CALCULATION OF SOME THERMODYNAMIC PROPERTIES OF WATER FOR THE BOUNDARY POINTS BETWEEN REGIONS 3 AND 4 OF INDUSTRIAL FORMULATION IAPWS-IF 97

In the calculations relating to the thermodynamic cycle of steam power plants, the enthalpy and the entropy of saturated liquid and saturated steam (hf, hg, sf, and sg) are needed. If one uses industrial formulation IAPWS-IF97, these four parameters (hf, hg, sf, and sg) should be determined by cumbersome numerical computations when the saturated pressure has a value between 16.529 MPa and 22.064 MPa. In this work, easy programmable equations are provided to calculate hf, hg, sf, and sg in this range of saturated pressure.


INTRODUCTION
To calculate the thermodynamic properties of water, the International Association for Properties of Water and Steam (IAPWS) has published a comprehensive formulation that can be utilized in the industrial applications with good accuracy.This formulation is called Industrial Formulation of IAPWS and is abbreviated as IAPWS-IF97.The formulation is accessible either on the website of IAPWS [1] or in the paper by Wagner et al. [2].IAPWS-IF97 can be considered as a replacement for the previously international known formulation IFC-67 for calculating the properties of water [2][3].Miyagawa [4] compared the computing times of IAPWS-IF97 and IFC-67 and determined that the computing times of the first is much faster than the latter.IAPWS-IF97 has been widely adopted by engineers such that it has been utilized in the development of a number of engineering softwares like STEAMEST [5], WSProps [6], Freesteam [7], Xsteam [8], and Ansys-CFX [9].It is notable that some authors have recently proposed other models as alternatives to IAPWS-IF97 (for example, Patek and Klomfar [10] and Wang et al. [11][12]).
In the numerical modeling of thermal power plants using IAPWS-IF97, there are some instances that the water pressure is known, and the properties of saturated liquid and saturated steam are needed.If the pressure value is between 16.529MPa and 22.064 MPa, the point of saturated liquid or saturated steam will sit on the boundary curve between regions 3 and 4 of IAPWS-IF97.This boundary curve has been illustrated in Figures 3 and 4 of reference [13] which is one of the supplementary releases on the initial IAPWS-IF97.The mathematical equations for these saturated-liquid and saturated-steam curves are provided by equations (10) and (11) of reference [13].These two boundary equations of P 3sat h and P 3sat s can be used to calculate the saturated pressure if the enthalpy or the entropy of saturated liquid (h f , s f ) or saturated steam (h g , s g ) are known.These two equations in dimensionless forms are repeated here: π η = ∑ n i η-1.02 (1) π σ = ∑ n i σ-1.03I i σ-0.699J i 10 i=1 (2) where η= h h * ⁄ , σ= s s * ⁄ , and π= P P * ⁄ with P * =22 MPa, h * =2600 kJkg 1 , and s * =5.2 kJkg 1 K 1 .The constants n i , I i , and J i are given in Reference [13].
For the case mentioned earlier in which saturated pressure is known and four properties of h f , h g , s f , s g are sought, equations ( 1) and ( 2) should be solved inversely in order to find the unknowns σ= s s * ⁄ and η= h h * ⁄ .
This inverse solving is difficult because these equations are nonlinear and complicated if and be considered as the unknown variables.In this work, simple equations are provided to calculate η f , η g , σ f , σ g directly as a functions of dimensionless pressure π=P/P * .These equations are very easy to program and do not require initial guess.They are derived based on the data obtained from equations ( 1) and (2).

DEVELOPMENT OF THE EQUATIONS
The formulas presented here are found by curve fitting on the thermodynamic data obtained directly from equations ( 1) and (2).In order to generate the thermodynamic data, the ranges of * ⁄ , , and in equations ( 1) and ( 2) are separately divided into a large number of points.Then, the dimensionless pressure is calculated at each point using equations (1)(2).Four series of data are generated using this procedure for (σ f -P), (σ g -P), (η f -P) and (η g -P).Then the curve fitting method is applied to each data series to find a mathematical formula as a function of π =P P * ⁄ .The P * , s * , and h * have the same values used in equations (1)(2).In order to fit a curve to these data, the program approx1.f90 has been used which is downloadable from the link given in [14].The program has been written in FORTRAN language and acts through minimizing the sum of squares errors to find a function which is a summation of orthogonal polynomials.The user chooses the degree of polynomials (k).

RESULTS
One common equation was derived to calculate all four thermodynamic properties (σ f , σ g , η f , η g ).This is a recursive equation and has the following dimensionless form: ϕ π =A 0 (6) in which represents any of these properties: σ f , σ g , η f , η g .In each case, the values of the coefficients , , and as well as (the degree of the fitted polynomial) vary.All equations meet the following minimum tolerance ϕ π 97 -ϕ 97 ϕ 97 <10 -4 (7) where is the dimensionless pressure obtained from IAPWS-IF97 (equations (1-2)) for any arbitrary value of .
is the corresponding value computed by equation (6).
Examination showed that it is not possible to fit a single curve both far from and very near to the critical point of water under accuracy of equation (7).Hence, the pressure domain was divided into three sections: 16.529 MPa ≤ P < 22 MPa, 22 MPa ≤ P < 22.0616 MPa, and 22.0616 MPa ≤ P < 22.064 MPa.This behavior of water near critical point has been also observed in original IAPWS-IF97 in which there are several subregions in the vicinity of critical point.
In the following, the values of coefficients of equations (3)(4)(5)(6) are presented for each property.To help the user in computer programming, a test value for every property is given in Table 13.

CONCLUSION
For the pressure range of [16.529, 22.064]MPa where the user of IAPWS-IF97 needs to employ either the trial and error method or nonlinear algebraic solution methods to find the saturated enthalpy and entropy of water for a given saturated pressure, a computational model were developed in this article.This model provides equations for direct calculation of the enthalpy and the entropy of saturated liquid and steam of water for the points on the boundaries between regions 3 and 4 of industrial formulation IAPWS-IF97.The model is easy to program with good accuracy and does not have the problems of initial guess and convergence.The application of the model combining with IAWPS-IF97 is in writing computer codes to compute the thermal efficiency of thermodynamic cycles of the power plant.However, due to the complex behavior of water properties near the critical point, the entire pressure range of the model was divided into three separate ranges, and one common recursive equation with different coefficients values was derived.
,α i , and β i are listed in Table 5 (with k = 17 .For : The coefficients γ i ,α i , and β i are listed in Table 6 (with k = 17 .For : The coefficients γ i ,α i , and β i are listed in Table 7 (with k = 17 .For : The coefficients γ i ,α i , and β i are listed in Table 8 (with k = 17).,α i , and β i are listed in Table 9 (with k = 17 .For : The coefficients γ i ,α i , and β i are listed in Table 10 (with k = 17 .For : The coefficients γ i ,α i , and β i are listed in Table 11 (with = 17).For : The coefficients γ i ,α i , and β i are listed in Table 12 (with k = 17)

Table 13 .
The values of properties for selective pressures.