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 Vejle, Denmark August 200 Supplementary Release on Backward Equations for the Functions (p,h), (p,h) and (p,s), (p,s) for Region of the IAPWS Industrial Formulation 9 for the hermodynamic Properties of Water and Steam 200 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 Koichi Watanabe Department of System Design Engineering Keio Uniersity -4- Hiyoshi, Kohoku-ku Yokohama , Japan Executie Secretary: Dr. R. B. Dooley Electric Power Research Institute 42 Hilliew Aenue Palo Alto, California 9404, USA his release contains 8 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 Vejle, Denmark, August, 200, 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 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( h, s ) to the Industrial Formulation 9 for the hermodynamic Properties of Water and Steam" (referred to here as IAPWS-IF-S0) [, 4]. Further details concerning the equations can be found in the corresponding article by H.-J. Kretzschmar et al. [5]. Further information concerning this supplementary release, IAPWS-IF, IAPWS-IF- S0, and other releases issued by IAPWS can be obtained from the Executie Secretary of IAPWS or from

2 2 Contents Nomenclature 2 2 Background Numerical Consistency Requirements 4 4 Structure of the Equation Set 5 5 Backward Equations (p,h) and (p,h) for Subregions a and b 7 5. he Equations (p,h) he Equations (p,h) 9 5. Numerical Consistency with the Basic Equation of IAPWS-IF 5.4 Consistency at Boundary Between Subregions 6 Backward Equations (p,s) and (p,s) for Subregions a and b 2 6. he Equations (p,s) he Equations (p,s) 4 6. Numerical Consistency with the Basic Equation of IAPWS-IF Consistency at Boundary Between Subregions 6 7 Computing ime in Relation to IAPWS-IF 7 8 References 7 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 = / * 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 Region a Subregion a b Subregion b ab Boundary between subregions a and b 4 Region 4 5 Region 5 B2 Boundary between regions 2 and c Critical point it Iterated quantity Maximum alue of a quantity RMS Root-mean-square alue of a quantity sat Saturation state olerated alue of a quantity a Note: denotes absolute temperature on the International emperature Scale of 990 (IS-90).

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 ( ph, ) and (, ) ps alid in the liquid region and the apor region 2; see Figure. IAPWS-IF was supplemented by "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" [, 4], which we will refer to as IAPWS- IF-S0, including equations for the backward function p( hs, ) alid in region and region 2. p / MPa p B2 ( ) IAPWS-IF 0 IAPWS-IF-S0 50 g p 0 ( p, ) ( ph, ) ( ps, ) ( hs, ) f (, ) ( ph, ) ( ps, ) ( ph, ) ( ps, ) g p ( p, ) ( ph, ) ( ps, ) ( hs, ) t c 4 p sat sat ( ) ( p) g ( p) 5, 5 Figure / K Regions and equations of IAPWS-IF, IAPWS-IF-S0, and the backward,,, ps, of this release equations ( ph ), ( ph ), and ( ps ), ( ) In modeling steam power cycles, thermodynamic properties as functions of the ariables ( ph, ) or (, ) IAPWS-IF, because they require two-dimensional iterations using the functions (, ) h(, ) or p(, ), (, ) equation (, ) ps are also required in region. It is difficult to perform these calculations with p, s that can be explicitly calculated from the fundamental region f. While these calculations are not frequently required in region, 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 ( ph, ), ( ph, ) and ( ps, ), (, ) ps, see Figure. With temperature and specific

4 4 olume calculated from the backward equations, the other properties in region can be calculated using the IAPWS-IF basic equation f ( ),. he numerical consistency with the IAPWS-IF basic equation f ( ), of and calculated from these backward equations is 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 f ( ) In these cases, the equations ( ph ), ( ph ) and ( ps ), ( ) calculating ery accurate starting alues. he backward equations ( ph ), ( ph ) and ( ps ), ( ), may be necessary. ps can be used for ps can only be used in their ranges of alidity described in Section 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 equations ( ph ), ( ph ) and ( ps ), ( ) corresponding alues by iterations from the basic equation of IAPWS-IF. Numerical Consistency Requirements he permissible alue for the numerical consistency functions ( ph, ) and (, ) f ps with the basic equation ( ) IAPWS [6, 7] as a result of an international surey. he permissible alue ( ) ps or to calculate the = 25mK of the backward, was determined by for the numerical consistency for the equations ( ) ps can be estimated from the total differentials = + h h h = + s and s s, ph 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 difference is for both functions ( ph ) and ( ) =0.0% At the critical point c = K, c = ( 22 kg m ) ps.. /, more stringent consistency requirements were arbitrarily set. hese were = 0.49mK and =0.000%.

