Generalized Temperature-Dependent Parameters of the Redlich-Kwong Equation of State for Vapor-Liquid Equilibrium Calculations Salah E. M. Hamam, W. K. Chung, 1. M. Elshayal, and Benjamin C.-Y. LU’ Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada K f N 6N5
The temperature-dependent parameters 9 , and f i b of the Redlich-Kwong equation of state evaluated from vapor pressures and saturated liquid volumes for 13 pure components were correlated in terms of T,. The coefficients of these correlations were further generalized in terms of w . The generalized correlations have been successfully used to compute the 9 , and fib values for seven arbitrarily selected components other than those included in the generalization. The applicability of the values computed from the generalized equations was further demonstrated by evaluation of pure-component properties and by the calculation and prediction of vapor-liquid equilibria data for eight binary systems at 32 isothermal conditions.
Introduction Equations of state are usually used for predicting vaporliquid equilibrium and volumetric properties in the absence of experimental data. Among the equations of state frequently used in chemical engineering is that proposed by Redlich and Kwong in 1949 (RK equation). I t is remarkably simple and can be employed in all property estimation systems (Model1 and Reid, 1974). I t has been the subject of much research in order to improve its predictive ability for both pure components and mixtures. Compilations of these efforts are available in the literature (Horvath, 1972; Harmens, 1975). I t is now generally accepted that the two parameters of the RK equation, R, and clb, are not universal constants. They vary from substance t o substance and are temperature dependent (see for examples, Zudkevitch and Joffe (1970);Joffe e t al. (1970); Chang and Lu (1970); Vogl and Hall (1970); De Mateo and Kurata (1975); Harmens (1975)). This investigation concerns the generalization of these parameters specifically for vapor--liquid equilibrium calculations. There are attempts in the literature to correlate and generalize one or both of these two parameters. Chandron e t al. (1973) correlated the parameters in terms of T , and W . A more recent correlation was proposed by Simonet and Behar (1976). These correlations placed their emphases on the representation and prediction of volumetric properties without the consideration of the equal fugacity criterion cfl = f ” ) which is a necessary condition for vapor-liquid equilibrium calculations. De Mateo and Kurata (1975) correlated the parameters for seven hydrocarbons a t temperatures below 183 K in their study of solubilities of solid hydrocarbons in liquid methane. However, no generalization of these parameters was proposed. Vogl and Hall (1970) correlated the two parameters for hydrogen and helium, but limited their correlation to the supercritical region and without any generalization. Wilson (1964) treated one of the parameters ( a ) temperature dependent, and subsequently (1966) generalized this parameter in terms of T , and w for enthalpy calculations. Another generalization of the same parameter was proposed by Soave (1972). However, when both of these correlations were tested for their capability in representing pure-component vapor pressures, the deviations obtained were higher than those obtained when both parameters were considered temperature dependent (See Applicability section below). Furthermore, Soave’s correlation gives large deviations in and sometimes negative values of saturated liquid volume (Beret and Prausnitz, 1975; Peng and Robinson, 1976).
It appears that there are no adequate expressions available in the literature which correlate and generalize the temperature-dependent parameters for the purpose of calculating vapor-liquid equilibrium data ( T-P-composition). The two temperature-dependent parameters of the RK equation may be evaluated from any two pure component properties at saturation at a given temperature. It appears that for vapor-liquid equilibrium calculations, vapor pressure and saturated liquid volume are the two desirable properties for the evaluation. Vapor pressure is used to satisfy the condition of equal fugacity in the liquid and vapor phases. In addition to this laboratory (Lu et al., 1969; Chang and Lu, 1970), this approach was used by Joffe et al. (1970),and recently adopted by Harmens (1975) in his computation program for low temperature vapor-liquid equilibrium and thermodynamic properties. The capability of the resulting parameters has been well demonstrated in these references as well as other
