ANALYSIS OF SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER USING THE LATTICE BOLTZMANN METHOD AND THE DISCRETE TRANSFER METHOD

Numerical Heat Transfer, Part A, 49: 279 299, 2006 Copyright# Taylor & Francis LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407780500359828 ANALYSIS OF SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER USING THE LATTICE BOLTZMANN METHOD AND THE DISCRETE TRANSFER METHOD 1. INTRODUCTION Rishi Raj, Amit Prasad, Pritish Ranjan Parida, and Subhash C. Mishra Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India This article deals with the analysis of solidification of a semitransparent material. The solidification was assumed to occur at a range of temperatures and thus the presence of a mushy zone was considered. The governing energy equation in terms of an enthalpy formulation was considered. The equivalence of the enthalpy-based governing equation in the continuum approach was derived for the lattice Boltzmann method (LBM). The radiative component of the energy equation in the LBM formulation was computed using the discrete transfer method. The LBM formulation was first validated by solving solidification of a radiatively opaque planar material. Next, effects of various parameters such as the extinction coefficient, the scattering albedo, the anisotropy factor, the conduction-radiation parameter, and the latent heat on temperature distribution in three zones and the location of the mushy zone were studied. These parameters were found to have significant bearing on the results. In recent years, there has been a rapid progress in developing and employing the lattice Boltzmann method (LBM) as an alternative computational fluid dynamics (CFD) approach for simulation of a large class of problems in science and engineering [1 19]. The LBM has been demonstrated to be successful in simulation of fluid flow and heat transfer problems and other types of complex physical systems [1 19]. Proponents of the LBM consider this method to have the potential to become a versatile CFD platform that is superior over the existing continuum-based CFD methods [1 4]. The LBM is a mesoscopic approach that incorporates microscopic physics with affordable computational expense. Unlike the conventional CFD methods that directly simulate the continuum-based governing equations (for example, the Navier Stokes equations), the LBM is based on a mesoscopic kinetic equation. This bottom-up approach of the LBM assures by construction the conservation of the relevant macroscopic quantities such as mass and momentum [1 8]. One hypothesis Received 6 May 2005; accepted 30 July 2005. Address correspondence to Subhash C. Mishra, Department of Mechanical Engineering, IIT Guwahati, Guwahati 781039, India. E-mail: scm iitg@yahoo.com 279

280 R. RAJ ET AL. NOMENCLATURE a anisotropy factor c p specific heat, kj=kg K C heat capacity, kj=m 3 K e i propagation speed in the direction i in the lattice, j~e i j, m=s j~e i j propagation velocity in the direction i in the lattice, m=s f l volume phase fraction of the liquid phase G incident radiation, W=m 2 I intensity, W=m 2 k thermal conductivity, W=m K L latent heat, kj=kg M total number of rays=intensities m number of lattice directions n i particle distribution function in the i direction, K n ð0þ i equilibrium particle distribution function in the i direction, K N conduction-radiation parameter ð¼ kb=4rtw 3 Þ q R radiative heat flux, W=m 2 S radiative source term, W=m 2 St Stefan number ½¼ C s ðt 0 T E Þ=LŠ t time, s T temperature, K w i weight factor corresponding to the direction i in a lattice x coordinate axis directions X length of the geometry, m a thermal diffusivity, m 2 =s b extinction coefficient, m 1 e emissivity h polar angle k a absorption coefficient, m 1 r Stefan-Boltzmann constant ð¼ 5:67 10 8 W=m 2 K 4 Þ r s scattering coefficient, m 1 s relaxation time, s U source term which affects the distribution function n i, Eq. (24) x scattering albedo X rate of change of f i due to collision, K=s Subscripts b boundary E, W east, west l liquid phase m melting mz mushy zone s solid phase 0 initial temperature Superscript K iteration level at a given time level for using this simplified kinetic-type method for macroscopic phenomena is that the macroscopic properties are the result of collective behavior of many microscopic particles and the macroscopic dynamics are not sensitive to the details of the microscopic mechanism. Because of the microscopic origin, the LBM has many advantages over conventional CFD methods. The advantages include, among others, a clear physical meaning, a simple calculation, simple implementation on a computer, parallel computation, easy handling of complex geometries and boundary conditions, and capability of stable and accurate simulation [1 8]. In comparison to its application to fluid mechanics problems [1 6], use of the LBM in conjugate-mode (conduction, convection, and=or radiation) problems has not been explored in much detail [9 17]. Ho et al. [11, 12] used the LBM to study non-fourier heat conduction in a planar layer. Kush et al. [18] solved heat conduction problems in 1-D, 2-D, and 3-D rectangular geometries and compared the LBM results with the finite-difference method (FDM) solutions for both temperature as well as flux boundary conditions. Jiaung et al. [10] did the analysis of solidification using the LBM. Mishra and Lankadasu [16] and Mishra et al. [17] used the LBM to solve energy equations of problems dealing with conduction and radiation. Gupta et al. [19] utilized the concept of variable relaxation time and solved the transient

