The International Association for the Properties of Water and Steam

Transcription

1 IAPWS SR7-9 he International Aociation for the Proertie of Water and Steam Doorwerth, he Netherland Setember 9 Sulementary Releae on a Comutationally Efficient hermodynamic Formulation for Liquid Water for Oceanorahic Ue 9 International Aociation for the Proertie of Water and Steam Publication in whole or in art i allowed in all countrie rovided that attribution i iven to the International Aociation for the Proertie of Water and Steam Preident: Dr. Daniel G. Friend hermohyical Proertie Diviion National Intitute of Standard and echnoloy 35 Broadway Boulder, CO, 835, USA Executive Secretary: Dr. R. B. Dooley Structural Interity Aociate, Inc. 66 Chelea Drive Charlotte, NC 89, USA bdooley@tructint.com hi releae contain 9 ae, includin thi cover ae. hi releae ha been authorized by the International Aociation for the Proertie of Water and Steam (IAPWS) at it meetin in Doorwerth, he Netherland, 6- Setember, 9, for iue by it Secretariat. he member of IAPWS are: Britain and Ireland, Canada, the Czech Reublic, Denmark, France, Germany, Greece, aan, Ruia, the United State of America, and aociate member Arentina and Brazil, Italy, and Switzerland. he equation of tate rovided in thi releae i a fundamental equation for the Gibb enery a a function of temerature and reure; detail can be found in the article A new extended Gibb thermodynamic otential of eawater and A Gibb function for eawater thermodynamic for 6 to 8 C and alinity u to k by R. Feitel [, ]. hi equation can be ued intead of the IAPWS-95 formulation [3, 4] for thermodynamic roertie of liquid water required for calculatin roertie of eawater accordin to the IAPWS eawater formulation [5]. It i retricted to the oceanorahic tandard rane of temerature and reure, within which it deviation from IAPWS-95 are well within the uncertaintie of IAPWS-95. Further information about thi ulementary releae and other document iued by IAPWS can be obtained from the Executive Secretary of IAPWS or from htt://

2 Content Nomenclature Introductory Remark and Secial Contant 3 3 he Equation of State 4 4 Relation of the hermodynamic Proertie to the Secific Gibb Enery 6 5 Rane of Validity and Brief Dicuion 7 6 Etimate of Uncertainty 7 7 Comuter-Proram Verification 8 8 Reference 8 Nomenclature Symbol Phyical quantity Unit c Secific iobaric heat caacity k K h Secific Helmholtz enery k Secific Gibb enery k * Reducin ecific Gibb enery, * = k k 76 Coefficient of the Gibb otential function, able h Secific enthaly k k Uncertainty coverae factor Abolute reure Normal reure, = 35 t IAPWS-95 calculated trile-oint reure, t = * Reducin reure, * = 8 Secific entroy k K Abolute temerature (IS-9) K Celiu zero oint, = 73.5 K K t rile-oint temerature, t = 73.6 K K * Reducin temerature, * = 4 K K t Celiu temerature C t min Lower temerature bound C u Secific internal enery k U u c Exanded uncertainty Combined tandard uncertainty

3 Symbol Phyical quantity Unit v Secific volume m 3 k w Seed of ound m α hermal exanion coefficient K β Ientroic temerature-reure coefficient K κ Ientroic comreibility κ Iothermal comreibility π Reduced reure, π = ( * ρ Denity k m 3 τ Reduced temerature, τ = ( * 3 Introductory Remark and Secial Contant he Releae on the IAPWS Formulation 8 for the hermodynamic Proertie of Seawater [5] decribe the difference between the ecific Gibb enerie of eawater and water. At iven temerature and reure, the water art mut earately be comuted from the Helmholtz function rovided by the IAPWS-95 formulation [3, 4], determinin the ure-water denity by numerical iteration. A a function of temerature and reure, the Gibb function for liquid water rovided in thi Sulementary Releae eliminate the need for iteration. It i retricted in it validity to the rane to M and t min ( ) = ( M ) C to 4 C, only lihtly exceedin the oceanorahic tandard rane to include the trile oint and the lowet oceanic freezin oint. hi Gibb function inificantly imlifie the imlementation and reduce the comutin time for oceanorahic alication uch a numerical circulation model or real-time rocein of in-itu data, while till ivin value for roertie whoe areement with IAPWS-95 i well within the uncertaintie aociated with IAPWS-95. ABLE Secial contant and value ued in thi releae Quantity Symbol Value Unit Reference rile-oint reure a t [3] Normal reure 35 [7] Reducin reure * 8 [] rile-oint temerature t 73.6 K [8] Celiu zero oint 73.5 K [8] Reducin temerature * 4 K [] a Numerical trile-oint reure value comuted from IAPWS-95 [3], bein well within the uncertainty of the exerimental value of 6.657() [6]

4 4 3 he Equation of State he equation of tate reented here i in the form of the ecific Gibb enery a a function of temerature and reure, (, ), correondin to a Gibb otential. he temerature are baed on the temerature cale IS-9 [8]. Reduced by * = k, the Gibb function i the dimenionle olynomial iven by Eq. () (, ) / * = j k jk, () j= k= with the reduced temerature τ = ( * and the reduced reure π = ( *. he reduced quantitie τ and π vary from to in the oceanorahic tandard rane. he contant,, *, and * are iven in able. he coefficient of Eq. () are iven in able. wo of thee 4 arameter ( and ) are arbitrary and are comuted from the reference-tate condition of vanihin ecific entroy,, and ecific internal enery, u, of liquid water at the trile oint, ( t, t ) =, () u( t, t ) =. (3) ABLE Coefficient of the Gibb function, a iven by Eq. (). Coefficient not contained in thi table have the value jk =. j k jk j k jk

