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.     

Optimal exponent heat balance and refined integral methods applied to Stefan problems

When using a polynomial approximating function the most contentious aspect of the Heat Balance Integral Method is the choice of power of the highest order term. In this paper we employ a method recently developed for thermal problems, where the exponent is determined during the solution process, to analyse Stefan problems. This is achieved by minimising an error function. The solution requires no knowledge of an exact solution and generally produces significantly better results than all previous HBI models. The method is illustrated by first applying it to standard thermal problems. A Stefan problem with an analytical solution is then discussed and results compared to the approximate solution. An ablation problem is also analysed and results compared against a numerical solution. In both examples the agreement is excellent. A Stefan problem where the boundary temperature increases exponentially is analysed. This highlights the difficulties that can be encountered with a time dependent boundary condition. Finally, melting with a time-dependent flux is briefly analysed without applying analytical or numerical results to assess the accuracy.     

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.     

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.     

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.     

A FIXED-GRID FINITE-DIFFERENCE METHOD FOR PHASE-CHANGE PROBLEMS

A simple finite-difference method was developed for solid-liquid phase-change problems. The present method is based on a fixed grid and implicit in time. A fictitious temperature concept is introduced to derive finite-difference equations to deal with the nodal points across the solid-liquid interface. The algorithm is applied to a one-dimensional Stefan problem for which exact solutions are available. The computational results are found to be in excellent agreement with the exact solutions. Further, the present method yields no oscillations of temperature and phase front, which are commonly observed with the typical enthalpy method.     

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.     

Mathematical modelling of phase change at the nanoscale

Standard mathematical models for phase change at the nanoscale involve an implicit assumption that the latent heat is released at the bulk phase change temperature. They also assume the latent heat to be constant (while the melt temperature decreases with decreasing size). There is clear experimental evidence that this is not the case. In this paper, we examine the formulation of the Stefan problem at the nanoscale and present a new form of Stefan condition which correctly reflects the latent heat release, including both melt temperature and latent heat depression. We go on to show that the standard formulation can lead to melt rates up to three times slower than in reality.     

Solidification of droplets on a cold surfaceSolidification de gouttelettes sur une surface froideDas erstarren von tropfen auf einer kalten oberflächeзaтвepдeвaниe кaпeль нa чoлoднoй пoвepчнocти

Title problem has been investigated theoretically and experimentally. In the theory a simple model of two-dimensional radial flow has been used. The degree of flattening ξ_m of a droplet depends upon the Weber, Reynolds and Péclet numbers, and upon the freezińg constant U, taken from the solution of a Stefan problem. The agreement of the theory with experiments is not bad if the constant U is taken for the Stefan problem with the isothermal cooling surface.     

Numerical simulation of the He II/He I phase transition in superconducting magnets

In this paper we simulate a special type of Stefan problem in large-scale superconducting magnet systems in which superfluid helium (He II) is used as the coolant for the system. Liquid helium in a narrow channel called a “cable-in-conduit” conductor (CICC) is used to remove the heat load from the conductors. Liquid helium exhibits a phase change transition to normal helium (He I) when its temperature rises above the lambda point (2.716 K under saturated vapor pressure). A simple one-dimensional model is described to analyze this special He II/He I Stefan problem. A moving mesh technique is used to solve this model to improve the numerical efficiency compared with front-tracking methods. The results illustrate the simplicity and efficiency of this model.     

The application of the homotopy perturbation method to one-phase inverse Stefan problem

In this paper, the application of the homotopy perturbation method for solving the inverse Stefan problem is presented. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of the functions describing temperature and heat flux on the boundary, when the position of the moving interface is known.     

Hybrid Laplace transform technique for Stefan problems with radiation-convection boundary condition

The hybrid application of the Laplace transform technique and the finite difference method (FDM) to one-dimensional Stefan problems involving the radiative and convective boundary condition is studied. The radiative term is linearized by Taylor's series approximation, and then the above hybrid method is used. This scheme is obtained by the use of the Laplace transform technique for the time-dependent terms and the fixed-grid FDM for space domain. It can be found from various illustrated examples that excellent agreement is obtained between the present results and those of early works. For the phase-change problem subjected to the nonlinear boundary condition, three or four iterations are required to obtain a convergent result at a specific time. The present analysis also demonstrates that the application of the Laplace transform technique is no longer limited to phase-change problems with the linear boundary condition.     

