168
Ind. Eng. Chem. Process Des. Dev. 1881, 20, 168-172
Inspection of eq 14-18 reveals that they express the boiloff rates as functions of auxiliary state variables and hence, indirectly, as functions of the state variables. These flows are the “balancing flows” needed to maintain the identity of the pressures given by eq 1 and 2. I t should be noted that eq 14 is linear, and hence the problem of determining w at each time step is relatively simple. Explicit solutions can be worked out for cases where the number of components does not exceed three, and after that, use of a linear equation solving package is required. At this stage it is possible, from a knowledge of the current system state, to evaluate the time derivatives dMl,,/dt from eq 4, dML/dt from eq 7, and dT,/dt from eq 9. Hence a solution has been found in the sense described in the Introduction, and a time-marching integration algorithm may be used to integrate the equations through some prescribed transient. Since the equations describe a general plate, it is possible to use the same procedure to build up a multi-plate simulation, by programming a self-contained simulation language MACRO block and calling it successively to calculate the behavior of each plate in the column. Conclusions The digital dynamic simulation of distillation processes has been reviewed, and a new approach has been presented which allows for consideration of the effect of varying vapor holdups. It has been found that calculation of the instantaneous component boil-off rates reduces to the problem of solving a set of linear simultaneous equations of the same order as the number of components present. Because a general plate is considered, this method is suitable for programming as a MACRO module in a continuous system simulation language. Nomenclature A = matrix as defined in eq 14 b = vector as defined in eq 14 f = function used in vapor equation of state Fl = liquid feed flow, kg-mol/s F, = vapor feed flow, kg-mol/s g, = vapor flow function, kg-mol/s g2 = Weir function, kg-mol/s g3 = liquid enthalpy function, kJ/kg-mol g4 = vapor enthalpy function, kJ/kg-mol h = liquid enthalpy, kJ/kg-mol
hf = feed liquid enthalpy, kJ/kg-mol H = vapor enthalpy, kJ/kg-mol Hf = feed vapor enthalpy, kJ/kg-mol K = vapor pressure function L = liquid flow, kg mol/s M I= liquid mass, kg-mol M, = vapor mass, kg-mol P = pressure, bar Ph = liquid head pressure, bar Q = heat input, kW S = total plate volume, m3 SI = liquid volume, m3 S, = vapor volume, m3 T = temperature, K u = liquid specific internal energy, kJ/kg-mol U = vapor specific internal energy, kJ/kg-mol V = vapor flow, kg-mol/s U I = liquid specific volume, m3/kg-mol w = vector of boil-off rates W = boil-off rate, kg-mol/s x = liquid fraction y = vapor fraction z1 = feed liquid fraction z, = feed vapor fraction A = diagonal matrix derived in eq 14 Subscripts i = plate number j = component number k = summation index
Literature Cited Augustln, D. C.; Straw. J. C.; Flneberg, M. S.; Johnson, B. B.; Linebarget, R. N.; Sansom, E. J. Simulation 1987, 21(6), 281-303. Doherty, M. F.; Pekins, J. D. Chem. Eng. Scl. 1978, 33, 281-301,
--- -.-. RRP-ri’lR
Huckaba, C. E.; May, F. P.; Franke. F. R. Chem. Eng. Prog. Symp. Ser. 1963, 59, NO. 46, 38-47. Lapidus, L.; Amundson, N. R. Ind. Eng. Chem. 1950, 42, 1071-1078. Mah, R. S. H.; Michaelson, S.; Sargent, R. W. H. Chem. Eng. Sci. 1962, 17, 619-639. Martln, E. N. Chem. Ags In& 1970, 27(2), 184-194. Peiser, A. M.; @over, S. S. Chem. €ng. Prcg. 1962, 58(9), 65-70. Rademaker, 0.; Rljnsdorp, J. E.; Maarleveld, A. ”Dynamlcs and Control of Continuous Distillation Units”, Elsevier, 1975. Waggoner, R. C.; Holland, C. D. AICbEJ. 1965, 77(1), 112-120.
