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Modeling of quasi-steady sodium droplet combustion in convective environment
International Journal of Heat and Mass Transfer 55 (2012) 734–743
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Modeling of quasi-steady sodium droplet combustion in convective environment P. Mangarjuna Rao a, V. Raghavan b,⇑, K. Velusamy a, T. Sundararajan b, U.S.P. Shet b a b
Indira Gandhi Center for Atomic Research, Department of Atomic Energy, Kalpakkam 603 102, India Thermodynamics and Combustion Engineering Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
a r t i c l e
i n f o
Article history: Received 14 October 2010 Received in revised form 25 May 2011 Accepted 9 September 2011 Available online 9 November 2011 Keywords: LMFBR Sodium droplet Droplet combustion Flame shape Flame stand-off distance d2-law Burning rate constant
a b s t r a c t In the event of accidental leakage of sodium from the systems of a Liquid Metal Fast Breeder Reactor (LMFBR), a spray of liquid sodium droplets may be formed which will burn by reacting with the surrounding atmospheric oxygen. In order to understand the burning characteristics of the complete spray, combustion of an individual sodium droplet forms the basis and this has been investigated in the present study. A comprehensive numerical model has been developed to analyze the isolated sodium droplet combustion in a mixed convective environment. The governing equations for mass, momentum, species and energy conservation have been solved in axisymmetric cylindrical coordinates using the Finite Volume Method (FVM). Finite rate kinetic mechanisms have been incorporated to simulate droplet burning, using available kinetics data for basic sodium oxidation reactions. Salient features of the numerical model include a global single-step reaction for sodium oxidation in air and the incorporation of property variations with temperature and concentration. An equilibrium mixture of sodium peroxide, sodium monoxide and sodium vapor is considered as the final reaction product, taking into account the effects of dissociation reactions. The numerical model has been validated with experimental results available in literature. Results for the fuel mass burning rates and flame shapes are presented for different sizes of droplets burning under different free-stream conditions. The model predicts the occurrence of envelope flame around a sodium droplet even at fairly high free-stream velocities. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Liquid sodium is widely used as a coolant to rapidly transfer heat from the reactor core to the steam generator of LMFBR because of its high thermal conductivity, high boiling point and low neutron absorption coefficient. However, in case of any accidental leakage, sodium can react rapidly with oxygen and water vapor available in the atmosphere, especially when its temperature exceeds a value of about 473 K. Depending upon the temperature and pressure of the system and also based on the size, shape and location of the leak, a sodium spray fire could be initiated. Analysis of sodium spray fire and the resultant consequences is an important aspect of the safety procedure associated with LMFBR. Generally spray fires are considered to be more severe than pool fires, with higher burning rate of sodium in droplet form due to larger total exposed surface area, than in the pool configuration. Extensive research work has been carried out for understanding droplet and spray combustion aspects of hydrocarbon fuels and many review papers are also available [1–6]. On the other hand, literature on the study of isolated sodium droplet combustion is
⇑ Corresponding author. Tel.: +91 44 22574712. E-mail address: [email protected] (V. Raghavan). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.10.036
limited; only a few theoretical studies [7–9] have been reported which deal with the fundamental combustion behavior of a sodium droplet in convective air environment. Krolikowski et al. [7] developed a mathematical model for the combustion of a spherical sodium droplet in air, based on the assumption that droplet burning rate is controlled by the diffusion of oxygen towards the reaction zone. This model has been extended to predict the quasi-steady burning rates of moving droplets. Furthermore, the spray burning rate and the resultant pressure rise in an enclosed volume have been simulated, and compared with data obtained from spray combustion experiments. Sagawa et al. [8] theoretically analyzed the combustion of a single sodium particle in convective air stream. Their model predicted that the rate of decrease in the square of particle diameter is about 0.1 mm2/s, which is one order of magnitude smaller than the values observed in hydrocarbon fuel droplets. Okano and Yamaguchi [9] investigated the combustion of a sodium droplet in forced convective air flow using a CFD code called COMET. Sodium oxidation reactions were considered to be infinitely fast and the product composition was evaluated based on chemical equilibrium. The droplet burning rate was determined using the convective heat, mass transfer correlation of Ranz and Marshall [10]. This model predicted the predominant product to be sodium monoxide. The first experimental study on the burning of an isolated sodium droplet was carried out by Richard et al. [11], in a
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Nomenclature Cp D g k l n p T t u, v Vcell X x, r Y Dt
specific heat (J/kg K) mass diffusivity (m2/s) acceleration due to gravity (m/s2) thermal conductivity (W/m K) length of the side (m) normal vector pressure (N/m2) temperature (K) time (s) axial, radial velocity components (m/s) volume of the cell (m3) mole fraction axial and radial coordinates mass fraction time step (s)
convective field with a free-stream velocity of 0.01 m/s. It was shown that sodium droplet combustion follows the classical d2-law. The flame radius was observed to be around 1.3 times the droplet radius. Morewitz et al. [12] carried out experiments on freely falling sodium droplets with diameters ranging from 3 mm to 9 mm. The droplets were allowed to fall from an elevation of 14.7 m, in a column of air. It was observed that droplets larger than 7.3 mm initial diameter break down further to an approximate size of 2 mm diameter, due to interfacial instabilities [13]. Droplet burning rate and the fraction of droplet mass burned were evaluated as functions of the initial droplet diameter. Makino and Fukada [14] estimated the burning rate constant of a sodium droplet and its flame size by conducting falling droplet experiments in a vertical chamber. The burning rate constant was reported as 1.0 mm2/s for a sodium droplet with an initial diameter of 2 mm, burning in quiescent atmospheric air. When a forced convection environment was introduced (relative air velocity of 2.6 m/s), the burning rate constant increased to a value of 3.4 mm2/s. It was also shown that sodium droplet combustion resembles the burning of hydrocarbon droplets in an oxidizer-rich environment. In the published literature, mass burning rate data of sodium droplets under different convective conditions and different sizes, is scarce. This data is very essential for the numerical analysis of sodium spray fire. The existing spray fire safety analysis codes such as SPRAY [15] and NACOM [16] were developed based on one-dimensional macroscopic models and experimental results, and they often use limited empirical correlations to approximately estimate the burning rate of a single droplet. The available theoretical models, on the other hand, do not account for the detailed heat and mass transfer phenomena around the droplet as well as the reaction kinetics including the effects of product dissociation. Results have also not been presented for the mass burning rate and flame shape, for a wide range of convective conditions and droplet sizes. The present research work attempts to develop a detailed numerical model to study isolated sodium droplet combustion in a mixed convective environment. Since buoyancy-driven convection is known to enhance the burning rate significantly in the case of large fuel droplets [17], mixed convective flow has been considered. A single-step global reaction with finite rate chemistry and product dissociation has been incorporated to simulate the rapid sodium combustion reactions. The model is able to predict the salient features of sodium droplet combustion such as smaller flame heights and stand-off distances, and the absence of partial flame extinction at the front portion of the droplet even at fairly high air velocities.
Greek symbols density (kg/m3) coefficient of viscosity (N s/m2) normal and shear stress fields (N/m2) / mixture fraction x_ i mass based reaction rate (kg/m3 s)
q l r, s
Subscripts i species i m gas mixture s droplet surface 1 free-stream conditions
2. Salient features of sodium droplet combustion Liquid sodium combustion exhibits features which are vastly different from that of a hydrocarbon fuel. For instance, the ignition temperature of liquid sodium is defined as the temperature at which it oxidizes rapidly enough to produce a self-generated temperature rise sustaining liquid evaporation and combustion subsequently. The range of ignition temperature for liquid sodium is 393–593 K [18], which is dependent on the mode of sodium combustion (pool or spray) and the oxygen content in the surrounding atmosphere. Also, ignition depends on the existence of a protective layer formed over the liquid surface during its oxidation. Ignition temperature is the lowest in the case of a liquid sodium droplet (393–423 K). The range of temperatures at which liquid sodium exists in reactor systems is fairly higher than this ignition temperature. Generally, liquid sodium burning exhibits surface combustion as well as the vapor-phase combustion. Once ignited, at lower temperatures (<723 K), flameless surface oxidation of sodium occurs due to its high chemical reactivity with oxygen. Since sodium is more volatile than its oxides, the surface oxidation will eventually lead to vapor-phase combustion as the bulk temperature of the liquid also increases rapidly. Liquid sodium, however, is not as volatile as hydrocarbon fuels. For instance, the boiling point of sodium at 1 atm pressure is 1156 K and its latent heat of vaporization is very high, around 3886 kJ/kg. For gasoline, the boiling point at 1 atm pressure is 353 K and the latent heat of vaporization is approximately twelve times smaller than that of sodium. The heat of combustion of sodium is around four times less than that of gasoline. Compared to hydrocarbon fuels, the mass based stoichiometric ratio between oxidizer and sodium is also low. Therefore, with properties such as large latent heat of vaporization, high boiling point, low heat of combustion, low stoichiometric air–fuel ratio and very high chemical reactivity with oxygen in the air, sodium combustion will generate reaction zones much closer to the liquid surface. In this study, an attempt has been made to capture these salient features of sodium droplet combustion using a detailed numerical model. Although sodium also reacts with the water vapor in the atmosphere, the contribution from this mechanism has been neglected due to the low concentration of water vapor (0.2% by mass) in atmospheric air.
3. Numerical model Combustion of a liquid sodium droplet involves multi-phase, transient, mass, momentum and energy transfer processes with
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complex chemical reactions. A strong coupling exists between the mass, momentum and heat transfer processes, in the liquid- and vapor-phases. During the rapid surface oxidation of liquid sodium, the surface temperature of the droplet increases quickly. Because of the high thermal conductivity of liquid sodium and the sheardriven liquid circulation within the droplet, the bulk temperature of the droplet also increases rapidly. At the end of ignition transients, which will be around 10% of the entire droplet lifetime, a steady vapor-phase sodium droplet combustion begins and the surface temperature reaches a value close to the boiling point temperature of sodium (1156 K) [18]. Thereafter, the heat transferred from the flame to droplet surface is mainly used for vaporizing the sodium and the burning rate is dictated by diffusion, convection and to some extent the rate of the reaction. This regime, called as the quasi-steady burning regime, has been numerically modeled in this study. A numerical model, which solves a set of transient governing equations in axisymmetric cylindrical polar coordinates, has been developed. Governing equations describing the steady, constant pressure burning of a spherical liquid sodium droplet, placed in a laminar free-stream flow of atmospheric air, have been solved using the Finite Volume Method. A global reaction step with finite reaction rate, described by 3-parameter Arrhenius type of rate equation, has been employed. Variable thermo-physical properties have been evaluated as functions of temperature using appropriate correlations. Ideal gas mixture rules are used to evaluate the properties for the gas-mixture. As discussed earlier, the temperature of the entire droplet is assumed to be spatially uniform during the quasi-steady combustion phase. Therefore, only the gas-phase transport equations are modeled in a decoupled way, with the simplification that there is no liquid phase heating. Radiative heat transfer from the flame to the droplet surface as well as to the surrounding gas medium has been neglected in comparison with the other modes of heat transfer. This is in view of the small emissivity value for the highly reflective droplet surface and the small absorption coefficient of the gas medium. Only spherical drops are considered here due to the small sizes involved. The governing equations for the gas-phase processes in conservative form subject to the above assumptions are given as follows: 3.1. x-momentum equation
@ @ 1 @ ðquÞ þ ðqu2 Þ þ ðr quv Þ @t @x r @r @ 1 @ q ¼ ðrxx Þ þ ðrsxr Þ g 1 q1 @x r @r
ð1Þ
@ @ 1 @ @ 1 @ rhh ðqv Þ þ ðquv Þ þ ðr qv 2 Þ ¼ ðsxr Þ þ ðr rrr Þ @t @x r @r @x r @r r The stress terms in the above equations can be written as:
v
rrr ¼ p þ 2l
rhh ¼ p þ 2l ; srx ¼ sxr ¼ l r
@v ; @r
@u @ v þ @r @x
q @u Dt
@x
þ
1 @ ðr v Þ r @r
ð2Þ
ð3Þ
After evaluating the pressure correction, p0 , the pressure field is corrected using the relation, pn+1 = p⁄ + p0 . The corrected velocity field is obtained using the gradient of pressure correction as follows:
u0 ¼
Dt @p0 ; q @x
v0 ¼
Dt @p0 ; q @r
unþ1 ¼ u þ u0 ;
v nþ1 ¼ v þ v 0
3.3. Finite rate chemistry model – evaluation of reaction rate Sodium vapor reacts with atmospheric oxygen to produce its oxides such as sodium monoxide (Na2O) and sodium peroxide (Na2O2), which are the main reaction products in sodium combustion. These oxides are released as aerosols (generally in condensed solid/liquid phase), which are very small in size. The oxide particles therefore, offer no resistance and they are easily carried away by the gas flow. Hence, from the flow point of view, instead of tracking the individual particles by an Eulerian–Lagrangian formulation, the aerosol particles are treated to be part of the gas phase. From an energy balance point of view, the corresponding phase enthalpy (solid or liquid) has been considered based on the local temperature. In the present study, a global single-step reaction has been employed for the sodium vapor reaction with oxygen in air using a finite rate chemistry model. However, there are no global reaction rate parameters available in literature for this purpose. Therefore, the rate constant for the single step global reaction has been chosen and evaluated from the limited rate constant data available for elementary sodium–oxygen reactions [19]. This has been done by imposing steady state condition for the concentration of an unstable oxide called sodium super oxide (NaO2). Hynes et al. [19] considered a reaction mechanism with sodium super oxide (NaO2) and sodium oxide (NaO) as immediate products. It has been observed that NaO2 plays an important pathway for the production of stable oxides. The most rapid reaction path for the formation of sodium peroxide has been proposed by Newman [20]. This depends upon the following elementary reactions: k1
3.2. r-momentum equation
@u ; @x
r2 p0 ¼
Na þ O2 þ X ! NaO2 þ X
The last term in Eq. (1) arises from the density variation (i.e., buoyancy effects) in the gas-mixture which could be significant in some cases, in addition to the forced convection effects. The positive and negative signs for this term signify the situations in which buoyancy aids or opposes the forced convective flow, respectively.
rxx ¼ p þ 2l
For coupling velocity and pressure, the SIMPLE algorithm is employed. From a guessed pressure field (p⁄), the guess velocity field (u⁄, v⁄) is calculated by integrating the momentum equations. These velocities would produce a residue when used in the continuity equation. A Poisson equation for pressure correction (p0 ) can be obtained with the source term as the continuity equation residue in the form:
k2
NaO2 þ Na þ X ! Na2 O2 þ X k3
Na2 O2 þ X ! Na2 O þ 1=2O2 þ X
ð4Þ ð5Þ ð6Þ
The rate coefficient for the global single-step reaction with Na2O2 as reaction product has been evaluated from the above elementary reactions (4) and (5) along with the assumption of steady state condition for the concentration of unstable NaO2. Therefore, it has been assumed that the rate of production of Na2O2 will depend on the rate coefficient of the first elementary reaction (k1) itself. The rate coefficient for the global single-step reaction with Na2O alone as product is evaluated from the reaction (6) given above. In the above three reactions, the rate controlling reaction is the NaO2 formation reaction (i.e., reaction (4)) and the rate constant data of this reaction [19] is used in the present analysis. Three types of global single-step reaction schemes are considered as shown below.
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3.3.1. Scheme 1: Sodium monoxide alone as reaction product Since sodium monoxide is the most stable oxide of sodium, initially a global single-step reaction has been considered with Na2O alone as the reaction product and it is given as k
2Na þ 0:5ðO2 þ 3:76N2 Þ ! Na2 O þ 1:88N2
ð7Þ
The overall reaction rate is estimated using a 3-parameter Arrhenius type rate equation, given in the following form:
Ea k ¼ AT m exp Ru T
ð8Þ
In Eq. (8), k is the rate coefficient, A is the pre-exponential factor, Ea is the activation energy and Ru is the universal gas constant. The values of Ea and m are taken from Newman [20] as Ea = 2.0 107 and m = 1.0. The pre-exponential factor A has been evaluated as 2.9861 1014 from the reactions (4)–(6). Following Newman [20], the overall rate of reaction is estimated as
x_ ¼ k½Na1 ½O2 1 kmol=m3 s
ð9Þ
Nitrogen in the chemical equation (7) is considered inert and it would participate only in the enthalpy transport process. The rates of consumption of sodium and oxygen, and the rate of generation of the monoxide are calculated as follows:
x_ i ¼ ðti Þx_ MW i kg=m3 s;
ð10Þ
where MWi and ti are the molecular weight and stoichiometric coefficient of the ith species in the single step global reaction. 3.3.2. Scheme 2: Both sodium peroxide and monoxide as reaction products As a second step, sodium peroxide has also been included as a reaction product and the global single-step reaction is given as: k
2Na þ O2 þ 3:76N2 ! Na2 O2 þ 3:76N2
ð11Þ
As mentioned earlier the rate coefficient k for the above reaction is evaluated from the elementary reactions (4) and (5) using the steady-state assumption for NaO2. The global reaction rate for sodium peroxide formation can be evaluated as given below:
d½Na2 O2 ¼ k½Na1 ½O2 1 ½X1 ; dt
ð12Þ
where X is taken as nitrogen. Sodium peroxide is unstable above the temperature value of about 950 K [21]. In this temperature range, it dissociates into sodium monoxide (Na2O) and oxygen, as given below:
Na2 O2 () Na2 O þ 1=2O2
3.3.3. Scheme 3: Both oxides as reaction products and with dissociation of monoxide As a third step, dissociation of monoxide into sodium and oxygen is also considered. This is because of the fact that above a certain temperature (>2100 K), sodium monoxide also dissociates into sodium vapor and oxygen. This can be written as follows:
Na2 O () 2Na þ 1=2O2
2Na þ O2 þ 3:76N2 ! aNa2 O þ bNa2 O2 þ cNa þ dO2 þ 3:76N2 ð16Þ The stoichiometric coefficients of equation (16) are evaluated by considering the atom balance of Na, O and N, and the equilibrium constants for the dissociation reactions given in Eqs. (13) and (15), as functions of temperature in each cell. The equilibrium mole fractions of the species (in Eq. (16)) evaluated at different temperatures are shown in Fig. 1. For all the three schemes described above, the oxide products are assumed to occur as aerosols and they are continuously removed by the convective air flow so that the liquid phase consists of liquid sodium only throughout the droplet life time. 3.4. Species conservation equation For a particular species ‘‘i’’, the species conservation equation is of the form,
@ @ 1 @ ðqY i Þ þ ðquY i Þ þ ðr qv Y i Þ @t @x r @r @ @Y 1 @ @Y i _ i; r qDim qDim i þ þx ¼ @x r @r @x @r
ð17Þ
where Dim is the binary mass diffusivity for the ith species in the _ i is the mass rate of production of species i per unit mixture and x volume. 3.5. Energy conservation equation With the finite rate chemistry model considered, the energy conservation equation is written in terms of the temperature as given below:
ð13Þ
ð14Þ
A similar type of rate equation as in Eq. (8) is employed in this case also; however, the values of the parameters are given as A = 7.25415 1013, Ea = 0.0 and m = 1.0. The rates of consumption of sodium and oxygen, and the rates of generation of the oxides are estimated as given in Eq. (10). The stoichiometric coefficients in Eq. (14) are evaluated by considering the atom balance of Na, O and N, and the equilibrium constant for the dissociation reaction given in Eq. (13), which is a function of temperature. The data for the Gibbs energy of formation of various species have been taken from JANAF tables [22].
