The International Association for the Properties of Water and Steam

IAPWS SR5-05(2016) he International Association for the Proerties of Water and Steam Moscow, Russia June 2014 Revised Sulementary Release on Backward Equations for Secific Volume as a Function of Pressure and emerature v(,) for Region 3 of the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam 2014 International Association for the Proerties of Water and Steam Publication in whole or in art is allowed in all countries rovided that attribution is given to the International Association for the Proerties of Water and Steam President: Professor amara Petrova Moscow Power Engineering Institute Moscow, Russia Executive Secretary: Dr. R. B. Dooley Structural Integrity Associates Southort, Merseyside, UK Email: bdooley@structint.com his revised sulementary release relaces the corresonding sulementary release of 2005, and contains 35 ages, including this cover age. his revised sulementary release has been authorized by the International Association for the Proerties of Water and Steam (IAPWS) at its meeting in Moscow, Russia, 22-27 June, 2014, for issue by its Secretariat. he members of IAPWS are: Britain and Ireland, Canada, the Czech Reublic, Germany, Jaan, Russia, Scandinavia (Denmark, Finland, Norway, Sweden), and the United States, and associate members Argentina & Brazil, Australia, France, Greece, Italy, New Zealand, and Switzerland. he backward equations v, for Region 3 rovided in this release are recommended as a sulement to "he IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam" (IAPWS-IF) [1, 2]. Further details concerning the equations of this revised sulementary release can be found in the corresonding article by H.-J. Kretzschmar et al. [3]. his revision consists of edits to clarify descritions of how to determine the region or subregion; the roerty calculations are unchanged. Further information concerning this sulementary release, other releases, sulementary releases, guidelines, technical guidance documents, and advisory notes issued by IAPWS can be obtained from the Executive Secretary of IAPWS or from htt://www.iaws.org.

2 Contents 1 Nomenclature 3 2 Background 4 3 Numerical Consistency Requirements 5 4 Structure of the Equation Set 5 5 Backward Equations v(,) for the Subregions 3a to 3t 6 5.1 Subregions 6 5.2 Backward Equations v(,) for the Subregions 3a to 3t 12 5.3 Calculation of hermodynamic Proerties with the v(,) Backward Equations 13 5.4 Numerical Consistency 14 6 Auxiliary Equations v(,) for the Region very close to the Critical Point 17 6.1 Subregions 17 6.2 Auxiliary Equations v(,) for the Subregions 3u to 3z 19 6.3 Numerical Consistency 20 7 Comuting ime in Relation to IAPWS-IF 21 8 Alication of the Backward and Auxiliary Equations v(,) 22 9 References 22 Aendix 24

hermodynamic quantities: c Secific isobaric heat caacity f Secific Helmholtz free energy h Secific enthaly Pressure s Secific entroy Absolute temerature a v Secific volume w Seed of sound Reduced temerature / * Reduced ressure, / * Reduced volume, v/v * Difference in any quantity 3 1 Nomenclature Subscrits: 1 5 Region 1 5 3a 3z Subregion 3a 3z 3ab Boundary between subregions 3a, 3d and 3b, 3e 3cd Boundary between subregions 3c and 3d, 3g, 3l, 3q, 3s 3ef Boundary between subregions 3e, 3h, 3n and 3f, 3i, 3o 3gh Boundary between subregions 3g, 3l and 3h, 3m 3ij Boundary between subregions 3i, 3 and 3j 3jk Boundary between subregions 3j, 3r and 3k 3mn Boundary between subregions 3m and 3n 3o Boundary between subregions 3o and 3 3qu Boundary between of subregion 3q and 3u 3rx Boundary between of subregion 3r and 3x 3uv Boundary between subregions 3u and 3v 3wx Boundary between subregions 3w and 3x B23 Boundary between regions 2 and 3 c Critical oint it Iterated quantity max Maximum value of a quantity RMS Root-mean-square value of a quantity sat Saturation state tol olerated value of a quantity Root-mean-square value: 1 x ( x ) RMS N N n 1 n 2 Suerscrits: Quantity or equation of IAPWS-IF 01 Equation of IAPWS-IF-S01 03 Equation of IAPWS-IF-S03rev 04 Equation of IAPWS-IF-S04 * Reducing quantity ' Saturated liquid state " Saturated vaor state where x n can be either absolute or ercentage difference between the corresonding quantities x; N is the number of x n values (10 million oints uniformly distributed over the range of validity in the - lane). a Note: denotes absolute temerature on the International emerature Scale of 1990 (IS-90).

4 2 Background he IAPWS Industrial Formulation 19 for the thermodynamic roerties of water and steam (IAPWS-IF) [1, 2] contains basic equations, saturation equations and equations for the frequently used backward functions, s valid in the liquid region 1 and the vaor region 2; see Figure 1. IAPWS-IF was sulemented by "Sulementary Release h and, on Backward Equations for Pressure as a Function of Enthaly and Entroy hs, to the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam" [4, 5], which will be referred to as IAPWS-IF-S01. hese equations are valid in region 1 and region 2. An additional "Sulementary Release on Backward Equations for the Functions, v s for Region 3 of the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam" [6, 7], which will be referred to as IAPWS-IF-S03rev, was adoted by IAPWS in 2003 and revised in 2004. In 2004, IAPWS-IF was sulemented by "Sulementary Release on Backward Equations hs, for Region 3, Equations as a Function of h and s for the Region Boundaries, and an Equation h, v, h and, s,, sat h, s for Region 4 of the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam" (referred to here as IAPWS-IF-S04) [8, 9]. / MPa 100 50 22.064 16.5292 10 0.000611 g 1 1 1 01 1 623.15 863.15 3 B23 f 3 v, 1 2 v3,, 03 g 2, 3, h, h 03, s 2, h, s h, s v v c 4 3 03 3 03 3 04 3, h, s h, s sat sat 04 sat, h s 2 01 2, s h, s IAPWS-IF 01 IAPWS-IF-S01 03 IAPWS-IF-S03rev 04 IAPWS-IF-S04 50 MPa g 5 5, Figure 1. 273.15 647.096 1073.15 2273.15 / K Regions and equations of IAPWS-IF, IAPWS-IF-S01, IAPWS-IF-S03rev, v of this release IAPWS-IF-S04, and the equations 3,