Philip J. Thomas
Control and Instrumentation Division UKAEA, Atomic Energy Establishment Winfrith, Dorchester Dorset DT2 8DH, Great Britain
Received for review June 8, 1979 Accepted August 1, 1980
Vapor-Liqukl Equilibria and Densities with the Martin Equation of State A Soave-type temperature function has been introduced into the Martin equation of state, and the resulting equation has been applied to the calculation of vapor-liquid equilibria and of molar volumes of coexisting phases. In comparing the Martin equation to the Soave-RedlbKwong equation,R was found that vapor-liquid equibrii are predicted equally weil by the two equations, but that the Martin equation Is considerably more accurate In predicting liquid volumes of mixtures considered in this study. Vapor volumes calculated with the Martin equation were likewise found to be more accurate than those obtained with the Soave-RK equation.
The Martin Equation of State After a detailed analysis of volume-cubic equations of state, Martin (1979) concluded that a four-parameter form of the Clausius equation of state (Clausius, 1881; Jeans, 1940) is the best two-term cubic. The equation is a p = - -RT (1) V - b (V+C)2 0196-4305/81/1120-0168$01.00/0
The equation may also be written in reduced form as PR
A TR = Z,VR - B (z,Vn - -_ C)2
+
(2)
where B = bP,/RT,, C = cPJRT,, and A = aPc/R2T,2.In the above equation, z, is the experimental compressibility factor at the critical point. 0 1980 American Chemlcal Society
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 1, 1981 169
To fit data along the critical isothermal at about twice the critical density, Martin has proposed that one should use the relation B = 0.8572, - 0.1674 (3) which is the relation used in the current study. For the parameter A, Martin recommends
A = (27/64)T~-" (4) where n is a constant for a given substance, to be determined from the slope M of the vapor pressure curve at the critical point, as detailed in Martin's paper (Martin, 1979). The parameter C is obtained from B through the relation C = 0.1250 - B (5) Martin has compared his equation, using P-V-T data for pure substances, with other frequently used equations, such as the Redlich-Kwong (1949),the Soave-RK (Soave, 1972),and the Peng-Robinson (1976) equations. He came to the conclusion that the Martin equation is the simplest and the best of the two-term cubics. If Z, represents the value of the critical compressibility factor calculated from the equation of state, then for the Martin equation 2, = 0.250 + B (6) Combining eq 3 and 6, it is seen that 2, varies with 2,. Thus, 2, is 0.332 for argon and 0.290 for ammonia. By contrast, the Redlich-Kwong and Soave-RK equations fix the value of Z , at 1/3 for all substances, and likewise the Peng-Robinson equation yields a fixed value of 2, = 0.3074 for all substances. Martin has argued that a variable Z , gives added flexibility to his equation which enables it to represent P-V-T properties of a variety of substances with greater accuracy than would be the case with the popular Soave-RK or Peng-Robinson equations. Application to Vapor-Liquid Equilibria It has been the aim of the current study to explore the application of the Martin equation to vapor-liquid equilibria. If an equation of state is to fulfill such a function, it should meet the criterion that the fugacities of the saturated vapor and of the saturated liquid are equal along the vapor pressure curve of any pure substance. I t follows from Martin's equation that the fugacity coefficient, 4, of a pure substance is given by In 4 = RT b ac 2a In P ( V - b) V - b RT(V+ C) RT(V+ c ) ~(7)
+--
+
When eq 1 is solved a t any given temperature and vapor pressure for the largest root (vapor) and the smallest root (liquid) and the values of the volume are substituted in turn into eq 7, the values obtained for the fugacity coefficient should be essentially the same. Martin's temperature function for parameters A and a (eq 4) was designed to fit the slope of the vapor pressure curve at the critical point but not necessarily to satisfy the criterion of equality of vapor and liquid phase fugacities along the vapor pressure curve. A few trials with the Martin equation indicated that the criterion is at least approximately met at the several points tested. Martin (1979) has pointed out that the Soave (1972) temperature function can easily be used instead of TR-" in eq 4. Since the Soave function (Soave, 1972; Peng and Robinson, 1976) was devised specifically to meet the criterion of equality of liquid and vapor fugacities along the vapor pressure curve, it was decided in the current study also to test the Soave modification of the Martin equation.
