On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions

Refined integral heat balance is developed for Stefan problem with time-dependent temperature applied to exchange surface. The method is applied to phase change in the half-plane and ordinary differential equation is obtained for the solid/liquid interface. The results are compared to those obtained by heat balance integral, perturbation and numerical methods.     

Applying the combined integral method to one-dimensional ablation

In this paper the combined integral method is applied to a simple one-dimensional ablation problem. One of the drawbacks of heat balance integral methods is how to choose the approximating function. It is common to use a polynomial form but even then it is not clear what the power of the highest order term should be. Previous studies have determined exponents either from exact solutions or from expansions valid over short time scales; neither approach is satisfactory nor very accurate for larger times. We combine the heat balance and refined integral methods to determine this exponent as part of the solution process, and conclude that it is in fact time-dependent in the ablation stage. From comparing the approximate solutions with numerical and exact analytical solutions whenever possible, we show that this new method greatly improves the accuracy on standard methods, without overcomplicating the method.     

A new look at the heat balance integral method

The heat balance integral method is a familiar technique for treating transport problems, particularly phase-change scenarios. Here a number of differences arising in the method's implementation are investigated that result in quantitatively distinct solutions. As a consequence some guidance is provided for selecting the appropriate implementation of the method.     

Analytic solutions to heat transfer problems on a basis of determination of a front of heat disturbance

With the use of additional boundary conditions in integral method of heat balance, we obtain analytic solution to nonstationary problem of heat conductivity for infinite plate. Relying on determination of a front of heat disturbance, we perform a division of heat conductivity process into two stages in time. The first stage comes to the end after the front of disturbance arrives the center of the plate. At the second stage the heat exchange occurs at the whole thickness of the plate, and we introduce an additional sought-for function which characterizes the temperature change in its center. Practically the assigned exactness of solutions at both stages is provided by introduction on boundaries of a domain and on the front of heat perturbation the additional boundary conditions. Their fulfillment is equivalent to the sought-for solution in differential equation therein. We show that with the increasing of number of approximations the accuracy of fulfillment of the equation increases. Note that the usage of an integral of heat balance allows the application of the given method for solving differential equations that do not admit a separation of variables (nonlinear, with variable physical properties etc.).     

An exact methodology for solving nonlinear diffusion equations based on integral transforms

This paper develops a new methodology for the solution of nonlinear diffusion equations. The solution technique is based on integral transforms and leads to exact numerical results. We apply the formal methodology to the problem of one-dimensional transient heat conduction. A new form of the heat equation is developed using a generalized expression for temperature-dependent thermal conductivity, based on a power-series expansion, for the three standard orthogonal coordinate systems. The resulting form of the heat equation suggests that the finite integral transform technique may reduce the dimensionality of the heat equation prior to the initiation of any numerical procedure. An example in a slab with linearly varying thermal conductivity is shown to produce exact results for the temperature distribution.     

A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

Optimal Control and Controllability of a Phase Field System with One Control Force

We investigate the relation between optimal control and controllability for a phase field system modeling the solidification process of pure materials in the case that only one control force is used. Such system is constituted of one energy balance equation, with a localized control associated to the density of heat sources and sinks to be determined, coupled with a phase field equation with the classical nonlinearity derived from the two-well potential. We prove that this system has a local controllability property and we establish that a sequence of solutions of certain optimal control problems converges to a solution of such controllability problem.     

On an inverse problem arising in continuous casting of steel billets

In continuous casting of steel, the control of the solidification front by means of the amount of water sprayed onto the strand is of great practical interest. We study the thermal history in a continuously cast cylindrical billet. The mathematical model is a two-dimensional nonlinear heat equation div[k(u)gradu] = ut subject to water-cooling and heat radiation boundary conditions. We establish existence, uniqueness and stability results for both the temperature field and the solidification front. We study the monotonicity behaviour of the temperature field and show that certain technically easy-to-realize cooling-strategies may generate double liquid fingers at the final stage of solidification. The inverse problem of determining the cooling strategy is an ill-posed problem. We therefore use Tikhonov regularization as a stable and convergent methodfor treating this problem.     

An analytical solution for conduction-dominated unidirectional solidification of binary mixtures

In this paper, a fully analytical solution technique is established for the solution of unidirectional, conduction-dominated, alloy solidification problems. By devising appropriate averaging techniques for temperature and phase-fraction gradients, governing equations inside the mushy region are made inherently homogeneous. The above formulation enables one to obtain complete analytical solutions for solid, liquid and mushy regions, without resorting to any numerical iterative procedure. Due considerations are given to account for variable properties and different microscopic models of alloy solidification (namely, equilibrium and non-equilibrium models) in the two-phase domain. The results are tested for the problem of solidification of a NH₄Cl–H₂O solution, and compared with those from existing analytical models as well as with the corresponding results from a fully numerical simulation. The effects of different microscopic models on solidification behaviour are illustrated, and transients in temperature and heat flux distribution are also analysed. A good agreement between the present solutions and results from computational simulation is observed.     

On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.     

On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.      

