The International Association for the Properties of Water and Steam

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1 IAPWS R16-17(018) The Intenatnal Asscatn f the Petes f Wate and Steam Pague, Czech Reublc Setembe 018 Revsed Release n the IAPWS Fmulatn 017 f the Themdynamc Petes f Heavy Wate 018 Intenatnal Asscatn f the Petes f Wate and Steam Publcatn n whle n at s allwed n all cuntes vded that attbutn s gven t the Intenatnal Asscatn f the Petes f Wate and Steam Please cte as: Intenatnal Asscatn f the Petes f Wate and Steam, IAPWS R16-17(018), Revsed Release n the IAPWS Fmulatn 017 f the Themdynamc Petes f Heavy Wate (018) Ths evsed elease elaces the cesndng elease f 017, whch elaced the evsed elease f 005 (IAPWS R3-84(005)), and cntans 17 ages, ncludng ths cve age. Ths Revsed Release has been authzed by the Intenatnal Asscatn f the Petes f Wate and Steam (IAPWS) at ts meetng n Pague, Czech Reublc, -7 Setembe, 018. The membes f IAPWS ae: Austala, Btan and Ieland, Canada, the Czech Reublc, Gemany, Jaan, New Zealand, Russa, Scandnava (Denmak, Fnland, Nway, Sweden), and the Unted States, and asscate membes Agentna and Bazl, Chna, Egyt, Fance, Geece, Italy, and Swtzeland. The Pesdent at the tme f adtn f ths dcument was Pf. Hans-Jachm Ketzschma f Gemany. Summay The fmulatn vded n ths Release s ecmmended f calculatng themdynamc etes f heavy wate, whch IAPWS defnes as wate whse hydgen atms ae entely the deuteum ste ( H D) and whse xygen atms have the stc cmstn f dnay wate [1]. Futhe detals abut the fmulatn can be fund n the atcle A Refeence Equatn f State f Heavy Wate by S. Heg et al. []. Ths fmulatn vdes the mst accuate eesentatn avalable at the tme ths Release was eaed f the themdynamc etes f the flud hases f heavy wate ve a wde ange f cndtns. A evsn t the gnal 017 Release was adted n 018 n de t educe the uncetanty f etes (atculaly densty) n the lqud egn. Futhe nfmatn abut ths Revsed Release and the dcuments ssued by IAPWS can be btaned fm the Executve Secetay f IAPWS (D. R.B. Dley, bdley@aws.g) fm htt://

2 Cntents 1 Nmenclatue Refeence Cnstants 3 3 The Fmulatn 3 4 Relatns f Themdynamc Petes t the Dmensnless Helmhltz Fee Enegy 5 5 Range f Valdty 9 6 Meltng and Sublmatn Cuves 10 7 Estmates f Uncetanty 11 8 Cmute-Pgam Vefcatn 11 9 Refeences 13 1 Nmenclatue Themdynamc quanttes: Suescts B Secnd val ceffcent deal-gas ety C Thd val ceffcent esdual c Mla sbac heat caacty satuated lqud state c v Mla schc heat caacty satuated va state f Mla Helmhltz fee enegy h Mla enthaly Subscts M Mla mass c ctcal nt Pessue m meltng R Mla gas cnstant n educng aamete s Mla enty subl sublmatn T Abslute temeatue t tle nt u Mla ntenal enegy va-lqud satuatn w Seed f sund s Dmensnless Helmhltz fee enegy, = f/(rt) Isentc thttlng ceffcent Reduced densty, = /c T Isthemal thttlng ceffcent Isthemal cmessblty T Jule-Thmsn ceffcent Reduced temeatue, = T/Tn Mla densty Invese educed temeatue, = Tc/T Nte: T dentes abslute temeatue n the Intenatnal Temeatue Scale f Ths s unaffected by the evsn t the SI system f unts scheduled t g nt effect n 019.

