Values required for the simulation of CO2-storage in deep aquifers

Transcription

1 Gesellschaft für Anlagenund Reaktorscherhet (GRS) mbh Values requred for the smulaton of CO-storage n deep aqufers GRS - 36

2 Gesellschaft für Anlagenund Reaktorscherhet (GRS) mbh Values requred for the smulaton of CO-storage n deep aqufers Klaus-Peter Kröhn May 008 Remark: Ths report was prepared under contract No. 03 G 06 B wth the German Federal Mnstry for Economcs and Technology (BMW). The work was conducted by the Gesellschaft für Anlagen- und Reaktorscherhet (GRS) mbh. The author s responsble for the content of ths report. GRS - 36 ISBN

3 Keywords: brne, CO -storage, deep aqufers, flud propertes, phase state, supercrtcal CO

4 Abstract The mathematcal descrpton of the flow of supercrtcal carbon doxde n deep salne aqufers s rather complex even f only the basc processes are consdered. In the lght of these complextes t s of vtal mportance to know the parameters and ther dependences as precse as possble. The present report comples therefore the parameters and the referrng approaches from the lterature. For each parameter the source, the range of valdty, at least one meanngful fgure, a comment where necessary and conclusons wth a vew to the applcablty of the formulaton are also gven. All requred coeffcents and auxlary functons ncludng the underlyng dmensons are lsted, too. Wth ths complaton an up-to-date set of mathematcal formulatons for flud parameters to be used for modellng CO -storage n deep aqufers s avalable allowng mmedate applcaton. Moreover, t gves nformaton about ther dependences on temperature, pressure and salnty. Searchng the lterature t became apparent that there are stll gaps n the knowledge of those formulatons. Ths report ncludes therefore a short dscusson of the state of knowledge referrng to each constant and ndcatng where further research appears to be necessary. I

5 Table of contents Abstract... I 1 Introducton Motvaton Conventons... 3 Propertes of the matrx Precedng remark Porosty Tortuosty Permeablty Depth-dependent temperature Propertes dependng on matrx and fluds at the same tme General remarks Two-phase flow equatons of state Hydrodynamc dsperson Propertes of water Vapour pressure Vapour pressure of pure water Vapour pressure of water wth dssolved NaCl Vapour pressure of water wth dssolved CO Vapour pressure of water wth dssolved NaCl and CO Densty Densty of pure water Densty of lqud water Densty of water vapour Densty of water wth dssolved NaCl Densty of water wth dssolved CO Densty of water wth dssolved NaCl and CO Vscosty Vscosty of pure water Vscosty of water wth dssolved NaCl Vscosty of water wth dssolved CO... 3 III

6 4.3.4 Vscosty of water wth dssolved NaCl and CO Thermal conductvty Thermal conductvty of pure water Thermal conductvty of water wth dssolved NaCl Thermal conductvty of water wth dssolved CO Thermal conductvty of water wth dssolved NaCl and CO Enthalpy Enthalpy of pure water Enthalpy of water wth dssolved NaCl Enthalpy of water wth dssolved CO Enthalpy of water wth dssolved CO and NaCl Heat capacty Heat capacty of pure water Heat capacty of water wth dssolved NaCl Heat capacty of water wth dssolved CO Heat capacty of water wth dssolved NaCl and CO Solublty Solublty of NaCl n water Solublty of NaCl n lqud water Solublty of NaCl n water vapour Solublty of CO n water Solublty of CO n water wth dssolved NaCl Dffusvty Propertes of carbon doxde Vapour pressure Vapour pressure of pure CO Vapour pressure of CO wth dssolved H O Helmholtz energy for CO Approach Dervatves Densty Densty of pure CO Densty of CO wth H O Densty of gaseous CO Densty of lqud CO IV

7 5.4 Vscosty Vscosty of pure CO Vscosty of CO wth H O Vscosty of gaseous CO Vscosty of lqud CO Thermal conductvty Thermal conductvty of pure CO Thermal conductvty of CO wth dssolved H O Enthalpy Enthalpy of pure CO Enthalpy of CO wth water Enthalpy of gaseous CO wth water Enthalpy of lqud CO wth water Heat capacty Heat capacty of CO Heat capacty of CO wth water Solublty Solublty of NaCl n CO Solublty of water n CO Solublty of water from a NaCl-soluton n CO Dffusvty Propertes of sodum chlorde Densty of NaCl Enthalpy of NaCl Heat capacty of NaCl Phase state Phase dagram for pure substances Phase dagram for mxtures of H O and CO Phase dagnostcs Prelmnary remarks Prmary and secondary varables Phase dagnostcs for a one-component flow Phase dagnostcs for the two-component flow of H 0 and CO Possble phase states durng normal and altered evoluton V

8 Phase dagnostcs for normal evoluton Phase dagnostcs for altered evoluton Combnaton of phase dagnostcs for normal and altered evoluton Phase dagnostcs for a NaCl-component Need for further research Normal and altered evoluton Two-phase flow propertes Densty-drven flow Thermodynamc flud propertes References Table of fgures Lst of tables Appendx A - Iteratve scheme for calculatng CO -densty Appendx B - Constants Appendx C - Currant knowledge about formulatons VI

9 1 Introducton 1.1 Motvaton The mathematcal descrpton of the movement of carbon doxde n deep aqufers s rather complex even consderng only the basc processes nvolved. These basc processes comprse mult-phase mult-component flow of water, carbon doxde and salt, transport of dssolved components, heat flow, and phase changes. The complexty orgnates from dfferent reasons. Already the well-known mathematcal descrpton of a smple two-phase flow nvolves two coupled hghly non-lnear mass balance equatons. In case of CO -storage n the underground an energy transport equaton s to be solved, too. Temperature effects are prmarly ntroduced by njectng the supercrtcal CO at a temperature dfferent from the surroundngs, by phase changes and by the geothermal gradent. Non-sothermal condtons mean that the temperature-dependence of the parameters used n the equatons must be taken nto account. By ths, several parameters ntroduce addtonal non-lneartes nto the mathematcal descrpton. The same apples also to the dependence on pressure and on the salt concentraton of the groundwater. Whle these complextes concern only the parameters n the equatons, dfferent sets of equatons are to be used n case of changes of the phase state. When a phase appears or dsappears the referrng balance equaton must be swtched to comply wth the physcal problem. And to control the phase state addtonal parameters must be montored. In the lght of all these complextes t s of vtal mportance to know the parameters and ther dependences as precse as possble. The present report comples the requred quanttes and some of the referrng approaches n order to provde a set of mathematcal formulatons. Despte the great effort that has gone nto the task of gettng a complete set contanng only the most accurate formulatons t must be acknowledged 1

10 1 Introducton that frstly, there are a lot more mathematcal formulatons n the lterature than can possbly got hold of or even assessed wthn a reasonable perod of pure research. Secondly, t became apparent, that there are stll gaps of knowledge. What s gven here, however, should more or less reflect the present state of the art n terms of CO -storage n the underground. The parameters addressed n ths report are lsted n Tab Tab. 1.1 Processes and parameters of the CO mgraton n the deep underground. Process Effect Constant/Equaton of state Chapters permeablty.3 relatve permeablty-saturaton relaton 3. advecton porosty. flow vscosty densty 4.3/5.4 4./5.3/6.1 capllary pressure capllary pressure-saturaton relaton 3. phase changes solublty vapour pressure 4.7/ /5.1 dffuson coeffcent 4.8/5.9 transport dffuson porosty tortuosty..3 dsperson (presently not enough knowledge [ 6 ]) - heat capacty 4.6/5.7/6.3 heat flow conducton thermal conductvty enthalpy 0/ /5.6/6. geothermal gradent.5 The parameters concernng the matrx are generally descrbed n chapter but no values for a specfc host rock are provded. In chapters 4, 5 and 6 the parameters and ther formulatons for water, carbon doxde, and sodum chlorde, respectvely, are descrbed and shortly evaluated. Chapter 7 begns wth a dscusson of the possble phase states of H O and CO. Crtera for phase changes and addtonal for ths purpose requred quanttes are gven further more. A dscusson of the approprateness of the descrbed parameters and an apprasal of a possble need for research are gven n chapter 8.

11 1 Introducton 1. Conventons The non-sothermal three-phase three-component system n a porous matrx s addressed n ths report. The components are water, carbon doxde, and salt. The components wll be denoted by the latn symbols, j, etc. These components consttute phases that are domnated by one of the components. The greek symbols,, etc. wll ndcate these phases. Latn as well as greek symbols thus denote H O, CO, or NaCl, respectvely, but have a dfferent meanng. If not stated otherwse component s related to the -phase, j to the -phase and so on. A phase domnated by the component wll be called the -phase. A quantty Q the partal pressure for nstance of a component n the phase s wrtten as Q, the superscrpt denotng the component and the subscrpt denotng the phase. Ths conventon does not show the state of aggregaton of the phase and thus ncludes the supercrtcal state. If not specfed otherwse the expresson water refers to lqud water whle vapour s used synonymeously for gaseous water. For the converson between bar and MPa 1 bar = 0.1 MPa s assumed f a more precse value not stated otherwse. The formulatons lsted n ths report are taken from a varety of sources. It has been tred to standardse the sectons for each constant by gvng them the followng structure: source formulaton ncludng an explanaton of the equatons range of valdty 3

12 1 Introducton graphcal plot comments conclusons The dmenson of nput and output quanttes may dffer from secton to secton. The vald range of nput data s hnted n the concernng paragraph range of valdty whle the dmenson of the calculated quantty s gven n the explanaton of the varables. Snce chapters 3 to 6 are exclusvely devoted to propertes of one component no specal ndces for the varables ndcatng the reference to the component are used f not necessary. The range of valdty for a constant s consdered to cover the relevant range for CO -sequestraton f the followng condtons are met: pressure: 0.1 < p < 30 [MPa] temperature: 10 < T < 100 [ C] salnty: 0 < NaCl mh O < 6.0 [mol/kg] ( 1.1 ) CO CO mass fracton: 0 < m < 1.0 [mol/kg] H O p - pressure [MPa] T - temperature [ C] m - molalty of NaCl n the water phase [mol/kg] NaCl H O m - molalty of CO n the water phase [mol/kg] CO H O 4

13 Propertes of the matrx.1 Precedng remark Whle sequestraton of CO nvolves carbon doxde and brne n any case the type of host rock envsaged wll be ste-specfc. For ths reason all parameters referrng to a host rock wll only be descrbed n general. No specfc data or functons wll be gven n ths respect.. Porosty Generally, the porosty s gven by the rato of the pore volume V p and the total volume V: V p (.1 ) V - porosty [-] V p V - pore volume [m³] - total volume [m³] Care and attenton are necessary wth regard to the pore volume. Some processes take place everywhere n the pore space lke dffusve solute transport. For other processes lke flud flow or advectve solute transport only parts of the total pore volume are accessble. It s therefore advsable to defne effectve porostes for each process consdered. Other processes lead to changes of the pore volume. Ths can ether be the result of changes of the matrx volume as n swellng clay partcles, or the pore volume s changed by precptaton or dssoluton. A thrd cause can be changes n the mechancal stress of the matrx. Porosty changes relatng to the precptated/dssolved mass can be descrbed by 5

14 Propertes of the matrx ms (. ) V s - porosty change due to precptaton/dssoluton [-] m s - change of sold mass due to precptaton/dssoluton [-] s - densty of the sold mass [kg/m³].3 Tortuosty On the pore scale the coeffcent of molecular dffuson suffces to correctly descrbe the process of dffuson. But for practcal purposes modellng can only be done on a macroscopc scale. Averagng the dffusonal fluxes n an REV (Representatve Elementary Volume) lets appear a second-rank tensor for the tortuosty whch accounts for the fact that there s no straght-lne dffusonal flow due to the sold matrx. Assumng an sotropc porous medum the tensor can be reduced to the well-known scalar tortuosty that takes nto consderaton the twstng of the flow paths. The tortuosty s theoretcally a property of the porous medum. However, a value cannot even vaguely be assgned to a specfc stuaton and so the tortuosty s rather a ft constant than a known quantty. It cannot exceed the value of 1 whch means no sold matrx s present, but t can theoretcally reach any value between 0 and 1..4 Permeablty The permeablty s generally assumed to be a functon of the porosty: k k( ) (.3 ) k - permeablty [m²] - porosty [-] Several models exst for the theoretcal dervaton of a permeablty-porosty relaton: Capllary bundle model Emprcal model wth porosty and permeablty lmts A smple model consderng pecewse constant but changng flow path cross-secton 6

15 Propertes of the matrx All these models yeld an exponental relaton between permeablty and porosty. In the capllary bundle model permeablty becomes zero when the porosty reaches zero. In the other two models, though, permablty vanshes already at a crtcal porosty c > 0. The thrd model has been used to smulate salt precptaton [ 45 ]. For the sake of smplcty t s assumed here that the flow channels consst of cylndrcal sectons wth two dfferent cross-secton areas. The secton wth the large cross-secton area A 1 represents the pore, and the secton wth the small cross-secton area A represents the bottleneck between two pores. In order to develop a model the sectons are arranged n such a way that all sectons wth the same sze are put together as ndcated n Fg..1. The total length of the flow path l s then subdvded nto two segments l 1 und l. r 1 r A 1 A l 1 l l Fg..1 Idealsed model of a flow channel wth varyng cross-secton area. Usng the standardsed length l 1 l (.4 ) - standardsed length for the wder segment [-] l 1 l - length of the wder segment [m] - total length of the flow path [m] 7

16 Propertes of the matrx the rato of the cross-sectonal areas A A 1 (.5 ) - rato of the cross-sectonal areas [-] A 1 - cross-secton area of the wder segment [m²] A - cross-secton area of the smaller segment [m²] and the standardsed porosty c 0 c (.6 ) - standardsed porosty [-] 0 - ntal porosty [-] c - crtcal porosty [-] the permeablty can be expressed as a functon of porosty [ 45 ]: k k (.7 ) k 0 - ntal permeablty [m²].5 Depth-dependent temperature Due to heat flow from earth s nteror the temperature n the earth crust rses wth depth. Ths ncrease s approxmately lnear from the surface down to a few klometres depth except for the upper 10 metres where seasonal temperature changes can be detected. For a formulaton for the depth-dependent temperature nformaton on the mean surface temperature and the temperature gradent s requred. 8

17 Propertes of the matrx In Germany the mean temperature at 10 m depth amounts to 8 C to 10 C [ 51 ] and the mean geothermal gradent to 30 C/km [ 47 ]. The mean global thermal gradent s specfed by [ 51 ] to be 5 C/km. The gradent depends on the local geology and can therefore devate consderably from the mean value. Wth approprate cauton the value of 30 C/km wll be adopted for a generc ste n Germany. 9

