Hans-Joachim Kretzschmar and Katja Knobloch

Transcription

1 Sulementary Backward Equatons for the Industral Formulaton IAPWS-IF of Water and Steam for Fast Calculatons of Heat Cycles, Bolers, and Steam Turbnes Hans-Joachm Kretzschmar and Katja Knobloch Deartment of Techncal Thermodynamcs, Zttau/Goerltz Unversty of Aled Scences, P.O. Box 455, D Zttau, Germany E-mal: The aer gves an overvew of the backward equatons whch were develoed as a sulement to the "IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam" (IAPWS-IF). Frst, backward equatons (h,s) for the lqud regon and vaor regon 2 were develoed and adoted as a Sulementary Release n 20 (IAPWS-IF-S). An nternatonal survey revealed that backward equatons n the crtcal and suercrtcal regons (regon 3) were also requred n rocess modelng. Thus, backward equatons T(,h), v(,h), T(,s), and v(,s) for regon 3 were develoed and adoted by IAPWS as a Sulementary Release n 20 (IAPWS-IF- S). An extended revson of ths release wll be adoted n 20. Further backward equatons (h,s) develoed for regon 3 have been evaluated successfully and wll be adoted as a Sulementary Release n 20. In addton, backward equatons v(,t) for regon 3 wll be roosed n 20. For steam-turbne calculatons, a backward equaton for the saturaton temerature as a functon of enthaly and entroy n the mortant art of two-hase regon 4 has been develoed. In order to determne f a gven state ont s located n one of the sngle-hase regons, 2, 3 or n the two-hase regon 4, teratons are necessary for the backward functons of the gven roertes (,h), (,s) or (h,s). For ths reason, secal boundary equatons were develoed. Usng the IAPWS-IF equatons, along wth the sulementary backward and boundary equatons here resented, all thermodynamc roertes from gven roerty ars (,T), (,h), (,s), and (h,s) can be calculated wthout teratons over the entre range of valdty ncludng the determnaton of the regon (excet for hgh temerature regon 5). Because the numercal consstency of the backward and boundary equatons s suffcent for most heat-cycle, boler, and steam-turbne calculatons, they wll sgnfcantly reduce the comutng tme of rocess modelng.. Introducton In 9, the Internatonal Assocaton for the Proertes of Water and Steam (IAPWS) adoted the "IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam" (IAPWS-IF) [, 2]. The IAPWS-IF contans fundamental equatons g(,t) for lqud regon, vaor regon 2 and hgh-temerature regon 5, a fundamental equaton f(v,t) for the crtcal and suercrtcal regons (regon 3) and for regon 4 an equaton ar for saturaton ressure sat (T) and for saturaton temerature T sat (). Fg. shows the regons of the IAPWS-IF. The IAPWS-IF equatons are marked by the suerscrt "". Usng the fundamental IAPWS-IF equatons, all thermodynamc roertes and dervatves can be calculated from gven ressure and temerature n regons, 2, 5, and from gven secfc volume or densty and temerature n regon 3. In addton, the IAPWS-IF contans backward equatons for the most commonly used mlct functons T(,h) and T(,s) n regons and 2. Further deendences have to be calculated teratvely from the fundamental equatons. Ths means that one-dmensonal or two-dmensonal teratons are necessary for determnng thermodynamc roertes n rocess modelng. For ths urose, the IAPWS workng grous "Industral Requrements and Solutons" (IRS) and "Thermohyscal roertes of Water and Steam" (TPWS) establshed the "Task Grou on Sulementary Backward Equatons for IAPWS-

2 IF" and sulementary backward equatons have been develoed over the last 5 years. The members of the task grou are A. Dttmann K. Knobloch H.-J. Kretzschmar (char) R. Mareš R. San I. Stöcker W. Wagner. Imortant contrbutons were made by J. R. Cooer D. G. Frend A. H. Harvey. Besdes these "develoers" the members of the evaluaton task grou J. Gallagher K. Myagawa (char). Okta I. Weber layed a consderable art n the qualty assurance of the develoed equatons. The sulementary releases lsted n Table are the results of ths work. 2. Structure of IAPWS-IF and Sulementary Backward Equatons Fgure shows, n addton to the IAPWS-IF equatons, the sulementary backward equatons for regons, 2, 3, and 4. Besdes the IAPWS-IF backward equatons, T (, h ), T (, s ) for lqud regon and T 2 (, h ), T 2 (, s ) for vaor regon 2, equatons hs and (, ) 2 ( hs, ) were develoed [4, 3]. They are ncluded n of the sulementary release IAPWS-IF-S [3]. For regon 3, IAPWS-IF dd not rovde backward equatons. Therefore, equatons T 3 (, h ), v 3 (, h ), T 3 (, s ), v 3 (, s ) and 3 ( hs, ) were develoed [7, 9, 3] and adoted by IAPWS as the sulementary releases IAPWS- IF-Srev [6] and IAPWS-IF-S [8]. In addton, IAPWS-IF-S contans an equaton Tsat ( h, s ) for the art of wet-steam regon 4 whch s mortant for steam turbne calculatons. The equatons develoed for hase and regon boundares are not ncluded n Fg.. / MPa B23 ( T ) f 3 ( vt, ) 2 v3 (, T ) g ( T, ) T3 (, h) g 2 ( T, ) T (, h) v T 2 (, h) 3 (, h ) T (, s) T T 2 (, s) 3 (, s ) ( hs, ) v ( s ) 2 ( hs, ) c 4 3, ( hs ) 3, sat sat T T ( T ) ( ) ( h s) sat, IAPWS-IF IAPWS-IF-S IAPWS-IF-Srev IAPWS-IF-S g ( T) 5, T / K Fg.. -T dagram wth regons and equatons of the IAPWS-IF, and sulementary backward equatons.