ð15Þ
This model also uses the same reaction rate as in the global singlestep reaction (11). However, considering reactions (13) and (15), which represent the dissociation of peroxide and monoxide respectively, the modified global reaction step with both the oxides and dissociation products is given as follows:
Taking this dissociation reaction into account in Eq. (11), a global single-step reaction having both the oxides can be written as
2Na þ O2 þ 3:76N2 ! aNa2 O þ bNa2 O2 þ cO2 þ 3:76N2
737
Fig. 1. Variation of equilibrium composition with temperature.
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@ @ 1 @ ðqC P TÞ þ ðquC P TÞ þ ðr qv C P TÞ @t @x r @r X n @ @T 1 @ @T þ ¼ x_ i Dhfi k rk @x @x r @r @r i¼1 n X @ @Y 1 @ @Y i rqDim C Pi T þ qDim C Pi T i þ @x r @r @x @r i¼1
mole fraction and mixture molecular weight (MWmixture) as follows:
Y Na ¼ X Na ð18Þ
In the above equation Dhfi is the enthalpy of formation for the ith species.
The computational domain along with boundary conditions is shown in Fig. 2. In the quasi-steady combustion model, only gasphase has been modeled by imposing suitable boundary conditions at the droplet surface as follows: (a) On the droplet surface, thermodynamic equilibrium is assumed to prevail between the liquid sodium and its vapor. The mole fraction of sodium vapor adjacent to the droplet surface is estimated as a function of the surface temperature using the Clausius–Clapeyron equation
X Na ¼
hfg MW Na 1 1 Psat ; ¼ exp T Tb Patm Ru
(b) The net mass flux of sodium vapor from the surface is equal to the sum of the mass fluxes due to convection and diffusion. The evaporation velocity, vs, at each location along the droplet surface is evaluated as
qs v s
3.6. Boundary conditions
ð19Þ
where hfg and Tb are the latent heat of vaporization and the boiling point of sodium, respectively. The mass fraction of sodium vapor is given in terms of its
MW Na ; MW mixture
Ds @Y@rNa s @Y Na ) vs ¼ ¼ qs v s Y Na þ qs Ds @r s Y Na 1
ð20Þ
(c) Assuming no internal heating after the attainment of quasisteady burning, the heat balance at the interface can be written as
k
dT ¼ qms hfg dr
ð21Þ
The above boundary conditions (Eqs. (19)–(21)) are solved iteratively in a coupled way to arrive at consistent values of the surface temperature, surface sodium vapor mass fraction, and the mass flux of sodium vapor at the droplet surface. 3.7. Evaluation of thermo-physical properties of species with temperature Mass densities of the species are estimated by using the ideal gas equation of state. Temperature dependent transport properties such as viscosity, thermal conductivity and mass diffusivity of sodium vapor and its oxides, oxygen and nitrogen are estimated with the help of Chapman–Enskog relations [23] for the range of temperatures of interest. These correlations use the Lennard-Jones parameters of species for estimating the transport properties, which are presented in Table 1 [9]. The data of specific heats of sodium and its oxides have been taken from JANAF tables [22]. Lennard-Jones parameter values and piecewise polynomial correlations for the data of specific heats of oxygen and nitrogen are taken from Ref. [24]. Gas mixture properties are evaluated in terms of species mass fractions using ideal gas mixing rules [23]. 3.8. Solution procedure The governing equations have been discretized using the Finite Volume Method. An explicit time marching scheme has been used for solving the transient governing equations. A non-orthogonal curvilinear grid with semi-collocated quadrilateral cells as shown in Fig. 3, has been employed in the numerical calculations following the work of Raghavan et al. [25]. Within each non-orthogonal cell, the variables are interpolated using general bilinear interpolation functions which are commonly used in Finite Element Method [17]. The velocity components are obtained by integrating the momentum equations with an assumed pressure field. The pressure field is updated by solving the pressure correction equation, which drives the mass residue in each continuity cell to acceptably small values. The velocities are also updated by solving the momentum equations again with the updated pressure field. The detailed solution procedure is available in Raghavan et al. [25] and hence it is not included here for the sake of brevity.
Table 1 Lennard-Jones parameter values used for sodium and its oxides.
Fig. 2. Computational domain.
Parameter
Na
Na2O [9]
Na2O2 [9]
e/k (K) r (Å)
1328.25 3.6682
2697.98 1.61
1820.16 1.62
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Fig. 3. (a) Semi-collocated grid, (b) discretized domain and (c) its close up view.
Species transport equations along with the energy equation are solved using the explicit time marching scheme with a smaller time step value than that used for flow equations for certain number of inner iterations. After evaluating the thermo-physical properties at the updated temperature, the chemical kinetic equations are solved to obtain the reaction rates. This transient marching procedure is carried out until either the steady state or a time independent oscillatory solution is obtained. The solution methodology outlined above has been implemented as a FORTRAN program and computations have been carried out on multi-core workstations. A grid sensitivity study with 46,000, 52,000 and 62,000 cells has shown that a grid with 52,000 cells gives a good balance between computational economy and solution accuracy. A very low time step value (nondimensional) of 5.0e05 has been used. Around 700,000 time steps (for the simulation case with monoxide as product) to 1,000,000 time steps (for the case with peroxide as product) are needed to have solutions converged to tolerance limit of 104. 4. Results and discussion The numerical model has been thoroughly validated first by comparing its predictions with the experimental results available in literature. Predictions in terms of burning rate constant and flame diameter have been compared against the corresponding experimental results of Makino and Fukada [14]. Further to validation, distributions of sodium vapor, oxygen mass fraction and
temperature around the droplet have been investigated to reveal the structure of diffusion flame around a sodium droplet. Parametric studies have been carried out using different droplet sizes and free-stream air velocities. 4.1. Validation Makino and Fukada [14] have studied the ignition and combustion behavior of an isolated freely falling sodium droplet in a vertical chamber filled with air. Variations of the square of droplet diameter and flame stand-off ratio (ratio of flame diameter to droplet diameter) with time have been reported in that study (Fig. 4). Since, Makino and Fukada [14] state that the classical d2-law is followed by sodium droplet combustion, a typical experimental case reported there has been considered for validation. Instantaneous quasi-steady burning of a 2 mm diameter sodium droplet in an atmospheric free stream of air flowing at 2.6 m/s has been simulated. The three reaction schemes discussed earlier have been employed. In the first scheme, sodium monoxide alone has been considered as the product; in the second, both the oxides (Na2O2 and Na2O) are present in the products and in the third scheme, both oxides are included along with the dissociation of monoxide. The simulations are carried out for combustion at atmospheric pressure (1 bar) and temperature (300 K). The predicted burning rate constant and flame stand-off ratio values are compared with the corresponding experimental results.
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P. Mangarjuna Rao et al. / International Journal of Heat and Mass Transfer 55 (2012) 734–743 Table 3 Predicted flame stand-off distances for 2 mm diameter sodium droplet. Reaction product
Flame stand-off distance/ droplet diameter
Na2O alone Na2O2 and Na2O Na2O2 and Na2O with Na2O dissociation
h1/d
h2/d
h3/d
0.2 0.255 0.196
0.365 0.465 0.313
1.255 1.81 0.763
Fig. 4. Definition of flame diameter, flame stand-off ratio and flame stand-off distances.
Table 2 presents the results obtained by the three reaction schemes along with the experimental results. It can be concluded from Table 2 that all the three schemes predict the burning rate constant and flame stand-off ratios reasonably well as compared to the experimental measurements in general. For the first scheme in which sodium monoxide alone is considered as product, the mass burning rate predicted by the model is 4.53 106 kg/s. This corresponds to a burning rate constant of 3.9 106 m2/s or 3.9 mm2/s, which is around 15% higher than the experimental value. However, the maximum flame temperature predicted by the model is around 2700 K, which is much higher than the adiabatic value of 2000 K reported in literature [26]. Using the second scheme with both the oxides as products and without dissociation of monoxide, the mass burning rate predicted is 4.26 106 kg/s. This corresponds to a burning rate constant of 3.66 106 m2/s or 3.66 mm2/s, which is around 8% more than the experimental value. Also in this case, the maximum flame temperature is highly over-predicted (2800 K). Using the third scheme, the predicted mass burning rate is 4.34 106 kg/s. This corresponds to a burning rate constant of 3.74 106 m2/s or 3.74 mm2/s, which is around 9% higher than the experimental value. However, the third scheme is able to predict a reasonable flame temperature of around 2100 K-closer to the value reported in literature [26]. This is presented in detail subsequently. Further to the validation, the structure of diffusion flame around a sodium droplet is studied. The third reaction scheme has been used for this purpose.
Fig. 5. Contours of sodium and oxygen mass fractions around the droplet.
of these flame stand-off distances are given in Table 3 for a droplet diameter of 2 mm and air velocity of 2.6 m/s. It is noted that for sodium droplet, the flame stand-off distances are much smaller than those for the flames around hydrocarbon or alcohol droplets [27]. In particular, the flame height (h3 in Fig. 4) of the sodium droplet combustion is very small when compared to alcohol droplet flames. The mass fraction contours of sodium vapor (left) and oxygen (right) around the droplet are shown in Fig. 5. It is observed from Fig. 5 that reactants diffuse towards each other from opposite directions and attain almost zero value around the flame (reaction) zone, thus rendering the problem diffusionally-controlled.
4.2. Further results The definitions of flame stand-off distances (h1, h2 and h3) at different angular positions, are shown in Fig. 4. The predicted values
Table 2 Comparison of numerical predictions with experimental results for the combustion of 2 mm diameter sodium droplet. Reaction scheme
Na2O alone Na2O2 and Na2O Na2O2 and Na2O with Na2O dissociation
Burning rate constant (mm2/s)
Flame stand-off ratio
Experimental [14]
Numerical simulation
Experimental [14]
Numerical simulation
3.4 3.4 3.4
3.90 3.66 3.74
1.1–1.9 1.1–1.9 1.1–1.9
1.73 1.93 1.63
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Fig. 6. Contours of reaction rates and mixture fraction around the droplet.
For the same case, the contours of reaction rate (right) along with the contours of mixture fraction (/) around the droplet (left) are shown in Fig. 6. The mixture fraction, which is a conserved scalar, is defined in terms of mass fractions of sodium and oxygen, as follows:
/¼
ðsY Na Y Ox Þ ðsY Na Y Ox Þ0 ; ðsY Na Y Ox Þ1 ðsY Na Y Ox Þ0
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where subscript 0 denotes the oxidizer stream, 1 denotes the sodium vapor stream and s is the stoichiometric mass ratio of oxygen to sodium vapor. Since the reaction rate has a finite value dictated by the Arrhenius rate equation, the flame zone has a finite thickness as well, and there is a small leakage of reactant species through the flame zone. The thickness of the reaction zone at the forward stagnation point (i.e., at 0° location) is approximately 0.28 mm. The locus of the midpoints of the reaction zone from the front to the rear stagnation points has been taken as the flame surface. The value of mixture fraction at the above locations is around 0.424, which is the stoichiometric mixture fraction. Based on this contour value of stoichiometric mixture fraction, flame diameter at a location of 90° from the forward stagnation point of the droplet is estimated as 3.26 mm, which corresponds to a flame stand-off ratio of 1.63. At the location of flame surface, the observed values of mass fraction of oxygen are in the range of 0.02–0.002 and this is consistent with the previous study [25,27]. The axial flame stand-off distance h3 (Fig. 4) is 1.53 mm, which corresponds to 0.76 times the droplet diameter. This value is much smaller than that reported for methanol droplet combustion [27] under similar convective conditions. For methanol, h3 has been reported in the range of 4–5 times of the sphere diameter. The short stand-off distance can be attributed to the high reactivity of sodium coupled with its low volatility. Such small flame heights might be observed in alcohol or hydrocarbon droplets only in the case of environments having high oxygen concentrations. The streamlines around the burning droplet are shown in Fig. 7a. It is observed that in the front portion of the sphere, the diffusion flame resembles an opposed flow stagnant flame. Due to availability of fresh oxidizer at the front stagnation point, the mass fraction gradient of fuel vapor at the surface is drastically increased, resulting in the highest mass burning rate at this location (Fig. 7b). Along the sphere surface from the front stagnation point towards the rear stagnation point, as the angle is increased, the fuel vapor has to travel a longer distance to mix with the required amount of oxidizer. This causes the sodium mass fraction gradient to decrease as the angle is increased from 0° (front stagnation point) to 180° (at rear stagnation point), resulting in a decrease of the local mass burning rate value with angle (Fig. 7b). It is observed from the streamlines that for this case, there is no vortex formation in the rear portion of the sphere due to the radial flow from the evaporating droplet surface.
Fig. 7. Stream lines (a) and evaporation velocity (b) of sodium vapor on the droplet surface.
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The temperature distribution around the droplet is shown in Fig. 8. As mentioned earlier, the surface temperature (1146 K) reaches a value very close to the boiling point (1156 K) under steady state burning conditions. The numerical model with the third kinetics scheme predicted a maximum flame temperature around 2120 K. Though precisely measured values of gas temperature around a burning sodium droplet are not available in literature, a maximum flame temperature value of around 2000 K has been reported [26]. The slight over-prediction by the present model can be attributed to the non-inclusion of radiative heat transfer between the flame and the surrounding gas phase. However, it is observed that the mass burning rates are not significantly affected by the neglect of radiation heat transfer to the droplet, implying that only the convection and conduction modes of heat transfer play a predominant role. Distribution of sodium peroxide and sodium monoxide around the droplet is shown in Fig. 9. As mentioned earlier, in the third scheme, sodium monoxide is formed from the dissociation of peroxide (at temperatures >950 K) and monoxide also dissociates into sodium and oxygen (at temperatures greater than around 2100 K) .The locations of formation and distribution of the oxides around the droplet shows this trend. It is clear that sodium monoxide is the predominant reaction product. The maximum value of monoxide mass fraction contour occurs almost along the flame surface. The locations of maximum sodium peroxide mass fraction are seen to occur a bit away from the flame surface, on the ambient side.
Fig. 9. Contours of sodium monoxide and peroxide mass fraction around the droplet.
4.3. Effect of free-stream velocity Simulations have been carried out for a 4 mm diameter droplet burning under free-stream velocity in the range of 0.5–5 m/s. In convective environment, the rate of evaporation of liquid from the droplet will increase with increase in the free-stream air velocity. The variation of mass burning rate predicted by the model as a function of free-stream velocity is shown in Fig. 10. The burning rate is seen to increase with air velocity. This is due to the increased transport of oxidizer towards the droplet surface, which
Fig. 8. Variation of temperature around the droplet.
Fig. 10. Variation of droplet mass burning rate with free-stream velocity.
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the third scheme which takes into account the dissociation of both sodium monoxide and peroxide, is able to predict a reasonable flame temperature as well. The specific features of sodium droplet combustion such as very small flame stand-off distance, high reactivity and absence of a wake flame, have been predicted quite well by the model. Parametric studies have been carried out for different free-stream velocities and different droplet diameters. The present model can be employed to analyze sodium droplet combustion under different convective conditions, to generate the data for droplet mass burning rate that can be used in the analysis of sodium spray fire. References
Fig. 11. Variation of droplet mass burning rate with droplet diameter.
results in higher mass burning rate. It should be noted here that a diffusion flame, which completely envelops the droplet, is observed for all the air velocities considered here. In the case of a hydrocarbon or an alcohol droplet, there will be a certain air velocity after which the diffusion flame cannot be sustained in the front portion of the droplet. This velocity has been termed as critical velocity [25,27]. For example, for a methanol sphere of 8 mm diameter, the critical velocity at which an envelope flame would not sustain is around 0.9 m/s [27]. However, for a sodium droplet even at an air velocity as high as 5 m/s, only an envelope flame is observed. This is one of the salient features of sodium droplet combustion, arising due to the high reactivity and almost zero activation energy of the sodium–air reaction. 4.4. Effect of droplet diameter Simulations have been carried out to evaluate the burning rates at different sizes of sodium droplets burning under an air velocity of 2.6 m/s. Droplet diameters have been increased from 1 mm to 8 mm. At the given air velocity, the mass burning rate increases with droplet diameter as shown in Fig. 11. This is due to the increase in surface area of the droplet as a result of increase in its diameter. This result is also consistent with that reported in literature for methanol droplets [25,27]. 5. Conclusions Simulation of sodium droplet combustion in a mixed convective atmospheric air environment has been carried out with a numerical model developed specifically for the purpose. The model incorporates three reaction schemes with either one or two main oxides of sodium. Rapid sodium–air oxidation reactions have been modeled using global single-step chemistry with finite reaction rate. Predictions from the numerical model have been validated against the experimental results available in literature in terms of sodium mass burning rate and flame stand-off ratio. All the three reaction schemes predict the mass burning rate and the flame stand-off ratio values reasonably close to the experimental results. However,
