A semi-analytical solution for time-varying latent heat thermal energy storage problems

Latent heat thermal energy storage (LHTES) problems include a lot of boundary conditions that could not be solved by exact solution, so new approaches to solving such problems could revolutionize the advanced energy storage devices. This paper focuses on reformulating the generalized differential quadrature method (GDQM) for a one-dimensional solidification/melting Stefan problem as a fundamental LHTES problem and solves some practical cases. Convergence and comparisons demonstrate that the proposed approach is sufficiently reliable. By checking the accuracy of the proposed approach for the LHTES problem (where Stefan number is below 0.2), it was demonstrated that for all Stefan numbers, the maximum error is less than 3.81% for temperatures. As the usual range of thermal energy storages, for Stefan numbers up to 0.2 the solution yields errors less than 0.2%. Then, the proposed approach is very ideal for such applications. In comparison, GDQM has a more accurate response than an integral solution for Stefan numbers less than 0.2. When this priority of GDQM comes with its low computational cost, it would undoubtedly be preferable.     

Numerical Solution of a Three-Phase Stefan Problem with High Power Input

The numerical solution of a one-dimensional, three-phase Stefan problem with a low Stefan number is presented. Joule heating and thermal radiation are demonstrated to be negligible compared to the high power input. The front tracking method is used along with a second-order Lagrangian interpolation of the temperature profile near the moving surface defined by the location of the phase change. Results are compared with analytical, numerical, and experimental solutions available in literature.     

基于等效比热的MPCMs平板层流传热理论模型

采用边界层的能量积分方程法,基于等效比热模型,对微胶囊相变悬浮液(Microencapsulated Phase Change Materials slurry, MPCMs)的热边界层进行理论建模,推导出了MPCMs外掠平板换热加热等壁温边界条件层流工况下,包含斯蒂芬数的MPCMs的对流换热关联式,然后与数值模拟结果进行比较。结果表明,对流传热系数的解析解与数值模拟结果趋势上相一致,修正后的解析解与数值模拟结果高度吻合。     

Parametric study and approximation of the exact analytical solution of the Stefan problem in a finite PCM layer in a steady periodic regime

In this paper a parametric study and an approximation of the exact analytical solution of the Stefan problem in steady periodic regime conditions (Mazzeo et al., 2015) is presented. The physical model describes the thermal behaviour of a PCM (phase change material) layer subject to phase transition, and the considered thermal regime ensures continuous cycles of phase changes under the action of the periodic boundary conditions.The exact analytical solution determines, through implicit transcendental equation with complex thermal parameters and unknowns, the bi-phase interface position, the thermal field and the sensible and latent stored energy in the layer. The dimensionless solution is a function of the Fourier and Stefan numbers calculated in the two phases, and of the dimensionless bi-phase interface position corresponding to the steady regime.For the parametric study, the thermophysical properties of the most commonly used PCMs and the variation range of attenuation and time lag between the temperature loadings operating on the internal and external surfaces were considered. This study has allowed for the identification of the thermal parameters that mainly influence the dynamic thermal behaviour of the PCM layer and the mathematical structure of the frequency response of the layer.Since an analytical expression in an explicit form of the position of the bi-phase interface in the layer is not available, an approximate analytical solution was obtained, which makes the bi-phase interface position depends on the product between the Fourier number and the Stefan number calculated in the two phases. The limits of validity of such a solution were determined evaluating the relative error, which is committed in the determination of the amplitude and of the argument of the oscillating component of the bi-phase interface position. Finally, the fields of variation of the thermal parameters that ensure a relative error value lower than 3% and the corresponding values of the maximum errors of the amplitude or the argument are determined. In PCM layer thermal analysis it is useful to have an expression in an explicit form of the oscillating component of the bi-phase interface position to obtain the mathematical expression of the temperature and heat flux field as a function of only dimensionless thermal parameters and boundary loadings.     