5 IAPWS-IF region 3 is covered by a basic equation for the Helmholtz free energy f v,. All thermodynamic roerties can be derived from the basic equation as a function of secific volume v and temerature. However, in modeling some steam ower cycles, thermodynamic roerties as functions of the variables, are required in region 3. It is cumbersome to erform these calculations with IAPWS-IF, because they require iterations of v from and using the function v, derived from the IAPWS-IF basic equation f v,. In order to avoid such iterations, this release rovides equations 3, With secific volume v calculated from the equations v3, region 3 can be calculated using the IAPWS-IF basic equation f v,. v ; see Figure 1., the other roerties in For rocess calculations, the numerical consistency requirements for the equations v, are very strict. Because the secific volume in the - lane has a comlicated structure, including an infinite sloe at the critical oint, region 3 was divided into 26 subregions. he first 20 subregions and their associated backward equations, described in Section 5, cover almost all of region 3 and fully meet the consistency requirements. For a small area very near the critical oint, it was not ossible to meet the consistency requirements fully. his nearcritical region is covered with reasonable consistency by six subregions with auxiliary equations, described in Section 6. 3 Numerical Consistency Requirements he ermissible value for the numerical consistency of the equations for secific volume with the IAPWS-IF fundamental equation was determined based on the required accuracy of the iteration otherwise used. he iteration accuracy deends on thermodynamic rocess calculations. o obtain secific enthaly or entroy from ressure and temerature in region 3 with a maximum deviation of 0.001 % from IAPWS-IF, and isobaric heat caacity or seed of sound with a maximum deviation of 0.01 %, a relative accuracy of vv 0.001 % is sufficient. herefore, the ermissible relative tolerance for the equations v, was set to vv 0.001 %. tol 4 Structure of the Equation Set he range of validity of the equations v he function B23 3, is region 3 defined by: 623.15 K < 863.15 K and < 100 MPa. B23 reresents the B23-equation of IAPWS-IF.

v 6 It roved to be infeasible to achieve the numerical consistency requirement of 0.001 % for 3, using simle functional forms in the region 3qu for 3rx 643.15 K 22.5 MPa ; see Figure 2. sat his limitation is due to the infinite sloe of the secific volume at the critical oint. In order to cover region 3 comletely, Section 6 contains auxiliary equations for this small region very close to the critical oint. Figure 2 shows the range of validity of the backward and auxiliary equations. 100 / MPa Backward Equations Auxiliary Equations B23 22.5 22.064 21.0434 sat 3qu c 3rx 16.5292 Figure 2. 646.919 650.1 623.15 643.15 647.096 / K 863.15 Range of validity of the backward and auxiliary equations. he area marked in gray is not true to scale but enlarged to make the small area better visible. 5 Backward Equations v(,) for the Subregions 3a to 3t 5.1 Subregions Preliminary investigations showed that it was not ossible to meet the numerical consistency requirement with only a few v, equations. herefore, the main art of region 3 was divided into 20 subregions 3a to 3t; see Figures 3 and 4.

/ MPa 40 3a 3ab 686.387 K 713.959 K 3ef 3b 732.505 K 3cd 3c 3d 3e 3f 3ab B23 7 25 23.5 22.5 23 22.064 21.0434 20.5 19.0088 16.5292 3qu sat 639.673 K 3s 634.659 K 3q 640.961 K 3t c 3r 3rx 652.8 K 3jk 3k Enlargement: Figure 4 623.15 643.15 647.096 663.15 683.15 703.15 723.15 Figure 3. Division of region 3 into subregions for the backward equations v 3, / K

25.5 / MPa 3d 3ab ( ) 3e 3ef 3f 25 24.5 3c 3cd ( ) 649.366 K 3ij 3gh 656.698 K 657.229 K 660.787 K 668.192 K 3jk ( ) 676.810 K 24 3g 3h 3i 3ef 3j B23 ( ) 8 23.5 645.6 K 651.570 K 3k 23 22.5 22.2 643.248 K 3qu ( ) 3l 3mn ( ) 3q 649.887 K 651.578 K 3o( ) 3n 3m 3o 3 646.919 K 650.1 K 3r 652.800 K 3rx ( ) : 648.071 K : 648.581 K : 648.721 K : 648.826 K : 649.298 K : 650.286 K : 650.800 K 633.15 643.15 653.15 663.15 673.15 / K 683.15 Figure 4. Enlargement from Figure 3 for the subregions 3c to 3r for the backward equation v,

9 he subregion boundary equations, excet for 3ab ( ), 3ef ( ), and 3o ( ), have the following dimensionless form: N I i * ni, (1) i1 where,, with 1K, 1MPa. he equations 3ab( ) and 3o ( ) have the form: and 3ef ( ) * N n ln i1 has the form: 3ef ( ) sat i Ii, (2) 3ef 22.064 647.096, (3) c where sat 3.727 888 004. c he coefficients n i and the exonents I i of the boundary equations are listed in able 1. able 1. Numerical values of the coefficients of the equations for subregion boundaries (excet 3ef ( ) ) Equation i I i n i i I i n i 3ab 3cd 3gh 3ij 3jk 3mn 3o 3qu 3rx he following descrition of the use of the subregion boundary equations is summarized in able 2 and Figures 3 and 4.

