numerical modeling of sodium fire—part i: spray combustion

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ARTICLE IN PRESS Model

ED-7683; No. of Pages 16

Nuclear Engineering and Design xxx (2014) xxx– xxx

Contents lists available at ScienceDirect

Nuclear Engineering and Design

j ourna l h om epa ge: www.elsev ier .com/ locate /nucengdes

umerical modeling of sodium fire—Part I: Spray combustion

ratap Sathiah ∗, Ferry Roelofsuclear Research and Consultancy Group (NRG), Westerduinweg 3, 1755ZG Petten, The Netherlands

i g h l i g h t s

A CFD based method is proposed for the simulation of sodium spray combustion.A pre-ignition model is proposed which is based on combustion mass transfer and reaction kinetics approach.The proposed method is validated against single droplet experiments of Miyahara and Ara.The predictions obtained using the proposed method is in good agreement with the experiments.

r t i c l e i n f o

rticle history:eceived 22 July 2013eceived in revised form 4 November 2013ccepted 12 November 2013

a b s t r a c t

A sodium cooled fast reactor is one of the fourth generation advanced reactor designs. Liquid sodium isused as a coolant in such a reactor as it has excellent thermophysical properties. However liquid sodiumcan react violently when exposed to air or water. A sodium-air reaction typically occurs in two dominantmodes: spray and pool. Typically, the spray mode of burning is considered as more severe than the poolmodel of burning. The focus of this paper is on sodium spray combustion.

For the safety of a sodium cooled fast reactor, sodium-air reactions should be avoided. To avoid andto mitigate the consequences if a sodium fire occurs, it is essential to understand various physicalphenomena involved in a sodium-air reaction. Computational fluid dynamics based numerical meth-ods can be used for this purpose as they are known to resolve all spatial and temporal scales and simulatevarious physical processes governing sodium-air reaction. The goal of the work presented within thispaper is to propose a numerical method to simulate sodium spray combustion and validate this method
against experiments.
A single sodium droplet combustion experiments is used for the validation. The model predictions offalling velocity and burned mass are in good agreement with experimental data. Additionally, parametricstudies were performed to investigate the effects of initial droplet diameter, temperature and oxygenconcentration on burning rate and on ignition time delay. Once sufficiently validated, the present method

luati
can be used for safety eva
. Introduction

A sodium cooled fast reactor (SFR) is under consideration as IVeneration advanced reactor. The sodium cooled fast reactor con-ept has a reasonable experience base, and large scale reactorsave been built and operated worldwide. Liquid sodium is useds a coolant in these reactors, since it has excellent thermophysicalroperties. In particular, it has a high thermal conductivity, a lowbsorption rate of fast neutrons which creates the possibility forreeding plutonium or minor actinides burning.

Please cite this article in press as: Sathiah, P., Roelofs, F., Numerical modehttp://dx.doi.org/10.1016/j.nucengdes.2013.11.081

However, liquid sodium also has shortcomings. Sodium whenxposed to air or water reacts violently, which can be a potential fireazard in a nuclear reactor. A sodium leak, which may result from

∗ Corresponding author.E-mail addresses: [email protected] (P. Sathiah), [email protected] (F. Roelofs).

029-5493/$ – see front matter © 2013 Published by Elsevier B.V.ttp://dx.doi.org/10.1016/j.nucengdes.2013.11.081

on of a sodium fast reactor.© 2013 Published by Elsevier B.V.

a pipe break up, releases into the containment in the form of sprayor jet. A part of the released sodium gets collected on the floor andmay form a sodium pool. Leaked sodium essentially burns in twodifferent modes, i.e. spray and pool modes. The spray mode of burn-ing is more severe than the pool mode of burning, since a sodiumspray burns at a higher rate. This is because it burns in a highlydivided state, i.e. in the form of the droplets. Furthermore, sodiumspray fires are difficult to extinguish in comparison to sodium poolfires (Saito et al., 2001). During a sodium spray or pool combus-tion, sodium reacts with air and water to form several sodiumby-products, e.g. sodium oxide (Na2O), sodium dioxide (Na2O2)and sodium hydroxide (NaOH), which release into the atmospherein the form of aerosols. These aerosols are particles of a diameter

ling of sodium fire—Part I: Spray combustion. Nucl. Eng. Des. (2014),

ranging from 0.1 �m to 10 �m, which can cause structural damageto the equipment and endanger the health of employees.

In the past, several reactor accidents were reported in the lit-erature (Anzieu and Martin, 2011; Takata., 2007). In the United

dx.doi.org/10.1016/j.nucengdes.2013.11.081


http://www.sciencedirect.com/science/journal/00295493

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2 P. Sathiah, F. Roelofs / Nuclear Engineering and Design xxx (2014) xxx– xxx

md fuel burning rate [kg/s]m droplet mass burning rate [kg/s]Ap area of the droplet [m2]B transfer numberBs [m/s]C1 turbulent model constantC2 turbulent model constantC3 turbulent model constantCd constantCd drag coefficientCg constantCp,g gas mixture heat capacity at constant pressure

[J/kg K]D diffusivity [m2/s]D droplet diameter [m]Dd diffusion coefficient of oxygen [m2/s]DNa,N2 diffusion coefficient [m2/s]Es activation energyf mixture fractionFD drag force [N]Gb production term due to buoyancy [kg m−1s−3]gi gravity acceleration [ms−2]Gk production term [kg m−1 s−3]H total enthalpy [J/kg]h heat transfer coefficient [W/m2 K]Hc heat of combustion [kJ/kg]hx heat of reaction [kJ/kg]hfg heat of evaporation [J/kg]K burning rate coefficients or evaporation constant

[m]kc mass transfer coefficient [m2/s]kg gas mixture thermal conductivity [W/m K]NO2 molar flux of oxygen species [moles/m2]ns reaction orderNu Nusselt numberp pressure [Pa]Pr Prandtl numberPrt turbulent Prandtl numberR universal gas constant [J/K mol]r stoichiometric ratioRep Reynolds numberSh heat loss or gain [W m−3]Sm mass source term [kg m−3 s−1]Smom momentum source term [kg m−3 s−1]Sc Schmidt numberSh Sherwood numbert time [s]Tg gas temperature [K]Tp droplet temperature [K]T∞ temperature at ambient [K]Tfilm film temperature [K]uj velocity in j-direction [m/s]up droplet velocity [m/s]vf stoichiometric coefficientWf molecular weight of fuel species [g mole−1]WO2 molecular weight of oxygen species [g mole−1]xj coordinates in j-direction [m]xp droplet location [m]

Greek symbols

�t turbulent viscosity [kg m−1 s−1]� density of gas [kg m3]�p droplet density [kg m3]� Stefan–Boltzmann constant�k turbulent model constant�t turbulent Schmidt number�ij viscous force tensor [Nm−2]ε turbulent dissipation rate [m2 s−3]εp emissivity of the droplet [m]k turbulent kinetic energy [m2 s−2]

AcronymsCFD computational fluid dynamicsDNS direct numerical simulationJNC Japan Nuclear Cycle Development InstituteJPDF joint probability distribution functionLFMBR liquid metal fast breeder reactorLTV large test vesselPDF probability distribution functionPFR prototype fast reactorPIRT phenomena identification and ranking tableSFR sodium fast reactor


˛ ˇ-PDF parameters ˇ-PDF parameters

� oxidizer-fuel coupling function

UDF user defined functions

Kingdom (UK), a major sodium leak was reported in 1960 in steamgenerator building of prototype fast reactor (PFR). In the UnitedStates (US), a sodium fire occurred in May 1970 in the 94 megawatt-electric (MWe) sodium cooled fast reactor (Fermi 1). In 1995, theJapanese Monju reactor was shut down after a sodium leakage acci-dent. The sodium leak occurred from the secondary circuit in apiping room of the reactor auxiliary building. The large scale Frenchfast neutron reactor, Superphenix was shut down in 1997 as a resultof sodium leaks(IAEA-Tecdoc, 2007). In Russia, several accidents insodium cooled fast reactors occurred in the BR-5/10, and BOR-60experimental reactors, in the BN-350 prototype reactor and in theBN-600 demonstration reactor.

To summarize, sodium leaks which might lead to sodium-airreactions are dangerous to the safety of reactor. For the safety of asodium cooled reactor, sodium-air reactions must be avoided andits possible consequences should be mitigated. Several mitigationsystems have been proposed in literature, for example, a Karlsruhetray (Huber et al., 1975) and a leak collection tray (Diwakar et al.,2008, 2011) to catch leaking sodium and avoid its contact with airor water. However, further work on optimization of the design ofsuch mitigation systems is necessary before these systems can beused in a real sodium leak scenario in a sodium fast reactor.

To understand these phenomena, detailed experimental andnumerical analyses of sodium-air reactions are important. Thisunderstanding is needed to develop computer codes, which canbe used for safety analyses of sodium fast reactors and will supportthe design of mitigation systems.