3 3. Backward and Boundary Equatons for Functons of Pressure and Enthaly (,h) 3. Subregons and Backward Equatons In rocess modelng, thermodynamc roertes of water and steam are requred for gven roertes (,h). Fg. 2 shows the range of valdty of IAPWS- IF and the regons/subregons of the rovded backward equatons n a -h dagram. In order to meet the extremely hgh numercal consstency requrements for the backward equatons, regon 2 was dvded nto three subregons, namely 2a, 2b, and 2c, and regon 3 nto subregons 3a and 3b. The IAPWS-IF contans equatons T (, h ) n lqud regon and T 2a (, h ), T 2b (, h ), T2c (, h ) n vaor subregons 2a, 2b, and 2c. Usng the temerature T calculated from these equatons and gven ressure, all thermodynamc roertes can be calculated from the dervatves of the fundamental equatons g (, T) or g 2 (, T ). For regon 3, equatons T 3a (, h ), v 3a (, h ) and T 3b (, h ), v 3b (, h ) for subregons 3a and 3b are contents of IAPWS-IF-Srev. Usng temerature T and secfc volume v calculated from these equatons, all other roertes can be determned from the IAPWS-IF fundamental equaton f3 (, vt ) of regon 3. The backward equatons T(,h) and v(,h) have the followng structures: I J = n * + a + b * * = I J = n * + a + b * * = T(, h) h T h, () v(, h) h v h. (2) The numbers of coeffcents and non-lnear arameters a and b of Eq. () and (2) are lsted n Table 2. The coeffcents n, exonents I, J, and reducng arameters T * * * or v,, and * h can be taken from [, 2] and [6, 7]. Table. Sulementary releases ncludng backward equatons for IAPWS-IF. Sulementary Release Equatons Status Ref. IAPWS-IF-S: Sulementary Release on Backward Equatons for Pressure as a Functon of Enthaly and Entroy (h,s) to the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam. (h,s) 2 (h,s) adoted n 20 [3, 4] IAPWS-IF-S: Sulementary Release on Backward Equatons for the Functons T(,h), v(,h) and T(,s), v(,s) for Regon 3 of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam. T 3,v 3 (,h) T 3,v 3 (,s) adoted n 20 [5] IAPWS-IF-Srev (Relacement for IAPWS-IF-S): Revsed Sulementary Release on Backward Equatons for the Functons T(,h), v(,h) and T(,s), v(,s) for Regon 3 of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam. T 3,v 3 (,h) T 3,v 3 (,s) 3sat (h) 3sat (s) adoted n 20 [6, 7] IAPWS-IF-S: Sulementary Release on Backward Equatons (h,s) for Regon 3, Equatons as a Functon of h and s for the Regon Boundares, and an Equaton T sat (h,s) for Regon 4 of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam. 3 (h,s) T sat (h,s) h'(s), h"(s) h B3 (s) T B23 (h,s) adoted n 20 [8, 9] Sulementary Release on Backward Equatons for Secfc Volume as a Functon of Pressure and Temerature v(,t) for Regon 3 of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam. (Draft) v 3 (,T) evaluated, to be adoted n 2005 [0, ]

4 In the two-hase regon 4, temerature can be drectly calculated from gven ressure usng the saturaton temerature equaton T sat ( ). The temerature has to be calculated teratvely from the IAPWS-IF fundamental equaton g5 (, T ) n hgh-temerature regon 5. Table 2. umbers of Coeffcents and non-lnear arameters of the backward equatons T(, h ), Eq. (), and v(, h ), Eq. (2). Equaton a b T ( ) 20 0 T 2a (, h ) T 2b ( ) T 2c ( ) T 3a (, h ) T 3b ( ) v3a (, h ) v3b (, h ) Boundary Equatons Before calculatng roertes, the regon/subregon has to be determned for gven roertes and h. Fg. 2 shows the boundares between regons or subregons to be calculated. Because ressure s gven, the boundary lnes x = 0 between regons 4 and, x = between regon 4 and subregons 2a, 2b and 2c, the 350 C sotherm and the 4 MPa sobare can be determned wthout teraton. The equaton TB23 ( ) s a converted olynomal of the second degree whle 2bc ( h ) and h 3ab ( ) are olynomals of the second and thrd degree. For calculatng the saturated lqud lne between two-hase regon 4 and subregon 3a and the saturated vaor lne between regon 4 and subregon 3b usng the IAPWS-IF fundamental equaton f 3 (, vt ) for gven roertes and h, an teraton would be necessary. Therefore, IAPWS- IF-Srev rovdes an equaton 3sat ( h ) whch s descrbed n [6, 7]. / MPa v 3a (, h) h 3ab ( ) v 3b(, h) T B23 ( ) 2bc ( h) MPa 0 C 350 C T (, h) T 3a (, h) 3a c 3sat T 3b (, h) ( h) 3b 2c 4 MPa T 2c (, h) 2b T 2b (, h) 4 2a T sat ( ) T 2a (, h) 800 C IAPWS-IF IAPWS-IF-Srev 0 MPa h /kjkg Fg. 2. -h dagram wth regons and subregons, backward equatons and boundary equatons for the functon of (,h).