able 2. Pressure ranges and corresonding subregion boundary equations for determining the correct subregion, 3a to 3t, for the backward equations v, Pressure Range Sub- For Sub- For region region 40 MPa < 100 MPa 3a 3ab ( ) 3b 3ab ( ) 10 25 MPa < 40 MPa 3c 3cd ( ) 3e 3ab ( ) 3ef ( ) ( ) ( ) 3f 3ef ( ) 3d 3cd 3ab 23.5 MPa < 25 MPa 3c 3cd ( ) 3i 3ef ( ) 3ij( ) 3g 3cd ( ) 3gh ( ) 3j 3ij( ) 3jk ( ) ( ) ( ) 3k 3jk ( ) 3h 3gh 3ef 23 MPa < 23.5 MPa 3c 3cd ( ) 3i 3ef ( ) 3ij( ) 3l 3cd ( ) 3gh ( ) 3j 3ij( ) 3jk ( ) ( ) ( ) 3k 3jk ( ) 3h 3gh 3ef 22.5 MPa < 23 MPa 3c 3cd ( ) 3o 3ef ( ) 3o ( ) sat 643.15 K 3l 3cd 3gh ( ) ( ) 3 3o ( ) 3ij( ) ( ) ( ) 3j 3ij( ) 3jk ( ) ( ) ( ) 3k 3jk ( ) 3m 3gh 3mn 3n 3mn 3ef < 22.5 MPa 3c 3cd ( ) 3r 3rx ( ) 3jk ( ) 20.5 MPa < b 3cd ( ) ( ) 3k 3jk ( ) 3q 3cd 3qu sat 643.15 K 3c 3cd ( ) 3r 3s 3cd sat sat ( ) ( ) ( ) ( ) 3k 3jk ( ) 20.5 MPa 3c 3cd ( ) 3t 3s ( ) ( ) < b 3cd sat 623.15 K 3c b 1 3cd = 1.900 881 189 173 929 10 MPa 3cd sat sat ( ) 3t sat ( ) sat ( ) he equation 3ab ( ) aroximates the critical isentroe from 25 MPa to 100 MPa and reresents the boundary equation between subregion 3a and subregion 3d. he equation 3cd ( ) ranges from 3cd = 1.900 881 189 173 929 10 1 MPa to 40 MPa. he ressure of 3cd = 1.900 881 189 173 929 10 1 MPa is given as sat ( ) 3cd ( ) 0. he equation 3cd ( ) reresents the boundary equation between subregions 3d, 3g, 3l, 3q or 3s, and subregion 3c. he equation 3gh ( ) ranges from 22.5 MPa to 25 MPa and reresents the boundary equation between subregions 3h or 3m and subregions 3g or 3l. 3 1 he equation 3ij ( ) aroximates the isochore v 0.0041 m kg from 22.5 MPa to 25 MPa and reresents the boundary equation between subregion 3j and subregions 3i or 3. he equation 3jk ( ) v v'' 20.5 MPa from 20.5 MPa to 25 MPa and reresents the boundary equation between subregion 3k and subregions 3j or 3r. aroximates the isochore 3jk

11 3 1 he equation 3mn ( ) aroximates the isochore v 0.0028 m kg from 22.5 MPa to 23 MPa and reresents the boundary equation between subregion 3n and subregion 3m. 3 1 he equation 3o ( ) aroximates the isochore v 0.0034 m kg from 22.5 MPa to 23 MPa and reresents the boundary equation between subregion 3 and subregion 3o. he equation 3qu ( ) aroximates the isochore v v'643.15k sat 643.15 K, where 1 from sat 643.15 K 2.104 336 732 10 MPa to 22.5 MPa and reresents the boundary equation between subregion 3q and subregion 3r (see Fig. 5). he equation 3rx ( ) aroximates the isochore v v'' 643.15 K sat 643.15 K, where 1 from sat 643.15 K 2.104 336 732 10 MPa, to 22.5 MPa and reresents the boundary equation between subregion 3r and subregion 3x (see Fig.5). he subregion boundary equation 3ef ( ) is a straight line from 22.064 MPa to 40 MPa having the sloe of the saturation-temerature curve of IAPWS-IF at the critical oint. It divides subregions 3f, 3i or 3o from subregions 3e, 3h or 3n. Comuter-rogram verification o assist the user in comuter-rogram verification of the equations for the subregion boundaries, able 3 contains test values for calculated temeratures. able 3. Selected temerature values calculated from the subregion boundary equations c Equation Equation MPa K MPa K 3ab ( 3jk ( ) 3cd ( 3mn ( 3ef ( 3o ( 3gh ( 3qu ( ) 3ij ( 3rx ( c It is recommended that rogrammed functions be verified using 8 byte real values for all variables.

