The International Association for the Properties of Water and Steam

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1 IAPWS SR3-03(2014) he International Association for the Properties of Water and Steam Moscow, Russia June 2014 Reised Supplementary Release on Backward Equations for the Functions (h), (h) and (s), (s) for Region 3 of the IAPWS Industrial Formulation 19 for the hermodynamic Properties of Water and Steam 2014 International Association for the Properties of Water and Steam Publication in whole or in part is allowed in all countries proided that attribution is gien to the International Association for the Properties of Water and Steam President: Professor amara Petroa Moscow Engineering Power Institute Moscow, Russia Executie Secretary: Dr. R. B. Dooley Structural Integrity Associates Southport, Merseyside, UK bdooley@structint.com his reised supplementary release replaces the corresponding reised supplementary release of 2004, and contains 22 pages, including this coer page. his reised supplementary release has been authorized by the International Association for the Properties of Water and Steam (IAPWS) at its meeting in Moscow, Russia, June, 2014, for issue by its Secretariat. he members of IAPWS are: Britain and Ireland, Canada, the Czech Republic, Germany, Japan, Russia, Scandinaia (Denmark, Finland, Norway, Sweden), and the United States, and associate members Argentina & Brazil, Australia, France, Greece, Italy, New Zealand, and Switzerland. he backward equations for temperature and specific olume as functions of pressure and enthalpy h, h and as functions of pressure and entropy s, s for region 3, and the equations for saturation pressure as a function of enthalpy p 3sath and as a function of entropy p 3sats for the saturation boundaries of region 3 proided in this release are recommended as a supplement to "IAPWS Industrial Formulation 19 for the hermodynamic Properties of Water and Steam" (IAPWS-IF) [1, 2]. Further details concerning the equations can be found in the corresponding article by H.-J. Kretzschmar et al. [3]. his reision consists of edits to clarify descriptions of how to determine the region or subregion; the property calculations are unchanged. Further information concerning this supplementary release, other releases, supplementary releases, guidelines, technical guidance documents, and adisory notes issued by IAPWS can be obtained from the Executie Secretary of IAPWS or from

2 2 Contents 1 Nomenclature 2 2 Background 3 3 Backward Equations h, h, s, and s for Region Numerical Consistency Requirements Structure of the Equation Set Backward Equations h and h for Subregions 3a and 3b Backward Equations s and s for Subregions 3a and 3b Computing ime in Relation to IAPWS-IF 16 4 Boundary Equations p sat (h) and p sat (s) for the Saturation Lines of Region Determination of the Region Boundaries for Gien Variables (h) and (s) Numerical Consistency Requirements Boundary Equations p sat (h) and p sat (s) Computing ime in Relation to IAPWS-IF 21 5 References 22 hermodynamic quantities: f Specific Helmholtz free energy h Specific enthalpy p Pressure s Specific entropy Absolute temperature a Specific olume Difference in any quantity Reduced enthalpy, h/h * Reduced temperature / * Reduced pressure, p/p * Density Reduced entropy, s/s * Reduced olume, / * x Vapor fraction Root-mean-square alue: N 1 2 xrms ( xn ) N n 1 where x n can be either absolute or percentage difference between the corresponding quantities x; N is the number of x n alues (100 million points uniformly distributed oer the range of alidity in the p- plane). 1 Nomenclature Superscripts: 01 Equation of IAPWS-IF-S01 Quantity or equation of IAPWS-IF * Reducing quantity ' Saturated liquid state " Saturated apor state Subscripts: 1 Region 1 2 Region 2 3 Region 3 3a Subregion 3a 3b Subregion 3b 3ab Boundary between subregions 3a and 3b 4 Region 4 5 Region 5 B23 Boundary between regions 2 and 3 c Critical point it Iterated quantity Maximum alue of a quantity RMS Root-mean-square alue of a quantity sat Saturation state olerance, range of accepted alue of a quantity a Note: denotes absolute temperature on the International emperature Scale of 1990 (IS-90).