[1] D.B. Spalding, The combustion of liquid fuels, in: Fourth Symposium (International) on Combustion, The Combustion Institute, 1953, pp. 847–864. [2] G.A.E. Godsave, Studies of the combustion of drops in a fuel spray – the burning of single drops of fuel, in: Fourth Symposium (International) on Combustion, The Combustion Institute, 1953, pp. 818–830. [3] A. Williams, Combustion of droplets of liquid fuels: a review, Combust. Flame 21 (1973) 1–21. [4] G.M. Faeth, Current status of droplet and liquid combustion, Prog. Energy Combust. Sci. 3 (1977) 191–224. [5] C.K. Law, Recent advances in droplet vaporization and combustion, Prog. Energy Combust. Sci. 8 (1982) 171–201. [6] W.A. Sirignano, Fuel droplet vaporization and spray combustion theory, Prog. Energy Combust. Sci. 9 (1983) 291–322. [7] T.S. Krolikowski, L. Leibowitz, R.E. Wilson, J.C. Cassulo, The reaction of a molten sodium spray with air in an enclosed volume, Part I. Experimental investigation, Part II. Theoretical model, Nucl. Sci. Eng. 38 (1969) 156–166. [8] N. Sagawa, S. Sakaguchi, A. Suzuoki, K. Akagane, M. Mochizuki, Energy release from sodium spray combustion, IAEA, IWGFR Specialists Meeting on Sodium Fires, Cadarache, 1978, pp. 95–111. [9] Y. Okano, A. Yamaguchi, Numerical simulation of a free-falling liquid sodium droplet combustion, Ann. Nucl. Energy 30 (2003) 1863–1878. [10] W.E. Ranz, W.R. Marshall Jr., Experimental and theoretical evaluation for droplet evaporation and combustion, Chem. Eng. Prog. 48 (3–4) (1952) 141– 146. 173–180. [11] J.R. Richard, R. Delbourgo, P. Laffitte, Spontaneous ignition and combustion of sodium droplets in various oxidizing atmospheres at atmospheric pressure, in: Twelfth Symposium (International) on Combustion, The Combustion Institute, 1969, pp. 39–48. [12] H.A. Morewitz, R.P. Johnson, C.T. Nelson, Experiments on sodium fires and their aerosols, Nucl. Eng. Des. 42 (1977) 123–135. [13] H.A. Morewitz, Sodium spray fires, Nucl. Eng. Des. 55 (1979) 275–281. [14] A. Makino, H. Fukada, Ignition and combustion of a falling, single sodium droplet, in: Proceedings of the Combustion Institute, vol. 30, 2005, pp. 2047– 2054. [15] P.R. Shire, SPRAY Code User’s Report, HEDL-TME 76-94, 1977. [16] S.S. Tsai, The NACOM Code for Analysis of Postulated Sodium Spray Fires in LMFBRs, BNL-NUREG-51180, 1981. [17] P. Balakrishnan, T. Sundararajan, R. Natarajan, Combustion of a fuel droplet in a mixed convective environment, Combust. Sci. Technol. 163 (2001) 77–106. [18] R.N. Newman, The ignition and burning behaviour of sodium metal in air, Prog. Nucl. Energy 12 (1983) 119–147. [19] A.J. Hynes, M. Steinberg, K. Schofield, The chemical kinetics and thermodynamics of sodium species in oxygen-rich hydrogen flames, J. Chem. Phys. 80 (1984) 2585–2597. [20] R.N. Newman, The vapour phase combustion of sodium, Berkeley Nuclear Laboratories Report, RD/B/N2965, 1974. [21] R.H. Lamoreaux, D.L. Hildenbrand, High temperature vaporization behavior of oxides – alkali metal binary oxides, J. Phys. Chem. 13 (1) (1984) 151–173. [22] M.W. Chase et al., JANAF thermochemical tables, J. Phys. Chem. Ref. Data 14 (Suppl. 1) (1985) 1565–1579. [23] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, McGraw Hill, New York, 1987. [24] R.S. Turns, An Introduction to Combustion: Concepts and Applications, International ed., McGraw-Hill Book Company, 2001. [25] V. Raghavan, V. Babu, T. Sundararajan, R. Natarajan, Flame shapes and burning rates of spherical fuel particles in a mixed convective environment, Int. J. Heat Mass Transfer 48 (2005) 5354–5370. [26] Y. Okano, A. Yamaguchi, Theoretical adiabatic temperature and chemical composition of sodium combustion flame, Nucl. Technol. 144 (2003) 388–399. [27] V. Raghavan, G. Gogos, V. Babu, T. Sundararajan, Effect of gravity on methanol diffusion flames burning within a forced convective environment, Int. C. Heat Mass Transfer 33 (2006) 686–697.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Modeling of quasi-steady sodium droplet combustion in convective environment P. Mangarjuna Rao a, V. Raghavan b,⇑, K. Velusamy a, T. Sundararajan b, U.S.P. Shet b a b
Indira Gandhi Center for Atomic Research, Department of Atomic Energy, Kalpakkam 603 102, India Thermodynamics and Combustion Engineering Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
a r t i c l e
i n f o
Article history: Received 14 October 2010 Received in revised form 25 May 2011 Accepted 9 September 2011 Available online 9 November 2011 Keywords: LMFBR Sodium droplet Droplet combustion Flame shape Flame stand-off distance d2-law Burning rate constant
a b s t r a c t In the event of accidental leakage of sodium from the systems of a Liquid Metal Fast Breeder Reactor (LMFBR), a spray of liquid sodium droplets may be formed which will burn by reacting with the surrounding atmospheric oxygen. In order to understand the burning characteristics of the complete spray, combustion of an individual sodium droplet forms the basis and this has been investigated in the present study. A comprehensive numerical model has been developed to analyze the isolated sodium droplet combustion in a mixed convective environment. The governing equations for mass, momentum, species and energy conservation have been solved in axisymmetric cylindrical coordinates using the Finite Volume Method (FVM). Finite rate kinetic mechanisms have been incorporated to simulate droplet burning, using available kinetics data for basic sodium oxidation reactions. Salient features of the numerical model include a global single-step reaction for sodium oxidation in air and the incorporation of property variations with temperature and concentration. An equilibrium mixture of sodium peroxide, sodium monoxide and sodium vapor is considered as the final reaction product, taking into account the effects of dissociation reactions. The numerical model has been validated with experimental results available in literature. Results for the fuel mass burning rates and flame shapes are presented for different sizes of droplets burning under different free-stream conditions. The model predicts the occurrence of envelope flame around a sodium droplet even at fairly high free-stream velocities. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Liquid sodium is widely used as a coolant to rapidly transfer heat from the reactor core to the steam generator of LMFBR because of its high thermal conductivity, high boiling point and low neutron absorption coefficient. However, in case of any accidental leakage, sodium can react rapidly with oxygen and water vapor available in the atmosphere, especially when its temperature exceeds a value of about 473 K. Depending upon the temperature and pressure of the system and also based on the size, shape and location of the leak, a sodium spray fire could be initiated. Analysis of sodium spray fire and the resultant consequences is an important aspect of the safety procedure associated with LMFBR. Generally spray fires are considered to be more severe than pool fires, with higher burning rate of sodium in droplet form due to larger total exposed surface area, than in the pool configuration. Extensive research work has been carried out for understanding droplet and spray combustion aspects of hydrocarbon fuels and many review papers are also available [1–6]. On the other hand, literature on the study of isolated sodium droplet combustion is
⇑ Corresponding author. Tel.: +91 44 22574712. E-mail address: [email protected] (V. Raghavan). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.10.036
limited; only a few theoretical studies [7–9] have been reported which deal with the fundamental combustion behavior of a sodium droplet in convective air environment. Krolikowski et al. [7] developed a mathematical model for the combustion of a spherical sodium droplet in air, based on the assumption that droplet burning rate is controlled by the diffusion of oxygen towards the reaction zone. This model has been extended to predict the quasi-steady burning rates of moving droplets. Furthermore, the spray burning rate and the resultant pressure rise in an enclosed volume have been simulated, and compared with data obtained from spray combustion experiments. Sagawa et al. [8] theoretically analyzed the combustion of a single sodium particle in convective air stream. Their model predicted that the rate of decrease in the square of particle diameter is about 0.1 mm2/s, which is one order of magnitude smaller than the values observed in hydrocarbon fuel droplets. Okano and Yamaguchi [9] investigated the combustion of a sodium droplet in forced convective air flow using a CFD code called COMET. Sodium oxidation reactions were considered to be infinitely fast and the product composition was evaluated based on chemical equilibrium. The droplet burning rate was determined using the convective heat, mass transfer correlation of Ranz and Marshall [10]. This model predicted the predominant product to be sodium monoxide. The first experimental study on the burning of an isolated sodium droplet was carried out by Richard et al. [11], in a
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Nomenclature Cp D g k l n p T t u, v Vcell X x, r Y Dt
specific heat (J/kg K) mass diffusivity (m2/s) acceleration due to gravity (m/s2) thermal conductivity (W/m K) length of the side (m) normal vector pressure (N/m2) temperature (K) time (s) axial, radial velocity components (m/s) volume of the cell (m3) mole fraction axial and radial coordinates mass fraction time step (s)
convective field with a free-stream velocity of 0.01 m/s. It was shown that sodium droplet combustion follows the classical d2-law. The flame radius was observed to be around 1.3 times the droplet radius. Morewitz et al. [12] carried out experiments on freely falling sodium droplets with diameters ranging from 3 mm to 9 mm. The droplets were allowed to fall from an elevation of 14.7 m, in a column of air. It was observed that droplets larger than 7.3 mm initial diameter break down further to an approximate size of 2 mm diameter, due to interfacial instabilities [13]. Droplet burning rate and the fraction of droplet mass burned were evaluated as functions of the initial droplet diameter. Makino and Fukada [14] estimated the burning rate constant of a sodium droplet and its flame size by conducting falling droplet experiments in a vertical chamber. The burning rate constant was reported as 1.0 mm2/s for a sodium droplet with an initial diameter of 2 mm, burning in quiescent atmospheric air. When a forced convection environment was introduced (relative air velocity of 2.6 m/s), the burning rate constant increased to a value of 3.4 mm2/s. It was also shown that sodium droplet combustion resembles the burning of hydrocarbon droplets in an oxidizer-rich environment. In the published literature, mass burning rate data of sodium droplets under different convective conditions and different sizes, is scarce. This data is very essential for the numerical analysis of sodium spray fire. The existing spray fire safety analysis codes such as SPRAY [15] and NACOM [16] were developed based on one-dimensional macroscopic models and experimental results, and they often use limited empirical correlations to approximately estimate the burning rate of a single droplet. The available theoretical models, on the other hand, do not account for the detailed heat and mass transfer phenomena around the droplet as well as the reaction kinetics including the effects of product dissociation. Results have also not been presented for the mass burning rate and flame shape, for a wide range of convective conditions and droplet sizes. The present research work attempts to develop a detailed numerical model to study isolated sodium droplet combustion in a mixed convective environment. Since buoyancy-driven convection is known to enhance the burning rate significantly in the case of large fuel droplets [17], mixed convective flow has been considered. A single-step global reaction with finite rate chemistry and product dissociation has been incorporated to simulate the rapid sodium combustion reactions. The model is able to predict the salient features of sodium droplet combustion such as smaller flame heights and stand-off distances, and the absence of partial flame extinction at the front portion of the droplet even at fairly high air velocities.
Greek symbols density (kg/m3) coefficient of viscosity (N s/m2) normal and shear stress fields (N/m2) / mixture fraction x_ i mass based reaction rate (kg/m3 s)
q l r, s
Subscripts i species i m gas mixture s droplet surface 1 free-stream conditions
2. Salient features of sodium droplet combustion Liquid sodium combustion exhibits features which are vastly different from that of a hydrocarbon fuel. For instance, the ignition temperature of liquid sodium is defined as the temperature at which it oxidizes rapidly enough to produce a self-generated temperature rise sustaining liquid evaporation and combustion subsequently. The range of ignition temperature for liquid sodium is 393–593 K [18], which is dependent on the mode of sodium combustion (pool or spray) and the oxygen content in the surrounding atmosphere. Also, ignition depends on the existence of a protective layer formed over the liquid surface during its oxidation. Ignition temperature is the lowest in the case of a liquid sodium droplet (393–423 K). The range of temperatures at which liquid sodium exists in reactor systems is fairly higher than this ignition temperature. Generally, liquid sodium burning exhibits surface combustion as well as the vapor-phase combustion. Once ignited, at lower temperatures (<723 K), flameless surface oxidation of sodium occurs due to its high chemical reactivity with oxygen. Since sodium is more volatile than its oxides, the surface oxidation will eventually lead to vapor-phase combustion as the bulk temperature of the liquid also increases rapidly. Liquid sodium, however, is not as volatile as hydrocarbon fuels. For instance, the boiling point of sodium at 1 atm pressure is 1156 K and its latent heat of vaporization is very high, around 3886 kJ/kg. For gasoline, the boiling point at 1 atm pressure is 353 K and the latent heat of vaporization is approximately twelve times smaller than that of sodium. The heat of combustion of sodium is around four times less than that of gasoline. Compared to hydrocarbon fuels, the mass based stoichiometric ratio between oxidizer and sodium is also low. Therefore, with properties such as large latent heat of vaporization, high boiling point, low heat of combustion, low stoichiometric air–fuel ratio and very high chemical reactivity with oxygen in the air, sodium combustion will generate reaction zones much closer to the liquid surface. In this study, an attempt has been made to capture these salient features of sodium droplet combustion using a detailed numerical model. Although sodium also reacts with the water vapor in the atmosphere, the contribution from this mechanism has been neglected due to the low concentration of water vapor (0.2% by mass) in atmospheric air.
3. Numerical model Combustion of a liquid sodium droplet involves multi-phase, transient, mass, momentum and energy transfer processes with
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complex chemical reactions. A strong coupling exists between the mass, momentum and heat transfer processes, in the liquid- and vapor-phases. During the rapid surface oxidation of liquid sodium, the surface temperature of the droplet increases quickly. Because of the high thermal conductivity of liquid sodium and the sheardriven liquid circulation within the droplet, the bulk temperature of the droplet also increases rapidly. At the end of ignition transients, which will be around 10% of the entire droplet lifetime, a steady vapor-phase sodium droplet combustion begins and the surface temperature reaches a value close to the boiling point temperature of sodium (1156 K) [18]. Thereafter, the heat transferred from the flame to droplet surface is mainly used for vaporizing the sodium and the burning rate is dictated by diffusion, convection and to some extent the rate of the reaction. This regime, called as the quasi-steady burning regime, has been numerically modeled in this study. A numerical model, which solves a set of transient governing equations in axisymmetric cylindrical polar coordinates, has been developed. Governing equations describing the steady, constant pressure burning of a spherical liquid sodium droplet, placed in a laminar free-stream flow of atmospheric air, have been solved using the Finite Volume Method. A global reaction step with finite reaction rate, described by 3-parameter Arrhenius type of rate equation, has been employed. Variable thermo-physical properties have been evaluated as functions of temperature using appropriate correlations. Ideal gas mixture rules are used to evaluate the properties for the gas-mixture. As discussed earlier, the temperature of the entire droplet is assumed to be spatially uniform during the quasi-steady combustion phase. Therefore, only the gas-phase transport equations are modeled in a decoupled way, with the simplification that there is no liquid phase heating. Radiative heat transfer from the flame to the droplet surface as well as to the surrounding gas medium has been neglected in comparison with the other modes of heat transfer. This is in view of the small emissivity value for the highly reflective droplet surface and the small absorption coefficient of the gas medium. Only spherical drops are considered here due to the small sizes involved. The governing equations for the gas-phase processes in conservative form subject to the above assumptions are given as follows: 3.1. x-momentum equation
@ @ 1 @ ðquÞ þ ðqu2 Þ þ ðr quv Þ @t @x r @r @ 1 @ q ¼ ðrxx Þ þ ðrsxr Þ g 1 q1 @x r @r
ð1Þ
@ @ 1 @ @ 1 @ rhh ðqv Þ þ ðquv Þ þ ðr qv 2 Þ ¼ ðsxr Þ þ ðr rrr Þ @t @x r @r @x r @r r The stress terms in the above equations can be written as:
v
rrr ¼ p þ 2l
rhh ¼ p þ 2l ; srx ¼ sxr ¼ l r
@v ; @r
@u @ v þ @r @x
q @u Dt
@x
þ
1 @ ðr v Þ r @r
ð2Þ
ð3Þ
After evaluating the pressure correction, p0 , the pressure field is corrected using the relation, pn+1 = p⁄ + p0 . The corrected velocity field is obtained using the gradient of pressure correction as follows:
u0 ¼
Dt @p0 ; q @x
v0 ¼
Dt @p0 ; q @r
unþ1 ¼ u þ u0 ;
v nþ1 ¼ v þ v 0
3.3. Finite rate chemistry model – evaluation of reaction rate Sodium vapor reacts with atmospheric oxygen to produce its oxides such as sodium monoxide (Na2O) and sodium peroxide (Na2O2), which are the main reaction products in sodium combustion. These oxides are released as aerosols (generally in condensed solid/liquid phase), which are very small in size. The oxide particles therefore, offer no resistance and they are easily carried away by the gas flow. Hence, from the flow point of view, instead of tracking the individual particles by an Eulerian–Lagrangian formulation, the aerosol particles are treated to be part of the gas phase. From an energy balance point of view, the corresponding phase enthalpy (solid or liquid) has been considered based on the local temperature. In the present study, a global single-step reaction has been employed for the sodium vapor reaction with oxygen in air using a finite rate chemistry model. However, there are no global reaction rate parameters available in literature for this purpose. Therefore, the rate constant for the single step global reaction has been chosen and evaluated from the limited rate constant data available for elementary sodium–oxygen reactions [19]. This has been done by imposing steady state condition for the concentration of an unstable oxide called sodium super oxide (NaO2). Hynes et al. [19] considered a reaction mechanism with sodium super oxide (NaO2) and sodium oxide (NaO) as immediate products. It has been observed that NaO2 plays an important pathway for the production of stable oxides. The most rapid reaction path for the formation of sodium peroxide has been proposed by Newman [20]. This depends upon the following elementary reactions: k1
3.2. r-momentum equation
@u ; @x
r2 p0 ¼
Na þ O2 þ X ! NaO2 þ X
The last term in Eq. (1) arises from the density variation (i.e., buoyancy effects) in the gas-mixture which could be significant in some cases, in addition to the forced convection effects. The positive and negative signs for this term signify the situations in which buoyancy aids or opposes the forced convective flow, respectively.
rxx ¼ p þ 2l
For coupling velocity and pressure, the SIMPLE algorithm is employed. From a guessed pressure field (p⁄), the guess velocity field (u⁄, v⁄) is calculated by integrating the momentum equations. These velocities would produce a residue when used in the continuity equation. A Poisson equation for pressure correction (p0 ) can be obtained with the source term as the continuity equation residue in the form:
k2
NaO2 þ Na þ X ! Na2 O2 þ X k3
Na2 O2 þ X ! Na2 O þ 1=2O2 þ X
ð4Þ ð5Þ ð6Þ
The rate coefficient for the global single-step reaction with Na2O2 as reaction product has been evaluated from the above elementary reactions (4) and (5) along with the assumption of steady state condition for the concentration of unstable NaO2. Therefore, it has been assumed that the rate of production of Na2O2 will depend on the rate coefficient of the first elementary reaction (k1) itself. The rate coefficient for the global single-step reaction with Na2O alone as product is evaluated from the reaction (6) given above. In the above three reactions, the rate controlling reaction is the NaO2 formation reaction (i.e., reaction (4)) and the rate constant data of this reaction [19] is used in the present analysis. Three types of global single-step reaction schemes are considered as shown below.