12 5.2 Backward Equations v(,) for the Subregions 3a to 3t he backward equations v, for the subregions 3a to 3t, excet for 3n, have the following dimensionless form: (, ) N v, c Ii d ni a b v i1 he equation for subregion 3n has the form: with vv v 3n (, ) 3n v,, and Ji I, ex n a b N i i Ji i1 e. (4), (5). he reducing quantities v,, and, the number of coefficients N, the non-linear arameters a and b, and the exonents c, d, and e are listed in able 4 for the equations of the subregions 3a to 3t. he coefficients n i and exonents I i and J i of these equations are listed in ables A1.1 to A1.20 of the Aendix. able 4. Reducing quantities v*, *, and *, number of coefficients N, non-linear arameters a and b, and exonents c, d, and e for the v, equations of the subregions 3a to 3t Subregion v* * * N a b c d e 3 1 m kg MPa K 3a 0.0024 100 760 30 0.085 0.817 1 1 1 3b 0.0041 100 860 32 0.280 0.779 1 1 1 3c 0.0022 40 690 35 0.259 0.903 1 1 1 3d 0.0029 40 690 38 0.559 0.939 1 1 4 3e 0.0032 40 710 29 0.587 0.918 1 1 1 3f 0.0064 40 730 42 0.587 0.891 0.5 1 4 3g 0.0027 25 660 38 0.872 0.1 1 1 4 3h 0.0032 25 660 29 0.898 0.983 1 1 4 3i 0.0041 25 660 42 0.910 0.984 0.5 1 4 3j 0.0054 25 670 29 0.875 0.964 0.5 1 4 3k 0.0077 25 680 34 0.802 0.935 1 1 1 3l 0.0026 24 650 43 0.908 0.989 1 1 4 3m 0.0028 23 650 40 1.00 0.9 1 0.25 1 3n 0.0031 23 650 39 0.6 0.9 - - - 3o 0.0034 23 650 24 0.4 0.996 0.5 1 1 3 0.0041 23 650 27 0.2 0.9 0.5 1 1 3q 0.0022 23 650 24 0.848 0.983 1 1 4 3r 0.0054 23 650 27 0.874 0.982 1 1 1 3s 0.0022 21 640 29 0.886 0.990 1 1 4 3t 0.0088 20 650 33 0.803 1.02 1 1 1 Comuter-rogram verification o assist the user in comuter-rogram verification of the equations for the subregions 3a to 3t, able 5 contains test values for calculated secific volumes.

13 able 5. Selected secific volume values calculated from the equations for the subregions 3a to 3t d Equation v Equation v MPa K 3 1 m kg MPa K 3 1 m kg v v v v v v v v v 3a, 3b, 3c, 3d, 3e, 3f, 3g, 3h, 3i, v 3j, v3k, v3l, v3m, v3n, v3o, v3, v3q, v3r, v3s, v3t, d It is recommended that rogrammed functions be verified using 8 byte real values for all variables. 5.3 Calculation of hermodynamic Proerties with the v(,) Backward Equations he v(, ) backward equations described in Section 5.2 together with IAPWS-IF basic equation f (, v ) make it ossible to determine all thermodynamic roerties, e.g., enthaly, entroy, isobaric heat caacity, seed of sound, from ressure and temerature in region 3 without iteration. he following stes should be made: - Identify the subregion (3a to 3t) for given ressure and temerature following the instructions of Section 5.1 in conjunction with able 2 and Figures 3 and 4. hen, calculate the secific volume v for the subregion using the corresonding backward equation v(, ). - Calculate the desired thermodynamic roerty from the reviously calculated secific volume v and the given temerature using the derivatives of the IAPWS-IF basic equation f ( v, ), where v v(, ) ; see able 31 in [1].

14 5.4 Numerical Consistency 5.4.1 Numerical Consistency with the Basic Equation of IAPWS-IF he maximum relative deviations and root-mean-square relative deviations of secific volume, calculated from the backward equations v(, ) for subregions 3a to 3t, from the IAPWS-IF basic equation f (, v ) in comarison with the ermissible tolerances are listed in able 6. he calculation of the root-mean-square values is described in Section 1. able 6 also contains the maximum relative deviations and root-mean-square relative deviations of secific enthaly, secific entroy, secific isobaric heat caacity, and seed of sound, calculated as described in Section 5.3. able 6. Maximum relative deviations and root-mean-square relative deviations of the secific volume, calculated from the backward equations for subregions 3a to 3t, and maximum relative deviations of secific enthaly, secific entroy, secific isobaric heat caacity and seed of sound, calculated as described in Section 5.3, from the IAPWS-IF basic equation f (, v ) Subregion v v h h s s c c w w % % % % % max RMS max RMS max RMS max RMS max RMS 3a 0.00061 0.00031 0.00018 0.00008 0.00026 0.00011 0.0016 0.0006 0.0015 0.0006 3b 0.00064 0.00035 0.00017 0.00008 0.00016 0.00008 0.0012 0.0003 0.0008 0.0003 3c 0.00080 0.00038 0.00026 0.00012 0.00025 0.00011 0.0059 0.0016 0.0023 0.0010 3d 0.00059 0.00025 0.00018 0.00008 0.00014 0.00006 0.0035 0.0010 0.0012 0.0004 3e 0.00072 0.00033 0.00018 0.00009 0.00014 0.00007 0.0017 0.0005 0.0006 0.0002 3f 0.00068 0.00020 0.00018 0.00005 0.00013 0.00004 0.0015 0.0003 0.0002 0.0001 3g 0.00047 0.00016 0.00014 0.00005 0.00011 0.00004 0.0032 0.0011 0.0010 0.0003 3h 0.00085 0.00044 0.00022 0.00012 0.00017 0.00009 0.0066 0.0018 0.0006 0.0002 3i 0.00067 0.00028 0.00018 0.00008 0.00013 0.00006 0.0019 0.0006 0.0002 0.0001 3j 0.00034 0.00019 0.00009 0.00005 0.00007 0.00004 0.0020 0.0006 0.0002 0.0001 3k 0.00034 0.00012 0.00008 0.00003 0.00007 0.00002 0.0018 0.0003 0.0002 0.0001 3l 0.00033 0.00019 0.00010 0.00006 0.00008 0.00005 0.0035 0.0015 0.0008 0.0004 3m 0.00057 0.00031 0.00015 0.00009 0.00011 0.00006 0.0062 0.0030 0.0006 0.0002 3n 0.00064 0.00029 0.00017 0.00008 0.00012 0.00006 0.0050 0.0013 0.0002 0.0001 3o 0.00031 0.00015 0.00008 0.00004 0.00006 0.00003 0.0007 0.0002 0.0001 0.0001 3 0.00044 0.00022 0.00012 0.00006 0.00009 0.00005 0.0026 0.0010 0.0002 0.0001 3q 0.00036 0.00018 0.00012 0.00006 0.00009 0.00005 0.0040 0.0016 0.0010 0.0005 3r 0.00037 0.00007 0.00010 0.00002 0.00008 0.00002 0.0030 0.0004 0.0002 0.0001 3s 0.00030 0.00016 0.00010 0.00005 0.00007 0.00004 0.0033 0.0015 0.0009 0.0005 3t 0.00095 0.00045 0.00022 0.00010 0.00018 0.00008 0.0046 0.0015 0.0004 0.0002 ermissible tolerance 0.001 0.001 0.001 0.01 0.01 able 6 shows that the deviations of the secific volume, secific enthaly, and secific entroy from the IAPWS-IF basic equation are less than 0.001 % and the deviations of secific isobaric heat caacity and seed of sound are less than 0.01 %. herefore, the values

