AN INTERFACIAL TRACKING MODEL FOR CONVECTION-CONTROLLED MELTING PROBLEMS

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1 Numerical Heat Transfer, Part B, 59: , 2011 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / AN INTERFACIAL TRACKING MODEL FOR CONVECTION-CONTROLLED MELTING PROBLEMS Qicheng Chen 1, Yuwen Zhang 2, and Mo Yang 1 1 College of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai, People s Republic of China 2 Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, Missouri, USA An interfacial tracking model is used to simulate melting in an enclosure in the presence of natural convection. It obtains the melting-front location by calculating the energy balance at the solid liquid interface and is a simple and convenient method to solve the solid liquid phase-change problem. It combines the advantages of the both deforming and fixed grids method and can handle natural convection controlled melting and solidification problems. It is demonstrated through comparisons with various experiments and numerical results that the interfacial tracking model can be used to solve melting problems controlled by natural convection at different Rayleigh numbers ranging from 10 5 to INTRODUCTION Modeling and numerical simulation for solid liquid phase-change problems has become an active area in the last several decades due to its wide application in energy systems [1] as well as thermal manufacturing, such as laser drilling [2], laser welding [3], and selective laser sintering [4], building systems [5], such as encapsulated materials used in plasterboard or packed beds [6], and in thermal energy storage systems [7 9]. Numerical simulation of convection-dominated phase-change problems is a challenging task. These problems are not only strongly nonlinear, i.e., the fluid flow in the melt dictates the heat transfer, they also involve moving boundaries. A variety of conventional numerical techniques have been developed for solving these problems: enthalpy [10, 11], apparent heat capacity [12], isotherm migration [13], and coordinate transformation methods [14 19]. These methods have been introduced by researchers to overcome the difficulties in handling moving boundaries. Previous works on multidimensional moving boundary problems include those of Received 28 July 2010; accepted 24 November Support for this work by the U.S. National Science Foundation under Grant CBET and the Chinese National Natural Science Foundation under Grants and is gratefully acknowledged. Address correspondence to Yuwen Zhang, Department of Mechanical and Aerospace Engineering, University of Missouri, E3411 Lafferre Hall, Columbia, MO 65211, USA. zhangyu@missouri.edu 209

2 210 Q. CHEN ET AL. NOMENCLATURE a nb coefficient in Eq. (22) a p coefficient in Eq. (22) A ratio of height and width (¼W=H) b term in Eq. (22) C p heat capacity, J=m 3 K d e coefficient in Eq. (23) d n coefficient in Eq. (24) f p liquid fraction Fo Fourier number (¼n t=h 2 ) g gravitational acceleration, m=s 2 Gr Grashof number (¼gbDTH 3 =n 2 ) h s latent heat of fusion, J=kg H height of wall, m k thermal conductivity in the liquid, W=mK K dimensionless thermal conductivity (¼k=k ) K s ratio of thermal conductivity (¼k s =k ) bk modified dimensionless thermal conductivity Nu Nusselt number at the heated wall (¼hH=k ) p pressure, Pa P dimensionless pressure [¼ðp þ q 1 gyþh 2 =qn 2 l ] Pr Prandtl number of the liquid PCM (¼n =a ) Ra Rayleigh number (¼Gr Pr) s location of solid liquid interface, m S dimensionless location of solid liquid interface (¼s=H) S 0 dimensionless location of solid liquid interface at the last time step Ste Stefan number [¼c (T h T m )=h s ] t time, s T temperature, K T c temperature of the cold wall, K T h temperature of the heated wall, K T i initial temperature, K melting point, K T m u velocity component in the x direction, m=s u I solid liquid interfacial velocity (¼qs=qt), m=s U dimensionless velocity in the x direction (¼uH=n ) U I dimensionless solid liquid interfacial velocity (¼u I H=n ) v velocity component in the y direction, m=s V dimensionless velocity in the y direction (¼vH=n ) Vol volume of the fusion Vol 0 volume of the enclosure W width of the enclosure, m x dimensional coordinate, m X dimensionless coordinate (¼x=H) y dimensional coordinate, m Y dimensionless coordinate (¼y=H) a thermal diffusivity, m 2 =s h dimensionless temperature [¼(T T m )= (T h T m )] m viscosity of the liquid PCM, kg=ms n kinematic viscosity, m 2 =s q density, kg=m 3 s dimensionless time (¼n t=h 2 ) Subscripts e east face of the control volume E east i initial I interface liquid m melting point n north face of the control volume N north s solid S south w west face of the control volume W west Duda et al. [20], Saitoh [21], Gong and Mujumdar [22], Cao et al. [23], Khillarkar et al. [24], Chatterjee and Prasad [25], and Beckett et al. [26]. These numerical models can be divided into two groups [27]: deforming-grid schemes (or strong numerical solutions) and fixed-grid schemes (or weak numerical solutions). Deforming-grid schemes transform the solid and liquid phases into fixed regions by using a coordinate transformation technique. The governing equations and boundary conditions are complicated due to the transformation. These schemes have been used successfully to solve multidimensional problems with or without natural convection. Ho and Viskanta [28] solved heat transfer during melting in a