Table I. Literature Sources of the Saturation Properties Used in the Investigation Component Argon
VI
Din (1961);Michels e t al. (1558) Methane Goodwin and Prydz (1972) Nitrogen Wilson et al. (1964) Ethylene Canjar and Manning (1967) Ethane Eubank (1972) Propane Das and Eubank (1973) * Propylene Canjar and Manning (1967) n-Butane Canjar and Manning (1967) Benzene Canjar and Manning (1967) n-Pentane Canjar and Manning (1967) n-Hexane Canjar and Manning (1967) n-Heptane Young (1910) n-Octane Young (1910) n-Nonane Young (1910) n-Decane Young (1910) Oxygen Webber (1970)
P Din (1961);Michels et al. (1958) Prydz and Goodwin (1972) Wilson et al. (1964) Canjar and Manning (1967) Eubank (1972) Das and Eubank (1973) Canjar and Manning (1967) Canjar and Manning (1967) Canjar and Manning (1967) Canjar and Manning (1967) Canjar and Manning (1967) Young (1910) Young (1910) Young (1910) Young(1910) Webber (1970)
Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 1, 1977
51
Table 11. Values of
Tr
0.55535 0.57862 0.62594 0.65756 0.70244 0.73565 0.79643 0.80306 0.80969 0.81632 0.82295 0.82958 0.83621 0.84283 0.84946 0.85609 0.86272 0.86935 0.87598 0.88261 0.88923 0.89586 0.90249 0.90912 0.91575 0.92238 0.92901 0.93564 0.94889 0.95552 0.96215 0.96878 0.97541 0.98204 0.98866 0.99529 0.99861 1,00000
0.65841 0.66204 0.67387 0.68570 0.69753 0.70936 0.72119 0.73302 0.74485 0.75667 0.76850 0.78033 0.79216 0.80399 0.81582 0.82765 0.83948 0.85131 0.86314 0.87497 0.88680 0.89862 0.91045 0.9 2 2 2 8 0.93411 0.94594 0.95777 1.00000
na
naand- ab Calculated from the Saturated Properties a h Tr R" ab Tr
Argon 0.40304 0.40652 0.41047 0.41162 0.41367 0.41525 0.41732 0.41699 0.41661 0.41617 0.41576 0.41530 0.41486 0.41448 0.41400 0.41353 0.41301 0.41249 0.41206 0.41160 0.41114 0.41067 0.41023 0.40971 0.40935 0.40890 0.40851 0.40822 0.40790 0.40779 0.40777 0.40785 0.40841 0.40957 0.41105 0.41563 0.42056 0.42557 n-Pentane 0.4354 7 0.43562 0.43544 0.43506 0.4347 3 0.43390 0.43289 0.4 3 194 0.43076 0.42969 0.42821 0.42675 0.42542 0.42377 0.42200 0.42048 0.41874 0.41669 0.41432 0.41210 0.41010 0.40789 0.40557 0.40323 0.40140 0.40005 0.39831 0.41887
0.08776 0.08784 0.08734 0.08675 0.08617 0.08583 0.08505 0.08489 0.08472 0.08453 0.08436 0.08417 0.08398 0.08381 0.08362 0.08343 0.08323 0.08302 0.08285 0.08266 0.08247 0.08228 0.08210 0.08190 0.08174 0.08155 0.08138 0.08124 0.08103 0.08094 0.08087 0 .O 808 3 0.08093 0.08121 0.0815 7 0.08286 0.08428 0.08577 0.07933 0.07940 0.07946 0.07951 0.07957 0.07954 0.07947 0.07943 0.07935 0.07929 0.07916 0.07902 0.07892 0.07875 0.07856 0.07844 0.07827 0.07802 0.07772 0.07744 0.07723 0.07697 0.07670 0.07643 0.07628 0.07630 0.07622 0.08406
0.49861 0.52485 0.55109 0.57734 0.58222 0.59681 0.60358 0.62982 0.64053 0.68425 0.75715 0.81551 0.83005 0.87388 0.90295 0.93224 0.97680 0.98431 0.99228 0.99538 1.00000
0.50040 0.52211 0.54208 0.56981 0.58994 0.61292 0.66403 0.69762 0.74556 0.78130 0.82274 0.87615 0.91759 1.00000
0.53830 0.54706 0.55801 0'56896
~:~~~~~ o.60180 0'61275 0.62370 0'63465 0.64560
0.65655 0.66750 0.67844 0.68939 0.70034
0.71129 0.72224 0.73319 0.76603 0.77698 0.78793
0.79888 0.80983 0.82077 0.83172
0.84267 0.85362
:::;:$ 0.88646 0.89741 0.90836 0.93026 0.94121 1.00000
52
Methane 0.39263 0.39671 0.40028 0.40337 0.40390 0.40537 0.40600 0.40818 0.40895 0.41136 0.41293 0.41212 0.41165 0.40963 0.40787 0.40595 0.40473 0.40558 0.40798 0.40999 0.42376 Nitrogen 0.41025 0.41365 0.41641 0.41795 0.41910 0.42040 0.42210 0.42251 0.42218 0.42202 0.42002 0.41646 0.41208 0.42477 n-Hexane 0.44070 0.44082 0.44 112 0.44081 0.44069 0.44055 0.44042 0.44013 0.43953 0.43907 0.43845 0.43821 0.43779 0.43587 0.4 3496 0.43403 0.43265 0.43159 0.43071 0.427 26 0.42549 0.424 32 0.42314 0.42106 0.41941 0.41770 0.41580 0.41376 0.41210 0.41005 0.40834 0.40611 0.40429 0.40152 0.39870 0.41937
0.08653 0.08646 0.08636 0.08622 0.08619 0.08610 0.08605 0.08583 0.08573 0.08525 0.08416 0.08302 0.08270 0.08165 0.08092 0.08021 0.07918 0.08005 0.08079 0.08140 0.08552 0.08732 0.08747 0.08759 0.08728 0.08712 0.08698 0.08653 0.08614 0.08545 0.08501 0.08419 0.08278 0.08147 0.08583 0.07719 0.077 26 0.07738 0.07739 0.07745 0.07751 0.07759 0.07 764 0.07764 0.07767 0.07768 0.07 77 8 0.07785 0.07761 0.07760 0.07758 0.07750 0.07746 0.07744 0.07729 0.07714 0.07707 0.07701 0.07685 0.07671 0.07657 0.07641 0.07622 0.07 607 0.07 589 0.07572 0.07548 0.07535 0.07537 0.07509 0.08421
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
Ra