Numerical Method for Calculation of Complete Casting Processes—Part II: Validation and Application

In this article, the numerical method proposed in Part I is validated by applying it to relatively simple validation test cases with phase change as well as to real-life problems. First, the results of the 1-D calculations are compared with the analytical solutions for the Stefan problem with mushy zone. Then, the 2-D and 3-D calculations of the Bridgman crystal growth are compared with available experimental results. The ability to predict the residual stresses is demonstrated on an academic 2-D example. Finally, the results of calculation for two real-life industrial cases, injection casting and solidification of a multicomponent metal drill head, are presented and discussed.     

Heat transfer in a reaction–diffusion system with a moving heat source

A general formulation is presented for a moving boundary problem in which heat is generated at the boundary due to an exothermic reaction involving a species which diffuses into a dispersed phase from an external medium of finite volume. The speed of the moving boundary is prescribed based on the solution of the mass diffusion problem and an analysis is presented of the thermal dynamics of the system. The set of equations describing heat transport leads to a Green’s function type problem with time dependent boundary conditions and the Galerkin finite element method is employed to develop a numerical solution. Transformations are introduced to freeze the moving boundary and partition the domain for ease of computation, and an iterative scheme is defined to satisfy the heat flux jump boundary condition and match the temperature field across the moving boundary. The numerical results are used to set the limits of applicability of an analytical perturbation solution. Essential aspects of thermal dynamics in the system are described and parametric regions resulting in a local temperature hot spot are delineated. Computed contour plots describing thermal evolution are presented for different combinations of parameter values. These may be of utility in the prediction of thermal development, for control and avoidance of hot spot formation, and in physical parameter estimation.     

One-dimensional ice growth due to incoming supercooled droplets impacting on a thin conducting substrate

A one-dimensional model for ice accretion due to incoming supercooled water impacting on a conducting substrate is developed, where the substrate is cooled from below by a liquid or gas. Both rime and glaze ice situations are considered. Non-dimensionalisation shows that conduction is the dominant method of heat transfer and so the heat equations are reduced to pseudo-steady forms. In this case the problem reduces to solving a single equation for the ice layer thickness. The water height and temperatures in the ice, water and substrate may subsequently be found. The asymptotic solution is validated by comparison with results from a numerical scheme which solves the full Stefan problem. This is an extension of a previously published solution method that involved simpler boundary conditions. For glaze ice, a comparison including water droplet energy either in the boundary conditions or as a source term in the heat equations, is also performed.     

On the one-phase reduction of the Stefan problem with a variable phase change temperature

The one-phase reduction of the Stefan problem, where the phase change temperature is a variable, is analysed. It is shown that problems encountered in previous analyses may be traced back to an incorrectly formulated Stefan condition. Energy conserving reductions for Cartesian, cylindrically and spherically symmetric problems are presented and compared with solutions to the two-phase problem.     

Application of the heat-balance integral to an inverse Stefan problem

Most phase change process controls are concerned with the inverse Stefan problem. In this paper, the heat-balance integral method is applied effectively to analyze the one-region and two-region inverse Stefan problems in Cartesian and spherical coordinates. It is shown that if the movement of the phase change boundary is specified arbitrarily the present technique to predict both the temperature and its gradient at the fixed boundary is simple and accurate. As numerical illustrations, the one-dimensional inward solidification problem in Cartesian and spherical coordinates are solved and discussed in detail when the movement of the phase change interface is specified as a power function. The accuracy of these approximate solutions, based on the heat-balance integral method, is demonstrated satisfyingly by comparison with the available exact and/or numerical solutions for the one-region and the two-region problems.     

Isotherm Migration Method in two dimensionsMethode de migration isotherme a deux dimensionsZweidimensionale,isotherme migrationsmethodeДвyмepный мeтoд изoтepмичecкoй мигpaции

The Isotherm Migration Method is extended to two dimensions. The equations are formulated and a convenient finite-difference method of solution is described for a variety of initial and boundary conditions. Particular attention is devoted to Stefan problems in which phase changes occur on a moving interface. As an example the solidification of a square prism of fluid is solved in detail and the numerical results are compared with those obtained by earlier authors.     

