The International Association for the Properties of Water and Steam

Transcription

1 IAP AN5-13(2016) he International Association for the Proerties of ater and team London, United Kindom etember 2013 Advisory Note No. 5: Industrial Calculation of the hermodynamic Proerties of eawater 2013 International Association for the Proerties of ater and team Publication in whole or in art is allowed in all countries rovided that attribution is iven to the International Association for the Proerties of ater and team Please cite as: International Association for the Proerties of ater and team, Advisory Note No. 5: Industrial Calculation of the hermodynamic Proerties of eawater (2013). his Advisory Note has been authorized by the International Association for the Proerties of ater and team (IAP) at its meetin in London, United Kindom, 1 6 etember, he Members of IAP are: Britain and Ireland, Canada, the Czech Reublic, Germany, Jaan, Russia, candinavia (Denmark, Finland, Norway, weden), and the United tates, lus Associate Members Arentina and Brazil, Australia, France, Greece, New Zealand, and witzerland. he President at the time of adotion of this document was Professor amara Petrova of Russia. ummary he calculation rocedure for the thermodynamic roerties of seawater for industrial use rovided in this Advisory Note is based on the IAP Formulation 2008 for the hermodynamic Proerties of eawater [2]. For calculatin the ure water contribution, the IAP Formulation 1995 for the hermodynamic Proerties of Ordinary ater ubstance for General and cientific Use [5] is relaced here by the IAP Industrial Formulation 1997 for the hermodynamic Proerties of ater and team [7]. For seawater in contact with ice, the Revised IAP Release on an Equation of tate 2006 for H2O Ice Ih [11] is used. Further details about the formulation can be found in the article by H.-J. Kretzschmar et al. [1]. Minor editorial chanes were made in 2016 to correct the last diits in a few calculated values in able A1 and to udate some references. his Advisory Note contains 24 aes, includin this cover ae. Further information about this Advisory Note and other documents issued by IAP can be obtained from the Executive ecretary of IAP (Dr. R.B. Dooley, bdooley@structint.com) or from htt://

2 2 Contents 1. Nomenclature 3 2. Introductory Remarks 5 3. hermodynamic Proerties of eawater he Fundamental Equation of tate for eawater ater Part aline Part 9 4. Colliative Proerties Phase Equilibrium between eawater and ater Vaor Phase Equilibrium between eawater and Ice rile-point emeratures and Pressures Osmotic Pressure Proerties of Brine-Vaor Mixture Proerties of Brine-Ice Mixture Reference tates Rane of Validity Uncertainty Comutin ime for the IAP eawater Functions Comuter-Proram Verification References 17 APPENDIX 19

3 3 1. Nomenclature ymbol b c c f v Physical quantity Molality of seawater ecific isobaric heat caacity of seawater ecific isochoric heat caacity of seawater ecific Helmholtz free enery of seawater ecific Gibbs free enery of seawater BI ecific Gibbs free enery of brine-ice mixture (sea ice) BV ecific Gibbs free enery of brine-vaor mixture Ih ecific Gibbs free enery of ice Ih aline art of the secific Gibbs free enery of seawater va ecific Gibbs free enery of water vaor ater art of the secific Gibbs free enery of seawater h h m m M b f osm ecific enthaly of seawater ecific enthaly of water Mass of seawater includin water vaor or ice Mass of sea salt in seawater Molar mass of sea salt Pressure Boilin ressure of seawater Freezin ressure of seawater Osmotic ressure of seawater t R Rm Re RM b f n rile-oint ressure of seawater ecific as constant of water Molar as constant Real art of a comlex number Root-mean-square value of a quantity alinity, mass fraction of sea salt in seawater includin water vaor or ice alinity of boilin brine alinity of freezin brine Normal salinity of seawater s ecific entroy of seawater s ecific entroy of water b Absolute temerature (I-90) Boilin temerature of seawater