Ethylene 0.60012 0.41514 0.60929 0.41485 0.62896 0.41643 0.64864 0.41788 0.66831 0.41802 0.68799 0.41730 0.70766 0.41613 0.72734 0.41543 0.74701 0.41648 0.76669 0.41634 0.78636 0.41660 0.84539 0.41400 0.86507 0.41287 0.88474 0.41018 0.90442 0.40782 0.92409 0.40561 0.94377 0.40298 0.96344 0.40077 1.00000 0.42211 Propane 0.62491 0.42683 0.64896 0.42674 0.67600 0.42645 0.70304 0.42581 0.73008 0.42473 0.75713 0.42331 0.78417 0.42146 0.81121 0.41919 0.83825 0.41655 0.86529 0.41356 0.89233 0.41013 0.91937 0.40644 0.94641 0.40281 0.95993 0.40134 0.99508 0.40494 0.99643 0.40630 0.99778 0.40836 0.99913 0.41190 1.00000 0.42282 n-Butane 0.64126 0.43245 0.65289 0.43262 0.66596 0.43201 0.67902 0.43168 0.69209 0.43113 0.70516 0.43043 0.71822 0.42957 0.73129 0.42913 0.74436 0.42757 0.75742 0.42636 0.77049 0.42504 0.78355 0.42371 0.79662 0.42205 0.80969 0.42006 0.82275 0.41805 0.84889 0.41507 0.86195 0.41381 0.87502 0.41200 0.88809 0.41007 0.90115 0.40795 0.91422 0.40569 0.92729 0.40438 0.94035 0.40309 0.99262 0.40694 1.00000 0.42149
0.08339 0.08325 0.08328 0.08325 0.08319 0.08300 0.08272 0.08245 0.08229 0.08200 0.0817 3 0.08093 0.08056 0.08001 0.07950 0.07907 0.07 859 0.07820 0.08503
0.65484 0.68758 0.72032 0.75306 0.78580 0.81854 0.85129 0.88403 0.91677 0.94951 0.95606 0.96916 0.97571 0.99208 0.99535 0.99699 0.99862 1.00000
0.61808 0.62395 0.63919 0.65442 0.66965 0.68488 0.70011 0.71534 0.73057 0.74580 0.76103 0.77626 0.79149 0.80672 0.82195 0.85241 0.86765 0.88288 0.89811 0.91334 0.92857 0.08100 0.94380 0.08102 0.95903 0.08098 1.00000 0.08094 0.08088 0.55309 0.08 08 0 0.56298 0.08070 0.57286 0.08062 0.58274 0.08048 0.59262 0.08033 0.60251 0.08017 0.61239 0,08001 0.62227 0.07978 0.62838 0 ,01941 0.63215 0 .O 79 16 0.64204 0.07881 0.65192 0.07 8 70 0.66180 0.07846 0.67168 0.07816 0.68157 0.07786 0.69145 0.07756 0.70133 0.07751 0.71121 0.07749 0.72110 0.08033 0.73098 0.08484 0.74086 0.75074 0.76063 0.77051 0.78039 0.79027 0.80016 0.81004 0.81992 0.82980 0.082 14 0.08205 0.08292 0.08176 0.08155 0.08128 0.08095 0.08056 0.08011 0.07962 0.07906 0.07848 0.07798 0.07786 0.07981 0.08025 0.08089 0.08197 0.08524
Ethane 0.41883 0.41922 0.41902 0.41824 0.41708 0.41531 0.41298 0.41010 0.40669 0.40322 0.40264 0.40181 0.40177 0.40534 0.40756 0.41018 0.41373 0.42350 Propylene 0.42451 0.42490 0.42531 0.42541 0.42538 0.42544 0.42551 0.42514 0.42444 0.42389 0.42315 0.42221 0.42107 0.41967 0.41811 0.41464 0.41308 0.41109 0.41019 0.40919 0.40797 0.40631 0.40358 0.42184 Benzene 0.42750 0.42784 0.42796 0.42816 0.42820 0.42820 0.42840 0.42841 0.42810 0.42815 0.42797 0.42766 0.42775 0.42776 0.42756 0.42729 0.42679 0.42640 0.42542 0.42445 0.42396 0.42353 0.42341 0.42213 0.42130 0.42077 0.41978 0.41865 0.41729 0.41630
0.08301 0.08276 0.08246 0.08208 0.08168 0.08119 0.08063 0.07998 0.07928 0.07866 0.07858 0.07853 0.07861 0.07993 0.08065 0.08145 0.08253 0.08544 0.08212 0.08219 0.08218 0.08215 0.08209 0.08206 0.08203 0.08192 0.08179 0.08165 0.08151 0.08134 0.08113 0.08087 0.08059 0.07994 0.07968 0.07931 0.07926 0.07919 0.07910 0.07890 0.07843 0.08495 0.07915 0.07922 0.07926 0.07930 0.07935 0.07938 0.07941 0.07942 0.07941 0.07940 0.07943 0.07944 0.07945 0.07944 0.07946 0.07944 0.07938 0.07936 0.07929 0.07919 0.07918 0.07915 0.07912 0.07900 0.07892 0.07887 0.07876 0.07865 0.07849 0.07837
Table I1 ( C o n t i n u e d ) T,
all
.~
a,
T Y
n-Heptane 0.50559 0.44760 0.52410 0.44837 0.54261 0.44806 0.561 12 0.44797 0.57963 0.44779 0.59814 0.44712 0.61665 0.44631 0.63516 0.44502 0.65367 0.44350 0.67218 0.44172 0.69069 0.43982 0.70902 0.43838 0.7 27 7 1 0.43607 0.74621 0.43389 0.76472 0.43094 0.78323 0.42835 0.80174 0.42512 0.82025 0.42222 0.83876 0.41916 0.85727 0.41614 0.87578 0.41254 0.89429 0.40878 0.91280 0.40604 0.93131 0.40147 0.94982 0.39734 1.ooooo 0.41911
ab
0.07572 0.07593 0.07610 0.07626 0.07640 0.07651 0.07661 0.07668 0.0767 3 0.07674 0.07674 0.07674 0.07670 0.07666 0.07657 0 ,07647 0.07631 0.07616 0.07596 0.075 76 0.07551 0.07526 0.07503 0.07487 0.07473 0.08414
publications in the literature (for example, Kat0 et al., 1976). The two temperature-dependent parameters obtained in this manner for pure components in the subcritical region are used in this investigation for correlation and generalization. The need of the proposed endeavor is apparent. In the absence of such a correlation, values of the two parameters would have to be determined from the saturation properties whenever a temperature is specified. Furthermore, the required saturation properties for the evaluation may not be available at the specified temperature condition. Values of 0, and f i b The parameters Q a and f i b are related t,o the quantities a and b of the RK equation (eq 1).