3 INTERFACIAL TRACKING MODEL FOR MELTING PROBLEMS 211 rectangular cavity heated from the side isothermally using a deforming-grid scheme. Their results clearly revealed that the rates of melting and heat transfer were greatly affected by the buoyancy-driven convection in the liquid domain. Gadgil and Gobin [29] used a deforming-grid scheme to analyze the two-dimensional melting in a rectangular enclosure in the presence of natural convection. The disadvantage of deforming grid schemes is that they require significant amounts of computational time. On the other hand, the fixed-grid schemes use one set of governing equations for the whole computational domain, including both liquid and solid phases, and the solid liquid interface is later determined from the temperature distribution. This simplicity makes the computation much faster than deforming-grid schemes, while it still provides reasonably accurate results [30]. There are two major methods in the fixed-grid schemes: the enthalpy method [31] and the equivalent heat capacity method [32, 33]. The enthalpy method can handle solid liquid phase changes at a fixed melting point as well as in a range of temperature. When it is used to model phase change at a fixed temperature, it has difficulty with temperature and interfacial location oscillation. The equivalent heat capacity method must assume that phase change occurs in a range of temperature. In order to simulate phase change at fixed melting point, the temperature range must be very small. However, the equivalent heat capacity method encounters convergence problems when the range of phase-change temperature is small. Cao and Faghri combined the advantages of the enthalpy and equivalent heat capacity methods and proposed a temperature-transforming model [34]. Similar to the equivalent heat capacity method, it is also assumed that solid liquid phase change occurs in a range of temperatures. However, it works for phase changes occurring in any ranges of phase-change temperature. Therefore, the model can properly handle phase change occurring over a temperature range or at a single temperature. For conduction-controlled solid liquid phase-change problems, the converged solution can be obtained with any grid size and time step. For convection-controlled phase-change problems, Ma and Zhang [35] suggested that in order to use the temperature-transforming model efficiently and obtain convergent and reasonable results, the grid size must be selected with a suitable time step (which should not be too small). In this article, a fixed-grid interfacial tracking method [36] is used to solve the solid liquid phase-change problem in a rectangular enclosure. The solid liquid phase change is a moving-boundary problem due to the existence of the moving solid liquid interface. The interfacial tracking method also combines the advantages of the deforming-grid and fixed-grid methods. The location of the solid liquid interface is obtained by energy balance at the solid liquid interface at every time step. The results obtained using the interfacial tracking method will be compared with existing experimental and numerical results. PHYSICAL MODEL Figure 1 shows the physical model of the problem: melting inside a rectangular enclosure. The top and bottom walls are insulated, whereas the left and right walls are kept at constant temperatures of T h and T c, respectively. The initial temperature is equal to T i, which is below the melting point, T m.

4 212 Q. CHEN ET AL. The following assumptions are made: 1. The phase-change material (PCM) is pure and homogeneous. 2. The liquid phase of the PCM is a Newtonian and incompressible fluid. 3. The thermophysical properties of the solid and liquid PCM are independent of temperature, except for the temperature-dependent liquid density in buoyancy term (Boussinesq approximation). 4. The volume change due to solid liquid phase change is negligible. 5. The natural convection in the liquid phase is laminar and two-dimensional. The dimensional governing equations are as follows. Continuity equation: Momentum equations: qðquþ qt qðqvþ qt Figure 1. Physical model. qu qx þ qv qy ¼ 0 þ qðqu2 Þ qx þ qðquvþ ¼ qp qy qx þ q qx þ qðquvþ qx þ qðqvvþ ¼ qg qp qy qy þ q qx m qu þ q qx qy m qu qy m qv þ q qx qy m qv qy ð1þ ð2þ ð3þ Energy equation: qc p qt qt þ u qt qx þ v qt qy ¼ q qx k qt qx þ q qy k qt qy ð4þ