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 281 conduction-radiation problem involving temperature-dependent thermal conductivity. In all the work done so far [9 17], the LBM has shown good promise for extending its application to more complex problems involving conduction and radiation heat transfer. Melting and solidification of semitransparent materials are important physical phenomena involved in many engineering areas such as crystal growth, laser material processing, and nuclear engineering. The problems become extremely challenging because of complex radiation heat transfer with internal absorbing, emitting, and scattering. In the present work, the use of the LBM has been extended for the simulation of the phase-change problems governed by heat conduction and radiation incorporated with enthalpy formulation. The discrete transfer method (DTM) has been used to compute the radiative information. To validate the LBM formulation, solidification of a planar layer without radiation has been considered first. Next the solidification problem has been solved by considering the effect of the volumetric radiation. Effects of various parameters such as the extinction coefficient, the scattering albedo, the anisotropy factor, the conduction-radiation parameter, and the latent heat on temperature distribution in three zones and the location of the mushy zone have been studied. 2. FORMULATION Consider solidification of a 1-D planar semitransparent material (Figure 1). Initially the material is at temperature T 0, which is higher than the melting temperature T m. At time t > 0, the east boundary is maintained at temperature T E, which is below T m, and the west boundary temperature T W is kept at T 0. As time passes, solidification starts from the east boundary. Since solidification of the material under consideration is considered over a range of temperatures, a mushy zone comes into the picture. Thus, unlike the movement of a distinct front, the mushy zone moves from the east boundary to the west boundary and with time its thickness also Figure 1. The 1-D planar geometry under consideration. D1Q2 lattice used in a 1-D planar geometry and an arrangement of lattices in the LBM and the control volumes in the DTM are also shown.

282 R. RAJ ET AL. changes. Further, since the material is a semitransparent one, volumetric radiation also plays a role, and it appears in the governing energy equation. qðqhþ qt ¼ q qx kqt qq R qx qx where H is the total enthalpy, q is the density, k is the thermal conductivity, and q R is the radiative heat flux. For a phase-change problem, the total enthalpy H can be written as [20] ð1þ H ¼ c P T þ f l L ð2þ where L is the latent heat and f l is the liquid fraction. In the solid region f l ¼ 0, while for the liquid region f l ¼ 1. In the mushy zone, 0 < f l < 1. With H defined as above, Eq. (1) can be written as qðqc P TÞ qt ¼ q qx kqt qx L q ð qf lþ qt qq R qx If the thermophysical property of the particular region is assumed constant and also independent of time, we can write Eq. (3) as qt qt ¼ aq2 T qx 2 L C qðqf l Þ qt 1 qq R C qx where a ¼ k=qc P is the thermal diffusivity and C ¼ qc p is the heat capacity. In the solid, mushy, and liquid zones, the liquid fraction f l and enthalpy are, respectively, related as [20] 8 >< 0 H < H s H H s f l ¼ H s H H l >: H l H s 1 H > H l In Eq. (5), subscripts s and l stand for the solid and liquid zones, respectively. For the problem under consideration, the initial and the boundary conditions are the following: ð3þ ð4þ ð5þ ð6þ Initial condition: Tðx; 0Þ ¼ T 0 Boundary conditions: Tð0; tþ ¼ T W ¼ T 0 TðX; tþ ¼ T E In the present work, the LBM was used to solve the energy equation and the DTM was used to compute the radiative information for the same. In the following pages, first we provide a brief formulation of the DTM and then formulation of the LBM to solve the energy equation is presented along with the solution procedure.

