Enthalpy of Mixtures by Modified BWR Equation Kenneth E. Starling' School of Chemical Engineering and Materials Science, The University of Oklahoma, Norman, Okla. 73069
John
E.
Powers
Department of Chemical and ilfetallurgieal Engineering, The Cniversity of Jfichigan, Ann Arbor, Jf ich. 48103
The enthalpy behavior of methane-propane mixtures i s predicted using a modified BWR equation for enthalpy developed to predict pure component enthalpy behavior. The use of geometric mixing rules for the three new parameters introduced into the BWR equation yields enthalpy predictions which are generally within 2 Btu per pound of experimental data for methane-propane mixtures. The use of specific values for unlike interaction parameters yields predicted mixture enthalpies that are somewhat more accurate than enthalpies calculated using geometric rules.
A
modified BWR equat'ion for enthalpy, developed from analysis of pure component enthalpy behavior, was tested for mixture enthalpy predictions. There were two objectives in this study. The first was t.0 determine if modificat,ioiis to the temperature dependence of the B K R equat'ioii which greatly improve pure component. eiit.halpy predictions also can significantly improve mixture enthalpy predictions. The second object,ive n-as to ascert'aiii the most appropriate parameter mixing rules for t'he new parameters in the modified BWR equation. Esperimeiital ent'halpy data for t'he met'hane-propane system obtained a t the Vniversit'y of 1Iichigan (Jones et al., 1963; lfanker, 1964; Mather, 1968; Yesavage, 1968) were utilized. The methane-propane system was chosen for st,udy because it is t'he only binary system for which highly accurate ent.halpy data over the eiit,ire composition range have been reported. Modified BWR Equation
The modified I3WR equation for pressure used in this study can be written in the form
The correy3onding modified BWR equation for enthalpy departure ha.. the form
H ( T , P ) - H(T,O)
=
4C T2
- 2A1u- O-
* T o whom correspondence should be sent.
+
-Although the density dependence of t'he modified BWR equation is identical to that of the original 13WR equation (Benedict et al., 1940), its temperature dependelice differs by virtue of the terms involving parameters Do, Eo,aiid d. The use of reciprocal temperature expansions in t'he modified BWR equat'ion is based on the perturbation theory of statistical mechanics (Zwanzig, 1954). There also is empirical ju3tificat,ioii for t'he particular modifications which were made. Benedict, Kebb, Rubin, and Friend (1951) found that parameter Co in t,he original E\YR equation may be varied t'o describe subatmospheric vapor preqsure data. St.arling (1966) noted that' the discrete values of C, required to describe vapor pressure data can be expressed as linear functions of reciprocal temperature for reduced t'emperatures above 0.4. This observation gave rise to the term involving parameter Do. Empirical analysis ( C o s , 1968) of methaiie eiithalpy data indicat.ed that the niodificatioii iiivolving parameter d improves enthalpy predictions in the critical region. Finally, the analysis (Starling, 1968) of the propane enthalpy data est'ending down to reduced temperat'ures below 0.4 iiidicated the need for the term iiivolviiig parameter E,. Pure component density and enthalpy data were used simultaiieously