Ind. Eng. Chem. Process Des. Dev. 1980, 79, 501-505 Feng, C. F., Stewart, W. E., Ind. Eng. Chem. Fundam., 12, 143 (1973). Feng, C. F., Kostrov, V. V., Stewart, W. E., Ind. Eng. Chem. Fundam., 13, 5 (1974). (;rover, S. S.,Chim. Ind. S n i e Chim., 103, 93 (1970). a n n , R. D., King, C. J.. AlChE J., 15, 507 (1969). Hite, R. H., Jackson, R., Chem. Eng. Sci., 32, 703 (1977). Hymn, M. H., Hydrocarboii Process., 47, 131 (1968). Kehoe, J. P. G., Aris, R., Chem. Eng. Sci., 28, 2094 (1973). Laborde, M. A., Gonzsiez, M. G., Williams, R . J. J., Proceedings of the ILatin American Petrochemical Congress, Instituto Petroquimico Argentino, Voi. 5, p. 323, 1976. Mason. E. A.. Maiinauskas A. P.. Evans. R. 0.. J. Chem. Phw., 48. 3199 (1967). Singh, Ch. P. P., Saraf, D. N., Ind. Eng Chem . Process Des. Dev., 18, 1 (1979).
.
50 1
Wiike, C. R., J. Chem. Phys., 18, 517 (1950).
Center of Research & Development Maria G . Gonztilez i n Catalytic Processes (CINDECA) Miguel A. Laborde (1900) La Plata, Argentina Roberto J. J. Williams* Department of Chemical Engineering Uniuersity of Mar del Plats (7600) Mar 'lata, Argentina Received f o r review August 31, 1979 Accepted February 21, 1980
Modified Soave! Equation of State for Phase Equilibrium Calculations Varicus forms of the Soave equation of state which could be used for hydrogen-containing mixtures are tested with (experimentalvapor-liquid equilibrium data. The original Soave method with properly determined values of the irtteraction constant C, appears to represent the data reasonably well, even for the systems containing heavy, nonparaffinic hydrocarbons. The values of C, are correlated with the solubility parameters. The modified Soave equation of Graboski and Daubert is found to work well only for the systems containing light hydrocarbons. Some comrnents are made on the results of comparison.
Introduction The Soave (1972) modification of the Redlich-Kwong equation of state (SRK) has found wide acceptance, particularly for predicting vapor-liquid equilibrium (VLE). The equation has the following form Cia p : = -R- -T u - b U(U + b ) where the adimensiclnal factor cy is a function of temperature cyi = (1 + mi(l - Tr/',5))2 (2) Soave (1972) correlated the slope m, against acentric factor wi. Recently, Graboelkiand Daubert (1978) redetermined
b,ased on a regression of vapor pressure the function for mi, data, and applied the result to correlate VLE data for hydrocarbon mixtures and for binary mixtures of hydrocarbons with COz, H2S, N2,and CO. They have also extended the Soave equation to describe the VLE behavior for systems containing hydrogen (Graboski and Daubert, 1979). However, they found that the basic Soave procedure failed to give accur,ate results for hydrogen K values. Subsequently, they recommended a new expression for cy of hydrogen &H[ = C1 exp(-CzTI) (3) with C1 = 1.202 and Cz = 0.30228. In the VLE calculations for mixtures, Graboski and Daubert applied the following mixing rules as suggested by Soave cy12 = CCxixJcyiJaij (4) i i
The cross parameter
cyiicyij
a..a.. V 11 == (1 -
in eq 4 is given by C i j ) ( f1f .1f f 1. a1. a . ) ' / 2
(6)
In general, the interaction parameter Cij was obtained from VLE data. For hydrocarbon mixtures, Soave recommended Cqi = 0. Graboski and Daubert (1978,1979) have found that the modified Soave equation worked well for representing the 0196-4305/80/1119-0501$01.00/0
