and preliminary estimates, the actual Afterdesign 'screening of a separation process requires a reasonably accurate representation of the important process variables. These are usually predictable if the effects of temperature, pressure, and composition on the phase equilibria can be adequately reproduced. It is desirable that the methods employed make efficient and complete use of the various types of available data. T-MV or higher order coefficients should be introduced o d y when binary data are inherently inadequate. The main advantage of the calculation method is lost if the data required for its application become eithu excessive or too difficult to obtain. Accordingly, methods are needed which are capable of supplying a minimum of readily obtainable information on the simple mixtures to predict reliable phase equilibria in the more complex multicomponent mixtures. For many applications such methods are already available. They will be illustrated here to predict phase equilibria of a quality suitable for process design. These methods have the important advantage that they may be used with approximate input data for preliminary design and only the quality of the input data needs to be improved to provide results suitable for final design. The general aspects of vapor-liquid equilibria are discussed first. This is followed by vapor phase fugacities, liquid phase activity coefficients, the correlation of binary data, and the development of multicomponent phase equilibria for separation processes. The results are applicable to hydrocarbon separations by straight fractionation or to extractive or azeotropic distillation in the presence of a polar solvent.
PREDICTION OF
FACTORS Part 2
PHASE EQUILIBRIA FOR PROCESS DESIGN CLINE BLACK
E. L. D E R R M. N . PAPADOPOULOS
and Vapor-Liquid Equilibria: General
The effects of temperature, pressure, and composition on phase equilibria are included in the imperfections of the vapor and liquid phases. Complete vapor-liquid equilibria in terms of the liquid molar composition xi, the vapor molar composition Y,,the total pressure P, and the temperature T provide more than the minimum data required for a check of the thermodynamic consistency. The pressure-temperature relationship at liquid composition xi and the vapor composition Y;at the corresponding x,, and T or P provide a double check on the consistency of the data (4). The two are related through the activity coefficient yi according to
P
= zygiPio/ei
(1 )
at, = yzt0e,/yp,%
= Y,x,ixiY,
(2)
The reference state for yi is the pure liquid component at its saturation pressure. Indeed, these two equations, involving the imperfection-pressure coefficient Bl and the liquid phase activity coefficient yc, form the basis for a valuable correlation procedure (4) in terms of two plots:
NO.1 (log y1)OJ
US.
(log yn)O.'
r1 US. x1 NO.2 log ya
Plot No. 1 and its combined use with Plot No. 2 for correlating activity coefficients was proposed by Black (3). The second, Plot No. 2, was proposed earlier by Herington (77) and by Redlich and Kister (29) as a
check of thenncdynamic consistency. For isothermal conditions the Gibbs-Duhem relation requires that:
J1 ): (log
dxl = 0
(3)
For isothermal data, consistency of the experimentally derived values on Plot No. 1 is an indication of consistency in the total pressure measurements. For isobaric data, consistency on this plot is an indication of consistency in the boiling point measurements. The second plot involves only the x, Y , T data. Scattering of the experimentally derived values in this plot indicates inaccuracies in the compmitions which may be reflected from the analytical procedures. The combined use of the two plots provides a double check on the thermodynamic consistency of the data. An adequate correlation of thermodynamically consistent i s o t h d data on the two plots will simultaneously satisfy Equations 1 and 2. The significanceof the two plots in establishing coefficients for the mcdiied van Laar equations has been described in detail earlier (3). Some illustrations will be given later in this paper. Equilibrium between a liquid and a vapor phase is given according to
(4)
in which pio and pi are the fugacity coefficientsat the saturation pressure P,O and at the total pressure P, respectively, and P,’ is the partial molal volume of component i in the liquid phase. A term involving the difference P“ - V,’ has been neglected as it cannot be evaluated separately except where it is insignificant. The imperfection-pressure coefficient 0, has been defined (2) according to:
(5) Except in the hypothetical liquid region in which the temperature is above the critical temperature of one pure component, a direct calculation of both ‘pio and ‘pi can AUTHOR Cline Black is an engineer with Shell DEvelopmcnt
Co., Emyville, Calif, and E . L. Dnr is a chemist with fhe sum comQany. M. N . PaQadoQoulos, a research suQnvisor with Shell Developmnt whm this article was prepared, is now a senior techlogist at Shell Oil Co.’s Norco, La.,r@wy.
be made with the aid of an equation of state. Above the critical temperature of a component i , both ‘pio and o :
J
( Pi’/RT)dP must be obtained by m e less
direct procedure. Both the liquid phase fugacity codficient ‘pio and the partial molar liquid volume 0,’ have been obtained empirically from binary vapor-liquid equilibria which extend into the region above the critical temperature of one component. For such calculations, F’rausnitz, Edmister, and Chao (27), Edmister (731, and Chao and Seader (9) correlated the liquid phase activity coefficients with the aid of solubility parameters and the equation of Scatchard and Hildebrand. A cross correlation of ‘pio and Pi’ with some pure component property furnishes values which can be used in subsequent calculations. Such values, however, are limited by the restrictions of the Scatchard-Hildebrand equations which are rdected in the empirical values obtained. Calculations to derive the hypothetical liquid phase fugacities have also been made assuming the validity of a particular equation for predicting the liquid volume P,‘. All quantities, q f 0 ,‘p, and P,’, as well as the liquid phase activity coefficient can be calculated directly from experimental vapor-liquid equilibrium data for the heavy component in a binary (2) even though the temperature is above the critical temperature of the more volatile component. The activity coefficient for the lighter component can be calculated from that of the heavy component with the aid of a two-coefficient integrated Gibbs-Duhem relation. This leaves qIo and PI’ to be derived empirically from experimental vapor-liquid equilibria. Such calculations have been successfully made with the aid of the van Laar equations for the activity coefficients. This procedure also furnishes reasonable values for ‘p,” and PI’ which can be used in subsequent calculations. Vapor Phon Fugacities
The vapor phase fugacity pt can be calculated with the aid of an equation of state. A modified van der Waals equation proposed by Black (2) furnishes approximate results for a nonpolar substance with the aid of the critical temperature T., the critical pressure P,, and the vapor pressure P,’. For a polar substance a minimum of one individual constant is required in addition to T,, Po, and Pio. A vapor density at one or two temperatures can furnish the required individual constants for each polar substance. Binary coefficients can be included where special “chemical effects” are present or where such coefficients are required for an adequate representation. The equation for the vapor volume (2) is given according to:
V,
=
+
I
R T / P Z b,Y, - [ Z ( a & o ) ~ ~Yil2/RT s [Z(&”)“.”Yi]*/RT 0.SC &jYiY,/RT
+
(6)
‘I
(Continued On next page) VOL 55
NO.
9
SEPTEMBER 1 9 6 3
39
where : RTc,/8P,, 27biRT,,/8
bi
=
ai
=
ti
= ,$io
+
F'P,; G'Pri2 K(YiP/P,")3
E