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 283 2.1. Discrete Transfer Method Formulation The divergence of radiative heat flux qq R =qx is given by [21] qq R qx ¼ k a 4p rt 4 p G ð7þ where k a is the absorption coefficient, G is the incident radiation which is given by, and in the DTM, for a planar medium, is numerically computed from [22] G ¼ 2p Z p h IðhÞsin h dh 4p XM j¼1 I h j sin hj sin Dh 2 ð8þ where I is the intensity, h is the polar angle, and M is the number of intensities I considered over the complete span of h ð0 h pþ. In Eq. (8), intensity I at any location in the discrete direction h j is computed from the following recursive relation: I nþ1 ¼ I n exp bdx þ S av 1 exp b Dx ð9þ cos h cos h where, in the above equation, for the given direction h j ; Dx=cos h is the physical distance between the downstream point ðn þ 1Þ and the upstream point n, and b Dx=cos h is the optical distance between the two points. In Eq. (9), S av ¼ ðs n þ S nþ1 Þ=2 is the source function at the exact middle of the path leg between the upstream and downstream points. For an absorbing-emitting and anisotropically scattering medium, the source term at any point is given by S ¼ k art 4 p þ r s ð 4p G þ a cos h q RÞ ð10þ where r s is the scattering coefficient. It should be noted that the expression of the source function given by Eq. (10) in terms of G and the net radiative heat flux q R results from approximating the anisotropic phase function by a linear anisotropic phase function pðh 0! hþ ¼ 1 þ a cos h cos h 0. The net radiative flux q R is given by, and in the DTM is numerically computed from, [22] q ¼ 2p Z p h IðhÞcos h sin h dh 2p XM j¼1 I h j cos hj sin h j sin Dh ð11þ It is to be noted that in a 1-D planar geometry, radiation is azimuthally symmetric. Hence, azimuthal angle does not appear in Eqs. (8) and (11). In these equations, the term 2p accounts for integration over the azimuthal direction. In the DTM, intensities are traced from the boundaries. In Eq. (9), for a given direction h j, if the upstream point lies on the boundary, then I n ¼ I o, and its values

284 R. RAJ ET AL. have to be computed from the radiative boundary condition. For a diffuse-gray boundary with temperature T b and emissivity e b, the boundary intensity I o is given by I o ¼ e brt 4 b p ð þ 1 e bþ p 2p XM=2 j¼1 I h j cos hj sin h j sin Dh ð12þ With the radiative informationqq R =qx known from the above, the equivalence of the energy equation [Eq. (4)] in the LBM is deduced below. 2.2. Lattice Boltzmann Method Formulation The starting point of the LBM is the kinetic equation, which, for a 1-D geometry (Figure 1), is given by [7, 8] qn i ð~x; tþ qt þ~e i rn i ð~x; tþ ¼ X i i ¼ 1 and 2 ð13þ where n i is the particle distribution function denoting the number of particles at the lattice node ~x and time t moving in direction i with velocity ~e i along the lattice link Dx ¼ e i Dt connecting the neighbors and m is the number of directions in a lattice (in the present case, m ¼ 2) through which the information propagates. The term X i represents the rate of change of n i due to collisions. The discrete Boltzmann equation with Bhatanagar-Gross-Krook (BGK) approximation [23] is given by [7, 8] qn i ð~x; tþ qt þ~e i rn i ð~x; t Þ ¼ 1 h i s n i ð~x; tþ n ð0þ ð~x; tþ where s is the relaxation time and n ð0þ i is the equilibrium distribution function. For a given application, relaxation time s is different for different lattices [7, 8, 18]. In the present work, for a 1-D planar medium, a D1Q2 lattice was used. In heat transfer applications, the relaxation time s for the D1Q2 lattice (Figure 1) is computed from [7, 8] s ¼ a j~e i j 2 þ Dt 2 For this lattice, the two velocities e 1 and e 2 and their corresponding weights w 1 and w 2 are given by i ð14þ ð15þ e 1 ¼ Dx=Dt e 2 ¼ Dx=Dt ð16þ w 1 ¼ w 2 ¼ 1 2 ð17þ It is to be noted that for any kind of lattice, weights always satisfy the relation X w i ¼ 1 ð18þ i¼1;2

