Journal of Energy Resources Technology Technical Brief Formulations for the Thermodynamic on mathematical integration of the correlations developed for spe- cific heat, and the property estimates obtained are remarkably Properties of Pure Substances accurate. The fundamental equation-of-state formulation typically re- quires several numerical constants to ensure accuracy. Besides, George A. Adebiyi subsequent determination of the complete list of thermodynamic Mechanical Engineering Department, Mississippi State properties from the fundamental equation of state requires obtain- University, Mississippi State, MS 39762 ing differentials of the characteristic function. The functional rep- e-mail: [email protected] resentation must be accurate, and great precision will be required if the estimates obtained for these other thermodynamic properties are to be reliable. These formulations are, as a result, not suitable for incorporation into programs that require repeated computation This article presents a procedure for formulation for the thermo- of system properties. dynamic properties of pure substances using two primary sets of The integrated approach is described in general terms in this data, namely, the pvT data and the specific heat data, such as the article. An illustration is provided using dry air at pressures as constant-pressure specific heat c p as a function of pressure and high as 50 bar ͑725 psia͒ and temperatures in the range 250 K temperature. The method makes use of a linkage, on the basis of ͑450 R͒ to 1000 K ͑1800 R͒. the laws of thermodynamics, between the virial coefficients for the pvT data correlation and those for the corresponding specific heat data correlation for the substance. The resulting equations of state Gibbs Function Fundamental Relation take on remarkably simple analytic forms that give accurate pre- Assume a Gibbs function fundamental equation of state has dictions over the range of input data employed. been determined as GϭG(p,T,N). This equation can be written ͓ ͔ DOI: 10.1115/1.1794695 in nondimensional form and in terms of reduced properties de- fined as follows: T p ͑V/N͒ ͑U/N͒ T ϵ , p ϵ ,vϵ u ϵ , Introduction r r r ¯ ͒ r ¯ Tc pc ͑RTc /pc RTc There is a growing need for formulations for the thermody- ͑H/N͒ ͑G/N͒ ͑A/N͒ namic properties of pure substances in accurate but simple ana- ϵ ϵ ϵ hr ¯ , gr ¯ , ar ¯ lytic forms for incorporation into system simulation programs that RTc RTc RTc repeatedly call for subroutines for the properties of the substances (1) ͑S/N͒ c c employed. The need is addressed in this article by integrating a ϵ ϵ P ϵ V ͑ ͓ ͔ sr , c Pr , cVr piecemeal approach see Smith and Van Ness 1 and Adebiyi and R¯ R¯ R¯ Russell ͓2͔, for example͒ with the fundamental equations-of-state method variously described in Haar et al. ͓3͔ for steam, Lemmon The Gibbs function fundamental equation of state can be written et al. ͓4͔ for a diverse set of fluids, and Jacobsen et al. ͓5͔ for as follows: cryogenic fluids, to mention a few. g ϭ f ͑p ,T ͒ (2) The key steps in the proposed integrated approach are as r r r follows: From this fundamental relation, all other thermodynamic prop- erties can be deduced as expressed by the following equations: 1. A virial form is assumed for the p-v-T relationship. From gץ this, a Gibbs function fundamental equation of state is de- ϭϪͩ r ͪ (3) ץ Entropy: sr rived for the substance. Tr 2. A direct relationship is established ͑based on the laws of pr gץ thermodynamics͒ between the virial coefficients and those ϭͩ r ͪ (4) ץ for the corresponding equation for correlating the specific Volume:vr pr heat data. It is thus possible to use correlations for the spe- Tr cific heat data to anticipate more accurately the virial form gץ of the p-v-T equation of state, and ultimately the fundamen- ϭ Ϫ ͩ r ͪ (5) ץ Enthalpy: hr gr Tr tal equation of state. Tr pr The equations for enthalpy and entropy in this method are based Constant pressure specific heat: 2ץ ץ Contributed by the Advanced Energy Systems Division for publication in the hr gr JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received