p = - -RT V-b
a
T0.5V(V
T,
aa
0.48020 0.49778 0.56810 0.58568 0.60326 0.62084 0.67358 0.69116 0.70874 0.72632 0.74390 0.76148 0.71906 0.79664 0.81422 0.83180 0.84938 0.86696 0.88454 0.90212 0.91970 0.93728
n-Octane 0.45256 0.45401 0.45363 0.45257 0.45124 0.44950 0.44446 0.44269 0.44008 0.43801 0.43558 0.43258 0.42972 0.42713 0.42439 0.42113 0.41751 0.41386 0.41045 0.40605 0.40208 0.39790 0.41791
1.00000
4 .-'.
0.07389 0.07416 0.07492 0.07506 0.07517 0.07527 0.07549 0.07553 0.07555 0.07556 0.07552 0.07545 0.07536 0.07529 0.07519 0.07507 0.07488 0.07469 0.07 451 0 .O 7 438 0.07419 0.07397 0.08378
T Y
na
0.83969 0.84957 0.85945 0.86933 0.87922 0.88910 0.89898 0.90886 0.91875 0.92863 0.93851 0.94839 0.95828
Benzene 0.41558 0.41450 0.41314 0.41153 0.41021 0.40897 0.40742 0.40603 0.40462 0.40258 0.40105 0.39962 0.39791 0.42101
1.00000
ab
0.07829 0.07814 0.07801 0.07785 0.07768 0.07755 0.07738 0.077 24 0.07711 0.07685 0.07674 0.07666 0.07651 0.08470
e-,
--'
.e>'
0..
I
j Colculoted ,,+1 -W w l t r
41
from V i and
Ps
"?\
from curve fittinq ewattions
- -Results
% \.
\
i
from q e w a l i z d equations
078
I
I
6
7
A
1 8
L
A 10
Tr
+ b)
Figure 1. Temperature dependence characteristics of R, and i2b of
as follows
propane.
(3) The procedure employed in this investigation for obtaining the values of these two parameters for pure components, using vapor pressure and saturated liquid volume as the chosen properties for the evaluation, is based on that proposed by Kat0 et al. (1976b). It should be mentioned that the same procedure was used for evaluating these two parameters at T < T , and T = T , conditions. At the critical point, experimental P , and V , values were treated as the vapor pressure and saturated liquid volume. In other words, the Q , and f i b values obtained at the critical point vary from substance to substance and differ from the classical values ( Q , = 0.42748, fib = 0.08664). In the literature, difficulties have been encountered in the evaluation of the parameters in the vicinity of the critical point due to either the divergence of the computation program or the critical anomaly of the RK equation. As a consequence, classical values have been used at the critical point. For example, Harmens (1975) described the parameters by a straight line between their known values a t T , = 0.96 and the classical critical values a t T , = 1. This limita-
tion is avoided in the approach we have adopted. Joffe et al. (1970) encountered difficulty in their evaluation of the parameters at the critical point because one of the equations used in their calculation procedure becomes indeterminate. This difficulty was not experienced in the procedure adopted in this investigation. Therefore, the equal fugacity criterion, which is a fundamental condition of equilibrium, has been satisfied at all temperature conditions investigated in the adopted calculation procedure. In this investigation, values of 9, and fib were evaluated for 13 pure components in the temperature region where reliable saturated properties are available. The critical properties of all hydrocarbons were taken from the API Technical Data Book (1971), with the exception of T , and P, for methane, which were taken from the values reported by Prydz and Goodwin (1972). For argon, the critical data given by Michels et al. (1958) were employed, while for nitrogen the critical data were taken from Din (1961).The sources of the saturation data used for the evaluation of the parameters are listed in Table I. Values of the parameters obtained together with the temperature ranges investigated are reported in Table 11. Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
53
PPPPPPPPPPPPP
O r l dI ~1 o o I o
!
I1
/I
/I
I1
I1 I1 I1
olnrilrjooo
! I
I
I1 /I I1 /I
11 1 I It
oojo?iooo
I
54
Ind. Eng. Chern., Process Des. Dev., Vol. 16,
No. 1,
1977
I
I
Table IV. A Comparison of fl, and Correlation Equations 4-7
flb
Values Obtained from the Saturated Properties with Those Computed from the
nu
6% in
A" Comuonent
A
B" Maximum
nb
~~
6% in
A"
B" Maximum
0.4924 Argon 2.2444 Methane 0.7655 1.4444 Nitrogen 1.0475 2.0544 Ethylene 0.5250 1.1802 Ethane 1.0324 0.4250 Propane 0.5742 0.3950 Propylene 0.7900 0.1887 n-Butane 1.5426 0.6809 Benzene 0.4615 0.8306 0.2388 0.5152 n-Pentane n-Hexane 0.4555 0.4077 n-Heptane 0.2781 0.1797 n-Octane 0.1146 0.3053 Overall av 0.7746 0.7000 " A: 0.85 6 T r Q 1.0; B: Tr(min) 6 T , 6 0.85.