3 3 2 Background he Industrial Formulation IAPWS-IF for the thermodynamic properties of water and steam [1, 2] contains basic equations, saturation equations and equations for the most often used backward functions h and s alid in the liquid region 1 and the apor region 2; see Figure 1. he IAPWS-IF was supplemented by "Backward Equations for Pressure as a Function of Enthalpy and Entropy phs, to the Industrial Formulation 19 for the hermodynamic Properties of Water and Steam" [4, 5], which is referred to here as IAPWS-IF-S01, including equations for the backward function p hs, alid in region 1 and region 2. p / MPa p B23 IAPWS-IF 01 IAPWS-IF-S g p h s h, s f c , h s h s p sat sat p g p h s h, s 50 MPa g 5 5, p / K Figure 1. Regions and equations of IAPWS-IF, IAPWS-IF-S01, and the backward equations h, h, and s, s of this release In modeling steam power cycles, thermodynamic properties as functions of the ariables h or s are also required in region 3. It is difficult to perform these calculations with p,, IAPWS-IF, because two-dimensional iteration is required using the functions h, or p,,, equation, s that can be explicitly calculated from the fundamental region 3 f. While these calculations are not frequently required in region 3, the relatiely large computing time required for two-dimensional iteration can be significant in process modeling. In order to aoid such iterations, this release proides equations for the backward functions p h, p h and p s, 3, 3, 3, p s, see Figure 1. With temperature and specific 3,

4 4 olume calculated from the backward equations, the other properties in region 3 can be calculated using the IAPWS-IF basic equation f In addition, boundary equations for the saturation pressure as a function of enthalpy p 3sath and as a function of entropy p 3sats for the saturated liquid and apor lines of region 3 are proided. Using these equations, whether a state point is located in the singlephase region or in the two-phase (wet steam) region can be determined without iteration. Section 4 contains the comprehensie description of the boundary equations. he numerical consistencies of all backward equations and boundary equations presented in Sections 3 and 4 with the IAPWS-IF basic equation are sufficient for most applications in heat cycle and steam turbine calculations. For applications where the demands on numerical consistency are extremely high, iterations using the IAPWS-IF basic equation may be necessary. In these cases, the backward or boundary equations can be used for calculating ery accurate starting alues. he time required to reach the conergence criteria of the iteration will be significantly reduced. he presented backward and boundary equations can only be used in their ranges of alidity described in Sections 3.2, 4.3, and 4.4. hey should not be used for determining any thermodynamic deriaties. In any case, depending on the application, a conscious decision is required whether to use the backward or boundary equations or to calculate the corresponding alues by iterations from the basic equation of IAPWS-IF. 3,. 3 Backward Equations h, h, s, and s for Region Numerical Consistency Requirements he permissible alue for the numerical consistency functions 3 h and 3, 25 mk of the backward f 3 p s with the basic equation IAPWS [6, 7] as a result of an international surey. he permissible alue 3,, was determined by for the numerical consistency for the equations p s can be estimated from the total differentials h h h and s s s 3,, p h and where,,, and are deriaties [8] calculated from the IAPWS- h h s s IF basic equation and h and s are alues determined by IAPWS for the adjacent

5 5 region 1 and subregion 2c [9], see able 1. he resulting permissible specific olume difference is 0.01 % p s. for both functions p h and. / [10], more stringent consistency requirements were arbitrarily set. hese were 0.49 mk and %. 3, At the critical point 3 c K, c kg m able 1. Numerical consistency alues 3 s, alues required for 3, h, s p h and p s Region 3 25 mk 80 3, h 3, of [6] required for 3, of [9], and resulting erances 1 Jkg 0.1 s Jkg 1 1 K p h and 0.01 % Critical Point 0.49 mk % 3.2 Structure of the Equation Set he equation set consists of backward equations h, h and s,, p s for region 3. Region 3 is defined by: K K and pb23 p 100 MPa, where p B23 represents the B23 equation of IAPWS-IF. Figure 2 shows the way in which region 3 is diided into the two subregions 3a and 3b. he boundary between the subregions 3a and 3b corresponds to the critical isentropic line 1 1 s s c kj kg K ; see Figure 2. For the functions s and s, input points can be tested directly to identify the subregion since the specific entropy is an independent ariable. In order to decide which h, h equation, 3a or 3b, must be used for gien alues of p and h, the boundary equation h3ab p, Eq. (1), has to be used; see Figure 2. his equation is a polynomial of the third degree and reads h3ab p 2 3 ( ) n1n2 n3 n4, (1) h 1 where hh and p p with h 1kJkg and p 1MPa. he coefficients n1 to n 4 of Eq. (1) are listed in able 2. he range of the equation h3ab p is from the critical point to 100 MPa. he related temperature at 100 MPa is K. Equation (1) does not exactly describe the critical isentropic line.

