ScienceDirect. Numerical modelling of debris bed water quenching

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1 Avaiabe onine at ScienceDirect Procedia IUTAM IUTAM Symposium on Mutiphase fows with phase change: chaenges and opportunities, Hyderabad, India December 08 December 11, 2014 Numerica modeing of debris bed water quenching D. Pavidis a,, J. L. M. A. Gomes b, P. Sainas a,c.c.pain a, A. A. K. Tehrani c, M. Moatamedi d, P. N. Smith a, A. V. Jones a, O. K. Matar e a Dept. Earth Science & Engineering, Imperia Coege London, London SW7 2AZ, UK b Schoo of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK c Office for Nucear Reguation, Boote L20 7HS, UK d Centre for Nucear Engineering, Imperia Coege London, London SW7 2AZ, UK e Dept. Chemica Engineering, Imperia Coege London, London SW7 2AZ, UK Abstract Debris beds may be formed during a nucear reactor severe accident and cooabiity of these beds is important to avoid reease of radioactive materias into the environment. However, debris bed water quenching is chaenging to understand, and mode, because of the compex muti-phase fow and heat transfer physics invoved which may incude boiing. This paper deveops a modeing method for boiing and demonstrates its abiities with some appications. The mode is based on a muti-fuid approach, in which one phase represents the iquid, the second phase the gas phase the third phase represents the soid debris bed. In each fuid phase a set of conservation equations for mass, energy and momentum are soved with appropriate inter-phase exchange terms couping the fuid phases. c The Authors. Pubished by Esevier by Esevier B.V. B.V. This is an open access artice under the CC BY-NC-ND icense Peer-review under responsibiity of Indian Institute of Technoogy, Hyderabad. Peer-review under responsibiity of Indian Institute of Technoogy, Hyderabad. Keywords: Boiing modeing; heat transfer; met poo; porous media fow. 1. Introduction In the event of proonged oss of cooing within a nucear reactor integrity of interna structures may be compromised and materias might met. This wi form a debris bed which consist of fragments from the cadding and peets that constitute the nucear fue. If it is not rapidy cooed, this mass wi begin to met and become harder to coo. Here a method for simuating such environments using a muti-fuid approach is presented. This incudes fooding of porous media and boiing. The remainder of this paper is organised as foows. A brief description of the mode is given in section 2 and preiminary resuts are presented in section 4. Finay, some concuding remarks are given in section 5. Corresponding author. Te.: E-mai address: dimitrios.pavidis@imperia.ac.uk The Authors. Pubished by Esevier B.V. This is an open access artice under the CC BY-NC-ND icense Peer-review under responsibiity of Indian Institute of Technoogy, Hyderabad. doi: /j.piutam

2 D. Pavidis et a. / Procedia IUTAM Mode Equations for Boiing In this section the governing muti-phase fuid equations and modeing assumptions are presented. The basic assumption is that the system can be modeed using a three-fuid mode, representing the iquid, steam and soid phases, in which the debris bed is the soid phase Conservation of mass The continuity equation for phase k is expressed by: α kρ k + α k ρ k v k =Γ k, where t is the time variabe, ρ k, α k and v k are the density, voume fraction and veocity of phase k, respectivey, and Γ is a source/sink term that accounts for mass exchange between phases Force baance equations The force baance equations are expressed by: α ρ v + v v = α p + τ + α ρ g +Σ g vg v +Σs v s v, 2 1 α g ρ g vg + v g v g = α g p + τ g + α g ρ g g +Σ g v v g +Σsg vs v g + FVM + F IP, 3 α s ρ s vs + v s v s = α s p + τ s + α s ρ s g +Σ s v v s +Σ sg vg v s Σss v s, 4 where τ is the deviatoric stress tensor, the Σ s are the inter-facia drag coefficients, p is the shared pressure of a phases; g is the gravitationa acceeration. Subscripts g, and s denote the gas, iquid and soid debris bed porous medium phases, respectivey. The absorption coefficient Σ ss is used here to ensure that the soid phase remains immobie. The virtua mass and inter-phase pressure terms are given by: F VM = α gρ v + v v v g v g v g and F IP = α 2 gρ vg v αg Interna energy conservation The equations for interna energy e, e g, e s expressed in terms of temperature T, T g, T s, in non-conservative form, are: T α C p ρ α + v T = p + α v + α κ T +Σ T g Tg T +Σ T s T s T +Σ vap T sat T +Γ h +Γ L e0 Γ C p T, 6 Tg αg C pg ρ g α g + v g T g = p + α g v g + α g κ g T g +Σ T g T T g +Σ T sg Ts T g +Σvapg Tsat T g +Γ g h g Γ g C pg T g, 7

