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TECHNICAL PAPERS: Thermodynamic Properties

# Supplementary Backward Equations for Pressure as a Function of Enthalpy and Entropy $p(h,s)$ to the Industrial Formulation IAPWS-IF97 for Water and Steam

[+] Author and Article Information
H.-J. Kretzschmar1

Department of Technical Thermodynamics,  University of Applied Sciences of Zittau and Gorlitz, P. O. Box 1455, D-02754 Zittau, Germanyhj.kretzschmar@hs-zigr.de

J. R. Cooper

Department of Engineering,  Queen Mary University of London, London, UK

A. Dittmann, J. Trübenbach, Th. Willkommen

Department of Technical Thermodynamics,  Technical University of Dresden, Dresden, Germany

D. G. Friend

Physical and Chemical Properties Division,  National Institute of Standards and Technology, Boulder, CO

J. S. Gallagher

Physical and Chemical Properties Division,  National Institute of Standards and Technology, Gaithersburg, MD

K. Knobloch, I. Stöcker

Department of Technical Thermodynamics,  University of Applied Sciences of Zittau and Gorlitz, P.O. Box 1455, D-02754 Zittau, Germany

R. Mareš

Department of Thermodynamics,  University of West Bohemia, Plzeň, Czech Republic

K. Miyagawa

Tokyo, Japan

W. Wagner

Lehrstuhl für Thermodynamik,  Ruhr-Universität Bochum, Bochum, Germany

Note: $T$ denotes absolute temperature on the International Temperature Scale of 1990 (ITS-90).

1

To whom correspondence should be addressed.

J. Eng. Gas Turbines Power 128(3), 702-713 (Jun 22, 2004) (12 pages) doi:10.1115/1.1915392 History: Received October 20, 2003; Revised June 22, 2004

## Abstract

In modeling steam power cycles, thermodynamic properties as functions of the variables enthalpy and entropy are required in the liquid and the vapor regions. It is difficult to perform these calculations with IAPWS-IF97, because they require two-dimensional iterations calculated from the IAPWS-IF97 fundamental equations. While these calculations are not frequently required, the relatively large computing time required for two-dimensional iteration can be significant in process modeling. Therefore, the International Association for the Properties of Water and Steam (IAPWS) adopted backward equations for pressure as a function of enthalpy and entropy $p(h,s)$ as a supplement to the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam (IAPWS-IF97) in 2001. These $p(h,s)$ equations are valid in the liquid region 1 and the vapor region 2. With pressure $p$, temperature $T(h,s)$ can be calculated from the IAPWS-IF97 backward equations $T(p,h)$. By using the $p(h,s)$ equations, the two dimensional iterations of the IAPWS-IF97 basic equations can be avoided. The numerical consistency of pressure and temperature obtained in this way is sufficient for most heat cycle calculations. This paper summarizes the need and the requirements for the $p(h,s)$ equations and gives complete numerical information about the equations. Moreover, the achieved quality of the equations and their use in the calculation of the backward function $T(h,s)$ is presented. The three aspects, numerical consistency with the IAPWS-IF97 basic equations, consistency along subregion boundaries, and computational speed important for industrial use are discussed.

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## Figures

Figure 1

Range of validity and equations of IAPWS-IF97, and presented backward equations p(h,s)

Figure 2

Numerical consistency relations of the backward functions p(h,s) and T(h,s) with the IAPWS-IF97 equations h97(p,T) and s97(p,T)

Figure 3

Numerical consistency of equation p(h,s), Eq. 3, with IAPWS-IF97 equation g197(p,T) for selected temperatures and along the saturated liquid line x=0

Figure 4

Division of the IAPWS-IF97 region 2 into three subregions 2a, 2b, 2c for the backward equations p(h,s)

Figure 5

Numerical consistency of equations p(h,s), Eqs. 5,6,7, with the IAPWS-IF97 equation g297(p,T) for selected temperatures and along the saturated vapor line x=1

Figure 6

Numerical consistency between p2a(h,s) and p2b(h,s) equations at the subregion boundary h2ab(s)

Figure 7

Numerical consistency between p2b(h,s) and p2c(h,s) equations at the subregion boundary s=5.85kJkg−1K−1

Figure 8

Numerical consistency of the temperature calculated by T197[p1(h97,s97),h97] with the IAPWS-IF97 equation g197(p,T) for selected temperatures and along the saturated liquid line x=0

Figure 9

Numerical consistency of the temperature calculated by T297[p2(h97,s97),h97] with IAPWS-IF97 equation g297(p,T) for selected temperatures and along the saturated vapor line x=1

Figure 10

Numerical consistency of T2(h,s) at the subregion boundary h2ab(s), Eq. 4

Figure 11

Numerical consistency of T2(h,s) at the IAPWS-IF97 subregion boundary line p=4MPa

Figure 12

Numerical consistency of T2(h,s) at the IAPWS-IF97 subregion boundary s=5.85kJkg−1K−1

Figure 13

Numerical consistency of T2(h,s) at the IAPWS-IF97 subregion boundary h2bc97(p)

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