Reply to G. J. Ausluender’s Correspondence on the Benedict- Webb-Rubin Equation
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Sir: T h e intent of our earlier correspondence was to point out that the conclusions reached by Shah and Thodos were in part based on calculations which were grossly in error. As a n example, Shah and Thodos present a plot for n-butane which indicates that the B-W-R equation predicts multiple roots in the vicinity of T , = 1.1. O u r note clearly demonstrates that this is not the case and that the equation follows closely the observed P V T behavior in this region. I t was not our intent to suggest that the equation will fulfill all possible needs. While it is a remarkably efficient and accurate tool for correlating and predicting a wide range of thermodynamic data, the B-Mi-R equation, as every other equation of state ever developed, is not without its limitations. Nevertheless, we cannot agree with some of the comments made by Mr. Auslaender. Mr. Auslaender takes exception to our statement that the B-W-R equation can represent both the gas and liquid phases adequately with a single set of coefficients. T h e original constants (6) for the low molecular weight hydrocarbons were derived by fitting the equation to vapor pressure data and to low- and high-density gas P V T data. The objective was to obtain an adequate description of the gas phase, the saturated liquid state, and the vapor pressure. Consequently, the coefficients represent a compromise, and whether this compromise is satisfactory depends on what is to be expected from the equation. Undoubtedly, a better fit could be obtained by fitting individual sets of coefficients to either phase, as Mr. Auslaender suggests. Kevertheless, it has been shown (6) that for n-butane the single set of coefficients leads to calculated saturated liquid densities (40” to 250’F) which differ from observed values with an average deviation of only 1.2%. Mr. Auslaender’s suggestion of using an experimental liquid density to find the predicted pressure is an exceedingly sensitive test because very small variations in liquid density correspond to very large changes in the pressure. I n our opinion, this is not a particularly significant test in evaluating the equation for practical calculations. A much more realistic approach is to compare predicted densities to experimental values at given pressures. For the low molecular weight hydrocarbons the B-W-R equation predicts saturated liquid densities which are in error by only 0 to 3%. It is important to recognize that the equation should not be extrapolated to temperatures below which the fit to vapor pressure was made unless suitable corrections are introduced (2,4). 82
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
State I n respect to Mr. Auslaender’s comments on the ability of the equation to predict the constant volume heat capacity, we feel that the deficiency referred to is relatively unimportant. I n an attempt to put the matter in proper perspective, we feel that potential users of the equation should not be discouraged because of this shortcoming. Even so, it is not as bad as Mr. Auslaender implies. The B-W-R equation yields the following expression for the derivative in question:
Using the original coefficients ( 3 ) , the density function enclosed in the brackets is positive u p to approximately the critical density, is negative between this density and about twice the critical density, and is again positive a t higher densities. Thus, the equation does predict isochores of negative curvature [negative (b2P/bT 2 ) d ]a t low density, isochores of positive curvature in the vicinity of the critical density, and isochores again of negative curvature a t still higher densities. These reversals in curvature agree well with experimental data, as may be seen in Benedict’s (6, Figure 1 ) . I t is true that the equation does not reproduce the very slight sigmoid behavior of single isochores, but we have yet to encounter situations where this limitation is of serious engineering importance. We must reiterate our conclusion that the B-W-R equation is capable of describing the saturated vapor and liquid phases within engineering accuracy with one set of coefficients. We may refer to several studies in the literature (3, 5, 7) where identical coefficients for vapor and liquid phases have led to satisfactory results. On the other hand, we certainly do not wish to imply that the equation is perfect. I n fact, we have recently discussed ( I ) the major limitations in some detail. REFERENCES (1) Barner, H. E Adler, S. B., paper presented a t the 47th Annual Convention of the Natural %as Producers Association, New Orleans (March 1968). ( 2 ) Barner, H. E., Schreiner, W. C., Hydrocarbon Processes Petrol. Refiner45 (6), (1966). (3) Benedict, M.,Statler, H. H., C. E. P . Sym. Serier No. 6 49,25 (1953). (4) Benedict,M., Webb, G.B., Rubin, L. C., Chcm. Eng. Progr. 47,419 (1951). (5) Ibid., p. 449. (6) Benedict, M., Webb, G, B., Rubin, L. C., J . Chem. Phys. 8,334 (1940). (7) Canjar, L. N., Schiller, F. C., C. E. P . Sym. Series N o . 7 49,67 (1953).
The M . W. Kellogg Co. Piscataw ay , N . J.
H. E. BARNER S. B. ADLER
