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Lecture

# Peng Robinson

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McGill University

Chemical Engineering

CHEE 621

Phillip Servio

Fall

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Palmer, HJ.,Berg, J. C., J. FIuidMech., 51, 385 (1972). Van Straien,S.J. D.,4thIntl. Heat Transfer Conf., Paris-Versailles, "Heat
Palmer, HJ..Berg,J. C.A./.Ch.J.,19, 1082 (1973). Transfer 1970"Vol. VIpaper B.7.b, 1970.
Plevan,R. E.QuinnJ.A,. A./.Ch.J.,12, 894 (1966).
Robb.I. D.Alexander, A. E., J. Co/Ioidlnterface Sci., 28, 1 (1968). Receivedfor reuiewDecember 11, 1974
Rosano,H.,LaMer. V. K.. J. Phys. Ch60,348 (1956). Accepted October 14,1975
Springer, T., PigforR.L., Ind.fng. Chem.. Fundam.,9, 458 (1970).
Van Stralen,.J.D.,Nether\.J.Agric. Sc4, 107(1956). This workwas supported by the Office of Saline Waterunder Con-
Van StralenS.J. 0.. lJ.Heat Mass TransferIO,1469 (1967). tractNo. 14-30-2964 and No. 14-30-2572.
A New Two-ConstantEquation of State
Ding-Yu Peng and Donald B. Robinson'
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada
The development of a new two-constant equation of state in which the attractive pressure term of the semiem-
pirical van der Waals equation has been modified is outlined. Examples of the use of the equation for predicting
the vapor pressure and volumetric behavior of singie-component systems, and the phase behavior and volu-
metric behavior of binary, ternary, and multicomponent systems are given. The proposed equation combines
simplicity and accuracy. It performs as well as or better than the Soave-Redlich-Kwong equation in all cases
tested and shows its greatest advantages inthe prediction of liquid phase densities.
Introduction Although one cannot expect a two-constant equation of
state to give reliable predictions for all of the thermody-
Ever since the appearance of the van der Waals equation namic properties, the demand for more accurate predic-
in 1873 (van der Waals, 1873), many authors have proposed tions of the volumetric behavior of the coexisting phases in
variations in the semiempirical relationship. One of the
most successful modifications was that made by Redlich VLE calculations has prompted the present investigation
into the possibility that a new simple equation might exist
and Kwong (1949). Since that time, numerous modified Re- which would give better results than the SRK equation. In
dlich-Kwong (RK) equations have been proposed (Redlich this paper, an equation is presented which gives improved
and Dunlop, 1963; Chueh and Prausnitz, 1967; Wilson,
1969; Zudkvitch and Joffe, 1970; and others). Some have liquid density values as well as accurate vapor pressures
introduced deviation functions to fit pure substance PVT and equilibrium ratios.
data while others have improved the equation's capability Formulation of the Equation
or vapor-liquid equilibrium (VLE) predictions. A review
of some of the modified RK equations has been presented Semiempirical equations of state generally express pres-
sure as the sum of two terms, a repulsion pressure PR and
(Tsonopoulos and Prausnitz, 1969).One of the more recent an attraction pressure PA as follows
modifications of the RK equation is that proposed by
Soave (1972). The Soave-Redlich-Kwong (SRK) equation P=PR+PA (1)
has rapidly gained acceptance by the hydrocarbon process-
ing industry because of the relative simplicity of the equa- The equations of van der Waals (1873),Redlich and Kwong
tion itself as compared with the more complicated BWRS (1949), and Soave (1972) are examples and all have the re-
equation (Starling and Powers, 1970; Lin et al., 1972) and pulsion pressure expressed by the van der Waals hard
because of its capability for generating reasonably accurate sphere equation, that is
equilibrium ratios in VLE calculations.
RT
However, there still are some shortcomings which the PR = V-b
SHK equation and the original RK equation have in com-
mon. lhe most evident is the failure to generatesatisfacto- The attraction pressure can be expressed as
ry density values for the liquid even though the calculated
vapor densities are generally acceptable. This fact is illus- (3)
trated in Figure 1which shows the comparison of the spe-
cific volumes of n-butane in its saturated states. The litera- where g(u) is a function of the molar volume u andthe con-
ture values used for the comparison were taken from Star- stant b which is related to the size of the hard spheres. The
ling (1973). It can be seen that the SRK equation always parameter a can be regarded as a measure of the intermo-
predicts specific volumes for the liquid which are greater lecular attraction force. Applying eq 1at the critical point
than the literature values and the deviation increases from where the first and second derivatives of pressure with re-
about 7%at reduced temperatures below 0.65 to about 27% spect to volume vanish one can obtain expressions for a
when the critical point is approached. Similar results have and b at the critical point in terms of the critical proper-
been obtained for other hydrocarbons larger than methane. ties. While b is usually treated as temperature indepen-
For small molecules like nitrogen and methane the devia- dent, a is constant only in van der Waals equation. For the
tions are smaller. RK equation andthe SRK equation, dimensionless scaling
Ind. Eng. Chem.,Fundam., Vol. 15,No. 1, 1976 5930I---- ---- 1 At temperatures other than the critical, we let
23 where cyT,,w) is a dimensionlessfunction of reduced tem-
Z
c perature and acentric factor and equals unity at the critical
4 temperature. Equation 12 was also used by Soave (1972)
> for his modified RK equation.
E 10 Applying the thermodynamic relationship
5
LU
y
0 to eq 4, the following expression for the fugacity of a pure
component can be derived
I 1 1 In-= Z - 1 -ln(Z- B) -- 24B In('Z-0.414B4B) (15)
08 09 10 P
REDUCED TEMPERATURE The functional form of a(T,, w) was determined by using
Figure 1. Comparison of predicted molar volumes forsaturated the literature vapor pressure values (Reamer et al., 1942;
n-butane. Rossini et al., 1953; Reamer and Sage, 1957; Starling, 1973)
and Newton's method to search for the values of cy to be
used in eq 5 and 15such that the equilibrium condition
factors are used to describe the temperature dependence of
the energy parameter.
A study of the semiempirical equations having the form is satisfied along the vapor pressure curve. With a conver-
of eq 1 indicates that by choosing a suitable function for gence criterion ofIf',- fvlI kPa about two to four it-
g(u), the predicted critical compressibility factor can be erations were required to obtain a value for cyat each tem-
made to approach a more realistic value. The applicability perature.
of the equation at very high pressures is affected by the For all substances examined the relationship between a
magnitude of blu, where u, is the predicted critical volume. and T, can be linearizedby the following equation
Furthermore, by comparing the original RK equation and
the SRK equation, it is evident that treating the dimen- G'"' = 1 + K(l - Trl'*) (17)
sionless scaling factor for the energy parameter as a func-
where K is a constant characteristic of each substance. As
tion of acentric factor in addition to reduced temperature shown in Figure 2, these constants have

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