VLE behavior of a variety of mixtures. However, the basic data used in their calculations were composed of only relatively light hydrocarbons. The Soave equation and its modifications have never been tested with the VLE data of mixtures containing heavy compounds. Chao and coworkers (1978) at Purdue University have recently reported experimental VLE data of a light gas (H2, CH4,and COz)in mixtures with heavy hydrocarbons at temperatures up to 430 OC and pressures to 250 atm. Data of this type are sensitive to the mixing rules and mixture parameters and, therefore, provide a severe test of the correlation methods. Much effort has also been made by this group to correlate those new data. I would like to report in this correspondence some of the calculated results from the Soave equation of state and to comment on certain arguable statements made by Graboski and Daubert. Results and Discussion The discussion here will concentrate on binary mixtures containing hydrogen. Table I shows the systems of interest and the data sources. Emphasis is placed on the data of hydrogen in heavy aromatic and naphthenic compounds taken a t Purdue. For light paraffins only representative systems (and representative data sets for a particular system) are reported. VLE data of this class are extensively available in the literature from several different studies. These are the basic data used by Graboski and Daubert, and also by Gray (1977), to test the Soave equation of state. The representative systems of hydrogen with light hydrocarbons shown in the table are those that cover a wide range of temperature or pressure. Also included in the table are the physical properties used in the calculations. These values are taken from the literature (essentially from Reid, Prausnitz, and Sherwood, 1977), if available. Otherwise, estimation methods described by Reid et al. (1977) were applied, while the acentric factors were calculated from the experimental vapor pressure data at TI = 0.7 according to the definition by Pitzer. Table I1 summarizes the calculated results from the original Soave procedure and alternative methods. The values reported in the table are the absolute average percent deviations (AAD) of the calculated K values from experiments. Graboski and Daubert used AAD to justify their @ 1980 American Chemical Society
502
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
Table I. Data Sources for Hydrogen-Containing Binary Mixtures ranges
no. of data points P,, atm
physical properties
T,K
P, atm
methane
116117 2
331272
13
45.4
190.6
carbon dioxide
2201290
111370
40
72.9
propane
2231348
131204
35
butane n-heptane
3 27 1394 4241498
271166 241177
n-octane
4631543
n-decane benzene m-xylene m-cresol toluene tetralin l-methylnaphthalene diphenylmethane bicyclohexyl quinoline thianaphthene
4261583 4331533 4621581 4621661 4611575 4631662 4621701 4631702 4621702 4621701 4611701
solvent
T,,K
6, ( C d /
cm3)0.5
references
0.008
5.68
304.2
0.225
6.0
41.9
369.8
0.152
6.4
60 26
37.5 27.0
425.2 540.2
0.193 0.351
6.73 7.43
91148
50
24.5
568.8
0.394
7.46
191250 201176 201250 201250 201250 201250 201250 201250 201250 201250 201250
26 49 27 41 25 24 27 27 28 27 26
21.3 48.3 35.0 45.0 40.6 33.1 35.2 28.2 25.8 45.0 38.3
617.5 562.1 617.0 705.9 591.7 716.5 772.0 770.2 731.4 782.0 752.1
0.490 0.21 2 0.331 0.465 0.257 0.315 0.302 0.434 0.388 0.333 0.283
7.72 9.16 8.84 10.78 8.93 9.5 10.02 9.6 8.55 10.9 10.56
Benham and Katz (1957) Yorizane et al. (1970); Spano et al. (1968) Trust and Kurata (1971) Mink et al. (1975) Peter and Reinhartz (1960) Connolly and Kandalic (1963) Chao et al. (1978) Connolly ( 196 2) Chao et al. (1978) Chao et al. (1978) Chao et al. (1978) Chao et al. (1978) Chao et al. (1978) Chao et al. (1978) Chao et al. (1978) Chao et al. (1978) Chao et al. (1978)
results. The methods used to obtain the results in Table I1 for comparisons are characterized in Table 111. A brief discussion of these methods is given below. SRK(a) represents the method originally suggested by Soave. The classical critical constants of hydrogen were used together with a value of -0.22 for acentric factor, w H , which was derived from experimental vapor pressure. Equation 2 was applied to extrapolate above the critical temperature for hydrogen gas. In this study, the critical properties were taken from McCarty (1975). The interaction parameter C..was treated as an adjustable paramfrom VLE data. The criterion used eter and to find Cjj was the minimization of the sum of squares in the relative deviations of K values of both the hydrogen (KH) and the solvent ( K s ) ,viz.