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 285 After discretization, Eq. (14) can be written as [7, 8] n i ð~x þ~e i Dt; t þ Dt Þ ¼ n i ð~x; tþ Dt h i n i ð~x; tþ n ð0þ i ð~x; tþ s This is the LB equation with BGK approximation that describes the evolution of the particle distribution function n i. The algorithm for Eq. (19) can be divided into two essential parts per time step: ð19þ The calculation of new distribution functions n i with respect to the right-hand side of Eq. (19), the so-called collision The streaming of the distribution functions to the next neighboring nodes, usually referred to as propagation In the case of heat transfer problems, the temperature is obtained after summing the particle distribution functions n i over all direction [7, 8], i.e., Tð~x; tþ ¼ X i¼1;2 n i ð~x; tþ ð20þ To process Eq. (19), equilibrium distribution functions n 0 i are required. For heat conduction problems, this is given by From Eqs. (18), (20), and (21), we also have X n ð0þ i i¼1;2 n ð0þ i ð~x; tþ ¼ w i Tð~x; tþ ð21þ ð~x; tþ ¼ X w i Tð~x; tþ ¼ Tð~x; tþ ¼ X i¼1;2 i¼1;2 n i ð~x; tþ ð22þ Equation (22), with definitions of temperature Tð~x; tþ and equilibrium distribution function n ð0þ i ð~x; tþ given in Eqs. (20) and (21), respectively, provides solution of a transient heat conduction problem in the LBM. To account for the liquid fraction (term 3) and volumetric radiation (term 4) in the energy equation [Eq. (4)], in the LBM formulation, Eq. (19) is modified to n i ð~x þ~e i Dt; t þ DtÞ ¼ n i ð~x; tþ Dt h i n i ð~x; tþ n ð0þ i ð~x; tþ s Dtw i U i Dtw iqq R C qx ð23þ where U i ¼ L f l ð~x; t þ DtÞ f l ð~x; tþ C Dt i ð24þ Equation (23) is the equivalent form of the energy equation [Eq. (4)] in the LBM formulation and it describes solidification of a semitransparent material taking place