5 3.3 umercal Consstency The numercal consstency of the backward equatons T(, h ) and v(, h ) wth the fundamental equatons of IAPWS- IF results from rocess modelng requrements. Usng results of an nternatonal survey [2], IAPWS has determned the ermssble tolerances ΔT tol for temerature, to be 25 for entroes less than or equal to 5.85 kj kg K (regon, subregons 3a, 3b, and 2c) and to be 0 for entroes greater than ths value (subregons 2a and 2b). The ermssble relatve tolerance Δv/v tol for secfc volume n subregons 3a and 3b was determned to be 0.. The numercal consstences of the equatons and the tolerated values are lsted n Table 3. As can be seen, the maxmum temerature dfferences ΔT max and the maxmum relatve dfferences of secfc volume Δv/v max are less than the ermssble tolerances. In order to avod numercal roblems, the backward equatons of subregons 3a and 3b reresent the crtcal ont exactly. At the subregon boundares = 4 MPa and 2bc ( h ) of regon 2, and at h 3ab ( ) of regon 3 the dfferences between the backward equatons of the adjacent subregons are smaller than ther numercal consstences wth the IAPWS-IF fundamental equatons. Therefore, numercal roblems at subregon boundares wll be avoded. Table 3. umercal consstences of the backward equatons T(, h) and v(, h ) wth IAPWS-IF. Equaton ΔT tol ΔT max T (, h ) T2a (, h ) T2b (, h ) T2c (, h ) T3a (, h ) T3b (, h ) Equaton Δv/v tol Δv/v max v3a (, h ) v h b (, ) 4. Backward and Boundary Equatons for Functons of Pressure and Entroy (,s) 4. Subregons and Backward Equatons Fgure 3 shows the range of valdty of IAPWS-IF and the regons/subregons of the rovded backward equatons as functons of (, s ) n a -s dagram. As was done for the functons of (, h ), regon 2 was also dvded nto three subregons, 2a, 2b, and 2c, and regon 3 nto subregons, 3a and 3b, n order to meet the numercal consstency requrements of rocess modelng. The IAPWS-IF contans equatons T (, s) n lqud regon and T 2a (, s ), T 2b (, s ), T2c (, s ) n vaor subregons 2a, 2b, and 2c. Usng the temerature T calculated from these equatons and gven ressure, all thermodynamc roertes can be calculated from the fundamental equatons g (, T) or g 2 (, T ). In regon 3, equatons T 3a (, s ), v 3a (, s) and T 3b (, s ), v 3b (, s ) for subregons 3a and 3b are rovded by IAPWS-IF-Srev. Usng T and v calculated from these equatons, all other roertes can be determned from the fundamental equaton f 3 (, vt ). The backward equatons T(, s ) and v(, s ) have the followng structures: I J = n * + a + b * * = T(, s) s T s, (3) I J v(, s) s = n * + a + b * * v = s. (4) The numbers of coeffcents and non-lnear arameters a and b are lsted n Table 4. The coeffcents n, exonents I, J, and reducng * * * arameters T or v,, and s * can be taken from [, 2] and [6, 7]. In the two-hase regon 4, temerature can be drectly calculated from gven ressure usng the saturaton temerature equaton T sat ( ). In hgh-temerature regon 5, on the other hand, temerature must be calculated teratvely usng the IAPWS-IF fundamental equaton g 5 (, T ). 4.2 Boundary Equatons Before calculatng roertes, the regon/subregon has to be determned for gven roertes and s. Fgure 3 shows the boundares between regons or subregons to be calculated. Because ressure and entroy are gven, the boundary lnes x = 0 between regons 4 and, x = between regon 4 and subregons 2a, 2b and 2c, the 350 C sotherm, sentroc lnes sc and s = 5.85 kj kg K, and the 4 MPa sobare can be calculated wthout teraton.