15 of secific volume, secific enthaly and secific entroy of IAPWS-IF are reresented with 5 significant figures, and the values of secific isobaric heat caacity and seed of sound with 4 significant figures by using the backward equations v (, ). 5.4.2 Consistency at Boundaries Between Subregions he maximum relative differences of secific volume between the v (, ) backward equations of adjacent subregions along the subregion boundary ressures are listed in the third column of able 7. able 8 contains these maximum relative differences along the subregion boundary equations. able 7. Maximum relative deviations of secific volume between the backward equations v, of adjacent subregions and maximum relative deviations of secific enthaly, secific entroy, secific isobaric heat caacity, and seed of sound, calculated as described in Section 5.3, along the subregion boundary ressures Subregion Between v v h h max max max Boundary Subregions c c max w w max % % % % % = 40 MPa 3a, 3c 0.00074 0.00021 0.00028 0.0018 0.0019 3a, 3d 0.00060 0.00017 0.00013 0.0013 0.0006 3b, 3e 0.00062 0.00015 0.00012 0.0009 0.0004 3b, 3f 0.00078 0.00018 0.00014 0.0004 0.0002 = 25 MPa 3d, 3g 0.00056 0.00015 0.00011 0.0031 0.0010 3d, 3h 0.00056 0.00015 0.00011 0.0021 0.0003 3e, 3h 0.00063 0.00017 0.00013 0.0014 0.0002 3f, 3i 0.00055 0.00014 0.00011 0.0011 0.0002 3f, 3j 0.00060 0.00015 0.00011 0.0015 0.0002 3f, 3k 0.00064 0.00013 0.00011 0.0011 0.0002 = 23.5 MPa 3g, 3l 0.00049 0.00015 0.00012 0.0033 0.0011 = 23 MPa 3h, 3m 0.00084 0.00023 0.00017 0.0074 0.0007 3h, 3n 0.00085 0.00022 0.00016 0.0047 0.0003 3i, 3o 0.00047 0.00012 0.00009 0.0006 0.0002 3i, 3 0.00059 0.00015 0.00012 0.0020 0.0002 = 22.5 MPa 3l, 3q 0.00033 0.00010 0.00008 0.0025 0.0008 3j, 3r 0.00035 0.00009 0.00007 0.0015 0.0002 sat 643.15 K 3q, 3s 0.00033 0.00010 0.00008 0.0036 0.0008 = 20.5 MPa 3k, 3t 0.00042 0.00009 0.00008 0.0019 0.0002 ermissible tolerance 0.001 0.001 0.001 0.01 0.01

16 able 8. Maximum relative deviations of secific volume between the backward equations v, of the adjacent subregions and maximum relative deviations of secific enthaly, secific entroy, secific isobaric heat caacity, and seed of sound, calculated as described in Section 5.3, along the subregion boundary equations Subregion Boundary Equation Between Subregions v v max h h s s c max max c max % % % % % w w max 3ab 3a, 3b 0.00075 0.00020 0.00020 0.0012 0.0010 3d, 3e 0.00061 0.00017 0.00013 0.0016 0.0005 3cd 3c, 3d 0.00089 0.00027 0.00021 0.0040 0.0016 3c, 3g 0.00029 0.00009 0.00007 0.0017 0.0007 3c, 3l 0.00059 0.00019 0.00014 0.0039 0.0015 3c, 3q 0.00056 0.00018 0.00014 0.0040 0.0015 3c, 3s 0.00039 0.00012 0.00010 0.0031 0.0011 3ef 3e, 3f 0.00060 0.00016 0.00012 0.0005 0.0001 3h, 3i 0.00061 0.00016 0.00012 0.0007 0.0001 3n, 3o 0.00031 0.00008 0.00006 0.0004 0.0001 3gh 3g, 3h 0.00083 0.00022 0.00016 0.0058 0.0006 3l, 3h 0.00083 0.00022 0.00016 0.0064 0.0006 3l, 3m 0.00052 0.00014 0.00011 0.0058 0.0006 3ij 3i, 3j 0.00034 0.00009 0.00007 0.0010 0.0002 3, 3j 0.00036 0.00009 0.00007 0.0020 0.0002 3jk 3j, 3k 0.00030 0.00007 0.00006 0.0008 0.0001 3r, 3k 0.00029 0.00007 0.00006 0.0018 0.0002 3m, 3n 0.00090 0.00024 0.00017 0.0070 0.0003 3mn 3o 3o, 3 0.00041 0.00011 0.00008 0.0013 0.0002 ermissible tolerance 0.001 0.001 0.001 0.01 0.01 For examle, the maximum relative difference between the backward equation of subregion 3a and the backward equation of subregion 3b along the subregion boundary 3ab was determined as follows: v v3a, 3ab v3b, 3ab. v v, max 3b 3ab In addition, ables 7 and 8 contain the maximum relative differences of secific enthaly, secific entroy, secific isobaric heat caacity and seed of sound, calculated as described in Section 5.3, along the subregion boundaries of the v (, ) backward equations. For examle, the maximum relative difference of secific enthaly along the subregion boundary 3ab was determined as follows: v3a, 3ab h v3b, 3ab h, 3 3 h h h v max 3 3b 3ab max where v v, and, 3a 3a 3ab v v. 3b 3b 3ab max