Heat flow rate at a bore-face and temperature in the multi-layer media surrounding a borehole

Assessment of the heat either delivered from high temperature rocks to the borehole or transmitted to the rock formation from circulating fluid is of crucial importance for a number of technological processes related to borehole drilling and exploitation. Normally the temperature fields in the well and surrounding rocks are calculated numerically by finite difference method or analytically by applying the Laplace-transform method. The former approach requires tedious and, in certain cases, non-trivial numerical computations. The latter method leads to rather bulky formulae that are inconvenient for further numerical evaluation. Moreover, in previous studies where the solution is obtained analytically, the heat interaction of the circulating fluid with the formation was treated on the condition of constant bore-face temperature. In the present study the temperature field in the rock formation disturbed by the heat flow from the borehole is modeled by a heat conduction equation, assuming the Newton model for the convective heat transfer on the bore-face, with boundary conditions that account for the thermal history of the borehole exploitation. The problem is solved analytically by the generalized heat balance integral method. Within this method the approximate solution of the heat conduction problem is sought in the form of a finite sum of functions that belong to a complete set of linearly independent functions defined at the finite interval bounded by the radius of thermal influence and that satisfy the homogeneous boundary conditions on the bore-face. In the present study first and second order approximations are obtained for the composite multi-layer domain. The numerical results illustrate that the second approximation is in a good agreement with the exact solution. The only disadvantage of this solution is that it depends on the radius of thermal influence, which is an implicit function of time and can only be found numerically by iterative algorithms. In order to eliminate this complication, in this study an approximate explicit formula for the radius of thermal influence and new close-form approximate solution are proposed on the basis of the approximate solution obtained by the integral-balance method. Employing the non-liner regression method the coefficients for this simplified solution are obtained. The accuracy of the approximate solution is validated by comparison with the exact analytical solution found by Carslaw and Jaeger for the homogeneous domain.     

HYPERBOLIC STEFAN PROBLEM WITH APPLIED SURFACE HEAT FLUX AND TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY

Abstract

The hyperbolic Stefan problem with an applied surface heat flux and temperature-dependent thermal conductivity is solved numerically for a semi-infinite slab using Mac-Cormack's predictor-corrector method. Solutions are presented for cases where the melt temperature is both below and above the instantaneous jump in surface temperature at time t = O⁺. The interface condition, surface temperature, and internal temperatures are presented for different Stefan numbers and melt temperatures, as well as thermal conductivity both increasing and decreasing with temperature. The results obtained from the hyperbolic solution are compared with those obtained from the parabolic solution.     

Solidification of a supercooled liquid in stagnation-point flow

The solidification of a thermally supercooled liquid in stagnation-point flow is investigated. Due to the advancing solidifying front, both the temperature and flow fields are time dependent. A numerical solution to the problem using an interface tracking method is compared to analytical solutions obtained for instantaneous similarity (short time solution) and quasi-steady state (long time solution). The results show that the velocity of the solid-liquid interface eventually reaches a constant value and that the magnitude of the interface velocity increases with greater thermal supercooling. The solution to this problem provides insight into more complicated solidification problems relating to crystal growth.     

FINITE ELEMENT ANALYSIS OF TRANSIENT HEAT CONDUCTION WITH MOVING CONVECTIVE BOUNDARIES

A numerical procedure is demonstrated for the solution of moving convective boundary problems using a general finite element program and a stationary mesh. The proposed technique will allow for the modeling of various time- and temperature-dependent heat transfer coefficient profiles and removes the restriction of a constant convective boundary velocity from the solution. Quenching of a hot surface that occurs during rewetting is effectively modeled for the case of a step function heat transfer coefficient profile. The accuracy of the numerical scheme is examined by comparing the results with an analytical rewetting solution. A final example demonstrates the general capability of this finite element technique in modeling the effects of the nonlinear temperature dependency of the material thermal conductivity and heat capacity on the rewetting solution.     