6 The equaton TB23 ( ) s a converted olynomal of the second degree. For calculatng the saturated lqud lne between two-hase regon 4 and subregon 3a and the saturated vaor lne between regon 4 and subregon 3b from IAPWS-IF fundamental Table 4. umbers of coeffcents and nonlnear arameters of the backward equatons T(, s ), Eq. (3), and v(, s ), Eq. (4). Equaton a b T ( ) T2a (, s ) T2b (, s ) T 2c ( ) T3a (, s ) T3b (, s ) v3a (, s ) v3b (, s ) equaton f 3 (, vt ) for gven roertes and s, an teraton would be necessary. Therefore, IAPWS- IF-Srev, descrbed n [6, 7], rovdes the boundary equaton s. 3sat () 4.3 umercal Consstency For the numercal consstences of the backward equatons T(, s ) and v(, s ) wth the fundamental equatons of IAPWS-IF, IAPWS has determned the ermssble tolerance ΔT tol = 25 for entroes less than or equal to 5.85 kj kg K (regon, subregons 3a, 3b, and 2c) and 0 for entroes greater than ths value (subregons 2a and 2b). The ermssble relatve tolerance for secfc volume n subregons 3a and 3b was determned as Δv/v tol = 0.. Table 5 shows the numercal consstences of the backward equatons n comarson wth the tolerated values. As can be seen, the maxmum temerature dfferences ΔT max and maxmum relatve dfferences of secfc volume Δv/v max are less than the ermssble tolerances. / MPa T 3a (, s) v 3a (, s) 00 MPa 350 C T (, s) 0 C T 3b (, s) v 3b(, s) s c 3a c T B23 ( ) 3b 3sat () s 2c T 2c (, s) 5.85 kj kg K 2b 4 MPa 800 C 0 MPa T 2b (, s) IAPWS-IF IAPWS-IF-Srev 2000 C 4 T sat ( ) 2a T 2a (, s) s / kj kg K Fg. 3. -s dagram wth regons and subregons, backward equatons and boundary equatons for the functon of (,s).

7 Table 5. umercal consstences of the backward equatons T(, s) and v(, s ) wth IAPWS-IF. Equaton ΔT tol ΔT max T (, s ) T2a (, s ) T2b (, s ) T2c (, s ) T3a (, s ) T3b (, s ) Equaton Δv/v tol Δv/v max v3a (, s ) v3b (, s ) In order to avod numercal roblems, the backward equatons of subregons 3a and 3b agan reresent the crtcal ont exactly. At the subregon boundares = 4 MPa and s = 5.85 kj kg K of regon 2, and s c of regon 3, the dfferences between the backward equatons of the adjacent subregons are smaller than ther numercal consstences wth the IAPWS- IF fundamental equatons. Ths avods numercal roblems at subregon boundares. 5. Backward and Boundary Equatons for Functons of Enthaly and Entroy (h,s) 5. Subregons and Backward Equatons The h-s dagram n Fg. 4 shows the range of valdty of IAPWS-IF and the regons/subregons of the backward equatons as functons of ( hs, ) resented here. As wth the other backward functons, regon 2 has also been dvded nto subregons 2a, 2b, and 2c and regon 3 nto subregons 3a and 3b to meet the numercal consstency requrements of rocess modelng. The sulementary release IAPWS-IF-S contans equatons ( hs, ) n lqud regon and 2a ( hs, ), 2b ( hs, ), 2c ( hs, ) n vaor subregons 2a, 2b, and 2c. Usng ressure calculated from these equatons, temerature T can be calculated by usng the equatons T (, h ) n lqud regon and T 2a (, h ), T 2b (, h ), T2c (, h ) n vaor subregons. h /kjkg h, 3b, 3a, B3 ( s ) h 2c, T B23, 3a ( s) s c 3b h 5.85 kj kg K c ( s) 3a 2c h Fg. 4. h-s dagram wth regons and subregons, backward equatons and boundary equatons for the functon of (h,s). h 2b, 2c3b ( s) s ''( K) 2b 2ab 4 T sat ( h, s) ( s) h 2ab ( s ) 2a 2a, 5 IAPWS-IF-S IAPWS-IF-S s / kj kg K

8 All thermodynamc roertes can then be calculated from the dervatves of the fundamental equatons g (, T) or g 2 (, T ). For regon 3, equatons 3a ( hs, ) and 3b ( hs, ) n subregons 3a and 3b are rovded by IAPWS-IF-S. Usng ressure calculated from these equatons, temerature T and secfc volume can be determned by usng the equatons T 3a (, h ), T 3b (, h ) and v 3a (, s ), v 3b (, s) of IAPWS-IF-S. ow, all other thermodynamc roertes can be calculated from the IAPWS- IF fundamental equaton f 3 (, vt ). The backward equatons ( hs, ) have the followng structure: I J hs (, ) h s = n * + a b * + *. (5) = h s The numbers of coeffcents, and non-lnear arameters a, b, and c are lsted n Table 6. The coeffcents n, exonents I, J