at the AES c ϭͩ ͪ ϭϪT ͩ ͪ (6) Pr r 2 ץ Tץ Division July 7, 2003; revised manuscript received June 30, 2004. Associate editor r p Tr H. Metghalchi. r pr Journal of Energy Resources Technology Copyright © 2005 by ASME MARCH 2005, Vol. 127 Õ 83 Table 1 Critical and reference property values for dry air ץ ץ gr gr Internal Energy: u ϭg ϪT ͩ ͪ Ϫp ͩ ͪ (7) pץ T rץ r r r r p r T Critical Parameters: r r Critical pressure, p ϭ37.7 bar c ϭ ͑␣ ͒2 Critical temperature, Tc 132.6 K Trvr Tc Molecular weightϭ28.97 kg/kmol Constant volume specific heat: c ϭc Ϫ Vr Pr ͑␬ p ͒ Reference State Property Values: T c Reference temperature, T ϭ298.15 K (8) ϭref Reference pressure, pref 1 atm Reference state enthalpy, h ϭ0 kJ/kg where refϭ Reference state entropy, sref 6.861 kJ/kg-K 2gץ v 1ץ 1 ͑␣ ͒ϭ ͩ r ͪ ϭ ͩ r ͪ ͑ ͒ Expansion Coefficient ץ ץ ץ Tc vr Tr vr pr Tr pr (9) nl͑!ϱ͒ 1 ϭ 0͑ ͒ϩ ϩ ␤ ͑ ͒ n ץ 1 vr gr gr Tr Tr ln pr ͚ n Tr •pr (18) ͑p ␬ ͒ϭϪ ͩ ͪ nϭ1 n pץ c T v r r T T2) to deduce the following for theץ/ 2gץ) r Use c ϭϪT Pr r r r pr 2ץ 1 gr constant-pressure specific heat: ϭϪ ͩ ͪ ͑Isothermal Compressibility͒ (10) nl͑!ϱ͒ 2 ץ v r pr 1 c ϭc0 ͑T ͒ϩ ͚ ␣ ͑T ͒ pn (19) Pr Pr r n r • r g nϭ1 nץ ϭ Ϫ ͩ r ͪ (11) ץ Helmholtz Function: ar gr pr pr where Tr 2 0͑ ͒ d gr Tr Pressure Series Virial Form of the pvT Equation of State c0 ͑T ͒ϵϪT (20) Pr r r 2 Next assume a pressure series virial form of the p-v-T equation dTr of state as follows: and ͑!ϱ͒ ץ ͒ ͑ ␤ nl p¯v prVr n Tr pr gr 2␤ ͑ ͒ ϵ ϭ ϭ ϩ nϭ ͩ ͪ d n Tr Z 1 ͚ pr (12) ␣ ͑ ͒ϵϪ (p n Tr Tr (21ץ T ϭ T • T ¯ RT r n 1 r r r T 2 r dTr Truncated Versions. The following truncated versions of Eq. Equation ͑21͒ defines the linkage between the virial coefficients ͑12͒ are frequently assumed and should be noted as follows: and those for the specific heat correlation. The use of this linkage is illustrated in the next section. • Pitzer et al. ͓6,7͔: ␤ ͑T ͒ ϭ ϩ 1 r Thermodynamic Formulation for the Properties of Dry Z 1 pr (13) Tr • Air „for Pressures to 50 Bar… 0.422 0.172 Table 1 gives the critical and reference property values assumed ␤ ͑T ͒ϭ0.083Ϫ ϩͩ 0.139Ϫ ͪ ␻ (14) 0 ͑ ͒ 1 r 1.6 4.2 for dry air. The c (Tr) in Eq. 19 is the zero-pressure specific T T Pr r r heat at constant pressure for the gas, and using data from Vargaftik ␻ is termed the acentric factor. The value for several substances is ͓10͔, the following correlation was obtained for dry air: given in the literature ͑see Reid et al. ͓8͔, for example͒. The cor- 6 1 relation is considered applicable only for nonpolar or slightly po- 0 ͑ ͒ϭ k c P Tr ͚ akTr (22) r 0.5 ϭ lar materials. Tr k 0 Tsonopoulos ͓9͔: The values of ak are given in Table 2. The correlation is valid for ␤t ͑ ͒ 1 Tr T in the range 250–1500 K. Zϭ1ϩ p (15) The truncated version of the virial equation of state, Eq. ͑13͒, T • r r gives an accurate correlation for the p-v-T relationship for dry air in the pressure range 0 to 50 bar and a temperature range of 0.422 0.172 a b ␤t ͑T ͒ϭ0.083Ϫ ϩͩ 0.139Ϫ ͪ ␻ϩͩ Ϫ ͪ 300–1000 K. A value of ␻ϭ0.078 for the acentric factor of dry air 1 r 1.6 4.2 6 8 ͓ ͔ Tr Tr Tr Tr was computed using the equation formulated by Pitzer et al. 7 . (16) Substitution of this estimated value of ␻ for dry air yields the following results for ␤ (T ) of Eq. ͑14͒ and ␣ (T ) of Eq. ͑21͒: a, b are constants that have been determined from correlations for 1 r 1 r several compound classes. This correlation is recommended for polar molecules. Table 2 Constants for the zero-pressure specific heat correla- tion Eq. „22… for dry air Gibbs Function Based on the Virial Form of the pvT Equa- tion of State. kak 0 1.685175 • Solve for gr in the following equation: 1 2.283617 2 Ϫ0.49965 nl͑!ϱ͒ ץ gr Tr Ϫ Tr 3 0.108826 ϭͩ ͪ ϭ ϩ ␤ ͑ ͒ n 1ͩ ϭ ͪ Ϫ ͚ n Tr •pr Z (17) 4 0.012765 ץ vr pr pr nϭ1 pr ϫ Ϫ4 Tr 5 7.471154 10 6 Ϫ1.732085ϫ10Ϫ5 The result is as follows: 84 Õ Vol. 127, MARCH 2005 Transactions of the ASME 0.422 0.013416 ␤ ͑ ͒ϭ Ϫ Ϫ 1 Tr 0.093842 1.6 4.2 (23) Tr Tr br br ␣ ͑ ͒ϭ 0 ϩ 1 1 Tr 2.6 5.2 (24) Tr Tr ϭ ϭ ϭ where br0 (0.422)(1.6)(2.6) 1.75552 and br1 (0.172)(4.2) ϫ(5.2)␻ϭ0.293005.