18 3 Propertes dependng on matrx and fluds at the same tme 3.1 General remarks The mathematcal descrpton of two-phase flow n a porous medum requres addtonal equatons of state (EOS) referrng to the effect of capllarty and to the mutual mpedment of flow of one phase by the other phase. Both can be formulated as a functon of the saturaton S of the -phase meanng the volumetrc fracton of pore space that s occuped by the -phase. The resultng capllary pressure-saturaton relaton and the relatve permeablty-saturaton relaton make the mathematcal formulaton hghly non-lnear. Relable predctons of two-phase flow depend therefore partcularly on a good representaton of the flow system by these two EOS. Note: In case of CO flowng upward to the surface a three-phase state wth water, lqud and gaseous CO s lkely to evolve [ 35 ]. We are not aware of equatons of state for ths specfc problem and wll therefore restrct the followng dscusson to two-phase flow problems. 3. Two-phase flow equatons of state Accordng to the Young-Laplace-equaton p c cos ( 3.1 ) r p c - capllary pressure [Pa] - nterfacal tenson [N/m] r - curvature radus [m] - wettng angle between sold surface and phase nterface [-] capllary pressure at a phase boundary depends on nterfacal tenson and on the radus r of a bubble or droplet, the radus r of a cylndrcal flow channel or half of the aperture d of a planar flow regme. 11

19 3 Propertes dependng on matrx and fluds at the same tme Even wthout consderng the wettng angle between sold surface and phase nterface, whch s often neglected, propertes of the fluds as well as propertes of the matrx are nvolved n an equaton of state for the capllary pressure. The nterfacal tenson depends on the combnaton of fluds that have contact at the nterface, on pressure and on temperature. In case of the combnaton brne-co the nterfacal tenson ncreases wth ncreasng temperature, wth ncreasng water salnty and wth decreasng pressure [ 3 ]. On pore sze scale capllary pressure can farly well be calculated. But for modellng purposes a formulaton on the macroscopc scale s requred. Though there exst several theoretcal approaches to explan the effect of capllary pressure - best known are the formulatons by Brooks and Corey [ 8 ] or by van Genuchten [ 18 ], a relable formulaton can be derved only emprcally from measurements. The geometry of the pore space as well as the characterstc pore sze dstrbuton for a certan rock type are generally more complex than the rather smple conceptual models underlyng the theoretcally derved formulatons. For the same reason t s necessary to measure relatve permeablty-data, too, even f theory allows to derve permeablty-saturaton relatons usng the same parameters as n the referrng capllary pressure-saturaton. Measurements for the two EOS are often not avalable for the porous medum n queston. In such a stuaton Leverett s dmensonless J-functon [ 9 ] k p ( S ) J ( S ) c w ( 3. ) w cos J - Leverett s J-functon [-] p c - capllary pressure [Pa] k - permeablty [m²] - porosty [-] s often used to transform a capllary pressure-relaton for a known porous medum to a new relaton for another one. Only permeablty and porosty of the unknown medum must be dentfed. Ths method s qute popular snce two-phase flow measurements are expensve. But t has some severe lmtatons. Leveretts scalng method s not vald for 1

20 3 Propertes dependng on matrx and fluds at the same tme certan ranges of wettng angles, and, addtonally, the wettng angle tself s dependent on the nterfacal tenson [ ]. Furthermore, Leveretts approach yelds only qualtatvely smlar EOS. But especally n case of CO storage where the storage strata can be located n a wde range of depths the n-stu condtons referrng to pressure, temperature and salnty may vary sgnfcantly from the condtons appled durng the laboratory measurements. Ths means dfferences n the nterfacal tenson whch n turn can not only change the scale of the EOS consderably but also the values for the resdual saturaton of brne and of CO [ 3 ]. Trappng of CO n solated bubbles at the resdual CO -saturaton happens n the wake of a CO -plume after shutdown of njecton and strongly nfluences further movement of the plume. Leverett s scalng method may therefore cause msleadng model results n the end. For all these reasons, the EOS for two-phase flow are deally measured n the laboratory and under n-stu condtons at that. If such data are not avalable - as s mostly the case - conclusons by analogy referrng to the EOS should be drawn wth extreme cauton. 3.3 Hydrodynamc dsperson The phenomenon of dsperson s well known n the context of sngle-phase transport problems. It s ntroduced by swtchng from a mcroscopc to a macroscopc observaton level. Dsperson descrbes the spreadng of transported matter by the spltted and tortuous flow channels on the mcroscopc level. For sngle-phase modellng the approach of [ 36 ] s generally used. Contrary to sngle-phase flow and transport ths phenomenon states a serous problem for two-phase flow. Approaches as well as referrng parameters are scarce [ 6 ] and hard to come by snce a saturaton-dependent tortuosty has to be consdered. It s therefore often neglected n two-phase flow smulatons. 13

21 4 Propertes of water 4.1 Vapour pressure Vapour pressure of pure water Source: [ 3 ] Formulaton: p s C p * B ( B 4AC) 1 4 ( 4.1 ) p s - vapour saturaton pressure [MPa] p - reference pressure (c.f. Appendx B) A, B, C - ansatz functons (q.v. eq. ( 4.)) A n n B n n n 3 C n n n ( 4. ) - ansatz functon (q.v. eq. ( 4.3 )) n 1 to n 6, - constants (c.f. Appendx B) T n9 T * T n T * 10 ( 4.3 ) T - temperature [K] T * - reference temperature (c.f. Appendx B) n 9, n 10, - constants (c.f. Appendx B) Range of valdty: temperature: K < T < K 15

22 4 Propertes of water 0.1 vapour saturaton pressure[mpa] temperature [ C] Fg. 4.1 Vapour pressure of pure water; after [ 3 ]. Conclusons: Eq. ( 4.1 ) s a relable formulaton wthn the ranges ( 1.1 ) Vapour pressure of water wth dssolved NaCl Source: [ 1 ], [ 4 ] Formulaton: e ew e3w e5 y ln p s e0 e4 ( 4.4 ) z z p s - vapour saturaton pressure [bar] e 0 to e 5 - constants (c.f. Appendx B) y - ansatz functon (q.v. eq. ( 4.5 )) z - ansatz functon (q.v. eq. ( 4.6 ) w - ansatz functon (q.v. eq. ( 4.7 )) 16

23 4 Propertes of water y T ( 4.5 ) 0 T 0 - temperature of pure water (referrng to brne temperature va eq. ( 4.8 )) [K] z T ( 4.6 ) w z e6 ( 4.7 ) e 6 - constant (c.f. Appendx B) ln T 0 f ln T ( 4.8 ) T - brne temperature [K] f - ansatz functon (q.v. eq. ( 4.9 )) f 1 a bt ( 4.9 ) a - ansatz functon (q.v. eq. ( 4.10 )) b - ansatz functon (q.v. eq. ( 4.11 )) a 3 1 a1m am a3m ( 4.10 ) m - molalty of NaCl n the water-phase [mol/kg] a - constants (c.f. Appendx B) b b1m bm b3m b4m b5m ( 4.11 ) b - constants (c.f. Appendx B) Range of valdty: temperature: K < T < K for eq. ( 4.4 ) and 6.15 K < T < K for eq. ( 4.8 ) 17

24 4 Propertes of water vapour saturaton pressure[mpa] 0.1 m= 0. mol/kg m= 1. mol/kg m=. mol/kg 0.08 m= 3. mol/kg m= 4. mol/kg m= 5. mol/kg 0.06 m= 6. mol/kg temperature [ C] Fg. 4. Vapour saturaton pressure of brne; after [ 1 ]. Comments: The formulaton for the vapour pressure of brne from [ 1 ] contans the physcally meanngful constant T 0. Ths constant s the temperature of pure water and s related to the brne temperature T x at the same vapour saturaton pressure by eq. ( 4.8 ). For two reasons ths approach s not entrely satsfyng n the framework of CO - sequestraton. Frstly, t s vald only n a temperature range that s clearly beyond the range of nterest. Secondly, the calculated vapour pressure wth a salt concentraton of 0 and a temperature of 100 C should result n a value of about 0.1 MPa but nstead yelds about 0.5 MPa. An mprovement s suggested by [ 4 ] startng wth eq. ( 4.8 ) but subsequently usng the formulatons of [ 5 ] for the vapour pressure of pure water. Fg. 4.3 shows a comparson of the formulaton of [ 1 ] and the equvalent vapour pressure usng the approach of [ 3 ] 1. The results wth the equvalent vapour pressure appear to be more credble than 1 Note that the formulatons n [ 3 ] are an updated verson of the formulatons suggested n [ 5 ]. 18

25 4 Propertes of water the approach of [ 1 ]. It has to be mentoned, though, that [ 1 ] as well as [ 4 ] clam that the computed vapour saturaton pressures are n good agreement wth measured data despte the sgnfcant dfferences n the sets of curves shown n Fg vapour saturaton pressure[mpa] m= 0. mol/kg m= 1. mol/kg m=. mol/kg m= 3. mol/kg m= 4. mol/kg m= 5. mol/kg m= 6. mol/kg m= 0. mol/kg m= 1. mol/kg m=. mol/kg m= 3. mol/kg m= 4. mol/kg m= 5. mol/kg m= 6. mol/kg after [ 1 ] after[ 4 ] temperature [ C] Fg. 4.3 Vapour saturaton pressure after [ 1 ] and the equvalent pure water vapour pressure after [ 4 ]. Conclusons: Eq. ( 4.8 ) n combnaton wth eq. ( 4.1 ) seems to be a relable formulaton wthn the ranges ( 1.1 ). It should be checked aganst measured data, though Vapour pressure of water wth dssolved CO Comments: We are not aware of formulatons concernng vapour pressure of water wth dssolved CO Vapour pressure of water wth dssolved NaCl and CO Comments: We are not aware of formulatons concernng vapour pressure of water wth dssolved NaCl and CO. 19

26 4 Propertes of water 4. Densty 4..1 Densty of pure water The approaches for the water densty presented n the next two sectons are gven n [ 3 ]. They actually provde data for the specfc volume of pure lqud water as well as of water vapour. The specfc volume s the recprocal of the densty Densty of lqud water Source: [ 3 ] Formulaton: 1 ( 4.1 ) - water densty [kg/m³] - specfc volume [m³/kg] p R T H O ( 4.13 ) p - pressure [MPa] T - temperature [K] R - specfc gas constant for water vapour (c.f. Appendx B) H O - standardsed pressure (q.v. eq. ( 4.14 )) [-] - ansatz functon (q.v. eq. ( 4.15 )) p ( 4.14 ) p * p - reference pressure (c.f. Appendx B) 0

27 4 Propertes of water 34 1 I 1 J ni ( 4.15 ) - standardsed temperature (q.v. eq. ( 4.16 ) [-] n, I, J - constants (c.f. Appendx B) T * ( 4.16 ) T - reference temperature (c.f. Appendx B) Range of valdty: temperature: K < T < K pressure: p s (T) < p < 100 MPa p s (T) - vapour saturaton pressure [MPa] (q.v. eq. ( 4.1 )) densty [kg/m³] T= 0. C T= 10. C T= 0. C T= 30. C T= 40. C T= 50. C T= 60. C T= 70. C T= 80. C T= 90. C pressure [MPa] Fg. 4.4 Densty of pure lqud water; after [ 3 ]. Conclusons: Eq. ( 4.13 ) s a relable formulaton wthn the ranges ( 1.1 ). 1

28 4 Propertes of water Densty of water vapour Source: [ 3 ] Formulaton: p R H T O 0 r ( 4.17 ) - specfc volume [m³/kg] (q.v. eq. ( 4.1 )) p - pressure [MPa] T - temperature [K] R - specfc gas constant for water vapour (c.f. Appendx B) H O r - ansatz functon (q.v. eq. ( 4.18 )) [-] - ansatz functon (q.v. eq. ( 4.19 )) 1 0 ( 4.18 ) - standardsed pressure (c.f. eq. ( 4.14 ) ) [-] p - reference pressure (c.f. Appendx B) 43 r I 1 n J I 0. 5 ( 4.19 ) 1 - standardsed temperature (q.v. eq. ( 4.16)) [-] - reference temperature (c.f. 0) n, I, J - constants (c.f. Appendx B) Range of valdty: temperature: pressure: K T K 0 MPa < p < p s (T) Comments: Eq.s ( 4.1 ), ( 4.14 ) and ( 4.16 ) are needed agan for calculatng the vapour densty from the specfc volume.

29 4 Propertes of water densty [kg/m³] T=100. C T=150. C T=00. C T=50. C T=300. C T=350. C pressure [MPa] Fg. 4.5 Densty of water vapour; after [ 3 ]. Water wth a pressure above 0.1 MPa and a temperature below 100 C exsts only n the lqud form as shown n Fg For temperatures above 100 C the vapour saturaton pressure p s (T) exceeds 0.1 MPa (c.f. Fg. 4.1) thus allowng water to take the vaporous form as long as pressures stay below p s (T). In Fg. 4.5 the change from the vaporous to the lqud phase s ndcated by the vertcal segment of a densty curve. Conclusons: Eq. ( 4.17 ) s a relable formulaton. However, water vapour as the only water phase s not to be expected wthn the ranges ( 1.1 ). 4.. Densty of water wth dssolved NaCl Sources: [ 1 ], [ 31 ] Formulaton: 3 A Bf Cf Df ( 4.0 ) - densty of water wth dssolved NaCl [g/cm³] A, B, C, D - constants (c.f. Appendx B) 3

30 4 Propertes of water f - ansatz functon (q.v. eq. ( 4.1 )) a1m at a p c1e ce c3e ( 4.1 ) f 3 m - molalty of NaCl n the water-phase [mol/kg] T - Temperature [ C] p - pressure [bar] a, c - constants (c.f. Appendx B) Range of valdty: salnty: 0.5 mol/kg < m < 5 mol/kg temperature: 10 C T 350 C pressure: p s (T) < p < 50 MPa p s (T) - vapour saturaton pressure [MPa] (q.v. eq. ( 4.1)) Accuracy: ± % n the range of valdty m=0.5 mol/kg, T= 10. C m=0.5 mol/kg, T= 50. C m=3.00 mol/kg, T= 10. C m=3.00 mol/kg, T= 50. C m=5.00 mol/kg, T= 10. C m=5.00 mol/kg, T= 50. C densty [kg/m³] pressure [MPa] Fg. 4.6 Densty of brne; after [ 1 ]. 4