and reducng * * arameters, h, and s * can be taken from [3, 4] and [8, 9]. In the art of the two-hase regon 4 whch s mortant for steam-turbne calculatons, IAPWS- IF-S rovdes an equaton T sat ( h, s ) ; see Fg. 4. The equaton s vald for entroes s s (350 C) = 5.2 kj kg K. Usng saturaton temerature T sat, saturaton ressure sat can be calculated from IAPWS-IF equaton sat ( T sat ) and vaor fracton x from x = ( h h )/( h h ), where h = h (, T ) and h = h (, T ). sat sat c 2 sat sat Table 6. umbers of coeffcents and nonlnear arameters of the backward equatons ( hs, ), Eq. (5). Equaton a b c ( hs, ) a ( hs, ) b ( hs, ) c ( hs, ) a ( hs, ) b ( hs, ) All other thermodynamc roertes n the twohase regon can then be determned from the fundamental equatons g (, T ) and g 2 (, T ). The backward equaton Tsat ( h, s) has the followng structure: 36 Tsat ( h, s) h s = n * * *.(6) T = h s The coeffcents n, exonents I, J and reducng arameters * * *, h and s can be taken from [8, 9]. In the other art of the two-hase regon 4, saturaton temerature, saturaton ressure and vaor fracton have to be calculated teratvely from the IAPWS-IF saturaton ressure equaton n combnaton wth the fundamental equatons. In hgh-temerature regon 5, temerature and ressure must be calculated usng the IAPWS- IF fundamental equaton g5 (, T ) va twodmensonal teraton. I 5.2 Boundary Equatons Before calculatng roertes, the regon/subregon has to be determned for gven roertes h and s. Fgure 4 shows the boundares between regons or subregons to be consdered. Because enthaly and entroy are gven, all regon boundares would have to be determned teratvely by usng IAPWS-IF fundamental equatons excet the subregon boundary lnes s c, s = 5.85 kj kg K, and h 2ab () s. Therefore, IAPWS-IF-S rovdes the followng boundary equatons: - h () s for saturated lqud lne x = 0 between regon 4 and regon - h 3a () s for saturated lqud lne x = 0 between regon 4 and subregon 3a - h 2c3b () s for saturated vaor lne x = between regon 4 and subregons 2c and 3b - h 2ab () s for saturated vaor lne x = between regon 4 and subregons 2a and 2b - hb3 () s for the sotherm T = 350 C between regons and 3 - T B23 ( h, s ) n addton to 2c ( hs, ) for the B23 ( T ) lne of IAPWS-IF between regons 2 and 3. The equatons and the determnaton of the subregons are descrbed comrehensvely n [8, 9]. 5.3 umercal Consstency Table 7 shows the numercal consstences of the backward equatons ( hs, ) and of the related backward functons Ths (, ) and vhs (, ) n comarson wth the tolerated values. As can be seen, the maxmum relatve ressure dfferences Δ/ max, the maxmum temerature dfferences ΔT max and J

9 the maxmum relatve dfferences of secfc volume Δv/v max n all regons and subregons are less than the ermssble tolerances. Ths s also true for saturaton ressure, saturaton temerature, and vaor fracton n two-hase regon Backward Equatons for Functons of Pressure and Temerature (,T) n Regon 3 6. Subregons and Backward Equatons Because an teraton s necessary to calculate roertes n regon 3 from the IAPWS-IF fundamental equaton f3 (, vt ) for gven roertes and T, the backward equatons v 3 (, T) are rovded for regon 3 [0, 3]. The -T dagrams n Fgs. 2 and 3 of the aer [] show the subregons (3a to 3t) nto whch regon 3 s dvded. All n all, 20 subregons were necessary n order to meet the extremely hgh rocess modelng numercal consstency requrements. Once we have the secfc volume v, calculated from the backward equatons, all other thermodynamc roertes can be determned usng the IAPWS-IF fundamental equaton f 3 (, vt ). For a small area very near the crtcal ont, t was not ossble to meet the numercal consstency requrements fully. Ths near-crtcal regon s covered wth reasonable consstency by sx subregons wth auxlary equatons. The backward equatons v(, T ) for subregons 3a to 3t, excet 3n, have the followng structure: I J c d v(, T) T n * a = * b * v = T (7) and for subregon 3n: I J v3n (, T) T ex * a = n * b *.(8) v = T Coeffcents n, exonents I, J, reducng * * arameters v,, and T *, and non-lnear arameters a, b, c, d, and e can be taken from [0, 3]. 