in multipropert'y regression aiialj 1967, Starling and Wolfe, 1966) to determine parameters in the modified BWR equation. Esperiment~al data from a number of literature sources were ut'ilized in t'hese calculations (Douslin et a?., 1964; Huaiig et al., 1966; Ros?ini, 1953; Sage and Lacey, 1950; van Itterbeek et al., 1963; Veiiiiis, 1966; Yesavage, 1968). The phi1osol)hy employed was to usc original IJWR parameter values to the extent' posqible within the stipulat'ioii t'hat uncertainties iii predicted enthalpies be less than 1 13tu per pound or roughly twice the esperimental uiicertaiiity. For methane, it, was p o s d ~ l eto ret,aiii sis of the origiiial BWR parameter value.;; for propane, four. The result's of pure component analysis to determine modified B K R parameters for methane and propane are summarized in Table I. Eiit,halpies predicted by the new equat'ioii are remarkably accurat,e, with deviations from esperimental data of 0.61 Btu per pound for methane and 0.85 I3t'u per pound for propane. Figures 1 a i d 2 show topographical plots of these deviations for the temperat,ure-pressure range of the dat8a.The legelid for deviations in Figures 1 and 2 aiid all other topographical plots in this paper is given in Figure 3. Ind. Eng. Chem. Fundam., Vol. 9, No. 4 , 1970
531
Table 1. Summary of Results of Pure Components Methane
No. of P V T data points KO.of enthalpy data points Temp. r a n g e o f P V T d a t a , O Temp. range of enthalpy data. O F Av. dev. - densities ( P < 2PJ, % Av. dev. - densities (P
25 75 -253 to +257 -250 to +280
58
F
-250 t o +50
-250 to +250
0.44
0.43
2.46
2.73
0 65 0 61
0.55 0.85
1.91
2.66
> 2pe)J %
Av. dev. - enthalpy departures, % Av. dev. - enthalpy, Btu/lb hIax. dev. - enthalpy, Btu/lb Modified BWR Parametersa
Propane
40
TEMPERATURE
Methane
OF
Figure 2. Comparison of enthalpy departures calculated by modified BWR equation with experimental values for propane Btu per second
Propane
6.82401b 11.7001 Bo X 10 21.2933 6 . 99525b A , x 10-3 8.67325b 57.7355b b x 10 24.9577b 5 . 11172b 01 x 10 252.478b 4. 9810gb c x 10-8 1.53961b 5.64524b Y 3.21834 c, x 1082.7999 1.33871 D o x 10-’0 88.8011 0.02974 E, x 58.9916 2.51441 48.3036 u x 10-3 1.50876 56.4493 d x a Parameter values consistent with use of R = 10.7335 psia-cu ft. J m.w. (methane) = 16.031, m.w. (propane) = 44.062. lb-mole-’R b Original BWR parameters.
( Hexptt.
Deviation = d
U
-
Hca~c.)l B + u ” b
O < d < 2
Exceeds limits of correlation
Figure 3. Legend for deviations presented in Figures 1, 2, and 4 to 8 Mixture Parameters Modified BWR Equation
Benedict, Webb, and Rubin (1942) proposed the following relations for BWR mixture parameters as functions of composition and the pure component parameters,
Bo
=
Cz Z ~ B ~ ~
(3) (4)
-250
-200
-150
-100
TEMPERATURE
-50
0
(5)
50
OF
(6)
.
Figure 1 Comparison of enthalpy departures calculated by modified BWR equation with experimental values for methane Btu per second
Errors in predicted densities average 0.44y0 for densities less than twice the critical density. Above twice the critical density predicted densities are too low by from 2 to 3%, a result which also occurs with the original B W R equation. The methane enthalpy data extend to the reduced temperature T R = 0.6 and reduced density p R = 2.5, while the propane enthalpy data extend to TR = 0.3, p R = 3.2. On the other hand, there are no highly accurate density data for these compounds for T , < 0.8 and p R > 2.2. Because of the inadequacy of high density PVT data and because emphaqis in this study was on enthalpy prediction, no attempt was made to modify the density dependence of the B K R equation. 532 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
