The International Association for the Properties of Water and Steam

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1 he International Association for the Properties of Water and Steam Kyoto, Japan September 2004 Reised Supplementary Release on Backward Equations for the Functions (p,h), (p,h) and (p,s), (p,s) for Region 3 of the IAPWS Industrial Formulation 9 for the hermodynamic Properties of Water and Steam 2004 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: Emeritus Professor Koichi Watanabe Keio Uniersity 2-2-7, Numabukuro, Nakano-ku okyo , Japan Executie Secretary: Dr. R. B. Dooley Electric Power Research Institute 300 West W.. Harris Bld. Charlotte, North Carolina 28262, USA his supplementary release is a reision of the corresponding supplementary release of It contains 22 pages, including this coer page. his supplementary release has been authorized by the International Association for the Properties of Water and Steam (IAPWS) at its meeting in Kyoto, Japan, 29 August- 3 September, 2004, for issue by its Secretariat. he members of IAPWS are: Argentina and Brazil, Britain and Ireland, Canada, the Czech Republic, Denmark, France, Germany, Italy, Japan, Russia, the United States of America, and associate member Greece. he backward equations for temperature and specific olume as functions of pressure and enthalpy ( p, h ), ( p, h ) and as functions of pressure and entropy ( p, s ), ( p, s ) for region 3, and the equations for saturation pressure as a function of enthalpy p ( h ) and as a function of entropy p ( s ) for the saturation boundaries of region 3 proided in this release are recommended as a supplement to "IAPWS Industrial Formulation 9 for the hermodynamic Properties of Water and Steam" (IAPWS-IF) [, 2] and to "Backward Equations for Pressure as a Function of Enthalpy and Entropy (, ) p hs to the Industrial Formulation 9 for the hermodynamic Properties of Water and Steam" (referred to here as IAPWS-IF-S0) [3, 4]. Further details concerning the equations can be found in the corresponding article by H.-J. Kretzschmar et al. [5]. Minor editorial changes were made to this reised supplementary release in July he only edits were of test points in ables 2 and 5; the property calculations are unchanged. Further information concerning this supplementary release, IAPWS-IF, IAPWS-IF- S0, and other documents issued by IAPWS can be obtained from the Executie Secretary of IAPWS or from

2 2 Contents Nomenclature 2 2 Background 3 3 Backward Equations (p,h), (p,h), (p,s), and (p,s) for Region Numerical Consistency Requirements Structure of the Equation Set Backward Equations (p,h) and (p,h) for Subregions 3a and 3b Backward Equations (p,s) and (p,s) for Subregions 3a and 3b 3.5 Computing ime in Relation to IAPWS-IF 6 4 Boundary Equations p sat (h) and p sat (s) for the Saturation Lines of Region Determination of the Region Boundaries for Gien Variables (p,h) and (p,s) Numerical Consistency Requirements Boundary Equations p sat (h) and p sat (s) Computing ime in Relation to IAPWS-IF 2 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 η Reduced enthalpy, η = h/h * θ Reduced temperature θ = / * π Reduced pressure, π = p/p * ρ Density σ Reduced entropy, σ = s/s * ω Reduced olume, ω = / * x Vapor fraction Root-mean-square alue: N 2 xrms = ( xn ) N n = where x n can be either absolute or percentage difference between the corresponding quantities x; N is the number of x n alues (00 million points uniformly distributed oer the range of alidity in the p- plane). Nomenclature Superscripts: 0 Equation of IAPWS-IF-S0 Quantity or equation of IAPWS-IF * Reducing quantity ' Saturated liquid state " Saturated apor state Subscripts: Region 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 990 (IS-90).

