carnegie Press, Pittsburgh, 1953. Sage, B. H., Lacey, W. N.,Znd. Eng. Chem. 40,1299 (1948). Scott, R. B., Myers, C. H., Rands, R. D., Brickwedde, F. G., Bekkedahl, N., J . Res. Natl. Bur. Std. 35,39 (1945). Smith, C. H., Thodos, G., A.1.Ch.E. J . 6 , 569 (1960). Su, G. S., Sc. D. thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1937. Su. G. S.. Viswanath. D. S.. A.1.Ch.E. J . 11.205- 11965’1. Stark, T.’M., J . Chem’. Eng.’Daia 11,570 (1966). Stark, T. M., Joffe, J., J . Chem. Eng. Data 9 , 327 (1964). Union Oil Co. of California, Research Department, unpublished data, 1944. Vohra, S. P., Kang, T. L., Kobe, K. A., McKetta, J. J., J. Chem. Eng. Data 7,150 (1962). Waals, J. D. van der, dissertation, Leiden, 1873. Zudkevitch, D., Kaufmann, T. G., A.Z.Ch.E. J . 12, 577 (1966). - 7 -
\ -
- I .
- .
RECEIVED for review March 9, 196 ACCEPTED July 25, 196
ESTIMATION OF RATE CONSTANTS FROM MULTIRESPONSE KINETIC DATA R E l J l M E Z A K I A N D J O H N B. B U T T Department of Engineering and Applied Science, Yale University, New Haven, Conn. Alternative procedures for the estimation of several parameters from multivariate observations are evaluated for a complex system. Primary attention is given to methods based on generalization of nonlinear least squares and Bayesian estimation as proposed by Box and Draper. The latter method provides an effective means for estimation even in very complicated cases, and demonstrates a parametric sensitivity which allows a precision of estimation not possible with other techniques.
HE
establishment of systematic procedures for the deter-
T mination of constants involved in many-parameter, nonlinear models, and for choosing between alternative models, has assumed considerable importance in the current literature. Various rate schemes for chemical reactions (both homogeneous and heterogeneous) have provided the favorite means for investigation of these procedures. Development and application of both linear and nonlinear estimation procedures have been documented (Box, 1960; Hougen and Watson, 1947; Kittrell et al., 1965; Lapidus and Peterson, 1966) and extensively discussed. Nonlinear estimation of several parameters from data ordinarily available for these reaction systems is sometimes difficult. These difficulties often stem from a lack of sufficient information regarding the reaction system; however, in many reactions of practical significance a number of reactants, intermediates, and products may be involved and a large amount of information is potentially available if only it is measured. It is, thus, of interest to investigate estimation procedures applicable when more than a single response variable is measured and to compare the results of specific multiresponse techniques with those obtained by alternative methods. 120
I&EC FUNDAMEN.TALS
Estimation Procedures and Criteria
T h e principal procedures for determination of parameters in multivariate systems which may be nonlinear have been summarized as follows (Hunter, 1967): Generalization of Nonlinear Least Squares. T h e best fit criterion is given by :
This criterion is the proper one only under certain restrictive conditions, as discussed later (Hunter, 1967). A corresponding weighted least squares criterion can be applied if the variance of the data is known. This ordinarily is not the case, so comparison with such a weighted criterion is not included in the present work. Bayesian Estimation. When the values of variance and covariance are unknown and a number of variables have been measured, best estimation criterion is given by :