31 4 Propertes of water Comments: The formulatons gven above can be found n [ 1 ] and n [ 31 ]. An approach for a wder range of applcaton n terms of temperature and pressure s provded by [ 3 ], but temperature dependent coeffcents are gven only for dscrete temperatures and thus are only condtonally useful for modellng purposes. [ 4 ] states that the compressblty of brne s almost the same as of pure water regardless of the salnty up to a temperature of 50 C. An algorthm comparson wth other formulatons has been performed n [ ]. Here, the approach of [ 31 ] was dscarded because of devaton from several other formulatons. However, these other formulatons yeld somewhat dubous values for hgh NaClconcentratons, namely 130 kg/m³ at atmospherc pressure, a temperature of 5 C and a salnty of about 5 mol/kg. Conclusons: Eq. ( 4.0 ) appears to be a relable formulaton wthn the ranges ( 1.1 ) except for the extreme values of the salnty. There are alternatve formulatons but presently no defnte conluson about the valdty of any of the approaches s possble Densty of water wth dssolved CO Source: [ 16 ] Formulaton: M c c V ( 4. ) H O CO H O - densty of water wth dssolved carbon doxde [kg/m³] M CO - molecular weght of CO [kg/mol] (c.f. Appendx B) H O - densty of pure water [kg/m³] (q.v. eq. ( 4.13 )) c - concentraton of dssolved CO [mol CO /m³ H O] V - molar volume of dssolved CO [m³/mol] (q.v. eq. ( 4.3 )) V T T T ( 4.3 ) T - temperature [ C] 5

32 4 Propertes of water Range of valdty: The range of valdty s not presented n [ 16 ]. The plots n [ 16 ] suggest: temperature: 0 C T 300 C pressure: p s (T) < p < 30 MPa (see Comments); p s (T) - (q.v. eq. ( 4.1)) CO -concentraton: c [mol/mol]: 0 < c < 0.05 mol/mol 100 densty [kg/m³] p= 0.1 MPa, T= 10. C p= 0.1 MPa, T= 40. C p= 0.1 MPa, T= 90. C p=15.0 MPa, T= 10. C p=15.0 MPa, T= 40. C p=15.0 MPa, T= 90. C CO-concentraton [mol/kg] Fg. 4.7 Densty of pure water wth dssolved CO up to CO -solublty; after [ 16 ]. Comments: The nfluence of dssolved carbon doxde on the water densty has been gnored n multphase flow modellng for a long tme (e.g. n [ 34 ] und [ 4 ]). The reason for that s that the densty ncrease due to salnty can be up to almost 0 % whereas carbon doxde content produces an ncrease n aqueous phase densty on the order of at most to 3 % [ 16 ]. Maxmum densty changes due to soluton of CO s estmated n [ 7 ] to be less than 0.6 % at temperatures between 0 and 5 C. However, for CO - sequestraton the densty effects may play a vtal role for the fnal fate of njected CO. Along wth the pure water densty (see secton 4..1) ths approach requres a second equaton for the molar volume V of dssolved CO whch s the volume occuped by one mole of a CO at a gven temperature and pressure. Accordng to [ 16 ] the molar volume 6

33 4 Propertes of water s only weakly dependent on the CO -concentraton and does not depend on pressure for temperatures below 300 C. The nfluence of salnty on the molar volume may be neglgble [ 16 ]. Note that pressure enters ths formulaton only va the densty of pure water whch can relably be calculated usng eq. ( 4.13 ) up to 100 MPa. Conclusons: Eq. ( 4. ) s a relable formulaton wthn the ranges ( 1.1 ) Densty of water wth dssolved NaCl and CO Source: [ 17 ] Formulaton: b M CO c cbv ( 4.4 ) - brne densty ncludng dssolved CO [kg/m³] b - brne densty wthout dssolved CO [kg/m³] M CO - molecular weght of CO [kg/mol] (c.f. 0) c - concentraton of dssolved CO [mol CO /m³ soluton] V - molar volume of dssolved CO [m³/mol] (q.v. eq. ( 4.3 )) Range of valdty: The range of valdty s not presented n [ 17 ]. Presumably, the ranges gven n secton 4..3 apply. Comments: Apparently, lttle s known about the densty of a soluton contanng CO as well as NaCl. [ 4 ] neglects the nfluence of CO on the brne densty on account of the low solublty of the CO. In [ 17 ] an ad hoc approach based on the formulaton n secton 4..3 for pure water and CO s used wthout justfcaton that smply exchanges the pure water densty wth the brne densty: Note: Nether eq. ( 4.0 ) nor eq. ( 4. ) offer a formulaton ncludng 0% salt content or 0% CO -content, respectvely. The brne densty for a salt molalty below 0.5 s therefore calculated as a lnear nterpolaton between the value of eq. ( 4.0 ) wth 7

34 4 Propertes of water m=0.5 and the value for pure water accordng to eq. ( 4.13 ). In case of low CO - concentratons the lower lmt for the applcablty of eq. ( 4. ) s gnored m NaCl = 3.0 mol/kg, p= 0.1 MPa, T= 10. C m 1000 NaCl = 3.0 mol/kg, p= 0.1 MPa, T= 40. C m NaCl = 3.0 mol/kg, p= 0.1 MPa, T= 90. C m NaCl = 3.0 mol/kg, p= 15.0 MPa, T= 10. C m NaCl = 3.0 mol/kg, p= 15.0 MPa, T= 40. C m NaCl = 3.0 mol/kg, p= 15.0 MPa, T= 90. C CO -concentraton [mol/kg] Fg. 4.8 Densty of brne wth 3 mol/kg NaCl and CO up to CO -solublty; after [ 17 ] m NaCl = 5.0 mol/kg, p= 0.1 MPa, T= 10. C m 1000 NaCl = 5.0 mol/kg, p= 0.1 MPa, T= 40. C m NaCl = 5.0 mol/kg, p= 0.1 MPa, T= 90. C m NaCl = 5.0 mol/kg, p= 15.0 MPa, T= 10. C m NaCl = 5.0 mol/kg, p= 15.0 MPa, T= 40. C m NaCl = 5.0 mol/kg, p= 15.0 MPa, T= 90. C CO -concentraton [mol/kg] Fg. 4.9 Densty of brne wth 5 mol/kg NaCl and CO up to CO -solublty; after [ 17 ]. 8

35 4 Propertes of water Conclusons: Eq. ( 4.4 ) s probably a good approxmaton but stll has to be valdated. The ranges ( 1.1 ) referrng to NaCl- and CO -concentratons are not fully covered. 4.3 Vscosty Vscosty of pure water Source: [ 4 ] Formulaton: T ) ( T, ) ( T, ) ( 4.5 ) 0( 1 - standardsed vscosty (q.v. eq.s ( 4.6 ) [-] T - standardsed temperature (q.v. eq.s ( 4.6 ) [-] - standardsed densty (q.v. eq.s ( 4.6 ) [-], - ansatz functons [-] (q.v. eq.s ( 4.7 ) to ( 4.9 )) 0 1, T T,, ( 4.6 ) T * * * T - temperature [K] - densty [kg/m³] - vscosty [Pa s] T*, p*, *, * - reference values (c.f. Appendx B) T H T ( 4.7 ) H - constants (c.f. Appendx B) e Hj 1 T 0 j0 j 1 ( 4.8 ) H j - constants (c.f. Appendx B) 9

36 4 Propertes of water 1 ( 4.9 ) Range of valdty: pressure: p < 500 MPa for 0 C < T < 150 C p < 350 MPa for 150 C < T < 600 C p < 300 MPa for 600 C < T < 900 C vscosty [Pa s] p= 0.10 MPa p= 0.00 MPa p= MPa p= MPa p= MPa p= MPa temperature [ C] Fg Vscosty of pure water as a functon of temperature; after [ 3 ]. Comments: For ndustral use eq. ( 4.9 ) may be used everywhere wthn the range of valdty. For general and scentfc use t becomes consderably more complex. Generally, water (and brne) vscosty s almost neglgbly dependent on pressure [ ]. Conclusons: Eq. ( 4.5 ) s a relable formulaton wthn the ranges ( 1.1 ). The nfluence of pressure on the water vscosty s apparently neglgble. 30

37 4 Propertes of water 4.3. Vscosty of water wth dssolved NaCl Source: [ 31 ] Formulaton: 1 am bm² cm³ dt (1 e w km ) ( 4.30 ) - vscosty [Pa s] w - vscosty of pure water [Pa s] (c.f. eq. ( 4.5 )) T - temperature [ C] m - molalty of NaCl n the water-phase [mol/kg] a, b, c, d, k - constants (c.f. Appendx B) Range of valdty: temperature: 10 C < T < 350 C pressure: 0.1 MPa < p < 50 MPa (for eq. ( 4.5 )) salnty: 0 mol/kg < m < 5 mol/kg Accuracy: Eq. ( 4.30 ) reproduces data to an average of ± % wthn the range of valdty. Comments: The approach of [ 31 ] provdes a departure functon that modfes the pure water vscosty by a factor accordng to the salnty. Note that there s a small nfluence of pressure whch s ntroduced by the formulaton ( 4.5 ) for the vscosty of pure water. An algorthm comparson wth other formulatons has been performed n [ ]. It appears that the dfferences between theses formulatons were small. NaCl-saturated soluton contans 359 g /l NaCl H O soluton at 5 C whch equals 6.1 mol /kg. NaCl H O 31

38 4 Propertes of water vscosty [Pa s] m=0.00 mol/kg, p= 0.1 MPa m=0.00 mol/kg, p=10.0 MPa m=0.00 mol/kg, p=30.0 MPa m=3.00 mol/kg, p= 0.1 MPa m=3.00 mol/kg, p=10.0 MPa m=3.00 mol/kg, p=30.0 MPa m=5.00 mol/kg, p= 0.1 MPa m=5.00 mol/kg, p=10.0 MPa m=5.00 mol/kg, p=30.0 MPa temperature [ C] Fg Vscosty of water wth dssolved NaCl as a functon of temperature; after [ 31 ]. Conclusons: Eq. ( 4.30 ) s a relable formulaton wthn the ranges ( 1.1 ) except for very hgh salntes Vscosty of water wth dssolved CO Source: [ 4 ] Comments: At the tme we are not aware of formulatons descrbng the vscosty of water wth dssolved CO. [ 4 ] neglects the nfluence of CO on the brne vscosty on account of the low solublty of the CO. Conclusons: No relable formulaton s known. Possbly t s neglgble. 3

39 4 Propertes of water Vscosty of water wth dssolved NaCl and CO Source: [ 7 ] Formulaton: NaCl NaCl CO CO a bt m ( c dt ) m ( e ft ) m ( g ht ) m H O H ( p 0.1) O ( T, p 0.1) H O H O H O ( 4.31 ) - vscosty [m(pa s)] H O - vscosty of pure water at T and p=0.1 MPa [m(pa s)] T - temperature [K] m - molalty referrng to NaCl [mol/kg] NaCl H O m - molalty referrng to CO [mol/kg] CO H O p - pressure [MPa] a to - constants (c.f. Appendx B) Range of valdty: temperature: 0 C < T < 5 C pressure: 0.1 MPa < p < 30 MPa salnty: 0 mol/kg < NaCl mh O < mol/kg CO CO -concentraton: 0 mol/kg < m < mol/kg H O Comments: The approach of [ 7 ] provdes a departure functon that modfes the pure water vscosty at 0.1 MPa by addng a summand accordng to salt- and CO -molalty. Note that there s a small nfluence of pressure whch s ntroduced by the formulaton ( 4.5 ) for the vscosty of pure water. Accordng to [ 17 ] n 003 the only source for data concernng the vscosty n the threecomponent-system of water, NaCl and CO was [ 7 ]. No other could be found n the meantme. But ths lterature ams at deep dsposal n the ocean and thus provdes only a formulaton for these specfc condtons. 33

40 4 Propertes of water vscosty [Pa s] m NaCl =0.0mol/kg,m CO =0.0mol/kg,p= 0.MPa m NaCl =0.0mol/kg,m CO =0.0mol/kg,p=30.MPa m NaCl =0.0mol/kg,m CO =0.9mol/kg,p= 0.MPa m NaCl =0.0mol/kg,m CO =0.9mol/kg,p=30.MPa m NaCl =0.8mol/kg,m CO =0.0mol/kg,p= 0.MPa m NaCl =0.8mol/kg,m CO =0.0mol/kg,p=30.MPa m NaCl =0.8mol/kg,m CO =0.9mol/kg,p= 0.MPa m NaCl =0.8mol/kg,m CO =0.9mol/kg,p=30.MPa temperature [ C] Fg. 4.1 Vscosty of water wth dssolved NaCl and CO ; after [ 7 ]. Conclusons: In Fg. 4.1 t seems as f the nfluence of dssolved CO on the brne vscosty decreases wth temperature compared to the nfluence of NaCl. The pressure nduced ncrease of vscosty n the three-component mxture s not sgnfcantly dfferent from the ncrease n pure water. Ths may justfy the assumpton of neglgble changes of the vscosty wth the CO -content especally n hgh-salnty brnes. The temperature range vald for eq. ( 4.31 ) actually excludes a use for CO - sequestraton n underground aqufers because these aqufers show temperatures above at least 10 C. However, ths formulaton mght be used anyway for a lack of better data. Therefore t s mportant to know the error that follows from an extrapolaton n the temperature drecton. Fg shows such extrapolatons for a brne wthout dssolved CO n comparson to the formulatons from [ 31 ]. Apparently, the calculated curves are not relable for hgher temperature and salntes. They cross the pure water curve as well as each other and some even fall below zero. The approach should certanly not be used above 40 C for low up to medum salntes and not above 0 C for medum up to hgh salntes. 34

41 4 Propertes of water vscosty [Pa s] m= 0.0 mol/kg, p= 0.1 MPa m= 0.0 mol/kg, p= 0.1 MPa m= 0.0 mol/kg, p=30.0 MPa m= 0.0 mol/kg, p=30.0 MPa m= 3.0 mol/kg, p= 0.1 MPa m= 3.0 mol/kg, p= 0.1 MPa m= 3.0 mol/kg, p=30.0 MPa m= 3.0 mol/kg, p=30.0 MPa m= 6.0 mol/kg, p= 0.1 MPa m= 6.0 mol/kg, p= 0.1 MPa m= 6.0 mol/kg, p=30.0 MPa m= 6.0 mol/kg, p=30.0 MPa c=0.0mol/kgnallcases lnes after [ 31 ] symbols after [ 7 ] temperature [ C] Fg Vscosty of water wth dssolved NaCl; after [ 7 ] and after [ 31 ]. 4.4 Thermal conductvty Thermal conductvty of pure water Source: [ 49 ] (cted n [ 31 ]) Formulaton: T ' T ' T' T' 3 4 ( 4.3 ) - thermal conductvty of pure water [J/(m K s)] T - normalsed temperature [-] T ' T ( 4.33 ) T - temperature [ C] 35

42 4 Propertes of water Range of valdty: temperature: 0 C < T < 330 C 0.7 thermal conductvty [J/(m K s)] temperature [ C] Fg Thermal conductvty of pure water; after [ 49 ], cted n [ 31 ]. Comments: The formulaton ( 4.3 ) s based on a comparson wth laboratory data. Addtonal data gven n [ 31 ] apparently confrm ths approach especally for the temperature range between 0 C and 80 C. Conclusons: Eq. ( 4.3 ) s a relable formulaton wthn the ranges ( 1.1 ) except for low temperatures Thermal conductvty of water wth dssolved NaCl Source: [ 49 ] (cted n [ 31 ]) 36