286 R. RAJ ET AL. over a range of temperatures. It is to be noted that using the Chapman-Enskog multiscale expansion, the energy equation [Eq. (4)] can be deduced from Eq. (23). Further details on this for the solidification problem without radiation can be found in [10] and for conduction-radiation problems without phase change, the same can be found in [16, 17]. 2.3. Implementation of Boundary Conditions in the LBM In the LBM, at the boundaries, the particle distribution functions n i going inside the medium are unknown and these are computed from the boundary conditions. For the present problem, from Eqs. (21) and (22), at the west and the east boundaries, the unknown particle distribution functions n 1 ð0; tþ and n 2 ðx; tþ, are respectively given by At the west boundary: n 1 ð0; tþ ¼ n 2 ð0; tþ þ T W ð25þ At the east boundary: n 2 ðx; tþ ¼ n 1 ðx; tþ þ T E Details on implementation of the temperature as well as flux boundary conditions for various geometries can be found in [18]. 3. SOLUTION PROCEDURE IN THE LBM The solution domain is divided into a finite number of lattices. To satisfy the boundary conditions, centers of the boundary lattices are taken at the boundaries and thus the lattices along the boundary extend beyond the boundary by a distance equal to half the lattice size, i.e., Dx=2 (Figure 1). The radiative information qq R =qx is computed at the lattice centers and thus the control surfaces of the control volumes for the DTM lie along lattice centers (Figure 1) and thus a lattice and a control volume have an overlap of Dx=2 [16, 17]. The procedure for solving the phase-change problem using the LBM is as follows. 1. With the temperature field known, for the two directions in the lattice considered, calculate the equilibrium particle distribution function n ð0þ i ð~x; tþ using Eq. (21). For the first time level, set the particle distribution function n i ð~x; tþ ¼ n ð0þ i ð~x; tþ. 2. Calculate the divergence of radiative heat flux qq R =qx using Eq. (7). 3. For the Kth iteration, at the new time level t þ Dt, a. Compute n K i ð~x þ~e i Dt; t þ DtÞ n K i ð~x þ~e i Dt; t þ DtÞ ¼ n i ð~x; tþ Dt h i sðxþ n i ð~x; tþ n ð0þ i ð~x; tþ L Dtw i CðxÞ fl K 1 ð~x; t þ DtÞ f l ð~x; tþ Dt Dtw i qq R CðxÞ qx ð26þ b. Compute temperature field T K ðx; tþ using Eq. (20). c. Compute the total enthalpy H K ¼ CT K þ f K 1 l L.

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 287 d. Update the liquid fraction fl K using 8 0 H K < H >< s fl K H ¼ K H s H s H K H l >: H l H s 1 H K > H l e. Check for convergence: min f l K fl K 1 fl K 1 ; T K T K 1 T K 1 10 6 ð27þ If converged, go to step 4. Else go to 3a. 4. At this time level t þ Dt, propagate the particle distribution function n i to the neighboring lattices and apply the boundary conditions. 5. Compute the new temperature field Tð~x; t þ DtÞ using Eq. (20). 6. Terminate the process when the desired time level is reached. Else go to step 1. If the steady-state values are required, temperature is checked for convergence: Tðt þ DtÞ TðtÞ TðtÞ 10 6 It is to be noted that the relaxation time s is a function of the thermal diffusivity a [Eq. (15)] and in a phase-change problem, a and C are different for different zones. Hence, in Eq. (26), the value of s is different for different zones. 4. RESULTS AND DISCUSSION The LBM formulation presented in Section 2 was first validated by analyzing solidification of a planar layer in which the effect of radiation was absent. Next, by keeping the temperature of both boundaries higher than the melting temperature of the material and incorporating the radiative component, the code was validated for conduction-radiation problems without phase change [16, 17] (comparisons not shown here). After that, the solidification problem involving radiation was considered. In the following pages, results on front movement, temperature field, and liquid fraction are presented and analyzed for different parameters. To solve the problems, 500 lattices in the LBM and the same number of control volumes in the DTM were used for the grid-independent situation. For the rayindependent situation, 18 directions over the complete span of the polar angle was found sufficient in the DTM. For all sets of parameters, the problem was analyzed over a period of 10 s, during which distinct variations in the results were observed. In Figure 2, results for the LBM formulation for the solidification in the absence of radiation have been validated against the exact results of Cho and Sunderland [24]. In order to make the comparison, material properties were assumed to be k l =k s ¼ 0:6; k mz =k s ¼ 0:76; C l =C s ¼ 1:2, and C mz =C s ¼ 1:12. The numerical values of a for the three regions were calculated from the knowledge of the above