5 The boundary conditions of Eqs. (1) (4) are as follows. Left vertical wall: Right vertical wall: Bottom horizontal wall: INTERFACIAL TRACKING MODEL FOR MELTING PROBLEMS 213 x ¼ 0 T ¼ T h u ¼ 0 v ¼ 0 ð5þ x ¼ W T ¼ T c u ¼ 0 v ¼ 0 ð6þ y ¼ 0 qt qy ¼ 0 u ¼ 0 v ¼ 0 ð7þ Top horizontal wall: Melting front: y ¼ H " x ¼ s 1 þ qs # 2 qy qt qy ¼ 0 u ¼ 0 v ¼ 0 ð8þ x ¼ s T ¼ T m ð9þ k s qt s qx k qt qx ¼ q h s qs qt Equation (10) is the energy balance at the solid liquid interface, and the term 1 þðqs=qyþ 2 is to account for the effect of inclination of the solid liquid interface. Introducing the following nondimensional variables, X ¼ x H s ¼ t n H 2 Y ¼ y H h ¼ T T m T h T m S ¼ s H K s ¼ k s k ð10þ U ¼ u H n V ¼ v H n P ¼ ðp þ q 1gyÞH 2 Ste ¼ c ðt h T m Þ h s qn 2 Sc ¼ T c T m T h T c ð11þ the governing equations can be nondimensionalized as qu qx þ qv qy ¼ 0 qu qs þ qu 2 qx þ qðuvþ qy ¼ q2 U qx 2 þ q2 U qy 2 qv qs þ quv qx þ qv 2 qy ¼ q2 V qx 2 þ q2 V qy 2 þ Gr h ð12þ ð13þ ð14þ

6 214 Q. CHEN ET AL. qh qs þ U qh qx þ V qh qy ¼ 1 Pr! q 2 h qx 2 þ q2 h qy 2 ð15þ where Gr is the Grashof number based on height, Gr ¼ gbdth 3 =n 2 The boundary conditions of Eqs. (12) (15) are X ¼ 0 h ¼ 1 U ¼ 0 V ¼ 0 ð16þ X ¼ A h ¼ S c U ¼ 0 V ¼ 0 ð17þ Y ¼ 0 Y ¼ 1 Ste Pr qh qy ¼ 0 U ¼ 0 V ¼ 0 ð18þ qh qy ¼ 0 U ¼ 0 V ¼ 0 ð19þ X ¼ S h ¼ 0 ð20þ " 1 þ qs # 2 qy qh qx K s qh s ¼ U I K qx NUMERICAL SOLUTION AND INTERFACIAL TRACKING Discretization of Governing Equations The above two-dimensional governing equations are discretized by applying a finite-volume method [37], in which conservation laws are applied over finite-sized control volumes around grid points and the governing equations are then integrated over the control volume. A staggered grid arrangement is used in the discretization of the computational domain in the momentum equations. A power-law scheme is used to discretize convection=diffusion terms in the momentum and energy equations. The main algebraic equation resulting from this control-volume approach is in the form of ð21þ a P / P ¼ X a nb / nb þ b ð22þ where / P represents the value of general variable / (U, V,orh) at the grid point P, / nb are the values of the variable at P s neighbor grid points, and a P, a nb,andb are corresponding coefficients and terms derived from the original governing equations. The numerical simulation is accomplished using the SIMPLE algorithm [37]. The velocity-correction equations for corrected U and V in the algorithm are U e ¼ U e þ d eðp 0 P P0 E Þ V n ¼ V n þ d nðp 0 P P0 N Þ ð23þ ð24þ