The relationships for Bo,do,C,, and y may also be expressed by the following general relation, which is of the form of statistical mechanical expressions for mixture second virial coefficients, B = CCXtxjBzj (11) $
3
Where B represents a general parameter for the mixturee.g., Bo, A,, C, or y while B,, is the corresponding parani-
Table II. Summary of Results of Regression on Mixture Enthalpy Data to Determine Do, E,, and Mixture Composition, Mole
5.2 88
3 2 1 0 2 2 0
D o X 10-l0, regression Do X geom. rule E, X regression E , x 10-12, geom. rule d x regression d x geom. rule Av. dev.-departure, 70
02278 51818 14398 31682 27086 13061 60
50.6
11.7
28.0
44
No. of Data Points 42
13.2496 12.0512 7.16008 5,17437 7.62802 6,85782 0.63
4.99799 4.51220 I . 99451 1.10438 3,46137 3.12215 0.61
76.6
39
44
30,6369 28.5137 19.3381 15,7733 19,1731 15.7685 0.98
57.7736 56.0865 37. 5306 35.0900 32.6263 32.9775 0.47
Table 111. Results of Enthalpy Calculations Using Geometric Mixing Rules for Do, E,, and Mixture Composition, Mole
AIolal av. molecular 1%t. Av. dev-departur es, 70 Av. de\-.-enthalpy, Btu/lb Max. dev.-enthalpy, Btu/lb
5.2
11.7
17 49 1 36 1 39 5 34
19 31 0 82 l 02 2 87
et'er for the i t h component. Parameter Bi3,i # j , is referred to in statistical mechanics as the unlike interaction parameter for interact,ioiis b e b e e n the ith and the j t h component. The assumption
B 2.3. -
1 - (Bii 2
+ B..) 33
(12)
d
70Propane
28 .o
23 1 2 7
d
% Propane
88 56 47 21
50.6
30 2 3 8
21 39 63 13
76.6
37 1 2 6
50 76 90 70
parameters Do, Eo, and d in tmhemodified 13TTR equation also may be expressed in the manner of Equat,ioiis 11 and 16. Only binary mist'ures will be considered, so these mixture parameter relat'ions can be expressed by the following equations, using the fact that' subscripts may be permut,ede.g., Dol, = Do,l,dl12= d121= dsn-
leads to the result
B
CxzBi
=
i
which corresponds to Equation 3 for Bo.The assumption
Bij
Bitl/PB,.1/2 33
=
(14)
leads t o t'he result
B
1
CxiBil/2
=
[ i
2
(15)
ivhich corresponds to Equations 4 to 6 for A,, eo, and y . Similarly, the relations for b, a, cy, and c iii Equations 7 t'o 10 may be expressed by the following general relation,
c=
i
j
CZiLjZK Cljk k
(16)
The assunipt,ion
Cijk =
c
cil/3
. 1 / 3 Ckl!3
(17)
leads to tmhe result
c == Cx.C
113
[ i r i
1
3
(18)
which corresponds to the relat'ions given in Equations 7 to 10 for b, a, cy, and c. Other prescriptions (Benedict et al., 1942; Stotler and Benedict, 1953) for unlike interaction parameters as functions of pure component parameters have been considered for replacing the linear average in Equation 12 and the geometric averages in Equations 14 and 17. Kumerical values of unlike interaction parameters also have been determined from mixture P V T data (Opfell et al.! 1959). However, very little use has been made of mixture parameters other than those given by Equat,ions 3 to 10. These relations were also used for the original BWR parameters in the present study.