288 R. RAJ ET AL. Figure 2. Comparison of the LBM solution with the analytic solution [24]: (a) variation of temperature T=T W at time t ¼ 1.0 s; (b) front location with time for St ¼ 0.1; (c) front location with time for St ¼ 1.0. ratios. The temperatures T=T E at the solid mushy and mushy liquid interfaces were set at 0.6 and 0.8, respectively. The initial temperature for this validation purpose was taken as T 0 =T E ¼ 1:0 and for t > 0; T W =T E ¼ 0:0. It is to be noted that for the sake of comparison, for this particular case only, T E was chosen as a reference temperature; for the remaining results, T W was taken as the reference temperature. In Figure 2a, at time t ¼ 1 s and for Stanton numbers St ½¼C s ðt 0 T E Þ=LŠ ¼ 0:1 and 1.0, temperature T=T W results of the LBM have been compared with the exact results [24]. In Figures 2b and 2c, movement of the fronts (i.e, the mushy zone) has been compared for St ¼ 0.1 and 1.0, respectively. It can be seen that the LBM results compare very well with the exact results. In Figures 3 9, the LBM results for the solidification over a period of 10 s, taking radiation into account, have been presented. For results in these figures,

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 289 Figure 3. Variation of (a) nondimensional temperature T=T W and (b) liquid fraction f l at different instants; x ¼ 0:0, b ¼ 1:0, a ¼ 0.0, N ¼ 1.0.

290 R. RAJ ET AL. Figure 4. Variation of (a) nondimensional temperature T=T W and (b) liquid fraction f l for different values of the extinction coefficient b at time t ¼ 10.0 s; x ¼ 0:0, a ¼ 0:0, N ¼ 0:1.

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 291 Figure 5. Variation of (a) nondimensional temperature T=T W and (b) liquid fraction f l for different values of the scattering albedo x at time t ¼ 10.0 s; b ¼ 1:0; a ¼ 0:0; N ¼ 0:1.

292 R. RAJ ET AL. Figure 6. Variation of (a) nondimensional temperature T=T W and (b) liquid fraction f l for different values of the anisotropy factor a at time t ¼ 10.0 s; b ¼ 1:0; x ¼ 0:5; N ¼ 0:1.

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 293 Figure 7. Variation of (a) nondimensional temperature T=T W and (b) liquid fraction f l for different values of the conduction-radiation parameter N at time t ¼ 10.0 s; x ¼ 0:0; a ¼ 0:0; b ¼ 1:0.

294 R. RAJ ET AL. Figure 8. Variation of (a) nondimensional temperature T=T W and (b) liquid fraction f l for different values of the latent heat L for N ¼ 10 at time t ¼ 10.0 s; b ¼ 0:1; x ¼ 0:0; a ¼ 0:0.

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 295 Figure 9. Variation of (a) nondimensional temperature T=T W and (b) liquid fraction f l for different values of the latent heat L for N ¼ 0.01 at time t ¼ 10.0 s; b ¼ 0:1; x ¼ 0:0; a ¼ 0:0:

296 R. RAJ ET AL. thermophysical properties of the material, and interfacial and initial temperatures were taken the same as for the results in Figure 2. The value of a s was taken as 0.2268 m 2 =s, and with this value, a for the mushy and liquid zones was calculated. This value of a s was obtained by the known values of the conduction-radiation parameter N ¼ kb=4rtw 3 and C s, and the extinction coefficient b. Since a temperature of 0.0 K is impractical, in the problems involving radiation, for t > 0, we have assumed the east boundary temperature T E =T W ¼ 0:5. In the radiative calculations, both the boundaries were assumed black. For an absorbing-emitting medium ðx ¼ 0:0Þ with extinction coefficient b ¼ 1:0, latent heat L ¼ 1.0 kj=kg, and conduction-radiation parameter N ¼ 1, temperature T=T W variations in the three zones and the liquid fraction f l ¼ ½ðH H s Þ=ðH l H s ÞŠ in the mushy zone at different instants over a period of 10 s have been plotted in Figures 3a and 3b, respectively. It can be seen from Figure 3b that the thickness of the mushy zone increases with time, and hence as seen from Figure 3a, the temperature gradient in the mushy-zone decreases with time. Further, it can be seen from Figures 3a and 3b that the movement of the fronts slows down as time progresses. In Figures 4 9, variations of temperature T=T W and liquid fraction f l have been presented at t ¼ 10 s. The effect of the extinction coefficient b on variations of T=T W and f l is shown in Figures 4a and 4b, respectively. For results in this figure, x ¼ 0:0, N ¼ 0.1, and L ¼ 1.0 kj=kg. For lower values of b, the temperature profile T=T W in every zone is almost linear (Figure 4a) and the mushy zone is thicker (Figure 4b). Further, with decrease in the values of b, front movement is faster. For the lower values of b, the medium is less participating and thus the radiation effect is less prominent, as a result of which front movement is faster. In Figure 5, effect of the scattering albedo x has been shown. For results in this figure, b ¼ 1:0, N ¼ 0.1, and L ¼ 1.0 kj=kg. For higher values of x, in every zone, nonlinearity in temperature decreases (Figure 5a) and the front moves faster. This trend is because of the fact that for a higher value of x, distribution of radiative energy is more uniform in the medium. The effect of anisotropy a on T=T W and f l is shown in Figures 6a and 6b, respectively. In these figures, results have been shown for 100% forward (a ¼ þ1), isotropic ða ¼ 0Þ, and 100% backward ða ¼ 1Þ scattering situations. For results in these figures, x ¼ 0:5, N ¼ 0.1, and L ¼ 1.0 kj=kg. For a ¼ þ1, at a given instant, in the present case at t ¼ 10 s, radiation effect is more and a ¼ 1, the effect is less. Accordingly, nonlinearity in the T=T W profile is more for a ¼ þ1 and also the front movement is slower. For x ¼ 0:0, b ¼ 1:0, and L ¼ 1.0 kj=kg, effects of the conduction-radiation parameter N on temperature distribution T=T W and f l have been shown in Figure 7. It is seen from Figure 7a that in the radiation-dominated situation (lower values of N), nonlinearity in temperature in the three zones is more. For N ¼ 10, conduction is more prominent and in this case, in every zone, the temperature profile is almost linear, but the gradient is different because of different values of a. From Figure 7b it is seen that in the conduction-dominated situation, movement of the front is fast and also the thickness of the mushy zone is more. A similar trend was also observed at different instants of time.

SOLIDIFICATION OF A SEMITRANSPARENT PLANAR LAYER 297 Figure 10. Locations of solid mushy zone and liquid mushy zone interfaces with time for latent heat L ¼ 1.0 and 10.0; b ¼ 1:0; x ¼ 0:0; a ¼ 0:0; N ¼ 1:0. Observations on temperature variation and front movement in Figures 4 7 are consistent. When radiative energy is more, nonlinearity in temperature in every zone is found to be more and the front movement was slow. The thickness of the mushy zone is also found to be less. The effect of the latent heat L on T=T W and f l is shown in Figures 8 and 9. For x ¼ 0:0 and b ¼ 0:1, these results are shown for the conduction-dominated N ¼ 10 and radiation-dominated N ¼ 0.01 situations in Figures 8 and 9, respectively. When L is large, more energy is required for the phase change to occur, and accordingly the front movement will be slow and thickness of the mushy zone will also be less. When the thickness of the mushy zone is less, the temperature gradient will be more. This trend is observed in Figures 8 and 9. By comparing Figures 8 and 9, we find that in the radiation-dominated case (Figure 9), for a given value of L, nonlinearity in temperature is more and front movement is slower. This observation is in line with that in Figure 7. The locations of the solid mushy and mushy liquid interfaces over a period of 10 s have been shown in Figure 10. For x ¼ 0:0, b ¼ 1:0, and N ¼ 1, these results have been shown for two values of L. It is seen from this figure that the thickness of the mushy zone increases with time. Further for a given time, thickness is more for the lower values of L. 5. CONCLUSIONS The LBM was used to analyze solidification of a semitransparent planar medium. The radiative information was computed using the DTM. Effects of

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