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Analytical Solutions to the Stefan Problem with Internal Heat Generation

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Analytical Solutions to the Stefan Problem with Internal Heat Generation Applied Thermal Engineering 103 (2016) 443–451 Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng Research Paper Analytical solutions to the Stefan problem with internal heat generation ⇑ David McCord a, John Crepeau a, , Ali Siahpush b, João Angelo Ferres Brogin c a Department of Mechanical Engineering, University of Idaho, 875 Perimeter Drive, MS 0902, Moscow, ID 83844-0902, United States b Department of Integrated Engineering, Southern Utah University, 351 W. University Blvd., Cedar City, UT 84720, United States c Departamento de Engenharia Mecânica, São Paulo State University, Ilha Solteira, SP 15385-000, Brazil highlights A differential equation modeling the Stefan problem with heat generation is derived. The analytical solutions compare very well with the computational results. The system reaches steady-state faster for larger Stefan numbers. The interface location is proportional to the inverse square root of the heat generation. article info abstract Article history: A first-order, ordinary differential equation modeling the Stefan problem (solid–liquid phase change) Received 15 February 2016 with internal heat generation in a plane wall is derived and the solutions are compared to the results Accepted 24 March 2016 of a computational fluid dynamics analysis. The internal heat generation term makes the governing equa- Available online 13 April 2016 tions non-homogeneous so the principle of superposition is used to separate the transient from steady- state portions of the heat equation, which are then solved separately. There is excellent agreement Keywords: between the solutions to the differential equation and the CFD results for the movement of both the Stefan problem solidification and melting fronts. The solid and liquid temperature profiles show a distinct difference Internal heat generation in slope along the interface early in the phase change process. As time increases, the changes in slope Solidification Melting decrease and the temperature profiles become parabolic. The system reaches steady-state faster for larger Stefan numbers and inversely, the time to steady-state increases as the Stefan number decreases. Ó 2016 Published by Elsevier Ltd. 1. Introduction Prusa [4] review the heat transfer of freezing and melting, and Vis- kanta [5] discusses in detail solid–liquid phase change, specifically Heat transfer driven by internal heat generation plays an impor- in metals. In many materials, a well-defined solidification front tant role in many engineering applications, including nuclear does not exist, but a so-called mushy zone, where the liquid and energy, systems involving joule or ohmic heating, geophysics and solid phases coexist. The mushy zone has been described by Wor- materials processing. Solid–liquid phase change, first described ster [6]. by Stefan [1] and known as the Stefan problem, is a nonlinear The presence of internal heat generation further complicates phenomenon, complicated by a time-dependent phase change the identification of a time-dependent solution. For materials interface. He solved the problem by assuming various temperature which generate internal heat, very limited analytical or approxi- profiles and used the heat balance equation to determine the mate solutions exist which relate the solidification or melting rate movement of the interface. to the magnitude of the internal heat generation. Clearly the inter- The Stefan problem is well-studied [2], and limited closed-form nal heat generation is coupled to the movement of the phase solutions exist. The text by Burmeister [3] provides an accessible change front, accelerating melting and opposing solidification. similarity solution to the Stefan problem and shows that the phase Chen et al. [7] proposed a simple transient conduction model change boundary increases as the square root of time. Yao and with phase change in a material with internal heat generation using the lumped parameter technique and averaged properties. Their goal was to calculate the time when a nuclear fuel pin begins to melt ⇑ Corresponding author. as well as the time required to completely melt the fuel. Later, E-mail addresses: saltfi[email protected] (D. McCord), [email protected] (J. Crepeau), [email protected] (A. Siahpush), [email protected] El-Genk and Cronenberg [8] used the successive approximation (J.A. Ferres Brogin). technique to study the effects of heat generation on the transient http://dx.doi.org/10.1016/j.applthermaleng.2016.03.122 1359-4311/Ó 2016 Published by Elsevier Ltd. 444 D. McCord et al. / Applied Thermal Engineering 103 (2016) 443–451 Nomenclature An, Bn Fourier coefficients Greek symbols cp specific heat a thermal diffusivity Dhf latent heat of fusion g nondimensional distance k