6.2 Subregon-Boundary Equatons Before calculatng the correct v(, T) equaton, the subregon has to be determned for gven roertes and T. The subregon-boundary equatons are descrbed n the aer []. The coeffcents are lsted n [0, 3]. e Table 7. umercal consstences of the backward equatons ( hs, ), of the related backward functons Thsand (, ) vhs (, ), and of sat ( h, s ) wth IAPWS-IF. Equaton Δ/ tol Δ/ max ΔT tol ΔT max Δv/v tol Δv/v max ( hs, ) 2.5MPa > 2.5MPa 5 kpa 4 kpa hs a (, ) 2b ( hs, ) hs c (, ) hs a (, ) 3b ( hs, ) Equaton ΔT tol ΔT max Δ/ tol Δ/ max Δx tol - Δx max - T sat ( ) K s>5.85 kj kg K

10 6.3 umercal Consstency The maxmum relatve devatons of secfc volume, enthaly, entroy, sobarc heat caacty and the seed of sound calculated from the backward equatons v(, T ) to the IAPWS-IF fundamental equaton f 3 (, vt ) of regon 3 are lsted n Table 3 of []. As can be seen, the devatons of the secfc volume, enthaly, and entroy calculated from the resented v(, T) equatons to the IAPWS-IF fundamental equaton are less than 0.0 and the devatons of sobarc heat caacty and seed of sound are less than 0.. Ths means that the values of secfc volume, enthaly and, entroy of IAPWS-IF are reresented wth 5 sgnfcant fgures, and the values of sobarc heat caacty and seed of sound wth 4 sgnfcant fgures by usng the backward equatons v(, T ). At subregon boundares, the dfferences between the calculated roertes are smaller than ther numercal consstences wth the fundamental equaton mentoned before. Therefore, numercal roblems at subregon boundares wll be avoded. 7. Comutng Tmes n Comarson wth IAPWS-IF Fundamental Equatons A very mortant motvaton for the develoment of the backward equatons was reducng the comutng tme needed to obtan thermodynamc roertes. Usng IAPWS-IF fundamental equatons, tme-consumng teratons such as the two-dmensonal ewton method are requred for backward functons. Table 8 shows the resultng comutng tme ratos (CTR), defned by Comtng tme of fundamental eq. CTR =. (9) Comutng tme of backward eq. The calculatons of the functons of gven roertes (, h ) by usng the backward equatons are between 4 tmes faster n regon 2 and 43 tmes faster n regon than the corresondng teratons usng the IAPWS-IF fundamental equatons [5]. The calculatons of functons of (, s ) by usng the backward equatons are between 7 tmes faster n regon 2 and 70 tmes faster n regon than the corresondng teratons [5]. The calculatons of functons of ( hs, ) by usng the backward equatons are between 0 tmes faster n regon 3 and 38 tmes faster n regon 2. The calculatons of functons of (, T ) n regon 3 by usng the backward equatons are 7 tmes faster than the corresondng teratons usng the IAPWS-IF fundamental equaton [6]. Table 8. Comutng tme ratos between backward equatons and teratons from IAPWS-IF fundamental equatons. Functon Reg. Backward Equaton(s) CTR (,h) T ( ) 43 2 T 2 ( ) 4 3 T 3 ( ) 3 ( ) 6 (,s) T ( ) 70 2 T 2 ( ) 7 3 T 3 ( ) 3 ( ) 8 (h,s) (, ) ( ) (, ) 2 ( ) (, ) 3 ( ) 0 & 3 ( ) 4 T sat ( ) sat ( ) h h & x = h h (,T) 3 v 3 (, T ) 7 An addtonal motvaton for the develoment of the equatons for regon boundares was reducng the comutng tme needed to determne the regon for a gven state ont. By usng boundary equatons, users can determne the regon wthout tme-consumng teratons of IAPWS-IF fundamental equatons. Table 9 shows the resultng CTR values. The calculatons of the hase boundary between two-hase regon 4 and regon 3 for gven roertes (, h) or (, s ) by usng the boundary equatons are more than 9 tmes faster than the corresondng teratons usng the IAPWS-IF fundamental equatons. The calculatons of the hase boundares and the boundares between regons and 2 or between regons 2 and 3 for gven roertes ( hs, ) by usng the boundary equatons are between 20 and 90 tmes faster than the corresondng teratons. In concluson, these factors show that the calculatons of backward functons, ncludng determnaton of the regon usng the backward equatons and boundary equatons, are between 5 and 20 tmes faster than the teraton of IAPWS- IF fundamental equatons.

