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
Ao,Bo,Co,Do,Eo, a,b,c,d = 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
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
The prediction of mixture thermodynamic properties which require different'iation with respect to composition, such as partial volumes or fugacity coefficients, is not yet sufficiently accurate without incorporating empirically fitted parameters from binary data. Mixtures of components which are highly nonconfornial because of strong polarity or hydrogen bonding, cannot be adequately represented b y a corresponding states theory a t this time. The corresponding states theory involves t.hree major approximations. The magnitude of the errors introduced can be minimized by refiiiements in the theory. The first two problem areas arise even for conformal mixtures where the simple intermolecular potential of Equation 1 is satisfied. The approximations occur iii the following: The defining equation for the pseudoforce parameters for the mixture. The combining rules for the intermolecular potential parameters for unlike pair interactions. Corrections to account for deviations for each component in the mixture from the simple intermolecular potential of Equation 1.
the number density of all components, as well as the temperat'ure and radial separation distance rij. The radial distribution function for the mixture must be approximated, as it cannot be evaluat'ed rigorously. The most' effective method of doing this is to assume that the radial distribution function for a pair in the mixture can be approximat,ed by a radial dist'ribution of the same form as an imaginary pure component a t the average mixture reduced density, p m s 3 , and t'emperature, T , with it's pair interactions described by the properties of an ij interaction potential. K i t h this assumption,
The approximation in Equation 5 places the distribution function for an ij pair interaction in a mixture in the same form as that for t'he pure component in Equation 3 but wit'h a3, e l , , and a i j as nondimensionalizing parameters. This approximation revises the complex density arid compositional dependence of gil and replaces it with an average density dependence. The compositional dependence now appears in 53.
Defining Equation for Pseudoforce Parameters
d mixture of components which are conformal with the reference can be represented by an equation of state of the form Z = f ( ~ / k TV/a3) ,
Kirkwood et al. (1952) have shown that the radial distribut'ion function of a pure component can be expanded about the hard sphere radial distribution function. Applying this procedure to the distribution function in Equation 3 gives
g(r*,
a3p, e / k T ) =
(2)
where any additional parameters have the same value for all components t,o satijfy the conformality requirement and thus can be treated as const,ants. The properties of the mixture can then be obtained from Equation 2 for a given temperature and volume after determining the pseudoforce parameters, ;and a, which represent the mixture. The pseudoforce parameters are functions of composition and the force paramet'ers for the individual components in the mixture. h detailed treatment of the development of the pseudoforce parameter defining equation is given b y Leland et a!. (1962), Reid and Leland (1965, 1966), and Leach e t a l . (1968). Since the properties of the mixture and reference are required to be equal when the pseudoforce parameters are substituted into the desired reference property equation, the right-hand side:: of the follo\ving equations must he equal:
where go(r*, a 3 p , d k T ) is the hard sphere radial distribution function, and and $ 2 are fuuctions of r* and p a 3 . Using this expansion in Equation 3 and also in Equation 4 after making the approximation of Equation 5 leads to t'he following after performing the integrations indicated: a3fo
+
83
kT
fl
+ ,. .
where fo, f l . are functions of p m a 3 . This equation cannot be satizfied identically, since there are only two adjustable parameters, 5 and ;. Leland et al. (1961) chose z and a3 to satisfy the two leading terms, so that
z=1-
n
n
53 =
ZtXlUij3
i=l j=1
16')
This approximation should be most accurate when the higher where r* = r / a and x i indicates the mole fraction of component i. Equation 3 is the general equation of state for the compressibility of the reference substance, and Equation 4 is the general equation of state for the compressibility of the mixture. Equation 3 contains the pseudoforce parameters, E and 8 , since it is t'hese paranieters which cause the reference compressibility to be equal to the mixture compressibility. g(r*, pa3, ; j k T ) is the radial distribution fuiiction for the pure reference with the pseudoforce parameters appearing as iioiidinien.;ioiializing factors. g i j ( r i j , p l , p e , . . . p n , T ) is the radial distribution function for the i, j pair in the mixture. I t is a function of the parametel,s of all molecular interactions! 538 Ind.
Eng.
Chem. Fundam., Vol. 9, No. 4, 1970
f2 order functions -
f3 -.