6 6 p / MPa a p h p h p s p s h3ab p s c K p s p h p s p h 3b p sat c p B / K Figure 2. Diision of region 3 into two subregions 3a and 3b for the backward equations h, h and s, s able 2. Numerical alues of the coefficients of the equation h its dimensionless form, Eq. (1), for defining the boundary between subregions 3a and 3b 3ab p in i n i i n i he imum specific entropy deiation was determined as 3ab 3 it 3ab it 3ab c 1, ( ),, ( ) 0.66Jkg K 1 s s p h p p h p s, where it and it of the IAPWS-IF basic equation for region 3. were obtained by iterations using the deriaties p, and s, 3 3

7 7 Equation (1) does not correctly reproduce the isentropic line p. Howeer, the calculated alues h s s at pressures lower than c 3ab p are not higher than the enthalpy on the saturated apor line and not lower than the enthalpy on the saturated liquid line. For computer-program erification, Eq. (1) gies the following p-h point: p MPa, 3 1 h p kj kg. 3ab 3.3 Backward Equations h and h for Subregions 3a and 3b he Equations h. he backward equation following dimensionless form: where 1 3a 31 i1 c p h for subregion 3a has the ( h) I Ji 3a, n i i 0.615, (2), p p, and hh, with 760 K, p 100 MPa, and h 2300 kj kg. he coefficients n i and exponents I i and J i of Eq. (2) are listed in able 3. he backward equation where 1 3b p h for subregion 3b reads in its dimensionless form 33 ( h) I Ji 3b, n i * i , (3) i1, p p, and hh, with 860 K, p 100 MPa, and h 2800 kj kg. he coefficients n i and exponents I i and J i of Eq. (3) are listed in able 4. able 3. Coefficients and exponents of the backward equation dimensionless form, Eq. (2) p h for subregion 3a in its i I i J i n i i I i J i n i

8 8 able 4. Coefficients and exponents of the backward equation dimensionless form, Eq. (3) p h for subregion 3b in its i I i J i n i i I i J i n i Computer-program erification. o assist the user in computer-program erification of Eqs. (2) and (3), able 5 contains test alues for calculated temperatures. able 5. Selected temperature alues calculated from Eqs. (2) and (3) a he Equations h. he backward equation following dimensionless form: 3a p h for subregion 3a has the, 32 i i Ji i1 ( h) 3a n I 0.727, (4) where, p p, and hh, with 1 Equation p / MPa h / 1 kj kg / K 3a h, Eq. (2) 3b h, Eq. (3) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables m kg, p 100 MPa, and h 2100 kj kg. he coefficients n i and exponents I i and J i of Eq. (4) are listed in able 6.

9 9 he backward equation 3b p h for subregion 3b reads in its dimensionless form, 30 i i Ji i1 ( h) 3b n I 0.720, (5) where, p p, and hh, with m kg, p 100 MPa, and h 2800 kj kg. he coefficients ni and exponents Ii and Ji of Eq. (5) are listed in able 7. able 6. Coefficients and exponents of the backward equation dimensionless form, Eq. (4) p h for subregion 3a in its i I i J i n i i I i J i n i able 7. Coefficients and exponents of the backward equation dimensionless form, Eq. (5) p h for subregion 3b in its i I i J i n i i I i J i n i

10 10 Computer-program erification. o assist the user in computer-program erification of Eqs. (4) and (5), able 8 contains test alues for calculated specific olumes. able 8. Selected specific olume alues calculated from Eqs. (4) and (5) a Equation p / MPa h / 1 kj kg / 3 1 m kg 3a h, Eq. (4) 3b h, Eq. (5) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. Numerical Consistency with the Basic Equation of IAPWS-IF. he imum temperature differences and related root-mean-square differences between the calculated temperature Eqs. (2) and (3) and the IAPWS-IF basic equation f 3 (, ) in comparison with the permissible differences are listed in able 9. he calculation of the root-mean-square alues is described in Section 1. able 9 also contains the imum relatie deiations and root-mean-square relatie deiations for specific olume of Eqs. (4) and (5) from IAPWS-IF. he critical temperature and the critical olume are met exactly by the equations h and h. able 9. Maximum differences and root-mean-square differences of the temperature calculated from Eqs. (2) and (3) and specific olume calculated from Eqs. (4) and (5) to the IAPWS-IF basic equation f 3 (, ) and related permissible alues Subregion Equation RMS 3a (2) 25 mk 23.6 mk 10.5 mk 3b (3) 25 mk 19.6 mk 9.6 mk Subregion Equation RMS 3a (4) 0.01 % % % 3b (5) 0.01 % % % Consistency at Boundary Between Subregions. he imum temperature difference between the two backward equations, Eq. (2) and Eq. (3), along the boundary h3ab p, Eq. (1), has the following alue h p h p 0.37mK. 3a 3ab 3b 3ab