3 66 D. Pavidis et a. / Procedia IUTAM C ps ρ s α s Ts αs + v s T s = p + α s v s Tg T s + S s, 8 + α s κ s T s +Σ T s T T s +Σ T sg where C p, κ and h are the heat capacity, therma conductivity and enthapy, respectivey. The Σ T s are inter-facia heat transfer coefficients and the Σ vap s are voumetric heat transfer coefficients due to vapourisation-condensation. T sat is the saturation temperature and L e0 = J/kg. The source term S s can be used to incude decay heat to the mode. It is now easy to see the effect of atent heat. On summing these two equations the interface heat exchange terms disappear, due to the interface Stefan condition see section 3.3, eaving the term L e0 Γ acting as a sink when boiing Γ g = Γ > 0 and reaxing the iquid temperature towards T sat. 3. Parametrisations In this section the parametrisations used are briefy outined. They are primariy taken from 2, Force baance equations The inter-facia drag coefficients are: Σ sg = 150 α2 gsμ g + 7 α sg d 2 p 4 α gs ρ g vg v s d p, Σ s = 150 α2 s μ + 7 α s d 2 p 4 α s ρ v v s, Σ g = 3 d p 4 C α g α g ρ vg v D α 2.65 g, 9 d b where d p is the debris bed partice diameter, d b is the average bubbe diameter and the drag coefficient C D is reated to the Reynods number Re g by: { 24 C D = α g Re g } α g Re g, if α g Re g < , if α g Re g 1000 and: Re g = ρ v g v db. μ The average bubbe diameter is cacuated using: d b = Weσ ρ vg v 2, where We = 5 and σ = 0.06N/m. The bubbe size is capped according to: d b = min { d p, max { 10 7, d b }}. 13 The normaised voume fractions are given by: α sg = α s α s + α g,α gs = 3.2. Heat transfer coefficients α g α s + α g,α s = α s α s + α,α s = α α s + α,α g = α α + α g,α g = α g α + α g. 14 The inter-facia heat transfer coefficients are cacuated using the Ranz-Marsha correation vaid for spherica bubbes or partices: Nu = hd κ = Re 1 2 Pr 1 3, 15

4 D. Pavidis et a. / Procedia IUTAM where Nu, Re [0, 200] and Pr [0, 250] are the dimensioness Nusset, Reynods and Prandt numbers, respectivey. Therefore: Σ T s = κ Re s d 2 Pr 1 3, Σ T sg = κ g Re sg p d 2 Pr 1 3 g, Σ T g = κ Re g p d 2 Pr 1 3, 16 b where: Re s = ρ v v s d p μ, Re sg = ρ g v g v s dp. 17 μ g 3.3. Interface Stefan condition for mass transfer from heat transfer The interface Stefan condition for mass transfer from heat transfer is determined by considering the therma energy equation over the interface between the iquid and vapour/gas and ignoring storage terms in the equations. The Stefan condition is parametrised here by summing the heat transfer and mass transfer terms for vapour in the energy equations and setting this baance to zero: Σ vap T sat T +Σ vapg T sat T g +Γ g L h = 0, 18 in which we have used the atent heat L h : Γ g h g +Γ h =Γ g h g h =Γ g L h, 19 and aso Γ g = Γ. Thus, the mass transfer rate Γ g is determined from: where: Γ g = Σ vap T sat T +Σ vapg T sat T g L h, 20 h = { Le0 + C p T + p ρ, if T sat < T L e0 + C p T sat + p ρ, if T sat T, C pg T sat + p ρ h g = g, if T sat < T g C pg T + p ρ g, if T sat T g. Based on 3, the heat transfer coefficient Σ vap is cacuated by: { κ max 12 d b π Σ vap = T sat T ρ C p ρ g L h β, { min Σ max vap, } F 5 L h ρ g ρ α g ρ ρ g κ d b α Re 0.5 } 3.6α b d b, if T sat < T, if T sat T, where β = 1, α b = max{α g, 10 5 }, { ϕC exp 45αb, if α F 5 = g < , if α g 0.25, 1, if vg v 0.61m/s ϕ = vg v 0.47, if vg v 25 > 0.61m/s, 24 C = { p 10 5, if p Pa p 1.418, if p > Pa, Σ max vap = max { } { 4.724, 472.4α g α max {0, min 1, α }} g