3 3 Refeence Cnstants Tc = K (1) c = ml dm 3 () R = J ml 1 K 1 (3) M = g ml 1 (4) The numecal value f the ctcal temeatue Tc s dentcal t that gven n the IAPWS elease n the ctcal aametes f dnay and heavy wate [3]. The value f the ctcal densty c s the massc ctcal densty gven n [3] cnveted t mla densty by the mla mass M, whch s that gven n [1]. The value f the mla gas cnstant R s that gven by the 014 CODATA evaluatn [4]. Due t the use f the mla gas cnstant, Eq. (5) cesnds t a mle-based fmulatn. In de t cnvet values f mla etes t mass-based etes, a mla mass must be used. In geneal, ths shuld be the value f M gven by Eq. (4). In sme secal cases (such as heavy wate that s enched n heavy xygen stes), secfc nfmatn may be avalable that the stc cmstn dffes fm the defntn f heavy wate gven n [1]; n such cases t may be aate t use a dffeent mla mass cesndng t that dffeent stc cmstn. 3 The Fmulatn The fmulatn s a fundamental equatn f the mla Helmhltz fee enegy f. Ths equatn s exessed n dmensnless fm, = f/(rt), and s seaated nt tw ats, an dealgas at and a esdual at, s that: f T, RT,,,, (5) whee = Tc/T and = /c wth Tc, c, and R gven by Eqs. (1), (), and (3). The deal-gas at f the dmensnless Helmhltz fee enegy s btaned fm a ft t the hgh-accuacy calculatns by Smkó et al. [5] f the sbac heat caacty n the deal-gas state. It s gven by: c 1. (6), a a ln c 1ln vln 1 exu T

4 4 Table 1 cntans the ceffcents and aametes f Eq. (6). Table 1. Numecal values f the ceffcents and aametes f the deal-gas at f the dmensnless Helmhltz fee enegy, Eq. (6) Paamete Value Paamete Value c a a u K u 1695 K u K u K The fm f the esdual at f the dmensnless Helmhltz fee enegy s: 6 1 d t d t l, n n ex d t n ex. All aametes (ceffcents n, temeatue exnents t, densty exnents d and l, and aametes f the Gaussan bell-shaed tems,,, and ) ae lsted n Table. The IAPWS efeence-state cnventn s that the ntenal enegy and the enty f the satuated lqud at the tle nt ae set equal t ze. Thus, at the tle-nt temeatue T t = K [6], t t (7) u 0, s 0. (8) In de t meet ths cndtn, the ceffcents a1 and a n Eq. (6) have been adjusted accdngly. Because f dffeences n cmute cdes (such as cnvegence tleances), a atcula cmute cde may duce values that dffe slghtly fm the zes n Eq. (8). If a use wants t educe Eq. (8) me ecsely, the cnstants a1 and a n Eq. (6) may be eadjusted by msng the cndtn ut 0, s t 0 wth the desed accuacy.

5 5 Table. Numecal values f the ceffcents and aametes f the esdual at f the dmensnless Helmhltz fee enegy, Eq. (7) n t d l Relatns f Themdynamc Petes t the Dmensnless Helmhltz Fee Enegy All themdynamc etes can be deved fm Eq. (5) by usng the aate cmbnatns f the deal-gas at, Eq. (6), and the esdual at, Eq. (7), f the dmensnless Helmhltz fee enegy and the devatves. Relatns between themdynamc etes and and and the devatves ae summazed n Table 3. All equed devatves f the deal-gas at and f the esdual at f the Helmhltz fee enegy ae gven n Table 4 and Table 5, esectvely. Besdes the sngle-hase egn, the fmulatn als descbes va-lqud equlba. F a gven temeatue T, slvng the hase-equlbum equatns (see Table 3) yelds the satuatn etes,, and at T. Wth these, all the satuatn etes can be deved fm Eq. (5). In ths way, the etes calculated n the satuatn cuve ae themdynamcally cnsstent wth the etes f the sngle-hase egn.