0.0700 0.0620 0.0000 0.0222 0.1607 0.0651 0.0867 0.0396 0.0253 0.0283 0.0515 0.0185 0.0250 0.0504
~
B
A
B
Maximum
0.1445 0.0203 0.0906 0.1510 0.0058 0.0078 0.0233 0.0374 0.0462 0.0408 0.0580 0.0479 0.1133 0.0605
0.3810 0.1258 0.0000 0.0447 0.4661 0.1486 0.2635 0.0988 0.0668 0.0884 0.2028 0.0309 0.0086 0.1482
Average
0.2467 0.0249 0.2392 0.1583 0.0128 0.0250 0.0530 0.1498 0.0730 0.0928 0.2210 0.0346 0.0397 0.1054
0.1342 0.0112 0.0689 0.0743 0.0060 0.0138 0.0249 0.0353 0.0256 0.0371 0.0445 0.0117 0.0168 0.0388
0.1036 0.0771 0.0000
0.0281 0.2172 0.0806 0.1257 0.0453 0.0241 0.0385 0.0645 0.0105 0.0033 0.0630
~
a,
6% in
A
B Average
0.5649 0.5279 0.7126 0.2959 0.3511 0.2368 0.3521 0.2366 0.2312 0.1119 0.2953 0.1938 0.0554 0.3204
Correlation and Generalization The values of the parameters obtained above were first of all correlated in terms of T , for all the pure components investigated individually in the following two ranges of temperature: 0.85 < T , < 1.0 and Tr(min) 6 T , < 0.85. The subscript (min) represents the lowest temperature investigated as reported in Table 111. In the temperature range T , = 0.85 to 1.0, the two parameters were fitted by means of eq 4 and 5.
+ U I ( I - Tr) + a ~ ( -1 T F ) " ~+ U 3 ( 1 - Tr)2'3 (4) f i b = CO + C l ( 1 - Tr) + CZ(1 - Tr)1'3 + C g ( 1 - T T ) " ~(5)
Qa
A
fib
Values Obtained from the Saturated Properties with Those Computed from the
~~
Component
B Average
0.2556 0.2986 Argon Methane 0.0994 0.0467 Nitrogen 0.0000 0.2466 Ethylene 0.0468 0.3522 0.3355 0.0126 Ethane 0.1175 0.0172 Propane 0.1854 0.0545 Propylene 0.0843 0.1237 n-Butane 0.0580 0.1156 Benzene n -Pentane 0.0720 0.0930 n -Hexane 0.1691 0.2032 n-Heptane 0.0538 0.1012 n -Octane 0.0659 0.2728 Overall avg. 0.1187 0.1491 ' A: 0.85 Q Tr Q 1.0; B: Tr(min) Q Tr Q 0.85. Table V. A Comparison of 0, and Generalized Ea 4-7 and 9-12
6% in
= a0
These equations are of the type suggested by Guggenheim (1945) for representing densities of saturated liquid argon, In the lower temperature range (Tr(min) 6 T , < 0.85), the two parameters were fitted by a different temperature function as shown in eq 6 and 7.
A
fib
B
A
Maximum
0.1919 0.8851 1.5295 0.6973 0.3965 0.2275 0.1170 0.4036 0.6072 0.4200 0.1841 0.0917 0.1314 0.4525
4.0649 2.2251 1.3434 0.5568 1.9078 0.8760 1.2000 2.2848 0.4942 0.5250 0.6835 0.2493 0.1422 1.2733
B Average
0.6668 1.2241 1.9209 0.8889 0.7834 0.1877 0.4719 0.5553 0.8400 0.7881 0.5050 0.5718 0.3200 0.7480
1.2450 1.2458 0.8781 0.4974 0.6425 0.3610 0.6033 0.3762 0.2759 0.2276 0.5145 0.2111 0.0717 0.5500
0.3854 1.0579 1.6029 0.7512 0.7416 0.0743 0.3858 0.3487 0.6779 0.6836 0.2737 0.2860 0.1609 0.5715
The coefficients a,, b,, c,, and d, were determined using the method of least-squares and are given in Table I11 for the 13 pure components investigated. The values of Ra and from eq 4-7 (Rcomp) are compared with those obtained from the saturated properties ( n c & d ) in terms of 6% as defined by the following expression:
6% =
Rcalcd
- ncomp
Rcalcd
I
x 100%
(8)
Both of the maximum and the average deviations for the 13 compounds are reported in Table IV. Futhermore, the coefficients a,, b,, c,, and d, were generalized in terms of Pitzer's acentric factor w , by the expressions 2
a, =
c a,,wJ J=o
(i = 0, 1 , 2 , 3)
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
(9)
55
Table VI. Comparison of the Computed Values of Q, and a b from the Generalized Eq 4-7 and 9-12 with Those Obtained from Saturation Properties for Compounds Not Included in the Generalization Component
TI
0,
Oxygen
0.492 0.453 0.743 0.832 0.536 0.527 0.539 0.561 10.566 0.621 0.913 0.523 0.607 0.719 0.747 0.831 0.503 0.577 0.611 0.665 0.719
0.399395 0.405495 0.420820 0.415890 0.428565 0.429701 0.430327 0.431174 0.431340 0.432024 0.405831 0.459692 0.455198 0.442273 0.437803 0.421421 0.466449 0.463759 0.459258 0.452949 0.444829
6%
0.61 0.086895 0.32 0.086746 Hydrogen sulfide" 0.12 0.082598 0.05 0.081239 Carbon tetrachloride" 0.13 0.079706 Cyclohexane" 0.86 0.079203 0.83 0.079344 0.78 0.079553 0.81 0.079598 0.74 0.079910 Carbon dioxide" 0.46 0.077043 n-Nonane 0.13 0.073867 0.07 0.074406 0.22 0.074626 0.32 0.074592 0.43 0.074277 n-Decane 2.72 0.073199 0.90 0.073441 0.27 0.073611 0.15 0.073707 0.61 0.073730 Overall average 0.55 Values of the parameters obtained from saturated properties were taken from the work of Hsi (1971). Table VII. The Relative Deviation of Vsl a t Specified Relative Deviations of 9 , and Temperatures (n/%alcd)
fi,
fib
32 O F
60 O F
1.015 1.005 1.005 1.000 0.995 0.995 0.985
1.015 1.005 0.995 1.000 1.005 0.995 0.985
1.50 0.50 0.69 0.00 0.69 0.50 1.50
2.11 1.11 0.13 0.00 1.30 0.10 0.91
Table VIII. Deviations in Calculated ( H * - H ) Values for Liquid Methane Temp, O -200 -220 -240
F
Pressure, psia
Av abs dev, %