5 68 D. Pavidis et a. / Procedia IUTAM The heat transfer coefficient Σ vapg is cacuated by: Σ vapg = α b d b. The fina vaue of the heat transfer coefficients for the iquid and gas are imited by a maximum of: Σ vap max = min{γρ g,α ρ }L h Δt ref max{ε, T sat T }, if T sat < T γα g ρ g L h Δt ref max{ε, T sat T }, if T, Σ max vapg = sat T min{γρ g,α ρ }L h Δt ref max{ε, T sat T g }, if T sat < T g γα g ρ g L h Δt ref max{ε, T sat T g }, if T sat T g, respectivey, where ε = 10 10, γ = 10 and Δt ref = 1s. 4. Resuts The mode is used to simuate boiing in a porous medium in 2D. A uniform porosity of 0.4 i.e. α s = 0.4 is assumed and the dimensions of the computationa domain are 1m 2m. The acceeration due to gravity is 9.81m/s 2 and gravity acts on the y direction. The materia properties of the three phases are given in Tabe 1. The saturation temperature in degrees Cesius is a function of pressure in Pa and is cacuated by: T sat = π arctan 0.5π p , 30 where: p = C p , 31 and: C = { 0.5, if p Pa 0.3, if p > Pa. 32 Tabe 1. Materia properties for the iquid, gas and soid phases. As far as the equations of state are concerned, the pressure is in Pa and the temperature is in degrees Cesius. ρ kg /m 3 C p J /kg K κ W /mk μ Pa s T g p/461.5t g s The domain is initiay saturated in water α = 0.6 and the background temperature of a three phases is 99.9 C. A perturbation in the iquid phase temperature fied initia condition is used to initiate boiing. Resuts from three such cases with perturbation magnitude 104, 106 and 110 C are presented here. Perturbations are circuar, centred around 0.5, 0.5 and have a radius of 0.15m. Free-sip boundary conditions are appied on the sides and bottom of the domain. The top boundary is assumed to be open and an atmospheric pressure condition is appied. A structured mesh of trianguar eements with 20 and 40 ayers in the x and y directions, respectivey, is used for the three simuations. Instantaneous maps of the vapour voume fraction at six time eves for the three simuations are shown in Fig. 1 and 2. The effect of the increased perturbation temperature on the vapourisation process is evident. For the 104 and 106 C cases the vapour starts to condense toward the end of the simuation see Fig. 2, eft and midde coumns. An additiona simuation is performed using anisotropic mesh adaptivity. The mode set-up is identica to the 106 C case except for the mesh. Instantaneous maps of the vapour voume fraction and vapour temperature aong with the adapted mesh bottom at three time eves are shown in Fig. 3

6 D. Pavidis et a. / Procedia IUTAM Fig. 1. Instantaneous maps of the vapour voume fraction at time eves t = 0.02, 0.4 and 0.8s from top to bottom. Left coumn: 104 C case. Midde coumn: 106 C case. Right coumn: 110 C case.

7 70 D. Pavidis et a. / Procedia IUTAM Fig. 2. Instantaneous maps of the vapour voume fraction at time eves t = 2, 4 and 8s from top to bottom. Left coumn: 104 C case. Midde coumn: 106 C case. Right coumn: 110 C case.

8 D. Pavidis et a. / Procedia IUTAM Fig. 3. Instantaneous maps of the vapour voume fraction top and vapour temperature aong with the adapted mesh bottom at time eves t = 0.2 eft coumn, 1 midde coumn and 2s right coumn. 5. Concusions A method for numericay simuating boiing in porous media has been briefy outined here. Some features, incuding anisotropic mesh adaptivity, have been demonstrated through the simuations presented here. Acknowedgements The authors woud ike to thank the EPSRC MEMPHIS muti-phase programme grant, the EPSRC Computationa modeing for advanced nucear power pants project, the EU FP7 projects THINS and GoFastR and ExxonMobi for heping to fund this work. References 1. Pain, C., de Oiveira, C., Umpeby, A.G.A.. Transient criticaity in fissie soutions compressibiity effects. Nucear Science and Engineering 2001;138: Gidaspow, D.. Mutiphase fow and fuidization. Academic Press; RELAP5-3D Code Manua Voume I: Code Structure, System Modes and Soution Methods; 2009.