11 11 hus, the temperature differences between the two backward functions, p h of the adjacent subregions are smaller than the numerical consistencies with the IAPWS-IF equations. he relatie specific olume differences between the two backward equations h of the adjacent subregions 3a and 3b are also smaller than the numerical consistencies of these equations with the IAPWS-IF basic equation. Along the boundary h imum difference between the corresponding equations was determined as:, p, p h p 3a p h3ab 3b p h3ab %. 3b 3ab 3ab p, Eq. (1), the 3.4 Backward Equations s and s for Subregions 3a and 3b he Equations (s). he backward equation following dimensionless form: 3a 33 i1 p s for subregion 3a has the ( s) Ii Ji 3a, ni , (6) where, p p, and s s, with 760 K, p 100 MPa, and 1 1 s 4.4 kj kg K. he coefficients n i and exponents I i and J i of Eq. (6) are listed in able 10. he backward equation where 1 1 s 5.3 kj kg K able 11. 3b Computer-program erification p s for subregion 3b reads in its dimensionless form 28 ( s) Ii Ji 3b, ni , (7) i1, p p, and s s, with 860 K, p 100 MPa, and. he coefficients n i and exponents I i and J i of Eq. (7) are listed in o assist the user in computer-program erification of Eqs. (6) and (7), able 12 contains test alues for calculated temperatures.

12 12 able 10. Coefficients and exponents of the backward equation dimensionless form, Eq. (6) p s for subregion 3a in its i I i J i n i i I i J i n i able 11. Coefficients and exponents of the backward equation dimensionless form, Eq. (7) p s for subregion 3b in its i I i J i n i i I i J i n i

13 13 able 12. Selected temperature alues calculated from Eqs. (6) and (7) a Equation p / MPa s / kj kg 1 1 K / K a s, Eq. (6) p s, Eq. (7) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. he Equations (s). he backward equation following dimensionless form: 3a 28 i1 p s for subregion 3a has the ( s) Ii Ji 3a, ni , (8) where, p p, and s s, with m kg, p 100 MPa, and 1 1 s 4.4 kj kg K. he coefficients n i and exponents I i and J i of Eq. (8) are listed in able 13. he backward equation 3b p s for subregion 3b reads in its dimensionless form 31 ( s) Ii Ji 3b, ni , (9) i1 where, p p, and s s, with 1 1 s 5.3 kj kg K able m kg, p 100 MPa, and. he coefficients n i and exponents I i and J i of Eq. (9) are listed in Computer-program erification. o assist the user in computer-program erification of Eqs. (8) and (9), able 15 contains test alues for calculated specific olumes.

14 14 able 13. Coefficients and exponents of the backward equation dimensionless form, Eq. (8). p s for subregion 3a in its i I i J i n i i I i J i n i able 14. Coefficients and exponents of the backward equation dimensionless form, Eq. (9) p s for subregion 3b in its i I i J i n i i I i J i n i

15 15 able 15. Selected specific olume alues calculated from Eqs. (8) and (9) a Equation p / MPa s / kj kg 1 1 K / 3 1 m kg a s, Eq. (8) p s, Eq. (9) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. Numerical Consistency with the Basic Equation of IAPWS-IF. he imum temperature differences and related root-mean-square differences between the temperatures calculated from Eqs. (6) and (7) and the IAPWS-IF basic equation f 3 (, ) in comparison with the permissible differences are listed in able 16. able 16 also contains the imum relatie deiations and root-mean-square relatie deiations for the specific olume of Eqs. (8) and (9) from IAPWS-IF. he critical temperature and the critical olume are met exactly by the equations s and s. able 16. Maximum differences and root-mean-square differences of the temperature calculated from Eqs. (6) and (7), and specific olume calculated from Eqs. (8) and (9) from the IAPWS-IF basic equation f3 (, ), and related permissible alues Subregion Equation RMS 3a (6) 25 mk 24.8 mk 11.2 mk 3b (7) 25 mk 22.1 mk 10.1 mk Subregion Equation RMS 3a (8) 0.01 % % % 3b (9) 0.01 % % % Consistency at Boundary Between Subregions. he imum temperature difference between the two backward equations, Eq. (6) and Eq. (7), along the boundary s c, has the following alue 3a ps ps,, 0.093mK. c hus, the temperature differences between the two backward functions, 3b p s of the adjacent subregions are smaller than their differences with the IAPWS-IF equations. c