3 3 2 Background he Industrial Formulation IAPWS-IF for the thermodynamic properties of water and steam [, 2] contains basic equations, saturation equations and equations for the most often used backward functions (, ) p s alid in the liquid region and the apor region 2; see Figure. he IAPWS-IF was supplemented by "Backward Equations for p h and (, ) Pressure as a Function of Enthalpy and Entropy p ( hs, ) to the Industrial Formulation 9 for the hermodynamic Properties of Water and Steam" [3, 4], which is referred to here as IAPWS-IF-S0, including equations for the backward function p ( hs, ) alid in region and region 2. p / MPa p B23 ( ) IAPWS-IF 0 IAPWS-IF-S0 50 g p 0 ( p, ) ( p, h) ( p, s) ( hs, ) f (, ) ( p, h) ( p, s) ( p, h) ( p, s) g p ( p, ) ( p, h) ( p, s) ( hs, ) c 4 p sat sat ( ) ( p) g ( p ) 5, 5 Figure / K Regions and equations of IAPWS-IF, IAPWS-IF-S0, and the backward,,, p, s of this release equations ( p h ), ( p h ), and ( p s ), ( ) In modeling steam power cycles, thermodynamic properties as functions of the ariables ( ph, ) or (, ) IAPWS-IF, because two-dimensional iteration is required using the functions (, ) h(, ) or p(, ), (, ) equation (, ) ps are also required in region 3. It is difficult to perform these calculations with p, 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 3 ( p, h ), 3 (, ) p h and ( p s ), ( ) 3, p s, see Figure. 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 ( ) s 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. p h and as a function of entropy p ( ) 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 (p,h), (p,h), (p,s), and (p,s) for Region 3 3. Numerical Consistency Requirements he permissible alue for the numerical consistency functions 3 ( p, 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 region and subregion 2c [9], see able. he resulting permissible specific olume

5 5 difference is for both functions ( p h ) and ( ) =0.0% 3, At the critical point 3 c = K, c = ( 322 kg m ) p s.. / [0], more stringent consistency requirements were arbitrarily set. hese were = 0.49 mk and =0.000%. able. Numerical consistency alues 3 ( p, s ), alues required for 3 (, ) h, s 3, p h and ( ) Region 3 25 mk 80 h 3, of [6] required for ( ) 3, of [9], and resulting erances p s Jkg 0. s Jkg K p h and 0.0 % Critical Point 0.49 mk % 3.2 Structure of the Equation Set he equation set consists of backward equations ( p, h ), ( p, h ) and ( p, s ), (, ) region 3. Region 3 is defined by: p s for K K and pb23 ( ) p 00 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 s = s c ; see Figure 2. For the functions ( p, s ) and ( p, s ), input points can be tested directly to identify the subregion since the specific entropy is an independent ariable. If the gien specific entropy s is less than or equal to s c = kj kg K, then the state point to be calculated is located in subregion 3a; otherwise it is in subregion 3b. In order to decide which ( p, h ), (, ) of p and h, the boundary equation h ( ) p h equation, 3a or 3b, must be used for gien alues 3ab p, Eq. (), has to be used; see Figure 2. his equation is a polynomial of the third degree and reads ( p) h3ab 2 3 = η( π) = n+ n2π + n3π + n4π, () h where η = hh and π = p p with h = kjkg and p = MPa. he coefficients n to n 4 of Eq. () are listed in able 2. he range of the equation h3ab( p ) is from the critical point to 00 MPa. he related temperature at 00 MPa is = K. Equation () 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 h) ( p s) ( p s) ( p h) 3b p sat ( ) c p B23 ( ) / K Figure 2. Diision of region 3 into two subregions 3a and 3b for the backward,,, p, s equations ( p h ), ( p h ) and ( p s ), ( ) able 2. Numerical alues of the coefficients of the equation h ( ) its dimensionless form, Eq. (), 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, ( ),, ( ) 0.66Jkg K 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 (, ) If the gien specific enthalpy h is greater than h3ab( p ) calculated from the gien pressure p, then the state point to be calculated is located in subregion 3b, otherwise it is in subregion 3a (see Figure 2). 3 3

