Numerical techniques for solving solidification and melting phase change problems

This article reviews several mathematical formulations including the corresponding boundary conditions for numerical predictions of solidification and melting. Emphasis is on techniques that are used in solving solid–liquid interface phenomena. The fixed grid enthalpy method is reviewed based on the solution techniques of conduction and convection related phase change problems. Variable grid methods are categorized and then analyzed based on their accuracy, computational efforts and convergence characteristics. The article concludes with some guidance for selecting the accurate solution techniques for solving solidification and melting problems.     

The UNIT algorithm for solving one-dimensional convection-diffusion problems via integral transforms

A unified approach for solving convection-diffusion problems using the Generalized Integral Transform Technique (GITT) was advanced and coined as the UNIT (UNified Integral Transforms) algorithm, as implied by the acronym. The unified manner through which problems are tackled in the UNIT framework allows users that are less familiar with the GITT to employ the technique for solving a variety of partial-differential problems. This paper consolidates this approach in solving general transient one-dimensional problems. Different integration alternatives for calculating coefficients arising from integral transformation are discussed. Besides presenting the proposed algorithm, aspects related to computational implementation are also explored. Finally, benchmark results of different types of problems are calculated with a UNIT-based implementation and compared with previously obtained results.     

Geometric design of solar-aided latent heat store depending on various parameters and phase change materials

The cylindrical latent heat storage tanks considered here are part of a domestic heating system. In this study, the performances of such energy storage tanks are optimized theoretically. Two different models describing the diurnal transient behaviour of the phase change unit were used. The first is suited to tanks where the phase change material (PCM) is packed in cylinders and the heat transfer fluid (HTF) flows parallel to it (mode 1). The second is suited to tanks where pipes containing the fluid are embedded in the PCM (mode 2). The problem (treated as two-dimensional) is tackled with an enthalpy-based method coupled to the convective heat transfer from the HTF. A series of numerical tests are then undertaken to assess the effects of various PCMs, cylinder radii, pipe radii, total PCM volume in the tank, mass flow rates of fluid, and inlet temperatures of the HTF on the storing time. In addition, optimal geometric design of the store depending on these parameters and PCMs is presented.     

A phase change processor method for solving a one-dimensional phase change problem with convection boundary

A simple yet accurate iterative method for solving a one-dimensional phase change problem with convection boundary is described. The one-dimensional model takes into account the variation in the wall temperature along the direction of the flow as well as the sensible heat during preheating/pre-cooling of the phase change material (PCM). The mathematical derivation of convective boundary conditions has been integrated into a phase change processor (PCP) algorithm that solves the liquid fraction and temperature of the nodes. The algorithm is based on the heat balance at each node as it undergoes heating or cooling which inevitably involves phase change. The paper presents the model and its experimental validation.     

Solving heat transfer problems with phase change via smoothed effective heat capacity and element-free Galerkin methods

This paper presents a smoothed effective heat capacity model that is combined with element-free Galerkin (EFG) method, to solve heat transfer problems with phase change. The Sigmoid function is employed to build a continuous and smooth effective heat capacity function, so as to avoid possible error caused by the step-jump. The proposed numerical model is verified via numerical examples, and the effects of arrangement of EFG nodes and parameters relevant with Sigmoid function on the solutions are investigated. Satisfactory results are achieved in comparison with analytical solutions.     

An enthalpy-based lattice Boltzmann formulation for unsteady convection-diffusion heat transfer problems in heterogeneous media

In this paper, we propose a direct extension of a previous work presented by Hamila et al. [1 R. Hamila, M. Nouri, S. Ben Nasrallah, and P. Perré, Int. J. Heat Mass Transfer, vol. 100, pp. 728–736, 2016.[Crossref], [Web of Science ®] , [Google Scholar]] dealing with the simulation of conjugate heat transfer by conduction in heterogeneous media. In [1 R. Hamila, M. Nouri, S. Ben Nasrallah, and P. Perré, Int. J. Heat Mass Transfer, vol. 100, pp. 728–736, 2016.[Crossref], [Web of Science ®] , [Google Scholar]] a novel enthalpy-based lattice Boltzmann (LB) formulation was successfully simulated in several conjugate heat transfer problems by conduction. We propose testing this enthalpic LB formulation in solving convection-diffusion heat transfer problems in heterogeneous media. The main idea of this formulation is to introduce an extra source term, avoiding any additional treatment of the distribution functions at the interface. Continuity of temperature and normal heat flux at the interface is satisfied automatically. The performance of the present method is successfully validated by comparison to the control volume methods (CVMs) solutions of several heat convection-diffusion problems in heterogeneous media.     