Table I. Pure Component Parametersa
m component methane ethane propane n-butane n-pentane carbon dioxide (I
zc
w
0.2874 0.2847 0.2803 0.2741 0.2627 0.2742
0.0115 0.098 0.152 0.193 0.251 0.225
(es9) 0.50594 0.64645 0.73349 0.80643 0.88566 0.84556
n (es4) 0.500
-----
0.762
---------
-----
Critical constants were taken from Reid e t al. (1977).
In order to use the function proposed by Soave (1972) and also adopted by Peng and Robinson (1976), eq 4 was replaced with a = (27/64) (R2T: / P,)a
(8)
where a = 11
+ m(1-
-112
(9)
The parameter m of eq 9 was established for each pure substance in the current study by successive trials to satisfy the criterion 4 L = 4v (10) at the normal boiling point (at the triple point of carbon dioxide) within a tolerance of better than 0.01% in most cases. Undoubtedly a better procedure, the one used by Peng and Robinson (1976), would be to evaluate m at several temperatures between the boiling point and the critical point and to take an average. However, since the current study is exploratory and of limited scope, a single point was deemed sufficient. The values of m used in the current work are shown along with other constants in Table I. In applying eq 1to mixtures, the same mixing rules were used as those presented by Graboski and Daubert (1978) for the Soave-RK equation. These are for a binary mixture a M = xl2a1 x2a2 2xlx2a12 (11)
+
+
where a12 = (1 - K12)(aia2)1'2
(12)
and bM = xlbl
+ x2b2
(13)
It was also assumed that CM xlcl + x2c2 (14) Based on these mixing rules and eq 1,it follows that the fugacity coefficient of component 1in the binary mixture is given by In 6,=
In the above equation, a, b, and c are mixture parameters, Le., the same as U M , bM, and CM. The fugacity coefficient of component 2 is obtained by interchanging subscripts 1 and 2 in eq 15, remembering that a12 = a21. Experimental data on coexisting phases in the methane-propane system (Reamer et al., 1950; Price and Kobayashi, 1959) were chosen to test the two forms of the Martin equation. The procedure consisted in substituting the experimental compositions of either phase into eq 11, 13, and 14 and substituting the calculated values of U M , bM, and C M for a, b, and c in eq 1. Equation 1 was then
170
Id.Eng. Chem. Process Des. Dev., Vol. 20, No. 1, 1981
Table 11. Methane (1)-Propane ( 2 ) System at 6895 kFa (1000 psia)"
A. vapor temp, K 227.59
255.37
277.59
310.93
cm3/mol _--_ 148.5 151.6 142.4
y1
obsd calcd A calcd B calcd C obsd calcd A calcd B calcd c ob4 calcd A calcd B dcdC
0.9458
O
0.6635
calcd A cdcd B 344.26
calcd obsd calcd calcd calcd
C 0.3558
A
B C
cm3/mol
XI
-_-_
1.285 1.290 1.293 1.322 1.715 1.709 1.718 1.742 1.942 1.912 1.925 2.118 2.028 1.943 1.956 1.908 1.271 1.252 1.255 1.229
65.96 69.91 63.02
___-
0.522
205.6 208.7 199.2 229.1 230.8 234.4 223.0 254.7 251.5 256.4 241.8 187.9 192.3 200.0 179.1
0.8208
equilibrium ratio
0.736
____
0.895
W
liquid
71.86 77.30 69.23 79.54 79.90 86.00 74.55 94.33 97.95 104.7 94.75 139.4 144.7 152.4 137.3
0.4226
0.3271
0.2800
0.2053 0.1935 0.1916 0.1537 0.2197 0.2123 0.2112 0.1836 0.3104 0.3060 0.3048 0.3169 0.5001 0.5176 0.5164 0.5125 0.8947 0.9003 0.8993 0.9075
B. % deviation in
temp, K
A
B
C
A
B
C
227.59 255.37 277.59 310.93 344.26 av abs % dev
0.35 -0.35 -1.53 -4.20 -1.45 1.58
0.63 0.16 -0.87 -3.56 -1.21 1.29
2.90 1.60 9.04 -5.94 -3.26 4.55
-5.77 -3.35 -1.39 3.50 0.62 2.93
-6.66 -4.00 -1.69 3.27 0.51 3.23
-25.14 -16.41 2.10 2.49 1.43 9.51
%-deviationin liquid volume
vapor volume
temp, K
A
B
C
A
B
C
277.59 310.93 344.26 av abs % dev
0.46 3.84 3.82 2.71
8.12 10.95 9.30 9.46
-6.27 0.45 -1.51 2.74
0.72 -1.24 2.34 1.43
2.31 0.65 6.44 3.13
-2.66 -5.06 -4.68 4.13