7 7 Equation () does not correctly reproduce the isentropic line p. Howeer, the calculated alues h ( ) c 3ab s = s at pressures lower than 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. () gies the following p-h point: 3 p = 25 MPa, ( ) = h p kj kg. 3ab 3.3 Backward Equations (p,h) and (p,h) for Subregions 3a and 3b he Equations (p,h). he backward equation ( ) following dimensionless form: where 3a 3 i= c p h for subregion 3a has the ( p, h) I Ji = θ 3a( π, η) = n ( 0.240) i i π ( η 0.65) +, (2) θ =, π = p p, and η = hh with 760 K, p 00 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 3b = = p h for subregion 3b reads in its dimensionless form 33 ( p, h) I Ji = θ 3b( π, η) = n ( ) i ( 0.720) * i π η, (3) i= θ =, π = p p, and η = hh with 860 K, p 00 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 (p,h). he backward equation ( ) following dimensionless form: 3a p h for subregion 3a has the, 32 i i Ji i= ( p, h) = ω 3a( πη) = n ( π 0.28) I ( η 0.727) +, (4) where ω =, π = p p, and η = hh with Equation p / MPa h / kj kg / K p h, Eq. (2) a (, ) p h, Eq. (3) b (, ) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. 3 = = m kg, p 00 MPa, and h = 200 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 i= ( p, h) = ω 3b( πη) = n ( π 0.066) I ( η 0.720) +, (5) where ω =, π = p p, and η = hh with 3 = = m kg, p 00 MPa, and h = 2800 kj kg. he coefficients n i and exponents I i and J i 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 0 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 / kj kg / 3 m kg p h, Eq. (4) a (, ) p h, Eq. (5) b (, ) 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. 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 ( p, h ) and ( p, 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 f3 (, ) and related permissible alues Subregion Equation RMS 3a (2) 25 mk 23.6 mk 0.5 mk 3b (3) 25 mk 9.6 mk 9.6 mk Subregion Equation RMS 3a (4) 0.0 % % % 3b (5) 0.0 % % % 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. (), has the following alue = ( p, h ( p) ) ( p, h ( p) ) = 0.37 mk. 3a 3ab 3b 3ab

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 ( p, 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 ( p ), Eq. (), the imum difference between the corresponding equations was determined as: (, ( p) ), ( p) p, h ( p) ( ) ( ) 3a p h3ab 3b p h3ab = = %. 3b 3ab 3ab 3.4 Backward Equations (p,s) and (p,s) for Subregions 3a and 3b he Equations (p,s). he backward equation ( ) following dimensionless form: 3a 33 i= p s for subregion 3a has the ( p, s) Ii Ji = θ 3a( π, σ) = ni ( π 0.240) ( σ 0.703) +, (6) where θ =, π = p p, and σ = s s with 760 K, p 00 MPa, and s = 4.4 kj kg K. he coefficients n i and exponents I i and J i of Eq. (6) are listed in able 0. he backward equation ( ) where 3b = = p s for subregion 3b reads in its dimensionless form 28 ( p, s) Ii Ji = θ 3b( π, σ) = ni ( π 0.760) ( σ 0.88) +, (7) i= θ =, π = p p, and σ = s s with 860 K, p 00 MPa, and s = 5.3 kj kg K able. Computer-program erification. 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 2 contains test alues for calculated temperatures. = =

12 2 able 0. 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. 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 3 able 2. Selected temperature alues calculated from Eqs. (6) and (7) a Equation p / MPa s / kj kg K / K p s, Eq. (6) a (, ) p s, Eq. (7) b (, ) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. he Equations (p,s). he backward equation ( ) following dimensionless form: 3a 28 i= p s for subregion 3a has the ( p, s) Ii Ji = ω 3a( πσ, ) = ni ( π 0.87) ( σ 0.755) +, (8) where ω =, π = p p, and σ = s s with 3 = = m kg, p 00 MPa, and s = 4.4 kj kg K. he coefficients n i and exponents I i and J i of Eq. (8) are listed in able 3. he backward equation ( ) 3b p s for subregion 3b reads in its dimensionless form 3 ( p, s) Ii Ji = ω 3b( πσ, ) = ni ( π 0.298) ( σ 0.86) +, (9) i= where ω =, π = p p, and σ = s s with s = 5.3 kj kg K able 4. Computer-program erification 3 = = m kg, p 00 MPa, and. he coefficients n i and exponents I i and J i of Eq. (9) are listed in o assist the user in computer-program erification of Eqs. (8) and (9), able 5 contains test alues for calculated specific olumes.

14 4 able 3. 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 4. 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 5 able 5. Selected specific olume alues calculated from Eqs. (8) and (9) a Equation p / MPa s / kj kg K / 3 m kg p s, Eq. (8) a (, ) p s, Eq. (9) b (, ) 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 6. able 6 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 ( p, s ) and ( p, s ). able 6. 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.2 mk 3b (7) 25 mk 22. mk 0. mk Subregion Equation RMS 3a (8) 0.0 % % % 3b (9) 0.0 % % % 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 6 he relatie specific olume differences between the two backward equations ( p, 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 ( p, h ), ( p, h ) and ( p, s ), ( p, s ) for region 3 was reducing the computing time to obtain thermodynamic properties and differential quotients from gien ariables ( p, h ) and ( p, 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 p, 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. 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 3 4. Determination of the Region Boundaries for Gien Variables (p,h) and (p,s) he boundaries between region 3 and the two-phase region 4 are the saturated liquid line x = 0 and saturated apor line x = ; 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 ( h ) and p ( s ), 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 and 3 can be calculated directly from a gien pressure p and from = K using the IAPWS-IF basic equation g ( p, ). he boundary between regions 2 and 3 can be calculated directly from gien pressure p and from the B23- equation = ( p ) of IAPWS-IF and using the IAPWS-IF basic equation g 2 ( p, ). B23