5 5 able. Numerical consistency alues ( ps, ), alues required for (, ) h, s ph and ( ps ) Region 25 mk 80 of [6] required for ( ) of [9], and resulting erances ph and h s Jkg 0. Jkg K 0.0 % Critical Point 0.49 mk % 4 Structure of the Equation Set he equation set consists of backward equations ( ph, ), ( ph, ) and ( ps, ), (, ) region. Region is defined by: B K 86.5 K and p ( ) p 00 MPa, ps for where p B2 represents the B2 equation of IAPWS-IF. Figure 2 shows the way in which region is diided into the two subregions a and b. p / MPa a ( ph) ( ph) ( ps) ( ps) h ab s c ( p) K ( ph) ( ps) ( ps ) ( ph) b p ( ) sat c p B2 ( ) / K Figure 2. Diision of region into two subregions a and b for the backward,,, ps, equations ( ph ), ( ph ) and ( ps ), ( )

6 6 able 2 shows the decisions which hae to be made in order to find the correct subregion for the functions ( ph, ), ( ph, ) and ( ps, ), (, ) ps. able 2. Criteria for finding the correct subregion, a or b, for the backward functions,,, ps, ( ph ), ( ph ) and ( ps ), ( ) Backward Functions ( ph, ), ( ph, ) Backward Functions ( ps, ), ( ps, ) for p < for p Subregion Subregion a b a b p c : ' '' h h ( p ) h ( ) p : h h ( p ) h > ( ) c ab h p for p < c hab p for p c p : ' '' s s ( p ) s s ( p ) p : s s c s > s c For pressures less than the critical pressure p c = MPa, the saturation line is the boundary between subregions a and b. hat means for the functions ( ph, ) and (, ) ' if the gien specific enthalpy h is less than or equal to h ( ) ph, p calculated from the gien pressure p on the saturated liquid line, then the point of state to be calculated is located in '' subregion a. If the gien enthalpy h is greater than or equal to h ( ) p calculated on the saturated apor line, then the point of state is located in subregion b. Otherwise, the point is in the two-phase region. In that case, the saturation temperature equation sat ( ) basic equation f ( ) p and the, of IAPWS-IF can be used to calculate the temperature and the specific olume from the gien pressure and the gien enthalpy. he decisions are analogous for the functions ( ps, ) and (, ) ps. For pressures greater than or equal to corresponds to the critical isentropic line (, ) p c, the boundary between the subregions a and b s s ps, and =, see Figure 2. For the functions ( ) c ps, 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 a; otherwise it is in subregion b. In order to decide which ( ph, ), (, ) of p and h, the boundary equation h ( ) ph equation, a or b, must be used for gien alues ab equation is a polynomial of the third degree and reads where η = hh and π h ( p) p, Eq. (), has to be used, see Figure 2. his ab 2 ηπ ( ) n n2π nπ n4π h = p p with = = + + +, () h = kjkg and p = MPa. he coefficients n to n 4 h p is from the critical point to of Eq. () are listed in able. he range of the equation ( ) 00 MPa. he related temperature at 00 MPa is = ab K. Equation () does not exactly describe the critical isentropic line. he imum specific entropy deiation was determined as

7 where 7 ( ( ) ( ) ), ( ),, ( ) 0.66Jkg K were obtained by iterations using the deriaties p (, ) and s (, ) s = s ph p ph p s =, it and ab it ab it ab c it of the IAPWS-IF basic equation for region. able. Numerical alues of the coefficients of the equation h ( ) its dimensionless form, Eq. (), for defining the boundary between subregions a and b a ab p in i n i i n i a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. If the gien specific enthalpy h is greater than hab( p ) calculated from the gien pressure p, then the state point to be calculated is located in subregion b, otherwise it is in subregion a (see Figure 2). Note, Eq. () does not correctly simulate the isentropic line p c. Howeer, the calculated alues hab( p ) are not higher than h ( ) ' h ( p ). For computer-program erification, Eq. () gies the following p-h point: p = 25 MPa, ( ) = kj kg h p. ab 5 Backward Equations ( ph, ) and (, ) he backward equation ( ) where a θ =, π 5. he Equations ( ph, ) s = s at pressures lower than c '' p and not lower than ph for Subregions a and b ph for subregion a has the following dimensionless form: ( ph, ) I Ji = θ a( πη, ) = n ( 0.240) i i π ( η 0.65) +, (2) = p p, and η i= = = = hh with 760K, p 00MPa, and h = 200kJkg. he coefficients n i and exponents I i and J i of Eq. (2) are listed in able 4. he backward equation ( ) b ph for subregion b reads in its dimensionless form ( ph, ) I Ji = θ b( πη, ) = ( ) i * ni π ( η 0.720), () i=