16 16 he relatie specific olume differences between the two backward equations s, Eqs. (8) and (9), of the adjacent subregions are also smaller than the differences of these equations with the IAPWS-IF basic equation. Along the boundary s c, the imum difference between the corresponding equations was determined as ps, c 3b ps, c ps, 3a %. 3b c 3.5 Computing ime in Relation to IAPWS-IF A ery important motiation for the deelopment of the backward equations h, h and s, s for region 3 was reducing the computing time to obtain thermodynamic properties and differential quotients from gien ariables h and s. In IAPWS-IF, time-consuming iterations, e.g., the two-dimensional Newton method, are required. Using the p h, p h, p s and 3, 3, 3, 3 s equations, the calculation speed is about 20 times faster than that of the two-dimensional Newton method with conergence erances set to the alues shown in able 1. he numerical consistency of and obtained in this way is sufficient for most heat cycle calculations. For users not satisfied with the numerical consistency of the backward equations, the equations are still recommended for generating starting points for the iteratie process. hey will significantly reduce the time required to reach the conergence criteria of the iteration. 4 Boundary Equations p sat (h) and p sat (s) for the Saturation Lines of Region Determination of the Region Boundaries for Gien Variables (h) and (s) he boundaries between region 3 and the two-phase region 4 are the saturated liquid line x 0 and saturated apor line x 1; see Figures 3 and 4. A one-dimensional iteration using the IAPWS-IF basic equation f 3 (, ) and the saturation-pressure equation psat is required to calculate the enthalpy or entropy from a gien pressure on the saturated liquid or saturated apor lines of region 3. he boundary equations p 3sath and p 3sats, proided in this release, make it possible to determine without iteration whether the gien state point is located in the two-phase region 4 or in the single-phase region 3. he boundary between regions 1 and 3 can be calculated directly from a gien pressure p and from K using the IAPWS-IF basic equation g 1 ( p, ). he boundary between regions 2 and 3 can be calculated directly from gien pressure p and from the B23- equation B23 p of IAPWS-IF and using the IAPWS-IF basic equation g 2 ( p, ).

17 17 p / MPa p3sath c p B23 h'( K) h c h''( K) h / kj kg 1 Figure 3. Illustration of IAPWS-IF region 3 and the boundary equation in a p-h diagram p3sat h p / MPa c p3sats p B23 s'( K) s c s''( K) s / kj kg 1 K 1 Figure 4. Illustration of IAPWS-IF region 3 and the boundary equation in a p-s diagram 4.2 Numerical Consistency Requirements he required consistency of the boundary equations for the saturation lines of region 3 result from IAPWS requirements on backward functions. herefore, the backward functions ( h ), ( h ), ( s ), and ( s ) hae to fulfill their numerical consistency requirements when using the boundary equations gien state point. p 3sath and p 3sat 4.3 Boundary Equations p sat (h) and p sat (s) he Equation p 3sat (h). he equation p p3sat s s for determining the region of a 3sat h describes the saturated liquid line and the saturated apor line including the critical point in the enthalpy range (see Figure 3): h' K h h" K, 3 1 where ' K kj kg 3 1 and h, h '' K kj kg.