4 4 ymbol f t Physical quantity Freezin temerature of seawater rile-oint temerature of seawater u Uncertainty of a roerty iven in able 3 u ecific internal enery of seawater u ecific internal enery of water v w x y α v β γ κ κ ecific volume of seawater eed of sound of seawater Mass fraction of the vaor hase in the brine-vaor mixture (vaor fraction) Mass fraction of the ice hase in the brine-ice mixture (ice fraction) Isobaric cubic exansion coefficient (thermal exansion coefficient) of seawater Haline contraction coefficient of seawater Reduced Gibbs free enery Isentroic exonent of seawater Isothermal comressibility of seawater µ Relative chemical otential of seawater µ Chemical otential of sea salt in seawater µ Chemical otential of water in seawater ξ π ρ τ φ Reduced salinity root Reduced ressure Mass density of seawater Reduced or inverse reduced temerature Osmotic coefficient of seawater

5 5 2. Introductory Remarks In 2008, IAP adoted the IAP Formulation 2008 for the hermodynamic Proerties of eawater [2, 3, 4], where the thermodynamic roerties of seawater are calculated from an equation of state, consistin of a water art and a saline art. he water art is comuted from the Helmholtz free enery equation of the IAP Formulation 1995 for the hermodynamic Proerties of Ordinary ater ubstance for General and cientific Use (IAP-95) [5, 6]. he saline art is formulated as a Gibbs free enery equation. However, the iterative calculation of required roerties from the IAP-95 Helmholtz free enery equation is comutationally intensive, makin its use less desirable in alications where comutational seed is imortant. In modelin industrial desalination and coolin rocesses, it is more reasonable to use the Gibbs free enery equation of Reion 1 of the IAP Industrial Formulation 1997 for the hermodynamic Proerties of ater and team (IAP-IF97) [7, 8] for the water art of the seawater formulation instead of IAP-95. In addition, the IAP- IF97 Industrial Formulation is used by industry for calculatin roerties of ure steam and water for ower lants and their comonents. he urose of this Advisory Note is to document the use of IAP-IF97 instead of IAP- 95 for calculatin thermodynamic roerties of the water art of the seawater formulation, to enable more efficient calculations. his industrial calculation of the thermodynamic roerties of seawater will be called the industrial formulation in this document. It should be noted that this usae differs from use of the simlified water Gibbs enery formulation adoted as the IAP ulementary Release on a Comutationally Efficient hermodynamic Formulation for Liquid ater for Oceanorahic Use [3, 9]. Reference [9] documents a formulation for liquid water thermodynamics to be used secifically for comutational efficiency in oceanorahic calculations, while the resent Advisory Note describes the alication of the existin IAP industrial formulation to calculate the water contribution to seawater roerties in industrial settins where the use of IAP-IF97 is desirable for other reasons. In this Advisory Note, the terms "seawater" and "brine" refer to a solution of sea salt in liquid water, the term "sea salt" refers to a mixed solute with the chemical Reference Comosition described in [10], "brine-vaor mixture" is a mixture of brine with water vaor, and "brine-ice mixture" is a mixture of brine with ice Ih. For water vaor, the IAP Revised Release on the IAP Industrial Formulation 1997 for the hermodynamic Proerties of ater and team (IAP-IF97) [7, 8] is used and for ice Ih, the IAP Revised Release on the Equation of tate 2006 for H 2 O Ice Ih (IAP-2006) [11, 12] is used. Note that salinity here refers to the socalled Absolute alinity of seawater which differs from the Practical alinity that is obtained from common salinometers or oceanorahic devices [2, 4, 10, 13].

6 6 he IAP Releases [2, 7, 11] referred to in this Advisory Note are available at the IAP website htt:// All details of the equations and alorithms should be taken from those Releases. 3. hermodynamic Proerties of eawater 3.1 he Fundamental Equation of tate for eawater he IAP equation of state for the liquid mixture "seawater" [2, 3, 4] is in the form of the secific Gibbs free enery as a function of ressure, temerature, and salinity, and reads ( ),, (, ) (,, ) As described in the followin sections, the water art (, ) formulation IAP-IF97, and the saline art (,, ) = +. (1) is comuted from the industrial from the IAP seawater formulation. he temerature is based on the International emerature cale I-90. alinity is the mass fraction of sea salt in seawater m = m, (2) where m is the mass of sea salt and m is the mass of the mixture seawater. he comosition of sea salt is assumed to be the Reference Comosition of tandard eawater [10]. For ure water with = 0, the saline art vanishes, (,,0) = 0. Usin salinity, the molality of seawater, defined as the quotient of moles of the solute sea salt er mass of the solvent water, is comuted as b = M ( 1 ), (3) where M is the molar mass of sea salt. Its value can be taken from able 1 of [2]. All thermodynamic roerties and derivatives can be calculated from Eq. (1) by usin the aroriate combinations of the Gibbs free enery equation and its derivatives with resect to,, and. Relations between relevant thermodynamic roerties and (,, ), Eq. (1), and its derivatives are summarized in able 1. 1 In addition, any thermodynamic derivative of the variables,, v, s, u, h, f, and, for examle, ( h/ ) v,, can be determined from alebraic combinations of v, s, c, α v, and κ as described in detail in IAP Advisory Note No. 3 [14]. 1 able 1 of [2] contains the value of the molar as constant R m used in the 2008 IAP formulation for seawater.