8 8 where θ =, π = p p, and η = = = hh with 860K, p 00MPa, and h = 2800kJkg. he coefficients n i and exponents I i and J i of Eq. () are listed in able 5. Computer-program erification o assist the user in computer-program erification of Eqs. (2) and (), able 6 contains test alues for calculated temperatures. able 4. Coefficients and exponents of the backward equation ( ) dimensionless form, Eq. (2) ph for subregion a in its i I i J i n i i I i J i n i able 5. Coefficients and exponents of the backward equation ( ) dimensionless form, Eq. () ph for subregion b in its i I i J i n i i I i J i n i

9 9 able 6. Selected temperature alues calculated from Eqs. (2) and () a he backward equation ( ) where ω =, π a 5.2 he Equations ( ph, ) ph for subregion a has the following dimensionless form:, 2 i i Ji i= ( ph, ) = ω a( πη) = n ( π 0.28) I ( η ), (4) = p p, and η = hh with = = m kg, p 00MPa, and h = 200kJkg. he coefficients n i and exponents I i and J i of Eq. (4) are listed in able 7. he backward equation ( ) where ω =, π b Equation p / MPa h / ph for subregion b reads in its dimensionless form, 0 i i Ji i= ( ph, ) = ω b( πη) = n ( π 0.066) I ( η ), (5) = p p, and η = hh with kjkg / K a ( ph, ), Eq. (2) b ( ph, ), Eq. () a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. = = m kg, p 00MPa, and h = 2800kJkg. he coefficients n i and exponents I i and J i of Eq. (5) are listed in able 8. Computer-program erification o assist the user in computer-program erification of Eqs. (4) and (5), able 9 contains test alues for calculated specific olumes.

10 0 able 7. Coefficients and exponents of the backward equation ( ) dimensionless form, Eq. (4) ph for subregion a in its i I i J i n i i I i J i n i able 8. Coefficients and exponents of the backward equation ( ) dimensionless form, Eq. (5) ph for subregion b in its i I i J i n i i I i J i n i

11 able 9. Selected specific olume alues calculated from Eqs. (4) and (5) a Equation p / MPa h / kjkg / m kg a ( ph, ), Eq. (4) b ( ph, ), Eq. (5) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. 5. 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 () and the IAPWS-IF basic equation f (, ) in comparison with the permissible differences are listed in able 0. he calculation of the root-mean-square alues is described in Section. able 0 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 ( ph, ) and ( ph, ). able 0. Maximum differences and root-mean-square differences of the temperature calculated from Eqs. (2) and () and specific olume calculated from Eqs. (4) and (5) to the IAPWS-IF basic equation f (, ) and related permissible alues Subregion Equation RMS a (2) 25 mk 2.6 mk 0.5 mk b () 25 mk 9.6 mk 9.6 mk Subregion Equation RMS a (4) 0.0 % % % b (5) 0.0 % % % 5.4 Consistency at Boundary Between Subregions he imum temperature difference between the two backward equations, Eq. (2) and Eq. (), along the boundary hab( ) ( ph ( p) ) ( ph ( p) ) p, Eq. (), has the following alue =,, = 0.7mK. a ab b ab