A fitting algorithm for solving inverse problems of heat conduction

The paper presents an algorithm for solving inverse problems of heat transfer. The method is based on iterative solving of direct and adjoint model equations with the aim to minimize a fitting functional. An optimal choice of the step length along the descent direction is proposed. The algorithm has been used for the treatment of a steady-state problem of heat transfer in a region with holes. The temperature and the heat flux density were known on the outer boundary of the region, whereas these values on the boundaries of the holes are to be determined. The idea of the algorithm consist in solving of Neumann problems where the heat flux on the outer boundary is prescribed, whereas the heat flux on the inner boundary is guessed. The guess is being improved iteratively to minimize the mean quadratic deviation of the solution on the outer boundary from the given distribution.The results obtained show that the algorithm provides smooth, non-oscillating, and stable solutions to inverse problems of heat transfer, that is, it avoids disadvantages inherent in other computational methods for such problems. The ill-conditioning of inverse problems in the Hadamard sense is exhibited in that a very quick convergence of the fitting functional to its minimum does not imply a comparable rate of convergence of the recovered temperature on the inner boundary to the true distribution.The considered method can easily be extended to nonlinear problems.Numerical calculation has been carried out with the FE program Felics developed at the Chair of Mathematical Modelling of the Technical University of Munich.     

A new adaptive algorithm for phase change heat transfer problems based on quadtree SBFEM and smoothed effective heat capacity method

A new adaptive algorithm is presented for solving the phase change heat transfer problems by integrating the advantages of quadtree SBFEM and smoothed effective heat capacity method. A new criterion of the refinement is addressed via an interval intersection, and a new nonerror estimation based criteria is presented to terminate the refinement. Only a local mesh refinement is required around the interface region, and the computational expense can be effectively reduced in comparison with uniform refinement in the whole domain. Numerical examples demonstrate the effectiveness of the proposed approach, and the potential for solving applied problems by simulating a finned enhanced latent heat storage system.     

A multi-zonal integral method for problems involving unsteady one-dimensional heat conduction with change of phase

A method is presented for calculating the temperature in a solidifying slab of molten material with greatly differing cooling rates on the inner and outer surfaces. For the purposes of analysis the slab is divided into a number of zones having time-varying boundaries, e.g. solidification or conduction fronts. Appropriate approximations are made for the temperature profile and the heat-balance integral equation applied separately in each zone. This leads to systems of first-order ordinary differential equations which can be integrated numerically by using simple standard methods. Results are presented which correspond to the industrial casting of polypropylene film.     

A three-dimensional numerical model for a convection-diffusion phase change process during laser melting of ceramic materials

Using a fixed-grid source-based method, a three-dimensional numerical model for a convection-diffusion phase change process during laser melting of ceramic materials has been studied. The model was applied to a realistic binary phase diagram containing a eutectic composition, including both an isothermal phase change occurring at a distinct temperature and a phase change taking place over a temperature range (the “mushy” region phase change). The effects of latent heat of fusion and fluid flow in the melt pool on the temperature, velocity fields and shape of the melt pool were analysed and compared. Results indicated that the effects of latent heat of fusion were more significant than those of fluid flow for two simulation cases, considered in the model. The best prediction accuracy for the profiles of the melt/solid interfaces was achieved from the developed model by considering both the latent heat of fusion and fluid flow in the melt pool.     

Particle Swarm Optimization-based algorithms for solving inverse heat conduction problems of estimating surface heat flux

An inverse analysis of estimating a time-dependent surface heat flux for a three-dimensional heat conduction problem is presented. A global optimization method known as Particle Swarm Optimization (PSO) is employed to estimate the unknown heat flux at the inner surface of a crystal tube from the knowledge of temperature measurements obtained at the external surface. Three modifications of the PSO-based algorithm, PSO with constriction factor, PSO with time-varying acceleration of the cognitive and social coefficients, and PSO with mutation are carried out to implement the optimization process of the inverse analysis. The results show that the PSO with mutation algorithm is significantly better than other PSO-based algorithms because it can overcome the drawback of trapping in the local optimum points and obtain better inverse solutions. The effects of measurement errors, number of dimensionalities, and number of generations on the inverse solutions are also investigated.     

相变蓄冷装置内传热及相变特性研究

建立了三种不同蓄冷球球径堆叠方式的相变蓄冷装置模型,对其进行了数值模拟以研究其内部传热及相变特性。结果表明：随着流速的提高、蓄冷球直径的减小,蓄冷结束后三种方案中装置内蓄冷球的凝固率提升显著;双球径方案与单球径方案装置内蓄冷球凝固率随时间变化的规律在蓄冷过程初段相类似,但两方案中相同球径部分蓄冷球相变结束,双球径方案中发生相变的蓄冷球由大直径转变为小直径时,其凝固率随时间变化的速度逐渐超过单球径方案。该研究可为相变蓄冷装置的实际设计及性能优化提供参考     

A technique for uncertainty analysis for inverse heat conduction problems

A technique is presented for the uncertainty analysis of the linear Inverse Heat Conduction Problem (IHCP) of estimating heat flux from interior temperature measurements. The selected IHCP algorithm is described. The uncertainty in thermal properties and temperature measurements is considered. A propagation of variance equation is used for the uncertainty analysis. An example calculation is presented. Parameter importance factors are defined and computed for the example problem; the volumetric heat capacity is the dominant parameter and an explanation is offered. Thoughts are presented on extending the analysis to include the non-linear problem of temperature dependent properties.     