Mixture Enthalpy Calculations
The first objective of this st,udy was to determine if t'he modifications to the temperature dependence of the BWR equation which improve pure component ent.halpy predictions also can improve mixture enthalpy predictions. T o make this assessment, parameters Do, E,, and d were determined for each misture by regression, using the enthalpy data (Mather, 1968; Yesavage, 1968) for iiitliridual niistures. The range of the dat'a for each mixture was -250" t o +250°F, 0 to 2000 psia. The eight original BWR parameter values given 113. Equations 3 t o I1 with the pure coniponeut paraineters in Table I were used in t'hese calculations:. These calculations necessarily are iterative in nature, by virtue of the fact t'hat for a given temperature and pressure, the density value used in Equation 2, t,he expression for enthalpy departure, must be determined by trial and error from Equation 1, t'he expression for the pressure. For a given iteration, the parameter values used in Equation 1 are the values determined in the previous it,eration. The results of t,hese regression calculations are summarized in Table 11. There can be no doubt' that the modified BWR equation given in Equation 2 is capable of describing the enthalpy behavior of the methanepropane system. The over-all average absolute deviation in mixture eiit'halpy depart'ures for t'he 257 data points referred to in Table I1 is 0.6470, compared to 0.59y0 for the 133 pure component data points referred t o in Table I. Thus, data for individual mixtures can be described by this equation form almost as accurately as pure component data. The second objective was to determine the most appropriate Ind. Eng. Chem. Fundarn., Vol. 9, No. 4, 1970
533
-150
-250
-50 50 TEMPERATURE O F
150
250
Figure 4. Comparison of enthalpy departures calculated b y modified BWR equation with experimental values for 5.2 mole propane in methane mixture
yo
TEMPERATURE OF
Figure 5. Comparison of enthalpy departures calculated b y modified BWR equation with experimental values for 1 1.7 mole propane in methane mixture
70
TEMPERATURE OF
Figure 6. Comparison of enthalpy departures calculated b y modified BWR equation with experimental values for 28.0 mole propane in methane mixture
70
534 Ind. Eng. Chem. Fundom., Vol. 9, No. 4, 1970
TEMPERATURE OF
Figure 7. Comparison of enthalpy departures calculated b y modified BWR equation with experimental values for 50.6 mole % propane in methane mixture
TEMPERATURE OF
Figure 8. Comparison of enthalpy departures calculated by modified BWR equation with experimental values for
76.6 mole yopropane in methane mixture
mixing rules for the new parameters in the modified B W R equation. Values of Do, Eo, and d resulting from use of geometric rules in Equations 19 to 21 are given in Table I1 to show the close agreement obt'ainable using geometric mixing rules. The predictive ability of the modified BWR equation for enthalpy departure using geomet'ric mixing rules for Do, Eo, and d is summarized in Table 111. Densities used in t'hese calculat'ions were determined using Equat'ion 1, the modified BWR equation for pressure. The over-all average absolute deviation in ent,halpy departures for the 257 mixture data points considered in Table I1 is 1.54y0, while the average deviat,ion of predicted enthalpies is 1.94 Btu per pound. To show the pressure-t'emperature regions in which the larger deviations occur, topographical plots of deviations in Btu per pound are given in Figures 4 to 8. The results in Figures 4 and 5 for the lower molecular weight mixtures containing 5.2 and 11.7 mole 7 0 propane are remarkably good, with virtually no regions of deviations greater t,haii 5 Utu per pound. Of the five binary mixtures considered, these two mixtures correspond most closely to mixtures encountered industrially-e.g., cryogenic