An effective conduction length model in the enthalpy formulation for the stefan problem

In the fixed-grid finite-volume formulation, so called the enthalpy formulation, for the Stefan problem, the temperature and the front movement show step-like history, which is a well-known characteristic of the enthalpy method. This paper presents an effective conduction length model to mitigate such an oscillatory behavior as well as to support the physical reasoning. The proposed model is based on the simple fact that the heat flux across the boundary of phase-change cells should be estimated with the distance between the phase front and the center of neighboring cell. The model is applied to one-dimensional Stefan problems with various Stefan numbers. The numerical results show that the proposed model can smooth the spurious oscillation of the history of temperature and the evolution of front movement.     

Meshless element free Galerkin method for unsteady nonlinear heat transfer problems

In this paper, meshless element free Galerkin (EFG) method has been extended to obtain the numerical solution of nonlinear, unsteady heat transfer problems with temperature dependent material properties. The thermal conductivity, specific heat and density of the material are assumed to vary linearly with the temperature. Quasi-linearization scheme has been used to obtain the nonlinear solution whereas backward difference method is used for the time integration. The essential boundary conditions have been enforced by Lagrange multiplier technique. The meshless formulation has been presented for a nonlinear 3-D heat transfer problem. In 1-D, the results obtained by EFG method are compared with those obtained by finite element and analytical methods whereas in 2-D and 3-D, the results are compared with those obtained by finite element method.     

Spreadsheet Solution of a Two-Dimensional Stefan Problem Using an Approximate Method

This article describes the use of a spreadsheet program for the numerical solution of a two-dimensional Stefan problem. A complete Lotus 1-2-3 macro is presented and numerical results are given and compared with other known results.     

Transient analytical solution to heat conduction in multi-dimensional composite cylinder slab

The analytical solution for the problem of transient heat conduction in multi-dimensional composite cylinder slab is developed for a time-dependent boundary condition. For such problems, numerical programs are needed to obtain eigenvalues and residues in most of the published papers. The numerical schemes may become unstable due to the existence of imaginary eigenvalues in multi-dimensional cases. In this paper, the proposed analytical method involves no numerical complications. By a novel application of the methods of the Laplace transform and separation of variables together with variable transformations, the residue calculation is avoided. The developed analytical method is powerful which represents extension of the analytical approach derived for the heat conduction problem in Cartesian coordinates. A closed form solution is provided. Calculation examples show that the analytical solutions predict good agreement with the numerical results.     

Numerical solution of stefan problemsResolution numerique des problemes de stefanNumerische lösung von stefan-problemenЧиcлeннoe peшeниe зaдaч cтeфaнa

The extension of the enthalpy method to multi-dimensional Stefan problems is outlined. The method is applied to the numerical solution of a problem involving the solidification of a square cylinder of fluid when the surface temperature is lowered at a constant rate. One aim of this paper is to demonstrate that the numerical method successfully predicts the experimental results which have been published for this problem. The same technique is then applied to a similar problem, in which the surface temperature is lowered discontinuously at the initial instant, and the results compared with those obtained by other authors.     

An Investigation of Numerical Modeling of Transient Heat Conduction in a One-Dimensional Functionally Graded Material

This article presents a closed form analytical solution for one-dimensional transient heat conduction in a material where the thermal conductivity varies linearly through the thickness but the thermal diffusivity is held constant. This solution is used to validate the results from finite-difference and finite-element approximations that account for this variation at the element level. This was motivated by a suggested limitation on the minimum time step used in the commercial finite-element software code ABAQUS for quadratic elements. Good agreement was found between the analytical and numerical approximations, indicating that conventional numerical techniques may be sufficiently robust to analyze heat conduction problems in functionally graded materials without the use of special elements. The minimum time step constraint was found to be unnecessary for a convective boundary condition for the one-dimensional elements and property variation used in this study.