T2' T 3
.
.are small a t higher temperatures. X
better aliproximation can be made setting a using the second two terms simultaneously after dividing by f l / k T . n
71
8'; =
~ a=1 j = 1
~
~
~
+ (
~ a 1~ ~ ~i~ tjj *~f 2 ii / f~i T )
would be deterniined em1)irically. This approach was used IJJ, Leach (Leach, 1967; Leach et al., 1968) and Prausnit'z and Gunu (1958) ; The theoretical significance of Equations 6 has been discussed by Leland et a l , (1 968, 1969).
f2/rl
Combining Rules for Unlike Pair Interactions
I n the defining equations for the pseudoforce parameters, (Equations 6), the force parameters in the intermolecular potential beta-een unlike molecules appear, u t j and t i j for i # j. The proper forms for u i j and t a l are not k n o m even for interactions between simple molecules. The assumptions usually made are the Lorentz-Berthelot espressions,
which can be obtained using the hard sphere repulsion approximation for u i j and London's description of the attraction interaction for nonpolar symmetrical molecules to obtain e l j (see discussion by Reid and Leland, 1965). Thus, Equations 7 involve considerable simplification and approximat ion. Empirical evidence indicates that these relations are inaccurate in some cases. Prausnitz and G u m (1958) suggest an empirical modificat'ion,
which should improve the prediction of t f j a t the expense of additional empirical information. Inaccuracies in the unlike pair interact,ions are probably the cause of errors in predicting partial thermodynamic properties such as fugacity coefficients on which vapor-liquid equilibrium factors depend. Differentiation of Equat'ions 6 with respect' to xi;for the calculation of partial thermodynamic properties of component k results in equations which are more heavily weighted in the unlike pair interactions, especially for the components that, have small mole fractioiis in the misture. Thus, the predictions of fugacity coefficients for heavy components in the vapor phase a t low temperatuies may be subject to Considerable error, a fact which is borne out by the calculated results. I n the total thermodynamic property predictions the unlike pair terms retain only their weight proportional to the mole fractions of t'he t'wo componeiits, z i and xj. Thus, total thermodynamic properties are not as sensitive to errors introduced by Equations 7 and are predicted more accurately. Mixtures of Nonconformal Components
Coinponent,s which do not satisfy the simple intermolecular potential function of Equation 1 because of nonsymnietrical potentials cause additional errors in the simple twoparameter corre~pondiiigstates theory. Pitzer et a l . (1955) accounted for these deviations b y introducing a third paranieter, the acentric factor. -4more theoretically based estension is to introduce temperature-dependeiit force parameters, a? first done by Cook and Roivlinson (1953). The condit'ions under rvhich a syninietrical potential with temperaturedependent pseudo force parameters can he a realistic replacement for an orientatioii-dependent poteiitial will be considered before exploring the temperature dependence further. The perturbation procedure developed b y Pople (1954) and discuosed by Rowlinson (1959) provides an analytical method for defining a factorable tlvo-parameter potential with pqeudo force parameters capable of serving as ail approximate replacement for a nonsymmetrical potential. Pople expaiided the orientation-dependent pair potential in spherical harmonic perturbation-; of the syninietrical potential in the Lennard-Jones form. The perturbing terms correct for interactions due to dipoles, nonisotropic polarizability, quadru-
poles, and steric effect's on the repulsive portion of the perturbed symmetrical potential. Pople (1954) showed that after integrat'ion over all orientat'ions in t'he configuration iiit'egral the first nonzero term involves the radial dependelice squared. Thus, a dipole-dipole interaction which perturbs the potential with an r e 3 dependence contributes a term with an r-6 dependence aft,er integration over all orientations. The coefficient of this r-6 term iiicludes a ( l j k T ) 2factor, so t,hat when it, is combined with the r-6 term in t'he unperturbed central force Lennard-Jones potential, the result is a new temperature-dependent coefficient. Similarly, the nonisotropic polarizability varying as r+ introduces a teniperaturedependent r-12 term in the symmetrical Leiinard-Jones potential. Thus, dipole-dipole and lionisotropic polarizability interactions can be accounted for by introducing a temperature dependence on the factorable central force potential of t,he Lennard-Jones form. This potential is, of course, confornial with all fluids which obey the simple corresponding states theory. Ciifortunately, there are other interactions which contribute powers of l / r which are not present in the unperturbed central force potential. The quadrupole interactions vary as rU5in the nonsyninietrical potential and came a temperaturedependent r-lo dependence in the symmetrical