where r ( z , p ) is the incomplete gamma function of second kind defined b y
r(z,p)
=
lm
uz-lexp ( - u ) du =
r(z) - y ( z , p )
or
(14)
For the case where m = n, the following solution is readily obtained without any approximation:
R(y,t) = Ro(y)exp ( - K y n t ) ; m = n (15) For the last case where m / n > 1, Equation 16 is obtained in the same manner used for deriving Equation 13.
I n Figure 1 (log-log grid), log&*(y,t) is plotted against Kynt for several values of m/n. As is readily seen from t h e figure, Equation 17 is closely approximated by Equation 18:
R(y,t) = expi - (pKynt)"} Ro(Y)
where ,U and v are the constants determinable from m / n as shown in Figure 2, in which the maximum discrepancy between Equations 17 and 18 is also given. The solution proposed by Harris (1968) can be written in the form
s.5
Equation 16 is rather inaccurate because the same approsimation applied to Equation 11 is used twice for this case. I n fact, Equation 13 reduces to the strict solution, Equation 15, when m / n approaches unity, but Equation 16 does not. Therefore, the second term in Equation 16 may originate mainly from the approximation used in the inverse transformation. I n any may, the second term dies out for xm >> y , or for large t . Then we may conclude t h a t the approximate solution of batch grinding equation for S(x)B(y,x) = Kxn( y / ~is)expressed ~ in the following equation:
where K ' , m', n', CY are the constants. Therefore the present solution being derived without the assumption of the time dependency of the selection function is virtually the same as Harris' solution escept for very short milling time. This suggests that Equation 17 is reasonable. literature Cited
Harris, C. C., Trans. AI:ME, 241 (12), 449 (1968) RECEIVED for review Xarch 31, 1972 July 24, 1972 ACCEPTED
Generalized Correlation for Fugacity Coefficients in Mixtures at Moderate Pressures Application of Chemical Theory of Vapor Imperfections Karl-Heinz Nothnagel,' Denis S. Abrams, and John M. Prausnitz2 Department os Chemical Engineering, C'niversity of California, Berkeley, Calif. 94720
I n vapor-liquid equilibrium calculations, liquid-phase nonidealities are described by activity coefficients while deviations from ideal gas behavior in the vapor phase are described b y fugacity coefficients (Prausnitz, 1969). I n this work, we present a generalized method for estimating fugacity coefficients for a wide variety of mixtures including those containing polar and hydrogen-bonded components. This method is applicable a t moderate pressures as commonly encountered in separations of petrochemical mistures; the upper limit of a moderate pressure varies with the temperature and with the
2
Present address, Roehm GmbH, Darmstadt, Germany. To whom correspondence should be addressed.
chemical nature of the mixture's components but for many typical industrial applications the method discussed here is useful a t pressures up to perhaps 5 or 8 atm. I n special casesfor example, if the vapor contains an excess of a light component such as hydrogen, nitrogen, or methane-the upper pressure limit may be considerably higher. The method presented here IS based on a chemical theory of vapor imperfections in contrast to the physical theory (equation of state) which forms the basis of most earlier work on fugacity coefficients (Leland and Reid, 1965; Chueh and Prausnitz, 1967; Redlich et al. 1965). Several authors (Tsonopoulos and Prausnitz, 1970; Sebastiani and Lacquaniti, 1967, lIarek, 1955) have used the chemical theory to calculate Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1,
1973 25
fugacity coefficients of carboxylic acids but no general chemical method, useful for engineeering work, has been given previously. The main purpose of this work is to develop a general engineering correlation applicable to a wide variety of vapors and vapor mixtures, including polar as well as nonpolar components and including also strongly hydrogen-bonded fluids such as alcohols, aldehydes, and acids. T o achieve such wide applicability, it is necessary t o extend theory with empiricism and to sacrifice high accuracy; as a result, the correlation given here is probably less accurate than that for nonpolar components (typically hydrocarbons) based on t h e equation of Pitzer and Curl (1957). However, it is probably more reliable than t h a t for polar and associating vapors based on empirical extensions of the Pitzer-Curl equation (O’Connell and Prausnitz, 1967). The correlation given here should be useful primarily for design of separation equipment as required in petrochemical plants dealing with a large variety of fluids more strongly nonideal than “normal” fluids as defined by Pitzer (1955).