12 2 hus, the temperature differences between the two backward functions (, ) ph 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 ( ph, ) of the adjacent subregions a and b 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) ph, ( p) b ab ( ) ( ) a phab b phab = = %. 6 Backward Equations ( ps, ) and (, ) he backward equation ( ) where s a θ =, π = 4.4kJkg able. K 6. he Equations ( ps, ) ab ps for Subregions a and b ps for subregion a has the following dimensionless form: ( ps, ) Ii Ji = θ a( πσ, ) = ni ( π 0.240) ( σ ), (6) = p p, and σ he backward equation ( ) where s i= = = = s s with 760K, p 00MPa, and. he coefficients n i and exponents I i and J i of Eq. (6) are listed in b θ =, π = 5.kJkg able 2. K ps for subregion b reads in its dimensionless form 28 ( ps, ) Ii Ji = θ b( πσ, ) = ni ( π 0.760) ( σ ), (7) = p p, and σ Computer-program erification i= = = = s s with 860K, p 00MPa, 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 contains test alues for calculated temperatures.

13 able. Coefficients and exponents of the backward equation ( ) dimensionless form, Eq. (6) ps for subregion a in its i I i J i n i i I i J i n i able 2. Coefficients and exponents of the backward equation ( ) dimensionless form, Eq. (7) ps for subregion b in its i I i J i n i i I i J i n i

14 4 able. Selected temperature alues calculated from Eqs. (6) and (7) a Equation p / MPa s / kjkg K / K a ( ps, ), Eq. (6) b ( ps, ), Eq. (7) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. he backward equation ( ) where ω s =, π a = 4.4kJkg able 4. K 6.2 he Equations ( ps, ) ps for subregion a has the following dimensionless form: 28 ( ps, ) Ii Ji = ω a( πσ, ) = ni ( π 0.87) ( σ ), (8) = p p, and σ he backward equation ( ) where ω s =, π = 5.kJkg able 5. K i= = s s with = = m kg, p 00MPa, and. he coefficients n i and exponents I i and J i of Eq. (8) are listed in b ps for subregion b reads in its dimensionless form ( ps, ) Ii Ji = ω b( πσ, ) = ni ( π 0.298) ( σ ), (9) = p p, and σ Computer-program erification i= = s s with = = m kg, p 00MPa, 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 6 contains test alues for calculated specific olumes.

15 5 able 4. Coefficients and exponents of the backward equation ( ) dimensionless form, Eq. (8). ps for subregion a in its i I i J i n i i I i J i n i able 5. Coefficients and exponents of the backward equation ( ) dimensionless form, Eq. (9) ps for subregion b in its i I i J i n i i I i J i n i

16 6 able 6. Selected specific olume alues calculated from Eqs. (8) and (9) a Equation p / MPa s / kjkg K / m kg a ( ps, ), Eq. (8) b ( ps, ), Eq. (9) a It is recommended that programmed functions be erified using 8 byte real alues for all ariables. 6. 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 (, ) in comparison with the permissible differences are listed in able 7. able 7 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 ( ps, ) and ( ps, ). able 7. 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) to the IAPWS-IF basic equation f (, ) and related permissible alues Subregion Equation RMS a (6) 25 mk 24.8 mk.2 mk b (7) 25 mk 22. mk 0. mk Subregion Equation RMS a (8) 0.0 % % % b (9) 0.0 % % % 6.4 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 a ( ps ) ( ps) =,, = 0.09mK. c b c

17 7 hus, the temperature differences between the two backward functions (, ) ps 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 ( ps, ), Eqs. (8) and (9), of the adjacent subregions are also smaller than the numerical consistencies 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) b ( ps, c) ( ps, ) a = = %. b c 7 Computing ime in Relation to IAPWS-IF A ery important motiation for the deelopment of the backward equations ( ph, ), ( ph, ) and ( ps, ), ( ps, ) for region was reducing the computing time to obtain thermodynamic properties and differential quotients from gien ariables ( ph, ) and ( ps, ). In IAPWS-IF, time-consuming iterations, e.g., the 2-dimensional Newton method, are required. Using the ( ph ), ( ph ), ( ps ) and ( ) is about 20 times faster than that of the 2-dimensional Newton method. ps equations, the calculation speed 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. 8 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 [] 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.

18 8 [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, submitted. [5] Kretzschmar, H.-J., Cooper, J. R., Dittmann, A., Gallagher, J., Friend, D. G., Harey, A. H., Knobloch, K., Mareš, R., Miyagawa, K., Okita, N., Stöcker, I., Wagner, W., and Weber, I., Supplementary Backward Equations for (p,h), (p,h), and (p,s), (p,s) for the Critical and Supercritical Regions to 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 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 th 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