18 18 he boundary equation p 3sat h has the following dimensionless form: 14 i i Ji i1 p3sat( h) p I n , (10) where p p and hh, with p 22 MPa and h n i and exponents I i and J i of Eq. (10) are listed in able 17. able 17. Coefficients and exponents of the boundary equation Eq. (10) 2600 kj kg p3sat 1. he coefficients h in its dimensionless form, i I i J i n i i I i J i n i Computer-program erification. o assist the user in computer-program erification of Eq. (10), able 18 contains test alues for calculated pressures. able 18. Selected pressure alues calculated from Eq. (10) a Equation h / 1 kj kg Equation p 3sat (s). he equation p p / MPa p 3sath, Eq. (10) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. 3sat s describes the saturated liquid line and the saturated apor line including the critical point in the entropy range (see Figure 4): s' K s s" K, where 1 1 ' K kj kg K and 1 1 '' K kj kg K s, s.

19 19 he boundary equation p 3sat s has the following dimensionless form: 10 i i Ji i1 p3sat() s p I n , (11) where p p and s s, with p 22 MPa and s 5.2 kj kg K coefficients n i and exponents I i and J i of Eq. (11) are listed in able 19. able 19. Coefficients and exponents of the boundary equation Eq. (11) p3sat 1 1. he s in its dimensionless form, i I i J i n i i I i J i n i Computer-program erification. o assist the user in computer-program erification of Eq. (11), able 20 contains test alues for calculated pressures. able 20. Selected pressure alues calculated from Eq. (11) a Equation s / kj kg Numerical Consistency with the Saturation-Pressure Equation of IAPWS-IF. he imum percentage deiation between the pressure calculated from the boundary equation p 3sath, Eq. (10), and the IAPWS-IF saturation-pressure equation p sat ( ) has the following alue sat p p3sat h p p p sat 1 1 K p / MPa p 3sats, Eq. (11) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables %.

20 20 he imum percentage deiation between the calculated pressure the IAPWS-IF saturation-pressure equation p 3sat p sat ( ) has the following alue sat p p3sat s p p p sat %. s, Eq. (11), and Consistency of the Backward Equations (h), (h), (s), and (s) with the Basic Equation of IAPWS-IF along the Boundary Equations p sat (h) and p sat (s). he imum temperature differences between the backward equations 3a h, Eq. (2), and 3b h, Eq. (3), and the IAPWS-IF basic equation f 3 (, ) along the boundary equation p 3sath, Eq. (10), in comparison with the permissible differences are listed in able 21. he temperature differences were calculated as 3 p3sath 3, h 3. he function 3 represents the calculation of h using the backward equations of subregions 3a and 3b, Eqs. (2) and (3). able 21. Maximum differences of temperature and specific olume calculated from Eqs. (2), (3), (4), and (5) from the IAPWS- IF basic equation f 3 (, ) along the boundary equation p h, Eq. (10), and related permissible alues 3sat Subregion Equation 3a (2) 25 mk 0.47 mk 3b (3) 25 mk 0.46 mk Subregion Equation / / 3a (4) 0.01 % % 3b (5) 0.01 % % Furthermore, able 21 contains the imum percentage differences of specific olume between the backward equations 3a h, Eq. (4), and 3b, IF basic equation f along the boundary equation p p h, Eq. (5), and the IAPWS- 3 (, ) 3sat h, Eq. (10). he relatie differences of specific olume were calculated as / 3 p3sath3, h 3. he function 3 represents the calculation of h using the backward equations of subregions 3a and 3b, Eqs. (4) and (5). he imum temperature differences and the imum relatie differences of specific olume are smaller than the permissible alues. herefore, the numerical consistency of the boundary equation h, Eq. (10), is sufficient. p 3sat

21 21 he imum temperature differences between the backward equations 3a, and 3b s, Eq. (7), and the IAPWS-IF basic equation 3 (, ) equation p s, Eq. (6), along the boundary p 3sat s, Eq. (11), in comparison with the permissible differences are listed in able 22. he temperature differences were calculated as 3 p3sats 3, s 3. he function 3 represents the calculation of s using the backward equations of subregions 3a and 3b, Eqs. (6) and (7). able 22. Maximum differences of temperature and specific olume calculated from Eqs. (6), (7), (8), and (9) to the IAPWS-IF basic equation f 3 (, ) along the boundary equation p s, Eq. (11), and related permissible alues 3sat Subregion Equation 3a (6) 25 mk 2.69 mk 3b (7) 25 mk 2.12 mk Subregion Equation / / 3a (8) 0.01 % % 3b (9) 0.01 % % Furthermore, able 22 contains the imum percentage differences of specific olume between the backward equations 3a s, Eq. (8), and 3b, IF basic equation f along the boundary equation p p s, Eq. (9), and the IAPWS- 3 (, ) 3sat s, Eq. (11). he relatie differences of specific olume were calculated as / 3 p3sats3, s 3. he function 3 represents the calculation of s using the backward equations of subregions 3a and 3b, Eqs. (8) and (9). he imum temperature differences and the imum relatie differences of specific olume are smaller than the permissible alues. herefore, the numerical consistency of the boundary equation s, Eq. (11), is sufficient. p 3sat 4.4 Computing ime in Relation to IAPWS-IF A ery important motiation for the deelopment of the equations for saturation lines of region 3 was reducing the computing time to determine the region for a gien state point h and s. In IAPWS-IF, time-consuming iterations, e.g., the Newton method, are required. By using equations p 3sath, Eq. (10), and p 3sats, Eq. (11), the calculation to determine the region is about 60 times faster than that of the two-dimensional Newton method.

