Technical Brief

Journal of Energy Resources Technology Technical Brief

Formulations for the Thermodynamic on mathematical integration of the correlations developed for specific heat, and the property estimates obtained are remarkably Properties of Pure Substances accurate. The fundamental equation-of-state formulation typically requires 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 properties 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 correlation 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. From these, the following equations were obtained for the constant-pressure specific heat and the compressibility factor: 6 1 br0 br1 c ͑T ͒ϭ a Tkϩͩ ϩ ͪ p (25) P r ͚ k r r r 0.5 ϭ 2.6 5.2 Tr k 0 Tr Tr 0.093842 0.422 0.013416 Zϭ1ϩͩ Ϫ Ϫ ͪ p (26) T 2.6 5.2 r r Tr Tr Equations ͑25͒ and ͑26͒ were tested against the data for dry air Fig. 2 A comparison of tabulated and calculated values of the in Vargaftik ͓10͔ over the pressure range of 0 to 50 bar and the specific enthalpy for dry air at 1 bar and 50 bar temperature range of 300 K to 1000 K. The maximum discrepancy between tabulated and calculated values was 0.59% for c P and 0.64% for Z. br0* A modified Pitzer model was developed by using the c p data to b ϭϪ ␣ 2 ͑ ͒͑ ͒ recalibrate 1(Tr). The improved correlation is as follows: 1.6 2.6 ϭϪ0.338 964 br0* br1* ␣ ͑T ͒ϭ ϩ (27) 1 r 2.6 1.6 and Tr Tr ϭ ϭ br* where br0* 1.410092 and br1* 0.137634. The corresponding ϭϪ 1 ϭϪ b3 0.143 369 correlation for c p is as follows: ͑0.6͒͑1.6͒ 6 The maximum discrepancy between calculated Z values using Eq. 1 br0* br1* c ͑T ͒ϭ a Tkϩͩ ϩ ͪ p (28) ͑29͒ and those obtained using the p-v-T data in Vargaftik ͓10͔ P r ͚ k r r r 0.5 ϭ 2.6 1.6 Tr k 0 Tr Tr was 0.52% in the pressure range of 0 to 50 bar and the tempera- ͑ ͒ ͑ ͒ ␤ ture range of 300 K to 1000 K. Equation 21 is then used with Eq. 27 to solve for 1(Tr), and hence Z. Two constants of integration emerge. These are de- Gibbs Function and Other Properties for Dry Air termined by using the p-v-T data in Vargaftik ͓10͔ to give a least-squares fit to the derived correlation for the compressibility Enthalpy. Solve Eq. ͑6͒ for enthalpy using Eq. ͑28͒ for the factor. The resulting correlation for the compressibility factor is as constant-pressure specific heat follows: 6 ak ϩ 2.6b2 1.6b3 h ͑T ,p ͒ϭͩ Tk .5ϩh ͪ ϩͩ b ϩ ϩ ͪ p b b b r r r ͚ r r0 1 r 1 2 3 ϭ kϩ.5 1.6 0.6 Zϭ1ϩͩ b ϩ ϩ ϩ ͪ p (29) k 0 Tr Tr 0 T 2.6 1.6 r r Tr Tr (30) where The first bracketed terms in Eq. ͑30͒ represent the ideal gas con- ϭϪ ϫ Ϫ4 ϭ tribution to the enthalpy for the real gas. The reference property b0 8.107 10 , b1 0.1629, values can be used to determine a value for hr0

Fig. 1 A comparison of tabulated and calculated values of the Fig. 3 A comparison of tabulated and calculated values of the constant-pressure speciﬁc heat for dry air at 1 bar and 50 bar speciﬁc entropy for dry air at 1 bar and 50 bar

Journal of Energy Resources Technology MARCH 2005, Vol. 127 Õ 85 6 a ϭ ͑ ͒Ϫ k kϩ.5 hr0 hr Trគref ,prគref ͚ Trគref kϭ0 kϩ.5

2.6b2 1.6b3 Ϫͩ b ϩ ϩ ͪ p គ 1 1.6 0.6 r ref Trគref Trគref ϭϪ9.098 244 (31) Entropy. Equation ͑6͒ serves equally as basis for the determination of entropy using the constant-pressure speciﬁc heat equation ͑28͒. The result can be written in the following form: 6 a ͑ ͒ϭͩ k kϪ.5Ϫ ͑ ͒ϩ ͪ sr Tr ,Pr ͚ Tr ln pr sr0 kϭ0 kϪ.5

1.6b2 0.6b3 ϩͩ Ϫb ϩ ϩ ͪ p (32) 0 2.6 1.6 r Tr Tr The first bracketed terms in Eq. ͑32͒ represent the ideal gas contribution to the entropy for the real gas. The reference property Fig. 4 A comparison of tabulated and calculated values of the compressibility factor for dry air at 1 bar and 50 bar values can be used to determine a value for sr0 : 6 a ϭ Ϫͩ k kϪ.5 Ϫ ͑ ͒ͪ ͑ ͒ ͑ ͒ sr0 srគref ͚ Trគref ln prគref the specific enthalpy h Fig. 2 , the specific entropy s Fig. 3 , and ϭ kϪ.5 k 0 the compressibility factor Z ͑Fig. 4͒. In each case the property values are for limiting pressures of 1 bar and 50 bar. The agree- 1.6b2 0.6b3 Ϫͩ Ϫb ϩ ϩ ͪ p គ ment between the tabulated values and the computed values is 0 2.6 1.6 r ref Trគref Trគref indeed excellent. ϭ 16.540 034 (33) Discussion and Conclusion Gibbs Function. The Gibbs Function is determined from the The approach in this paper basically employs thermodynamic definition of the function in terms of enthalpy, entropy, and tem- laws to affirm a connection between the p-v-T equation of state perature and the corresponding equation of state for the constant-pressure specific heat c as a function of pressure p and temperature T. For g ͑T ,p ͒ϵh ͑T ,p ͒ϪT s ͑T ,p ͒ϭg0͑T ,p ͒ϩ⌬g ͑T ,p ͒ p r r r r r r r r r r r r r r r r many substances, the minimum available set of data generally (34) includes the p-v-T and the c p-p-T data. Starting with a truncated Equation ͑34͒ gives the Gibbs function as a sum of an ideal gas version of the virial form of the p-v-T equation of state, one can Gibbs function and a term that accounts for the real gas condition. anticipate the form of the c p-p-T equation of state. These two terms are given as follows: The technique here is to then develop a least-squares regression 0͑ ͒ϭ͑ Ϫ ͒ϩ ͑ ͒ equation for c p based on the anticipated form of the equation and gr Tr ,pr hr0 Tr•sr0 Tr• ln pr using available experimental data for the specific heat of the sub- 6 stance. Performing the necessary integration yields remarkably a Ϫ k kϩ0.5 consistent and accurate equations for enthalpy h and entropy s ͚ ͑ Ϫ ͒͑ ϩ ͒ Tr (35) kϭ0 k 0.5 k 0.5 from which the Gibbs Function fundamental equation of state can be derived. Determination of the other properties from the Gibbs b1 b2 b3 Function fundamental equation of state yields equally consistent ⌬g ͑T ,p ͒ϭͩ b ϩ ϩ ϩ ͪ T p (36) r r r 0 2.6 1.6 r• r and accurate equations for computing these other properties. Tr T T r r It is noteworthy that the resulting equations of state take a Internal Energy. The equation for internal energy can be de- rather simple analytic form with relatively few constants com- termined by using the definition of enthalpy in terms of u, p, and pared to the more complex fundamental equations of state that v have been formulated for pure substances. On the other hand, no attempt has been made in this work to have one equation that will ͑ ͒ϭ ͑ ͒Ϫ ͑ ͒ϭ 0͑ ͒ϩ⌬ ͑ ͒ ur Tr ,pr hr Tr ,pr pr•vr Tr ,pr ur Tr ur Tr ,pr apply to all the principal phases of a substance. A wider range of (37) conditions can be covered by developing separate equations of The ideal gas component in Eq. ͑37͒ is as follows: state for the different phases along the lines proposed in this paper. Care must be taken to ensure that these equations are consis- 6 a tent in the regions where they overlap. 0͑ ͒ϭ k kϩ.5ϩ Ϫ ur Tr ͚ Tr hr0 Tr (38) kϭ0 kϩ0.5 Nomenclature The additional contribution for the real gas condition is as ͓ ͔ ϭ follows: a kJ/kg specific Helmholtz function A ͓kJ͔ ϭ Helmholtz function ϭ ͑ ͒ 1.6b2 0.6b3 ak coefficients in the regression equation 22 ⌬u ͑T ,p ͒ϭͩ Ϫb ϩ ϩ ͪ p T (39) r r r 0 2.6 1.6 r• r for the zero-pressure specific heat at con- Tr Tr stant pressure ϭ ͑ ͒ Figures 1–4 give a comparison between tabulated ͑from bi coefficients in Eq. 29 for the compress- Vargaftik ͓10͔͒ property values and those calculated using the ibility factor ϭ equations based on the fundamental equation of state for dry air. brj , br*j coefficients in the truncated virial forms of ͑ ͒ The properties are the constant-pressure specific heat c p Fig. 1 , the p-v-T equation of state