43 4 Propertes of water Formulaton: w T T T S 8 T S ( 4.34 ) - thermal conductvty of water wth dssolved NaCl [J/(m K s)] - thermal conductvty of pure water [J/(m K s)] (q.v. eq. ( 4.3 )) T - temperature [ C] S - ansatz functon [-] (q.v. eq. ( 4.35 )) m S ( 4.35 ) m m - molalty of NaCl n the water-phase [mol/kg] Range of valdty: temperature: 0 C < T < 330 C pressure: 0 mol/kg < m< 5 mol/kg 0.7 thermal conductvty [J/(m K s)] m= 0.0 mol/kg m= 1.0 mol/kg m=.0 mol/kg m= 3.0 mol/kg m= 4.0 mol/kg m= 5.0 mol/kg temperature [ C] Fg Thermal conductvty of water and dssolved NaCl; after [ 49 ], cted n [ 31 ]. 37

44 4 Propertes of water Comments: The formulaton ( 4.34 ) s based on a comparson wth laboratory data. Addtonal data gven n [ 31 ] apparently confrm ths approach especally for the temperature range between 0 C and 80 C. Conclusons: Eq. ( 4.34 ) s a relable formulaton wthn the ranges ( 1.1 ) except for low temperatures and for very hgh salntes Thermal conductvty of water wth dssolved CO Comments: We are not aware of formulatons concernng the thermal conductvty of water wth dssolved CO Thermal conductvty of water wth dssolved NaCl and CO Comments: We are not aware of formulatons concernng the thermal conductvty of water wth dssolved NaCl and CO. 4.5 Enthalpy Enthalpy of pure water Source: [ 3 ] Formulaton: h R T H O ( 4.36 ) h - enthalpy [kj/kg] R HO - specfc gas constant for water vapour (c.f. Appendx B) T - temperature [K] - ansatz functon (q.v. eq. ( 4.38)) - standardsed temperature (q.v. eq. ( 4.37)) [-] 38

45 4 Propertes of water T * ( 4.37 ) T - reference temperature (c.f. Appendx B) 34 I J 1 n 7.1 J( 1.) ( 4.38 ) 1 n, I, J - constants (c.f. Appendx B) - standardsed pressure [-] p ( 4.39 ) p * p - reference pressure (c.f. Appendx B) p - pressure [MPa] Range of valdty: temperature: pressure: K < T < K p s (T) < p < 100 MPa p s (T) - vapour saturaton pressure [MPa] (q.v. eq. ( 4.1)) Comments: A formulaton for the enthalpy can be derved from the specfc Gbbs free energy (eq. ( 4.40)): g h g T ( 4.40 ) T p g - specfc Gbbs free energy [kj/kg] Conclusons: Eq. ( 4.36 ) s a relable formulaton wthn the ranges ( 1.1 ). 39

46 4 Propertes of water p= 0.1 MPa p= 10.0 MPa p= 0.0 MPa p= 30.0 MPa enthalpy [kj/kg] temperature [ C] Fg Enthalpy of pure water; after [ 3 ] Enthalpy of water wth dssolved NaCl Source: [ 30 ] Formulaton: h X h X h mh ( 4.41 ) H O H O H O NaCl H O NaCl NaCl H O h - enthalpy of water wth dssolved NaCl [kj/kg] X - mass fracton of water [-] (q.v. eq. ( 4.4 )) H O H O X - mass fracton of salt [-] (q.v. eq. ( 4.43 )) NaCl H O h - enthalpy of pure water [kj/kg] (q.v. eq.s ( 4.36) or ( 4.45 )) H O h NaCl - enthalpy of pure salt [kj/kg] (q.v. eq. ( 6.3 )) h NaCl H O - enthalpy of mxng [kj/kg] (q.v. eq. ( 4.44 )) T - temperature [ C] m - molalty of NaCl n the water-phase [mol/kg] 40

47 4 Propertes of water X 1000 ( 4.4 ) m H O H O 58.44m m X NaCl H O ( 4.43 ) 3 NaCl j h H O ajt m ( 4.44 ) m 0 j0 a j - constants (c.f. Appendx B) h O 4 3 H T T T ( 4.45 ) Range of valdty s not specfed, but secondary lterature gves hnts: temperature: 0 C < T < 350 C salnty: 0 to full NaCl saturaton (as n the tabulated data from [ 1 ]) enthalpy after [ 30 ] after [ 3 ] at p= 0 MPa after [ 3 ] at p= 30 MPa enthalpy [kj/kg] temperature [ C] Fg Enthalpy of pure water; after [ 30 ] and after [ 3 ]. 41

48 4 Propertes of water enthalpy [kj/kg] m= 0.0 mol/kg m= 1.0 mol/kg m=.0 mol/kg m= 3.0 mol/kg m= 4.0 mol/kg m= 5.0 mol/kg m= 6.0 mol/kg temperature [ C] Fg Enthalpy of water wth dssolved NaCl at 0.1 MPa; after [ 30 ], usng the modfcatons descrbed n the text. Comments: Formulaton ( 4.41) s derved from tabulated data from [ 1 ] and [ 37 ]. The approach follows the general prncple that the enthalpes of the components of a soluton add proportonally to the mass of the components. An addtonal term for the heat of dssoluton s consdered, too. Formulaton ( 4.45 ) for the enthalpy of pure water hh O s less extensve than eq. ( 4.36 ) but t does not nclude the admttedly small effect of pressure. Addtonally, t appears to be optmsed for hgher temperatures as has already been observed n [ 0 ]. For pure water there are notceable devatons n the lower temperature range from the more accurate data of [ 3 ] as shown n Fg The curves n Fg are therefore calculated usng eq. ( 4.36 ) nstead of eq. ( 4.45 ). Problems wth the correlaton for the enthalpy of salt h NaCl after [ 30 ] are dscussed n secton 6.. Fnally, the summands n eq. ( 4.41 ) are not consstent n terms of dmensons. In the defntons ( 4.45 ), ( 6. ) and ( 4.44) all three enthalpes are gven n [kj/kg] but n eq. ( 4.41 ) the enthalpes h 1 and h are multpled by the referrng mass fracton 4

49 4 Propertes of water whle the enthalpy of mxng s multpled by the molalty of the soluton. So ether correlaton ( 4.44) for the enthalpy of mxng contans another error, or t s not gven n the rght dmensons, or the enthalpy of mxng s multpled n ( 4.41 ) by the salt mass fracton nstead of the molalty. The frst two assumptons are rather hard to check. The curves n Fg are thus calculated usng salt mass fractons nstead of the molaltes n ( 4.41 ) as has been done for practcal model calculatons descrbed n [ 6 ]. Conclusons: Formulaton ( 4.41 ) s vald wthn the ranges ( 1.1 ). However, the approach of [ 30 ] s obvously not too relable - nether related to the sngle enthalpes of pure water h 1, of pure salt h as well as of mxng h nor related to the formulaton as such. A bg problem here s that data to check the formulatons are hard to come by. The formulatons for the enthalpes h 1 and h from [ 30 ] can be replaced by eq.s ( 4.45 ) and ( 6.3 ), respectvely. But the formulaton for the enthalpy of mxng ntroduces an nconsstency. The reason for that s unknown and must be checked snce the dfferent explanatons to resolve these nconsstences lead to a hghly dfferent relevance of ths term for the enthalpy of brne Enthalpy of water wth dssolved CO Source: adapted from [ 14 ] Formulaton: h X h X h X h ( 4.46 ) H O H O H O CO H O CO CO H O CO H O h - enthalpy of water wth dssolved CO [kj/kg] X - mass fracton of water [-] H O H O X - mass fracton of CO [-] CO H O h - enthalpy of pure water [kj/kg] (q.v. eq. ( 4.36)) H O h CO - enthalpy of pure CO [kj/kg] (q.v. eq. ( 5.5 )) CO h H O - enthalpy of dssoluton [kj/kg] (q.v. eq. ( 4.47 )) 43

50 4 Propertes of water h CO H O (-RT c c - T ) 3 c T 4 c 5 c 7 p c8 p - c p T T T 630 T 630 T 9 c 10 p ( 4.47 ) p - Druck der Wasserphase [bar] T - temperature [K] R - unversal gas constant [J/(K mol)] c - constants (c.f. Appendx B) Range of valdty: temperature: 0 C < T < 60 C pressure: 0 MPa < p < 00 MPa salnty: 0 mol/kg < m < 4.87 mol/kg Comments: Formulaton ( 4.46 ) can be regarded as a smplfed verson of formulaton ( 4.48 ). Due to the mass proportonalty of the three contrbutory enthalpes the range of valdty for formulaton ( 4.46 ) s equal to the common range of valdtes of all these three enthalpes. For the range of valdty: see Comments n secton Conclusons: Formulaton ( 4.46 ) s vald wthn the ranges ( 1.1 ) Enthalpy of water wth dssolved CO and NaCl Source: [ 4 ], [ 14 ] Formulaton: h (1 X h X h X h ( 4.48 ) CO H ) O b CO H O CO CO H O CO H O h - enthalpy of water wth dssolved NaCl and dssolved CO [kj/kg] h b - enthalpy of water wth dssolved NaCl [kj/kg] (q.v. eq. ( 4.41 )) h CO - enthalpy of pure CO [kj/kg] (q.v. eq. ( 5.5 )) CO h H O - enthalpy of dssoluton [kj/kg] (q.v. eq. ( 4.49 )) X - mass fracton of CO [-] CO H O 44

51 4 Propertes of water ln K T hb p hs R( T 73.15) M CO ( 4.49 ) K hb - Henry s constant for brne and gaseous CO [Pa] (q.v. eq. ( 4.50 )) T - temperature [ C] R - unversal gas constant [J/(mol K)] M CO - molecular weght of CO [kg/mol] m k b ( 4.50 ) K 10 hb K h K hb - Henry s constant for brne and gaseous CO [Pa] K h - Henry s constant for pure water [Pa] (q.v. eq. ( 4.51 ) ) k b - saltng-out coeffcent [kg/mol] (q.v. eq. ( 4.5 ) ) m - molalty of NaCl n the water-phase [mol/kg] 5 K B( ) T ( 4.51 ) h 0 B() - constants (c.f. Appendx B) 4 k C( ) T ( 4.5 ) b 0 C() - constants (c.f. Appendx B) As a knd of valdaton of the approach ( 4.49 ) formulaton ( 4.46 ) for the enthalpy of dssoluton for pure water from [ 14 ] was checked aganst the approach from [ 4 ] as shown n Fg Range of valdty: temperature: 0 C < T < 300 C salnty: 0 mol/kg < m < halte saturaton further clues: (see secton 4.5.) 45

52 4 Propertes of water Henry-coeff. [MPa] m= 0.0 mol/kg m= 1.6 mol/kg m=.9 mol/kg m= 3.9 mol/kg m= 6.1 mol/kg halte saturated at 0.1 MPa temperature [ C] Fg Henry s constant for dfferent salntes, after [ 4 ]. dssoluton enthalpy [kj/kg] m= 0.0 mol/kg m= 1.6 mol/kg m=.9 mol/kg m= 3.9 mol/kg halte saturated at 0.1 MPa (ncl. temp. effect) m= 0.0 mol/kg at 1 bar after [ 14 ] temperature [ C] Fg. 4.0 Dssoluton enthalpy of CO ; after [ 4 ], approach for pure water after [ 14 ]. 46

53 4 Propertes of water enthalpy [kj/kg] p= 0.1 MPa,m= 0.0 mol/kg,xco= 0.0 [-] p= 30.0 MPa,m= 0.0 mol/kg,xco= 0.0 [-] p= 0.1 MPa,m= 6.0 mol/kg,xco= 0.0 [-] p= 30.0 MPa,m= 6.0 mol/kg,xco= 0.0 [-] p= 0.1 MPa,m= 0.0 mol/kg,xco= 0.9 [-] p= 30.0 MPa,m= 0.0 mol/kg,xco= 0.9 [-] p= 0.1 MPa,m= 6.0 mol/kg,xco= 0.9 [-] p= 30.0 MPa,m= 6.0 mol/kg,xco= 0.9 [-] temperature [ C] Fg. 4.1 Enthalpy of water wth dssolved NaCl and dssolved CO ; after [ 4 ]. Comments: Formulaton ( 4.48 ) s based on an already known brne enthalpy as well as the mass proportonalty of the enthalpes of brne and of CO. The enthalpy of dssoluton of gaseous CO s not gven drectly n [ 4 ] but as a functon of the temperature- and salnty-dependent Henry s constant as well as on the exclusvely temperature-dependent saltng-out coeffcent. It s ftted to data n the temperature range of 100 C up to 350 C. Consderng ths, the sharp bendg of the curves below 0 C n Fg. 4.0 looks suspcous, especally so, because the formulaton n [ 14 ] for fresh water looks almost lnear n ths range. The addtonal data used n [ 14 ] confrms a more or less lnear trend even down to 30 C. No comparson wth measured data under any salnty s gven n [ 4 ]. Conclusons: The only quantty n eq. ( 4.48 ) not already evaluated n other sectons s CO H O the dssoluton enthalpy h. Snce the enthalpy of dssoluton s multpled by the mass fracton of CO n brne t has no sgnfcant meanng for practcal purposes. The referrng formulatons ( 4.49) or ( 4.46 ) appear to be relable and vald wthn the ranges 47

54 4 Propertes of water ( 1.1 ) except for temperatures below 0 C where the formulatons dffer strongly. Here, t cannot be decded whch formulaton s more realstc. Otherwse the conclusons concernng brne enthalpy from secton 4.5. apply. 4.6 Heat capacty Heat capacty can be derved from the enthalpy as the partal dervatve wth respect to temperature: h c ( 4.53 ) T c h T - heat capacty [kj/(kg K)] - enthalpy [kj/kg] - temperature [K] Heat capacty of pure water Source: [ 3 ], [ 30 ] Formulaton after [ 3 ]: c p R ( 4.54 ) c p - sobarc heat capacty [kj/(kg K)] R - specfc gas constant for water vapour [kj/( kg K )] (c.f. Appendx B) - ansatz functon - standardsed temperature [-] (q.v. eq. ( 4.16 )) 34 1 I J J J n ( 4.55 ) - standardsed pressure [-] (q.v. eq. ( 4.13 )) n, I, J - constants for eq. ( 4.38 ) (c.f. Appendx B, eq. ( 4.15 )) 48