Table IV. Results OF Regression to Determine Do and E, Using Geometric Rule for Mixture Composition, Mole
D o x 1O-lo, regression Do,, x E , x lo-'*, regression E,,, x 10-12 Av. dev.-departures, yo
d
% Propane
5.2
11.7
28.0
50.6
76.6
2,96056 15.3897 1 ,00864 9.11947 0.60
4.80607 12.3435 1.63645 3.89990 0.66
12.9137 13.0422 6.51587 4.70128 0.74
29.5143 12.9047 17.1335 4.04568 1.24
57.8661 15.8671 37.7336 8.69860 0.48
processing or liquefaction of natural gas. On the ot'her hand, the deviations in Figures 6 to 8 for the higher molecular weight mixtures generally exceed 5 B t u per pound only a t teniperat,ures below - 150°F. It is doubtful that' hydrocarbon niixt.ures having molecular weights above 24 are processed at teniperatures below -150°F a t the present time. Thus, the use of geometric mixing rules for D,,Eo, and d in the modified BWR equat'ion for enthalpy is appropriate from a practical point of view. From an academic point of view, on the other hand, it is reasonable to seek mixing rules which yield predictions approaching more closely the accuracy quoted in Table 11, using parameters determined by regression. T o i n v e 4 g a t e paramet'er mixing rules furt'her, advantage is taken of the fact that geometric mixing rules are iiumerically close to the values of paranieters determined by regression. For example, the fact that the larger deviations in enthalpies predicted using the geometric rules generally occur a t lower teniperatures is a11 indicatioii that deviations froni t'he geomet'ric rule for parameters D, aiid E , markedly affect the accuracy of predicted enthalpies. O n the ot'lier hand, the fact that predictions are accurat'e for pseudoreduced t'emperatures near unity indicates that use of the geoniet'ric rule for paranieter d does not adversely affect enthalpy predictions. T o verifj- this observation, the results of regression calculations to determine D,and E , for each mixture, using the geometric niisiiig rule for d, are given in Table IV. .lvei,age tleviations iii Table IV for enthalpy departures using the values of Do and E , from regression with d fixed by the geometric rule are o d y slightly greater than the average deviations in Talile I1 result'ing froni using values of D,, E,, aiid d detennined hy regression. Because of this agreement, oiily the geometric rule is recornmended for paraniet~erd . On the other hand, fixing both Do and d by the geoinetric rules nnd regressing to determiiie E , offered very littlc i~iiprovenieiitover use of geometric rules for all three new p:irmieters ( D o , E,, a n d d ) . Therefore, attention iiiust be focu>ed 011 both D, aiid E,, but' iiot on d. The unlike interaction paranieters Do,? and E,!? in Table IV wei'e calculated for each mixture using Equations 19 aiid 20 and the taliulated values of Do and E, for each mixture deterniiiied liy regression. Albhough, in principle, DUl2and E,,? should be indepeiident of composition, t'here is considerahle vai,iatioii in these parameters with mole fraction (Table IT:). However, there is good agreement for both D,,, and I:',,? a~iiongthe 11.7, 28.0, and 50.6 inole 7, propane mixture.;. The unlike interaction terms in Equat,ioiis 19 and , go through rnasima a t x1 = 0.5. of nul? ant1 Eol,given in Table IT.' aiie are the recoinmeiided unlike iiiteractioii ~inrariietervalues for the riiethaiie-liroi)ane Using these values of D,,? and E,,?, eiit'halpy predictions for all fi\-e mixtures were iiiade (Table V ) ,
Table V.
D,,?
Results of
Enthalpy
Calculations
= 12.9047 X lolo,EUl2= 4.04568 X Mixture Composition, Propane
5.2
11.7
Using
loL2 Mole %
28.0 50.6 76.6
Av. dev.-departures, yo 0 81 1 16 0 86 1 24 1 03 av. dev.-enthalpy, Btu/lb 0 59 1 52 0 97 1 35 1 54 N a x . dev-enthalpy, Btu/lb 2 59 8 09 5 92 8 65 3 49