7 7 able 1 Relations between the relevant thermodynamic roerties of seawater and (,, ) Eq. (1), and its derivatives a, Proerty Relation ecific volume v v( ) = Density ρ = 1 v,, ρ (,, ) = 1 ecific internal enery u u(,, ) = ecific enthaly = + h,, h u v ( ) = ecific entroy s s( ) = ecific Helmholtz free enery = ecific isobaric heat caacity ecific isochoric heat caacity c,, f,, f u s ( ) = c h = v u =, v, 1 v Cubic isobaric exansion coefficient αv = v Isothermal comressibility eed of sound Isentroic exonent w= v v 1 v κ = v s, v κ = v s, Relative chemical otential µ =, Chemical otential of water µ = µ Chemical otential of sea salt µ = µ + µ Osmotic coefficient φ =,, (,, ) = c 2 cv (,, ) = α v (,, ) = κ (,, ) ( ) = w,, = 2 ( ) 1 κ (,, ) = ( ) µ,, = 2 ( ) µ ( ),, = ( ) ( ) µ = + ( µ ) br Haline contraction coefficient m 1 ρ β = ρ, φ β,, 1 (,, ) (,, ) = ( ) = br m

8 8 a Definitions of derivatives used in able 1: = = +,,, = = ,,, = = +,,, = = ,,, = = +, = =,,, 2 2 = = Backward functions are calculated by iteration from equations listed in able 1. For examle, for iven ressure, secific enthaly h, and salinity, the temerature is calculated from the equation for h(,, ) by iteration. 3.2 ater Part he water art of Eq. (1) is calculated from the secific Gibbs free enery equation of IAP-IF97 reion 1 34 (, ) Ii Ji = γ( πτ, ) = ni ( 7.1 π) ( τ ), R (4) i= 1 where the reduced Gibbs free enery is γ = /( R ), the reduced ressure is π = / * and the inverse reduced temerature is τ = */. he secific as constant R of water, the reducin arameters * and *, the coefficients n i and the exonents I i and J i are iven in ection 5.1 of the IAP-IF97 Release [7]. able 2 contains the relations for determinin (, ) derivatives from the reduced Gibbs free enery equation γ( πτ) equations for the derivatives of γ( πτ) and its ressure and temerature, and its derivatives. he, can be taken from able 4 in [7]. Usin the equations of able 2, the water arts in the equations of able 1 can be determined. Note that for temeratures less than the meltin temerature of ure water [15, 16], or reater than the saturation temerature of ure water [6, 7], the IAP-IF97 Gibbs free enery equation of liquid water reion 1, Eq. (4), is evaluated at conditions where the liquid hase of ure water is metastable. Investiations have shown that Eq. (4) can be reasonably extraolated into this metastable reion and even below K, the minimum temerature of IAP-IF97. he accuracy of the thermodynamic roerties of seawater calculated from IAP-IF97 is sufficient for industrial calculations. ection 7 contains a descrition of the deviations of IAP-IF97 from IAP-95.