17 ables 7 and 8 show that the relative secific volume differences between the backward equations v (, ) of the adjacent subregions and the maximum relative deviations of secific enthaly, secific entroy, secific isobaric heat caacity, and seed of sound along the subregion boundary ressures and along the subregion boundary equations are smaller than the ermissible numerical tolerances of these equations with the IAPWS-IF basic equation. 6 Auxiliary Equations v(,) for the Region very close to the Critical Point 6.1 Subregions he auxiliary equations v, for the subregions 3u to 3z are valid from 3qu for 3rx 643.15 K 22.5 MPa ; see Figure 5. sat 22.5 / MPa 3q 646.919 K 3uv 3v 3ef 3wx 3w 650.1 K 22.11 22.064 647.182 K c 3y 647.385 K 3r 21.9316 21.9010 3qu 3u 3z 3rx sat 3x 3jk 3k 21.0434 Figure 5. 643.15 647.096 Division of region 3 into subregions 3u to 3z for the auxiliary equations / K

18 he subregion boundary equation 3uv ( ) has the form of Eq. (1) and 3wx ( ) has the form of Eq. (2). he coefficients n i and the exonents I i of the boundary equations are listed in able 9. able 9. Numerical values of the coefficients of the equations 3uv ( ) and 3wx ( ) for subregion boundaries Equation i I i n i i I i n i 3uv 3wx he following descrition of the use of the subregion boundary equations is summarized in able 10 and Figure 5. able 10. Pressure ranges and corresonding subregion boundary equations for determining the correct subregion, 3u to 3z, for the auxiliary equations v, Suercritical Pressure Region Pressure Range Sub- For Sub- For region region 22.11 MPa < 22.5 MPa 3u 3qu ( ) 3uv ( ) 3v 3uv ( ) 3ef ( ) 3w 3ef 3wx ( ) ( ) 3x 3wx ( ) 3rx ( ) 22.064 MPa < 22.11 MPa 3u 3qu ( ) 3uv ( ) 3y 3uv ( ) 3ef ( ) Subcritical Pressure Region emerature Pressure Range Range sat ( ) 3 1 e sat 3z 3ef 3wx ( ) ( ) 3x 3wx ( ) 3rx ( ) 0.00264 m kg 22.064 MPa sat643.15 K sat ( ) 3 1 f sat 3 sat 1 e 0.00264 m kg 0.00385 m kg 22.064 MPa sat643.15 K 3 sat 1 f 0.00385 m kg e 3 1 1 sat 0.00264 m kg 2.193 161 551 10 MPa f 3 1 1 sat 0.00385 m kg 2.190 096 265 10 MPa Subregion For 3u 3qu 3uv ( ) ( ) 3y 3uv ( ) 3u 3qu ( ) 3z 3wx ( ) 3x 3wx 3rx ( ) ( ) 3x 3rx ( )

19 he equation 3uv ( ) aroximates the isochore 3 1 v 0.00264 m kg from sat0.00264 m 3 kg 1 3 1 1, where sat0.00264 m kg 2.193 161 55110 MPa, to 22.5 MPa and reresents the boundary equation between subregions 3v or 3y and subregion 3u. he equation 3wx ( ) aroximates the isochore 3 1 v 0.00385 m kg from sat0.00385 m 3 kg 1 3 1 1, where sat0.00385 m kg 2.190 096 26510 MPa, to 22.5 MPa and reresents the boundary equation between subregion 3x and subregions 3w or 3z. Comuter-rogram verification o assist the user in comuter-rogram verification of the equations for the subregion boundaries, able 11 contains test values for calculated temeratures. able 11. Selected temerature values calculated from the subregion boundary equations 3uv ( ) and 3wx ( ) Equation MPa K 3uv ( 3wx ( g It is recommended that rogrammed functions be verified using 8 byte real values for all variables. g 6.2 Auxiliary Equations v(,) for the Subregions 3u to 3z he auxiliary equations v, for the subregions 3u to 3z have the dimensionless form of Eq. (4). he reducing quantities v*, *, and *, the number of coefficients N, the non-linear arameters a and b, and the exonents c, d, and e are listed in able 12 for the auxiliary equations of the subregions 3u to 3z. he coefficients n i and exonents I i and J i are listed in ables A2.1 to A2.6 of the Aendix. able 12. Reducing quantities v*, *, and *, number of coefficients N, non-linear arameters a and b, and exonents c, d, and e for the auxiliary equations v, of the subregions 3u to 3z Subregion v* * * N a b c d e 3 1 m kg MPa K 3u 0.0026 23 650 38 0.902 0.988 1 1 1 3v 0.0031 23 650 39 0.960 0.995 1 1 1 3w 0.0039 23 650 35 0.959 0.995 1 1 4 3x 0.0049 23 650 36 0.910 0.988 1 1 1 3y 0.0031 22 650 20 0.996 0.994 1 1 4 3z 0.0038 22 650 23 0.993 0.994 1 1 4