55 4 Propertes of water Range of valdty: temperature: K < T < K pressure: p s (T) < p < 100 MPa p s (T) - vapour saturaton pressure [MPa] (q.v. eq. ( 4.1)) Another formulaton for the heat capacty can be derved startng wth eq. ( 4.45 ) from [ 30 ] for the enthalpy of pure water. The temperature dervatve of ths approach reads as follows: c w h T T T ( 4.56 ) c w - heat capacty of pure water [kj/(kg C)] h 1 - enthalpy of pure water (q.v. eq. ( 4.45 )) T - temperature [ C] Range of valdty: (see secton 4.5. ) heat capacty [kj/(kg K)] after [ 30 ] p= 0.1 MPa; after [ 3 ] p= 30.0 MPa; after [ 3 ] temperature [ C] Fg. 4. Heat capacty of pure water; after [ 3 ] and [ 30 ]. 49

56 4 Propertes of water Comments: Followng the formulaton from [ 3 ] for the heat capacty the effects of pressure and temperature appear to be neglgble n the constant range of nterest. In contrast the approach based on the enthalpy of pure water provded by [ 30 ] shows a clear dependency on temperature whle the effect of pressure s not ncluded. Conclusons: Eq. ( 4.54 ) s a relable formulaton wthn the ranges ( 1.1 ). Instead of usng the approach ( 4.56 ) a representatve constant value based on eq. ( 4.54 ) should be consdered Heat capacty of water wth dssolved NaCl Source: [ 3 ], [ 30 ] Formulaton: c X X H O H O H O H O h1 X T c w X NaCl H O NaCl H O c h X T NaCl X NaCl H O NaCl H O h T hm T m ( 4.57 ) c - heat capacty of water wth dssolved NaCl [kj/(kg K)] c w - heat capacty of pure water [kj/(kg K)] (q.v. eq. ( 4.54 )) c NaCl - heat capacty of NaCl [kj/(kg K)] (q.v. eq. ( 6.4 )) X - mass fracton of water [-] (q.v. eq. ( 4.4 )) H O H O X - mass fracton of salt [-] (q.v. eq. ( 4.43 )) NaCl H O h 1 - enthalpy of pure water [kj/kg] (q.v. eq. ( 4.36)) h - enthalpy of pure salt [kj/kg] (q.v. eq. ( 6.3 )) h - enthalpy of mxng [kj/kg] (q.v. eq. ( 4.44 )) T - temperature [ C] 3 h ( 1) j T m 0 j0 a T j m ( 4.58 ) m - molalty of NaCl n the water-phase [kg/mol] 50

57 4 Propertes of water Range of valdty: (see secton 4.5.) heat capacty [kj/(kg K)] p= 0.1 MPa, m= 0.0 mol/kg p= 0.1 MPa, m= 6.0 mol/kg p=30.0 MPa, m= 0.0 mol/kg p=30.0 MPa, m= 6.0 mol/kg temperature [ C] Fg. 4.3 Heat capacty of water wth dssolved NaCl; derved from [ 30 ]. Comments: Analogously to the enthalpy the heat capacty for water wth dssolved NaCl s the mass weghted sum of the sngle heat capactes plus enthalpy of mxng and/or enthalpy of dssoluton. Formulaton ( 4.41 ) for the enthalpy of water wth dssolved NaCl s nserted n eq. ( 4.53 ) ncludng the modfcatons descrbed n the comments n secton The approach conssts of three summands that can be dfferentated separately. Conclusons: (see secton 4.5.) 51

58 4 Propertes of water Heat capacty of water wth dssolved CO Source: - Formulaton: CO h h CO h CO H O CO CO H O c ( 1 X H O) X H O X ( 4.59 ) H O T T T c - heat capacty of water wth dssolved NaCl [kj/(kg K)] X - mass fracton of CO [-] CO H O h - enthalpy of pure water [kj/kg] (q.v. eq. ( 4.36)) H O h CO - enthalpy of pure CO [kj/kg] (q.v. eq. ( 5.5 )) CO h H O - enthalpy of dssoluton [kj/kg] (q.v. eq. ( 4.47 )) Comments: The heat capacty gven here s derved from the enthalpy ( 4.46 ) usng ( 4.53 ). Conclusons: Formulaton ( 4.59 ) s vald wthn the ranges ( 1.1 ) Heat capacty of water wth dssolved NaCl and CO Source: [ 4 ] Formulaton: Approach ( 4.53 ) usng eq. ( 4.48 ) can be used. Snce a specal formulaton would be very complex, the partal dervatve of eq. ( 4.48 ) wth respect to temperature s calculated numercally. Range of valdty: (see secton 4.5.3) Comments: - Conclusons: (see secton 4.5.3) 5

59 4 Propertes of water heat capacty [kj/(kg K)] p= 0.1 MPa,m= 0.0 mol/kg,xco= 0.0 [-] p= 0.1 MPa,m= 0.0 mol/kg,xco= 0.9 [-] p= 0.1 MPa,m= 6.0 mol/kg,xco= 0.0 [-] p= 0.1 MPa,m= 6.0 mol/kg,xco= 0.9 [-] p= 30.0 MPa,m= 0.0 mol/kg,xco= 0.0 [-] p= 30.0 MPa,m= 0.0 mol/kg,xco= 0.9 [-] p= 30.0 MPa,m= 6.0 mol/kg,xco= 0.0 [-] p= 30.0 MPa,m= 6.0 mol/kg,xco= 0.9 [-] temperature [ C] Fg. 4.4 Heat capacty of water wth dssolved NaCl and CO ; derved from [ 4 ]. 4.7 Solublty Solublty of NaCl n water Solublty of NaCl n lqud water Source: [ 33 ] cted n [ 9 ] Formulaton: L NaCl water T T ( 4.60 ) 3 Solublty s essentally the rato of the mass of solute to the mass of solvent. However, solublty can be gven n dfferent unts lke [g/100g solvent], [kg/kg], [mol/mol], [mol/kg], etc.. 53

60 4 Propertes of water NaCl L water - solublty of NaCl n lqud water 4 [kg/kg] (q.v. eq. ( 4.61 )) T - temperature [ C] Converson to molalty: L NaCl water 1 L water NaCl NaCl L M water NaCl ( 4.61 ) NaCl L water - solublty of NaCl n lqud water [mol/kg] NaCl M - molecular weght of NaCl [kg/mol] (c.f. Appendx B) Range of valdty: temperature: 0 < T < 800 C clamed by [ 33 ] 0 < T < 400 C recommended by [ 9 ] 6.7 I:\projekte\co_ugs_rsks\rechnungen\eos\xmax-ho-nacl.lay max. salt concentraton [mol/kg] temperature [ C] 4 In order to dfferentate the phase state of H O the term water wll be used for lqud H O and vapour for a H O rch gasphase. 54

61 4 Propertes of water Fg. 4.5 Solublty of NaCl n water; after [ 33 ]. Comments: Tabulated data also n [ 31 ]. Conclusons: Eq. ( 4.60 ) s a relable formulaton wthn the ranges ( 1.1 ) Solublty of NaCl n water vapour Source: [ 1 ] Formulaton : L 0 ( 4.6 ) NaCl vapour NaCl L vapour - solublty of NaCl n water vapour [mol/kg] Comments: The concentraton of salt n water vapour amounts to mol percent at 350 C [ 1 ]. Wthn the temperature range ( 1.1 ) t s even lower. Conclusons: Solublty of NaCl n water vapour s neglgble wthn the ranges ( 1.1 ) Solublty of CO n water Source: [ 41 ] Formulaton: ˆ CO H O LH O p B L ( ) CO 1 ˆ ( 4.63 ) L - Solublty of CO n pure water [mol/mol] ˆCO H O( p) ˆ - Solublty of H O n CO -phase [mol/mol] (q.v. eq. ( 4.65 )) H O LCO B - constant (q.v. eq. ( 4.69 )) 55

62 4 Propertes of water Converson of solublty L to molalty: ˆCO H O L Lˆ CO CO H O( p) ˆ H O( p) H O( p) CO H O 1 L M ( 4.64 ) L - Solublty of CO n pure water [mol/kg] CO H O( p) H O M - molecular weght of H O [kg/mol] (c.f. Appendx B) ˆ L H O CO 1 B 1 B A ( 4.65 ) A,B - ansatz functons (q.v. eq. ( 4.67 )) Converson of solublty ˆ to molalty: H O LCO L H O H O CO ˆ CO L - Solublty of H O n the CO -phase [mol/kg] CO H O Lˆ ( 4.66 ) CO H O CO 1 L M CO M - molecular weght of CO [kg/mol] (c.f. 0) K A H O 0 H O e p H p p V O 0 RT ( 4.67 ) H O K 0 - equlbrum constant [bar], (q.v. eq. ( 4.68)) H O - fugacty coeffcent for water [-] (q.v. eq. ( 4.71 )) p - total pressure [bar] p - reference pressure [bar], here: p 1bar 0 H O V - partal molar volume of CO n water at nfnte soluton [cm³/mol] (c.f. Appendx B) T - temperature [K] R - unversal gas constant [(bar cm³)/(k mol)] 0 56

63 4 Propertes of water K0 et H O 3 4 a bt ct dt ( 4.68 ) a,b,c,d,e - constants (c.f. Appendx B) CO p B K CO 0 e CO p p V 0 RT ( 4.69 ) CO - fugacty coeffcent for carbon doxde [-] (q.v. eq. ( 4.71 )) CO K - equlbrum constant [bar], (q.v. eq. ( 4.70 )) 0 V CO - partal molar volume of CO n water at nfnte soluton [cm³/mol] (c.f. Appendx B) K CO 0 a bt ( 4.70 ) ct a,b,c - constants (c.f. Appendx B) k ln V ln V b a RT mx bk b mx 1.5 mx bk V b mx V b ln V mx n xco a RT bmx bmx V b mx k V b ln V mx pv ln RT for k=co : ( 4.71 ) ln CO V ln V b x a amxb 1.5 RT b mx CO mx bco V b x H O CO H OCO 1.5 RT bmx V b ln V mx CO CO mx a CO V b ln V bmx V b mx mx pv ln RT 57

64 4 Propertes of water for k= H O: ln H O V ln V b x CO CO a amxbh 1.5 RT b O mx mx bh V b x H OCO 1.5 RT bmx V b ln V O mx H O CO mx a H O V b ln V bmx V b mx mx pv ln RT k - fugacty coeffcent of component k n mxtures wth other components [-] V - partal molar volume of the gas phase [cm³/mol] (q.v. eq. ( 4.75 )) a - measure for attractve molecular force of component [bar cm 6 K 0.5 mol - ]; (q.v. eq. ( 4.74 )) b k - measure for molecular sze of component k [cm³/mol] (c.f. Appendx B) a mx - constant a for mxtures [bar cm 6 K 0.5 mol - ] (q.v. eq. ( 4.7 )) b mx - constant b for mxtures [cm³/mol] (q.v. eq. ( 4.73 )) xco - mole fracton of component n the CO -phase [-] a k - measure for attractve molecular force of components and k [bar cm 6 K 0.5 mol - ] (c.f. Appendx B) n - number of nvolved components [-] - ndex standng for H O and CO a mx H O H O CO CO xco a H O xco x CO a H O CO x CO a CO ( 4.7 ) b mx x b x b H O CO CO H O CO CO ( 4.73 ) 4 a CO T ( H O s not requred) a ( 4.74 ) 3 RT RTb a ab V V V b ( 4.75 ) p p pt pt 58

65 4 Propertes of water Eq. ( 4.75 ) can yeld more than one soluton for the Volume V for a gven subcrtcal state expressed n temperature T and pressure p. Maxmum root depcts the volume V gas of a gas phase, mnmum root the volume V lqud of a lqud phase. Whch phase state s actually stable can be decded from the crterum c 0 stable gas phase c 0 two-phase state ( 4.76 ) c 0 stable lqud state c - crterum for a stable subcrtcal phase state [-] (q.v. eq. ( 4.77 )) c w w 1 ( 4.77 ) w 1, w - work expressons (q.v. eq.s ( 4.78 ) and ( 4.79 )) w1 p V gas V lqud ( 4.78 ) V gas - partal molar volume of the gas phase [cm³/mol] V lqud - partal molar volume of the lqud phase [cm³/mol] V gas b Vlqud Vlqud b Vgas V gas b a w RT ln ln 0. 5 ( 4.79 ) Vlqud b T b Range of valdty: temperature: 1 C < T < 100 C pressure: 0 MPa < p < 60 MPa 59

66 4 Propertes of water 3 I:\projekte\co_ugs_rsks\rechnungen\eos\l-ho-sp.lay.5 p= 0.1 MPa p=10mpa p=30mpa solublty [mol/kg] temperature [ C] Fg. 4.6 Solublty of CO n pure water; after [ 41 ]. Comments: Henry s law consttutes a smple lnear relaton between the mole fracton of dssolved gas n a soluton and the partal pressure of ths gas over the soluton: x p x K ( 4.80 ) CO CO CO CO H O CO H H O CO The proportonalty factor K n ths equaton s called Henry s constant. It s rather H H O accurate for strongly dluted deal solutons and deal gases at constant temperature and pressures up to 10 MPa. Apparently, ths approach s not approprate n the context of CO -sequestraton because of varyng temperature and rather hgh pressures. Addtonally, the dssolved substance s not allowed to react wth the solvent as n the case of carbon doxde that partly becomes carbonc acd n water. Henry s law must therefore be substtuted by a more accurate relaton. Conclusons: Eq. ( 4.63 ) s a relable formulaton wthn the ranges ( 1.1 ). 60

67 4 Propertes of water Solublty of CO n water wth dssolved NaCl Source: [ 14 ], [ 40 ] Formulaton after [ 14 ]: L ln CO vapour CO mh O p RT 1(0) CO ln CO CO Na m Na water CO NaCl m Na water m Cl water ( 4.81 ) CO L vapour - solublty of CO n the vapour phase [mol/kg] p - total pressure [bar] CO m - molalty of CO n the water-phase [mol/kg] H O 1(0) CO - chemcal potental 1(0) RT - ftted quantty [-] (q.v. eq. ( 4.8 )) CO CO - fugacty coeffcent for CO [-] CO Na - nteracton constant [kg/mol](q.v. eq. ( 4.8 )) - nteracton constant [(kg/mol)²] (q.v. eq. ( 4.8 )) CO NaCl m - molalty of Na the water-phase [mol/kg] Na H O m - molalty of Cl the water-phase [mol/kg] Cl H O c3 c5 ( quantty) c1 ct c4t c6 p c7 plnt c T 630 T p p c9 c10 c11t ln p 630 T 630 T 8 p T ( 4.8 ) quantty - stands for 1(0) CO RT, CO and Na CO NaCl c1 to c constants; dfferent constants for dfferent quanttes (c.f. Appendx B) T - temperature [K] 61