dissociation (Equation 2) is rare a t normal temperatures. However, small positive deviations from ideal gas behavior are commonly found for stable common gases (e.g., methane) at moderate pressures and at temperatures well above t h e critical. These deviations cannot realistically be ascribed to dissociation but must be accounted for by the concept of excluded volume-the inaccessibility of some parts of space owing to the finite size of the molecules. In this work, therefore, we use t h e equation of state
(3) where V is the total volume, nT is the true total number of moles, and nTb, is the excluded volume due to the finite size of the molecules (a mixture of monomers and dimers). The parameter b, in Equation 3 is not identical to t h a t of van der Waals but is closely related t o it, as briefly discussed in the Appendix. The total number of moles depends on the equilibrium constant and on the pressure; for a pure component A which can dimerize, the equation of chemical equilibrium is
Chemical Theory of Vapor Imperfections
Many years ago it was pointed out t h a t deviations from ideal gas behavior can be “explained” by postulating the existence of several molecular species which are in chemical equilibrium. Thus, the molar volume of a vapor which is less than that corresponding t o a n ideal gas a t the same temperature and pressure (negative deviation) can be accounted for by equilibrium association where two molecules A form a dimer A P
2A*Ag
(1)
This explanation is supported by spectroscopic evidence for those molecules (e.g., acetic acid) where strong intermolecular hydrogen bonding exists. For weak molecular interactions (e.g., argon), a stable dimer does not exist, b u t for short periods of time two argon molecules form a temporary cluster. The formal similarity between physical and chemical explanations of vapor-phase imperfection is reviewed by Mason and Spurling (1969). Volumes larger than those of an ideal gas a t t h e same temperature and pressure (positive deviation) can be accounted for by equilibrium dissociation of a molecule A into tfio other molecules A‘ and A”
A *A’ + A ”
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
where f A 2 is the fugacity of the dimer, fA is the fugacity of the monomer, and where z stands for the true mole fraction and ( for the true fagacity coefficient. At modest densities, as discussed by Hirschfelder e t al. (1942), it is reasonable to assume that the excluded volume for the dimer is the same as that for the monomer; therefore,
nTbm
= (%A
+n ~ J b
(5)
where b is the excluded volume for the monomer. From Equation 3 and standard thermodynamics, we obtain
When b and K are known, z A and z A , can be found from Equations 4 and 6 coupled with a material balance. Consider a substance 1 whose molecules can dimerize according to 2 A AP. Let nl be the number of moles of substance 1 and let nA be the number of moles of monomer A and nA, the number of moles of dimer A z . Then
(2)
Chemical equilibrium (association or dissociation) is characterized by a n equilibrium constant which depends only on temperature. If we assume the “true” species (Le., A and A B or A , A’, and A ” ) form a n ideal gas mixture, i t is then possible to calculate the fugacity coefficient a t any desired temperature and pressure where the assumed chemical equilibria are uniquely responsible for deviations from ideal gas behavior. This treatment is readily generalized to mixtures; in a mixture of components A and B it is now necessary to assume chemical equilibria for pure A , for pure B , and for solvation; i.e , for formation of cross dimers AB. If the chemical equilibria assumed are limited to a stoichiometry of two (dimers), then the chemical theory is necessarily limited to moderate pressures. To apply this theory t o higher pressures it is necessary to take into account chemical equilibria for the formation of such molecules as trimers and tetramers. While there is much experimental evidence to support the existence of association (Equation 1) in vapor mixtures, 26
(4)
At equilibrium, let CY equal the fraction of molecules dimerized; 2 nA,/nl.The true mole fractions are given by t h a t is, cy
arid the equation of equilibrium (Equation 4) becomes
biP PK exp - =
RT
(1 - a/2) 2 (1 - CY)’
CY -
Relationship Between Chemical Theory and Virial Equation
Xumerous authors (Hirschfelder et al., 1942; Lambert et al., 1949; Hill, 1960; Mason and Spurling, 1969) have shown that the chemical theory of gas imperfections leads to an equation of state of the virial form in the limit as a -+ 0. I n
At moderate pressures, the chemical theory of vapor-phase imperfections ascribes deviations from the ideal gas law to dimerization-i.e., to association of similar molecules and to solvation of dissimilar molecules. Fugacity coefficients in mixtures are readily calculated when the dimerization equilibrium constants are known as a function of temperature. The chemical theory i s particularly applicable to nonideal vapors containing polar and hydrogen-bonding componenis where the extent of dimerization i s large, since the commonly used virial equation holds only in the limit of very small extent of dimerization Association constants for 178 pure fluids are reported, and reasonable mixing rules are established for estimating solvation constants in mixtures. The results of this work are useful for calculation of multicomponent vapor-liquid equilibria, especially for strongly nonideal systems as frequently encountered in separations of petrochemicals.
particular, the second virial coefficient, B , is related to the excluded volume b and to the dimerizatioii constant K by Limit B
=
b - RTK
a-0
I t is therefore possible to find dimerization equilibrium constants from experimental second virial coefficients, provided these coefficients were obtained a t lo^ densities where a is much less than unity. For highly polar fluids, and especially for strongly hydrogen-bonded fluids, the condition a