9 9 able 2 Relations for determinin reduced Gibbs free enery equation ( ) R = R γ, = πγ π ( ) = R γ τγ τ (, ) and ressure and temerature derivatives from the γ πτ, and its derivatives of IAP-IF97 reion 1 a 2 R 2, = τ γ 2 ττ 2 R 2, = π γ 2 2 ππ 2 R π γ, = ( τγ ) π, πτ a γ 2 γ γ 2 γ 2 γ γ =, γ =, γ =, γ =, γ = π τ π τ π ππ 2 τ ττ 2 πτ τ π τ π τ π 3.3 aline Part he saline art of Eq. (1) is calculated from the Gibbs free enery equation of the IAP formulation for seawater roerties [2] (,, ) 2 i j k = 1 jkξ lnξ + ijkξ τ π *, (5) k= 0 j= 0 i= 2 where the reduced ressure is π = ( ), the reduced temerature is ( ) 0 / * τ = 0 / *, and the square root of the reduced salinity is ξ = / *. he constants *, *, *, *, 0, and 0 are iven in able 1 of the IAP eawater Release [2], and the coefficients ijk are iven in able 2 of [2]. able 6 of [2] contains the relations for determinin the followin derivatives,, = =,, 2 2 2, = =, from Eq. (5) which are used in able Colliative Proerties, =, Phase Equilibrium between eawater and ater Vaor, =, 2 For comutation of the equilibrium between seawater and water vaor, the required thermodynamic condition is the equality of the chemical otential of H2O in seawater µ with the secific Gibbs enery of water vaor va. Here, chemical otentials are exressed on a mass basis, which is the usual (molar) chemical otential divided by molar mass. =,

10 10 For the hase equilibrium between seawater and water vaor, the followin condition must be fulfilled: ( ) va ( ) µ,, =,, (6) which can be written equivalently in terms of the osmotic coefficient φ as va ( ) ( ) ( ) brm φ,, =,,, (7) where µ ( ) and (,, ),, φ are from able 1, and (,) is from Eq. (4). he molality b is determined from Eq. (3) and the value of the molar as constant Rm is taken from able 1 of [2]. he Gibbs free enery of water vaor va (,) is calculated from the Gibbs free enery equation of the IAP-IF97 reion 2, va R (, ) with the ideal as art, o o r (, ) (, ) (, ) = γ πτ = γ πτ + γ πτ, (8) 9 i= 1 o i J o i γ = lnπ + n τ, (9) and the residual art, r 43 I i J ni ( 0.5) i. (10) γ = π τ i= 1 va In Eqs. (8) to (10), γ = /( R ) is the reduced Gibbs free enery, π = / * is the reduced ressure, and τ = */ is the inverse reduced temerature. he secific as constant R of water, the reducin arameters * and *, the coefficients ni o and n i, and the exonents I i, J, and J i are iven in ection 6.1 of the IAP-IF97 Release [7]. o i Usin Eq. (6) or (7), the boilin temerature = b can be calculated by iteration from ressure and salinity, or boilin ressure = b from temerature and salinity, or brine salinity = b from and. At a iven equilibrium state between seawater and water vaor, the roerties of the seawater (brine) hase are calculated from Eq. (1) and the roerties of the water vaor result from Eq. (8), see ection 4.5. At brine-vaor equilibrium, vaor is suerheated, i.e., at iven ressure, the temerature is hiher than the saturation temerature of ure water or the ressure at iven temerature is below the saturation ressure of ure water. Note that the Gibbs free enery of liquid water, (, ) in Eq. (7), is evaluated from IAP-IF97 at conditions where the liquid hase of ure water is metastable. Due to salinity, the boilin temerature elevation can be u to 2 K. able A2 shows selected boilin temeratures for iven ressures and salinities.