7 INTERFACIAL TRACKING MODEL FOR MELTING PROBLEMS 215 where e and n represent the control-volume faces between grid P and its east neighbor E and grid P and its north neighbor N, respectively. In the present work, the governing equations are used for the entire computational domain. The velocity in the solid region is set to zero by letting a p ¼ and b ¼ 0 in Eq. (22) for the momentum equation. Interfacial Tracking Method Wang and Matthys [38] proposed an effective interface tracking method by introducing an additional node at the interface, which divides the control volume containing the interface into two small control volumes. In this work, an interfacial tracking method [36] that was developed for conduction-controlled melting of metal film under irradiation by a femtosecond laser will be extended to be able to handle convection-controlled solid liquid phase-change problems. This method is an alternative approach that does not require dividing the control volume containing the interface but can still accurately account for the energy balance at the interface. For a control volume that contains a solid liquid interface, the dimensionless temperature h P is numerically set as the interfacial temperature (h I ¼ 0) by letting a P ¼ and b ¼ 0 in Eq. (22) with / ¼ h. The above treatment yields an accurate result when the solid liquid interface is exactly at grid point P, as shown in Figure 2a. When the interfacial location within the control volume is not at grid point P, there are two scenarios as shown in Figure 2b: The interface is on the right side of the grid point; or (c) the interface is on the left side of the grid point. With scenario (b), a modified dimensionless thermal conductivity, bk w, at the face of the control volume w is introduced by equating the actual heat flux across the face of the control volume w, based on the position and temperature of the main grid point P [39], Considering h P ¼ h I, Eq. (25) becomes K w ðh W h I Þ K ¼ b w ðh W h P Þ þðf P 0:5ÞðDXÞ P ð25þ bk w ¼ þðf P 0:5ÞðDXÞ P K w ð26þ Similarly, a modified thermal conductivity at face e of the control volume can be obtained as bk e ¼ ðdxþ e ðdxþ e ðf P 0:5ÞðDXÞ P K e ð27þ With the scenario shown in Figure 2c, a modified dimensionless thermal conductivity, bk w, at the face of control volume w is obtained by a similar procedure: bk w ¼ ð0:5 f P ÞðDXÞ P K w ð28þ

8 216 Q. CHEN ET AL. Figure 2. Grid system near the liquid solid interface. A modified thermal conductivity at face e of the control volume is bk e ¼ ðdxþ e ðdxþ e þð0:5 f P ÞðDXÞ P K e ð29þ Combining the situations of Figures 2b and 2c, the universal modified dimensionless thermal conductivity can be expressed as bk w ¼ ð0:5 f P ÞðDXÞ P K w ð30þ

9 INTERFACIAL TRACKING MODEL FOR MELTING PROBLEMS 217 ðdxþ bk e ¼ e K e ðdxþ e þð0:5 f P ÞðDXÞ P ð31þ The modified thermal conductivities defined by Eqs. (30) and (31) are used to obtain the coefficients for grid points W and E, which allows the temperature at the main grid P to be used in the computation regardless of the location of the interface within the control volume. To determine the interfacial location, the energy balance at the interface, Eq. (21), can be discretized, and the solid liquid interfacial velocity can be obtained as ( U I ¼ Ste 1 þ S S ) 2 S Pr ðdyþ s K w ðh W h I Þ K e ðh I h E Þ ð0:5 f P ÞðDXÞ P ðdxþ e þð0:5 f P ÞðDXÞ P ð32þ where S S is the interfacial location at the grid at the south of grid P. The interfacial location is then determined using S ¼ S 0 þ U I Ds and the liquid fraction in the control volume that contains the interface is f ¼ S X P ðdxþ P =2 ðdxþ P Numerical Solution Procedure The numerical solution starts from time s ¼ 0. Once the temperature at the first control volume from the heated surface obtained exceeds the melting point, the temperature of the first control volume sets at the melting point by letting a P ¼ and b ¼ 0inEq.(22) with / ¼ h. After melting is initiated, the following iterative procedure is employed to solve for the interfacial velocity and the interfacial location at each time step. ð33þ ð34þ 1. Assume an interface velocity U I, using the velocity U I of the last time step as initial value. 2. Determine the new interface location, S, from Eq. (33). 3. Obtain the modified dimensionless thermal conductivities, bk w and bk e, at the faces of control volumes w and e from Eqs. (30) and (31). 4. Solve Eqs. (12) (15) to obtain the temperature distributions and obtain the new interface velocity U I by using Eq. (32). 5. Compare the newly obtained U I and the assumed value in step 1. If the difference is less than 10 5 and the maximum difference between the temperatures obtained from two consecutive iterative steps is less than 10 5, the interfacial location for the current step is obtained. If not, the process is repeated until the convergence criterion is satisfied.

