GAS RESOLUTION AND RELEASED MODELS IN REACTOR THERMAL-HYDRAULICS. E. Zaltsgendler, W. S. Liu, R. K. Leung

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1 CA GAS RESOLUTION AND RELEASED MODELS IN REACTOR THERMAL-HYDRAULICS E. Zaltsgendler, W. S. Liu, R. K. Leung Ontari Hydr Nuclear, Reactr Safety and Operatinal Analysis Department 700 University Avenue, Trnt, Ontari, CANADA, M5G 1X6 ABSTRACT The preliminary analysis f the nn-cndensable gas reslutin and release phenmena in reactr thermalhydraulics is presented. The transprt equatin fr the disslved gas cntent is prpsed. The mdel fr gas release and reslutin rate is develped fr the bubbly flw. The apprach t calculate this rate fr ther tw-phase flw regimes is discussed. Frm the analysis f experimental data, the crrelatins fr the equilibrium cncentratin fr air cmpnents have been develped. 1. INTRODUCTION The clant in the Heat Transprt Systems (HTS) f a CANDU reactr cntains disslved gas in a slutin. Under transient cnditins, the disslved gas may be released when the cncentratin f the disslved gas is higher than the equilibrium cncentratin (gaseus cavitatin). This phenmenn has t be distinguished frm the vaprus cavitatin, which happens when the lcal liquid pressure falls belw the saturated vapur pressure [Reference 1 ]. Vaprus cavitatin (r flashing) takes place almst instantly while gaseus cavitatin is a much slwer prcess. As a result f released gas, bubbles appear and the flw regime changes frm single phase t tw-phase. In additin, vaprizatin int gas bubbles will take place. If during this tw-phase system transient the pressure in the HTS becmes higher than the disslved gas pressure, reslutin f the gas bubbles will ccur. These phenmena may play an imprtant rle in reactr thermal-hydraulics, fr example, in the analysis f a large LOCA and the water level measurements in cmpnents such as the bleed cndenser. The gas reslutin and release can be prperly analyzed with a tw-fluid mdel like that in TUF [Reference 2]. Several physical aspects shuld be cnsidered: (1) the cnservatin equatins fr gas phase, (2) the equatin f state fr a mixture f tw cmpnents where cmpnent water may cntain liquid and vapur, (3) the transprt equatin fr disslved gas cntent, (4) the release and reslutin rates. The first tw items have already been cnsidered in TUF. The last tw subjects are analyzed in this paper. 2. TRANSPORT EQUATION FOR DISSOLVED GAS CONTENT T simplify the physical mdel, the disslved gas in the liquid is cnsidered as a part f the liquid with equal temperatures and velcities. Let us define a mass fractin fr the disslved gas c as C-^.^ (!)

2 where M c is the ttal mass f the disslved gas in a cntrl vlume, M, is the ttal mass f pure liquid (fluid) and M, is the ttal mass f liquid (M, = M c + M f ). The mass cnservatin equatins fr the fluid and disslved gas are x-i w «- w < ~ F * Ot in ut (2) ^ = EW C - W e - F t (3) F t where W f is the fluid flw rate, W c is the disslved gas flw rate, F, and F Y are the ttal gas release and vapur generatin rates, respectively, within the cntrl vlume. By cmbining Equatins (2) and (3) and using the mass fractin c definitin, ne can btain dt -f where W, (= W f + W c ) is the ttal liquid flw rate. The ttal rate f gas release cnsists frm tw parts F, = F,' + F," (6) The first part F,' is the rate f gas release due t a gas diffusin prcess frm liquid t bubbles with n fluid vaprizatin. The secnd part F," is the rate f gas release during vaprizatin when all gas cntained in the evaprated liquid is released. It can be shwn that F." = -*- Fv (7) 1-c Substituting Equatins (6) and (7) int Equatin (5), it prduces the transprt equatin fr nn-cndensable gas cntent during vaprizatin (F, > 0) (1-c) F.' (8) It is apparent frm Equatin (8) that vaprizatin des nt change the nn-cndensable gas mass fractin which agrees with the physics f the prcess. During the cndensatin prcess nly the vapur is cndensed. Therefre F," = 0 and the transprt equatin fr gas cntent fr cndensatin (F, < 0) becmes