potential. Other interactions introduce similarly higher order r dependences. The combination of these terms with the LeniiardJones type central force potential n i g h t still he factorable as in Equation 1, but it would contain additioiial parameters and could not be made conformal with other fluid.: obeying the two-parameter corresponding state; priiiciple by introducing a temperature dependence into the parameters. There is an additional coinplicatioii arising from threebody and higher order molecular interactions n-hich cannot be expressed as a sum of pair potentials. These terms cannot, of course, be included in the ternl)erature-depelldeiit symmetrical potential, since only the pairwise additive contributions to the configurational integral are included. This effect would not be involved in predicting the second virial coefficient, since it involves oiily t\vo-bod\- interactions. This fact provides a check on the validity of the temperaturedependent pseudoforce parameters in representing fluid mixtures without the complication of three-body and higher interactions. Cook and Rowhison (1953) and Leach (1967) have thus developed temperature-del)eiideiit pseudoforce parameters to predict second virial coefficients. Because of these limitations of a tn-o-parameter potential with teiiil,erature-depeiideiit peudoforce parameters to serve aq a replacement for an actual iioiisyinmetrical potential, it is desirable to define the pseudoparameters empirically, These empirical pseudoforce parameters are mainly teniperaturedependeiit, but also contain a density dependence in some regions. The pseudoforce paraineters defined in thih manner have 110 relation to the perturbation of a simple potential. They are defined by equating two dinien~ionlessthermodynamic properties for a pure component and reference such a i +
-
I
- +j
.ii = At
Vi (_ _J kTi lij
(7,i) kTi
V
ti3
Qtj
(8)
which are solved simultaneously for pi? and s i j a; a fiuiction of temperature and volume. Subscript i deiiotei the properties for the pure component for which the pcudoforce liaranieters are defined relative to the reference j . Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
539
Shape Factors
It is convenient to relate the pseudoforce parameters to critical constants
(9) where a , and b j are uiiiversal constants for all fluids conformal with reference j . Oil and $ i jare termed shape factors, which correct for deviations of component i from conformality with the reference j. Substitute Equations 9 int’o Equations 8, and the defining equations for the shape factors become
Consequent’ly, the shape factors are multipliers for crit,ical constants which correct for deviations from the two-parameter corresponding states principle. When i aiid j are conformal, t8he shape factors are bot’h unity. The shape factors are determined only from pure component empirical information. Therefore, shape factor correlat’ions are relatively easy to obtain. For gases at moderate densit’y, the shape factors are functions of temperature only. This is shown by using t,he reduced second and third virial coefficieiitmsto define the shape factors as in Equations 11.
For pure components the arguments in Equation 14 reduce to Tri* = T / T C tand V7** = V / V c i . I n Equations 13 the value of wI for the methane reference was taken as 0.005. The Z c t term is the critical compressibility factor. of the methane reference. At V,(* values less than 0.5, Equat’ion 13 becomes independent of density and V7$*is set equal to the constant’ value 0.5. Likewise, for V r I *values greater t’han 2.0, the shape factors again become independent of density and V r I *is set equal to the constant value 2.0. The equations in 13 give shape factors which correct the components of a mixture for noaeonformality with methane. The T,’ and V,’ in Equation 14 are the pseudocritical parameters for the mixture given in Equation 15. For mixtures, the shape’ factor equations in 13 become nonlinear and require a n iterative solution. I n the case of pure components, no iterat’ion is necessary. Subst,itut,ionof Equations 9 into the defining equations for the pseudoforce parameters, Equations 6, leads to the definition of pseudocritical parameters for the mixture
T,’ n
= n
i=l j=1
Xizj
+
l/BiirTciBjrcTcj[ 1 / 2 ( + t ~ : V ~ J ~1/2($j~V~j)’~~l~ /~
Vc’ n
v,’= i = l
n
+
zi~,[1/2($ii;~cij1/3i / 2 ( $ j k v c j ) l ~ 3(15) ~3 j=1
where subscript 12 denotes the reference. Subscripts i and j are comp0nent.s in the mixture of n components. Actually, in this work the following equation was used as the defining equation for Vc’ since it gave similar results aiid is simpler: n
Ti,’
n
zizj(1/2)
= .j=l j = 1
For gases a t densitlei requiring more than three ~ i r i a l and s for liquids Equations 11 do not serve to deterinine the shape factors uniquel) . h convenient choice is then to use the compres.ibilitj factor and the residual chemical potential to define the shape factors. The residual chemical potential function is that which defines the logarithmic fugacity-pressure ratio.
Shape factors defined by Equations 12 do not show a density dependence for low densit’y gases, as suggested b y Equations 11, and also for compressed liquids. Leach (1967) found that the density dependence existed for reduced volumes in the range 0.5 5 V r t5 2.0 for isomeric and paraffin hydrocarbons. He also correlated the shape factors for these hydrocarbons using methane as a reference as follows:
eil
=
1
+
(wi
+
- w ~ ) [ ~ . ~8 90.8493 2 111 T , ~ *
- 0.5)] ( u t - w1)[O.3903(Vri* - 1.0177) 0.9462 (V,**- 0.7663) In T ~ ~ * ] } Z , I / Z(13) ,~
(0.3063 - 0.4506/Tri*)(Vri* $11
=
(I
+
The equations are applicable to Tri* values less than 2.0 where
T,,* 540 Ind.
=
T B n ~ T cand ’ Vri* = T’$ii/Tic’
Eng. Chem. Fundam., Vol. 9, No. 4, 1970
(14)
[(+ikVCi
+ $jkVCJ 1
The critical parameters, T,’ and V,‘, depend on the choice of reference and thus are not directly related to the actual critical constants for the mixture. They are the values which make the reference properties,Z and ( p - po))/RT,equal to the mixture properties. The pseudocritical parameters are temperature- and density-dependent from the dependences of the shape factors. Even though the shape factors are defined in terms of t’he volunietric properties of the pure components and the reference, it, is possible to calculate energy fuiictioiis for the mixture, Like Chueh and Praueiiitz (1967), Leach (1967) derived the following expression for the residual internal energy function:
Thus, with Equation 16 and the compressibility factor aiid ( p - po)!RT functions for the reference, all thermodynamic properties of the total mixture can be obtained. Equations for some component properties have been derived by Leach (1967), such as partial volumes and fugacity coefficients. Partial properties are subject to greater error than total properties, probably because of errors in the unlike pair combining rules. ThuR, the shape factors provide a convenient and logical method for introducing temperature-dependent parameters to correct for deviations from coiiforniality. It is relatively easy to convert shape factors evaluated from one reference to shape factors i.elative t o another reference. The relations
for thii; operation are given by Leach (1967) aiid Leach et al. (1968). Thus, it is possible to use shape factors based on one reference to calculate fluid properties from another reference which may be more similar to the fluid of interest. Reference Equations for Methane
The therrnodynarnic properties can be calculated most simply from an equation of state of the reference substance. For accurate property prediction, a very accurate tabulation of data or an accurate equation of state is required. The prediction of enthalpy, for example, depends on the derivative, (bZ;bT)i., integrated from zero pressure bo system pressure. Since differentiation tends to reduce accuracy, errors in the equation of 5tate r a i l be magnified in enthalpy prediction. Vapor-liquid equilibrium ratios are even more sensitive to errors in the reference properties. Siiice the most accurate volumetric data for hydrocarbons are available for methane, methane was used as the prime reference iii this work. The methane equation of state developed hy T'ennis et a l . (1970)) based on the data of Veniiix (1966) and Douslin et al. (1963), was used as the reference. All properties in the vapor phase and supercritical regions were obtained from this equation either directly as in the case of compressibility or by differentiation and integration as required by enthalpy and fugacity. Entropy is a combination of all three properties. The predicted error in the equation of state of Vennix (T'eiiiiix et al., 1970) is less t,han 0.17, oyer the raiige of V , > 0.45 and 0.6 5 T , 5 3.3. Properties in the liquid phase were predicted differently to take into account the two-phase region. Since the equation of state does iiot accurately predict kothernis in the twophase region, integration through this region for the enthalpy and fugacity calculatiotis iii the liquid was not poshible. The vapor pressure data of Keyes et al. (1922) aiid T'eiiiiix (1966) were correlated. The Paturated liquid aiid vapor volumes were obtained from the equation of state, using the vapor pressure correlation. The saturated vapor fugacity was similarly obtained aiid the saturated liquid fugacity was set equal to the saturated vapor fugacity. The subcooled liquid fugacity was then obtained by adding the correction to account for integration from the saturated liquid volume to the subcooled volume. Enthalpy wab baqed on the saturated liquid enthalpy of Jones (1961), followed by an integration to correct for the subcooled enthalpy difference. Integration along isotherms was performed by 8-point Gaussian quadrature. X comparison was made with integrations by 4-point Gaussiaii quadrature to eiisure accurate integration. There was no discernible difference in the two rebult?;. Equation of State Convergence
The Veiinix-Kobayashi equation of state (as in most equations of state) has temperature arid volume (or density) as independent variables, P = f ( T , V ) . Since pressure and temperature are the standard specifications of process condition., implicit solution for volume is required. The iniplicit solution in the vapor phabe and in the supercritical regions for P,< 1 for the reference equation can be obtained from several simple methods, direct iteration, Newton-Raph?on, Kegstein, etc., as the radius of coiivergence is large. I n the liquid region none of the standard techniques work unless the initial estimate of the volume is very close to the correct answer. Of course, the interval halving technique would work, but excessive computer time is required for sufficient accuracy.
-1procedure which performed exceptionally well in accelerating convergence to the accuracy required was the use of a ic'ewtoii-Raphson determined coefficient in a log-log function approsirnation of an isotherm. This technique leads t o the following equation for prediction of V 2based on a calculated P I a t the previous estimated volume VI aiid Po is the desired pressure:
where
as required by a Newton-Raphson technique. Thiq convergence scheme would normally converge to ail error in pressure of 10-5 with three to five iterations in the liquid aiid vapor regions. This technique bhould work for all coiivergence problems where
I n I'
=
d
111
X
+B
approximates the function to be solved implicitly. Pentane Reference
Siiice the methane triple point occurs near T , = 0.45, the methane reference could iiot be relied upoii for T , 5 0.5. A 1)eiitane reference was therefore used. -is iiidicated iii the discussion of shape factor, the shape factors must he relative to the same component as the reference. Thus, the >hape factors were modified iteratively according to the relations of Leach (1967) for chaiiging from methane to the 1)eiitaiie reference. The pentane reference equation of state used was correlated by Leach (1967) for the liquid region according to the Tait equation, using the data of Brydoii et ril. (1953), Li and Caiijar (1963), Sage aiid Lacey (1950), and Sage et 01. (1953). Comparisons with Experimental Data
The thermodynamic properties of light' hydrocarbon niixtiirez have been calculated over a wide raiige of coiitiitioiis for cornparisoiih with available exl)erimental data. The enthalpy data of Polyers (1969) and others have lieen used extensively to cornpare with calculat,ed rcsults, bhce these data Irovidetl an accurate independent check 011 the c,alculated results for the light hydrocartion s y s t e m . 'The other data ubed were the tabulatioii of Sage and Lacey (1950) for pentane, nitrogen, niethaiie-l)roi)ane, methane-decane, and methalie-peiitaiie. Comiiarisoiis with these data are given in Tables I and I1 for enthalpy, I l l for fugacity aiid conipressibility factors, and I S r for entropy. The enthalpy was reported in terms of the eiithalpy deviation, H* - H , to remove any base difference$ ailti ideal gas correlation errors. The average deviation with the esperimental results was about 2 Btu per pound. T o eliminate the tliffereiices in reference states aiid ideal gas heat capiicitiei betlveen the entropy values calculated and those reported t?y 13hirud (1969), the calculated value at O O F and 14.7 p i a was arbitrarily set equal to the esperimeiital value. The value,< at higher pressure were then tabulated, using the calculated differences. Table IV shows the comparison with experirneiitally based results for the mixture. The experimeiital and calculated results are graphed for the enthalpy deviation of the mixture, 49.47, methane and 50.6y0 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
541
Table 1. Enthalpy Calculations for Binary Mixtures and Pure Componentsa Mixture c1-c3
NC5
c1-xc5
NB
Calcd.