10 218 Q. CHEN ET AL. RESULTS AND DISCUSSION The interfacial tracking method will be validated by comparing its results with experimental results as well as other numerical results. The first numerical simulation is performed under conditions that are the same as the experiments carried out by Okada [40]. The subcooling parameter is 0.01, which means that the initial temperature is very close to the melting point. The Rayleigh number is , the Stefan number is 0.045, and the Prandtl number is After a grid number and time step test, the grid number used in the simulation was and the time step was 0.1. Figures 3 and 4 show the positions of melting fronts obtained by the interfacial tracking method compared with experimental results by Okada [40] at different dimensionless times. At the early time of s ¼ 39.9, the melting interfacial velocity is slower than the experimental result at the bottom of the enclosure; however, the melting interfacial velocity obtained by the numerical solution is faster than the experimental results at the top of the enclosure. The volume faction of the liquid obtained by the interfacial tracking method is about 3.3% lower than that of the experimental results. At a longer time of s ¼ 78.68, the interfacial tracking method gives results very close to the experimental results. The difference between the predicted and measured liquid volume fraction is only 0.2%. The results in Figures 3 and 4 demonstrate that the interfacial tracking method can correctly solve melting problems at low Rayleigh number. In order to make sure that the interfacial tracking method is also valid at high Rayleigh numbers, an additional numerical simulation was performed based on the conditions specified by Ho and Viskanta [28]. The aspect ratio of the enclosure is A ¼ The subcooling parameter is equal to 0.01, the Rayleigh number is , the Stefan number is 0.09, and the Prandtl number is Since the top surface of the PCM is a free surface in [28], the velocity boundary condition at the top Figure 3. Comparison of the locations of the melting fronts (Ra ¼ , s ¼ 39.9).

11 INTERFACIAL TRACKING MODEL FOR MELTING PROBLEMS 219 Figure 4. Comparison of the locations of the melting fronts (Ra ¼ , s ¼ 78.68). specified in Eq. (19) is changed to qu Y ¼ 1 qy ¼ 0 ð35þ Since the boundary layer at the heated wall and the solid liquid interface decreases with increasing Rayleigh number, a finer grid must be used for the case of higher Rayleigh number to capture the behavior of the natural convection. After a grid number test, the grid number used in the simulation was 80 80, which was different from Ho and Viskata s Figure 5 shows the comparison between the locations of solid liquid interfaces obtained using the present model and that of Ho and Viskanta [28]. At early dimensionless time of s ¼ , the agreement is very good at the bottom and top of the rectangular cavity. As melting continues to the dimensionless time of s ¼ , the effect of the natural convection makes the top portion of the liquid region wider and the results obtained by the numerical and experimental results once again agree very well. When the dimensionless time s ¼ , the effect of natural convection on the melting becomes very strong and the interfacial tracking method is capable of simulating the strong natural convection during melting. Comparisons among all the numerical results obtained by the interfacial tracking method and experimental results indicate that the interfacial tracking method can obtain very good results even at higher Rayleigh number. The molten volume fraction of liquid is a very important parameter to assess the overall melting rate. The predicted instantaneous molten volume fraction, Vol=Vol 0, is evaluated from the solid liquid interface position by a numerical

12 220 Q. CHEN ET AL. Figure 5. Comparison of the locations of the melting fronts (Ra ¼ 10 8 ). integration of the instantaneous position and is defined as Vol=Vol 0 ¼ 1 Z H sdy¼ 1 Z A SdY ð36þ HW 0 A 0 Figure 6 shows a comparison of the molten volume fraction obtained from the interfacial tracking method and the experiments [28]. It can be seen that the liquid volume fraction obtained by numerical simulation agrees very well with the measured results. This comparison once again demonstrates that the interfacial tracking method can be used to solve melting problems at high Rayleigh number. Additional simulation was carried out under the same conditions as Gadgil and Gobin s [29] numerical solution based on a deformed-grid approach. Subcooling was not considered, and the Rayleigh number was The Stefan number was 0.2 and the Prandtl number was 50. The grid number was and the time step was A larger grid number was used because the Stefan number of 0.2 is higher than the 0.09 for Figure 5, and natural convection under higher Stefan number is stronger. Figure 7 shows a comparison of results obtained from the interfacial tracking method and Gadgil and Gobin s [29] numerical solution based on a deformed grid. The results obtained by the present numerical solution are represented by various lines, while the results obtained by Gadgil and Gobin s [29] are represented by symbols. At early dimensionless time of s ¼ 0.54, the agreement of the results is excellent. As time progresses to s ¼ 0.76, the natural convection plays a more important role

13 INTERFACIAL TRACKING MODEL FOR MELTING PROBLEMS 221 Figure 6. Predicted and experimental molten volume fractions as a function of dimensionless time. and the two results are still very close to each other. At later dimensionless time of s ¼ 1.10, the result obtained by the interfacial tracking method is slower than that of the Gadgil and Gobin s [29] numerical result in the lower portion of the rectangular. Gadgil and Gobin carried out their simulation by dividing the transient process into a large number of quasi-static steps. In each quasi-static step, steady-state natural convection in the liquid phase was calculated. On the other hand, the present interfacial tracking method does not require the quasi-steady-state assumption, Figure 7. Time evolution of the melting-front position.