5 5 ABLE 3 Relation of the thermodynamic roertie to the equation for the Gibb enery of liquid water, Eq. (), and it - derivative a Proerty Relation Unit Eq. Denity ( ) ρ, v = ( / ) (, ) = Secific entroy (, ) = ( / ) (, ) = Secific iobaric heat caacity c (, ) = ( / ) c (, ) = Secific enthaly (, ) = h(, ) = h + Secific internal enery u k m ρ = 3 (, ) = + v u(, ) = Secific Helmholtz enery f (, ) = v f (, ) = hermal exanion coefficient ( ) = v α, ( v / ) α (, ) = / Ientroic temerature-reure coefficient, adiabatic lae rate β (, ) = ( / ) β (, ) = / Iothermal comreibility κ (, ) = v ( v / ) κ (, ) = / Ientroic comreibility (, ) = v ( v / ) κ κ (, ) = ( ) ( ) Seed of ound w ( t ) = ( / ρ ), w(, ) = ( ) k K k K k k k K K m (4) (5) (6) (7) (8) (9) () () () (3) (4) a,,,,

6 4 Relation of the hermodynamic Proertie to the Secific Gibb Enery 6 hermodynamic roertie can be derived from Eq. () by uin the aroriate combination of the ecific Gibb enery and it derivative. Relation between thermodynamic roertie and (, ) and it derivative with reect to and are ummarized in able 3. All required derivative of the aline art of the ecific Gibb enery are exlicitly iven in able 4. ABLE 4 Equation for the Gibb enery, Eq. (), and it derivative a Equation Unit (, ) = * jk j= k= j k with * = k, ( * τ =, = 73.5 K, * = 4 K, ( * π =, = 35, * = 8 * j k (, ) = j jk K * j= k= k k (, ) = * jk * j= k= 7 * ( *) 6 k j k (, ) = j( j ) jk j= k= τ j k π m 3 k k K m 3 * j k (, ) = jk jk k K * * j= k= * j k (, ) = k( k ) jk j= k= ( *) m 3 k a,,,,

7 7 5 Rane of Validity and Brief Dicuion he equation of tate, Eq. (), i valid for liquid water only within the reure and temerature rane 8 and ( ) K 33.5 K, includin the reion of ambient ocean water. he lower temerature limit deend on the reure roortional to the exerimental Clauiu-Claeyron coefficient [9] and cover the lowet temerature found in the lobal ocean: the freezin temerature of about 4.5 C off the Antarctic helf at 3 m deth. he deviation of roertie derived from thi Gibb function from thoe calculated with IAPWS-95 are inificantly le than the exerimental uncertainty for each roerty (able 5). Within it rane of validity, it i intended to be ued a a imler ubtitute for IAPWS-95, rovidin the water art of the Gibb function of eawater [5] without inificant lo in accuracy. hi formulation for the thermodynamic roertie of liquid water wa develoed in cooeration with the SCOR/IAPSO Workin Grou 7 on hermodynamic and Equation of State of Seawater. 6 Etimate of Uncertainty Here, etimated combined tandard uncertaintie u c [] are reorted, from which exanded uncertaintie U = k u c can be obtained by multilyin with the coverae factor k =, correondin to a 95% confidence level. he term uncertainty ued in the followin refer to combined tandard uncertaintie or to relative combined tandard uncertaintie. Deviation of elected roertie of thi ulementary releae from IAPWS-95 are reorted in able 5. hee deviation are well within the etimated uncertaintie of IAPWS-95. ABLE 5 Combined tandard uncertaintie from IAPWS-95 [3, 4] of elected quantitie in certain reion of the - ace, comared with the root-mean-quare (r.m..) and the maximum deviation between IAPWS-95 and the formulation of thi releae. he reure-deendent lower temerature bound i tmin = ( M ) C. For ρ and w the uncertaintie and deviation are relative and therefore dimenionle. Quantity interval interval C M IAPWS-95 Uncertainty r.m.. Deviatio n Maximum Deviation Unit c k K α a K α t min () ρ ρ t min () ρ t min () w t min () a Etimate adoted from eawater data [] a..9 K

8 8 7 Comuter-Proram Verification o ait the uer in comuter-roram verification, able 6 with tet value i iven. It contain value for the ecific Gibb enery, (, ), toether with the correondin derivative and ome thermodynamic roertie. 8 Reference [] Feitel, R., Pror. Oceanor. 58, 43 (3). [] Feitel, R., Dee-Sea Re. I 55, 639 (8). [3] IAPWS, Revied Releae on the IAPWS Formulation 995 for the hermodynamic Proertie of Ordinary Water Subtance for General and Scientific Ue (9). Available from htt:// [4] Waner, W., and Pruß, A.,. Phy. Chem. Ref. Data 3, 387 (). [5] IAPWS, Releae on the IAPWS Formulation 8 for the hermodynamic Proertie of Seawater (8). Available from htt:// [6] Guildner, L. A., ohnon, D. P., and one, F. E.,. Re. Natl. Bur. Stand. 8A, 55 (976). [7] ISO, ISO Standard Handbook: Quantitie and Unit (International Oranization for Standardization, Geneva, 993). [8] Preton-homa, H., Metroloia 7, 3 (99). [9] Feitel, R., and Waner, W.,. Phy. Chem. Ref. Data 35, (6). [] ISO, Guide to the Exreion of Uncertainty in Meaurement (International Oranization for Standardization, Geneva, 993).

9 9 ABLE 6 Numerical check value for the Gibb function and it derivative, able 4, he numerical function evaluated here at iven oint (, ) are defined in able 3 and 4. Quantity Value Value Value Unit K k ( / ) k K ( / ) m 3 k ( / ) k K / m 3 k K ( / ) m 3 k h k f k u k k K ρ k m 3 c k K w m