The over-all average absolute deviation in enthalpy departures for the 257 mixture data points considered in Table V is 0.987,, while the average deviation of predicted enthalpies is 1.09 B t u per pound. As not'ed with reference to Table 111, these over-a11 averages are 1.547, and 1.94 13tu per pound when geometric rules are used for Do and E,. The corresponding over-all averages using the D, and E , values in Table IV det'erniined by regression are O.7ly0 and 0.62 Dtu per pouiid. Thus, use of the values for t'he unlike iiiteract'ion parametera, Do,? and E,,? in Table Tr, offers improvenient over use of geometric rules, with ent,halpy predictions only slightly less accurat'e t'han when D, and E , values determined by regression for each mixture are used. I t t'herefore is worthwhile tto seek eq)ressioiil; for Do,?and E,,? as fuiictioiis of the pure cornlmiient parameter.. I11 seeking expressions for D,]? and E,,?, advantage is t,akeii of the fact t'hat t'he values of D,and E, from regression fall betweeii the linear and geometric rule values. To show this result, the values of D, and E , iii Table IT', determined for each mixture by regression, are plotted in Figures 9 and 10, along nit'h curves resulting froni use of linear and geometric rules. The difference bet'neen these curves for D, at, a given composit'ion is 2zlx2
[: -
1
(DUl f Do>)- (Do,Do,)1/2, \r.it,h a
corresponding relation obtained for E,. The niixt.ure 1)aranieter values determined by regression fall only a fract,ion of this distance above the geometric rule curve; t.hus, it is convenient to express the composition dependence of D, and Eo bl- the relations
+
(D, DuJ - ( D u l D u J 1 ~ z (22) ] E,
=
Z X ~ E , ~+~ ' ~ [ i
1
(E,,
1
+ E,>) + (Eu1Eu2j1112 (23)
where h(D,,J aiid h(E,,,) are the appropriate fractions of the diff erelice between the linear and geomet,ric rule paraniet,ers. Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
535
0
10
20
30
40
50
60
70
,
80
I
90
100
MOLE PERCENT PROPANE
Figure 9 . Comparison of modified BWR parameter Do from regression on mixture enthalpy data with linear and geometric mixing rules 900
-
800 700 600 500 GEOMETRIC MIXING RULE
' 2 400. no 300 200 -
0 DETERMINED B Y REGRESSION
100 -
0 L-
A
0
IO
20
30
40
50
60
70
80
90
I
100
M O L E PERCENT PROPANE
Figure 10. Comparison of modified BWR parameter E, from regression on mixture enthalpy data with linear and geometric mixing rules
Using Equations 22 aiid 23 in Equations 19 aiid 20 s h o w that Do,? aiid E,], may be expressed as the folloning average- of linear and geometric rules,
The values of /z(Dol?) = 0.05858 and h(Eol?)= 0.09651, when uqed in Equations 24 and 25, yield the values of Do and EO,? used in Table V. 536 Ind.
Eng. Chem. Fundam., Vol.
9, No. 4, 1970
Discussion
The study demonstrates that. a niodified 131S'K equation for eiitha1l)y developed to predict pure co~npoiieiitenthalpy behavior accurately also is capable of describing the enthalpy behavior of mixtures. Even inore striiigent, t'ests of t'he modified BWR equation can be obtained by attempting its use for vnl~orpressure and multicompo~ientphase equilibrium predictions. .Int,ici11ating these te propaiie >yst,em was studied, since this is the only biliary system for which extensive accurate PV T , enthalpy, and phase equilibrium data esist. The use of the modified I3WR equation for 1)haPe equilibrium predictions vi11 be reported a t a future date. Acknowledgment
We thank J. L. Uoiio, S . F. Cariinhaii, K. W.Cox, Y. C. K n o k , nnd F. D. Robert3 as well a3 Pamela Pe*ek for their
help in this study, which is part of a continuing effort in equation of state developiiieiit.
Jones, lX,L., Jr., IIage, D. T.. Faiilkner. R. C.. .Tr.. K a t z C‘henz. Eno. Proor. Sunin.