6 6 Table 3. Relatns f themdynamc etes t the deal-gas at and the esdual at f the dmensnless Helmhltz fee enegy and the devatves a Pety Relatn Pessue, 1 f/ RT Mla ntenal enegy T u ft f/ T Mla enty s f/ T Mla enthaly h f Tf/ T f/ T Mla schc heat caacty cv u/ T Mla sbac heat caacty c h/ T Seed f sund w M 1 / 1/ Jule-Thmsn ceffcent T/ h Isthemal thttlng ceffcent h/ T T Isentc temeatue-essue ceffcent T/ s s s u, RT s, R h, 1 RT c v, R 1 c, R 1 w, RT M R T 1 1 s R 1 1 Secnd val ceffcent BT lm / RT / 0 Thd val ceffcent 1 CT lm / RT/ 0 T Phase-equlbum cndtn (Maxwell cten) T t c lm, t B n n ex 0,3,7,8,1 13,15,17,18,,3,4 t 7 c lm, t t C 7 n n n ex 0 4,5,10,11 19,0,1 t 4 n ex 13,15,17,18,,3,4 1, ; 1, RT RT 1 1 ln,, RT,,,,,,. a

7 7 Table 4. The deal-gas at f the dmensnless Helmhltz fee enegy and ts devatves a = ln + a1 + a + c 1ln u Tc v ln 1 e = 1/ = 1/ = a + c 1/ vu T c 1 e c u T 1 = c vu T 1/ = u Tc e c 1 u Tc 1 e a,,,,.

8 Table 5. The esdual at f the dmensnless Helmhltz fee enegy and ts devatves a l 4 d t d t d e t e n n n l d1 t d1 t l d 1 e t nd n d l n e d nd d n d l d l l n d l 1 1 t d t l l l t e e d d d d d d 1 d l d 1 1 t d t d e t e t nt nt n d 1 e l t d t d t e t t nt t nt t n l d1 t1 d1 t1 l d e t d e t ndt nt d l n a,,,,.

9 9 5 Range f Valdty IAPWS has tested the fmulatn and endses ts valdty n the fllwng way: (1) The fmulatn s vald n the ente stable flud egn f D O fm the meltng-essue cuve (see Sectn 6) t 85 K at essues fm the tle-nt essue t 100 MPa; the lwest temeatue n the meltng-essue cuve s K (at.40 MPa). At essues belw the tle-nt essue and temeatues abve 100 K, t s als vald f the stable va at temeatues fm the sublmatn cuve 1 u t 85 K. In ths ente egn, Eq. (5) eesents the exemental data avalable at the tme the Release was eaed (excet f vey few data nts) t wthn the uncetantes. Althugh Eq. (5) s als n satsfacty ageement wth the lmted exemental data n the ctcal egn, the equatn, lke any analytcal Helmhltz enegy fmulatn, s nt able t educe the nnclasscal values f ctcal exnents n the mmedate vcnty f the ctcal nt. () In the stable flud egn, the fmulatn can als be extalated beynd the lmts gven unde tem (1). Tests shw that Eq. (5) behaves easnably when extalated t essues u t abut 100 GPa and temeatues u t abut 5000 K. Ths hlds at least f the densty and enthaly f undsscated D O. (3) The fmulatn behaves easnably when extalated nt the metastable egns. Equatn (5) s n geneal ageement, n mst cases wthn exemental uncetantes, wth the cuently avalable exemental data f the subcled lqud (metastable wth esect t sld). N elable exemental data ae avalable f ethe the sueheated lqud (metastable wth esect t va) the suecled va (metastable wth esect t lqud), but the behav f Eq. (5) when extalated nt these egns s hyscally easnable. F futhe detals, see []. 1 Nte that the equatn f the sublmatn cuve n Sectn 6 shuld nt be extalated belw 150 K.