At NRG, our main focus is to develop a numerical tool based onCFD to simulate simultaneous sodium spray and pool combustion.We do this in three steps. In the first step, we develop and validatea code to simulate sodium spray combustion. This is the subject ofthe present work. In the second step, we developed and validatea code to simulate sodium pool combustion. In the third step, wecombine these codes to simulate simultaneous sodium spray andpool combustion and demonstrate the feasibility of our code. This


work will be presented in a future publication.Extensive research has been performed in the past to under-

stand sodium spray combustion. Richard et al. (1968) investigatedthe burning of a stationary single sodium droplet. They measured


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ARTICLEED-7683; No. of Pages 16

P. Sathiah, F. Roelofs / Nuclear Eng

he droplet burning rate photographically and investigated theffects of the droplet size, the oxygen mole fraction and the gasemperature on the burning rate. Important conclusion from theirork is that the burning of sodium follows the D2-law proposed

y Spalding (1955) for hydrocarbon fuels. Furthermore, they high-ighted that the mass transfer by diffusion to the droplet surfaces the rate determining step in the sodium droplet combustion.

orewitz et al. (1977) performed experiments to investigate theurning of a single sodium droplet falling from a height of 14.7 m.hey measured fall distance to ignition and burned mass fractions.

free falling single sodium droplet experiment was conducted byiyahara and Ara (1998). They observed burning behavior of a sin-

le sodium droplet. In particular, burnt weight and falling velocitiesere measured.

Yuasa (1984) experimentally investigated the burning of aodium droplet in air with different moisture concentrations. Hebserved that droplet temperature increased gradually in theeginning, because of surface reaction and once ignition startsodium temperature increases rapidly due to vapor phase com-ustion. One of the important findings in his work was the

dentification of the existence of a pre-ignition phase in sodiumombustion.

Krolikowski et al. (1969) developed the diffusion controlledapor phase combustion model to simulate burning of sodiumroplet. In their approach, the reaction rate is assumed to be con-rolled by diffusion of oxygen from the surroundings to the flameurface. It was concluded that burning rate and flame temperatureecreases with decrease in oxygen content. If oxygen concentration

s less than 4%, a vapor phase combustion is not possible. They high-ighted that the spray particle size is the most important parameterffecting the pressure rise due to spray combustion in an enclosedolume.

Tsai (1979) developed the NACOM code for the analysis ofodium spray fire in SFRs and indicated that the burning of a sodiumroplet can be divided into a pre-ignition and a post-ignition phase.

n the pre-ignition phase, the burning of a single droplet is con-rolled by diffusion of oxygen at the droplet surface, while in theost-ignition phase, the vapor-phase combustion theory (Spalding,955) was used to model the burning of a single droplet. The codeas validated against experiments of Johnson et al. (1977).

Newman (1983) presented a review which describes variousechanisms involved in the ignition and combustion of a sodium

pray and pool. He pointed out that the ignition of liquid sodium cane described by the ignition temperature. This is defined as the tem-erature of liquid sodium at which it starts to oxidize sufficientlyapid that a self-generated temperature rise is produced. A wideange of ignition temperatures of liquid sodium was reported with

lowest ignition temperature of 397 K for a pool mode of burningnd a highest value of 723 K for a spray mode of burning. The wideange of ignition temperatures is due to the presence of a protectivexide layer on the sodium surface.

A computer program SOFIA-II was developed by Kawabe et al.1982) to predict pressure and temperature transients during theroplet burning. Malet et al. (1981) developed the PULSAR code toredict consequences of sodium spray fire.

Murata et al. (1993) modified the CONTAIN code, which has aodel for droplet burning rate based on Tsai’s work. The modified

ode was validated against single droplet experiments of Richardt al. (1968) and Morewitz et al. (1977).

The SPHINCS (Yamaguchi et al., 2001; Yamaguchi and Tajima,003) code developed by Japan Nuclear Cycle Development Insti-ute (JNC) is used for safety evaluation of sodium cooled fast


eactors. The code solves coupled phenomena of thermal hydraulicsnd sodium fire and is based on a multi zonal model. The code uses

spray model developed by Tsai (1979) with a submodel to sim-late the pre-ignition and the post-ignition phase. The gas phase

PRESSg and Design xxx (2014) xxx– xxx 3

combustion is obtained by assuming chemical equilibrium, whichis calculated using the BISHOP code.

AQUA-SF developed by Takata. (2007), Yamaguchi et al. (2001),Takata et al. (2003), simulates the burning of sodium in both thespray and pool mode. The code is developed to evaluate spatial dis-tributions of gas temperature and chemical species during a sprayand pool combustion. The code was validated against experimentsof Ohno (1994), Nagai (1999).

The COMET code, developed by JNC is used for direct numeri-cal simulation (DNS) of combustion of a free falling single sodiumdroplet by Okano and Yamaguchi (2003). The flow was solved withan unsteady Navier–Stokes equation and chemistry was assumedto be in chemical equilibrium. The sodium mass flux at the dropletsurface was assumed to be proportional to the oxygen consump-tion rate. The code was validated against a free falling single dropletsodium experiment of Miyahara and Ara (1998).

Ignition of a falling sodium droplet is experimentally investi-gated by Makino and Fukada (2005a), Makino and Fukada (2005b).They concluded that the dominant parameters affecting the igni-tion of sodium droplets are initial droplet temperature, dropletdiameter, oxygen concentration and relative velocity of the droplet.The burning rate constant was reported to be 1.0 mm2/s when asodium droplet of initial diameter of 2 mm burns in a quiescentatmosphere. However, when a sodium droplet burns in a convec-tive environment the burning rate constant increases to 3.4 mm2/s.

Recently, Saravanan et al. (2011) numerically investigatedeffects of the pre-ignition model on ignition time delay. Theyevaluated ignition time delay by using mass transfer and reac-tion kinetics limited pre-ignition models. They concluded that thereaction kinetics limited model predicts the limit of ignitability ofsodium droplet under different initial and convective conditions,while the mass transfer limited model predicts the limit of ignitabil-ity of a sodium droplet under very low oxygen concentration.

Recently, Rao et al. (2012) proposed a model to simulate quasi-steady sodium droplet combustion in a convective environment.The mass burning rate and flame stand-off ratio was predicted wellusing the model.

Recently, Ohno et al. (2012) constructed a phenomena identifi-cation and ranking table (PIRT) identifying important phenomenato be modeled in sodium fire analyses codes. According to them, thephenomenon of droplet combustion was identified as highly impor-tant while pool combustion was ranked as moderately important.

In the available literature, the effects of a pre-ignition model onsodium spray combustion have not been addressed. This will be themain objective of this paper. In this article, we will report the devel-opment of a sodium spray combustion code. A new pre-ignitionmodel based on combined mass transfer and reaction kineticslimited approach is proposed. The proposed pre-ignition model isvalidated against single droplet experiments of Miyahara and Ara(1998). The sensitivity of the initial droplet diameter, droplet tem-perature, air temperature, and initial oxygen concentration on theburning of a single droplet is also investigated.

The article is organized as follows. In Section 2, the com-putational fluid dynamics (CFD) model is presented. Governingequations for Eulerian and Lagrangian phase are described here.In addition, combustion, turbulence and spray surface reac-tion/evaporation models are also presented. Section 3, presents thedescription of the experiment used for the validation. Section 4,presents the validation and additional results of the sensitivitystudy. Finally, summary and conclusions are presented in Section 5.Appendix A provides physical properties of liquid sodium.


2. CFD model

We have utilized the commercial CFD code ANSYS (2008)as the basis for development of the sodium spray combustion


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olver. Within the code, the gaseous phase was simulated in anulerian phase. The unsteady Favre-averaged (density-averaged)avier–Stokes equations are solved using the standard k − ε turbu-

ence model. The spray droplets are tracked in a Lagrangian phase.he equations for the position, the velocity, the mass and the tem-erature of the droplet are solved.

The models and the governing equations are described below inollowing subsections.

.1. Eulerian phase governing equations

Within the code, the following governing equations for the con-ervation of mass, momentum, energy and mixture fraction areolved.

Mass:

∂�

∂t+ ∂�uj

∂xj

= Sm. (1)

Velocity:

∂�ui

∂t+ ∂�uiuj

∂xj

= − ∂p

∂xi

+ ∂∂xj

(�ij − �˜u′′iu′′

j) + �gi + Smom + Fdrag.