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3.3.1. Scheme 1: Sodium monoxide alone as reaction product Since sodium monoxide is the most stable oxide of sodium, initially a global single-step reaction has been considered with Na2O alone as the reaction product and it is given as k
2Na þ 0:5ðO2 þ 3:76N2 Þ ! Na2 O þ 1:88N2
ð7Þ
The overall reaction rate is estimated using a 3-parameter Arrhenius type rate equation, given in the following form:
Ea k ¼ AT m exp Ru T
ð8Þ
In Eq. (8), k is the rate coefficient, A is the pre-exponential factor, Ea is the activation energy and Ru is the universal gas constant. The values of Ea and m are taken from Newman [20] as Ea = 2.0 107 and m = 1.0. The pre-exponential factor A has been evaluated as 2.9861 1014 from the reactions (4)–(6). Following Newman [20], the overall rate of reaction is estimated as
x_ ¼ k½Na1 ½O2 1 kmol=m3 s
ð9Þ
Nitrogen in the chemical equation (7) is considered inert and it would participate only in the enthalpy transport process. The rates of consumption of sodium and oxygen, and the rate of generation of the monoxide are calculated as follows:
x_ i ¼ ðti Þx_ MW i kg=m3 s;
ð10Þ
where MWi and ti are the molecular weight and stoichiometric coefficient of the ith species in the single step global reaction. 3.3.2. Scheme 2: Both sodium peroxide and monoxide as reaction products As a second step, sodium peroxide has also been included as a reaction product and the global single-step reaction is given as: k
2Na þ O2 þ 3:76N2 ! Na2 O2 þ 3:76N2
ð11Þ
As mentioned earlier the rate coefficient k for the above reaction is evaluated from the elementary reactions (4) and (5) using the steady-state assumption for NaO2. The global reaction rate for sodium peroxide formation can be evaluated as given below:
d½Na2 O2 ¼ k½Na1 ½O2 1 ½X1 ; dt
ð12Þ
where X is taken as nitrogen. Sodium peroxide is unstable above the temperature value of about 950 K [21]. In this temperature range, it dissociates into sodium monoxide (Na2O) and oxygen, as given below:
Na2 O2 () Na2 O þ 1=2O2
3.3.3. Scheme 3: Both oxides as reaction products and with dissociation of monoxide As a third step, dissociation of monoxide into sodium and oxygen is also considered. This is because of the fact that above a certain temperature (>2100 K), sodium monoxide also dissociates into sodium vapor and oxygen. This can be written as follows:
Na2 O () 2Na þ 1=2O2
2Na þ O2 þ 3:76N2 ! aNa2 O þ bNa2 O2 þ cNa þ dO2 þ 3:76N2 ð16Þ The stoichiometric coefficients of equation (16) are evaluated by considering the atom balance of Na, O and N, and the equilibrium constants for the dissociation reactions given in Eqs. (13) and (15), as functions of temperature in each cell. The equilibrium mole fractions of the species (in Eq. (16)) evaluated at different temperatures are shown in Fig. 1. For all the three schemes described above, the oxide products are assumed to occur as aerosols and they are continuously removed by the convective air flow so that the liquid phase consists of liquid sodium only throughout the droplet life time. 3.4. Species conservation equation For a particular species ‘‘i’’, the species conservation equation is of the form,
@ @ 1 @ ðqY i Þ þ ðquY i Þ þ ðr qv Y i Þ @t @x r @r @ @Y 1 @ @Y i _ i; r qDim qDim i þ þx ¼ @x r @r @x @r
ð17Þ
where Dim is the binary mass diffusivity for the ith species in the _ i is the mass rate of production of species i per unit mixture and x volume. 3.5. Energy conservation equation With the finite rate chemistry model considered, the energy conservation equation is written in terms of the temperature as given below:
ð13Þ
ð14Þ
A similar type of rate equation as in Eq. (8) is employed in this case also; however, the values of the parameters are given as A = 7.25415 1013, Ea = 0.0 and m = 1.0. The rates of consumption of sodium and oxygen, and the rates of generation of the oxides are estimated as given in Eq. (10). The stoichiometric coefficients in Eq. (14) are evaluated by considering the atom balance of Na, O and N, and the equilibrium constant for the dissociation reaction given in Eq. (13), which is a function of temperature. The data for the Gibbs energy of formation of various species have been taken from JANAF tables [22].
ð15Þ
This model also uses the same reaction rate as in the global singlestep reaction (11). However, considering reactions (13) and (15), which represent the dissociation of peroxide and monoxide respectively, the modified global reaction step with both the oxides and dissociation products is given as follows:
Taking this dissociation reaction into account in Eq. (11), a global single-step reaction having both the oxides can be written as
2Na þ O2 þ 3:76N2 ! aNa2 O þ bNa2 O2 þ cO2 þ 3:76N2
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Fig. 1. Variation of equilibrium composition with temperature.
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@ @ 1 @ ðqC P TÞ þ ðquC P TÞ þ ðr qv C P TÞ @t @x r @r X n @ @T 1 @ @T þ ¼ x_ i Dhfi k rk @x @x r @r @r i¼1 n X @ @Y 1 @ @Y i rqDim C Pi T þ qDim C Pi T i þ @x r @r @x @r i¼1
mole fraction and mixture molecular weight (MWmixture) as follows:
Y Na ¼ X Na ð18Þ
In the above equation Dhfi is the enthalpy of formation for the ith species.
The computational domain along with boundary conditions is shown in Fig. 2. In the quasi-steady combustion model, only gasphase has been modeled by imposing suitable boundary conditions at the droplet surface as follows: (a) On the droplet surface, thermodynamic equilibrium is assumed to prevail between the liquid sodium and its vapor. The mole fraction of sodium vapor adjacent to the droplet surface is estimated as a function of the surface temperature using the Clausius–Clapeyron equation
X Na ¼
hfg MW Na 1 1 Psat ; ¼ exp T Tb Patm Ru
(b) The net mass flux of sodium vapor from the surface is equal to the sum of the mass fluxes due to convection and diffusion. The evaporation velocity, vs, at each location along the droplet surface is evaluated as
qs v s
3.6. Boundary conditions
ð19Þ
where hfg and Tb are the latent heat of vaporization and the boiling point of sodium, respectively. The mass fraction of sodium vapor is given in terms of its
MW Na ; MW mixture
Ds @Y@rNa s @Y Na ) vs ¼ ¼ qs v s Y Na þ qs Ds @r s Y Na 1
ð20Þ
(c) Assuming no internal heating after the attainment of quasisteady burning, the heat balance at the interface can be written as
k
dT ¼ qms hfg dr
ð21Þ
The above boundary conditions (Eqs. (19)–(21)) are solved iteratively in a coupled way to arrive at consistent values of the surface temperature, surface sodium vapor mass fraction, and the mass flux of sodium vapor at the droplet surface. 3.7. Evaluation of thermo-physical properties of species with temperature Mass densities of the species are estimated by using the ideal gas equation of state. Temperature dependent transport properties such as viscosity, thermal conductivity and mass diffusivity of sodium vapor and its oxides, oxygen and nitrogen are estimated with the help of Chapman–Enskog relations [23] for the range of temperatures of interest. These correlations use the Lennard-Jones parameters of species for estimating the transport properties, which are presented in Table 1 [9]. The data of specific heats of sodium and its oxides have been taken from JANAF tables [22]. Lennard-Jones parameter values and piecewise polynomial correlations for the data of specific heats of oxygen and nitrogen are taken from Ref. [24]. Gas mixture properties are evaluated in terms of species mass fractions using ideal gas mixing rules [23]. 3.8. Solution procedure The governing equations have been discretized using the Finite Volume Method. An explicit time marching scheme has been used for solving the transient governing equations. A non-orthogonal curvilinear grid with semi-collocated quadrilateral cells as shown in Fig. 3, has been employed in the numerical calculations following the work of Raghavan et al. [25]. Within each non-orthogonal cell, the variables are interpolated using general bilinear interpolation functions which are commonly used in Finite Element Method [17]. The velocity components are obtained by integrating the momentum equations with an assumed pressure field. The pressure field is updated by solving the pressure correction equation, which drives the mass residue in each continuity cell to acceptably small values. The velocities are also updated by solving the momentum equations again with the updated pressure field. The detailed solution procedure is available in Raghavan et al. [25] and hence it is not included here for the sake of brevity.
Table 1 Lennard-Jones parameter values used for sodium and its oxides.
Fig. 2. Computational domain.
Parameter
Na
Na2O [9]
Na2O2 [9]
e/k (K) r (Å)
1328.25 3.6682
2697.98 1.61
1820.16 1.62
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Fig. 3. (a) Semi-collocated grid, (b) discretized domain and (c) its close up view.
Species transport equations along with the energy equation are solved using the explicit time marching scheme with a smaller time step value than that used for flow equations for certain number of inner iterations. After evaluating the thermo-physical properties at the updated temperature, the chemical kinetic equations are solved to obtain the reaction rates. This transient marching procedure is carried out until either the steady state or a time independent oscillatory solution is obtained. The solution methodology outlined above has been implemented as a FORTRAN program and computations have been carried out on multi-core workstations. A grid sensitivity study with 46,000, 52,000 and 62,000 cells has shown that a grid with 52,000 cells gives a good balance between computational economy and solution accuracy. A very low time step value (nondimensional) of 5.0e05 has been used. Around 700,000 time steps (for the simulation case with monoxide as product) to 1,000,000 time steps (for the case with peroxide as product) are needed to have solutions converged to tolerance limit of 104. 4. Results and discussion The numerical model has been thoroughly validated first by comparing its predictions with the experimental results available in literature. Predictions in terms of burning rate constant and flame diameter have been compared against the corresponding experimental results of Makino and Fukada [14]. Further to validation, distributions of sodium vapor, oxygen mass fraction and
temperature around the droplet have been investigated to reveal the structure of diffusion flame around a sodium droplet. Parametric studies have been carried out using different droplet sizes and free-stream air velocities. 4.1. Validation Makino and Fukada [14] have studied the ignition and combustion behavior of an isolated freely falling sodium droplet in a vertical chamber filled with air. Variations of the square of droplet diameter and flame stand-off ratio (ratio of flame diameter to droplet diameter) with time have been reported in that study (Fig. 4). Since, Makino and Fukada [14] state that the classical d2-law is followed by sodium droplet combustion, a typical experimental case reported there has been considered for validation. Instantaneous quasi-steady burning of a 2 mm diameter sodium droplet in an atmospheric free stream of air flowing at 2.6 m/s has been simulated. The three reaction schemes discussed earlier have been employed. In the first scheme, sodium monoxide alone has been considered as the product; in the second, both the oxides (Na2O2 and Na2O) are present in the products and in the third scheme, both oxides are included along with the dissociation of monoxide. The simulations are carried out for combustion at atmospheric pressure (1 bar) and temperature (300 K). The predicted burning rate constant and flame stand-off ratio values are compared with the corresponding experimental results.
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Flame stand-off distance/ droplet diameter
Na2O alone Na2O2 and Na2O Na2O2 and Na2O with Na2O dissociation
h1/d
h2/d
h3/d
0.2 0.255 0.196
0.365 0.465 0.313
1.255 1.81 0.763
Fig. 4. Definition of flame diameter, flame stand-off ratio and flame stand-off distances.