20 Comuter-rogram verification o assist the user in comuter-rogram verification of the auxiliary equations for the subregions 3u to 3z, able 13 contains test values for calculated secific volumes. able 13. Selected secific volume values calculated from the auxiliary equations for the subregions 3u to 3z h Equation v Equation v MPa K 3 1 m kg MPa K 3 1 m kg v 21.5 644.6 2.268 366 647 v v 3u, 3v, 3w, v3x, 22.11 648.0 4.528 072 649 22.0 646.1 2.296 350 553 22.3 649.0 4.556 905 799 22.5 648.6 2.832 373 260 v3y, 22.0 646.84 2.698 354 719 22.3 647.9 2.811 424 405 22.064 647.05 2.717 655 648 22.15 647.5 3.694 032 281 v3z, 22.0 646.89 3.798 732 962 22.3 648.1 3.622 226 305 22.064 647.15 3.701 940 010 h It is recommended that rogrammed functions be verified using 8 byte real values for all variables. 6.3 Numerical Consistency 6.3.1 Numerical Consistency with the Basic Equation of IAPWS-IF he maximum relative differences and root-mean-square relative deviations of secific volume, calculated from the auxiliary equations v(, ) for subregions 3u to 3z, to the IAPWS-IF basic equation f3 (, v ) are listed in able 14. For the calculation of the rootmean-square values, which is described in Section 1, one million oints uniformly distributed over the range of validity in the - lane have been used. able 14 shows that the deviations of the secific volume from the IAPWS-IF basic equation are better than 0.1 %. Only in a small region for ressures less than 22.11 MPa (see Figure 5) do the deviations of the secific volume from the IAPWS-IF basic equation aroach 2 %. As a result, the secific volume values of saturated liquid and saturated vaor lines calculated with the auxiliary equations are not monotonically increasing; they oscillate around the values calculated from the basic equation f v, by iteration. able 14. Maximum relative deviations and root-mean-square relative deviations of the secific volume, calculated from the auxiliary equations for subregions 3u to 3z from the IAPWS-IF basic equation Subregion v v Subregion v v % % max RMS max RMS 3u 0.0 0.058 3x 0.090 0.050 3v 0.082 0.040 3y 1.77 1.04 3w 0.065 0.023 3z 1.80 0.921

21 6.3.2 Consistency at Boundaries Between Subregions he maximum relative differences of secific volume between the v (, ) auxiliary equations of adjacent subregions along the subregion boundary ressures are listed in able 15. able 16 contains these maximum relative differences along the subregion boundary equations. able 15. Maximum relative deviations of secific volume between the auxiliary equations v, of the adjacent subregions along the subregion boundary ressures Subregion Boundary Between Subregions v v max % 22.5 MPa 3l, 3u 0.096 3m, 3u 0.096 3m, 3v 0.035 3n, 3v 0.046 3o, 3w 0.019 3, 3w 0.021 3, 3x 0.042 3j, 3x 0.043 22.11 MPa 3v, 3y 1.7 3w, 3z 1.7 able 16. Maximum relative deviations of secific volume between the auxiliary equations v, of the adjacent subregions along the subregion boundary equations Subregion Boundary Between v v max Equation Subregions % 3q, 3u 0.0 3qu 3rx 3x, 3r 0.045 3uv 3u, 3v 0.14 3u, 3y 1.8 3ef 3v, 3w 0.080 3y, 3z 3.5 3wx 3w, 3x 0.049 3z, 3x 1.8 7 Comuting ime in Relation to IAPWS-IF A very imortant motivation for the develoment of the backward equations v, was reducing the comuting time to obtain thermodynamic roerties and differential quotients from given variables, in region 3. Using IAPWS-IF, time-consuming iteration is

22 required. Using the v, backward equations, iteration can be avoided. he calculation seed is about 17 times faster than iteration with IAPWS-IF. If iteration is used, the time to reach convergence can be significantly reduced by using the backward equations v, to calculate very accurate starting values. 8 Alication of the Backward and Auxiliary Equations v(,) he numerical consistency of the secific volume v calculated from the main backward equations v described in Section 5 with the IAPWS-IF basic equation f v 3, sufficient for most alications in rocess modeling. 3, is In comarison with the backward equations, the corresonding numerical consistency of the auxiliary equations v, is clearly worse. Nevertheless, for many calculations, the numerical consistency of the auxiliary equations described in Section 6 is satisfactory in the region very close to the critical oint. For alications where the demands on numerical consistency are extremely high, iteration using the IAPWS-IF basic equation f v, may be necessary. In these cases, the backward and auxiliary equations v, can be used for calculating very accurate starting values. he backward and auxiliary equations v, should only be used in their ranges of validity described in Section 4. hey should not be used for determining any thermodynamic derivatives. hey should also not be used together with the fundamental equation in iterative calculations of other backward functions such as, s. Iteration of backward functions can only be erformed by using the fundamental equations. h or, In any case, deending on the alication, a conscious decision is required whether to use the backward and in articular the auxiliary equations v, or to calculate the corresonding values by iteration from the basic equation of IAPWS-IF. 9 References [1] IAPWS, R7-(2012), Revised Release on the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam (2007), available from: htt://www.iaws.org. [2] Wagner, W., Cooer, 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 Proerties of Water and Steam, ASME J. Eng. Gas urbines Power 122, 150-182 (2000).