(2)

Mixture fraction:

∂�f

∂t+ ∂�uif

∂xi

= ∂∂xi

(�D

∂f

∂xi

)− ∂�˜u′′

if ′′

∂xi

+ Sm. (3)

Variance of mixture fraction:

∂�f ′′2)∂t

+ ∂�uif ′′2

∂xi

= ∂∂xi

(�Df

∂f ′′2

∂xi

)+ 2f ′′ ∂

∂xi

(�Df

∂f

∂xi

)

− ∂�˜

u′′if ′′2

∂xi

− 2�u′′if ′′ ∂f

∂xi

− 2�D∂f ′′

∂xi

∂f ′′

∂xi

(4)

Total enthalpy:

∂�H

∂t+ ∂�uiH

∂xi

= ∂∂xj

(kg

Cp

∂H

∂xj

)− ∂�˜u′′

iH′′

∂xi

+ Sh. (5)

Here, . and . are the Reynolds averaged and Favre-averageduantities, respectively. ui, f and H are the Favre-averaged veloc-

ty, the mixture fraction and the total enthalpy, D is the mixtureraction diffusivity, �t is the turbulent viscosity, �t is the turbulentchmidt number, kg is the laminar thermal conductivity and kt ishe turbulent thermal conductivity. Fdrag is the drag force due to theroplets, Sm and Smom are the mass and the momentum transfer dueo evaporation to the gas phase from the liquid droplets. Sh accountsor source/sink term due to radiation heat transfer through walloundaries, heat exchange with the liquid droplets or pool, t is theime, � is the density, p is the mean pressure, Cp,g is the specific heatapacity and gi is the gravitational acceleration.

The total enthalpy is defined as

˜ =∑

YjHj where Hj =∫ T

Cp,jdT + hj(Tref ). (6)


j Tref

here Yj and Hj are the Favre-averaged mass fraction and enthalpyf jth species and Tref is the reference temperature.

PRESSg and Design xxx (2014) xxx– xxx

The mixture fraction is defined as the ratio of the mass of mate-rial having its origin in the fuel stream to the mass of the mixture(Bilger et al., 1990) that is:

f = � − �ox,0

�fu,0 − �ox,0(7)

where, � is the oxidizer-fuel coupling function, � = Yfu − Yox/S, S isthe stoichiometric oxygen to fuel mass ratio. If f = 0, the mixtureconsists of only air, while f = 1 means the mixture consists of onlyfuel. Here, fuel refers to sodium and oxidiser refers to air. In Eq. (4),

f ′′2 = (f − f )2

is the variance of mixture fraction. The term �˜u′′iu′′

jin

Eq. (2), is the Reynolds stress, which needs a closure and is describedin Section 2.3. The combustion model is described next.

2.2. Combustion model

A mixture fraction based non-premixed combustion model isused here to simulate sodium spray combustion. This model iswidely used to simulate turbulent diffusion hydrocarbon flames.The model is well validated in different applications of hydrocarboncombustion, for example in piloted diffusion flames (Merci et al.,2001; Pitsch and Steiner, 2000), hydrogen–air diffusion flames(Forkel and Janicka, 2000) and in gas turbine applications (Ghenai,2010). In addition, it is also used to simulate fires (Xue et al., 2001;Kang and Wen, 2004; Consalvi et al., 2012) and diesel engine appli-cations (Cook and Riley, 1998; Demoulin and Borghi, 2002). Becauseof its extensive validation, this model is also available in commer-cial codes, for example, in ANSYS FLUENT (ANSYS, 2008) and ANSYSCFX (ANSYS, 2007). The authors have therefore selected this modelto simulate sodium spray combustion. It is worth mentioning thatin sodium combustion modeling literature this model has neverbeen used.

In this model, it is assumed that the reaction takes place in aninfinitely thin flame sheet, where both the fuel and the oxygenspecies meet each other. The combustion in turn is controlled bythe rate of mixing of fuel and oxygen. Due to this assumption, itis possible to represent all the species relevant to the combustionprocess by one single parameter, i.e. mixture fraction. Then, thecombustion modeling is reduced to:

1 The problem of mixing, which means finding a location of theflame surface (where fuel and oxygen mix together). This is doneby solving a mixture fraction equation.

2 The calculation of the distribution of the species, which isobtained by assuming that the instantaneous thermochemicalstate (i.e. density, temperature and species mass fractions) of thefluid is related to the mixture fraction.

For the latter, the chemistry can be treated using a chemi-cal equilibrium approach or a steady laminar flamelet approach(Poinsot and Veynante, 2005). In the chemical equilibriumapproach, it is assumed that chemical reactions are fast and reachesequilibrium as soon as they meet each other. To consider the effectsof the finite rate chemistry, the steady laminar flamelet model canbe used. In the present work, we used the chemical equilibriumapproach as proposed by Doda et al. (2001), Yamaguchi and Tajima(2003), Doda et al. (2003), Yamaguchi and Tajima (2009). Further-


more, to consider the effect of heat loss/addition to the chemicalequilibrium, non-adiabatic equilibrium tables were constructed.This means that temperature, density and the mass fraction of thespecies are tabulated as a function of mixture fraction and enthalpy.


IN PRESSG Model

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500

1000

1500

2000

Mixture Fraction [−]

Tem

pera

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[K]

NASA CEAANSYS FLUENT

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Once the value of the mixture fraction in a computational cell isnown, the mean value of temperature and species mass fractionn a computational cell is then obtained by the following equation:

˜ =∫ 1

0

∫ 1

0

T(f, H)JPDF(f, f ′′, H)dfdH (8)

nd

k =∫ 1

0

∫ 1

0

Yk(f, H)JPDF(f, f ′′, H)dfdH. (9)

Here, T(f, H) and Yk(f, H) are the temperature and the mass frac-ion of the kth species which is obtained from the non-adiabaticquilibrium probability distribution function (PDF) table. In Eqs.8) and (9), JPDF(f, f′, H) is the joint probability distribution func-ion (JPDF) of the mixture fraction and the enthalpy. It describeshe joint probability that the flow field takes a certain value forhe mixture fraction and enthalpy at a given point. The JPDF cane obtained as a product of the PDF of the mixture fraction PDF(f,

′) and PDF of the enthalpy PDF(H), if statistical independence ofixture fraction and enthalpy is assumed. In the literature Poinsot

nd Veynante (2005), the PDF of the mixture fraction is presumedo be ˇ-PDF, since the shape of this PDF is able to change contin-ously from single delta, double delta to a Gaussian shapes. Whilehe PDF of the enthalpy is assumed to be a single delta function [8],hich means that the fluctuations in the enthalpy are neglected.

he ˇ-PDF can be estimated from the first and second moment ofhe mixture fraction i.e. the mean mixture fraction and the variancef the mixture fraction as follows:

DF(f ) = f ˛−1(1 − f )ˇ−1∫f ˛−1(1 − f )ˇ−1

(10)

here

= f

[f (1 − f )

f ′′2− 1

](11)

nd

= (1 − f )

[f (1 − f )

f ′′2− 1

](12)

re the ˇ-PDF parameters.Once the value of the PDF is known using Eq. (10), the mass

raction and the temperature are obtained by integrating the stateunction (mass fraction and temperature) with the PDF over theomplete physical range of the mixture fraction. Finally, the meanensity can be calculated as follows:

1�

=∫ 1

0

JPDF(f, f ′′, H)�(f, H)

dfdH. (13)

here, �(f, H) is obtained from the non-adiabatic equilibriumDF table. Further details of the model are mentioned elsewhereANSYS, 2008; Poinsot and Veynante, 2005).

To generate the non-adiabatic equilibrium PDF table, the equi-ibrium calculations were performed using ANSYS FLUENT. Forhis purpose, equilibrium PDF tables were generated for a Na–airystem at a temperature of 298 K and a pressure of 1 atm. Theollowing gas phase species N, NO, NO2, NO3, N2, N2O, N2O3,

2O4, N2O5, N3, Na, NaNO2, NaNO3, NaO, Na2, Na2O, Na2O2, O, O2,3 and condensed phase species Na(cr), Na(l), NaO2(cr), NaO2(l),a2O(c), Na2O(b), Na2O(a), Na2O(l), Na2O2(b), Na2O2(a), Na2O2(l),


aNO2(a), NaNO2(b), NaNO2(l), NaNO3(a) NaNO3(b) and NaNO3(l)ere considered for the equilibrium calculation. The letters in thearentheses for condensed phase species indicate their physicalorm: l = liquid; cr = crystalline; a, b, and c = various solid forms. In

Fig. 1. The variation of (a) temperature and (b) mass fractions of Na with mixturefraction.

order to compare the results obtained from the ANSYS FLUENT,separate calculations were performed using the NASA CEA code(Gordon and McBride, 1994). The NASA CEA code calculates the adi-abatic flame temperature and equilibrium composition by using theGibb’s free energy minimization (Turns, 2000) procedure based onthe element potential method. The NASA CEA code is used here forcomparison as this code is the benchmark code for equilibrium cal-culations Glassmann and Yetter (1996). The results obtained usingthe ANSYS FLUENT code were compared with the NASA CEA code.Figs. 1 and 2 show the variation of temperature and the mass frac-tion of Na, Na2O and Na2O2 with the mixture fraction, respectively.Fig. 3 shows the variation of mass fraction of NaO and O2 with mix-ture fraction. Similarity of the results between these codes verifiesthat the ANSYS FLUENT results are accurate. Hence, the equilibriumPDF table generated using ANSYS FLUENT can be used for furthersimulations of sodium spray combustion. It is worth mentioningthat it is difficult to import a NASA CEA equilibrium solution toANSYS FLUENT. That is why we have made sure the ANSYS FLU-ENT results are accurate and can be used for further simulations ofsodium combustion.