* A = Martin equation, Soave temperature function, eq 8 and 9. B = Soave-RK equation (Graboski and Daubert, 1978). C = Martin equation, Martin temperature function, eq 4. solved for the volume twice, to obtain the vapor volume and the liquid volume. The fugacity coefficients of each phase, &v and @tL, were then calculated with eq 15. The vapor-liquid equilibrium ratio, Ki, of each component was calculated from the relation Kl = 4lL/4N (16) and compared with the experimental yJx, ratio. At three temperatures, at which dew-point and bubble-point volumes were given, the volumes calculated with eq 1 were compared with experimental values. For comparison, similar calculations were carried out with the Soave-RK equation modified by Graboski and Daubert (1978). In the methanepropane system the binary interaction constant, K ~ in~ eq , 12 was assumed to be zero for both forms of the Martin equation, and likewise for the Soave-RK equation, as recommended for the latter by Graboski and Daubert (1978). Table IIA gives the data used in the calculations, namely temperature, pressure, and compositions, as well as observed and calculated volumes and observed and calculated equilibrium ratios. Table ID3 lists the percent deviations in the calculated equilibrium ratios and in the calculated volumes. It is evident from in-
spection of the results that both the Soave form of the Martin equation and the Soave form of the RedlichKwong equation give much more reasonable results for the equilibrium ratios than the Martin equation with the Martin temperature function. Because of the discouraging results obtained in the methane-propane system, no further tests were carried out with the original Martin temperature function. However, the Soave form of the Martin equation was tested on two more binary systems: the ethane-n-pentane system (Reamer et al., 1960) and the n-butane-carbon dioxide system (Olds et al., 1949). Similar calculations were carried out with the Soave-RK equation as modified by Graboski and Daubert (1978). In the case of the ethane-n-pentane system the binary interaction constant, K ~ of~ eq, 12 was assumed to be zero both for the Martin and the Soave-RK equations, as likewise recommended by Graboski and Daubert (1978) for hydrocarbon systems. In the case of the n-butane-carbon dioxide system a value of K~~ = 0.140 was used with the Martin equation, while a value of K~~ = 0.1474 was assumed for the Soave-RK equation, as recommended by Graboski and Daubert (1978). For this system Peng and Robinson (1976) recommend a value of
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 1, 1981
171
Table 111. Ethane (1)-n-Pentane (2) System at 4137 kPa (600 psis)'"
A. vapor temp, K 310.93
cm3/mol 359.6 374.1 377.7 449.5 472.6 478.5 507.6 536.1 545.1 506.9 534.2 549.0 433.3 480.0 502.1
YI
obsd calcd calcd o bsd calcd calcd o bsd calcd calcd obsd calcd calcd obsd calcd calcd
344.26 377.59 410.93 444.26
liquid
0.9782
A B 0.9032
A B 0.7938
A B 0.5874
A B 0.3299
A B
equilibrium ratio
cm3/mol 95.27 95.30 103.57 107.0 104.2 120.0 121.9 122.4 142.5 143.2 148.1 171.8 193.5 191.7 218.6
XI
0.8503 0.5804 0.4188 0.2842 0.1606
K,
K,
1.150 1.140 1.140 1.556 1.533 1.535 1.895 1.841 1.845 2.067 1.947 1.952 2.054 1.739 1.744
0.1456 0.1592 0.1596 0.2307 0.2306 0.2308 0.3548 0.3767 0.3767 0.5764 0.5950 0.5949 0.7983 0.8347 0.8345
B. % deviation in
liquid volume A B 0.04 8.71 -2.60 12.17 0.40 16.93 3.45 19.96 -0.93 12.95 1.48 14.14
temp, K 310.93 344.26 377.59 410.93 444.26 av abs % dev a
K,
vapor volume A B 4.03 5.14 5.62 5.38 10.78 6.19
5.04 6.46 7.40 8.30 15.89 8.62
Kl
A
B
A
B
-0.91 -1.48 -2.89 -5.82 -15.36 5.29
-0.92 -1.36 -2.66 -5.54 -15.12 5.12
9.32 -0.04 6.16 3.23 4.56 4.66
9.58 0.04 6.16 3.20 4.54 4.70
A = Martin equation, Soave temperature function, eq 8 and 9. B = Soave-RK equation (Graboski and Daubert, 1978).