Table 2 presents the results obtained by the three reaction schemes along with the experimental results. It can be concluded from Table 2 that all the three schemes predict the burning rate constant and flame stand-off ratios reasonably well as compared to the experimental measurements in general. For the first scheme in which sodium monoxide alone is considered as product, the mass burning rate predicted by the model is 4.53 106 kg/s. This corresponds to a burning rate constant of 3.9 106 m2/s or 3.9 mm2/s, which is around 15% higher than the experimental value. However, the maximum flame temperature predicted by the model is around 2700 K, which is much higher than the adiabatic value of 2000 K reported in literature [26]. Using the second scheme with both the oxides as products and without dissociation of monoxide, the mass burning rate predicted is 4.26 106 kg/s. This corresponds to a burning rate constant of 3.66 106 m2/s or 3.66 mm2/s, which is around 8% more than the experimental value. Also in this case, the maximum flame temperature is highly over-predicted (2800 K). Using the third scheme, the predicted mass burning rate is 4.34 106 kg/s. This corresponds to a burning rate constant of 3.74 106 m2/s or 3.74 mm2/s, which is around 9% higher than the experimental value. However, the third scheme is able to predict a reasonable flame temperature of around 2100 K-closer to the value reported in literature [26]. This is presented in detail subsequently. Further to the validation, the structure of diffusion flame around a sodium droplet is studied. The third reaction scheme has been used for this purpose.
Fig. 5. Contours of sodium and oxygen mass fractions around the droplet.
of these flame stand-off distances are given in Table 3 for a droplet diameter of 2 mm and air velocity of 2.6 m/s. It is noted that for sodium droplet, the flame stand-off distances are much smaller than those for the flames around hydrocarbon or alcohol droplets [27]. In particular, the flame height (h3 in Fig. 4) of the sodium droplet combustion is very small when compared to alcohol droplet flames. The mass fraction contours of sodium vapor (left) and oxygen (right) around the droplet are shown in Fig. 5. It is observed from Fig. 5 that reactants diffuse towards each other from opposite directions and attain almost zero value around the flame (reaction) zone, thus rendering the problem diffusionally-controlled.
4.2. Further results The definitions of flame stand-off distances (h1, h2 and h3) at different angular positions, are shown in Fig. 4. The predicted values
Table 2 Comparison of numerical predictions with experimental results for the combustion of 2 mm diameter sodium droplet. Reaction scheme
Na2O alone Na2O2 and Na2O Na2O2 and Na2O with Na2O dissociation
Burning rate constant (mm2/s)
Flame stand-off ratio
Experimental [14]
Numerical simulation
Experimental [14]
Numerical simulation
3.4 3.4 3.4
3.90 3.66 3.74
1.1–1.9 1.1–1.9 1.1–1.9
1.73 1.93 1.63
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Fig. 6. Contours of reaction rates and mixture fraction around the droplet.
For the same case, the contours of reaction rate (right) along with the contours of mixture fraction (/) around the droplet (left) are shown in Fig. 6. The mixture fraction, which is a conserved scalar, is defined in terms of mass fractions of sodium and oxygen, as follows:
/¼
ðsY Na Y Ox Þ ðsY Na Y Ox Þ0 ; ðsY Na Y Ox Þ1 ðsY Na Y Ox Þ0
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where subscript 0 denotes the oxidizer stream, 1 denotes the sodium vapor stream and s is the stoichiometric mass ratio of oxygen to sodium vapor. Since the reaction rate has a finite value dictated by the Arrhenius rate equation, the flame zone has a finite thickness as well, and there is a small leakage of reactant species through the flame zone. The thickness of the reaction zone at the forward stagnation point (i.e., at 0° location) is approximately 0.28 mm. The locus of the midpoints of the reaction zone from the front to the rear stagnation points has been taken as the flame surface. The value of mixture fraction at the above locations is around 0.424, which is the stoichiometric mixture fraction. Based on this contour value of stoichiometric mixture fraction, flame diameter at a location of 90° from the forward stagnation point of the droplet is estimated as 3.26 mm, which corresponds to a flame stand-off ratio of 1.63. At the location of flame surface, the observed values of mass fraction of oxygen are in the range of 0.02–0.002 and this is consistent with the previous study [25,27]. The axial flame stand-off distance h3 (Fig. 4) is 1.53 mm, which corresponds to 0.76 times the droplet diameter. This value is much smaller than that reported for methanol droplet combustion [27] under similar convective conditions. For methanol, h3 has been reported in the range of 4–5 times of the sphere diameter. The short stand-off distance can be attributed to the high reactivity of sodium coupled with its low volatility. Such small flame heights might be observed in alcohol or hydrocarbon droplets only in the case of environments having high oxygen concentrations. The streamlines around the burning droplet are shown in Fig. 7a. It is observed that in the front portion of the sphere, the diffusion flame resembles an opposed flow stagnant flame. Due to availability of fresh oxidizer at the front stagnation point, the mass fraction gradient of fuel vapor at the surface is drastically increased, resulting in the highest mass burning rate at this location (Fig. 7b). Along the sphere surface from the front stagnation point towards the rear stagnation point, as the angle is increased, the fuel vapor has to travel a longer distance to mix with the required amount of oxidizer. This causes the sodium mass fraction gradient to decrease as the angle is increased from 0° (front stagnation point) to 180° (at rear stagnation point), resulting in a decrease of the local mass burning rate value with angle (Fig. 7b). It is observed from the streamlines that for this case, there is no vortex formation in the rear portion of the sphere due to the radial flow from the evaporating droplet surface.
Fig. 7. Stream lines (a) and evaporation velocity (b) of sodium vapor on the droplet surface.
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The temperature distribution around the droplet is shown in Fig. 8. As mentioned earlier, the surface temperature (1146 K) reaches a value very close to the boiling point (1156 K) under steady state burning conditions. The numerical model with the third kinetics scheme predicted a maximum flame temperature around 2120 K. Though precisely measured values of gas temperature around a burning sodium droplet are not available in literature, a maximum flame temperature value of around 2000 K has been reported [26]. The slight over-prediction by the present model can be attributed to the non-inclusion of radiative heat transfer between the flame and the surrounding gas phase. However, it is observed that the mass burning rates are not significantly affected by the neglect of radiation heat transfer to the droplet, implying that only the convection and conduction modes of heat transfer play a predominant role. Distribution of sodium peroxide and sodium monoxide around the droplet is shown in Fig. 9. As mentioned earlier, in the third scheme, sodium monoxide is formed from the dissociation of peroxide (at temperatures >950 K) and monoxide also dissociates into sodium and oxygen (at temperatures greater than around 2100 K) .The locations of formation and distribution of the oxides around the droplet shows this trend. It is clear that sodium monoxide is the predominant reaction product. The maximum value of monoxide mass fraction contour occurs almost along the flame surface. The locations of maximum sodium peroxide mass fraction are seen to occur a bit away from the flame surface, on the ambient side.
Fig. 9. Contours of sodium monoxide and peroxide mass fraction around the droplet.
4.3. Effect of free-stream velocity Simulations have been carried out for a 4 mm diameter droplet burning under free-stream velocity in the range of 0.5–5 m/s. In convective environment, the rate of evaporation of liquid from the droplet will increase with increase in the free-stream air velocity. The variation of mass burning rate predicted by the model as a function of free-stream velocity is shown in Fig. 10. The burning rate is seen to increase with air velocity. This is due to the increased transport of oxidizer towards the droplet surface, which
Fig. 8. Variation of temperature around the droplet.
Fig. 10. Variation of droplet mass burning rate with free-stream velocity.
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the third scheme which takes into account the dissociation of both sodium monoxide and peroxide, is able to predict a reasonable flame temperature as well. The specific features of sodium droplet combustion such as very small flame stand-off distance, high reactivity and absence of a wake flame, have been predicted quite well by the model. Parametric studies have been carried out for different free-stream velocities and different droplet diameters. The present model can be employed to analyze sodium droplet combustion under different convective conditions, to generate the data for droplet mass burning rate that can be used in the analysis of sodium spray fire. References
Fig. 11. Variation of droplet mass burning rate with droplet diameter.
results in higher mass burning rate. It should be noted here that a diffusion flame, which completely envelops the droplet, is observed for all the air velocities considered here. In the case of a hydrocarbon or an alcohol droplet, there will be a certain air velocity after which the diffusion flame cannot be sustained in the front portion of the droplet. This velocity has been termed as critical velocity [25,27]. For example, for a methanol sphere of 8 mm diameter, the critical velocity at which an envelope flame would not sustain is around 0.9 m/s [27]. However, for a sodium droplet even at an air velocity as high as 5 m/s, only an envelope flame is observed. This is one of the salient features of sodium droplet combustion, arising due to the high reactivity and almost zero activation energy of the sodium–air reaction. 4.4. Effect of droplet diameter Simulations have been carried out to evaluate the burning rates at different sizes of sodium droplets burning under an air velocity of 2.6 m/s. Droplet diameters have been increased from 1 mm to 8 mm. At the given air velocity, the mass burning rate increases with droplet diameter as shown in Fig. 11. This is due to the increase in surface area of the droplet as a result of increase in its diameter. This result is also consistent with that reported in literature for methanol droplets [25,27]. 5. Conclusions Simulation of sodium droplet combustion in a mixed convective atmospheric air environment has been carried out with a numerical model developed specifically for the purpose. The model incorporates three reaction schemes with either one or two main oxides of sodium. Rapid sodium–air oxidation reactions have been modeled using global single-step chemistry with finite reaction rate. Predictions from the numerical model have been validated against the experimental results available in literature in terms of sodium mass burning rate and flame stand-off ratio. All the three reaction schemes predict the mass burning rate and the flame stand-off ratio values reasonably close to the experimental results. However,
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