Heat transfer analysis for the solidification of a binary eutectic system under imposed movement of the material
Tóm tắt
The current article deals with a moving boundary problem describing solidification of a eutectic alloy in a semi-infinite medium. The process of solidification of a eutectic alloy is considered under imposed movement of material in the mushy zone in place of liquidus zone due to imposing an insulated boundary condition at liquidus front. It is assumed that solid fraction
$$f_{\text {s}}$$
has a linear, quadratic and cubic relationship with distance within the mushy zone between the solidus and liquidus. An exact solution of the problem is obtained with the help of similarity transformation. A numerical example of the solidification of Al–Cu alloy is presented to demonstrate the application of the current analysis. Solidification of eutectic system is discussed in the absence of material movement and in the presence of material movement in each case of solid fraction distribution. Thus, the temperature profile and moving interfaces in each region are shown for different Peclet numbers Pe. In addition, heat extraction Q from the surface
$$x=0$$
is shown with respect to the time for different Pe. The novelty of the current study is transition process becomes fast when material moves in the direction of freeze and hence time for complete freezing of the alloy reduces. Moreover, mushy zone becomes thinner when material moves in the direction of freeze. A comparative study and error analysis between the present work and Tien and Geiger (ASME J Heat Transf 89:230–233, 1967)[9] in linear case of solid fraction are presented in figure and tables. The application of the present analysis is useful for both eutectic and solid solution alloys.
Tài liệu tham khảo
Viskanta R. Heat transfer during melting and solidification of metals. ASME J Heat Transf. 1988;110(4b):1205–19.
Fukusako S, Seki N. Fundamental aspects of analytical and numerical methods on freezing and melting heat-transfer problems. Ann Rev Heat Transf. 1987;1:351–402.
Rabin Y, Shitzer A. Numerical solution of the multidimensional freezing problem during cryosurgery. J Biomech Eng. 1998;120:32–7.
Crank J. Free and moving boundary problems. Clarendon, Oxford; 1984.
Gupta SC. The classical Stefan problem, basic concepts, modelling and analysis. Elsevier; 2003.
Carslaw HS, Jaeger JC. Conduction of heat in solids. Oxford Science Publication; 1986.
Barry GW, Goodling JS. A Stefan problem with contact resistance. ASME J Heat Transf. 1987;109:820–5.
Cho SH, Sunderland JE. Heat-conduction problems with melting or freezing. ASME J Heat Transf. 1969;91:421–6.
Tien RH, Geiger GE. A heat-transfer analysis of the solidification of a binary eutectic system. ASME J Heat Transf. 1967;89:230–3.
Tien RH, Geiger GE. The unidimensional solidification of a binary eutectic system with a time-dependent surface temperature. ASME J Heat Transf. 1968;90:27–31.
Ozisik MN, Uzzell JC Jr. Exact solution for freezing in cylindrical symmetry with extended freezing temperature range. J Heat Transf. 1979;101:331–4.
Juaifer HJA, Ayani MB, Poursadegh M. Melting process of paraffin wax inside plate heat exchanger: experimental and numerical study. J Therm Anal Calorim. 2020;140:905–16.
Shakeri F, Dehghan M. Solution of delay differential equations via a homotopy perturbation method. Appl Math Comput. 2008;48:486–98.
Zielinski DP, Voller VR. A control volume finite element method with spines for solutions of fractional heat conduction equations. Numer Heat Transf Part B: Fundam. 2016;70:503–16.
Myres TG, Font F. On the one-phase reduction of the Stefan problem with a variable phase change temperature. Int Commun Heat Mass Transf. 2015;61:37–41.
Rajeev, Rai KN, Das S. Numerical solution of a moving boundary problem with variable latent heat. Int J Heat Mass transf. 2009;52:1913–7.
McCue SW, Wu B, Hill JM. Classical two-phase Stefan problem for spheres. Proc R Soc A?: Math Phy Eng Sci. 2008;464:2055–76.
Li X, Xu M, Jiang X. Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition. Appl Math Comput. 2009;208:434–9.
Ahmed SG, Meshrif SA. A new numerical algorithm for 2D moving boundary problems using a boundary element method. Comput Math Appl. 2009;59:1302–8.
Yadav S, Kumar D, Rai KN. Finite element Legendre wavelet Galerkin approach to inward solidification in simple body under most generalized boundary condition. Zeit fur Naturf. 2014;69:501–10.
Ribera H, Myers TG, MacDevette MM. Optimising the heat balance integral method in spherical and cylindrical Stefan problems. Appl Math Comput. 2019;354:216–31.
Singh J, Jitendra, Rai KN. Legendre wavelet based numerical solution of variable latent heat moving boundary problem. Math Comput Simul. 2020;178:485-500.
Chaurasiya V, Kumar D, Rai KN, Singh J. A computational solution of a phase-change material in the presence of convection under the most generalized boundary condition. Thermal Sci Eng Proc. 2020;20:100664.
Bhowmick D, Randive PR, Pati S, Agrawal H, Kumar A, Kumar P. Natural convection heat transfer and entropy generation from a heated cylinder of different geometry in an enclosure with nonuniform temperature distribution on the walls. J Therm Anal Calorim. 2020;141:839–57.
Khalid MZ, Zubair M, Ali M. An analytical method for the solution of two phase Stefan problem in cylindrical geometry. Appl Math Comput. 2019;342:295–308.
Kakitani R, De Gouveia GL, Garcia A, Cheung N, Spinelli JE. Thermal analysis during solidification of an AlCu eutectic alloy: interrelation of thermal parameters, microstructure and hardness. J Therm Anal Calorim. 2019;137:983–96.
Ali HM. Recent advancements in PV cooling and efficiency enhancement integrating phase change materials based systems: a comprehensive review. Solar Energy. 2020;197:163–98.
Tariq SL, Ali HM, Akram MA, Janjua MM, Ahmadlouydarab M. Nanoparticles enhanced phase change materials (NePCMs): a recent review. Appl Therm Eng. 2020;176:115305.
Turkyilmazoglu M. Stefan problems for moving phase change materials and multiple solutions. Int J Therm Sci. 2018;126:67–73.
Ceretani AN, Tarzia DA. Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model. Comput Appl Math. 2018;37:2201–17.
Parhizi M, Jain A. Solution of the phase change Stefan problem with time-dependent heat flux using perturbation method. ASME J Heat Transf. 2019;141:1–5. https://doi.org/10.1115/1.4041956.
Barannyk L, Williams SDV, Ogidan OI, Crepeau JC, Sakhnov A. On the Stefan problem with internal heat generation and prescribed heat flux conditions at the boundary. Paper No: HT2019-3703, V001T10A014. 2019. p. 10. https://doi.org/10.1115/HT2019-3703.
Assuncao M, Vynnycky M, Mitchel SL. On small-time similarity-solution behaviour in the solidification shrinkage of binary alloys. Eur J Appl Math. 2020. https://doi.org/10.1017/S0956792520000091.
Voller VR. A similarity solution for the solidification of a multicomponent alloy. Int J Heat Mass Transf. 1997;40(12):2869–77.
Voller VR. A numerical scheme for solidification of an alloy. Can Met Quart. 1998;37:169–77.
Rehmann T, Ali HM, Janjua MM, Sajjad U, Yan WM. A critical review on heat transfer augmentation of phase change materials embedded with porous materials/foams. Int J Heat Mass Transf. 2019;135:649–73.