3 It indicates that the mass fractin f nn-cndensable gas decreases during cndensatin. Calculatin f the rate f vaprizatin F, is a well-established prcedure [Reference 2]. Therefre the fllwing analysis is devted t gas release (r reslutin) term F.'. 3. GAS RELEASE AND RESOLUTION MODELS Gas release is a diffusin cntrlled prcess f interfacial mass transfer. The cncentratin gradient is the main driving frce in this prcess. The interfacial mass transfer rate fr the cntrl vlume V can be express by the fllwing equatin: F,' = VAH e (C-CJ where A is the interfacial area per unit vlume, H c is the interfacial mass transfer cefficient fr gas cntent, C is the cntent cncentratin in slutin in kg/m 3, and C $- is the equilibrium (saturated) cncentratin which is a functin f pressure and temperature. It is apparent frm the definitin f C and Equatin (1) that we have Substituting Equatin (11) int Equatin (10), it prduces C = *. ^ (11) V V F.' = ^ (c-cj (12) where x c is the s-called relaxatin time fr gas release which is defined as A The interfacial mass transfer cefficient as well as the interfacial area are flw regime dependent parameters. There are six majr regimes fr a tw-phase system in the TUF cde: bubbly, slug, annular, churn-turbulent, mist and stratified flws. The detailed analysis fr calculatin f t c fr bubbly flw is presented belw. Fr ther regimes, an apprach t calculate x c is als discussed. Fr a bubbly flw the analysis begins with a single bubble submerged in surrunding liquid. The grwth prcess is in general cmplicated, invlving simultaneus mass, mmentum and energy transfers. The mvement f the gas-liquid interface depends n the gas pressure inside the bubble and the bubble grwth (r cllapsed) dynamics. On the ther hand, the gas pressure and the bubble radius are related t each ther and t the rate f gas diffusin thrugh the cnservatin f mass. Als, the rate f gas diffusin itself depends n the mvement f the interface thrugh the diffusin equatin. Hence the three equatins, namely the mmentum, the mass balance and the diffusin, gverning the mass diffusin-induced grwth f a bubble are highly cupled and nn-linear. Due t these cmplexities, there is n knwn analytical slutin t predict the bubble grwth even fr a bubble submerged in a stagnant liquid [Reference 3]. T slve the prblem f a mre cmplex case f a bubble submerged int a flwing liquid, which is f interest fr present analysis, Levich [Reference 4] has intrduced sme simplificatins. He assumed that: a) the prcess is steady state and isthermal, b) a bubble size is fixed, c) a cncept f a cncentratin bundary layer can be applied. As a result nly the equatin f cnvective diffusin has t be slved. Levich fund that fr

4 mderate Reynlds numbers (Re - 10) the ttal diffusin flw I t the bubble surface is expressed by = 4 i * """ p2//i_i^» (14) where D is the diffusin cefficient, U is the bubble relative velcity (slip velcity), and R is the bubble radius. If the ttal number f bubbles per unit vlume is equal t N and their cncentratin bundary layers d nt interact each ther, the ttal mass release rate in the cntrl vlume V is Intrducing the interfacial mass transfer area A = 4JI R 2 N and taking int accunt that C V = p,cv = M, c, ne can get It is apparent frm the cmparisn f Equatins (16) and (13) that the cefficient f the mass transfer fr a bubbly flw is equal t r~ (17) The diffusin cefficient D fr diffusin f air in water is abut 2.0x10 ' 9 mvs fr a base temperature 25 C [Reference 5]. This cefficient may be crrected frm the base temperature T, t anther temperature T 2 by using the Einstein relatin [Reference 6] D, = D, -±^1 (18) where p is the viscsity f liquid. Fr a bubbly flw the interfacial mass transfer area is equal t 3 a/r where a is the vid fractin. The relaxatin time x c can be estimated with the fllwing practical values. Fr a vid fractin f 5xlO" 3 with a slip velcity 0.1 m/s and a bubble radius f 10" 4 m, the interfacial mass transfer area per unit vlume is abut 150 m" 1 and the mass transfer cefficient at T = 265 C is abut 5xlO' 3 m/s. Thus the relaxatin time fr gas release fr bubbly flw is abut 1 s. The relaxatin time fr gas reslutin is much larger due t bubble radius decrease when gas disslves. Fr ther tw-phase flw regimes, the cefficient f mass transfer may be fund by using the mass transfer crrelatin fr a turbulent flw [Reference 7] H c = Re" 3 Sc a44 (19) a where Sc = ft/pd is the Shmidt number and d is pipe diameter. The interfacial mass transfer area is equal t