68 4 Propertes of water 6 Z - ansatz functon [-] (q.v. eq. ( 4.86 )) T r - reduced temperature [-] (q.v. eq. ( 4.84 )) V r - reduced volume [-] (q.v. eq. ( 4.85 )) a 1 to a 15 - constants (c.f. Appendx B) T c - crtcal temperature of CO [K] p r - reduced pressure [-] (q.v. eq. ( 4.87 )) ln 1 ln V r a r r r r r r r r r r r r r r e V a a a a T a V T a T a a V T a T a a V T a T a a V T a T a a Z Z ( 4.83 ) c r T T T ( 4.84 ) c c r p RT V ( 4.85 ) V r a r r r r r r r r r r r r r r r r r r e V a a V T a V T a T a a V T a T a a V T a T a a V T a T a a T p V Z ( 4.86 )

69 4 Propertes of water p pr ( 4.87 ) p c p c - crtcal pressure of CO [bar] Formulaton after [ 40 ]: CO CO L H O( p) LH O ( 4.88 ) * * - actvty coeffcent after [ 14 ] [-]; (q.v. eq. ( 4.90 )) L - solublty of CO n pure water [mol/kg]; (q.v. eq. ( 4.89 )) CO H O( p) L - solublty of CO n the water-phase [mol/kg] CO H O CO Converson of L to mole fracton: H O Lˆ CO H O 1 H O M L CO H O m NaCl H O L CO H O ( 4.89 ) L - solublty of CO n the water-phase [mol/mol]; (q.v. eq. ( 4.63 )) ˆCO H O H O M - molecular weght of H O [kg/mol] (c.f. Appendx B) m - molalty of NaCl n the water-phase [mol/kg] NaCl H O - stochometrc number of ons contaned n the dssolved salt [-]; n case of NaCl Na Na Cl CO Na m H O CO Na Cl m H O m H O * e ( 4.90 ) CO Na - nteracton constant [kg/mol] (q.v. eq. ( 4.8 )) - nteracton constant [(kg/mol)²] (q.v. eq. ( 4.8 )) CO NaCl m - molalty of Na the water-phase [mol/kg] Na H O m - molalty of Cl the water-phase [mol/kg] Cl H O 63

70 4 Propertes of water Range of valdty: [ 14 ]: [ 40 ] temperature: 0 C < T < 60 C 1 C < T < 100 C pressure: 0 MPa < p < 00 MPa 0 MPa < p < 60 MPa salnty: 0 mol/kg < m < 4.3 mol/kg 0 mol/kg < m < 6 mol/kg.5 solublty [mol/kg] temperature [ C] Fg. 4.7 Solublty of CO n NaCl-solutons; lne codng: lne thckness: thn lnes - after [ 14 ], thck lnes - after [ 40 ]; lne type: sold - p=0.1 MPa, dashed - 10 MPa, dotted - 30 MPa; lne colour: blue - NaCl mh O = 0 mol/kg, green - 3 mol/kg, red - 6 mol/kg. Comments: V r n eq. ( 4.86 ) s to be solved teratvely befor Z can be calculated. was set to zero bevor fttng parameters were calculated. CO Na Whle components of a gas mxture do not nfluence each other wth respect to solublty, the solublty of gases n NaCl-solutons depends strongly on the salt concentraton [ 10 ]. 64

71 4 Propertes of water The relaton derved n [ 14 ] s ftted to an extensve lot of solublty experments. Snce t ncludes the salnty of the water phase t s presently one of the best adapted and most general formulatons. However, they provde only solublty of CO n aqueous salt solutons but not solublty of water n CO. The formulatons n [ 14 ] ncludes an teratve calculaton of constant V r. [ 17 ] and [ 40 ] state that calculatng the teratve soluton requres too much computatonal tme. On the other hand, [ 40 ] seems to prefer another formulaton for the actvty coeffcent x that s apparently of smlar accuracy as the approach of [ 14 ] but requres an teratve procedure as well. Whle [ 14 ] clams that ther formulatons are accurate up to 4.3 mol/kg NaCl, [ 40 ] attests even a suffcent accuracy up to 6 mol/kg NaCl f pressure and temperature are restrcted to a range typcal for CO -sequestraton. Conclusons: Eqs. ( 4.81 ) and ( 4.88 ) are relable formulatons wthn the ranges ( 1.1 ) up to 6 mol/kg NaCl. 4.8 Dffusvty Sources: data: [ 8 ], [ 11 ] formulaton: present report Formulaton: CO CO m D H O T ( 4.91 ) D - Dffuson coeffcent of CO n water [m²/s] CO H O T - Temperature [ C] m - molalty of CO n the water-phase [mol/kg] CO H O L - solublty of CO n the water-phase [mol/kg] (q.v. eq. ( 4.64 )) CO H O H O T 100 T 1 0. CO LH O Range of valdty: temperature: 0 C < T < 100 C 65

72 4 Propertes of water pressure: 0 MPa < p < 100 MPa dffuson coeffcent [m²/s] 1E-08 8E-09 6E-09 4E-09 E-09 source 363b *) source 391 *) source 391 *) source 391 *) source 336 *) source 46 *) source 391 *) source 398 *) source 333 *) [ 8 ] source 393 *) *) n ad-hoc approach temperature [ C] Fg. 4.8 Self-dffusvty of H O (probably at atmospherc pressure); data from [ 8 ]. dffuson coeffcent [m²/s] 1E-08 8E-09 6E-09 4E-09 E-09 H O-CO ; source 373a *) H O-CO ; source 374a *) H O-CO ; source 38a *) H O-CO ; source 38a *) H O-CO ; source 40a *) H O-CO ; varous sources *) H O-CO ; source 379a *) *) n [ 8 ] ad-hoc approach temperature [ C] Fg. 4.9 Dffusvty of CO n H O for dfferent temperatures; data from [ 8 ]. 66

73 4 Propertes of water dffuson coeffcent [m²/s] 1E-08 8E-09 6E-09 4E-09 E-09 T= 10 C T= 5 C T= 45 C T= 50 C pressure [MPa] Fg Self-dffusvty of H O dependng on pressure; data from [ 8 ], [ 11 ]. dffuson coeffcent [m²/s] E E E E-09 1.E-09 H O-NaCl; T=15 C; source 350 *) H O-NaCl; T=18.5 C; source 43 *) H O-NaCl; T=5 C; source 385 *) H O-NaCl; T=5 C; sources 57,58 *) H O-NaCl; T=5 C; source 370a *) *) n [ 8 ] 1E salnty [mol/l] Fg Dffusvty of NaCl n H O for dfferent temperatures; data from [ 8 ]. 67

74 4 Propertes of water Comments: We are not aware of any formulatons for the dffusvty n H O as a functon of temperature, pressure and/or salnty. However, a good complaton of measured data exsts. The formulaton presented above s an ad hoc approach based on the data of [ 8 ] and [ 11 ] to provde a frst approxmaton. Dffusvty of water s apparently nether sgnfcantly dependent on pressure nor on salnty n the ranges ( 1.1 ) (c.f. Fg and Fg. 4.31). Temperature, however, changes the dffuson coeffcent consderably. The coeffcent of self-dffuson s dstnctly lowered by dssolved CO. Conclusons: A rough estmate for the dffusvty n water can be concluded from the data. However, there s no data concernng the pressure-dependency of the dffuson of NaCl or CO n water or CO n brne. 68

75 5 Propertes of carbon doxde 5.1 Vapour pressure Vapour pressure of pure CO Source: [ 39 ] Formulaton: p s Tc ln p c T 4 1 a 1 T T c t ( 5.1 ) p s - vapour pressure of carbon doxde [Pa] T - temperature [K] p c - vapour pressure of CO at the crtcal pont [Pa] (c.f. Appendx B) T c - temperature of CO at the crtcal pont [K] (c.f. Appendx B) a, t - parameters (c.f. Appendx B) Range of valdty: temperature: 10 K < T < T c Comments: - Conclusons: Eq. ( 5.1 ) s a relable formulaton wthn the ranges ( 1.1 ) up to the crtcal temperature. 69

76 5 Propertes of carbon doxde 8 vapour saturaton pressure[mpa] temperature [ C] Fg. 5.1 Vapour saturaton pressure of CO [kg/m³]; after [ 39 ] Vapour pressure of CO wth dssolved H O Source: - Comments: We are not aware of formulatons concernng the vapour pressure CO of wth dssolved H O. 5. Helmholtz energy for CO 5..1 Approach Several thermodynamc propertes can be derved from the dmensonless Helmholtz energy. An elaborate formulaton s provded by [ 39 ]. 70

77 5 Propertes of carbon doxde A ( 5. ) R T CO A - specfc Helmholtz energy [J/kg] - dmensonless Hemholtz energy [-] R CO - specfc gas constant for CO [J/(kg K)] (c.f. Appendx B) T - temperature [K] r 0 ( 5.3 ) 0 - dmensonless Helmholtz energy from deal gas behavour [-] (q.v. eq. ( 5.4 )) r - dmensonless Helmholtz energy from resdual flud behavour [-] (q.v. eq. ( 5.7 )) a1 a a3 ln a ln1 exp 0 ln ( 5.4 ) 4 a 0, 0 - parameters (c.f. Appendx B) - standardsed densty [-] (c.f. eq. ( 5.5 )) - nverse standardsed temperature [-] (c.f. eq. ( 5.6 )) ( 5.5 ) c - densty of carbon doxde [kg/m³] c - densty of CO at the crtcal pont [kg/m³] (c.f. Appendx B) T c ( 5.6 ) T c - temperature at the crtcal pont [K] (c.f. Appendx B) 71

78 5 Propertes of carbon doxde r d t d t c d t n n e n e 4 b n ( 5.7 ) b, c, d, n, t,,,, - parameters of Helmholtz energy (c.f. Appendx B) - dstance functon (q.v. eq. ( 5.8 )) - exponental functon (q.v. eq. ( 5.10 )) a B 1 ( 5.8 ) a, A, B - parameters of Helmholtz energy (c.f. Appendx B) - auxlary functon (q.v. eq. ( 5.9 )) 1 1 A 1 ( 5.9 ) C 1 1 D e ( 5.10 ) C, D - parameters of Helmholtz energy (c.f. Appendx B) 5.. Dervatves In [ 39 ] the wrtng conventon x x y wth x, and y, ( 5.11 ) - dmensonless Helmholtz energy [-] (q.v. eq. ( 5.3 )) - nverse standardsed temperature [-] (c.f. eq. ( 5.6 )) - standardsed densty [-] (c.f. eq. ( 5.5 )) s used. In the followng, several dervatves of the dmensonless Helmholtz energy are gven that wll be used n subsequent sectons. 7

79 5 Propertes of carbon doxde 73 Dervatve 0 a 0, 0 - constants (c.f. Appendx B) Dervatve r a, b, c, d, n, t,,,,, A, B, C, D - parameters of Helmholtz energy (c.f. Appendx B) - dstance functon (q.v. eq. ( 5.8 )) - exponental functon (q.v. eq. ( 5.10 )) b - auxlary functon (q.v. eq. ( 5.15 )) - auxlary functon (q.v. eq. ( 5.16 )) e a a a ( 5.1 ) e e a a ( 5.13 ) b b t d c t d t d r c n d e n c d n e n d ( 5.14 ) a b b B a A b ( 5.15 ) 1 C ( 5.16 )

80 5 Propertes of carbon doxde 74 Dervatve r b - auxlary functon (q.v. eq. ( 5.18 )) - auxlary functon (q.v. eq. ( 5.0 )) - auxlary functon (q.v. eq. ( 5.19 )) b b b d d d d t c c c t d t d r c n d d d e n c c d c d n e d n d ( 5.17 ) 1 1 b b b b b ( 5.18 ) a a B A ( 5.19 ) C C 1 1 ( 5.0 )

81 5 Propertes of carbon doxde 75 Dervatve r b - auxlary functon (q.v. eq. ( 5. )) - auxlary functon (q.v. eq. ( 5.3 )) Dervatve r b - auxlary functon (q.v. eq. ( 5.5 )) b b t d t d t d r c n t e n e n t n t ( 5.1 ) 1 b b b ( 5. ) 1 D ( 5.3 ) b b b t d t d t d r c n t t e n e t n t t n t ( 5.4 )

82 5 Propertes of carbon doxde 76 - auxlary functon (q.v. eq. ( 5.6 )) Dervatve r b - auxlary functon (q.v. eq. ( 5.8 )) - auxlary functon (q.v. eq. ( 5.9 )) b b b b b b ( 5.5 ) D D 1 1 ( 5.6 ) b b b b t d c t d t d r c n t d e n c d t n e n d t ( 5.7 ) b b b b b Ab ( 5.8 ) D C ( 5.9 )

83 5 Propertes of carbon doxde 5.3 Densty Densty of pure CO Source: [ 39 ] Formulaton: p R T CO r 1 ( 5.30 ) p - pressure of the carbon doxde [Pa] - densty of carbon doxde [kg/m³] R CO - specfc gas constant for CO [J/(kg K)] (c.f. Appendx B) T - temperature [K] - standardsed densty [-] (c.f. eq. ( 5.5 )) r - partal dervatve of the Helmholtz energy (q.v. eq. ( 5.14 )) Range of valdty: temperature: pressure: 16 K < T < 1100 K 0 MPa < p < 800 MPa Comments: Pressure - usually a prmary varable n a numercal scheme - s replaced by the densty. Unfortunately, [ 39 ] does not provde an equaton for the back calculaton of the densty as a functon of temperature and pressure. The same problems are apparently encountered n [ 17 ] when t comes to applcaton n a numercal model. There, the problem s solved by calculatng the densty of CO at dscrete ponts n the p- T-doman and lnear nterpolaton durng run tme of the program. An teratve scheme to calculate the densty from temperature and pressure s descrbed n Appendx B. Conclusons: Eq. ( 5.30 ) s a relable formulaton wthn the ranges ( 1.1 ). 77

84 5 Propertes of carbon doxde pressure[mpa] = 50. kg/m³ =100.kg/m³ =150.kg/m³ =00.kg/m³ =50.kg/m³ =300.kg/m³ =350.kg/m³ =400.kg/m³ =450.kg/m³ =500.kg/m³ =550.kg/m³ =600.kg/m³ =650.kg/m³ =700.kg/m³ =750.kg/m³ =800.kg/m³ =850.kg/m³ =900.kg/m³ =950.kg/m³ =1000. kg/m³ two-phase doman temperature [ C] Fg. 5. Densty of CO ; after [ 39 ] Densty of CO wth H O Densty of gaseous CO Source: e.g. [ 17 ] Formulaton: CO H O CO CO CO ( 5.31 ) CO - densty of gaseous carbon doxde [kg/m³] - partal densty of carbon doxde n the gas phase [kg/m³] CO CO - partal densty of water vapour n the gas phase [kg/m³] H O CO Range of valdty: temperature: K T K pressure: 0 MPa < p < p s (T,m); p s - vapour pressure of CO 78