86 Õ Vol. 127, MARCH 2005 Transactions of the ASME ϭ c p ͓kJ/kg-K͔ constant-pressure specific heat Superscripts ͓ ͔ ϭ ¯c p kJ/kmol-K molar constant-pressure specific heat 0 ϭ zero-pressure or ideal-gas condition c ͓kJ/kg-K͔ ϭ constant-volume specific heat v Greek Symbols ¯c ͓kJ/kg-K͔ ϭ molar constant-volume specific heat r ␣ ϭ ͑ ͒ g ͓kJ/kg͔ ϭ specific Gibbs function constant-pressure specific heat, Eq. 19 ␤ ϭ ͑ ͒ G ͓kJ͔ ϭ Gibbs function compressibility factor, Eq. 12 ␬ ϭ isothermal compressibility h ͓kJ/kg͔ ϭ specific enthalpy T ␻ ϭ acentric factor H ͓kJ͔ ϭ enthalpy N ͓kmol͔ ϭ amount ͑number of moles͒ References p ͓kPa͔ ϭ pressure ͓1͔ Smith, J. M., and Van Ness, H. C., 1987, Introduction to Chemical Engineer- R ͓kJ/kmol-K͔ ϭ universal gas constant ing Thermodynamics, Fourth Edition, McGraw-Hill, New York. ͓2͔ Adebiyi, G. A., and Russell, L. D., 1992, ‘‘Computer Codes for the Thermo- s ͓kJ/kg-K͔ ϭ specific entropy dynamic Properties of Common Substances Encountered in Engineering Ap- S ͓kJ/K͔ ϭ entropy plications,’’ ASME Conference Proceedings, HTD-Vol. 225, pp. 1–7. ͓ ͔ ϭ ͓3͔ Haar, L., Gallagher, J. S., and Kell, G. S., 1984, NBS/NRC STEAM TABLES: T K absolute temperature Thermodynamic and Transport Properties and Computer Programs for Vapor u ͓kJ/kg͔ ϭ specific internal energy and Liquid States of Water in SI Units, Hemisphere, Washington, D.C. U ͓kJ͔ ϭ internal energy ͓4͔ Lemmon, E. W., Jacobsen, R. T., Penoncello, S. G., and Beyerlein, S. W., 3 1992, ‘‘Software for the Calculation of Thermodynamic Properties of Several v ͓m /kg͔ ϭ specific volume Fluids of Engineering Interest,’’ ASME Conference Proceedings, HTD-Vol. ¯v ͓m3/kmol͔ ϭ molar specific volume 225, pp. 19–24. ͓ ͔ ͓ 3͔ ϭ 5 Jacobson, R. T., Penoncello, S. G., and Lemmon, E. W., 1997, Thermodynamic V m volume Properties of Cryogenic Fluids, Plenum Press, New York. Z ϭ compressibility factor ͓6͔ Pitzer, K. S., 1955, ‘‘The Volumetric and Thermodynamic Properties of Fluids, I. Theoretical Basis and Virial Coefficients,’’ J. Am. Chem. Soc., 77͑13͒, pp. Subscripts 3427–3433. ͓7͔ Pitzer, K. S., Lippman, D. Z., Curl, R. F., Huggins, C. M., and Petersen, D. E., c ϭ critical state parameters 1955, ‘‘The Volumetric and Thermodynamic Properties of Fluids, II. Com- i ϭ index counter pressibility Factor, Vapor Pressure and Entropy of Vaporization,’’ J. Am. ϭ Chem. Soc. 77͑13͒, pp. 3433–3440. j index counter ͓8͔ Reid, R. C., Prausnitz, J. M., and Poling, B. E., 1987, The Properties of Gases k ϭ index counter and Liquids, Fourth Edition, McGraw-Hill, New York. n ϭ index counter ͓9͔ Tsonopoulos, C., 1974, ‘‘An Empirical Correlation of Second Virial Coeffi- ϭ cients,’’ AIChE J., 20͑2͒, pp. 263–272. r reduced properties ͓10͔ Vargaftik, N. B., 1975, Tables on the Thermophysical Properties of Liquids ref ϭ reference state properties and Gases, Second Edition, Hemisphere, Washington, D.C.

George A. Adebiyi is a Professor of Mechanical Engineering at Mississippi State University. He obtained his Bachelor’s degree and a Doctorate from the Victoria University of Manchester, England. He is co- author of Classical Thermodynamics, a textbook published by Oxford University Press.

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