5 the interfacial heat transfer area which is rutinely calculated fr the tw-fluid mdel [Reference 2]. Using the crrelatin in Equatin (19) fr H c and the value A frm the cde calculatin, the relaxatin time fr nncndensable gas mass transfer can be fund fr ther flw regimes. 4. EQUILIBRIUM CONCENTRATION OF THE DISSOLVED GAS It is apparent frm Equatin (12) that the release / reslutin rate depends upn the difference between the existence cncentratin f the nn-cndensable gas in liquid and its lcal saturated (equilibrium) cncentratin. Therefre, the crrelatins fr the equilibrium cncentratin as a functin f the system parameters i.e., pressure and temperature, have t be knwn. Accrding t Henry's law, at the cnstant temperature the mass f any gas that can be disslved in a given vlume f liquid is directly prprtinal t the pressure that gas exerts abve the liquid x = (20) where x is the mle fractin slubility, p is the partial pressure f the gas, and k is the Henry's law cnstant fr the gas at the given temperature. By the definitin, the mle fractin slubility is expressed as x = "^ (21) n(a)+n(f) where n(a) is the number f gas mles in the vlume and n(f) is the number f fluid mles. Fr slightly sluble gases like air in water, the mle fractin slubility is apprximately expressed by x - *& (22) n(f) Fr many gases, Henry's law hlds quite well when the partial pressure f slute gas is less than 101 kpa. Fr a partial pressure f slute gas greater than 101 kpa, k is seldm independent f the partial pressure f the slute gas. We are interested in the air cntent in D 2 O fr the primary side f the HTS and in H 2 O fr the secndary side system. There are few experimental data available fr slubility in the light and heavy water ver a wide range f the thermdynamic parameters, pressure up t 10 MPa and temperature up t 310 C, which is f interest fr the reactr thermal-hydraulics. But the experimental data and sme crrelatins fr slubilities f the air cmpnents, nitrgen and xygen, are available in the literature. Frm Equatin (22) the slubility f air can be expressed as (23) where x a, x N and x 0 are the mle fractin f air, nitrgen and xygen, respectively. Battin, Rettich and Tminaga [Reference 8] have reviewed the slubility f nitrgen in liquids. Slubility data were critically evaluated. Recmmended equatins were presented as smthing equatins. Using their recmmendatins and data frm Reference [9], we have derived the fllwing crrelatins fr equilibrium cncentratin f nitrgen in water:

6 fr T < 350 K In x = In x In p p (24) fr T > 350 K In x = ^ In T In p p (25) where x =T[K]/100 and p is measured in bar. Figure 1 illustrates the pressure dependence f the equilibrium cncentratin f nitrgen in water. The slubility f xygen in water has prbably been the mst intensively studied gas slubility system. In Reference [10] ver frty papers with data n an Oj/F^O system were screened. As a result, the crrelatins f Bensn, Krause and Petersn [Reference 11] has been recmmended t express slubility f xygen as a functin f temperature. We used their crrelatins tgether with the experimental data frm Reference [10] t derive the fllwing crrelatins fr slubility f xygen as a functin f temperature and pressure: fr T < 373 K fr T > 373 K In x = tap p (26) v x 2 In x = In p p (27) x x 2 x 3 Figure 2 illustrates the pressure dependence f the equilibrium cncentratin f xygen in water. It shuld be nted that the pressure in Equatins (24) t (27) is the partial pressure f the gas. Therefre, we have p=0.8 p. fr Equatins (24) and (25) and p=0.2 p. fr Equatins (26) and (27), where p, is the air pressure (it is assumed that air cntent is 80% f N 2 and 20% f O 2 ). Little wrk has been dne n the slubility f gases in heavy water. Csgve and Walkley [Reference 12] have reprted the slubility f nitrgen in D 2 O fr a narrw range f temperatures (278 K<T<313K)ata nitrgen partial pressure f 101 kpa. The crrespnding mle fractin was ranged frm 7% t 10% higher in D 2 O than that in H 2 O. But due t lack f experimental data fr wide range f temperature and pressure, we can assume that the mle fractin slubilities in D 2 O are the same as that in H 2 O fr engineering applicatins. The equilibrium cncentratin f air c tu which is needed t slve Equatin (12) can be fund frm the mle fractin slubility by using the definitin c frm Equatin (1). Taking int accunt that M, = n, m,, ne can btain where m., m, are the mlecular weights f air and fluid.

7 5. CONCLUSION T simulate the gas reslutin and release, a transprt equatin fr the disslved cntent f a nn-cndensable gas is prpsed. The transprt equatin includes a surce term fr the rate f the gas generatin ( releasereslutin). Simple mdels fr the gas reslutin and release are develped. The time cnstants, incrprated in these mdels, are evaluated by using the slutin f diffusin f the disslved gas int the bubbles. Anther parameter in these mdels is the equilibrium cncentratin f the disslved gas. Crrelatins tr this cncentratin as a functin f the partial pressure and temperature are prpsed fr the mst interesting case f air-d 2 O (H 2 O) systems. This mdel is currently being testing in the TUF cde fr statin applicatins, particularly in the ECI system, shutdwn cling system and system cmpnents such as bleed cndenser. REFERENCES 1. D.C. Wiggert., M.J. Sundquist., "The Effect f Gaseus Cavitatin n Fluid Transients", J. Fluid Engineering, 101, 79 (1979). 2. W. S. Liu et al., "TUF: A Tw-Fluid Cde fr Thermal-Hydraulics Analysis",!Oth Annual Cnference f Canadian Nuclear Sciety, Ottawa, Canada (1989). 3. A. Arefmanesh, S.G. Advani, E.E. Michaelides, "An Accurate Numerical Slutin fr Mass Diffusin Induced Bubble Grwth in Viscus Liquids Cntaining Limited Disslved Gas", Int. J. Heat Mass Transfer, 35, 1711 (1992) 4. V.G. Levich, "Physicchemical Hydrdynamics", Prentice-Hall (1962). 5. "CRC Handbk f Chemistry and Physics", 71th Editin, CRC Press (1981). 6. "Perry's Chemical Engineers' Handbk", 6th Editin, McGraw-Hill Bk Cmp. (1984). 7. CJP. Kthandaraman, S. Subramanyan, "Heat and Mass Transfer Data Bk", J. Willey & Sns (1975). 8. R. Buttin, T.R. Rettich, T. Tminaga, "The Slubility f Nitrgen and Air in Liquids", J. Phys. Chem. Ref. Data, 13, 566 (1984). 9. "Slubility Data Series. Nitrgen and Air", Int. Unin Pure Appl. Chem., 10 (1982). 10. "Slubility Data Series. Oxigen and Ozne", Int. Unin Pure Appal. Chem, 7 (1981). 11. BB. Bensn, D. Krause Jr., M.A. Petersn, "The Slubility and Istpic Fractin f Gases in Dilute Aqueus Slutin. I. Oxygen", J. Slutin Chem., 8, 655 (1979). 12 B.A. Csgrve, J. Walkley,"Slubility f Gases in H 2 O and 2 H 2 O", J. Chrmatgr., 216, 161 (1981)

8 3.0 NITROGEN SOLIBILITY IN WATER tf 1-5H, 0 H Id 0.5H CO 0.0, ' " Partial Pressure, [bar] u ' *** r ~ T = 100 C T=265C T=300 C Figure 1 OXYGEN SOLIBILITY IN WATER I " * CO Partial Pressure, [bar] T = 100C T=265C T=300 C Figure 2