2.3. Turbulence model

The closure for Reynolds stress term in Eq. (2) was achievedwith the standard k − ε turbulence model. Within this model, thefollowing equations for turbulent kinetic energy, k and turbulentdissipation rate, ε are solved:

∂�k

∂t+ ∂�uik

∂xi

= ∂∂xj

(� + �t

�k

) ∂k

∂xj

+ Gk + Gb − �ε (14)

and


∂�ε

∂t+ ∂�uiε

∂xi

= ∂∂xi

[(� + �t

�ε

) ∂ε

∂xi

]+ ε

k(C1Gk + C1C3Gb − C2�ε)

(15)



ARTICLE ING Model


6 P. Sathiah, F. Roelofs / Nuclear Engineerin

0.1 0.2 0.3 0.4 0.5 0.6 0.710

−6

10−4

10−2

100


Mas

s F

ract

ions

[−]


(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.710

−6

10−4

10−2

100


Mas

s F

ract

ions

[−]


(b)

Fig. 2. The variation of mass fraction of (a) Na2O and (b) Na2O2 with mixture frac-tion.

0.1 0.2 0.3 0.4 0.5 0.6 0.710

−6

10−4

10−2

100


Mas

s F

ract

ions

[−]


(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.710

−6

10−4

10−2

100


Mas

s F

ract

ions

[−]


(b)

Fig. 3. The variation of mass fraction of (a) NaO and (b) O2 with mixture fraction.


where C1, C2 and C3 are model constants. Here, k and ε are Favre-averaged quantities, for simplicity˜is ignored. The production term,Gk in Eqs. (14) and (15) is given by

Gk = �˜u′′iu′′

j

∂ui

∂xj

=[

�t

(∂ui

∂xj

+ ∂uj

∂xi

− 23

∂uk

∂xk

ıij

)+ 2

3�k

]∂ui

∂xj(16)

where, the turbulent viscosity �t is evaluated as follows:

�t = �C�k2

ε(17)

where, C� is a constant.The total enthalpy flux in Eq. (5) and the mixture fraction flux in

Eq. (3) are closed using the classical gradient assumption as follows:

−�˜u′′iH′′ = �t

Prt

∂H

∂xi

and − �˜u′′if ′′ = �t

�t

∂f

∂xi

. (18)

Here, Prt, �t are the turbulence Prandtl number and Schmidtnumber. The mixture fraction variance flux, its production and dis-sipation is closed as follows:

−∂�˜

u′′if ′′2

∂xi

= ∂∂xi

(�t

�t

∂f ′′2

∂xi

), (19)

2�u′′if ′′ ∂f

∂xi

= −Cg�t

(∂f

∂xi

)2

(20)

and

2�D∂ ˜f ′′

∂xi

∂ ˜f ′′

∂xi

= Cd�ε

kf ′′2. (21)

The default values of the constants Prt, �t, Cg and Cd are0.85, 0.85, 2.86, and 2.0, respectively (ANSYS, 2008; Poinsot andVeynante, 2005). The effect of buoyancy on the turbulent kineticenergy and the turbulent dissipation rate is given by underlinedterms in Eqs. (14) and (15). The term Gb is given as follows:

Gb = �′u′igi = −gi

�

�t

Prt

∂�

∂xi

(22)

where �′, u′i

and gi are fluctuations in density, velocity and thegravitational acceleration, Prt is the turbulent Prandtl number.

2.4. Modeling of sodium spray burning/evaporation inLagrangian phase

The models used for sodium spray evaporation are describedin this section. Firstly, the equations for the position, the velocity,the mass and the temperature of the droplets are described. Sec-ondly, a spray evaporation/surface reaction models for pre-ignitionand post-ignition phases are discussed. Three sub models for pre-ignition phenomena are described in detail, which are based onmass transfer, reaction kinetics and combined mass transfer andreaction kinetics approaches.

In this approach, the droplets are assumed as discrete particleswith a spherical shape and are tracked in a Lagrangian methodwithin the code. The particle motion is obtained by integratingthe force (drag and gravity) balance on the particle. Other forcesfor example Brownian, lift and rotational forces are neglected. The


integration leads to the following equation:

dup

dt= FD(u − up) + gi(�p − �)

�p(23)


IN PRESSG Model

N


ff

F

wa

C

R

wudi

w

oibmigm

ptbit

a

m

Hopoemaivcmibhttlr

Fig. 4. Schematic of pre-ignition phenomena. The red color denotes droplet withhigh temperature while, blue droplet denotes low droplet temperature. (For inter-



or the velocity of the particle. Here, the term FD(u − up) is the dragorce per unit particle mass and is given as follows:

D = 18�

�pD2

CdRe

24(24)

here, Cd is the drag coefficient for sodium (Tsai, 1979) and is givens:

d =

⎧⎪⎪⎨⎪⎪⎩24/Re if Re < 0.1,

2.6 + 23.71/Re if 0.1 < Re < 6,

18.5/Re0.6 if 6 < Re < 500,

4/9 if Re > 500.

(25)

Here, Rep is the Reynolds number, defined as

ep = �pD|up − u|�

(26)

here, �p is the density of the droplet, up is the droplet velocity,˜ is the Favre-averaged gas velocity and D is the diameter of theroplet. The droplet position is obtained by integrating the follow-

ng equation:

dxp

dt= up (27)

here xp is the position of the droplet.Turbulent dispersion of the droplets, i.e. the effects of turbulence

n the droplet is taken into account. In this approach, the effect ofnstantaneous velocity on the droplet motion is taken into accounty using a stochastic model (Gosman and Ioannides, 1983). Further-ore, the effects of droplets on the turbulence and droplet–droplet

nteraction are not considered in this work. For the present sin-le droplet validation, this is not important. Further details of theodel are described in ANSYS (2008).The liquid sodium droplet undergoes surface reaction and vapor

hase combustion depending on droplet temperature. To calculatehe mass burning rate during surface reaction and vapor phase com-ustion, it is essential to track the mass and temperature of the

ndividual droplet. Therefore, the equations for the mass and theemperature of the droplet must be solved, which are given below:

dmd

dt= md (28)

nd

dCpdTp

dt= hAp(T∞ − Tp) − dmd

dthfg + dmd

dtHc + εpAp�(4 − T4

p ).

(29)

ere, md is the mass of the droplet, md is the mass burning ratef the droplet, Cp is the specific heat of the droplet at a constantressure, AP is the area of the droplet, T∞ is the local temperaturef the gas phase, h is the heat transfer coefficient, εp is the dropletmissivity, is the radiation temperature, � is the Stefan Boltz-ann constant and hfg is the enthalpy of vaporisation. Hc is the

mount of heat generated during the surface reaction of sodiumn J/kg. It is worth mentioning that Eq. (29) ignores the spatialariation of the droplet temperature, because of the high thermalonductivity of liquid sodium (i.e. Biot number (hL/k) « 0.1). Thiseans that the droplet temperature is spatially uniform, but vary-

ng with time. Furthermore, Eq. (29) describes the heat exchangeetween the droplet and the gas phase through convection, latenteat and radiation, which are given by the first, second and fourth


erm, respectively. The third term describes the heat transfer tohe droplet due to the surface reaction. The models for the massoss rate of the droplet md, due to the droplet evaporation/surfaceeaction are described in the next subsection.

pretation of the references to color in this figure legend, the reader is referred to theweb version of the article.)

2.4.1. Spray evaporation/burning modelTsai (1979) suggested that the combustion of the sodium

droplet can be divided into two phases namely pre-ignition andpost-ignition. When the droplet temperature is less than igni-tion temperature, the droplet undergoes pre-ignition phenomena.Once the droplet temperature is equal to the ignition tempera-ture, the droplet starts to evaporate, and vapor phase combustion isinitiated. The corresponding submodels for pre-ignition and post-ignition phases are explained below.

1. Pre-ignition modelThe pre-ignition stage is important and must be accounted in

the modeling of sodium spray combustion. Existence of the pre-ignition phase is also confirmed experimentally by Yuasa (1984),Richard et al. (1968), Makino and Fukada (2005b). In the pre-ignition phase, a film of oxide is formed on the droplet surface bythe surface oxidation. The heat produced by the oxidation is fedback to the droplet surface, thereby increasing the droplet tem-perature. The phenomena are schematically shown in Fig. 4. Thedroplet ignition starts when the droplet ignition temperature isreached.