determines
W
The values of Cij calculated from eq 9 are about 0.04-0,13 larger than those from eq 8. Slight reduction in AAD was obtained with eq 9 for the K values of hydrogen, as compared with the results from eq 8. The comparisons are shown in Table I1 under the column SRK(b). Graboski and Daubert (1979) developed an alternative expression, eq 3, for the temperature dependence of a for hydrogen. They recommended using the “effective” critical constants ( T , = 41.67 K and P, = 20.75 atm) for hydrogen together with WH = 0 and C , = 0. The deviations listed in the column SGD(a) are the results obtained by using this original Graboski-Daubert version of the Soave method. Chueh and Prausnitz (1967) proposed a temperaturedependent pseudocritical temperature and pressure for quantum gases. For hydrogen, the expressions are
T, = The values of C , so determined were then correlated with the solubility parameters C,= 0 A6 < 2.4 = 0.5187 - 0.6156(A6) + 0.2079(A6)’ 2.4 I A6 I7.0 (8) 0.01663(A6)3 = 0.7
A6
> 7.0
where A6 = - aHI. 6H (= 3.25) and are the solubility parameters at 25 “C of hydrogen and the solvent, respectively. Equation 8 was obtained from the best fit to all the VLE data of hydrogen-containing mixtures we have compiled, including those not reported in the table. A detailed description of the determination for eq 8 will be given in a separate paper which is in preparation. In the method of SRK(b), the correlated Cijwas used. The results were found to be comparable with those from SRK(a) for most of the systems studied. It is also interesting to note that a better correlation for the systems containing hydrogen with heavy aromatic and naphthenic compounds, reported by Chao and co-workers, is with Cij
= -0.2233A6
+ 0.1273(A6)’ - 0.01134(A6)3
2.4 5 A6 < 7.6 (9)
P, =
43.6 1 + 10.81/T 20.47
1
+ 21.92/T
The critical constants used by Graboski and Daubert are approximately the high temperature limits of eq 10 and 11. Applying these equations to the Graboski and Daubert procedure for the determination of C1 and C2 in eq 3 led to C1 = 1.610 and C2 = 0.31368. The deviations calculated using this method are reported in Table I1 in the column headed SGD(b). The VLE data used in the regressions for SGD(b) are composed essentially of light hydrocarbons, similar to those used by Graboski and Daubert to obtain eq 3. The results show slight improvement over SGD(a). The improvement is significant only at lower temperatures. VLE data of hydrogen in heavy hydrocarbons have recently become available due to Chao and co-workers. If those data are included in the least-squares regressions to redetermine C1 and C2 in eq 3, the result leads to the methods designated as MGD(a) and MGD(b). In MGD(a), the critical constants of Graboski and Daubert were used, while eq 10 and 11 were applied in MGD(b). As shown in Table 11, these new versions of the Graboski and Daubert method reproduced the K values of hydrogen with better accuracy.