23 [3] Kretzschmar, H.-J., Harvey, A. H., Knobloch, K., Mareš, R., Miyagawa, K., Okita, N., San, R., Stöcker, I., Wagner, W., and Weber, I., Sulementary Backward Equations v(,) for the Critical and Suercritical Regions (Region 3) of the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam, ASME J. Eng. Gas urbines Power 131, 043101 (2009). [4] IAPWS, SR2-01(2014), Revised Sulementary Release on Backward Equations for Pressure as a Function of Enthaly and Entroy (h,s) for Regions 1 and 2 of the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam (2014), available from htt://www.iaws.org. [5] Kretzschmar, H.-J., Cooer, J. R., Dittmann, A., Friend, D. G., Gallagher, J. S., Knobloch, K., Mareš, R., Miyagawa, K., Stöcker, I., rübenbach, J., Wagner, W., and Willkommen, h., Sulementary Backward Equations for Pressure as a Function of Enthaly and Entroy (h,s) to the Industrial Formulation IAPWS-IF for Water and Steam, ASME J. Eng. Gas urbines Power 128, 702-713 (2006). [6] IAPWS, SR3-03(2014), Revised Sulementary Release on Backward Equations for the Functions (,h), v(,h) and (,s), v(,s) for Region 3 of the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam (2014), available from: htt://www.iaws.org. [7] Kretzschmar, H.-J., Cooer, J. R., Dittmann, A., Friend, D. G., Gallagher, J. S., Harvey, A. H., Knobloch, K., Mareš, R., Miyagawa, K., Okita, N., Stöcker, I., Wagner, W., and Weber, I., Sulementary Backward Equations (,h), v(,h), and (,s), v(,s) for the Critical and Suercritical Regions (Region 3) of the Industrial Formulation IAPWS-IF for Water and Steam, ASME J. Eng. Gas urbines Power 129, 294-303 (2007). [8] IAPWS, SR4-04(2014), Revised Sulementary Release on Backward Equations (h,s) for Region 3, Equations as a Function of h and s for the Region Boundaries, and an Equation sat ( h,s) for Region 4 of the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam (2014), available from: htt://www.iaws.org. [9] Kretzschmar, H.-J., Cooer, J. R., Dittmann, A., Friend, D. G., Gallagher, J. S., Harvey, A. H., Knobloch, K., Mareš, R., Miyagawa, K., Okita, N., San, R., Stöcker, I., Wagner, W., and Weber, I., Sulementary Backward Equations (h, s) for the Critical and Suercritical Regions (Region 3), and Equations for the wo-phase Region and Region Boundaries of the IAPWS Industrial Formulation 19 for the hermodynamic Proerties of Water and Steam, ASME J. Eng. Gas urbines Power 129, 1125-1137 (2007). [10] IAPWS, R2-83(1992), Release on the Values of emerature, Pressure and Density of Ordinary and Heavy Water Substances at their Resective Critical Points (1992), available from htt://www.iaws.org.

24 Aendix A1 Coefficients for Backward Equations able A1.1. Coefficients and exonents of the backward equation v 3a, for subregion 3a able A1.2. Coefficients and exonents of the backward equation v 3b, for subregion 3b

able A1.3. Coefficients and exonents of the backward equation v 25 3c, for subregion 3c able A1.4. Coefficients and exonents of the backward equation v 3d, for subregion 3d

able A1.5. Coefficients and exonents of the backward equation v 26 3e, for subregion 3e able A1.6. Coefficients and exonents of the backward equation v 3f, for subregion 3f

able A1.7. Coefficients and exonents of the backward equation v 27 3g, for subregion 3g able A1.8. Coefficients and exonents of the backward equation v 3h, for subregion 3h

able A1.9. Coefficients and exonents of the backward equation v 28 3i, for subregion 3i able A1.10. Coefficients and exonents of the backward equation v 3j, for subregion 3j

able A1.11. Coefficients and exonents of the backward equation v 29 3k, for subregion 3k able A1.12. Coefficients and exonents of the backward equation v 3l, for subregion 3l

able A1.13. Coefficients and exonents of the backward equation v 30 3m, for subregion 3m able A1.14. Coefficients and exonents of the backward equation v 3n, for subregion 3n

able A1.15. Coefficients and exonents of the backward equation v 31 3o, for subregion 3o able A1.16. Coefficients and exonents of the backward equation v 3, for subregion 3 able A1.17. Coefficients and exonents of the backward equation v 3q, for subregion 3q

able A1.18. Coefficients and exonents of the backward equation v 32 3r, for subregion 3r able A1.19. Coefficients and exonents of the backward equation v 3s, for subregion 3s

able A1.20. Coefficients and exonents of the backward equation v 33 3t, for subregion 3t A2 Coefficients for Auxiliary Equations able A2.1. Coefficients and exonents of the auxiliary equation v 3u, for subregion 3u

able A2.2. Coefficients and exonents of the auxiliary equation v 34 3v, for subregion 3v able A2.3. Coefficients and exonents of the auxiliary equation v 3w, for subregion 3w

able A2.4. Coefficients and exonents of the auxiliary equation v 35 3x, for subregion 3x able A2.5. Coefficients and exonents of the auxiliary equation v 3y, for subregion 3y able A2.6. Coefficients and exonents of the auxiliary equation v 3z, for subregion 3z