10 10 6 Meltng and Sublmatn Cuves Whle IAPWS has nt made an ffcal ecmmendatn f the meltng and sublmatn cuves f heavy wate, t s necessay t have a eesentatn f these cuves n de t cmute the bundaes f the ange f valdty as gven n Sectn 5. F that use, the celatns gven n [] ae adted, and summazed hee. Meltng-essue equatn f DO ce Ih (temeatue ange fm K t K): m,ce Ih n , (9) wth educed temeatue = T/Tn and the educng aametes Tn = K and n = MPa. F checkng cmute mlementatns, at 70 K Eq. (9) gves a meltng essue f MPa. Meltng-essue equatn f DO ce III (temeatue ange fm K t K): m,ce III 0 33 n , (10) wth = T/Tn, Tn = K, and n =.41 MPa. F checkng cmute mlementatns, at 55 K Eq. (10) gves a meltng essue f MPa. Meltng-essue equatn f DO ce V (temeatue ange fm K t K): m,ce V n , (11) wth = T/Tn, Tn = K, and n = MPa. F checkng cmute mlementatns, at 75 K Eq. (11) gves a meltng essue f MPa. Meltng-essue equatn f DO ce VI (temeatue ange fm K t 315 K): m,ce VI 1 4 n , (1) wth = T/Tn, Tn = K, and n = MPa. F checkng cmute mlementatns, at 300 K Eq. (1) gves a meltng essue f MPa. Sublmatn-essue equatn f DO ce Ih (temeatue ange fm 10 K t K): subl ln , (13) n wth = T/Tn, Tn = K, and n = MPa. F checkng cmute mlementatns, at 45 K Eq. (13) gves a sublmatn essue f MPa. Equatn (13) extalates easnably dwn t at least 150 K.

11 11 7 Estmates f Uncetanty Estmates have been made f the uncetanty f the densty, seed f sund, sbac heat caacty, and va-lqud satuatn essue and cexstng denstes when calculated fm the fmulatn, Eq. (5). These estmates wee deved fm cmasns wth the vaus sets f exemental data tgethe wth the judgment f the Wkng Gu n Themhyscal Petes f Wate and Steam f IAPWS. IAPWS cnsdes these as estmates f the exanded uncetantes wth cveage fact k =, ughly cesndng t a 95 % cnfdence nteval. F the sngle-hase egn, these values ae ndcated n Fgs. 1 3, whch gve the estmated uncetantes n vaus egns. Wth egad t the uncetanty f the seed f sund and the secfc sbac heat caacty, see Fgs. and 3, t shuld be nted that the uncetantes f these etes ncease dastcally when aachng the ctcal nt. The statement n defntve uncetanty estmates ssble f sme egns s based n the lack f exemental data n the egn. Hweve, because the fmulatn was valdated aganst exemental data and wth egad t cect hyscal behav ve ts ente ange f valdty, the esults n these egns shuld be hyscally easnable. As nted n the catns f Fgs. 1 3, the uncetantes f etes f the va at lw essues becme small. Ths s because the lw-densty va aaches the deal-gas lmt; the deal-gas behav s knwn essentally exactly f the densty and t a elatvely small uncetanty f the seed f sund and sbac heat caacty. F the satuatn etes, the estmates f the exanded uncetantes f va essue, satuated lqud densty, and satuated va densty ae shwn n Fg Cmute-Pgam Vefcatn T assst the use n cmute-gam vefcatn, thee tables wth test values ae gven. Table 6 cntans values f the deal-gas at and the esdual at f the dmensnless Helmhltz fee enegy tgethe wth the cesndng devatves. Table 7 lsts values f the essue, the secfc schc heat caacty c v, the seed f sund w, and the secfc enty s calculated at selected values f temeatue T and densty. Table 8 gves values f the va essue, values f the densty, enthaly h, and enty s f the satuated lqud, and values f the densty, enthaly h, and enty s f the satuated va. All these satuatn values have been calculated wth Eq. (5) by usng the hase-equlbum cndtn (see the cesndng cmment n Sectn 4).

12 1 Table 6. Values f the deal-gas at, Eq. (6), and f the esdual at, Eq. (7), f the dmensnless Helmhltz fee enegy tgethe wth the cesndng devatves a f T = 500 K and = 46.6 ml dm 3 1 = = = = = = = = = = = 0 1 = a F the abbevated ntatn f the devatves f and see the ftntes f Tables 4 and 5, esectvely. Table 7. Themdynamc ety values n the sngle-hase egn f selected values f T and T/K /(ml dm 3 ) /MPa cv/(j ml 1 K 1 ) w/(m s 1 ) s/(j ml 1 K 1 ) a a In the lqud-wate egn at lw essues, small changes n densty alng an sthem cause lage changes n essue. F ths easn, due t an accumulatn f small es, a atcula cmute cde a atcula cmute may fal t educe the essue value wth nne sgnfcant fgues.