17 7 p / MPa p ( h) c p B23 ( ) h'(623.5 K) h c h''(623.5 K) h / kj kg Figure 3. Illustration of IAPWS-IF region 3 and the boundary equation p ( h ) in a p-h diagram p / MPa c p ( s) p B23 ( ) s'(623.5 K) s c s''(623.5 K) s / kj kg K Figure 4. Illustration of IAPWS-IF region 3 and the boundary equation p ( s ) 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 ( p, h ), ( p, h ), ( p, s ), and ( p, s ) hae to fulfill their numerical consistency requirements when using the boundary equations ( ) gien state point. p h and p ( ) 4.3 Boundary Equations p sat (h) and p sat (s) he Equation p (h). he equation p ( ) s for determining the region of a 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, ( ) ( ) where 3 '( K) = kj kg and 3 ''( K) = kj kg h, h.

18 8 he boundary equation p ( ) h has the following dimensionless form: p 4 i i Ji i= ( h) = π η = η η p I ( ) n (.02) ( 0.608), (0) = where π = p p and η = hh with p 22 MPa and h n i and exponents I i and J i of Eq. (0) are listed in able 7. = 2600 kj kg able 7. Coefficients and exponents of the boundary equation p ( ) Eq. (0). he coefficients h in its dimensionless form, i I i J i n i i I i J i n i If the gien pressure p is greater than p ( ) h calculated from the gien enthalpy h, then the state point to be calculated is located in region 3, otherwise it is in the two-phase region 4 (see Figure 3). Computer-program erification. o assist the user in computer-program erification of Eq. (0), able 8 contains test alues for calculated pressures. able 8. Selected pressure alues calculated from Eq. (0) a Equation h / kj kg Equation p (s). he equation p ( ) p / MPa p h, Eq. (0) ( ) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. 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 '( K) kj kg = K and s ''( K) = kj kg K. s,

19 9 he boundary equation p ( ) s has the following dimensionless form: 0 i i Ji i= p() s = π σ = σ σ p I ( ) n (.03) ( 0.699), () where π = p p and σ = 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. () are listed in able 9. able 9. Coefficients and exponents of the boundary equation p ( ) Eq. () =. he s in its dimensionless form, i I i J i n i i I i J i n i If the gien pressure p is greater than p ( ) s calculated from the gien entropy s, then the state point to be calculated is located in region 3, otherwise it is in the two-phase region 4 (see Figure 4). Computer-program erification. o assist the user in computer-program erification of Eq. (), able 20 contains test alues for calculated pressures. able 20. Selected pressure alues calculated from Eq. () a Equation s / Numerical Consistency with the Saturation-Pressure Equation of IAPWS-IF. he imum percentage deiation between the pressure calculated from the boundary equation p ( h ), Eq. (0), and the IAPWS-IF saturation-pressure equation p sat ( ) has the following alue ( ) sat( ) ( ) p p h p = = %. p p sat kj kg K p / MPa p s, Eq. () ( ) 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 p ( ) the IAPWS-IF saturation-pressure equation ( ) sat( ) ( ) sat p sat ( ) has the following alue p p s p = = %. p p s, Eq. (), and Consistency of the Backward Equations (p,h), (p,h), (p,s), and (p,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 ( p, h ), Eq. (2), and 3b ( p, h ), Eq. (3), and the IAPWS-IF basic equation f 3 (, ) along the boundary equation p( h ), Eq. (0), in comparison with the permissible differences are listed in able 2. he temperature differences were calculated as = 3 p( h 3 ), h 3. he function 3 represents the calculation of ( p, h ) using the backward equations of subregions 3a and 3b, Eqs. (2) and (3). able 2. 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. (0), and related permissible alues ( ) Subregion Equation 3a (2) 25 mk 0.47 mk 3b (3) 25 mk 0.46 mk Subregion Equation / / 3a (4) 0.0 % % 3b (5) 0.0 % % Furthermore, able 2 contains the imum percentage differences of specific olume between the backward equations 3a ( p, h ), Eq. (4), and 3b (, ) IF basic equation f along the boundary equation p ( ) 3 (, ) p h, Eq. (5), and the IAPWS- h, Eq. (0). he relatie ( ) differences of specific olume were calculated as ( ) function 3 represents the calculation of ( ) / = p h, h. he p, 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 p ( ) h, Eq. (0), is sufficient.