n
T
Ph.D. thesis, University of lXic E., Ph.D. the&, University of AIic Opfell, J. B., Pings, C. J., $age, B. H., “Equa6ons of State for Hydrocarbons,” pp. 1-184, American Petroleum Institute, New York, 1959. Rossini, F. D., ed., “Selected Yalues of Physical and Thermod p a m i c Properties of Hydrocarbons and Related Compounds,” pp. 290-300, Carnegie Pres.s, Pittsburgh, Pa., 1953. Sage, B. H., Lacey, IT. S . , “Thermodynamics of the Lighter Paraffin Hydrocarboil? and Nitrogen,” pp. 33-S, API Research Project 37, New York, 1950. Starling, K. E., Satural Gaq Processor, As,sociation Enthalpy Project Progress Report, Sept. 30, 1968. Starling, K. E., SOC.Petrol. Eng. J . 6 (4),363 (December 1966). Starling, IC. E., “Use of 3luItiproperty Thermodynamic Data in Equation of State Development,” Research P r o p o d to KSF, AIarch 1967. Starling, Iityof llichignn, 1968. Zwanzig, R. IT., J . Cheni. Phys. 22, 1420 (1954). ,
Nomenclature
parameters in modified B K R equation DOI?,E0,2!d112&2 = unlike iiiteractioii parameters H = enthalpy, B t u per pound Jz(Do,,)!h(E,,,)= fractions defined in Equations 24 and 25 P = pressure, p i a R = gas constant T = absolute temperature, OR x i = mole fraction of itti component
Ao,Bo,Co,Do,Eo, a,b,c,d
=
GREEKLETTERS p
a,-/
= =
molar density, 1bmoles:cu ft parameters in modified BWR equation
SCBSCRIPTS i = ith co~npo~ieiit R = reduced property Literature Cited
Benedict, AI., Rebb, G. B., Rubin, L. C., J . Chem. Phys. 8, 334 (1940). Benedict, AI,, TTebb, G. B., Rubin, L. C., J . Chem. P h y s . 10, 747 (1942): Benedict, AI,, Webb, G. B., Rubin, L. C., Friend, L., Chem. Eng. Progr. 47,419 (1951). Cox, K. W,)31. S.Ch. E. thesis, University of Oklahoma, 1968. I)oiwlin, D. I?,)Harrison, R . H., AIoore, R. T., AIcCullough, J. P., J . Cheni. Eng. Data 9, 358 (1964). Ruang, E. T. S.,Swift, G. IV.,Kurata, F., A.I.Ch.E. J . 12 ( 5 ) , 932 (1966).
RIXEIVEDfor review Febrliary 24, 1970 ACCT:PTI:DJuly 30, 1970 Symposium on Enthalpy of AIixtures, Division of Indii-trial and Engineering ChemistrJ,: 159th AIeeting, ACS, Hoiiqton, Tex., Febrliary 22 to 27, 19iO. ” ITork supported i n part by the University of Oklahoma, the Natural Gas ProcePsor the Sational Science Foundation (Grant GK-2211).
Corresponding States Principle Using Shape Factors Gary D. Fisher The ChemShare Carp., Houston, Tex.770.27
Thomas W. Leland, Jr.’ Chemical Engineering Department, Rice L‘niversitg. Houston, Tex. Y700l
The simple corresponding states principle provides for predicting properties of pure fluids and mixtures that are conformal with a reference. Slightly nonconformal substances (require an extended CSP for satisfactory representation. This paper considers an extension of the simple CSP involving additional parameters called shape factors, which modify the critical properties of nonconformal fluids so that they conform to the reference. Theory and comparisons with experimental data indicate that total thermodynamic properties of a mixture (compressibility, enthalpy, fugacity, entropy) can be accurately calculated even at low temperatures b y CSP. Prediction of partial thermodynamic properties involves differentiation of CSP parameters with respect to composition and requires much greater accuracy in specifying the unlike pair interactions. Prediction of partial thermodynamic properties i s limited to reduced temperatures above 0.6 and mixtures having no large differences in molecular properties.
T h e principle of corresponding states enables the properties of complex niixtures t o tie determined from the properties of a .suitable reference or references. If all the mixture components and the reference conform t o t,he same intermolecular potential function
C7(r) =
Ef(T/r)
To whom correspondence should be sent.
(1)
a simple two-parameter corresponding states theory repre.qeiits the mixture. If all the components are not conformal with respect t o the reference because of ionc central force fields, small differences in polarizaliility, or weak dipole moments, the mistnre can be represented with hiifficieiit accuracy by an extended corresponding states theory. This theory can be used t o predict accurately total thermodynamic properties of the misture, such as eathalpy, entropy, and compreisiliility. Ind. Eng. Chem. Fundam., Vol.
9, No. 4, 1970 537