200-2000 200-2000 200-2000
0.80 0.60 1.11
For the hydrocarbons, the revised values of w reported by Passut and Danner (1973) were used in the correlation. The w values of argon and nitrogen were taken from the work of Prausnitz (1969). The w values of the 13 pure components investigated cover a range of w of 0 to 0.3942. The coefficients a , . b . . c',.,. and dij were evaluated using the least-squares method and are also reported in Table 111. Similarly, the computed values of R, and fib from the generalized equations are compared with the calculated values of R, and R b obtained from the saturation properties in terms of 6% as expressed by eq 8. The results of the comparison are given in Table V. The temperature-dependent characteristics of R, and f i b are depicted in Figure 1for propane using the proposed eq 4-7 and 9-12. I t should be mentioned that other combinations of the fitting equations were investigated. For example, the polynomial equations, as expressed in eq 6 and 7, were used for representing Q, and Qb in two ranges of temperature (e.& Trcmin, C T , C 0.85, 0.80 < T, < 1.00) with the coefficients subserJ,
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Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
6% in V,' 140 O F
1.50 0.49 0.83 0.00 0.83 0.51 1.51
6%
nb
nb
0.26 0.15 0.05 0.55 0.75 1.44 1.36 1.24 1.27 1.14 1.02 0.74 0.33 0.36 0.57 0.24 2.92 1.74 0.79 0.59 0.76 0.87
for n-Hexane a t Different
190 O F
250 OF
1.35 0.35 1.07
0.87 0.12 1.77 0.00 0.57
0.00
0.97 0.65 1.64
1.11
2.10
quently represented by the generalized equations (eq 9-12). It was found that the representation is unfortunately restricted to T , < 0.97. Much larger 6% values were obtained a t higher Tr values. Another drawback of this combination is that the parameters at the critical point cannot be accurately predicted. If the upper temperature range was taken to be T , = 0.80 to 0.97 the 6% values in R, and f i b are 0.300 and 0.498, respectively. These deviations are somewhat lower than those obtained by the proposed equations. Another approach investigated was to use one function such as that shown in eq 4 and 5 to represent the two parameters over the whole temperature range (Tr(min) < T,S 1.0) with the coefficients subsequently generalized by means of eq 9 and 11).The 6% values in R, and R b obtained by this approach are 0.518 and 0.595, respectively. Although these values are not significantly different from those obtained from the proposed combination, the deviations are mainly contributed from those obtained a t lower temperatures. This is the reason why eq 6 and 7 are preferable for the lower temperature range ( Tr(,,,in) < T , < 0.85). Applicability of the Generalized Correlation For testing the applicability of the generalized correlation, seven components other than those included in the generalization were selected. These components are either of different chemical nature (carbon dioxide, hydrogen sulfide, carbon tetrachloride, cyclohexane, and oxygen) or having w values greater than those covered in the generalization (n-nonane,
I
1 aI
N
I
b m
f
5
Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 1, 1977
57
w = 0.4437, and n-decane, w = 0.4902). The values of the parameters R, and &i computed from the generalized correlation a t arbitrary selected temperatures are compared with those obtained from saturation properties and shown in Table VI in terms of 6%. The overall average 6% values are 0.55 and 0.87, respectively, for R, and f i b . With the exception of one point (n-decane a t T , = 0.503), the maximum deviations are less than 1%and 2% for a, and ab, respectively. In order to evaluate the effect of the deviation in 9, and f i b values caused by the generalization on the prediction of saturated molal volume of liquids, n-hexane was selected for the testing purpose. The calculated V,' values obtained a t different temperatures with different deviations in R, and a b values are compared in Table VI1 with the literature values in terms of 6% by an expression similar to eq 8. The results indicated that the deviation in V,' of n-hexane is 0.5% when the relative errors in both Q, and f i b are +0.5%, while for relative error of -0.5% and +0.5% in $2, and R b , respectively, the deviation is 0.69% a t 32 OF. These levels of errors are similar to those obtained from the generalized correlation for $I(, and n b (eq 4-7 and 9-12). The capability of the generalized correlation for generating reasonably accurate vapor pressure data is expected. Nevertheless, vapor pressures were calculated and compared with those compiled by Canjar and Manning (1969) for 19 compounds. The overall deviation in terms of I M / P l av is only 1.5%.This compares favorably with the deviations obtained by the method of Soave (1972) (2.2%) and the method of Wilson (1966) (4.3%),when the same literature data are used for the comparison. I t should be mentioned that there are 20 compounds included in the compilation of Canjar and Manning (1969). The only compound not included in the comparison is hydrogen. The generalized correlation was also tested for its capability for generating enthalpy data. Heats of vaporization a t the normal boiling point for normal paraffins from methane to n-decane were calculated and compared with the literature values (API, 1971). The average absolute deviation obtained is only 1.4%. In addition, the enthalpy departure, (H*- H), of liquid methane was calculated a t three temperatures and compared with the data reported by Jones et al. (1963). A summary of the satisfactory results is given in Table VIII. The applicability of the generalized correlation to vaporliquid equilibrium calculations is of utmost importance as far as this investigation is concerned. For this reason, isothermal vapor-liquid equilibria have been calculated for seven arbitrarily selected binary systems a t 31 isothermal conditions. The generalized correlation permits the computation of R, and R b with only the knowledge of w and T , of the pure components concerned. In the calculation of vapor-liquid equilibria data, the same mixing rules as those proposed by Chueh and Prausnitz (1969) were employed. Values of the binary interaction constant k I 2 ,which is involved in the mixing rules, were taken from the literature. The overall absolute deviation in y for a total of 237 data points is 0.0053 mole fraction compared to 0.0068 mole fraction when the parameters R, and R b were calculated from the saturation properties. The calculation results are given in Table IX. An attempt was made to predict binary vapor-liquid equilibrium data when one of the components is in the supercritical state. For such a component, values of R, and n b may simply be taken to be those obtained a t the critical temperature. The experiment results reported by Sage and Lacey (1950) for the binary system methane-n-butane a t 100 O F were selected for this purpose. At this temperature, methane is in the supercritical state. Therefore, the values of R, and R b for methane were taken from Table I1 a t T , = 1.0. The generalized correlation was used to obtain the values of R, and f i b for n-butane. In the calculation, k . 1 2 was taken to be 0.027,
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Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
which was the value used in the calculations for the same system a t lower temperatures as shown in Table IX. The calculated results compared very favorably with the experimental data. The deviations obtained are summarized as follows:
I Ay I av = 0.0078 lAP/Plavx 100% = 1.21 I t may be mentioned that the same data were used by Soave (1972) for testing his correlation, but the deviations obtained from his method as indicated below
1 Ay I
= 0.0139
I AP/P(av x 100% = 3.93 are somewhat higher.
Conclusion In this investigation, generalized correlations were established for the temperature dependence of the parameters Q, and Q b of the Redlich-Kwong equation of state in terms of T , and acentric factor w in the subcritical region. The expressions are simple, adequate, and lend themselves easily to computer evaluation. They can be used, for example, in Harmen's program for the evaluation of liquid compressibility and vaporliquid equilibria up to acentric factors of 0.4902 for n-decane. They produced favorable results over the range of study and satisfy the critical point.
Acknowledgment The authors are indebted to the National Research Council of Canada for financial support, and to M. Kat0 for his assistance in some of the calculations.
Nomenclature ai, bi = coefficients in eq 4 and 6 a, b = constants of the Redlich-Kwong equation of state c j , dj = coefficients in eq 5 and 7 ai,, bi;, c j j , dj, = coefficients in eq 9-12 f = fugacity H = enthalpy k I 2 = binary interaction constant P = pressure R = universal gas constant T = temperature V = molalvolume y = vapor phase mole fraction
Greek Letters A = deviation 6 = relative absolute deviation defined by eq 8 w = Pitzer's acentric factor R,, R b = parameters of the Redlich-Kwong equation of state Superscripts * = idealgasstate 1 = liquidstate v = vapor state
Subscripts calcd = evaluated from saturation properties comp = computed from correlation equations c = critical properties i j or 1,2 = component identification r = reduced s = saturated
Literature Cited American Petroleum Institute, "Technical Data Book-Petroleum Refining", 2d ed, Washington, D.C., 1971. Beret, S.,Prausnitz,J. M., Paper presented at 54th Annual GPA Convention, Mar 10-12, Houston, Texas, 1975.