H*-H, Btu/Lb Exptl.
- 200
2000 1500 1000 500 250
208.4 210.1 211.9 213.6 214.5
207.6 209.4 211.1 212.8 213.4
- 100
2000 1500 1000 500
180.6 182.1 183.6 185.0
183.7 185.3 186.7 188.1
-3.1 -3.2 -3.1 -3.1
100
2000 1500 1000 400
126.2 120.2
-0.7 0.4
22.5
126.9 119.8 Two-phase 25.5
150
2000 1500 1000 750 500 250
105.7 90.6 56.3 38.4 23.3 10.8
106.8 91.7 59.7 41.3 25.5 11.8
-1.1 -1.1 -3.4 -2.9 -2.2 -1 .o
250
2000 1500 1000 750 500 250
68.9 53.2 34.7 25.4 16.5 8.0
71.1 55.1 36.4 26.9 17.2 8.5
-2.2 -1.9 -1.7 -1.5 -0.7 -0.5
1.000
160
0.4-0.6
160
2000 2500 3000 2000 2500 3000
140.5 139.3 138.0 131.1 130.7 130.1
141.1 140.0 138.8 126.9 126.3 125.1
-0.6 -0.7 -0.8 4.2 4.4 4.0
0.5-0.5
160
2000 2500 3000
127.7 128.0 127.8
126.1 126.6 126.1
1.6 1.4 1.7
1.000
0 0 0 0
1000 2000 3000 4000
8.8 15.8 20.7 23.4
7.4 13.2 17.2 19.7
1.4 2.6 3.5 3.7
Mole Fraction
Temp., OF
Pressure,
0.494-0.506
PSlA
Difference
0.8 0.7 0.8 0.8 1.1
-3.0
C1-C3 experimental data from Yesavage (1968). N2,Ct-nC6, and YC, data from Sage and Lacey (1950).
Table 11. Enthalpy Comparisons for Ternary Mixture C1-C2-C3a Mole Fraction
TI
Pressure, PSlA
CoIcd.
( H T ~- H d , Btu/Lb Exptl.
Difference
0.0 0.3
0.361-0.309-0.330
127.0 127.1
151.9 20.18
2000 1998
22.6 68.4
22.6 68.1
0.365-0.313-0.322
127.3 127.3
152.3 201.9
1000 1002
23.8 60.8
24.2 62.2
-0.4 -1.4
0.366-0.313-0.321
51.8 51.8 51.8
62.3 91.9 126.5
1750 1750 1750
9.4 33.3 64.2
8.0 33.0 65.0
1.4 0.3 -0.8
2.6 19.8 52.6
2002 1999 1998
16.2 27.3 50.2
16.7 28.5 52.1
-0.5 -1.2 -1.9
2000 2000
44.7 19.8
41.6 19.7
0.372-0.312-0.316
-22.6 -22.6 -22.6
0.366-0.311-0.323
a
T2
-234.6 -234.6 Experimental data from Powers (1969).
542 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
-160.6 -199.6
3.1 0.1
Table 111. Fugacity and Compressibilitya Temp.,
Component
Mole Fraction
NZ
1.000
0
Pressure, PSlA
Calcd.
Fugacity, PSlA Exptl.
1000 2000 3000 4000
957.8 1868.0 2786.9 3762
961.9 1881.2 2812.4 3804
__
% Deviation -0.4
-0.7 -0.9 -1.1
1.000
0
1000 2000 3000 4000
0.9654 0.9684 1 ,0099 1.0789
Compressibility Factor 0.9686 0,9738 1.0185 1.0899
Ci-NCio
0.2-0.8
100
695 1000 1500 2000 2500 3000
0,3197 0,4588 0.6852 0.9098 1.1327 1.3840
0.3111 0.4460 0.6653 0.8824 1.0976 1.3111
C,-Ca
0.3-0.7
100
200 293 1500 2000 2500
0.8677 0.7924 0.3472 0.4462 0.5430
0.8690 0.7748 0.3494 0.4478 0.5454
1.0 2.4 -0.6 -0.4 -0.4
NCs
1.000
160
2000 2500 3000
0.5854 0.7256 0.8639
0.5869 0 7277 0 8667
-0 3 -0 3 -0 3
KZ
a
O F
-0.3 -0.6 -0 8 -1.0
.