85 5 Propertes of carbon doxde Comments: In case of a gaseous CO -phase coexstng wth a lqud water phase equlbrum of the water vapour n the gaseous CO -phase and the lqud water s assumed. The partal pressure of water vapour equals the vapour saturaton pressure whch s dscussed n secton for pure water and n secton 4.1. for water wth dssolved NaCl. The partal densty of water vapour can be calculated wth the help of formulaton ( 4.17 ) usng the approprate partal pressure of water vapour. The densty of the gas mxture can fnally be calculated as the sum of the partal gas denstes. Conclusons: Eq. ( 5.31 ) s a relable formulaton wthn the ranges ( 1.1 ) Densty of lqud CO Source: [ 4 ], [ 17 ] Comments: [ 4 ] and [ 17 ] assume that soluton of water n lqud carbon doxde can be descrbed lke an evaporaton process. Ths leads to the same relaton as n secton , agan wth the addtonal assumpton that the pressure of dssolved water equals the vapour saturaton pressure. Conlusons: Apparently, very lttle s known about the relaton between CO -densty and the amount of dssolved water n the CO -domnated phase. 5.4 Vscosty Vscosty of pure CO Source: [ 15 ], [ 46 ] Formulaton: 0 ( 5.3 ) c - vscosty of carbon doxde [µpa s] - vscosty n the zero-densty lmt [µpa s] (q.v. eq. ( 5.33 )) 79

86 5 Propertes of carbon doxde - excess vscosty [µpa s] (q.v. eq. ( 5.36 )) c - crtcal enhancement n the vcnty of the crtcal pont [µpa s] T ( 5.33 ) 0 * * - reduced effectve cross-secton (q.v. eq. ( 5.34)) T - Temperature [K] 4 0 * * ln a ln T ( 5.34 ) T* - reduced Temperature [K] (q.v. eq. ( 5.35)) a - constants (c.f. Appendx B) * T T ( 5.35 ) d d d11 d1 d ( 5.36 ) * 3 81 * T T d j - constants (c.f. Appendx B) - densty of carbon doxde [kg/m³] c 4 1 e ( 5.37 ) e - constants (c.f. Appendx B) Range of valdty: temperature: 00 K < T < 1500 K pressure: p< 300 MPa for T < 1000 K densty: < 1400 kg/m³ 80

87 5 Propertes of carbon doxde Accuracy s ±0.3% for the vscosty of dlute gas near room temperature and ±5% at the hghest pressures. vscosty [Pa s] p= 0.1 MPa p= 1. MPa p=. MPa p= 3. MPa p= 4. MPa p= 5. MPa p= 6. MPa p= 7. MPa p= 8. MPa p= 9. MPa p= 10. MPa p= 0. MPa p= 30. MPa 5E temperature [ C] Fg. 5.3 Vscosty of CO ; after [ 15 ]. Comments: The thrd summand of eq. ( 5.3 ) s effectve only n the vcnty of the crtcal pont. Vcnty s defned n [ 15 ] by 300 K < T < 310 K and 300 kg/m³ < < 600 kg/m³. In ths range the quotent c / exceeds 1%. c s not gven n [ 15 ] but the authors refer to [ 46 ] for formulaton ( 5.37 ). Values of the vscosty at specfed temperatures, pressures and denstes are gven n [ 15 ] as well and can be found under n Appendx B. An addtonal backup for the formulatons from [ 15 ] comes from [ 17 ]. The correlatons from [ 15 ] are compared wth an approach from [ 1 ] whch yelds a maxmum dfference of less than %. Conclusons: Eq. ( 5.3 ) s a relable formulaton wthn the ranges ( 1.1 ). 81

88 5 Propertes of carbon doxde 5.4. Vscosty of CO wth H O Vscosty of gaseous CO Source: [ 4 ] Formulaton: ( 5.38 ) g CO CO H O H O X g g X g g g - vscosty of the mxture of gaseous CO and water vapour [Pa s] CO g - vscosty of gaseous CO [Pa s] (q.v. eq. ( 5.3 )) HO g - vscosty of water vapour [Pa s] (q.v. eq. ( 4.5 )) CO X g - mass fracton of CO n the gas phase [-] HO X g - mass fracton of water vapour n the gas phase [-] Range of valdty: pressure: H O H O pg psat Comments: At sngle gas phase condtons the vscosty of a mxture of water vapour and carbon doxde s assumed to be mass proportonal. Conclusons: Apparently, no measurements are known to confrm approach ( 5.38 ) Vscosty of lqud CO Comments: At the tme we are not aware of formulatons descrbng the vscosty of lqud carbon doxde wth dssolved H O. A frst approxmaton may be to neglect the nfluence of the dssolved water on the lqud CO on account of the low solublty of water n carbon doxde. Conclusons: Apparently, no measurements are known to back up the suggested approach. 8

89 5 Propertes of carbon doxde 5.5 Thermal conductvty Thermal conductvty of pure CO Source: [ 46 ] ncludng modfcatons from [ 38 ] Formulaton 0 ( 5.39 ) c - thermal conductvty [W/(m K)] - thermal conductvty of the deal gas wth neglgble densty [W/(m K)] (q.v. eq. ( 5.40 )) - correcton for the nfluence of densty [W/(m K)] (q.v. eq. ( 5.44 )) c - crtcal enhancement [W/(m K)] (q.v. eq. ( 5.45 )) * T 1 r 0 ( 5.40 ) 1000 T - temperature [K] r - ansatz functon (q.v. eq. ( 5.41 )) * - ansatz functon (q.v. eq. ( 5.4 )) T T r 1 e c ( 5.41 ) c - constants (c.f. Appendx B) 7 * b * ( 5.4 ) T 0 * T - normalsed temperature [-] (q.v. eq. ( 5.43 )) b - constants (c.f. Appendx B) 83

90 5 Propertes of carbon doxde * T T ( 5.43 ) d ( 5.44 ) d - constants (c.f. Appendx B) - densty [kg/m³] * R kt ~ ~ c c p 0 ( 5.45 ) 6 c p - sobarc heat capacty [J/(kg K)] (q.v. eq. ( 5.55 )) R* - constant; R*=1.01 [-] k -Boltzmann constant;k = [J/K] - Ludolph s number [-] - vscosty of carbon doxde [Pa s] (q.v. eq. ( 5.3 )) - correlaton length [m] (q.v. eq. ( 5.46 )) ~ - auxlary functon (q.v. eq. ( 5.50 )) ~ - auxlary functon (q.v. eq. ( 5.51 )) 0 ~ ( 5.46 ) reference length [m]; 0 = m ~ - auxlary functon [-] (q.v. eq. ( 5.47 )) - constant [-]; = constant [-]; = constant [-]; =

91 5 Propertes of carbon doxde Tr ~ ~ (, T ) ~ (, Tr ) T ( 5.47 ) ~ - symmetrsed compressblty [-] (q.v. eq. ( 5.49 )) T r - reference temperature [K] (q.v. eq. ( 5.48 )) T 1. 5 ( 5.48 ) r T c T c - crtcal temperature of CO [K] ~ p c ctc T p T ( 5.49 ) p c - crtcal pressure of CO [Pa] c - crtcal densty of CO [Pa] p - specally defned compressblty [kg / (m³ Pa)] (q.v. comments ) T ~ c p cv c arctan ~ ~ D cp cp v q q D ( 5.50 ) c v - sochorc heat capacty [J/(kg K)] (q.v. eq. ( 5.56 )) q ~ D - constant [m]; q ~ D =1/ /m 1 1 Dc ~ q~ D 3 e ( 5.51 ) q~ Range of valdty: temperature: pressure: densty: 00 K < T < 450 K 0 MPa < p < 100 MPa 5 kg/m³ < < 1000 kg/m³ 85

92 5 Propertes of carbon doxde thermal conductvty [W/(K m)] p= 0.1 MPa p= 1.0 MPa p= 3.0 MPa p= 4.0 MPa p= 5.0 MPa p= 6.0 MPa p= 7.0 MPa p= 8.0 MPa p= 9.0 MPa p= 10.0 MPa p= 15.0 MPa p= 0.0 MPa p= 5.0 MPa p= 30.0 MPa temperature [ C] Fg. 5.4 Thermal conductvty for CO as a functon of temperature; after [ 39 ]. Comments: As dscussed n the comments to secton referrng to the formulatons n [ 39 ], [ 13 ] does not provde an equaton for the back calculaton of the densty as a functon of temperature and pressure. In the framework of the formulatons presented here, Eq. ( 5.49 ) can therefore not provded exactly but only by means of teraton as descrbed n Appendx B and by calculatng the dervatve numercally. Conclusons: Eq. ( 5.39 ) s a relable formulaton wthn the ranges ( 1.1 ) Thermal conductvty of CO wth dssolved H O Comments: We are not aware of formulatons concernng the thermal conductvty of CO wth dssolved H O. 86

93 5 Propertes of carbon doxde 5.6 Enthalpy Enthalpy of pure CO Source: [ 39 ] Formulaton: R h T CO 1 0 r r ( 5.5 ) h - enthalpy of carbon doxde [J/kg] R CO - specfc gas constant for CO [J/(kg K)] (c.f. Appendx B) T - Temperature [K] - standardsed densty [-] (c.f. eq. ( 5.5 )) - nverse standardsed temperature [-] (c.f. eq. ( 5.6 )) 0, r r, - partal dervatves of the dmensonless Helmholtz energy, (c.f. eqs. ( 5.1 ), ( 5.1 ), ( 5.14 )) Range of valdty: temperature: pressure: 16 K < T < 1100 K 0 MPa < p < 800 MPa Comments: Formulaton ( 5.5 ) s vald over an exceptonal range of pressure and temperature. However, t s too complex to be used n a numercal code. Instead, nterpolaton of dscrete data s often used. Conclusons: Eq. ( 5.5 ) s a relable formulaton wthn the ranges ( 1.1 ). 87

94 5 Propertes of carbon doxde enthalpy [kj/kg] p= 0.1 MPa p= 1. MPa p= 5. MPa p= 10. MPa p= 0. MPa p= 30. MPa temperature [ C] Fg. 5.5 Enthalpy for CO as a functon of temperature; after [ 39 ] pressure [MPa] temperature [ C] Fg. 5.6 Enthalpy for CO as a functon of temperature and pressure; after [ 39 ]. 88

95 5 Propertes of carbon doxde 5.6. Enthalpy of CO wth water Enthalpy of gaseous CO wth water Source: [ 4 ] Formulaton: ( 5.53 ) CO CO H O H O h X CO h X CO h h - enthalpy of the mxture of gaseous CO and water vapour [J/kg] CO h - enthalpy of gaseous CO [J/kg] (q.v. eq.( 5.5 )) H O h - enthalpy of water vapour [J/kg] (q.v. eq.( 4.36 )) X - mass fracton of CO n the gas phase [-] CO CO H O X CO - mass fracton of water vapour n the gas phase [-] Comments: At sngle gas phase condtons the enthalpy of a mxture of water vapour and carbon doxde s assumed to be mass proportonal. Conclusons: Apparently, no measurements are known to back up approach ( 5.53 ) Enthalpy of lqud CO wth water Source: - Formulaton: ( 5.54 ) CO CO H O H O H O H O hco X CO h X CO h X CO h CO h - enthalpy of the mxture of lqud CO and water [J/kg] h CO - enthalpy of lqud CO [J/kg] (q.v. eq.( 5.5 )) h HO - enthalpy of lqud water [J/kg] (q.v. eq.( 4.36 )) X - mass fracton of CO n the lqud CO -phase [-] CO CO 89

96 5 Propertes of carbon doxde H O X CO H O hco - mass fracton of water n the lqud CO -phase [-] - enthalpy of dssoluton of water n lqud CO [J/kg] Comments: ( 5.54 ) s an analogon to ( 4.46 ), the enthalpy of water wth dssolved CO. At the tme we are not aware of formulatons descrbng the soluton enthalpy for water n CO. Conclusons: Even beng theoretcally correct formulaton ( 5.54 ) s not ready for use because of the mssng formulaton for the soluton enthalpy for water n CO. 5.7 Heat capacty Heat capacty of CO Source: [ 39 ] Formulaton: Isobarc heat capacty cp R r r 0 r 1 r 1 ( 5.55 ) r c p - sobarc heat capacty [J/(kg K)] R - specfc gas constant for CO [J/(kg K)] (c.f. Appendx B) - standardsed densty [-] (c.f. eq. ( 5.5 )) - nverse standardsed temperature [-] (c.f. eq. ( 5.6 )) 0, r, r, r r, - partal dervatves of the dmensonless Helmholtz energy, (c.f. eq.s ( 5.13 ), ( 5.4 ), ( 5.14 ), ( 5.17 ), ( 5.7 )) Isochorc heat capacty cv R 0 r ( 5.56 ) 90

97 5 Propertes of carbon doxde c v - sochorc heat capacty [J/(kg K)] sobarc heat capacty [kj/(kg K)] p= 0.1 MPa p= 10.0 MPa p= 0.0 MPa p= 30.0 MPa temperature [ C] Fg. 5.7 Isobarc heat capacty of CO ; after [ 39 ]. sochorc heat capacty [kj/(kg K)] p= 0.1 MPa p= 10.0 MPa p= 0.0 MPa p= 30.0 MPa temperature [ C] 91

98 5 Propertes of carbon doxde Fg. 5.8 Isochorc heat capacty of CO ; after [ 39 ]; other scale than Fg Comments: Formulaton ( 5.5 ) s vald over an exceptonal range of pressure and temperature. However, t s too complex to be used n a numercal code. Instead, nterpolaton of dscrete data s often used. Conclusons: Eq.s ( 5.55 ) and ( 5.56 ) are a relable formulatons wthn the ranges ( 1.1 ) Heat capacty of CO wth water Comments: We are not aware of formulatons concernng the heat capacty of CO wth dssolved H O. 5.8 Solublty Solublty of NaCl n CO Source: [ 4 ] Formulaton: NaCl L CO 0 ( 5.57 ) NaCl LCO - solublty of NaCl n water CO [kg/kg] Comments: For thermodynamc reasons every substance s soluable n an other substance at temperatures above 0 C [ 47 ]. For practcal purposes some of these solubltes are neglgbly small. Whle salt s well soluble n polar lquds lke water solublty n non-polar substances lke lqud carbon doxde s very low. Salt solublty n CO s thus commonly neglected as n [ 4 ]. Conclusons: Solublty of NaCl n CO s neglgble wthn the ranges ( 1.1 ). 9

99 5 Propertes of carbon doxde 5.8. Solublty of water n CO Source: [ 41 ] Formulaton: (q.v. eq. ( 4.65 )) Range of valdty: temperature: 1 C < T < 100 C pressure: 0 MPa < p < 60 MPa 1 I:\projekte\co_ugs_rsks\rechnungen\eos\l-co-ho-sp.lay 0.8 p= 0.1 MPa p=10mpa p=30mpa solublty [mol/kg] temperature [ C] Fg. 5.9 Solublty of water n CO ; after [ 41 ]. Comments: The solublty of water n CO s calculated as part of the formulaton ( 4.63 ) for the solublty of CO n water. Conclusons: Eq. ( 4.65 ) s a relable formulaton wthn the ranges ( 1.1 ) (see secton 4.7.). 93