In sodium spray combustion modeling literature, two differ-ent models based on namely (a) the mass transfer and (b) thereaction-kinetics approaches are available to model pre-ignition.In addition to these approaches, (c) combined mass-transfer andreaction kinetics approaches will be described.

(a) Mass transfer based modelThis model is suggested by Tsai (1979) and is available in the

NACOM code. The model invokes the following assumptions:• the oxide film formed on the droplet surface is thin and does


not stop diffusion of oxygen to the droplet surface,• the change of droplet diameter during the surface reac-

tion/oxidation is negligible,


IN PRESSG Model

N

8 ineering and Design xxx (2014) xxx– xxx



• the heat transfer from the sodium droplet to the gas phasecan be neglected, because of high thermal conductivity ofthe liquid sodium,

• the oxygen concentration at the droplet surface is zero, thismeans that the sodium-air reaction is infinitely fast and

• viscous forces within the liquid droplets are neglected.During the surface reaction, sodium reacts with oxygen to

form sodium oxide Na2O, which is a single step reaction givenas follows:

4Na(l) + O2 → 2Na2O(s) Hc = 418 kJ/mol Na. (30)

In this model, the burning of the sodium droplet is propor-tional to the consumption rate of oxygen at the droplet surfacewhich in turn is proportional to the rate of diffusion of oxygento the droplet surface from the surroundings. Then, the massburning rate md, of the fuel droplet is given by:

md = NO2

rApWO2 . (31)

Here, r is the stoichiometric ratio, NO2 is the molar flux ofoxygen into the sodium surface, and WO2 is the molecularweight of the oxygen species. The molar flux of oxygen canbe evaluated as follows:

NO2 = kc(˜YO2,∞ − ˜YO2,s) = kc˜YO2,∞ (32)

where, kc is the mass transfer coefficient, ˜YO2,∞ is the mass

fraction of oxygen in the surroundings, and ˜YO2,s is the massfraction of oxygen at the droplet surface which is assumedto be zero, since sodium-oxygen reaction is assumed tobe infinitely fast. The mass transfer coefficient kc, is evalu-ated using the Ranz–Marshall correlation (Ranz and Marshall,1952) for free falling droplets as follows:

Sh = kcD

Dd= 2 + 0.6Re1/2Sc1/3. (33)

Here, Sh is the Sherwood number, Dd is the diffusion coef-ficient and Sc is the Schmidt number. The above correlationtakes into account effects of forced convection on the massburning rate, which increases due to the change in the shapeof the droplet as it moves in a convective environment. Afterrearranging, the mass burning rate of the fuel droplet is givenby:

md = �WO2 DdDSh

(˜YO2,∞

r

)(34)

where, � is the mean gas density.(b) Reaction kinetics based model

Makino (2003) highlighted that the reaction kinetics of sur-face oxidation is the rate controlling step in the pre-ignitionphase. It is assumed that droplet temperature is spatiallyuniform but temporally varying because of the high ther-mal conductivity of sodium as shown in Fig. 5. In addition,it is assumed that sodium and air react at the droplet surface(see Eq. (35)). He proposed a model to simulate pre-ignitionphenomena and suggested the following expression:

md = D2�f Wf

(�˜YO2,∞

WO2

)ns

Bsexp−Es/RTp (35)

to calculate the mass burning rate. Here, �f,Wf are the stoichi-ometric coefficient, molecular weight of the fuel and R is the


gas constant. The frequency factor, Bs is 320 m/s, the activa-tion energy, Es/R is 6350 K and the reaction order ns = 1 (seeEq. (30)). These values are obtained by fitting the experimen-tal data (Makino, 2003). Makino (2003), Alao et al. (2011)

Fig. 5. Schematic of pre-ignition phenomena.

theoretically obtained the ignition time delay of a sodiumdroplet using this model and compared it with experiments.The variation of the ignition time delay with the initial droplettemperature, the oxygen concentration, the droplet diameterand the Nusselt number were predicted well in their studyusing this model.

Recently, Saravanan et al. (2011) numerically calculated theignition time delay of a sodium droplet using two differentpre-ignition models namely a mass transfer based model, i.e.Eq. (34) and a reaction kinetics based model, i.e. Eq. (35). Thevariation of the ignition time delay with the initial droplettemperature, the diameter, the oxygen concentration and theNusselt number were compared. It was concluded in theirstudy that the reaction kinetics based model predicts the limitof ignitability for different initial droplet diameter, droplettemperature and Nusselt number. Whereas the mass transferbased model could predict it under very low oxygen concen-tration.

Hence, it can be concluded that pre-ignition models basedon reaction kinetics and mass transfer based approaches areboth essential to describe the limit of ignitability under dif-ferent conditions.

(c) Combined (diffusion/kinetics limited) modelA model which combines both mass transfer and reaction

kinetics based approaches is proposed here by us. The modelenables us to describe the limit of ignitability under all condi-tions. The mass burning rate in this model is given as follows:

md = D2 �˜YO2,∞r

(DiRi

Di + Ri

)= D2 �˜YO2,∞

r�i (36)

where1�i

= 1Di

+ 1Ri

= Di + Ri

DiRi. (37)

Here, Di and Ri are obtained from the mass transfer (Eq.(34)) and the reaction kinetics based (Eq. (35)) model given asfollows:

Di = DdSh

Dand Ri = Bsexp−Es/RTp . (38)

A Damkohler number Da can be defined, which is the ratio


of kinetic rate and diffusion rate and is given as follows:

Da = Ri

Di= DBsexp−Es/RTp

DdSh. (39)




P. Sathiah, F. Roelofs / Nuclear Engineering and Design xxx (2014) xxx– xxx 9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

3

Oxygen Mass Fraction [%]

Igni

tion

Tim

e D

elay

[s]

773 K723 K693 K

Fig. 6. The variation of ignition time delay with oxygen mass fraction for three dif-fco

Fdtr

0 2 4 6 8x 10

−3

0

0.5

1

1.5

2

Initial Droplet Diameter [m]

Igni

tion

Tim

e D

elay

[s] Nu−8

Nu−14Nu−17

Fig. 8. The ignition time delay with inlet droplet diameter for three different Nusselt

from the vapor phase combustion theory (D2-law) of Spalding(1955). This theory is well established for spray combustionof hydrocarbons. However, it can also be applied for sodium

2

erent initial temperatures of 773 K, 723 K and 693 K. Solid line represent theoreticalurve of Makino and Fukada (2005a) and dashed line indicate simulation resultsbtained using combined pre-ignition model (Eq. (36)).

When the kinetics rates are large compared to the diffusionrate (i.e. Di � Ri or Da � 1), then �i = Di, when the diffusionrates are large compared to the kinetic rate (i.e. Ri � Di orDa � 1), then �i = Ri.

The above equation has an analogy with electrical circuits.It is worth mentioning that Eqs. (36)–(38) describe both themajor rate controlling process of oxygen diffusion and thesodium surface oxidation, respectively. The model is also usedto describe the oxidation of a char particle (Glassmann andYetter, 1996; ANSYS, 2008; Baum and Street, 1971). Now, themodel can describe the limit of ignitability in low oxygen con-centration and different initial and convective conditions.

To demonstrate the capability of this model to describe thelimit of ignitability of a sodium droplet, numerical simula-tions (as performed by Saravanan et al. (2011)) are performedto calculate the ignition time delay. For these simulations,a droplet diameter of 1.6 mm, an air temperature of 300 Kand a Nusselt number of 8 are used. The ignition time delayis defined as the time a droplet takes to reach its ignitiontemperature from an initial droplet temperature. The igni-tion temperature is assumed to be 1156 K. The variationsof ignition time delay with initial droplet diameter, droplettemperature, oxygen concentration and Nusselt number areshown in Fig. 6–9. The solid lines indicate the analytical solu-


tions obtained by Makino and Fukada (2005a) and the dashedlines correspond to simulation results obtained using thecombined pre-ignition model (see Eq. (36)). It is worth men-tioning that differences in ignition time delay values obtained

500 600 700 800 9000

0.5

1

1.5

2

Initial Temperature [K]

Igni

tion

Tim

e D

elay

[s] 0.43

0.230.11

ig. 7. The variation of ignition time delay with initial droplet temperature for threeifferent oxygen concentrations of 0.43, 0.23 and 0.11 (by vol.%). Solid line representheoretical curve of Makino and Fukada (2005a) and dashed line indicate simulationesults obtained using combined pre-ignition model.

numbers of 8, 14 and 17. Solid line represent theoretical curve of Makino and Fukada(2005a) and dashed line indicate simulation results obtained using combined pre-ignition model.

by our simulation and analytical solution are due to the useof a different definition to calculate ignition time delay. How-ever, the trends in ignition time delay obtained by Makinoand Fukada (2005a) in their experiments and analytical solu-tions, and our simulations are in agreement. In particular,analytical results of Makino and Fukada (2005a) show that theignitable range expands with increase in droplet temperatureand oxygen concentration and decrease in Nusselt number.These trends are reproduced well by our simulations.