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980 503
N Y 9 *?1? Y 11E? 11 Y E? 9 1Y m d Q, (1) m rl w m (0 w w m w 0 e- Q, w rl rl
c9 "I Y Q? ? ? Y Y E? 1 dl 1? ? "I p: Y rl m 7 4 CO 0 CD m w m w Q, w 0 w r? w
cy
rl
Tests of the SGD and the MGD methods were made with the PVT data of hydrogen (McCarty, 1975) over the temperature range 100-2800 K and pressures up to 1020 atm. The results are found to be comparable among these methods, which all show reasonable agreement with the literature data. The data below 100 K were not used to avoid complications due to quantum effects. The bottom line of Table I1 presents the average AAD totals for all the systems shown. However, note that both H2 + n-octane and H2 + thianaphthene systems require a Cij in SRK(a) greater than unity to minimize root mean square errors. The reason for this deficiency in H2 + octane is not known, although Gray stated that the problem was due to the data a t temperatures relatively close to the critical temperature of octane. The critical constants estimated from Lyderson's method are much in doubt for thianaphthene. As a result, the deviations for the H2 + thianaphthene system are exceedingly large for all the methods other than SRK(a). The overall AAD shown in the parentheses of Table I1 are those found by excluding H2 thianaphthene mixtures. Some comments about the comparisons given in the table can be made. (1) The Soave equation and its modifications, SRK(a) and (b), represent the VLE behavior for hydrogen-containing mixtures reasonably well. The results are very impressive for the systems of hydrogen with heavy aromatic and naphthenic hydrocarbons, particularly, considering the uncertainties involved in the correlation of the vapor pressures by the Soave equation for these types of heavy compounds and in the estimations of their critical constants. Graboski and Daubert (1978a) showed that the Soave equation correlated the vapor pressures of naphthenes and aromatics with an average error of 4.2% and 4.070,respectively. The errors are significantly larger than those for paraffins (1.7%). The heaviest aromatic and naphthenic compounds used in their correlations were Clo. Further improvement in the ability of the Soave equation to represent the vapor pressures of these classes of compounds is necessary in order to improve the results of predicting the VLE behavior of their mixtures. Experimental vapor pressure data for these compounds are in demand for this purpose. Furthermore, the critical constants, which are required in the VLE calculations, are only estimates for most of the heavy compounds reported in Table I. Their accuracy is uncertain. Accurate information for these properties is also needed to improve the correlated results. (2) The original method suggested by Soave, SRK(a), is found to work the best among the methods studied. Even with the correlated values for Cij from eq 8 or eq 9, the SRK(b) still proved superior to the modified Soave method of Graboski and Daubert (SGD). This conclusion also applies to systems not reported in Table 11. A particularly noteworthy advantage of SRK over SGD is that the deviations of the calculated results from experimental values for SRK methods do not appear to increase as the molecular weight of the hydrocarbon solvent is increased, while the SGD methods fail to represent the K values of hydrogen in heavy hydrocarbons as accurately as in the lighter solvents. Gray has also compared the root mean square deviations in K values calculated from the original Soave procedure with those from the Graboski and Daubert method for 17 representative binary mixtures. The data used by Gray are essentially mixtures of hydrogen with paraffins. His results also confirmed the superiority of SRK(a) over SGD(a). The average root mean square deviations in K values of hydrogen and solvent for the 17 systems are 9% from SRK(a) and 13% from SGD(a). This
+
E???
"Y""?p:?YY???E?1
Q,wmdmmmwOr?O4rlrlCOwm rl
ri
rlrlrlrl
N
EQ?," Im? Ed? d? *w0 9d? m " I EQ? ?, Er?l0w9 ?O? ?w" N N w w t rl
4
rlrlrlrl
N
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ow~~ww~wr?Q,c-owt-Q,orl rl
rl
rl
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504
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
Table 111. Characterization of Methods Shown in Table I1 hydrogen properties
T,,K P,, atm w
SRK( a)
33.19 12.98 -0.22 best fit to data eq 2
Cij
__
01
__
c, c2
SRK( b)
SGD(a)
SGD( b)
MGD( a)
MGD(b)
33.19 12.98 -0.22 eq 8 or 9 eq 2
41.67 20.75 0
eq 10 eq 11 0 0 eq 3
41.67 20.75
eq 10 eq 11
0 0 eq 3
0 0
__
I
is contrary to the statement made by Graboski and Daubert that the basic Soave method with interaction coefficient Cij did not predict hydrogen K values accurately, but that their modified method correlated the data very well. (3) K values of hydrogen are fairly sensitive to the interaction constant Cij used in the SRK method. Without use of this constant, the overall average errors of the predicted K values of hydrogen and the heavy component were found to be 22 and 9%, respectively, for all the mixtures reported in Table 11. Both SGD and MGD methods were applied with C , = 0. Additional calculations were thus performed for these methods to adjust C, to obtain the best fit of data. For light paraffins, adjusting Cij in the Graboski and Daubert methods did not significantly improve the results. Reduction in the errors of hydrogen K values was observed with the optimum C,for the systems containing heavy hydrocarbon compounds, but the overall errors in K values of the heavy component were slightly increased. In all cases, the Graboski and Daubert methods represent the data no better than the original Soave methods. Further calculations for two ternary diphenylmethane and mixtures, hydrogen tetralin hydrogen tetralin m-xylene (Oliphant et al., 1979), appear to support this conclusion. The comparisons shown in Table I1 also indicate that the original Graboski and Daubert method, SGD(a), predicted the K values of hydrogen with satisfactory results only for the mixtures of hydrogen in light hydrocarbon solvents. Extension of this method to systems containing heavy aromatic and naphthenic compounds did not meet with much success, although the results are not unacceptable. The modified Graboski and Daubert methods (MGD) are found to slightly improve the accuracy, but are no better than the original Soave methods. This implies that eq 3, even with the redetermined values of C1and C2, seems to be not flexible enough to describe the temperature dependence for CY for hydrogen. (4) Graboski and Daubert tested their modified Soave equation with data of Chao et al. for binary mixtures of hydrogen in 1-methylnaphthalene, bicyclohexyl, tetralin, and diphenylmethane. The average deviations were found to be 18.6% and 10.2% in K values of hydrogen and the heavy component, respectively. Their proposed method also reproduced the K values of hydrogen in phenanthrene by DeVaney et al. (1977) to 32%. They concluded, “based on the anticipated errors in the available data, a much better correlation is probably not possible and it is proposed that the modified CY function (for hydrogen) be used directly for hydrogen systems without interaction coefficients”. They furthermore stated that their method is “capable of extrapolation to new mixtures without need for data to determine interaction coefficients”. These conclusions are debatable. The success of extrapolating the SGD(a) method to predict the VLE data for hydrogen in heavy hydrocarbon compounds is uncertain, as illustrated in Table 11. However, further improvement of the correlations requires more extensive data on VLE of
+
+ +
+
0 eq 3
1.202 0.30228
1.610 0.31368
1.993 0.40197
eq 3
2.372 0.40310
mixtures and the volumetric properties of their constituent compounds. After all, the original Soave method was already shown to represent the VLE behavior reasonably well. The modified method of Graboski and Daubert makes no appreciable improvement. (5) The interaction parameter Cij in eq 6 corrects for the deviation from the geometric mean rule for combining aiai. Strictly speaking, Cij here is not the same parameter k.. as described by Chueh and Prausnitz (1967). Chueh ana Prausnitz defined kij as a factor to correct for the geometric mean rule for the cross critical temperature (TCJand suggested that the best sources to obtain kij’s were from cross second virial coefficients, Bij. From a theoretical