## Structure of the Formulation

The IAPWS Industrial Formulation 1997 consists of a set of equations for different regions which cover the following range of validity:

273.15 K < T < 1073.15 K p < 100 MPa 1073.15 K < T < 2273.15 K p < 10 MPa .

Figure 1 shows the five regions into which the entire range of validity of IAPWS-IF97 is divided. The boundaries of the regions can be directly taken from Fig. 1 except for the boundary between regions 2 and 3; this boundary is defined by the so-called B23-equation given in Section 4. Both regions 1 and 2 are individually covered by a fundamental equation for the specific Gibbs free energy g(p, T), region 3 by a fundamental equation for the specific Helmholtz free energy f ( p, T), where p is the density, and the saturation curve by a saturation-pressure equation ps( T). The high-temperature region 5 is also covered by a g( p, T) equation. These five equations, shown in rectangular boxes in Fig. 1, form the so-called basic equations.

10 0

10 0

1 |
3 2 |

9(PJ) |
f(pJ)\\g(pJ) |

T(p,h) T(p,s) |
273.15 623.15 1073.15 2273.15 Fig. 1. Regions and equations of IAPWS-IF97. 77 K Regarding the main properties specific volume v, specific enthalpy h, specific isobaric heat capacity cp, speed of sound w, and saturation pressure ps, the basic equations represent the corresponding values from the "IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use" [3] (hereafter abbreviated to IAPWS-95) to within the tolerances specified for the development of the corresponding equations; details of these requirements and their fulfillment are given in the comprehensive paper on IAPWS-IF97 [2]. The basic equations for regions 1 and 3 also yield reasonable values for the metastable states close to the stable regions. For region 2 there is a special equation for the metastable-vapor region. Along the region boundaries the corresponding basic equations are consistent with each other within specified tolerances; for details see Section 10. In addition to the basic equations, for regions 1, 2, and 4 so-called backward equations are provided in form of T ( p, h ) and T ( p, s ) for regions 1 and 2, and Ts( p ) for region 4. These backward equations are numerically consistent with the corresponding basic equations and make the calculation of properties as functions of p, h and of p, s for regions 1 and 2, and of p for region 4 extremely fast. In this way, properties such as T ( p, h ), h ( p, s), and h'( p) can be calculated without any iteration from the backward equation alone or by combination with the corresponding basic equation, for example, h( p,s) via the relation h( p, T( p,s)). As a consequence, the calculation of the industrially most important properties is on average more than five times faster than the corresponding calculation with IFC-67; for details see Section 11. The estimates of uncertainty of the most relevant properties calculated from the corresponding equations of IAPWS-IF97 are summarized in Section 12. |

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