100 5 Propertes of carbon doxde Solublty of water from a NaCl-soluton n CO Source: [ 40 ] Formulaton: ˆ L H O CO NaCl CO A LH O x H O 1 ˆ ( 5.58 ) ˆ - Solublty of H O n CO -phase [mol/mol] (q.v. eq. ( 5.60 )) H O LCO L - Solublty of CO n the water-phase [mol/mol] (q.v. eq. ( 4.89 )) ˆCO H O A - (q.v. eq. ( 4.67 )) x - mole fracton of NaCl n the water-phase [mol/mol] (q.v. eq. ( 5.59 ) ) NaCl H O m NaCl xh O m NaCl H O NaCl H O L CO H O ( 5.59 ) - stochometrc number of ons contaned n the dssolved salt [-]; NaCl H O n case of NaCl m - molalty of NaCl n the water-phase [mol/kg] L - Solublty of CO n the water-phase [mol/kg] (q.v. eq. ( 4.88 )) CO H O Converson of solublty L to molalty: ˆCO H O L Lˆ m CO NaCl CO H O H O H O ˆCO 1 LH O ( 5.60 ) L - Solublty of CO n the water-phase [mol/kg] CO H O Range of valdty: 94

101 5 Propertes of carbon doxde temperature: 1 C < T < 100 C pressure: 0 MPa < p < 60 MPa 10 3 /fsbraperm/a401/projekte/co_ugs_rsks/rechnungen/eos/l-co-ho-nacl-sp.lay p= 0.1 MPa, m= 0.0 mol/kg solublty [mol/kg] p= 0.1 MPa, m= 3.0 mol/kg p= 0.1 MPa, m= 6.0 mol/kg p=10.0 MPa, m= 0.0 mol/kg p=10.0 MPa, m= 3.0 mol/kg p=10.0 MPa, m= 6.0 mol/kg p=30.0 MPa, m= 0.0 mol/kg p=30.0 MPa, m= 3.0 mol/kg p=30.0 MPa, m= 6.0 mol/kg temperature [ C] Fg Solublty of water from a NaCl-soluton n CO, after [ 40 ], p=0.1 MPa (sold lnes), 10 MPa (dashed lnes) and 30 MPa (dotted lnes); NaCl mh O = 0 mol/kg (blue lnes), 3 mol/kg (green lnes), 6 mol/kg (red lnes). Comments: (see secton 4.7.3) Conclusons: (see secton 4.7.3) 5.9 Dffusvty Source: [ 8 ] Formulaton: - Comments: We are not aware of a formulaton for the self-dffusvty of CO as a functon of temperature and pressure. However, a good complaton of measured data exsts. 95

102 5 Propertes of carbon doxde Apparently, the values for self-dffusvty of supercrtcal CO le between the values for gaseous and lqud CO n the ranges ( 1.1 ). dffuson coeffcent [m²/s] g g g crtcal pressure g g g ggg gg g gg gg g gg gg g ggss g g s s gggsssss s s s s sss s l l ss s l g g g s g s crtcal pressure T= 5 C lqud T= 5 C gas T= 3 C gas T= 50 C gas T= 50 C supercrtcal T= 75 C gas T= 75 C supercrtcal l s l s l s l s ll pressure [MPa] Fg Self-dffusvty of CO for dfferent temperatures and phase states; data from [ 8 ]. More approprate would have been data on dffusvty of water n CO. Snce we haven t found such data self-dffusvty ndcates at least the order of magntude for ths value. Conclusons: Only a rough estmate for the dffuson coeffcent of water n supercrtcal CO can be provded. 96

103 6 Propertes of sodum chlorde 6.1 Densty of NaCl Generally, the NaCl densty s dependng on pressure and temperature: ( p, T ) ( 6.1 ) p T - densty of sodum chlorde[kg/m³] - pressure [Pa] - temperature [K] However, the densty of NaCl s only rarely consdered (e.g. [ 34 ]). Densty of sold NaCl changes only n the range of tenths of percent wth temperatures between 0 C and 100 C as shown n Fg Due to the hgh Young modulo of at least 10 GPa for rock salt the same apples to the dependency on pressure. A constant value of = 165 kg/m³ appears therefore adequate n case of the CO -sequestraton. Fg. 6.1 Temperature-dependent densty of rock salt; from [ 6 ], cted n [ 51 ]. 97

104 6 Propertes of sodum chlorde 6. Enthalpy of NaCl Source: [ 30 ], [ 1 ] Formulaton: from [ 30 ]: h T T T ( 6. ) from [ 1 ]: T 6.77 T h / T / T 4 / ( 6.3 ) h T - enthalpy of pure NaCl [kj/kg] - temperature [ C] Range of valdty s not specfed, but secondary lterature gves hnts: temperature: 0 C < T < 350 C salnty: 0 to full NaCl saturaton (as n the tabulated data from [ 1 ]) Comments: In comparson the correlaton for the enthalpy of salt h looks wrong. Followng eq. ( 6. ) the enthalpy of pure salt actually decreases wth temperature whch t shouldn t do. The formulaton from [ 1 ] shows nstead an ncreasng enthalpy that s n the order of the values from the formulaton from [ 30 ]. Whle several typos n [ 30 ] are corrected n [ 0 ] ths has not been reported. Conclusons: Formulaton ( 6.3 ) s vald wthn the ranges ( 1.1 ). 98

105 6 Propertes of sodum chlorde enthalpy [kj/kg] 0-50 enthalpy after enthalpy after [ 1 ] [ 30 ] temperature [ C] Fg. 6. Enthalpy of NaCl; after [ 30 ] and [ 1 ]. 6.3 Heat capacty of NaCl Source: usng [ 1 ] from secton 6. 3 hnacl T T T c ( 6.4 ) T Formulaton: c - heat capacty of NaCl [kj/(kg K)] h NaCl - enthalpy of pure salt [kj/kg] (q.v. eq. ( 6.3 )) T - temperature [ C] Range of valdty: (see secton 4.5.) 99

106 6 Propertes of sodum chlorde Comments: Heat capacty can be derved from the enthalpy as the partal dervatve wth respect to temperature (c.f. secton 4.6). Here, the formulaton of the NaCl-enthalpy from [ 1 ] s used. Conclusons: (see secton 4.5.) 0.9 heat capacty [kj/(kg K)] temperature [ C] Fg. 6.3 Heat capacty of NaCl; derved from [ 1 ]. 100

107 7 Phase state 7.1 Phase dagram for pure substances The state of a substance or the phase n whch t resdes can be charactersed by three state varables: pressure p, temperature t and volume V. State means here not only the three basc states of aggregaton sold, lqud and gaseous but also the coexstng states lke sold-lqud or lqud-gaseous. Each combnaton of the state varables s related to a specfc state. But only two of the three state varables are ndependent from each other. A plot of all possble states n a three-dmensonal p-v-t-dagram yelds thus a complex area wth a specal property: projectons of ths area along each axs yeld a dagram of two ndependent state varables (p-t, p-v or T-V), n whch the phase state s unquely defned. The two- or three-dmensonal drawngs are called phase dagrams. Two examples for a 3-D phase dagram of a pure substance are gven n Fg Dagram a) from [ 4 ] vsualses most clearly the three-dmensonal plane n the p-v-t-coordnate system whle dagram b) from [ 6 ] ponts out the three possble projectons on the p-t-, p-v- or T-V-plane, respectvely, whch helps greatly to understand these otherwse hardly comprehensble plots. In the p-t-dagram of Fg. 7.1 a) some parts of the three-dmensonal plane are just seen as lnes due to the projecton along the V-axs. These lnes represent states of two coexstng phases. Fg. 7. shows a p-t-dagram for carbon doxde, n whch the lnes are drawn n colour. Two characterstc states are hghlghted as well: the trple pont at whch even three phases coexst and the crtcal pont at the end of the red vapour pressure curve. 101

108 7 Phase state a) Schmelzgebet - meltng regon Naßdampfgebet - wet steam regon Sublmatonsgebet - sublmaton regon Schmelzdruckkurve - meltng pressure curve Sublmatonskurve - sublmaton curve Dampfdruckkurve - vapour pressure curve Flüssgket - lqud Gas - gas Sl melt lne El soldfcaton lne Vl evaporaton lne Kl condensaton lne Tl trpel lne tr trpel pont kr crtcal pont b) Fg. 7.1 Phase dagrams of a pure substance, a) after [ 4 ] and b) from [ 6 ]. 10

109 7 Phase state Fg. 7. p-t-dagram for carbon doxde; from [ 17 ]. The crtcal pont of a substance can be reached owng to pressure ncrease n the gas phase as well as owng to temperature ncrease n the lqud phase. At the crtcal pont the dfference between densty of the lqud and densty of the gas vanshes. Increase of temperature or pressure beyond ths pont means change nto the supercrtcal state where the dstncton between lqud and gas cease to exst. The supercrtcal state s often descrbed as a state n whch gas cannot be lquefed by any pressure ncrease. Pressure, temperature and densty at the crtcal pont for water and carbon doxde are compled n Tab Tab. 7.1 Values of the state varables for CO and H O at the crtcal pont. temperature [ C/K] pressure [Mpa] densty [kg/m³] water 374,15 / 647,3,1 317 carbon doxde 31,04 / 304, 7,38 468,16 The supercrtcal state can be reached ether from the lqud phase or from the gaseous phase. The transton s contnuous as shown for carbon doxde n Fg. 5., Fg. 5.3 and 103

110 7 Phase state Fg. 5.6 wth respect to densty, vscosty and enthalpy. Therefore the supercrtcal state s not clearly defned n a phase dagram and does not consttute an own phase. Ths means that, curously, a lqud can be transformed n to a gas wthout an actual phase change. However, n the lterature the boundary to the supercrtcal state s often assumed to concde wth parallels to the axes n a p-t-dagram startng at the crtcal pont (e.g. Fg. 7.). Ths approach s adopted n the followng. 7. Phase dagram for mxtures of H O and CO Phase dagrams exst for mxtures of two substances or components, too, only an addtonal state varable x s requred that defnes the mass fractons of the components. In ths case three state varables are ndependent from each other so that a graphcal representaton of the phase states of a bnary mxture has to be truly three-dmensonal. Other than n the p-v-t-dagram for a pure substance where only a complcated area s constructed n space, every pont n a p-t-x-space s defned. The effort to comple the data and to plot a comprehensble dagram n space s consderable. One of the very few three-dmensonal phase dagrams for a H O-CO - mxture s provded by [ 43 ] and shown n Fg Accordng to [ 43 ] the dagram s dstorted, though, to underlne certan propertes of the H O-CO -mxture: The spatal three-phase zone marked n yellow s strongly magnfed n Fg In contrast to realty the crtcal pressure of water has about the same value as carbon doxde. The temperature axs s not to scale and compresses the supercrtcal zone between the three-phase zone and the crtcal temperature for water. The temperature n Fg. 7.3 les between 0 C and 370 C, the pressure s gven up to 350 MPa. The bold lnes represent the vapour pressure curves of the pure components whch end n the crtcal ponts C HO and C CO, respectvely. The fne lnes represent sotherms. The crtcal curves connectng the crtcal pont of the respectve component wth horzontal tangents of the sotherms are drawn as dashed lnes. The crtcal curves do not le n planes but are curved n space. For the more volatle component the crtcal curve s often called lower crtcal curve LC. LCEP marks the lower crtcal end pont. 104

111 7 Phase state The orgnal fgure n [ 43 ] s only a lne drawng whch s not easy to nterpret. Therefore some characterstc areas representng dfferent phase states have been coloured n the present fgure. But not every phase state has been ponted out n Fg Ths becomes apparent by lookng at the red marked area more closely. For hgh temperatures and pressures t represents a state where water does not exst as a separate phase but s totally dssolved n the supercrtcal CO. But f the pressure s lowered below the vapour pressure of the water, water vapour evolves and changes the sngle-phase carbon doxde nto a bnary gas mxture. Ths phase change s not dsplayed n Fg. 7.3 but ndcated by fadng from red to cyan. Fg. 7.3 Phase dagram for a HO-CO mxture, not to scale; from [ 43 ]. Selected areas are coloured to mprove understandng. The colour codng used n Fg. 7.3 s appled for Fg. 7.4, Fg. 7.5 and Fg. 7.6, too. Fg. 7.4 shows two cross-sectons n the p-t-plane. Unfortunately, the mass fracton x s not gven n ether case. The temperature range apparently exceeds the range covered n Fg Below 10 C to 1 C the dagrams show whte areas where a CO-hydratrate phase forms. Included s even the temperature range where ce s formed. In order to avod problems arsng from hydrate formaton [ 41 ] choses 1 C to be a lower temperature boundary for developng formulatons for mutual solubltes of water and carbon doxde. 105

112 7 Phase state Fg. 7.4 p-t phase dagrams for a H O-CO mxture; a) after [ 19 ] (cted n [ 44 ]), b) after [ 41 ]. Colour codng corresponds to Fg. 7.3; I Ice(H O), H CO -hydrate, L 1 lqud H O, L lqud CO, V gas phase, Q 1 quadruple pont; crcles n b) represent measured data from the lterature. The phase dagram n Fg. 7.5 s gven as a p-x-plane at a temperature level of 5 C. At about 64 bar and a very low H O mole fracton the spatal, yellow marked three-phase zone of Fg. 7.3 can be found agan. A cut-out of ths regon s gven n ths fgure as well. It shows that the sze of ths regon are actually very small, about 3 n x-drecton and 0,3 bar n p-drecton. [ 41 ] concludes that the spatal three-phase regon s not relevant n case of CO -sequestraton because of ts sze and of ts poston n the p-t-x-dagram. Fg. 7.6 fnally shows two cross-sectons n T-x-plane for very hgh pressures, 0 MPa and 35 MPa. The frst one les below, the other one above the crtcal pont for water. Due to the hgh temperatures consdered the carbon doxde s manly n the supercrtcal state. However, here n the dagrams t s called vapour 5. 5 Apparently, labellng of the supercrtcal state s ambguously done n the lterature. 106

113 7 Phase state Fg. 7.5 p-x phase dagram for a H O-CO mxture at 5 C; after [ 41 ]. Colour codng corresponds to Fg. 7.3; for the explanaton of abbrevatons, see Fg. 7.4; symbols represent measured data from the lterature. Fg. 7.6 T-x phase dagram for a H O-CO mxture at 00 and 350 bars; after [ 7 ] (cted n [ 17 ]). Colour codng corresponds to Fg. 7.3; symbols represent measured data from the lterature. 107