11 Phase Equilibrium between eawater and Ice For the hase equilibrium between seawater and ice Ih, the followin condition has to be fulfilled: ( ) Ih ( ) µ,, =,, (11) or equivalently: Ih ( ) ( ) ( ) brm φ,, =,,, (12) with µ ( ) and φ (,, ) from able 1 and (, ),, from Eq. (4). he molality b is determined from Eq. (3) and the value of the molar as constant Rm is taken from able 1 of [2]. Ih he Gibbs free enery of ice Ih (, ) is calculated from the Gibbs free enery equation of the corresondin IAP Release [11]: Ih ( ) 0 0 t, = ( ) s he functions ( ) 0 τ 2 τ + t Re rk ( tk τ) ln ( tk τ) + ( tk + τ) ln ( tk + τ) 2tk ln tk k = 1 tk and ( ) 2. (13) rk, the trile-oint temerature of water t, and the reduced temerature τ are iven in [11]. he real constant s0 as well as the comlex constants t1, r1, and t2 are listed in able 2 of [11]. Usin Eq. (11) or (12), the freezin temerature = f can be calculated by iteration from ressure and salinity, or the freezin ressure = f from and, or brine salinity = f from and. At a iven equilibrium state between seawater and ice, the roerties of the seawater (brine) hase are calculated from Eq. (1) and the roerties of the ice hase result from Eq. (13), see ection 4.6. At brine-ice equilibrium, the temerature is lower than the meltin temerature at a iven ressure of ure ice or the ressure at a iven temerature is below the meltin ressure of ure ice. Note that the Gibbs free enery of liquid water (, ) in Eq. (12) is evaluated from IAP-IF97 at conditions where the liquid hase of ure water is metastable. Due to salinity, the freezin-oint deression can be u to 8 K. able A3 shows selected freezin temeratures for iven ressures and salinities. 4.3 rile-point emeratures and Pressures he trile-oint temeratures = t and trile-oint ressures = t of seawater are calculated for iven salinity by iteration from both Eqs. (6) and (11) or from both Eqs. (7) and (12).

12 12 able A4 contains selected trile-oint temeratures and trile-oint ressures for iven salinities. 4.4 Osmotic Pressure On the two sides of a membrane ermeable to water but not to sea salt, equilibrium between liquid water and seawater causes an excess ressure of seawater, the osmotic ressure osm, comuted by iteration from the condition ( + ) = ( ) µ osm,,,, (14) or equivalently from ( + ),, = ( + ), (, ) brm φ osm osm, (15) with µ and φ from able 1 and from Eq. (4). he molality b is determined from Eq. (3) and the value of the molar as constant Rm is taken from able 1 of [2]. Usin Eq. (14) or (15), the osmotic ressure osm is calculated by iteration for ressure, temerature, and salinity. 4.5 Proerties of Brine-Vaor Mixture he roerties for the mixture of brine and water vaor, termed brine-vaor mixture, can be comuted from the combined Gibbs free enery equation BV va (,, ) ( 1 ) (,, ) (, ) = x + x, (16) b where is the iven ressure, is the iven temerature, and is the iven salinity of the seawater mixture includin water vaor. he boilin-brine salinity b = b (, ) is calculated for ressure and temerature usin Eq. (6) by iteration. he mass fraction of water vaor in the seawater mixture (vaor fraction) x can be determined from the equation x = 1 (, ). (17) b he Gibbs free enery of the brine (,, b ) va enery of water vaor (, ) from Eq. (8). BV Brine-vaor roerties can be comuted from the equation (,, ) derivatives in analoy to able 1. 2 va he derivatives of the included equation (, ) is calculated from Eq. (1) and the Gibbs free, Eq. (16), and its, Eq. (8), 2 he artial derivatives of BV b, via the chain rule, makin use of Eq. (6). he related terms vanish identically for the first derivatives BV BV (, ). his is true, e.., for the roerties v, h, u, and s. But they rovide the dominant hasechane contributions such as latent heat to the second derivatives (,, ) [17, 18] which BV BV BV are used, e.., for the cubic isobaric exansion coefficient α v. with resect to or include differentiation of ( )