thermal conductivity / initial temperature profile L distance from the centerline to the edge of the plane U nondimensional initial temperature profile wall kn; n characteristic eigenvalues q_ volumetric internal heat generation h nondimensional temperature Q nondimensional internal heat generation q density s(t) distance to the phase change front s nondimensional time St Stefan number (solid) f(s) nondimensional distance to phase change front t time T temperature Subscripts T0 constant surface temperature liq liquid Tm melt temperature sol solid x distance ss steady-state tr transient freezing of molten nuclear fuel through a reactor shield plug. They For the solution to the problem, the following assumptions found that their transient analysis was more accurate than previous were made: (1) the internal heat generation is constant and the steady-state analyses by more than a factor of two. For laminar flow same in both the solid and liquid phases; (2) the material proper- of a fluid with internal heat generation in tubes, Kikuchi and Shige- ties in both phases are constant, uniform and equal; (3) heat is masa [9] determined the heat transfer characteristics and calculated transferred solely by conduction and there is no convection in the Nusselt number and its behavior along the length of the tube. the liquid phase; and (4) the phase change occurs at a single, con- Numerical studies performed by Cheung et al. [10] detailed melting stant temperature, Tm, so there is no mushy zone at the interface and freezing in a heat generating slab bounded by two semi-infinite between the phases. cold walls. They stated that due to the highly nonlinear nature of the The governing equation in both the solid and liquid phases is Stefan problem with internal heat generation, it was not feasible to given by the heat conduction equation [21], seek even approximate analytical solutions. Chan et al. [11] @2Tðx; tÞ q_ 1 @Tðx; tÞ described a phase change model where internal melting was induced þ ¼ 0 6 x 6 1; 0 6 t 6 1ð1Þ @ 2 @ by radiative transfer in semi-transparent materials. Chan and Hsu x k a t [12] used the enthalpy method to model the mushy zone in the phase with the boundary and initial conditions in the liquid phase given change problem with uniform internal heat generation. They by, claimed that the mushy zone thickness increases as the solid melts. @ ð ; Þ Crepeau and Siahpush [13] found approximate solutions to the Tliq 0 t ¼ ð ð Þ; Þ¼ @ 0 Tliq s t t Tm ð Þ Stefan problem with internal heat generation using quasi-static x 2 ð ; Þ¼ ð Þ methods, valid for Stefan numbers less than one. Crepeau et al. Tliq x 0 /liq x [14,15] then compared those solutions to results from the compu- and in the solid phase the boundary and initial conditions are, tational analysis of Spotten [16] and showed good agreement for Stefan numbers less than one. Yu et al. [17] used perturbation T ðsðtÞ; tÞ¼T T ðL; tÞ¼T sol m sol 0 ð3Þ methods to solve the Stefan problem with internal heat generation, ð ; Þ¼ ð Þ Tsol x 0 /sol x and showed good results for Stefan numbers less than one and rea- sonable results Stefan numbers one and higher. An and Su [18] There is conduction heat transfer along the interface between used a lumped parameter model obtained through two-point Her- the solid and liquid phases, as well as latent heat. The governing mite approximations for integrals. They found excellent agreement equation along the interface is given as [22], with computational models using the enthalpy method. An et al. @T ðx; tÞ dsðtÞ @T ðx; tÞ [19] used a finite difference technique to model the melting pro- k liq þ Dh ¼ k sol ð4Þ liq @ qsol f sol @ cess, driven by internal heat generation, in a nuclear fuel rod. x x¼s dt x x¼sðtÞ Shrivistava et al. [20] performed experimental studies of melting with internal heat generation and compared their results with The governing equations are then nondimensionalized in order computational modeling and showed favorable correlation. to simplify the solution process. The following terms are introduced, x at Tðx; tÞT sðtÞ 2. Problem description g ¼ ; s ¼ ; hðg; sÞ¼ 0 ; fðsÞ¼ ; 2 À L L Tm T0 L The solid–liquid phase change problem with internal heat gen- c ðT À T Þ qL_ 2 St ¼ p m 0 ; Q ¼ ð5Þ eration in a symmetric, plane-wall geometry is considered. A sche- Dhf kðTm À T0Þ matic of the geometry, cut along the centerline is presented in Fig. 1. The phase change is driven by internal heat generation, q_ , Here, g is the nondimensional distance, s is the Fourier number in both the solid and liquid phases. The temperature at which (nondimensional time), h(g, s) is the nondimensional temperature, the phase change occurs, i.e., the fusion temperature, is denoted fðsÞ is the nondimensional distance from the centerline to the phase by Tm, and the wall is held at a constant temperature, T0, which change front, St is the Stefan number of the solid, and Q is the is below the fusion temperature.
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