11 Takng n consderaton the frequency of use for the varous backward equatons n rocess modelng, the calculatons of heat-cycles, bolers, and artcularly of steam turbnes can be exected to be 2 to 3 tmes faster when usng the sulementary backward and boundary equatons. Table 9. Comutng tme ratos between boundary equatons and teratons from IAPWS-IF fundamental equatons. Funct. Bound. Reg.-Reg. Bound. Eq. CTR (,h) x = 0 x = 3-4 3sat ( ) 2 (,s) x = 0 x = 3-4 3sat () 9 (h,s) x = 0-4 h () h 3a () s 90 x = 2-4 h 2ab () s, 20 h 2c3b () s 3-4 h 2c3b () s K - 3 h B3 () 37 B23 ( T ) 2-3 T B23 ( h, s ), 20 2c ( hs, ) 8. Alcaton of the Backward and Boundary Equatons The numercal consstency of the backward equatons and boundary equatons descrbed n Sectons 3 to 6 wth the IAPWS-IF fundamental equatons s suffcent for most alcatons n heat cycle, boler, and steam turbne calculatons. For users not satsfed wth the numercal consstency of the backward equatons or boundary equatons, they are stll recommended for generatng good startng onts for an teratve rocess. It wll sgnfcantly reduce the tme requred to meet the convergence crtera of the teraton. The equatons resented can only be used wthn ther ranges of valdty. They should not be used to determne any thermodynamc dervatves. The determnaton of thermodynamc dervatves from IAPWS-IF fundamental equatons s descrbed n [4]. The backward and boundary equatons should also not be used together wth the fundamental equatons n teratve calculatons of other backward functons. Iteratons of backward functons can only be erformed usng the fundamental equatons. In any case, deendng on the alcaton, a conscous decson s requred whether to use the backward or boundary equatons or to calculate the corresondng values by teraton from the IAPWS- IF fundamental equatons. 9. Conclusons The aer has resented backward equatons and equatons for regon boundares for calculatng the functons of (, h ), (, s ), ( hs, ) and (, T ) n rocess modelng. The equatons can be used as sulement to the Industral Formulaton IAPWS- IF for water and steam. The numercal consstences of the backward and boundary equatons wth the IAPWS-IF fundamental equatons are suffcent for most alcatons n heat-cycle, boler, and steam-turbne calculatons. Therefore, by usng the backward and boundary equatons, the roertes as functons of (, T ), (, h ), (, s ), and ( hs, ), ncludng determnaton of the regon, can be calculated wthout teratons. As a result, rocess calculatons wll be between 2 and 3 tmes faster when usng the sulementary backward and boundary equatons. For alcatons where the demands on numercal consstency are extremely hgh, teratons usng the IAPWS-IF fundamental equatons may stll be necessary. In these cases, the backward or boundary equatons resented here can be used for calculatng very accurate startng values. Users who are nterested n the equatons here resented can receve the source code uon request; see htt://thermodynamcs.hs-zgr.de. Acknowledgements The authors are ndebted to the charman of the Evaluaton Task Grou, K. Myagawa, and hs colleagues J. Gallagher,. Okta, and I. Weber for ther excetonal efforts when testng the equatons as to ther fulfllment of requrements. We are ndebted to all members of the IAPWS Workng Grous "Industral Requrements and Solutons," and "Thermohyscal Proertes of Water and Steam," and to other colleagues who contrbuted to the roject of develong the sulementary equatons for IAPWS-IF. We are artcularly grateful to the Saxony State Mnstry of Scence and Art of Germany for ts fnancal suort.