21 2 he imum temperature differences between the backward equations 3a (, ) and 3b ( p, s ), Eq. (7), and the IAPWS-IF basic equation 3 (, ) equation p ( ) p s, Eq. (6), along the boundary s, Eq. (), in comparison with the permissible differences are listed in able 22. he temperature differences were calculated as = 3 p( s 3 ), s 3. he function 3 represents the calculation of ( p, 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. (), and related permissible alues ( ) Subregion Equation 3a (6) 25 mk 2.69 mk 3b (7) 25 mk 2.2 mk Subregion Equation / / 3a (8) 0.0 % % 3b (9) 0.0 % % Furthermore, able 22 contains the imum percentage differences of specific olume between the backward equations 3a ( p, s ), Eq. (8), and 3b (, ) IF basic equation f along the boundary equation p ( ) 3 (, ) p s, Eq. (9), and the IAPWS- s, Eq. (). he relatie ( ) differences of specific olume were calculated as ( ) function 3 3a and 3b, Eqs. (8) and (9). represents the calculation of ( ) / = p s, s. he p, s using the backward equations of subregions he imum temperature differences and the imum relatie differences of specific olume are smaller than the permissible alues. herefore, the numerical consistency of the p s, Eq. (), is sufficient. boundary equation ( ) 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 (, ) p h and (, ) p s. In IAPWS-IF, time-consuming iterations, e.g., the Newton method, are required. By using equations p( h ), Eq. (0), and p( s ), Eq. (), the calculation to determine the region is about 60 times faster than that of the two-dimensional Newton method.

22 22 5 References [] International Association for the Properties of Water and Steam, Release on the IAPWS Industrial Formulation 9 for the hermodynamic Properties of Water and Steam, IAPWS Secretariat, Electric Power Research Institute, Palo Alto, CA, 9. [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 9 for the hermodynamic Properties of Water and Steam, Journal of Engineering for Gas urbines and Power, Vol. 22, 2000, pp [3] International Association for the Properties of Water and Steam, Supplementary Release on Backward Equations for Pressure as a Function of Enthalpy and Entropy p(h,s) to the IAPWS Industrial Formulation 9 for the hermodynamic Properties of Water and Steam, IAPWS Secretariat, Electric Power Research Institute, Palo Alto, CA, 200. [4] Kretzschmar, H.-J., Cooper, J. R., Dittmann, A., Friend, D. G., Gallagher, J., Knobloch, K., Mareš, R., Miyagawa, K., Stöcker, I., rübenbach, J., Wagner, W., and 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, Journal of Engineering for Gas urbines and Power, in press. [5] Kretzschmar, H.-J., Cooper, J. R., Dittmann, A., Gallagher, J. S., Friend, D. G., Harey, A. H., Knobloch, K., Mareš, R., Miyagawa, K., Okita, N., Stöcker, I., Wagner, W., and Weber, I., Supplementary Backward Equations (p,h), (p,h), and (p,s), (p,s) for the Critical and Supercritical Regions (Region 3) of the Industrial Formulation IAPWS-IF for Water and Steam, Journal of Engineering for Gas urbines and Power, in preparation. [6] Kretzschmar, H.-J., Specifications for the Supplementary Backward Equations (p,h) and (p,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 200, ed. by B. Dooley, Electric Power Research Institute, Palo Alto, CA, 200, 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 994, ed. by B. Dooley, Electric Power Research Institute, Palo Alto, CA, 994, 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 3th 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 99, ed. by B. Dooley, Electric Power Research Institute, Palo Alto, CA, 99, pp [0] International Association for the Properties of Water and Steam, IAPWS Release on the Values of emperature, Pressure and Density of Ordinary and Heay Water Substances at their Respectie Critical Points (Reised 992), IAPWS Secretariat, Electric Power Research Institute, Palo Alto, CA. 992.