The combined model proposed here will be used for ourvalidation.

Once the mass burning rate is known, the droplet tem-perature is obtained by solving Eq. (29). Since there is noevaporation during the pre-ignition phase, the second term inEq. (29) (i.e. the term with hfg) is neglected, while the fourthterm is considered to simulate heat exchange due to the sur-face reaction of the droplet. The final form of Eq. (29) solvedin the pre-ignition phase is given by:

mdCpdTp

dt= hAp(T∞ − Tp) + dmd

dtHc + εpAp�(4 − T4

p ). (40)

The post ignition model is described next.2. Post-ignition model

Once the droplet ignites, it undergoes a post-ignition stage.The mass burning rate in the post ignition stage is obtained


combustion, since a sodium droplet also follows the D -law

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

1.2

1.4

Relative Speed [m/s]

Igni

tion

Tim

e D

elay

[s]

653 K693 K723 K

Fig. 9. Ignition time delay versus relative velocity for three different initial temper-atures of 653 K, 693 K and 723 K. Solid line represent theoretical curve of Makino andFukada (2005a) and dashed line indicate simulation results obtained using combinedpre-ignition model.


ING Model

N

1 ineerin


0 P. Sathiah, F. Roelofs / Nuclear Eng

as confirmed experimentally by Richard et al. (1968), Yuasa(1984), Makino and Fukada (2005a).

In the vapor-phase combustion theory, it is assumed that aspherical burning flame is surrounded by a stationary droplet.The evaporated fuel, when it reaches the burning flame, isconsumed by the flame instantaneously under steady stateconditions. Under this assumption, it can be postulated thatthe burning rate of the fuel is controlled by the evaporationof the fuel, which in turn is controlled by the heat transfer tothe droplet.

The mass burning rate is given by Tsai (1979):

m = �DK

4(41)

where, K is the burning rate coefficient (Takata., 2007; Takataet al., 2003; Spalding, 1955) or the evaporation constant whichis given by:

K = 8kg

Cp,g�ln(1 + B) (42)

where, kg is the gas mixture thermal conductivity, Cp,g is thegas mixture specific heat capacity and B is the transfer numberis defined as follows:

B = 1hfg

[Cp,g(Tg − Ts) + hc

˜YO2,∞r

]. (43)

Here, Tg , Ts are the gas and the surface temperature, hfg is thelatent heat of evaporation and hc is the heat of combustion.The mass burning rate of a single droplet burning under aconvective environment is obtained as follows:

md = mNu

2= kg

Cp,gDNu[ln(1 + B)]. (44)

Here, Nu is the Nusselt number which is obtained from theRanz and Marshall (1952) correlation as follows:

Nu = 2 + 0.6Re1/2p Pr1/3. (45)

Once the droplet undergoes post-ignition the droplettemperature remains constant and equal to the ignition tem-perature. The pre-ignition model is invoked when the droplettemperature is lower than the ignition temperature Tign.When the droplet temperature is higher than the ignitiontemperature, a gas phase reaction is initiated, and hence thepost-ignition model is used. Here, an ignition temperature of1156 K is assumed. This value is recommended by Tsai (1979),Takata et al. (2003), Saravanan et al. (2011), Rao et al. (2012).

The Reynolds number, Schmidt number and Prandtl num-ber used for droplet property calculation (for example in Eq.(45)) are evaluated at a film temperature which is calculatedusing Sparrows 1/3rd rule (Turns, 2000) as follows:

Tfilm = 2Tp + Tb

3(46)

where, Tp and Tb are the droplet temperature and bulk tem-perature. The gas mixture density is estimated using the idealgas equation of state and the mixture molecular weight. Thebinary diffusivity (needed in Eq. (33)) is estimated using thefollowing expression:

Dd = 0.00266e−04T3/2

PMAB1/2�AB

2�D

m2/s (47)


where MAB is the molecular weight of the mixture. In theabove expression the Lennard–Jones parameters �AB and �D

are extracted from Bruce et al. (2001). The viscosity and ther-mal conductivity are calculated using CANTERA (2011) and


fitted as a function of temperature. The specific heat capacitiesof species are fitted as piecewise polynomials in temperature.

To summarize, at every flow time step Eqs. (1)–(5) aresolved to obtain the velocity, the pressure, the mean mixturefraction, the variance of mixture fraction and the tempera-ture. At every droplet time step Eqs. (23), (27), (28) and (29)are solved to obtain the velocity, the position, the mass and thetemperature of the droplet. The mass burning rate in Eq. (28)can be obtained either by the pre-ignition model i.e. Eq. (34),(35) or (36) or the post-ignition model i.e. Eq. (44). The con-tribution of the mass, the momentum and the energy sourcesfrom the droplet, i.e. Sm, Smom and Sh are appropriately calcu-lated and added to the Eulerian phase equations.

2.5. Applied numerical scheme and numerical implementation

A second order upwind scheme and second-order implicit timestepping numerical scheme, are used for the spatial and temporaldiscretization of the flow, mixture fraction, energy and turbulenceequations were performed with. The SIMPLE method is used forpressure-velocity coupling. The discretized equations are solvedusing a segregated solver in an iterative manner. The droplet posi-tion and its trajectories are obtained by integrating the equation offorce balance, which is achieved by using a 5th order Runge–Kuttascheme (Cash and Karp, 1990). The Lagrangian particle trackingis fully coupled with the Eulerian gas flow in a Lagrangian fash-ion i.e. heat, mass and momentum exchanges are treated as thesource/sink term in the gas phase equation.

User defined functions (UDF) in the code are used to imple-ment temperature dependent sodium properties (e.g. diffusivity,enthalpy of vaporization and vapor pressure), drag laws and sprayevaporation/burning models. The droplet source terms for mass,momentum, mixture fraction and energy (Sm, Smom and Sh in Eqs.(1), (3) and (5)) are also implemented using UDFs.

3. Experiment used for validation of droplet combustion

A free falling single sodium droplet experiment conductedby Miyahara and Ara (1998) is used for the validation. In theirexperiment, a single droplet of uniform diameter 3.8 mm with atemperature of 773 K was injected into an ambient atmosphere.The experimental facility used by them is shown in Fig. 10. Thedroplet falls from a height of 2.7 m and it burns during its down-ward motion. The pressure is 1 atm and air temperature is 298 K.The air temperature and oxygen concentration are kept constantduring the experiment. The burning behavior of a single sodiumdroplet was observed. In particular, the burned mass and fallingvelocities were measured at two different locations of 0.1 m and2.4 m, respectively. The initial conditions and the measurementresults reported by them are provided in the Tables A.2 and A.3,respectively.

4. Results and discussion

The spray CFD model described above is validated against sin-gle droplet experiments of Miyahara and Ara. The results of thevalidations are reported below.

A three dimensional computational domain is used to performthe simulations. The domain height is 6 m, while the depth andwidth are 1.5 m. A longer computational domain is used to makesure that droplet completely evaporates during the drop of 6 m.


The gas phase is solved in an Eulerian phase, using a standardk-ε turbulence model and non-premixed mixture fraction basedcombustion model. The effects of buoyancy on velocity, turbulentkinetic energy and turbulent dissipation rate are taken into account.






Fig. 10. Experimental set-up used by Miyahara and Ara (1998).The figure is taken from Ohno et al. (2012).

0 0.2 0.4 0.6 0.8 1700

800

900

1000

1100

1200

Time [s]

Dro

plet

Tem

pera

ture

[K]

123

(a)

0 0.2 0.4 0.6 0.8 13.6

3.65

3.7

3.75

3.8

3.85

3.9

Time [s]

Dro

plet

Dia

met

er [m

m]

123

(b)

Fig. 11. The variation of (a) droplet temperature and (b) droplet diameter dp withtime, t. Curves 1–3 are obtained using three different pre-ignition models based onmass transfer, reaction kinetics and combined mass transfer and reaction kineticsapproach.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

Time [s]

Fal

ling

Dis

tanc

e [m

]

123

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

Time [s]

Fal

ling

Vel

ocity

[m/s

]

123Experiment

(b)

Fig. 12. The variation of (a) falling distance and (b) falling velocity with time,
t. Curves 1–3 are obtained using three different pre-ignition models based onmass transfer, reaction kinetics and combined mass transfer and reaction kineticsapproach.
Unsteady simulations are performed and the droplets are trackedwithin a Lagrangian frame of reference by solving unsteady equa-tions for the position, velocity, mass and temperature of the droplet.To simulate the droplet surface reaction/evaporation, pre-ignitionand post-ignition models are used. Radiative heat transfer fromthe droplet surface to the flame and to the surrounding gases hasbeen neglected. The flow is initially assumed to be quiescent. Theboundaries of the domain are defined as walls, a no-slip and adia-batic boundary condition is applied for velocity and temperature,respectively. A single droplet with a diameter of 3.8 mm and a tem-perature of 773 K is injected at a location of 0.2 m from the top. Anon-adiabatic equilibrium PDF table is generated for a fuel temper-ature of 773 K and an air temperature of 298 K, which is then usedin the simulations. The results of the simulations are comparedwith the experiments. In addition, effects of pre-ignition modelson falling velocity and burned mass are determined. Furthermore,additional sensitivity studies are performed.