2.8 2.9 3.0 3.1 3.2 3.3
Experimental data from Sage and Lacey (1950).
220
Table IV. Comparison between Corresponding States Calculated Entropy and Values Reported from Experimental Calorimetric Data" for a Mixture of Approximately 94.8% CH, and 5.2% CzHg Entropy from Corresponding States, Btu/Lb O R
Pressure, PSlA
T
=
14.7 250 400 500 650 800 1000 1500 2000
T
=
1000 1200 1500 2000
0°F (Vapor) (2.510) 2.174 2.111 2.078 2.037 2.000 1,957 1.862 1.794
=
2,510 2.171 2.110 2.075 2,033 1.994 1.954 1.860 1.793
-90°F (Liquid) 1.615 1.596 1.578 1.556
T
Entropy from Calorimetric Data, Btu/Lb O R
n
200
\
\
u,
I80
ai
n
n
"
.
u
"
n
3 -200°F
u
U
u -IOO'F
\
\
\
\
160
\ \
1
\
140
80
\
-
0
400
800
1200
1600
2000
PRESSURE - P S I A
1.619 1.599 1.580 1.559
100°F (Vapor)
14.7 2.610 2.609 ,500 2.191 2.190 1000 2.092 2.091 1500 2.025 2,025 2000 1.974 1.974 Entropy from calorimetric data reported by Bhirud (1969). Reported reference state is entropy of 0.0 for each pure component in a pei,fect crystalline state at 0"R.
Figure 1. Calculated and experimental enthalpies for a C1-C3 mixture Nominal 51 % propane, 49% methane
-- Experimental (Yesavage, 1968) Points calculated
propane, in Figure 1. Thiq mixture should shmv the 1argc.t errors for the niethaiie-l)rol)aiie s;\-5tem, since the erroi'< due to the inaccuracy of the unlike pair combining rulcs a r t the largest for a 50%-50% mixture. The conipl'es,sibilit;\- factor coniparisom in Table I11 nlso illustrate another trend. The errors teiid to iiiereasc a b the molecules become more dis3imilar. The Inethan~-decaiie Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
543
mixture shows about 3% error, while the methane-propane misture shows an average error of about 1%. Longer chain molecules could be accurately calculated using a reference more conformal t,o the long-chain molecules, such as propane or butane. Unfortunately, sufficiently accurate reference data are not available for these hydrocarbons. Investigations are now being conducted toward improving the unlike pair interaction term in the defining equations for T,’ and Vc’. These results should improve the property predictions in mixtures involving long-chain hydrocarbons. Conclusions
The corresponding states theory using shape factors satisfactorily predicts total thermodynamic properties of mixtures which do not deviate too far from conformality with t,he reference, in this case methane. -1dditional correlational effort is required to obtain shape factors for components other t’haii the light paraffin hydrocarbons. =Idditioiial experimental data and correlations for another reference such as propane would be useful to extend accurate predictions to longer chain molecules; however, the moat important need is for better prediction of the interaction between unlike molecules. Nomenclature
proportionality constant second virial coefficient Boltzmann’s constant temperature proportionality constant dimensionless thermodynamic function dimensionless thermodynamic function gas constant pressure dependent variable independent variable enthalpy of a n ideal gas state pair radial distribution function iiondimeiisioiial separation dist’ance between molecules intermolecular potential energy mole fraction of a n indicated component shape factor of substance i relative to reference j used t,o multiply the critical temperature of i shape factor of substance i relatire to reference j used to multiply the critical volume of i pseudocritical temperature molal volume pseudocritical volume reduced temperature of substance i reduced volume of substance i entropy of a n ideal gas state iiiternal energy of a n ideal gas state compressibility factor pot,ential energy minimum in intermolecular potential separation distance between molecules a t zero potentia 1 energy chemical potential density of mixture