point of view, the physical significance of kij is different than that of C,. The explanation for this point can be found from the treatment of Bij in statistical mechanics. In brief, k’.corresponds to a factor which corrects for the deviation from the geometric mean rule for combining the energy parameters ti and e, of intermolecular forces d e t u ) . The parameter a in eq 1, which reflects the contributions due to intermolecular attractive forces, has the dimension of energy X volume. It is related to t X u3 (e.g., see the examples given by Reed and Gubbins, 1973). Consequently, the correction factor Cij in eq 6 accounts for the effects of both e (energy) and u (size). The interpretation of Cijand CYUby Graboski and Daubert in view of molecular theory is not adequate. (6) Graboski and Daubert suggested using Cij = 0 for hydrocarbon mixtures. However, we have found that the parameter Cij is necessary for mixtures of methane with heavy hydrocarbon compounds. For instance, the average AAD in K values of methane for the binary systems of CHI with benzene, toluene, rn-xylene, m-cresol, tetralin, 1methylnaphthalene, and quinoline is 4.8% with the optimum Cij, as compared with the result of 12.3% obtained with Cij = 0. No appreciable improvement was found in the K values of the solvents by adjusting C,. Similar results were observed in two ternary mixtures: hydrogen + methane + 1-methylnaphthalene (Sebastian et al., 1980) and hydrogen + methane + tetralin (Simnick et al., 1980). For light paraffins, the values of Cij are extremely small and Cij = 0 is a good approximation. I certainly agree with the conclusions from many other studies that at least one correction factor is required to correct for the deviations from such simple mixing rules as the geometric or arithmetic mean rules used to account for the dissimilarity (particularly the molecular sizes) of the constituent molecules in the mixtures. With this one adjustable parameter, the Soave equation of state reproduces the VLE data for all the binary mixtures of hydrocarbons with hydrogen, methane, and carbon dioxide, which we have studied, with reasonable accuracy. Graboski and Daubert found a significant improvement in K values if C, was adjusted to best fit the VLE data for systems containing COz. The values of Cij so obtained were correlated with the solubility parameters. Their method for correlating Cij was also found to work well for hydrogencontaining and methane-containing systems.
505
Ind. Eng. Chem. Process Des. Dev. 1980, 19, 505-507
Literature Cited Benham, A. L., Katz, D. L., AIChE J., 3, 33 (1957). Chao, K. C., Lin, H. M., Simnick, J. J., Sebastain, H. M., references for H, hydrocarbon and CH, hydrocarbon systems were ctted in Annual Report, EPRI Project RP 367-2, 19'78; results for COS-containingmixtures are unpublished. Chueh, P. L., Prausnitz, J. M., Ind. Eng. Chem. Fundam., 6, 492 (1967). Connolly, J. F., J. Chem. Phys., 36, 2897 (1962). Connolly, J. F., Kandalic, G. A., API 28th Midyear Meeting, Philadelphia, PA, May 13, 1963 (Paper 14-63), DeVaney, W. E., Kao, P. L., Borryman, J. M., Progress Report to GPA, 1977. Graboski, M. S., Daubert, T. E., Ind. Eng. Chem. Process Des. Dev., 17,443 (1978a). Graboski, M. S., Daubert, T. E., Ind. Eng. Chem. Process Des. D e v . , I ? , 448 (1978b). Graboski, M. S., Daubert, T. E., Ind. Eng. Chern. Process D e s . Dev., 18, 300 (1979). Gray, R. D., "Correlation of H,/Hydrocarbon VLE Using Redlich-Kwong Variants", presented at 70th Annual AIChE Meeting, New York, 1977. Kiink, A. E., Cheh, H. Y., Amick, E. H., AIChE J., 21, 1142 (1975). McCarty, R. D., "Hydrogen Technological Survey-Thermophysical Properties", NASA-SP-3089 (1975).
+
+
Oiiphant, J. L., Lin, H. M., Chao, K. C., Fluid Phase Equilib., 3 , 35 (1979). Peter, S., Reinhartz, K., Z. Phys. Chem., 24, (1960). Reed, T. M., Gubbins, K. E., "Applied Statistical Mechanics", Chapters 7 and 9, McGraw-Hiii, New York, 1973. Reid, R. C., Prausnltz, J. M., Sherwood, T. K., "The Properties of Gases and Liquids", 3rd ed, Appendix A, pp 629-665, McGraw-Hill, New York, 1977. Sebastian, H. M., Lin, H. M., Chao, K. C., submitted to J. Chem. €no. Data for publication (1980). Simnlck, J. J., Sebastian, H. M., Lin, H. M., Chao, K. C., J. Chem. Eng. Data, in press (1980). Soave, G., Chem. Eng. Sci., 27, 1197 (1972). Spano, J. O., Heck, C. K., b r i c k , P. L., J. Chem. Eng. Data, 13, 168 (1968). Trust, D. 9.. Kurata, F., AIChE J., 17, 86 (1971). Yorizane, M., Yoshimura, S., Masouka, H., Kagaku KOgakU, 34, 953 (1970).