Fig. 11a shows the variation of the droplet temperature withtime. Curves 1–3 are obtained using three different pre-ignitionmodels based on mass transfer, reaction kinetics and a combinedmass transfer and reaction kinetics approach. The figure clearlyindicates that until time t = 0.3 s, the droplet temperature increasesgradually and reaches the boiling point of sodium, i.e. 1156 K.The temperature increase from initial temperature of 773–1156 Kis because of pre-ignition (surface reaction) phenomena. Oncethe droplet reaches its boiling point, the droplet temperature is


unchanged, and it undergoes a post-ignition phase wherein a vaporphase combustion is initiated.

Fig. 11b illustrates the change in droplet diameter with timeduring the pre-ignition and post-ignition phase. A slight increase




12 P. Sathiah, F. Roelofs / Nuclear Engineering and Design xxx (2014) xxx– xxx

0 0.2 0.4 0.6 0.8 117

19

21

23

25

Time [s]

Dro

plet

Mas

s [m

g]

123Experiment

(a)

0 0.2 0.4 0.6 0.8 11.3

1.35

1.4

1.45

1.5

1.55x 10−5

Time [s]

Squ

are

of D

ropl

et D

iam

eter

[m2 ]

123

(b)

Fig. 13. The variation of (a) droplet mass and (b) square of droplet diameter withtoa

ioTeadd

vitiHtiop

wf5a

oempti

300 320 340 360 380 4000.5

1

1.5

2

2.5

3

Air Temperature [K]

Bur

ning

Rat

e C

onst

ant [

mm

2 /s]

(a)

300 320 340 360 380 4005

5.5

6

6.5

Air Temperature [K]F

allin

g V

eloc

ity [m

/s]

(b)

300 320 340 360 380 400

0.1

0.2

0.3

0.4

Air Temperature [K]

Igni

tion

Tim

e D

elay

[s]

(c)

ime, t. Curves 1–3 are obtained using the three different pre-ignition models basedn mass transfer, reaction kinetics and combined mass transfer and reaction kineticspproach.

n the droplet diameter during the pre-ignition phase is becausef the increase in volume of the droplet due to thermal expansion.his trend of the increase in droplet diameter is also observed inxperiments performed by Makino and Fukada (2005a), Makinond Fukada (2005b) (see Fig. 10). In the post-ignition phase, theroplet diameter decreases linearly due to the evaporation of theroplet.

Fig. 12a and b shows the variation of falling distance and fallingelocity with time. Curves 1–3 are obtained using the three pre-gnition models. The falling distance increases exponentially withime in the beginning (t = 0–0.4 s). This is because the gravity forces dominant in comparison to the drag force in the initial phase.owever, at a later stage both these forces are equal in magni-

ude, and they counteract each other. Hence, the falling distancencreases slowly with time. It is worth noting that the variationf falling velocity and distance with time are the same for all there-ignition model.

The falling velocities obtained using simulations are comparedith experiments at falling distances of 0.1 m and 2.4 m. The

alling velocities with the three pre-ignition models are 1.6 m/s and.8 m/s, which agrees with the experimental values of 1.3 ± 0.33nd 5.5 ± 0.48 m/s.

In Fig. 13a, the results of variation of droplet mass with timebtained using numerical simulations are compared with thexperiments. Curves 1–3 are obtained using the three pre-ignition


odels. The droplet mass decreases with time because of the vaporhase combustion of the droplet. During the post-ignition phase,he droplet mass decreases linearly with time for all three pre-gnition models. The droplet mass predicted at a falling distance

Fig. 14. The variation of (a) burning rate constant, (b) falling velocity and (c) ignitiontime delay with the initial air temperature.

of 2.7 m is 20.41 mg. This corresponds to a decrease of the dropletmass of 3.59 mg which agrees well with an experimental valueof 3.37 ± 0.69 mg. It is worth noting that Okano and Yamaguchi(2003) calculated values of 1.3 and 5.4 m/s and 3.39 mg, using theCOMET code. The COMET code is based on a direct numerical simu-lation (DNS) approach. Hence, it is computationally expensive andcannot be used to simulate multiple droplet sodium spray combus-tion which is likely to occur in a real scale accident scenario. Theresults obtained using the code based on a DNS approach and ourRANS based approach are within the experimental uncertainty. Thisproves the suitability of this code to simulate spray combustion ina real accident scenario.

Fig. 13b shows the variation of the square of the droplet diame-ter with time. Curves 1–3 are obtained using the three pre-ignitionmodels. During the post-ignition phase, the droplet diameter varieslinearly with time. This essentially means that the droplet fol-lows the D2-law as shown by Richard et al. (1968), Yuasa (1984),


Makino (2003) in their experiments. The slope of the dropletdiameter with time is the burning rate constant (Tsai, 1979)which can be extracted from simulation results. The burning rate





2 3 4 5 6x 10

−3

0.5

1

1.5

2

2.5

3

Droplet Diameter [m]

Bur

ning

Rat

e C

onst

ant [

mm

2 /s]

(a)

2 3 4 5 6x 10

−3

5

5.5

6

6.5


Fal

ling

Vel

ocity

[m/s

]

(b)

2 3 4 5 6x 10

−3

0.1

0.2

0.3

0.4


Igni

tion

Tim

e D

elay

[s]

(c)

Ft

c2

iegurc

4

drtci

ar

700 750 800 850 9000.5

1

1.5

2

2.5

3

Droplet Temperature [K]

Bur

ning

Rat

e C

onst

ant [

mm

2 /s]

(a)

700 750 800 850 9005

5.5

6

6.5

Droplet Temperature [K]F

allin

g V

eloc

ity [m

/s]

(b)

700 750 800 850 900

0.1

0.2

0.3

0.4

Droplet Temperature [K]

Igni

tion

Tim

e D

elay

[s]

(c)

ig. 15. The variation of (a) burning rate constant, (b) falling velocity and (c) ignitionime delay with an initial droplet diameter.

onstants obtained for the three pre-ignition models are.03 mm2/s, 1.94 mm2/s and 1.99 mm2/s, respectively.

To summarize, the simulation results compare well with exper-ments, which indicate that the model is able to reproducexperiments. Results obtained using all pre-ignition models are inood agreement with experiments. However, it is recommended tose the pre-ignition model based on combined mass transfer andeaction kinetics based approach as this model can describe rateontrolling processes of mass transfer and surface reaction.

.1. Parametric studies

In order to investigate the effects of initial droplet diameter,roplet and air temperature and oxygen concentration on burningate, falling velocity and ignition time delay, additional simula-ions were performed. From the simulation results, the burning rateonstant and falling velocity at a falling distance of 2.4 m and the


gnition time delay were obtained.Fig. 14a–c shows the variation of burning rate, falling velocity

nd ignition time delay with initial air temperature. The burningate, falling velocity and ignition time delay increase slightly with

Fig. 16. The variation of (a) burning rate constant, (b) falling velocity and (c) ignitiontime delay with the initial droplet temperature.

the increase in air temperature. A slight increase in ignition timedelay with the increase in air temperature is because of the slightdecrease in air density. This leads to a decrease in surface reac-tion rate, which increases the heating time of the droplet and thusincreases the ignition time delay.

The variation of burning rate with an initial droplet diameteris shown in Fig. 15a. The burning rate increases three fold with athreefold increase in droplet diameter. This is due to an increasein the surface area of the droplet which leads to enhanced dropletevaporation which thus increases the burning rate constant. Thefalling velocity increases with the increase in initial droplet diam-eter as shown in Fig. 15b. This is attributed to the increase in thegravity forces due to the increase in mass of the droplet with theincrease in initial droplet diameter. A three fold increase in ignitiontime delay with an increase in initial droplet diameter is shown inFig. 15c. It is attributed to an increase in the volume–surface arearatio of the droplet with an increase in diameter of the droplet. This


increase in volume–surface area ratio of the droplet increases theheating time of the droplet due to a surface reaction and therebyincreases the ignition time delay.


