Multiproperty Analysis. Modified BWR Equation for Methane from PVT and Enthalpy Data K. W. COX,^ J. 1. Bono,* Y. C. K ~ o kand , ~ K. E. Starling School of Chemical Engineering and Xaterials Science, The University of Oklahoma,Xorman, Okla.
The principles of multiproperty analysis were applied in this study to develop a modified BWR equation of state for methane. This new equation of state accurately predicts all thermodynamic properties for methane in the cryogenic region from - 150" to -250°F. Enthalpy data were utilized to discern that modification of only the temperature dependence of the BWR equation can dramatically improve its predictive ability at low temperatures. Enthalpy data then were used to test the adequacy of trial forms for the modified BWR equation. Finally, density and enthalpy data were used simultaneously in multiproperty regression analysis to determine the optimal values of parameters in the modified BWR equation.
I t is \vel1 known that a n accurate description of fluid PVT behavior by an analytical equation of state does not ensure equally accurate predictions of other thermodynamic properties such as enthalpy, Gibbs free energy, and entropy, especially in the liquid region. Nevertheless, most equations of state proposed to date have been developed exclusively from PVT data. The work by Benedict, Webb, and Rubin (1940) and Martin (1962) are notable exceptions. The former investigators determined certain of the parameters in the BM7R equation by demanding equality of the fugacities of the coexisting vapor and liquid phases along the vapor pressure curve. This approach greatly enhanced the capability of the BWR equation for phase behavior predictions. Despite the successful use of PVT and vapor pressure data by Benedict and coworkers, regression methods simultaneously employing several properties were not utilized in equation of state development until recently (Starling and Rolfe, 1966). I n fact, research presented in this paper constitutes the first instance in uThich density and enthalpy data have been utilized simultaneously in regression calculations. By virtue of well known thermodynamic relations, accurate predictions of density and enthalpy in a given temperaturepressure region, as well as fugacity along the vapor pressure curve, enhances the accuracy of predictions of the Gibbs free energy and entropy for the region. For this reason, the primary objective in this research was the simultaneous description of P V T , enthalpy, and vapor pressure data. It should be noted that a logical goal in equation of state development is to define adequately the temperature, density, and composition dependence of the Helmholtz free energy for use as the generating function for the various other thermodynamic properties. Since the pressure is directly related to the density derivative of the Helmholtz free energy, PVT data provide more information concerning t h e density dependence of the Helmholtz free energy than its temperature dependence. On the other hand, enthalpy is related to the temperature derivative of the Helmholtz free energy, so that enthalpy data should provide information concerning the temperature dependence of the Helmholtz free energy. When a n equation of state developed from PVT data yields
Present address, Celanese Chemical Co., Bay City, Texas. Present address, Procter and Gamble Co., Cincinnati, Ohio. Present address, Amoco Chemical Co., Naperville, 111.
poor predictions of enthalpy departures Ivhile predicting accurate densities, one has implicit proof that the temperature dependence of the Helmholtz free energy is inadequate. This inadequacy may occur only in certain regions, such as low reduced temperatures in the case of the B R R equation. Because it was believed that enthalpy data can help to define the temperature dependence of the equation of state, a preliminary objective of this work was to determine the feasibility and practical advantages of utilizing enthnlpy data to this end in equation of state development. An additional objective was to determine the feasibility of developing the equation of state solely from PVT and enthalpy data. This was a practical objective since the planned portion of the multiproperty regression program using vapor pressure data had not been developed a t the time of this work. Methane was chosen for study because of the availability of extensive and accurate experimental PVT, enthalpy, and phase data for wide ranges of temperature and pressure. The BWR equation was chosen for study to test the above ideas regarding use of enthalpy data and also to demonstrate that only the temperature dependence of the BWR equation requires modification to improve predictions of enthalpy behavior. Because space does not permit a detailed discussion of the general framework for multiproperty analysis (Starling, 1967; Starling and Wolfe, 1966), only an outline of the procedure is given here. Simultaneous treatment of compressibility factor data and enthalpy departure data requires minimization of the following function to obtain optimal estimates of the parameters in an assumed equation of state
I n this relation Z E , and ZC, are the experimental and calculated compressibility factors, respectively, a t the j t h P V T data point, while ( H E - H*)k and ( H C - H * ) x are the experimental and calculated enthalpy departures a t the kth enthalpy data point. W X is a weighting factor for enthalpy relative to a weighting factor of unity for compressibility factor. I n general, solution for the minimum in the regression function Q in Equation 1 requires a nonlinear regression procedure such as the Gauss-Newton method. Space does not permit discussion of the rather complex computer program required for multiproperty analysis, However, i t should be Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
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ENTHALPY Deviation - d:(AHexp-AHcalc), ETU/lb
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