Table IV. n-Butane (1)-Carbon Dioxide ( 2 ) Systema
A. vapor temp, K 310.93
press., kPa 3447
310.93
6895
344.26
3447
344.26
6895
377.59
3447
377.59
6895
Y1
0.133
obsd calcd A calcd B obsd calcd A calcd B obsd calcd A calcd B o bsd calcd A calcd B obsd calcd A calcd B o bsd calcd A calcd B
0.056 0.306 0.216 0.589 0.432
liquid
cm 3/mol 579.7 577.2 584.5 186.3 184.7 192.4 645.1 641.0 650.5 238.7 239.1 249.3 636.5 635.5 648.2 239.0 239.7 253.2
equilibrium ratio
cm3/mol 88.41 85.37 98.28 77.99 91.99 101.75 105.11 103.66 118.05 100.05 105.25 117.18 124.21 128.50 143.67 132.00 136.67 150.50
XI
0.622 0.129 0.778 0.457 0.876 0.607
K2
Kl
0.2138 0.2105 0.2116 0.4341 0.5222 0.5359 0.3933 0.3997 0.4000 0.4726 0.4783 0.4788 0.6724 0.6792 0.6791 0.7117 0.7175 0.7158
2.294 2.320 2.363 1.084 1.068 1.O 67 3.126 3.048 3.108 1.444 1.432 1.439 3.315 3.133 3.187 1.445 1.424 1.431
B. % deviation in
liquid volume temp, K
press., kPa
310.93 310.93 344.26 344.26 377.59 377.59 av abs % dev
3447 6895 3447 6895 3447 6895
vapor volume
K,
Kl
A
B
A
B
A
B
A
B
-3.44 17.95 -1.38 5.20 3.45 3.54 5.83
11.16 30.47 12.31 17.12 15.66 14.02 16.79
-0.42 -0.86 -0.64
0.83 3.24 0.84 4.43 1.83 5.94 2.85
-1.56 20.28 1.62 1.20
-1.06 23.44 1.69 1.30 1.00 0.58 4.85
1.15 -1.46 -2.50 -0.82 -5.47 -1.50 2.15
3.02 -1.58 -0.57 -0.35 -3.84
0.18
-0.17 0.29 0.43
1.01
0.82 4.42
-1.01
1.73
A = Martin equation, Soave temperature function, eq 8 and 9. B = Soave-RK equation (Graboski and Daubert, 1978). K~~ = 0.130 for use with the Peng-Robinson equation. Tables IIIA and IVA give the data used in the calculations
and also show observed and calculated equilibrium ratios and volumes. Tables IIIB and IVB present the results as
172
Ind. Eng. Chem. Process Des. Dev. 1981, 20, 172-175
deviations of calculated from observed values. Discussion of Results The results of all comparisons are summarized in Tables IIB, IIIB, and IVB. It is seen that the Soave form of the Martin equation and the Soave-RK equation predict vapor-liquid equilibrium ratios equally well in the three systems tested. However, the Martin equation is much better than the Soave-RK equation in predicting liquid volumes and measurably better in predicting saturated vapor volumes. Superior P-V-T behavior of the Martin equation was argued by Martin (1979) in his article on volume-cubic equations. Martin also pointed out in his article that the Peng-Robinson equation should do relatively well with substances whose experimental critical compressibility factors fall approximately in the range of 0.26 to 0.27. This is demonstrated in the work of Peng and Robinson (1976), who showed the superiority of their equation over the Soave-RK equation in predicting liquid volumes of n-butane. Conclusions drawn from the current study are tentative because of its limited scope. The indications are that the Martin equation, when equipped with a Soave-type temperature function, can predict vapor-liquid equilibria equally as well as the Soave-RK equation and that it performs considerably better than the Soave-RK equation in predicting liquid volumes. Nomenclature a , b, c = parameters in Martin equation A , B, C = parameters in reduced Martin equation K = vapor-liquid equilibrium ratio m = characteristic constant of Soave-type temperature function M = slope of vapor pressure curve at the critical point n = exponent of temperature in eq 4 P = absolute pressure R = ideal gas constant T = absolute temperature V = volume