544
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
acentric factor
w
=
$
= dimensionless thermodynamic function
n
=
dimensionless thermodynamic function
literature Cited
Bhirud, T’, L., Ph.D. thesis, University of hlichigan, Ann Arbor, JIich., 1969. Brydon, J. W.,Walen, X., Canjar, L. N., Chem. Eng. Progr. Symp. Ser. 49, 131 (1953). Chueh, P. L., Prausnitz, J . \I., A.I.Ch.E. J . 13, 1099 (1967). Cook, I]., Rowlinson, J. S., Proc. Roy. SOC.(London) A219, 405 (1933). Douslin, D. R., Harrison, R . H., hloore, R . T., AIcCullough, J. P., J . Chem. Eng. Data 9 , 3.58 (1964). Jones, AI., Ph.D. thesis, University of Michigan, Ann Arbor, AIich., 1961. Keyes, F. G., Taylor, 11. S., Smith, L. B., J . -lIath. Phys. 1, 211 (1922). Kirkwood, J . G., Lewinson, V. A., Alder, B. J., J . Chem. Phys. 20, 929 (1932). Ph.D. thesis, Rice University, Houston, Tex., 1967. Leach, J. W., Leach, J. W.,Chappelear, P. S., Leland, T. W., A.I.Ch.E. J . 14, 368 (1968). Leland, T. W,, Chappelear, P. S.,Gamson, B. W., A.I.Ch. J . 8 , 4’12 (1962). Kobayashi, Riki, LIueller, R.H., A.I.Ch. J . 7,535 Leland, T. W., (\ -1- 9- -6, 1 ~
Leland, T. W,, Rowlinaon, J. S.,Sather, G. A., Tmns. Fataday SOC. 64, 1447 (1968). Leland, T. W.,Sather, G. A,, Itowlinson, J. S., Watson, I. D., Trans. Faraday SOC.6 5 , 2034 (1969). Li, K., Canjar, L. X., Chem. Eng. Progr. Symp. Scr. 7(49), 147 I1963 I . Pit‘zer, K . S.,Lippmann, D. Z., Curl, R . F., Huggins, C. ll., Peterson, I). E., J . Amer. Chon. SOC.77, 3433 (1955). Pople, J. A , , Proc. Roy. SOC.(London)A221, 498 (1954). Powers, J. E., Cniversitg of Michigan, Ann Arbor, hlich., private communication, 1969. Prausnitz, J. JI.,Gunn, R. I)., A.I.Ch.E. J . 4, 430, 494 (1938). Reid. R.C.. Leland, T. W., A.Z.Ch.E. J . 11. 228 11965): . , , 12. 1227 (1966). ’ Rowlinson, J. S., “Liquids and Liquid Mixtures,” p. 270, Butterworths, London, 1959. Sage, B. H., Lacey, W.S . , “Thermodynamic Properties of the Lighter Paraffin Hydrocarbons and Nitrogen,” 4 P I Project 37,7 (1950). Sage, B. H., Schaafsma, J. G., Lacey, W.N., Ind. Eng. Chem. 27, 48 (1933). T’ennix, A . J., Ph.D. thesis, Itice University, Houston, Tex., 1966. T’ennix, A. J., Kobayashi, It.,Leland, T. W., J . Chem. Eng. Data 15, 238 (1970). Yesavage, T’. F., Ph.D. thesis, University of 3Iichigan, h I l Arbor, X c h . , 196s. I
Acknowledgment
The authors gratefully acknowledge the financial assistance of the Katural Gas Processors Xssociatioii, which contributed t o this paper in the area of the enthalpy predictions and investigations. RECI:IVI,;D for review February 27, 1970 ACCI’.PTEDAugust, 12, 1970 Division of Industrial and Engineering Chemistry, Symposium on Enthalpy of Alixtures, 159th Meetiiig A4CSHoiiston, Tex., February 1970. A computer program for calculating thermodynamic properties of a wide range of mixture, and conditions is being marketed by the ChemShare Corp. Iiiformation concerning the program can be obtained from Gary I). Fisher, The ChemShare Corp., 4141 Southwest Freeway, Houston, Tex., 77027.