ARTICLE ING Model


14 P. Sathiah, F. Roelofs / Nuclear Engineerin

10 15 20 25 300.5

1

1.5

2

2.5

3

Oxygen Concentration [%mass]

Bur

ning

Rat

e C

onst

ant [

mm

2 /s]

(a)

10 15 20 25 305

5.5

6

6.5


Fal

ling

Vel

ocity

[m/s

]

(b)

10 15 20 25 30

0.1

0.2

0.3

0.4


Igni

tion

Tim

e D

elay

[s]

(c)

Ft

tstibp

tfTafitroei

scenario has to be performed.The present method once sufficiently validated can be employed

ig. 17. The variation of (a) burning rate constant, (b) falling velocity and (c) ignitionime delay with the initial oxygen concentration.

The burning rate constant decrease slightly with the increase inhe droplet temperature as shown in Fig. 16a and b. This trend is alsoeen in the ignition time delay which decreases significantly withhe increase in the droplet temperature (see Fig. 16c). The decreasen ignition time delay with the increase in droplet temperature isecause of decrease in the heating time of the droplet during there-ignition phase.

Fig. 17a–c shows that the burning rate constant increases withhe increase in oxygen concentration, while ignition time delay andalling velocity decrease with the increase in oxygen concentration.his trend is also observed by Makino and Fukada (2005b), Makinond Fukada (2005a) in their experiments. In particular, a three-old increase in burning rate is observed for a threefold increasen initial oxygen concentration. The burning rate increases withhe increase in oxygen concentration because of the increase inate of vapor phase combustion due to the availability of more


xygen. This trend is also observed in single droplet combustionxperiments by Richard et al. (1968) and in pool combustion exper-ments by Newman and Payne (1978). Fig. 17b shows a slight


decrease in falling velocity with the increase in oxygen concen-tration. This is because of the decrease in droplet diameter andmass, which leads to decrease in the gravity force. The ignitiontime delay decreases with the increase in oxygen concentrationas shown in Fig. 17c. A threefold increase in initial oxygen concen-tration leads to a threefold decrease in ignition time delay. Thisis due to the fact that the surface reaction rate increases with theincrease in oxygen concentration because of availability of moreoxygen. This increase in surface reaction rate causes the droplet toreach its ignition temperature quickly and hence the ignition timedelay is lower.

It is concluded from the sensitivity studies, that the dropletdiameter and initial oxygen concentration are the dominantparameters affecting the burning rate constant and ignition timedelay.

5. Summary and conclusion

A method based on CFD is proposed here to model sodiumspray combustion. The method consists of solving density-averagedNavier–Stokes equation with a standard k-ε turbulence model inan Eulerian frame of reference. The effects of buoyancy on momen-tum, turbulent kinetic energy and turbulent dissipation rates aretaken into account. Furthermore, a mixture fraction based non-premixed combustion model is employed to determine the effectsof turbulence on combustion. In this model, a chemical equilib-rium approach is used, which assumes that chemical reactionsare faster than fluid flow and species diffusion. Moreover, a non-adiabatic equilibrium PDF table is generated using ANSYS FLUENTand results are compared with another code (NASA CEA). The simi-larity of the results verifies that ANSYS FLUENT results are accurateand the equilibrium PDF table can be used for the further valida-tion.

Spray droplets are tracked in a Lagrangian frame of referencewherein equations of the position, velocity, temperature and massof the droplets are solved. The surface reaction/evaporation of thedroplets is modeled using pre-ignition and post-ignition models.A pre-ignition model based on combined mass transfer and reac-tion kinetics is proposed. The trends of initial droplet temperature,diameter, relative velocity and oxygen concentration on ignitiontime delay obtained using the combined pre-ignition model arein good agreement with experiments. The post-ignition phase ismodeled using the vapor phase combustion theory of Spalding(1955). In this model, the burning rate of the fuel is controlled bythe evaporation of the fuel which in turn is controlled by the heattransfer to the droplet. The proposed method is implemented intocommercial CFD code ANSYS FLUENT using UDFs.

The developed spray model was validated against single dropletexperiments of Miyahara and Ara (1998). In particular, the fallingvelocity at falling distances of 0.1 and 2.4 m and burned mass arein good agreement with experiments.

Furthermore, parametric studies have been carried out to deter-mine the effects of droplet diameter, droplet and air temperatureand oxygen concentration on burning rate, falling velocity, ignitiontime delay. It can be concluded that droplet diameter and initialoxygen concentration are the dominant parameters which has sig-nificant influence on burning rate constant and ignition time delay.This essentially means that the size distribution of the dropletsemanating from a pipe crack will play a larger role when accuratesimulations of sodium spray combustion in a real scale accident


to simulate sodium spray combustion in a real scale accident sce-nario. However, additional validation for experimental data whichis currently not available in open literature is recommended.


IN PRESSG Model

N


A

CMe

A

o(t

�

f

wg

�

w

H

f

S

l

f

u

C

f

v

C

f

L

Table A.1Lennard Jones potential parameters.

Species ε/k � (×10−10 m)

Na 712.28 1.192O2 106.7 3.46N2 71.4 3.79

Table A.2Initial conditions of free falling droplet experiments (Miyahara and Ara, 1998).

Phase Variable Value

Droplet Diameter 3.8 mmTemperature 775 K

Air Temperature 290 KPressure 1 atm

Table A.3Results of free falling droplet experiment (Miyahara and Ara, 1998).



cknowledgments

The work presented here was funded by the FP7 Europeanommission Collaborative Project ESFR No. 232658, and the Dutchinistry of Economic Affairs. ANSYS FLUENT customer support

specially Dr. Ren Liu for this work is highly appreciated.

ppendix A. Physical properties of sodium

The physical properties of sodium are described. Recently, datan sodium liquid and vapor are reviewed by Fink and Leibowitz.1995) in their report. Following correlations were reported byhem for various physical properties.

Liquid density (kg m−3):

= �c + f(

1 − T

Tc

)+ g(

1 − T

Tc

)h

(A.1)

or 371 K ≤ T ≤ 2503.7 K

here, critical density �c = 219 kg m3, constants f = 275.32, = 511.58, h = 0.5 and critical temperature, Tc = 2503.7 K.

Surface tension (Nm−1):

= �o

(1 − T

Tc

)m

(A.2)

here, �o = 240.5 and m = 1.126 are the constants.Enthalpy of vaporization (J/kg):

fg = 393.37(

1 − T

Tc

)+ 4398.6

(1 − T

Tc

)0.29302(A.3)

or 371 K ≤ T ≤ 2503.7 K.

aturation pressure (MPa):

n(p) = 11.9463 − 12, 633.73T

− 0.4672ln(T) (A.4)

or 864 K ≤ T ≤ 2499 K.

Specific heat capacity at constant pressure (kJ kg−1 K−1) for liq-id sodium:

p = 8.8556 × 10−18T6 − 6.9168 × 10−14T5 + 2.1333 × 10−10T4

− 3.2958 × 10−7T3 − 2.667 × 10−4T2 − 0.10636T + 17.5167.

(A.5)

or 400 K ≤ T ≤ 2400 K.

Specific heat capacity at constant pressure (kJ/kg K) for sodiumapor:

p = 9.8922 × 10−18T6 − 7.7734 × 10−14T5 + 2.4188 × 10−10T4

− 3.7555 × 10−7T3 + 2.9948 × 10−4T2 − 0.11133T + 16.105.

(A.6)

or 400 K ≤ T ≤ 2400 K


The above polynomials are obtained by fitting the data (Fink andeibowitz., 1995) into 6th order polynomial.

Thermal conductivity (J m−1s−1 K−1):

Droplet velocity Falling distance at 0.1 m 1.3 ± 0.33 m/sFalling distance at 2.4 m 5.5 ± 0.48 m/s

Droplet burnt mass Falling distance at 2.7 m 3.37 ± 0.69 mg

� = 124.67 − 0.11381T + 5.5226 × 10−5T2 − 1.1842 × 10−8T3

for 373 K ≤ T ≤ 1500 K. (A.7)

Okano and Yamaguchi (1999) provided expression to calculate dif-fusion coefficients (m2/s) of sodium vapor in nitrogen

DNa,N2 = 1.8583 × 10−7

√T3(1/MNa + 1/MN2 )

P�2NaN2

˝NaN2

(A.8)

where, MNa and MN2 are the molecular weight of Na and N2 species,respectively. Here, collision integral ˝NaN2 is given as

˝NaN2 = f

( BT

εNaN2

)(A.9)

where, εNaN2 and �NaN2 are given by

εNaN2 =√

εNaεN2 and �NaN2 = �Na + �N2

2. (A.10)

The values of �Na, �N2 , εNa and εN2 are provided in Tables A.1–A.3and are extracted from (Okano and Yamaguchi, 2003), while thevalue of collision integral is extracted from Bird et al. (2002).Rewriting Eq. (A.8), we obtain

DNa,N2 = 8.396 × 10−6 T3/2

Pf (T/225.51). (A.11)

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numerical modeling of sodium fire—part i: spray combustion

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