13 13 for water vaor can be taken from [7]. As an examle, the secific enthaly of brine-vaor mixture is calculated from the equation BV va (,, ) ( 1 ) (,, ) (, ) h = x h + xh (18) b va where the determination of h(,, ) is iven in able 1 and of (, ) BV Further roerties such as secific volume v (,, ) BV BV u (,, ), and secific entroy s (,, ) 4.6 Proerties of Brine-Ice Mixture b h in [7]., secific internal enery are comuted analoously. he roerties for the mixture of brine and ice Ih, termed brine-ice mixture or sea ice, can be comuted from the combined Gibbs free enery equation BI Ih (,, ) ( 1 ) (,, ) (, ) = y + y, (19) f where is the iven ressure, is the iven temerature, and is the iven salinity of the seawater mixture includin ice Ih. he freezin-brine salinity f = f (, ) is calculated for ressure and temerature usin Eq. (11) by iteration. he mass fraction of ice Ih in the seawater mixture (ice fraction) y can be determined from the equation y = 1 (, ). (20) f he Gibbs free enery of the brine (,, f ) Ih enery of ice Ih (, ) from Eq. (13). BI Brine-ice mixture roerties can be comuted from the equation (,, ) its derivatives in analoy to able 1. 3 Ih he derivatives of the included equation (, ) is calculated from Eq. (1) and the Gibbs free, Eq. (19), and, Eq. (13), for ice Ih can be taken from [11]. As an examle, the secific enthaly of brine-ice mixture is calculated from the equation BI Ih (,, ) ( 1 ) (,, ) (, ) h = y h + yh, (21) f Ih where the determination of h(,, ) is iven in able 1 and of (, ) BI Further roerties such as the secific volume v (,, ) BI BI u (,, ), and secific entroy s (,, ) f h in [11]., secific internal enery are comuted analoously. BI with resect to or include differentiation of ( ) 3 he artial derivatives of f, via the chain BI BI rule, makin use of Eq. (11). he related terms vanish identically for the first derivatives (, ). his is true, e.., for the roerties v, h, u, and s. But they rovide the dominant hase-chane BI BI BI contributions such as latent heat to the second derivatives (,, ) [17, 18] which are used, e.., for the cubic isobaric exansion coefficient α. v

14 14 5. Reference tates Accordin to the IAP Formulation 2008 for the hermodynamic Proerties of eawater [2], the secific enthaly and the secific entroy of seawater calculated from Eq. (1) have been set to zero at = MPa, = K, and = n = k k 1 : h = 0 and s = 0, where n is the normal salinity of seawater [10]. he reference state of Eq. (1) for ure liquid water, water vaor, and ice corresonds to the followin values. he secific internal enery and the secific entroy of saturated liquid water at the trile oint ( = K, = MPa, and = 0 ) are: u = 0 and s = 0. From u = 0 it follows for the secific enthaly 1 h = kj k at this state oint. hese values corresond to the reference state of IAP-IF Rane of Validity he equation of state, Eq. (1), is valid for tandard eawater with sea salt of the Reference Comosition [10] in certain reions inside the followin ressure, temerature, and salinity ranes kpa 100 MPa, 261 K 353 K, and k k. All roerties of able 1 can be calculated with hih accuracy in the temerature rane f (, ) 313 K 1 for MPa 100 MPa and k k (Reion A in [2]) or 1 for 0.3 kpa MPa and k k (Reion B in [2]). Outside the above two reions, roerties of able 1 excet those deendin on ressure derivatives can be calculated with hih accuracy near standard atmosheric ressure ( = MPa ). 1 for f (, ) 353 K and k k (Reion C in [2]). Density is very well extraolated by this formulation u to the hihest salinity values at low temeratures. In the rane of hiher salinity and hiher temerature near standard atmosheric ressure, density is described with lower accuracy and quantities comuted from its extraolated derivatives may even be invalid. Due to the lack of exerimental data, no statements could be made in the release on the IAP formulation 2008 [2] about the accuracy at hih ressures for temeratures reater than 313 K and salinities reater than Restrictions reardin certain roerties outside these ranes are described in detail in ection 6 of the IAP Release [2]. Reference [2] also contains a rahical reresentation of the rane of validity in a -- lot.

15 15 7. Uncertainty A summary of estimated combined standard uncertainties u (coverae factor k = 1) of selected quantities usin IAP-95 for the water art in certain reions of the -- sace is iven in able 7 in the IAP release on the IAP formulation 2008 [2]. By usin IAP- IF97 instead of IAP-95, some deviations caused by the difference between IAP-IF97 and IAP-95 must be added to the uncertainties. he total uncertainty usin IAP-IF97 can be estimated by the equation RM u= u +, (22) where u08 reresents the uncertainty of the 2008 IAP seawater formulation [2] (which uses IAP-95 for ure-water roerties), and RM reresents the root-mean-square of the deviations caused by the difference between IAP-IF97 and IAP-95 and is comuted from 1 = ( x ) RM N N n = 1 2 n, where xn can be either absolute or ercentae difference between the corresondin quantities x ; N is the number of xn values ( oints uniformly distributed over the resective rane of validity were used to calculate able 3). A summary of estimated values u08, RM, and u is iven in able Comutin ime for the IAP eawater Functions One imortant reason to use IAP-IF97 instead of IAP-95 for calculatin the water art of the IAP seawater functions is the comutin-seed difference between these two standards. he relations of the comutin seed of the seawater functions of the 2008 IAP seawater formulation [2] (which uses IAP-95) in comarison with the industrial formulation for seawater iven herein (usin IAP-IF97) were investiated and reorted in [1], where imrovements in seed by factors on the order of 100 or 200 were reorted. hile the observed imrovement will deend on the efficiency with which IAP-95 is rorammed (and also on details of the machine and comiler used), the use of IAP-IF97 clearly roduces a sinificant reduction in comutin time.