22 22 5 References [1] IAPWS, Reised Release on the IAPWS Industrial Formulation 19 for the hermodynamic Properties of Water and Steam (2007), aailable from: [2] Wagner, W., Cooper, J. R., Dittmann, A., Kijima, J., Kretzschmar, H.-J., Kruse, A., Mareš, R., Oguchi, K., Sato, H., Stöcker, I., Šifner, O., anishita, I., rübenbach, J., and Willkommen, h., he IAPWS Industrial Formulation 19 for the hermodynamic Properties of Water and Steam, ASME J. Eng. Gas urbines Power 122, (2000). [3] Kretzschmar, H.-J., Cooper, J. R., Dittmann, A., Friend, D. G., Gallagher, J. S., Harey, A. H., Knobloch, K., Mareš, R., Miyagawa, K., Okita, N., Stöcker, I., Wagner, W., and Weber, I., Supplementary Backward Equations (h), (h), and (s), (s) for the Critical and Supercritical Regions (Region 3) of the Industrial Formulation IAPWS-IF for Water and Steam, ASME J. Eng. Gas urbines Power 129, (2007). [4] IAPWS, Reised Supplementary Release on Backward Equations for Pressure as a Function of Enthalpy and Entropy p(h,s) for Regions 1 and 2 of the IAPWS Industrial Formulation 19 for the hermodynamic Properties of Water and Steam (2014), aailable from: [5] Kretzschmar, H.-J., Cooper, J. R., Dittmann, A., Friend, D. G., Gallagher, J. S., Knobloch, K., Mareš, R., Miyagawa, K., Stöcker, I., rübenbach, J., Wagner, W., Willkommen, h., Supplementary Backward Equations for Pressure as a Function of Enthalpy and Entropy p(h,s) to the Industrial Formulation IAPWS-IF for Water and Steam, ASME J. Eng. Gas urbines Power 128, (2006). [6] Kretzschmar, H.-J., Specifications for the Supplementary Backward Equations (h) and (s) in Region 3 of IAPWS-IF, in: Minutes of the Meetings of the Executie Committee of the International Association for the Properties of Water and Steam, Gaithersburg 2001, ed. by B. Dooley, IAPWS Secretariat (2001), p. 6 and Attachment 7- Item #6. [7] Rukes, B., Specifications for Numerical Consistency, in: Minutes of the Meetings of the Executie Committee of the International Association for the Properties of Water and Steam, Orlando 1994, ed. by B. Dooley, IAPWS Secretariat (1994), pp [8] Kretzschmar, H.-J., Stöcker, I., Klinger, J., and Dittmann, A., Calculation of hermodynamic Deriaties for Water and Steam Using the New Industrial Formulation IAPWS-IF, in: Steam, Water and Hydrothermal Systems: Physics and Chemistry Meeting the Needs of Industry, Proceedings of the 13th International Conference on the Properties of Water and Steam, ed. by P. R. remaine, P. G. Hill, D. E. Irish, and P. V. Balakrishnan, NRC Press, Ottawa, 2000, pp [9] Rukes, B., and Wagner, W., Final Set of Specifications for the New Industrial Formulation, in: Minutes of the Meetings of the Executie Committee of the International Association for the Properties of Water and Steam, okyo 1991, ed. by B. Dooley, IAPWS Secretariat (1991), pp [10] IAPWS, Release on the Values of emperature, Pressure and Density of Ordinary and Heay Water Substances at their Respectie Critical Points (1992), aailable from: