SIMULATION OF DIRECT CONTACT CONDENSATION OF STEAM JETS SUBMERGED IN SUBCOOLED WATER BY MEANS OF A ONE-DIMENSIONAL TWO-FLUID MODEL

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1 HEFAT th International Conference on Heat Transfer, Flui Mechanics an Thermoynamics July 014 Orlano, Floria SIMULATION OF DIRECT CONTACT CONDENSATION OF STEAM JETS SUBMERGED IN SUBCOOLED WATER BY MEANS OF A ONE-DIMENSIONAL TWO-FLUID MODEL a Mechanical Engineering, Kernkraftwerk Gunremmingen GmbH, Dr.-August-Weckesser-Str. 1, Gunremmingen, Germany Heinze D. a,b,*, Schulenberg T. b an Behnke L. a * Author for corresponence avi.heinze@partner.kit.eu b Institute for Nuclear an Energy Technologies, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, Eggenstein-Leopolshafen, Germany ABSTRACT A one-imensional simulation moel for the irect contact conensation of steam in subcoole water is presente. The moel allows to etermine major parameters of the process such as the jet penetration length an the axial evelopment of the temperature. Entrainment of water by the steam jet is moele accoring to the turbulent entrainment assumption, which can be erive from the Kelvin-Helmholtz instability theory. The steamwater two-phase flow obtaine uring the mixing process is simulate base on a one-imensional two-flui moel. An interfacial area transport equation is use to track changes of the interfacial area ensity ue to roplet entrainment an steam conensation. Interfacial heat an mass transfer rates uring conensation are calculate using the two-resistance moel. The resulting two-phase flow equations constitute a system of orinary ifferential equations which is iscretize by means of an explicit Runge-Kutta metho. The moel shows goo agreement with publishe ata of pool irect contact conensation experiments at low steam mass flux. INTRODUCTION The irect contact conensation DCC) of a high-velocity steam jet in subcoole water offers a highly efficient means of steam conensation an is therefore use in many inustrial applications, such as thermal egasification, irect contact heat exchangers or the epressurization systems of current light water reactors. Furthermore, the phenomenon is of particular importance for the operation of steam-riven jet pumps, where efficient steam conensation is crucial for stable operation. The DCC process can be ivie into two parts. First, the interface between the steam jet an the water pool is isrupte ue to the high velocity ifference between the two phases. The large interfacial area ensity obtaine by this turbulent mixing process then establishes the basis for rapi steam conensation with heat transfer coefficients up to 10 6 W/m K). Accoringly, the initial evelopment of the two-phase flow is mainly governe by the momentum transfer from the high-velocity steam to the entraine roplets, while mass an heat transfer ominate with growing interfacial area ensity. PREVIOUS WORK Numerous experimental an theoretical stuies of irect contact conensation have been performe in the past to gain a better unerstaning of the occurring physical phenomena. However, experimental ata is mostly limite to the global flow structure. Reliable information regaring the local flui-ynamic properties is limite ue to the complex two-phase flow which impees experimental measurements. Experimental observations In general, three ifferent DCC conensation moes can be istinguishe: chugging, bubbling an jetting [1]. The oscillating flow moes of chugging an bubbling occur at low steam mass fluxes, while a stable jet flow appears as soon as the steam flow is choke, i. e. for sonic or supersonic steam injection []. The present paper focuses on the stable jetting regime. Three flow regimes have been observe uring stable jet conensation [] [4]. First, a vapor core also calle steam cavity) in the immeiate proximity of the injection nozzle, where the flow velocity is almost constant [5]. This region is followe by the mixing region, where liqui roplets are entraine into the vapor core an provie a large interfacial area for steam conensation. The thir region is the conensation-inuce turbulent liqui jet, which has been shown to be in goo agreement with turbulent jet 1571

2 NOMENCLATURE a if [m /m 3 ] interfacial area ensity B [ ] conensation riving potential c [m/s] velocity c p [J/kg K)] specific heat capacity at constant pressure [m] iameter E 0 [ ] entrainment coefficient h [J/kg] specific enthalpy h lg [J/kg] specific conensation enthalpy L [ ] imensionless jet penetration length l [m] jet penetration length La [ ] Laplace number Ma [ ] Mach number ṁ [kg/m s)] mass flux ṅ [1/m s)] particle flux Nu [ ] Nusselt number p [Pa] pressure Pr [ ] Prantl number q [W/m ] heat flux R [m] jet raius Re [ ] Reynols number S m [ ] transport moulus T [K] temperature We [ ] Weber number z [m] axial istance Special characters α [W/m K)] heat transfer coefficient ε [ ] volume fraction η [kg/m s)] ynamic viscosity Γ [kg/m 3 s)] volumetric mass source term λ [W/m K)] thermal conuctivity Φ [1/m s)] interfacial area source term ρ [kg/m 3 ] ensity σ [kg/s ] surface tension Subscripts critical value 0 stagnation conition ambient property 0 surface-average mean value 30 volume-average mean value 3 Sauter-average mean value c continuous phase isperse phase crit critical conition at sonic velocity) en entrainment e nozzle exit conition g gas phase if interface property l liqui phase m jet mixture property max maximum value sat saturation theory [5], [6]. Gas ynamic effects, namely oblique shocks an expansion waves have been observe in over- an uner-expane jets [7], [8]. These phenomena influence the flow structure near the nozzle exit an become more pronounce with increasing water temperature. One of the major parameters to characterize the DCC flow is the imensionless jet penetration length L, which is efine as the ratio of the jet penetration length l [m] an the nozzle exit iameter e [m]: L = l/ e 1) Most measurements of L rely on visual observation an are therefore subject to a large experimental bias [9]. Nevertheless, various investigators have shown that L is mainly epenent on the steam mass flux an the temperature of the water pool [3]. Moeling approaches Kerney et al. [10] erive a semi-empirical correlation for the jet penetration length base on the nozzle iameter, the steam mass flux at the nozzle exit ṁ e [kg/m s)] an the rate of subcooling: ) 0.5 ṁe L = l/ e = S m B 1 ) ṁ crit Here, the transport moulus S m is an empirical parameter analogous to the Stanton number of convective heat transfer, an ṁ crit is the critical steam mass flux at ambient water pressure. The conensation riving potential B is efine as B = c p T sat T ) h lg, 3) where c p is the specific heat capacity at constant pressure [J/kg K)], T sat an T are the saturation temperature [K] an the temperature of the ambient water, respectively, an h lg is the specific conensation enthalpy [J/kg]. This correlation was later revise [11] base on a single-flui moel in orer to account for the influence of water pressure, roplet entrainment an bubble formation. Nevertheless, most subsequent authors have relie on the original formulation when eriving similar correlations [4], [6] [8], [1]. In general, these empirical correlations agree well with the experimental ata that was use to erive the correlation. However, there is substantial isagreement when applying the various correlations to a single experimental ata set [9]. More etaile analyses have been performe using computational flui ynamics CFD). In most moels, the voi istribution an the rate of conensation are estimate base on local turbulence values [9], often in conjunction with a probability ensity function [13]. Aitional information can be foun in a review article by Gulawani et al. [14]. In summary, it can be sai that the available empirical correlations on the one han are limite in their range of applicability. On the other han, CFD moels offer a better moeling accuracy, but at a high computational cost. Therefore, the object of the present work is the evelopment of a one-imensional simulation moel which accounts for the ominant physical processes entrainment of water roplets into the jet core, formation of a steam-water interface, conensation of steam) an which is capable of preicting major jet parameters such as the jet penetration length an the axial evelopment of the temperature. DIRECT CONTACT CONDENSATION MODEL Theoretical moel Immeiately after steam injection, the jet consists of a conical vapor core surroune by the pool water, similar to annular twophase flow. However, the flow is by no means fully evelope: Initially, there is a sharp raial velocity graient at the bounary between the vapor core an the surrouning stagnant water. Waves are forme at this bounary an liqui ligaments are entraine into the gas core, rapily breaking up into small roplets. These roplets will cause a quick eceleration of the gas phase ue to their high inertia. Accoringly, the raial velocity profile 157

3 will flatten with increasing istance from the nozzle. At the same time, steam conenses upon the entraine roplets an the volume fraction ε ecreases, finally resulting in a isperse bubbly flow with negligible slip. There exists little experimental ata regaring the local flow structure of a turbulent conensing two-phase jet. Therefore, some simplifying assumptions have been mae in the moel evelopment where necessary, in particular regaring the jet profile an the changes in the flow regime. In contrast, appropriate physical moel accuracy has been sought regaring the ominant processes of water entrainment an steam conensation. p T ṁ e p e r c en z roplet flow bubbly flow ε g = 0.5 ε g = 0 L Rz) Figure 1 The DCC flow moel ivies the two-phase jet into a isperse roplet flow regime an a isperse bubbly flow regime which are surroune by the stagnant water. As can been seen from Figure 1, the moel ivies the jet region into two areas: The two-phase jet an the surrouning, stagnant water. The two-phase jet flow is initially consiere as a isperse roplet flow, which turns into a isperse bubbly flow at ε = 0.5. Mass entrainment from the water pool into the jet is consiere to be the ominant exchange mechanism, therefore heat conuction an momentum transfer ue to interfacial shear are neglecte. Entrainment is moele accoring to the turbulent entrainment assumption explaine in etail below), an is assume to be perpenicular to the flow axis. Accoringly, the entraine mass is ae to the two-phase jet without momentum in the axial irection. In the roplet flow regime, the iameter of entraine roplets is obtaine base on the initial velocity ifference between the entraine roplet an the jet. In the two-phase jet, no slip has been assume between the isperse an the continuous phase. This assumption hols that entraine mass is immeiately accelerate to the jet velocity. Conservation equations The conservation equations for the two-phase jet are erive uner the following assumptions: The flow is stationary, oneimensional an in mechanical equilibrium no slip). Gravity, issipation, shear stresses an heat conuction are neglecte. Due to the high ensity ifference between gas an liqui phase at ambient pressure, the kinetic energy of the vapor is small with respect to the inertia of the entraine water. Accoringly, kinetic energy is neglecte with respect to enthalpy. The pressure p is assume to be constant an equal to the pool pressure p, while the jet raius R is a function of the axial istance z. Then, the mass conservation equations for the gas phase g an the liqui phase l have the form z z ε g ρ g c z R ) = Γ g R, 4) ε l ρ l c z R ) = Γ l + Γ en,l ) R, 5) where ρ is the ensity [kg/m 3 ], c z is the axial velocity, Γ en,l is the volumetric mass source term [kg/m 3 s)] ue to entrainment, an Γ g an Γ l are the volumetric mass source terms ue to phase change in the gas an the liqui phase, respectively. Introucing the mixture ensity ρ m as ρ m = ε g ρ g + ε l ρ l, 6) the mixture momentum equation can be written as ρ m c z R ) = 0. 7) z The energy conservation for the gas an the liqui phase are written as ε g ρ g c z h g R ) = ) Γ g h g,if + a if q g,if R, 8) z ε l ρ l c z h l R ) = ) Γ l h l,if + Γ en,l h + a if q l,if R, 9) z where h g an h l are the gas an liqui phase enthalpy [J/kg], respectively, h g,if an h l,if are the enthalpies on the gas an liqui sie of the phase interface, respectively, h is the ambient water enthalpy, a if is the interfacial area ensity [m /m 3 ], an q g,if an q l,if represent the heat flux [W/m ] on the gas an the liqui sie of the interface, respectively. The interfacial transfer conitions are given by Γ g + Γ l = 0, 10) Γ g h g,if + a if q g,if + Γ l h l,if + a if q l,if = 0, 11) an the enthalpies on the interface h g,if an h l,if are assume to be equal to the gas an liqui saturation enthalpies. Interfacial area transport In aition to the conservation equations, an interfacial area transport equation [15] for the isperse phase is use to track the change of the interfacial area ensity a if ue to roplet entrainment, roplet growth an bubble conensation: a if c R ) = z Φ en + 3 a if ρ ) ) Γ ρ c R 1) ε z Here, the interfacial area source term ue to roplet entrainment Φ en [1/m s)] is calculate as Φ en = Rṅenπ 0,en, 13) 1573

4 with the particle flux ṅ [1/m s)] of entraine roplets across the jet bounary ṅ en = 6 π c en 3 30,en. 14) In equations 13) an 14), 0 an 30 are the surface an volume mean iameters, respectively. Combining equations 13) an 14) an introucing the Sauter mean iameter 3 = 3 30 / 0 yiels Φ en = 1 R c en 3,en. 15) Turbulent entrainment The mixing of two miscible fluis flowing with ifferent velocities can be escribe by the turbulent entrainment assumption, initially erive for hot gases rising in air [16], [17]. The theory has been extene to miscible gases with high ensity ifferences [18], resulting in the following equation for the entrainment velocity c en : c en = E 0 ρm ρ c 16) Here, ρ is the ensity of the entraine flui an ρ m is the local mean ensity of a jet flowing with the velocity c. The empirical entrainment coefficient E 0 has been experimentally etermine in the range of 0.06 to 0.1 with a recommene value of Equation 16) has been successfully applie to gas an vapor jets in subcoole liquis [11], [19]. This extension from miscible to immiscible fluis can be corroborate using the Kelvin- Helmholtz instability theory [0]. Here, flui entrainment is escribe by the formation of a capillary wave at the phase bounary which breaks up into a rige of liqui, yieling the following relationship for the entrainment velocity: c en ρm ρ ρ ρ + ρ m c 17) Equation 17) reuces to equation 16) for ρ ρ m an a proportionality constant E 0. Using equation 17) with an appropriate entrainment coefficient allows to calculate the volumetric mass source term ue to entrainment: Γ en,l = R ρ c en. 18) Size of entraine roplets In the roplet flow regime, the entraine liqui ligaments will break up uner the impact of the aeroynamic rag force ue to the jet velocity. The maximum iameter of entraine roplets max,en can be etermine implicitly as a function of the critical Weber number We [1]: max,en = We ) max,en σ ρ m c z c ) 19) We ) ) max,en = La 0,37 0) La,en = ρ σ max,en η,en 1) In equations 19) to 1), σ is the surface tension [kg/s ] of the roplet, c is the axial roplet velocity immeiately after entrainment assume to be zero as escribe above), La,en is the Laplace number of a roplet having the maximum stable iameter an η is the roplet ynamic viscosity [kg/m s)]. Assuming an upper log-normal size istribution, the ratios between the surface, volume an Sauter mean iameters 0, 30, 3 ) an the maximum roplet iameter are taken as [1], [] 0 max = 0.11, 30 max = 0.14, 3 max = 0.5. ) Interfacial heat an mass transfer Interfacial heat an mass transfer has been moele with the two-resistance moel for the phase change in pure substances. This approach consiers the heat transfer processes on each sie of the phase interface, where the heat flux q can be written as q l,if = α l T sat T l ) + ṁ g l h l,if, 3) q g,if = α g Tsat T g ) ṁg l h g,if. 4) Here, α is the heat transfer coefficient [W/m K)] an ṁ g l is the mass flux from the gas to the liqui phase. Then, the mass flux can be etermine from the heat flux balance q l + q g = 0) as ṁ g l = α l T sat T l ) + α g Tsat T g ) h g,if h l,if. 5) In the bubbly flow regime, the heat transfer coefficient across interface an the isperse phase α is set to α g = 10 4 W/m K) [4]. In effect, the bubble temperature will quickly approach the interface temperature T sat. Due to the small size of the entraine roplets, the same approach can be use to moel the heat transfer in the isperse roplet regime, thus: α = 10 4 W m K 6) The heat transfer coefficient in the continuous phase α c α g in the roplet flow regime, α l in the bubbly flow regime) is etermine base on the Nusselt number Nu: α c = λ cnu c 7) The Nusselt number for 0 Pr c 50 is calculate accoring to Hughmark [5]: { + 0.6Re 0.5 c Nu c = Pr0.33 c ; 0 Re c < Rec 0.6 8) Pr0.33 c ; Re c In equation 8), the relative Reynols number between the isperse an the continuous phase Re c an the Prantl number of the continuous phase Pr c are efine as Re c = ρ c c c c η c, 9) 1574

5 Title Stagnation Nozzle throat Nozzle exit Ref. pressure iameter mass flux iameter mass flux p 0 /bar crit /mm ṁ crit /kg/m s) e /mm ṁ e /kg/m s) WU07A [3] WU07A [3] WU07B [3] WU07B [3] WU [4] WU [4] Table 1 Parameters of selecte DCC experiments taken from the literature. All experiments have been performe at ambient pool conitions p 1bar). Pr c = c p η c /λ c. 30) Therefore, equation 8) reuces to Nu c = when no phase slip is assume c = c c ). Finally, the volumetric mass source term can be obtaine as Γ l = a if ṁ g l. 31) Simulation moel The conservation equations 4), 5) an 7) to 9) an the interfacial transport equation 1) constitute a system of six orinary ifferential equations for the variables R, c z, ε, h g, h l an a if. The system is solve using an explicit fourth-orer Runge-Kutta algorithm. Gas-ynamic phenomena ue to over- an uner-expansion are neglecte an the effective-aapte-jet approximation is applie as bounary conition at the nozzle exit. This approach is wiely use in treating two-phase jets with an without conensation [6] an assumes isentropic aaptation to the ambient water pressure p. The nozzle exit velocity, ensity an iameter are then replace by the aapte values. Aitionally, a maximum voi fraction of ε g = is enforce at the nozzle exit to avoi numerical errors ue to ivision by zero. Subsequently, each solver step consists of the following major sub-steps: 1. Thermoynamic properties are calculate using the IAPWS-IF97 equation of state [7] as a function of the pool pressure p an the gas an liqui phase enthalpies h g an h l.. The entrainment velocity c en an volumetric mass source term Γ en,l are calculate using equations 17) an 18) with an entrainment coefficient of E 0 = Equations 19) to 1) are iteratively solve to obtain the maximum iameter of entraine roplets max,en, which is then use to calculate the Sauter mean iameter 3,en an the interfacial area source term ue to roplet entrainment Φ en accoring to equation ) an equation 15), respectively. 4. Interfacial heat an mass transfer q g,if, q l,if, ṁ g l, Γ l ) is solve using equations 3) to 31). 5. The conservation equations an the interfacial transport equation equations 4), 5), 7) to 9) an 1)) are solve using the explicit Runge-Kutta-Fehlberg algorithm. Initially, the solver is invoke for isperse roplet flow liqui phase l = isperse phase ). The solver procees until ε g = 0.5 is reache, where the solver is re-initialize for isperse bubbly flow gas phase g = isperse phase ) an continues until a minimum voi fraction of ε g = 10 6 is reache. The axial istance at this point ) correspons to the preicte penetration length: z ε g = 10 6 = l. SIMULATION RESULTS The simulation moel has been compare to various experiments taken from the literature, which cover a wie range of parameters nozzle exit iameter, mass flux an pressure, pool water temperature). Details on the selecte experiments are given in Table 1. The stagnation state has been etermine using the stagnation pressure p 0 provie in the literature while assuming a saturate steam state. Non-equilibrium effects uring expansion cf. [8]) have been neglecte, as not all literature sources provie sufficient information about the nozzle geometry. Accoringly, the nozzle exit state has been etermine assuming isentropic equilibrium expansion. The simulate nozzle exit conitions obtaine in this manner are given in Table an are in goo agreement with the experimental ata. Jet penetration length In Figures a an b, the simulate imensionless jet penetration length L for ifferent nozzle exit conitons an pool temperatures is compare with the experimental measurements from [4], [3]. Figure a shows the results for low stagnation pressures an accoringly low steam mass fluxes. Here, the simulation is in goo agreement with the experimental ata. However, L is unerpreicte for higher mass fluxes, as can be seen in Figure b. Axial temperature profile In aition to the jet penetration length, the axial temperature profile of the two-phase jet has been measure in [4]. However, it was not possible to etermine whether this ata shoul 1575

6 L 4 Tl/ C ṁ e = 49 kg/m s) 134 kg/m s) 7 kg/m s) T / C a) Low steam mass flux: Experimental values WU07A-; WU07B-; WU10-3) an respective simulation results ; ; ). T = 0 C 30 C 40 C 50 C z/mm a) Experiment WU10-3 ṁ e = 7.410kg/m s)) L 8 6 Tl/ C ṁ e = 487 kg/m s) 6 kg/m s) 374 kg/m s) T / C b) High steam mass flux: Experimental values WU07A-4; WU07B-4; WU10-5) an respective simulation results ; ; ). Figure Dimensionless penetration length L for ifferent pool temperatures T. T = 0 C 30 C 40 C 50 C z/mm b) Experiment WU10-5 ṁ e = kg/m s)) Figure 3 Temperature profile T l along the jet axis z for the conensation-inuce single phase jet: Experimental values for ifferent pool temperatures T 0 C; 30 C; 40 C; 50 C) an respective simulation results ; ; ; ). 1576

7 Title Nozzle exit Mach number mass flux pressure Ma e ṁ e /kg/m s) p e /bar WU07A %) 0.6 WU07A %) 1. WU07B %) 0. WU07B %) 0.4 WU %) 0.4 WU %) 0.7 Table Simulation results for the flow conitions at the nozzle exit. Values in parentheses inicate the eviation from the literature ata. be correlate to the simulate temperature in the gas or the liqui phase. These ifficulties o not arise in the conensation-inuce single-phase jet region. Accoringly, comparisons of the axial temperature profile were limite to this region z > l). For this purpose, the simulation was continue from the en of the twophase flow region by setting ε l = 1 an Γ l = q l,if = 0, which converts equations 5), 7) an 9) into the conservation equations for a single-phase jet with turbulent entrainment. The results obtaine in this manner for the axial temperature profile are shown in Figures 3a an 3b for experiments WU10-3 an WU10-5, respectively. Again, the simulation moel is capable of preicting the experimental ata at low mass flux, while no aequate agreement coul be achieve for the experiments at high mass flux. Interpretation of results The jet penetration length is mainly epenent on the water temperature an the steam mass flux. The simulation results regaring the influence of the pool water temperature are in goo agreement with experimental ata. However, the increase in penetration length when increasing the steam mass flux coul not be preicte by the evelope moel. Therefore, the moel unerestimates the penetration length of the two-phase jet an the axial temperature profile of the conensation-inuce single phase jet at high steam mass fluxes, but matches closely with experimental ata for low steam mass fluxes. The two major physical processes in the two-phase jet region are the mass entrainment at the jet bounary an the conensation in the jet core, while only the former is relevant in the singlephase jet region. Since the simulation ata is in goo agreement with the experimental ata in the single-phase jet region, the entrainment moel on the one han equations 17) an 18)) is consiere to be vali. On the other han, it is believe that aitional research is require to improve the conensation moel, particularly with respect to the calculation of the interfacial area ensity. CONCLUSION In the present work, turbulent entrainment an the evelopment of the interfacial area ensity are consiere to be the ominant processes uring irect contact conensation of a steam jet in subcoole water. Accoringly, it has been attempte to moel these phenomena in a physically soun manner, while applying appropriate simplifications regaring the evelopment of the jet profile an the changes in the flow regime. The simulation results are in goo qualitative agreement with experimental ata, supporting the valiity of the moeling approach. Quantitative accorance is achieve for steam injection at low mass fluxes. 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