16 16 able 3 Uncertainties of selected quantities in certain reions of the -- sace estimated for Eq. (1) Quantity interval k k 1 interval K interval MPa u 08 RM u ρ ρ α v K K K 1 w w va va f mk mk 2 mk b mk 1.2 mk 2.3 mk φ a φ a a c J k 1 K 1 - a 0.5 J k 1 K 1 c J k 1 K J k 1 K J k 1 K 1 a he quantity is only deendent on the saline art of Eq. (1).

17 17 9. Comuter-Proram Verification o assist the user in comuter-roram verification, test values are iven in able A1. It contains values for the secific Gibbs enery, (,, ), toether with the corresondin derivatives and some thermodynamic roerties. For an easy check of the Gibbs free enery functions of the water art (, ) and of the saline art (,, ), the results of both arts are reorted searately in able A References [1] Kretzschmar, H.-J., Feistel, R., aner,., Miyaawa, K., Harvey, A.H., Cooer, J.R., Hieemann, M., Blanetti, F., Orlov, K., eber, I., inh, A., and Herrmann,., he IAP Industrial Formulation for the hermodynamic Proerties of eawater, Desalin. ater reat. 55, (2015). [2] IAP, R13-08, Release on the IAP Formulation 2008 for the hermodynamic Proerties of eawater (2008). Available at htt:// [3] Feistel, R., A New Extended Gibbs hermodynamic Potential of eawater. Pror. Oceanor. 58, (2003). [4] Feistel, R., A Gibbs Function for eawater hermodynamics for 6 to 80 C and alinity u to 120 k l. Dee-ea Res. I 55, (2008). [5] IAP, R6-95(2014), Revised Release on the IAP Formulation 1995 for the hermodynamic Proerties of Ordinary ater ubstance for General and cientific Use (2014). Available at htt:// [6] aner,. and Pruß, A., he IAP Formulation 1995 for the hermodynamic Proerties of Ordinary ater ubstance for General and cientific Use, J. Phys. Chem. Ref. Data 31, (2002). [7] IAP, R7-97(2012), Revised Release on the IAP Industrial Formulation 1997 for the hermodynamic Proerties of ater and team (he Revision only Relates to the Extension of Reion 5 to 50 MPa) (2007). Available at htt:// [8] aner,., Cooer, J. R., Dittmann, A., Kijima, J., Kretzschmar, H.-J., Kruse, A., Mareš, R., Ouchi, K., ato, H., töcker, I., Šifner, O., akaishi, Y., anishita, I., rübenbach, J., and illkommen, h., he IAP Industrial Formulation 1997 for the hermodynamic Proerties of ater and team, J. En. Gas urbines & Power 122, (2000). [9] IAP, R7-09, ulementary Release on a Comutationally Efficient hermodynamic Formulation for Liquid ater for Oceanorahic Use (2009). Available at htt:// [10] Millero, F.J., Feistel, R., riht, D.G., and McDouall,.J., he Comosition of tandard eawater and the Definition of the Reference-Comosition alinity cale. Dee-ea Res. I 55, (2008).