On the perturbation method for the Stefan problem with time-dependent boundary conditions

Perturbation methods are developed for Stefan problems with time-dependent boundary conditions. The methods are applied to melting of ice in the half-plane, outward spherical solidification and outward cylindrical solidification of a saturated liquid. The results are shown to compare well with those obtained by other numerical methods.     

Homotopy Perturbation Method for Solving the Two-Phase Inverse Stefan Problem

In this article, the possibility of the application of the homotopy perturbation method for solving the two-phase inverse Stefan problem is presented. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of the functions describing the temperature and the heat flux on the boundary when the position of the moving interface is known. The validity of the approach is verified by comparing the results obtained with the exact solution.     

The inward solidification of cylinders with a slightly perturbed temperature distribution at the boundary

This paper is concerned with the inward solidification of infinite liquid cylinders, where the boundary values (the cases of specified temperature distribution and specified heat flux are considered) vary slightly with position around the cylinder. This paper is therefore the cylindrical geometry analogue of the spherical problems considered in Gammon and Howarth [1]. The problems are solved analytically by means of a large Stefan number approximation.     

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.     

Phase Change Heat Transfer Simulation for Boiling Bubbles Arising from a Vapor Film by the VOSET Method

This article presents a numerical method directed towards the simulation of flows with changes of phase. The volume-of-fluid level set (VOSET) method, which is a new interface capturing method and combines the advantages of both volume-of-fluid (VOF) and level set methods, is used for interface tracking. A difficulty occurs for the problems studied here: the discontinuous velocity field due to the difference between mass-weighted velocity and volume weighted velocity caused by the phase change at the interface. In this article, some special treatment is made to overcome this difficulty. The VOSET method and the developed treatment for the difference between mass-weighted and volume-weighted velocities are adopted to simulate a one-dimensional Stefan problem, two-dimensional horizontal film boiling, and horizontal film boiling of water at near critical pressure. The predicted results in both Nusselt number and flow patterns are agreeable with experimental results available in the literature.     

Using genetic algorithms for the determination of an heat transfer coefficient in three-phase inverse Stefan problem

In the paper, an example is presented of the application of a genetic algorithm to a design inverse Stefan problem. The problem consists in the reconstruction of the function which describes the heat transfer coefficient, where the positions of phase change moving interfaces are well-known. In numerical calculations, the Tikhonov regularization, a genetic algorithm and a generalized alternating phase truncation method were used. The featured examples of calculations show a very good approximation of the exact solution.     

AN ALTERNATIVE FORMULATION OF THE APPARENT HEAT CAPACITY METHOD FOR PHASE-CHANGE PROBLEMS

Abstract

In the apparent heat capacity method (AHCM) for heat transfer problems with phase change, the conventional time discretization is subject to severe restrictions on time steps in spite of the various approximation techniques that have been developed so far. To improve the conventional AHCM we propose an alternative formulation. By introducing a nominal heat capacity, the new time discretization can better approximate the time derivative of enthalpy while maintaining the same form as the conventional AHCM. The new formulation also establishes an equivalent relation between the full enthalpy formulation and the AHCM. A one-dimensional (I-D) Stefan problem is used as a test problem, and comparison is made between the solutions of the conventional and the new AHCM formulations. It is found that for implicit schemes with large time steps, the new formulation performs much better than the conventional AHCM     

Experimental Verification of Selected Artificial Intelligence Algorithms Used for Solving the Inverse Stefan Problem

The article presents a comparative study of three procedures applied for solving the inverse Stefan problem. The investigated problem consists of reconstruction of the unknown boundary condition on the basis of measurement data, and the procedures of solution differ in the way of minimizing the proper functional—in each approach considered, one of three artificial intelligence algorithms (Ant Colony Optimization, Artificial Bee Colony, and Harmony Search) is used. Methods applying the respective algorithms are compared with regard to their velocity and the precision of results.