School of Chemical Engineering Purdue University West Lafayette, Indiana 47907
Ho-mu Lin
Received for review August 10, 1979 Accepted M a r c h 24, 1980
Distillation Calculation for Cases of Specified Heat Duty by Modification of Existing Computer Programs Some algorithms applied to solve problems of distillation in which heat duty of the reboiler or the condenser is specified, but the reflux ratio is not, are discussed. A useful and flexible algorithm is presented, in which an existing computer program for an operating column analysis is employed without change, and a useful correction loop for the reflux ratio is added externally. This loop is obtained from the relationship between the reflux ratio and an overall energy balance. This algorithm is readily applied because It is independent of the calculation technique used with an existing computer program. Numerical examples indicate the effectiveness and reliability of this algorithm.
Introduction In an operating distillation column analysis, the reflux ratio is usually specified. However, we must sometimes solve problems of multicomponent distillation in which the heat duty of the reboiler or the condenser is specified instead. Some algorithms, which are applicable to problems like this, are discussed. We then propose a simple algorithm in which an existing computer program for an operating column analysis is employed without any change. In this algorithm, the correction loop for the reflux ratio, which is obtained from the relationship between the reflux ratio and the energy balance of a distillation column, is added to the outside of an existing computer program for an operating column analysis. This algorithm is independent of the calculation technique of the existing computer program, e.g. the Newton-Raphson method (Naphtali and Sandholm, 1971; Kubichek et al., 1976; Kawase et al., 1978), the tridiagonal matrix method (Wang and Henke, 1966), and so on. Therefore, this algorithm is flexible and readily applied. Its effectiveness is discussed by numerical examples for the lower column of an air separation plant. Algorithm and Calculation Procedure For solving distillation problems in which the heat duty of the reboiler or the condenser is specified but the reflux ratio is not, some algorithms may be available. First, consider algorithms in which it is necessary to revise an existing computer program. In the case of the NewtonRaphson method, we must replace the dependent variable Q, or QR with L1(or R ) (Kubichek et al., 1976). When the dependent variable Q, is replaced by L1,this problem is easily solved by using the Thomas algorithm without revision because of the block tridiagonal form of its Jacobian matrix. However, we must change the computer program 0196-4305/80/1119-0505$01.00/0
according to the replacement of the dependent variable. When QR is specified and replaced as a dependent variable by L1,an off-band element occurs in the Jacobian matrix. Therefore, any existing computer program must be drastically changed because it is impossible to apply the Thomas algorithm directly. Instead, a modified Thomas algorithm to solve this problem must be employed (Kubichek et al., 1976; Waggoner and Loud, 1977; Hofeling and Seader, 1978). In the case of the tridiagonal matrix method (Wang and Henke, 19661, the procedure for evaluating the vapor flow rate Vi and the liquid flow rate Lj by the energy balance equation and the overall material balance equation must be changed. A schematic representation of the j t h plate of the distillation column is shown in Figure 1. When Q, is specified but L1(or R ) is not, the computational procedure for obtaining new values of V; and Li is (step 1) Ll = -iQc + V1Wl - H2)l/(hl - H2)
(1)
(step 2) v, = v1 + L1 (2) (step 3) Vj+l = [(Vj+ Wj)(Hj- hj) - Lj-l(h;-l- hj) - Fj(H., hi) + Qjl/(Hj+~ - hj) (3) 0'=2,N-l) (4)
0'=2,N-l) where F1= W1 = 0 and D = Vl + U1. Values for Liand Vj are successively obtained from plate 1 to plate N by using eq 1 to 4. In a usual operating column analysis, where L1(or R) is specified, eq 1 is not necessary. 0 1980 American
Chemical Society