18 18 [11] IAP, R10-06, Revised Release on the Equation of tate 2006 for H 2O Ice Ih (2009). Available at htt:// [12] Feistel, R. and aner,., A New Equation of ate for H 2 O Ice Ih. J. Phys. Chem. Ref. Data 35, (2006). [13] IOC, COR, IAPO, he International hermodynamic Equation of eawater : Calculation and Use of hermodynamic Proerties. Interovernmental Oceanorahic Commission, Manuals and Guides No. 56, UNECO, Paris (2010). Available at htt:// [14] IAP, AN3-07(2014), Revised Advisory Note No. 3: hermodynamic Derivatives from IAP Formulations (2014). Available at htt:// [15] IAP, R14-08(2011), Revised Release on the Pressure alon the Meltin and ublimation Curves of Ordinary ater ubstance (2011). Available at htt:// [16] aner,., Riethmann,., Feistel, R., and Harvey, A.H., New Equations for the ublimation Pressure and Meltin Pressure of H 2 O Ice Ih. J. Phys. Chem. Ref. Data 40, (2011). [17] Feistel, R., riht, D.G., Kretzschmar, H.-J., Haen, E., Herrmann,., and an, R., hermodynamic Proerties of ea Air. Ocean ci. 6, (2010). [18] Feistel, R., riht, D.G., Jackett, D.R., Miyaawa, K., Reissmann, J.H., aner,., Overhoff, U., Guder, C., Feistel, A., and Marion, G.M., Numerical Imlementation and Oceanorahic Alication of the hermodynamic Potentials of Liquid ater, ater Vaour, Ice, eawater and Humid Air Part 1: Backround and Equations. Ocean ci. 6, (2010).

19 19 APPENDIX able A1 Numerical check values for the water art comuted from (, ) saline art comuted from (,, ) function (,, ), Eq. (1) and its derivatives and for selected seawater roerties of able 1 at iven oints (,, ) Proerties at, Eq. (4), and its derivatives, for the, Eq. (5), and its derivatives, for the seawater roerties comuted from the Gibbs 1 = MPa, = K, = n = k k Quantity ater art aline art Proerty of seawater Unit J k 1 ( / ), m 3 k 1 ( 2 / 2 ), m 3 k 1 Pa 1 ( / ), J k 1 K 1 ( 2 / 2 ), J k 1 K 2 ( 2 / ) m 3 k 1 K 1 ( / ), J k 1 ( 2 / ) m 3 k 1 v m 3 k 1 u kj k 1 h kj k 1 s kj k 1 K 1 c kj k 1 K 1 w a m s 1 µ kj k 1 a his value cannot be comuted from alone because it is a nonlinear exression in

20 20 able A1 continued Proerties at = MPa, = 353 K, = 0.1 k k 1 Quantity ater art aline art Proerty of seawater Unit J k 1 ( / ), m 3 k 1 ( 2 / 2 ), m 3 k 1 Pa 1 ( / ), J k 1 K 1 ( 2 / 2 ), J k 1 K 2 ( 2 / ) m 3 k 1 K 1 ( / ), J k 1 ( 2 / ) m 3 k 1 v m 3 k 1 u kj k 1 h kj k 1 s kj k 1 K 1 c kj k 1 K 1 w a m s 1 µ kj k 1 a his value cannot be comuted from alone because it is a nonlinear exression in

21 21 able A1 continued Proerties at 1 = 100 MPa, = K, = = k k n Quantity ater art aline art Proerty of seawater Unit J k 1 ( / ), m 3 k 1 ( 2 / 2 ), m 3 k 1 Pa 1 ( / ), J k 1 K 1 ( 2 / 2 ), J k 1 K 2 ( 2 / ) m 3 k 1 K 1 ( / ), J k 1 ( 2 / ) m 3 k 1 v m 3 k 1 u kj k 1 h kj k 1 s kj k 1 K 1 c kj k 1 K 1 w a m s 1 µ kj k 1 a his value cannot be comuted from alone because it is a nonlinear exression in

22 22 able A2 Boilin temeratures b in K of seawater for selected oints (,) calculated by iteration from Eq. (6) or from Eq. (7) Pressure in MPa alinity in k k

23 23 able A3 Freezin temeratures f in K of seawater for selected oints (,) calculated by iteration from Eq. (11) or from Eq. (12) Pressure in MPa alinity in k k a - a - a - a a - a - a - a a - a - a - a a - a - a - a a - a - a - a a his state oint is out of rane of validity of Eq. (1).

24 24 able A4 rile-oint temeratures t and related trile-oint ressures t of seawater for selected salinities calculated by iteration from both Eqs. (6) and (11) or from both Eqs. (7) and (12) alinity in 1 k k rile-oint temerature t in K rile-oint ressure t in kpa