12 References and otes [] Release on the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam, IAPWS Secretarat, Electrc Power Research Insttute, Palo Alto, CA (9). [2] W. Wagner et al., The IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam, J. of Eng. for Gas Turbnes and Power, 22, 50 (2000). [3] Sulementary Release on Backward Equatons for Pressure as a Functon of Enthaly and Entroy (h,s) to the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam, IAPWS Secretarat, Electrc Power Research Insttute, Palo Alto, CA (20). [4] H.-J. Kretzschmar et al., Sulementary Backward Equatons for Pressure as a Functon of Enthaly and Entroy (h,s) to the Industral Formulaton IAPWS- IF for Water and Steam, J. of Eng. for Gas Turbnes and Power (20), n ress. [5] Sulementary Release on Backward Equatons for the Functons T(,h), v(,h) and T(,s), v(,s) for Regon 3 of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam, IAPWS Secretarat, Electrc Power Research Insttute, Palo Alto, CA (20). [6] Revsed Sulementary Release on Backward Equatons for the Functons T(,h), v(,h) and T(,s), v(,s) for Regon 3 of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam, IAPWS Secretarat, Electrc Power Research Insttute, Palo Alto, CA (20) to be adoted. [7] H.-J. Kretzschmar et al., Sulementary Backward Equatons T(,h), v(,h), and T(,s), v(,s) for the Crtcal and Suercrtcal Regons (Regon 3) of the Industral Formulaton IAPWS-IF for Water and Steam, J. of Eng. for Gas Turbnes and Power (20), to be submtted. [8] Sulementary Release on Backward Equatons (h,s) for Regon 3, Equatons as a Functon of h and s for the Regon Boundares, and an Equaton T sat (h,s) for Regon 4 of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam, IAPWS Secretarat, Electrc Power Research Insttute, Palo Alto, CA (20), to be adoted. [9] Kretzschmar et al., Sulementary Backward Equatons (h,s) for the Crtcal and Suercrtcal Regons (Regon 3), Equatons for the Regon Boundares, and an Equaton for the Two-Phase Regon of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam, J. of Eng. for Gas Turbnes and Power, to be submtted. [0] Sulementary Release on Backward Equatons for Secfc Volume as a Functon of Pressure and Temerature v(,t) for Regon 3 of the IAPWS Industral Formulaton 9 for the Thermodynamc Proertes of Water and Steam, IAPWS Secretarat, Electrc Power Research Insttute, Palo Alto, CA, to be adoted. [] K. Knobloch, I. Stoecker, H.-J. Kretzschmar, A. Dttmann, Sulementary Backward Equatons v(,t) for the Crtcal and Suercrtcal Regons (Regon 3) of the Industral Formulaton IAPWS-IF for Water and Steam, Proceedngs of the 4th Internatonal Conference on the Proertes of Water and Steam (4th ICPWS), Kyoto (20) [2] B. Rukes, Secfcatons for umercal Consstency, n: Mnutes of the Meetngs of the Executve Commttee of the Internatonal Assocaton for the Proertes of Water and Steam, Orlando 994, ed. by B. Dooley, Electrc Power Research Insttute, Palo Alto, CA, (994). [3] K. Knobloch, Glechungen für thermodynamsche Umkehrfunktonen von Wasser und Wasserdamf m krtschen und überkrtschen Zustandsgebet für energetechnsche Prozessberechnungen (Equatons for thermodynamc backward functons of water n the crtcal and suercrtcal regons for ower cycle calculatons), VDI Fortschrttsberchte, Rehe 6, n rearaton. [4] H.-J. Kretzschmar, I. Stöcker, J. Klnger, and A. Dttmann, Calculaton of Thermodynamc Dervatves for Water and Steam Usng the ew Industral Formulaton IAPWS-IF, n: Steam, Water and Hydrothermal Systems: Physcs and Chemstry Meetng the eeds of Industry, Proceedngs of the 3th Internatonal Conference on the Proertes of Water and Steam, ed. by P. R. Tremane et al., RC Press, Ottawa, (2000). [5] K. Myagawa, Comutng Tme Test of Backward Functons T(,h) and T(,s) for Regons and 2 of IAPWS-IF, IAPWS Secretarat, Electrc Power Research Insttute, Palo Alto, CA (20). [6] K. Myagawa, J. Gallagher,. Okta, and I. Weber, Test Reort of the Proosal of Sulementary Release on v(,t) Backward Equatons for Regon 3 of IAPWS-IF, IAPWS Secretarat, Electrc Power Research Insttute, Palo Alto, CA (2005).