Canjar. L. N., Manning, F. S.,"Thermodynamic Properties and Reduced Correlations for Gases", Gulf Publishing Co., 1967. Chandron, J., Asselineau, L., Rennon, H., Chem. Eng. Sci., 23, 839 (1973). Chang, S. D.. Lu, B. C.-Y., Can. J. Chem. Eng., 48, 261 (1970). Chueh, P. L., Prausnitz, J. M., lnd. Eng. Chem., Fundam. 6, 492 (1967). Das, T. R., Eubank, P. T., Adv. Cryog. fng., 18, 208 (1973). De Mateo, A., Kurata, F., lnd. Eng. Chem., Process Des. Dev., 14, 137 (1975). Din, F. "Thermodynamic Function of Gases", Vol. 3, Butterworths. London, 1961. Elliot, D. G., Chen, R. J. J., Chappelear, P. S.,Kobayashi, R., J. Chem. Eng. Data, 19, 71 (1974). Eubank, P. T., Adv. Cryog. Eng. 17, 270 (1972). Goodwin, R. D., Prydz. R., NBS J. Res., 75A, 81 (1972). Guggenheim, E. A., J. Chem. Phys., 13, 253 (1945). Hamam, S.E. M., Lu, B. C.-Y., Can. J. Chem. Eng., 52, 238 (1974); J. Chem. Eng. Data, 21, 200 (1976). Harmens. A., Cryogenics, 15, 217 (1975). Horvath, A. L., Chem. fng. Sci., 27, 1185 (1972). Hsi, C., Ph.D. Thesis, University of Ottawa, 1971. Joffe, J., Schroeder. G. M., Zudkevitch, D., AlChE J., 16, 496 (1970). Jones, M. L., Jr., Mage, D. T., Faulkner, R. C., Jr., Katz. D. L., Chem. Eng. Progr. Symp. Ser. No. 44, 59, 61 (1963). Kato, M., Chung, W. K., Lu, B. C.-Y. Chem. Eng. Sci., 31, 733(1976a): Can. J. Chem. Eng., in press (1976b). Lu, B. C.-Y.. Chang, S.-D., Elshayal, I. M., Yu, P., Gravelle, D., Poon, D. P. L., Proceedings, First International Conference on Calorimetry and Thermodynamics, pp 755-766 Warsaw, 1969. Michels, A., Levelt, J. M., De Graff. W., Physica, 24, 659 (1958). Modell, M., Reid, R. C., "Thermodynamics and its Applications", p 535, Prentice-Hail, Englewood Cliffs, N.J., 1974. Parrish, W. R.. Hiza. M. J.. Adv. Cryog. Eng., 19, 300 (1974).
Passut, C. A., Danner, R. P. lnd. Eng. Chem., Process. Des. Dev., 12, 365 (1973). Peng, D.-Y.. Robinson, D. B. lnd. Eng. Chem., Fundam., 15, 59 (1976). Prausnitz, J. M. "Molecular Thermodynamics of Fluid-Phase Equilibria" Prentice-Hall, Englewood Cliffs, N.J., 1969. Prydz. R., Goodwin, R. D., J. Chem. Thermodyn., 4, 126 (1972). Rediich, O., Kwong, J. N. S.,Chem. Rev., 44, 233 (1949). Robinson, D. E., Kaka, H., Proceedings of 53rd Annual Convention of GPA, pp 14-20, Denver, March 1974. Sage, B. H., Lacey, W. N., API Research Project No. 37 "Thermodynamic Properties of the Lighter Paraffin Hydrocarbons and Nitrogen" (1950); "Some Properties of the Lighter Hydrocarbons Hydrogen Sulphide and Carbon Dioxide", (1955). Simonet, R.. Behar. E. Chem. Eng. Sci., 31 37 (1976). Soave. G.. Chem. Eng. Sci., 27, 1197 (1972). Stryjek, R., Chappelear. P. S.,Kobayashi. R., J. Chem. Eng. Data, 19, 334 (1974). Toyama, A.. Chappelear, P. S., Kobayashi, R., Adv. Cryog. Eng., 7, 125 (1962). Vogi, W. F., Hail, K. R.. AlChE J., 16, 1103 (1970). Webber, L. A., NBS J. Res., 74A, 93 (1970). Wichterle, I., Kobayashi, R.. J. Chem. Eng. Data, 17, 4 (1972a); 17, 9 (1972b). Wilson, G. M., Silverberg, P. M., Zeilner. M. G., U.S. Departeent of Commerce, Technical Documentary Report No. APL TDR -64-64, Apr 1964. Wilson, G. M., Adv. Cryog. fng., 11, 392 (1966). Young, S.,Sci. Proc. Royal Dublin SOC., 12, 374, (1910). Zudkevitch, D., Joffe, J., AlChEJ.. 16, 112 (1970).
Received /or reuieub December 8, 1975 Accepted August 4, 1976
A Simple Method for Determining the Reaction Rate Constants of Monomolecular Reaction Systems from Experimental Data Feg-Wen Chang, Thomas J. Fitzgerald," and Jin Yong Park Department of Chemical Engineering, Oregon State University, Corvallis, Oregon 9 733 I
A new method is presented for obtaining reaction rate constants for any set of simultaneous first-order chemical reactions. For k independent reactions the required data consist of measurements on k species at k 1 regularly spaced points in time. The method is exact and gives precise results for error-free data. Redundant data are treated by a simple least-square regression technique.
+
Introduction The monomolecular reaction system is of particular interest to chemical engineers because many important chemical processes are first-order or approximately first-order reactions. Wei and Prater (1962) used the characteristic direction method to determine the reaction rate constants of monomolecular reaction systems. Although the method shown in their work demonstrated a better way for determining the reaction rate constants than the conventional curve-fitting techniques (Haag and Pines, 1960), it is still tedious and complicated, particularly in reaction systems with many chemical species. The object of this paper is to develop a method which provides a more accurate and much easier way for determining the reaction rate constants from experimental data than other previously published methods.
Determination of Reaction Rate Constants from Experimental Data
this system is shown in Figure 1.The ith species of this system is designated by A, and the concentration by c,. The reaction rate constant of the ith species to the j t h is denoted by k,,, i.e.
From the mass balance of the j t h species at time t , we find dt
Define
Consider a reversible (or irreversible) monomolecular reaction system of n chemical species. A schematic diagram of Ind. Eng. Chern., Process Des. Dev., Vol. 16, No. 1, 1977
59