x , = mole fraction of component i in liquid phase y i = mole fraction of component i in vapor phase z = experimental compressibility factor Z = compressibility factor calculated from equation of state
Greek Letters a = temperature function in eq 8 K = binary interaction constant 4 = fugacity coefficient w = acentric factor
Subscripts c = critical i = component i L = liquid phase M = mixture R = reduced V = vapor phase 1, 2 = component 1, component 2 12 = pertaining to interaction of components 1 and 2 Literature Cited Clausius, R. Ann. Phys. Chem. 1881, I X , 337. Graboski, M. S.; Daubert, T. E. Ind. Eng. Chem. Process Des. Dev. 1078, 17, 443-54. Jeans, J. “An Introduction to the Kinetic Theory of Gases”,Cambridge University Press: Cambridge, England, 1940; pp 100-102. Martin, J. J. Ind. Eng. Chem. Fundam. 1970, 78, 81-97. Olds, R. H.; Reamer, H. H.; Sage, B. H.; Lacey, W. N. Ind. Eng. Chem. 1949, 41, 475-82. Peng. D. Y.; Robinson, D. 8. Ind. Eng. Chem. Fundam. 1978, 15, 59-64. Price, A. R.; Kobayashi, R. J. Chem. Eng. Data 1059, 4 , 40-52. Reamer, H. H.; Sage, B. H.; Lacey, W. N. Ind. Eng. Chem. 1950, 4 2 , 534. Reamer, H. H.; Sage, B. H.; Lacey, W. N. J . Chem. Eng. Data 1060, 5 , 44-50. Redlich, 0.; Kwong, J. N. S. Chem. Rev. 1040, 44, 233-44. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. “The Properties of Gases and Liquids”, 3rd 4.;McGraw-HUI: New York, 1977. Soave, G. Chem. Eng. Sci. 1972, 2 7 , 1197.
Department of Chemical Engineering and Chemistry New Jersey Institute of Technology Newark, New Jersey 07102
Joseph Joffe
Received for reuiew February 11, 1980 Accepted July 24, 1980
Simulation Study of Double-Annulus Center-Line Light Source Photochemical Reactor The photochemical reaction between 9,lO-phenanthrenequinoneand I ,4diixane produces an adduct product which absorbs light in the same wavelength range as the reactant. A double-annulus center-line light source reactor with the appropriate flow configuration should result in improved product yield and lamp utilization as compared with a single-annulus reactor. The results of a simulation study for such a reactor showed that although improved performance was obtained, the improvement to be expected would be marginal.
Introduction Photochemical reactions possess a number of attractive features that include specificity of the reaction, the possibility of performing reactions whose equilibrium yields would be low if activated by thermal means, and, in some cases, the absence of side reactions. Although photochemical reactions are usually studied with a monochromatic light source (Calvert and Pitts, 1966),a convenient industrial technique for carrying out a photochemical reaction would be an annular reactor with a polychromatic fluorescent lamp as the center-line light source. Lamps which have a relatively constant output in the near-ultraviolet region over useful lives in the 1000-h range are readily produced. Atlas et al. (1976) developed design equations for such an annular reactor with a polychromatic center-line light 0196-4305/81/1120-0172$01 .OO/O
source. The design equations included the case where not only the reactant but also the product species absorb light in the emission range of the light source. In their experimental work they studied the photochemical reaction of 1,4-phenanthrenequinone (PQ)with 1,4-dioxane(solvent) to give the 1:l adduct
0
solvent
OH
PQ
CJ
1: 1 adduct
The solvent absorbance was nil in the wavelength range 0 1980 American Chemlcal Society