14 222 Q. CHEN ET AL. Figure 8. Heat transfer from the heated wall as a function of time. and transient natural convection was considered. Therefore, the location of the melting front obtained by the interfacial tracking method is slower than that obtained by Gadgil and Gobin [29]. However, the overall melting rates obtained by the present method and that of [29] for all three different instances agreed very well. The Nusselt number at the heated wall was further calculated. The Nusselt number gives the heat transfer from the heated wall to the liquid domain, and it can be obtained as Nu ¼ hh k where the heat transfer coefficient h can be obtained from k qt qx ¼ hðt h T m Þ x¼0 Substituting Eq. (38) into Eq. (37), the dimensionless Nusselt number can be obtained as Nu ¼ qh qx X¼0 ð37þ ð38þ ð39þ The average Nusselt number on the heated wall is Z 1 qh Nu ¼ qx dy X¼0 0 ð40þ Figure 8 shows a quantitative comparison of the average Nusselt number at the heated wall obtained by different methods. During early time, two results are close to each other. After the dimensionless time reaches 1, the Nusselt number obtained by Gadgil and Gobin [29] becomes constant very quickly. On the contrary, the Nusselt number obtained by the present method decreases slowly after reaching

15 INTERFACIAL TRACKING MODEL FOR MELTING PROBLEMS 223 dimensionless time 1. This different trend may also be attributed to the different ways that the transient effect is treated. CONCLUSIONS An interfacial tracking method was proposed and tested for two dimensional convection-controlled melting problems at different Rayleigh numbers and Stefan numbers. The interfacial tracking method combines the advantages of both fixedand deforming-grid schemes. It can successfully solve multidimensional problems with or without natural convection. It retains the simplicity of grid construction of fixed-grid schemes while maintaining the accuracy of the energy balance at the solid interface of deforming grids. This simplicity makes the computation much faster than using the deforming-grid method. It also eliminates the need for of assuming a range of phase-change temperature as in other fixed grid methods. It does not product nonlinear oscillation of the temperature and interfacial location as enthalpy and equivalent heat capacity methods do. The interfacial tracking method can solve melting problems accurately at different Rayleigh and Stefan numbers. The results are much closer to the experimental results and other numerical results, and it also yields better results for cases with higher Rayleigh number. However, the interfacial tracking method requires more grids to deal with energy balance at the solid liquid interface. Although the method is presented for 2-D melting in a rectangular cavity, it can be extended to any other geometric configuration as long as the shape of the interface can be described by a continuous differentiable function. REFERENCES 1. Y. Zhang and A. Faghri, Heat Transfer Enhancement in Latent Heat Thermal Energy Storage System by Using an External Radial Finned Tube, J. Enhanced Heat Transfer, vol. 3, pp , Y. Zhang and A. Faghri, Vaporization, Melting and Heat Conduction in the Laser Drilling Process, Int. J. Heat Mass Transfer, vol. 42, pp , K. Mundra, T. DebRoy, and K. M. Kelkar, Numerical Prediction of Fluid Flow and Heat Transfer in Welding with Moving Heat Source, Numer. Heat Transfer A, vol. 29, pp , Y. Zhang, A. Faghri, C. W. Buckley, and T. L. Bergman, Three-Dimensional Sintering of Two-Component Metal Powders with Stationary and Moving Laser Beams, ASME J. Heat Transfer, vol. 122, pp , V. V. Tyagi and D. Buddhi, PCM Thermal Storage in Buildings: A State of Art, Renewable Sustainable Energy Rev., vol. 11, pp , A. F. Regin, S. C. Solanki, and J. S. Saini, Heat Transfer Characteristics of Thermal Energy Storage System Using PCM Capsules: A Review, Renewable Sustainable Energy Rev., vol. 12, pp , A. Sharma, V. V. Tyagi, C. R. Chen, and V. Buddhi, Review on Thermal Energy Storage with Phase Change Materials and Applications, Renewable Sustainable Energy Rev., vol. 13, pp , M. M. Farid, A. M. Khudhair, S. A. K. Razack, and S. Al- Hallaj, A Review on Phase Change Energy Storage: Materials and Applications, Energy Conversion Management, vol. 45, pp , 2004.

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