13 13 Table 8. Themdynamc ety values f va-lqud satuatn states at selected temeatues a T = 80 K T = 450 K T = 65 K /MPa / (ml dm 3 ) / (ml dm 3 ) h/(j ml 1 ) h/(j ml 1 ) s/(j ml 1 K 1 ) s/(j ml 1 K 1 ) a All these test values wee calculated fm the Helmhltz fee enegy, Eq. (5), by alyng the hase-equlbum cndtn (Maxwell cten). 9 Refeences [1] IAPWS, G5-01(016), Gudelne n the Use f Fundamental Physcal Cnstants and Basc Cnstants f Wate (001). Avalable fm htt:// [] Heg, S., Thl, M., Havey, A.H., and Lemmn, E.W., A Refeence Equatn f State f Heavy Wate, J. Phys. Chem. Ref. Data 47, (018). [3] IAPWS, R-83(199), Release n the Values f Temeatue, Pessue, and Densty f Odnay and Heavy Wate Substances at The Resectve Ctcal Pnts (199). Avalable fm htt:// [4] Mh, P.J., Newell, D.B., and Tayl, B.N., CODATA Recmmended Values f the Fundamental Physcal Cnstants: 014, J. Phys. Chem. Ref. Data 45, (016) [als ublshed n Rev. Md. Phys. 88, (016)]. [5] Smkó, I., Futenbache, T., Hubý, J., Zbv, N.F., Plyansky, O.L., Tennysn, J., Gamache, R.R., Szdavszky, T., Dénes, N., and Császá, A.G., Recmmended Ideal-gas Themchemcal Functns f Heavy Wate and ts Substtuent Istlgues, J. Phys. Chem. Ref. Data 46, (017). [6] Makó, L., Jákl, G., and Jancsó, G., Vau essue f heavy wate at ts tle nt, J. Chem. Themdyn. 1, (1989).

14 14 Fg. 1. Exanded elatve uncetantes n densty, /, estmated f Eq. (5). In the enlaged ctcal egn (tangle), the uncetanty s gven as ecentage uncetanty n essue, /. Ths egn s bdeed by the tw sches 8 ml dm 3 and 9 ml dm 3 and by the 30 MPa sba. The stns f the lnes seaatng the uncetanty egns ae axmate. At lw essues f the va, the uncetantes becme much smalle than ndcated because the va s nealy an deal gas.

15 15 Fg.. Exanded elatve uncetantes n seed f sund, w/w, estmated f Eq. (5). F the defntn f the tangle aund the ctcal nt, see the Fg. 1 catn. The stns f the lnes seaatng the uncetanty egns ae axmate. At lw essues f the va, the uncetanty becmes small because the va aaches the deal-gas lmt.

16 16 Fg. 3. Exanded elatve uncetantes n secfc sbac heat caacty, c /c, estmated f Eq. (5). F the defntn f the tangle aund the ctcal nt, see the Fg. 1 catn. The stns f the lnes seaatng the uncetanty egns ae axmate. The uncetanty n the va hase at lw essues aaches the uncetanty f the deal-gas heat caacty, whch s less than 0.0 %.

17 17 Fg. 4. Uncetantes n va essue, / n satuated lqud densty,/, and n satuated va densty, /, estmated f Eq. (5). The uncetantes f the satuated denstes ncease t 1.5 % at the ctcal temeatue.

The International Association for the Properties of Water and Steam

The International Association for the Properties of Water and Steam IAPWS R6-95(016) The Intenatnal Asscatn f the Ppetes f Wate and Steam Desden, Gemany Septeme 016 Revsed Release n the IAPWS Fmulatn 1995 f the Themdynamc Ppetes f Odnay Wate Sustance f Geneal and Scentfc