Application of program generation technology in solving heat and flow problems

Based on a new DIY concept for software development,an automatic program-generating technology attached ona software system called as Finite Element Program Generator(FEPG)provides a platform of developing pro-grams,through which a scientific researcher can submit his special physico-mathematical problem to the systemin a more direct and convenient way for solution.For solving flow and heat problems by using finite elementmethod,the stabilization technologies and fraction-step methods are adopted to overcome the numerical difficul-ties caused mainly due to the dominated convection.A couple of benchmark problems are given in this paper asexamples to illustrate the usage and the superiority of the automatic program generation technique,including theflow in a lid-driven cavity,the starting flow in a circular pipe,the natural convection in a square cavity,and theflow past a circular cylinder,etc.They are also shown as the verification of the algorithms.     

A modular phase change heat exchanger for a solar oven

A modular energy storing heat exchanger designed to use pentaerythritol for thermal storage (solid-solid phase change at 182°C) is tested in an oven by circulating heat transfer oil which is electrically heated in a manner to simulate a concentrating solar collector. Three efficiencies for heating the system under controlled and measured power input are determined—the heat exchanger efficiency, the efficiency of the heater with distribution lines, and the total system efficiency. Thermal energy retention times and cooking extraction times are determined, and along with the efficiencies, are compared with the results previously reported for a nonmodular heat exchanger. The modular configuration provides a highly improved extraction rate for cooking due to its wrap-around character and its increased surface-to-volume ratio. A full scale glass model of the copper tubing of the system is described and flow observations reported demonstrating how uniformly the parallel pumping branches perform and how trapped air pockets affect pumping power. A technique for measuring pumping power is described and its application to our system is quantified to show that less than 1 watt is required to circulate the heat transfer oil even when the system includes the solar collector and its longer connecting tubes.     

An investigation about the relations between the results of heat conduction problems with and without phase change

There are physical relations between the velocities of the isotherms for two heat conduction problems having the same initial and boundary conditions, but one of them has phase change while the other one has not. The main idea of this study is to find these relations mathematically, which stand on a physical base, depending on the physical properties of the phases and the common parameters of these two different problems. If such kind of relations are known, then it will be possible to find the position of solid-liquid interface by using the analytical solutions and the relation itself, not solving the phase change problem. This idea is applied to heat conduction problem in semi infinite media and one dimensional phase change problem is solved for three different kind of boundary conditions via the Enthalpy Method. The results are correlated with the analytical solutions of the problem having same geometry and conditions but without phase change and obtained some relations having correlation coefficients changing between 0.91 and 0.98. This result shows that the regression analysis made is statistically meaningful.     

Iterative algorithms for solving inverse problems of heat conduction in multiply connected domains

The presented paper displays a method of solving the inverse problems of heat transfer in multi-connected regions, consisting in iterative solving of convergent series of the direct problems. For known temperature and flux values at the outer boundary of the region the temperature and flux values at the inner boundaries are sought (the cauchy problem for the Laplace equation). In case of such a formulation of the problem, the solution does not always exist, one of the conditions is met in the mean-square sense, providing the optimization criterion. The idea of the process consists in solving the direct problem in which the boundary condition is subject to iterative changes so as to attain minimum of the optimization criterion (the square functional). Two algorithms have been formulated. In the first of them the heat flux at the inner boundaries of the region, while in the other the temperature were subject to changes. Convergence of both the algorithms have been compared.The numerical calculation has been made for selected examples, for which an analytical solution is known. The effect of random disturbance of the boundary conditions on the solution obtained with iterative algorithms has been checked. Moreover, a function was defined, serving as convergence measure of the solution of the inverse problem solved with the algorithms proposed in the paper. The properties of the function give evidence that it tends to the value exceeding unity.     

相变储热研究进展及综述

清洁能源大多存在固有的间歇性和波动性,可能导致能源供应和能源消费之间的不平衡。储能是必不可少的,相变储能具有熔融凝固循环温度恒定,相对较高的能量密度,体积小,控制相对容易,安全可靠等优点。并针对相变材料的分类、相变储能系统强化传热技术、相变储能技术应用等方面进行了综述。     

A simplified solution technique for some fluid flow and heat transfer problems

A new simple solution technique is presented for fluid flow and heat convection problems that incorporate the similarity solution. Finite-difference methods and numerical integration are used to solve the describing equations using spreadsheets. Numerical solution by spreadsheets results in a solution of good accuracy and it requires much less effort than conventional programming languages. This paper also presents the current status of using spreadsheets in solving energy-related problems © 1998 John Wiley & Sons, Ltd.