Determination of functional gradient in an optimal control problem related to metal solidification

The gradient of the cost functional in the discrete optimal control problem of metal solidification in casting is exactly evaluated. The mathematical model describing the solidification process is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. Formulas determining exact gradient determination are derived using the fast automatic differentiation technique.     

Application of the combined integral method to Stefan problems

In this paper we present a new, accurate form of the heat balance integral method, termed the combined integral method (CIM). The application of this method to Stefan problems is discussed. For simple test cases the results are compared with exact and asymptotic limits. In particular, it is shown that the CIM is more accurate than the second order, large Stefan number, perturbation solution for a wide range of Stefan numbers. In the initial examples it is shown that the CIM reduces the standard problem, consisting of a PDE defined over a domain specified by an ODE, to the solution of one or two algebraic equations. The latter examples, where the boundary temperature varies with time, reduce to a set of three first order ODEs.     

Functional gradient evaluation in the optimal control of a complex dynamical system

The gradient of the cost functional in a discrete optimal control problem for metal solidification in metal casting is exactly calculated. In contrast to previous studies, the object under analysis has a complex geometric shape. The mathematical model for describing the solidification process is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. Formulas for exact gradient evaluation are derived using the fast automatic differentiation technique.     

An inverse diffusion problem with nonlocal boundary conditions

This article considers the inverse problem of identification of a time‐dependent thermal diffusivity together with the temperature in an one‐dimensional heat equation with nonlocal boundary and integral overdetermination conditions when a heat exchange takes place across boundary of the material. The well‐posedness of the problem is studied under some regularity, and consistency conditions on the data of the problem together with the nonnegativity condition on the Fourier coefficients of the initial data and source term. The inverse problem is also studied numerically by using the Crank–Nicolson finite difference scheme combined with predictor‐corrector technique. The numerical examples are presented and discussed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 564–590, 2016     

Method of Decreasing the Order of a Partial Differential Equation by Reducing to Two Ordinary Differential Equations

Using additional unknown functions and additional boundary conditions in the integral method of heat balance, we obtain approximate analytic solutions to the non-stationary thermal conductivity problem for an infinite solid cylinder that allow to estimate the temperature state practically in the whole time range of the non-stationary process. The thermal conducting process is divided into two stages with respect to time. The initial problem for the partial differential equation is represented in the form of two problems, in which the integration is performed over ordinary differential equations with respect to corresponding additional unknown functions. This method allows to simplify substantially the solving process of the initial problem by reducing it to the sequential solution of two problems, in each of them additional boundary conditions are used.     

Control of substance solidification in a complex-geometry mold

The control of metal solidification in a mold of complex geometry is studied. The underlying mathematical model is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the optimal control problem for the solidification process is presented. This problem was solved numerically using gradient optimization methods. The gradient of the cost function was computed by applying the fast automatic differentiation technique, which yields the exact value of the cost function gradient for the chosen discrete version of the optimal control problem. The results of the study are described and analyzed. Some of the results are illustrated as plots.     

在纵向自由薄膜凝固期内温度分布的数值分析

考虑到薄膜表面的热通量主要是来自辐射，因而采用一个依赖时间的拟二维拟线性扩散方程的Ｓｔｅｆａｎ问题（混合初边值问题）作为该问题的数学模型。用一种具有Ｃｒａｎｋ－Ｎｉｃｈｏｌｓｏｎ格式的无条件稳定的有限差分析来求解抛物型偏微分方程的定解问题。用追赶法求解离散化的三对角方程组，然后用预估校正法求解拟线性扩散方程，从而避免了示解非线性差分方程组，琥珀亚硝酸酯在纵向自由薄膜凝固期内的温度分布数值计算结果和     

Influence of electromagnetic stirrer position on fluid flow and solidification in continuous casting mold

A computational study of the effect of stirrer position on fluid flow and solidification in a continuous casting billet mold with in-mold electromagnetic stirring has been carried out. The numerical investigation uses a full coupling method in which alternating magnetic field equations are solved simultaneously with the governing equations of fluid flow and heat transfer. An enthalpy-porosity technique is used for the solidification analysis while the magnetohydrodynamics technique is used for studying the fluid flow behavior under the electromagnetic field. The streamline, liquid fraction, and solid shell thickness at the mold wall have been predicted with and without EMS application at different positions along the length of the mold. Recirculation loops are seen to be formed above and below the stirrer position when fluid flow and electromagnetic field equations were solved, without incorporating the solidification model. Application of the solidification model interestingly resulted in the reduction of the size of the recirculation loops formed. The tangential component of velocity of the fluid near the solidification front, stirring intensity and the effective length of stirring below the stirrer decrease as the stirrer position is moved downwards. Significant changes in characteristics of solid shell formation like delay in initiation of solidification at the mold wall and formation of a gap in the re-solidified shell have been observed with change in stirrer position.     

Synthesis scheme for hybrid methods of solving boundary problems

A scheme is proposed for the development of a numericoanalytic boundary problem technique that can synthesize the mesh-projective method with the integral transform method to yield an effective hybrid method for the determination of the temperature field in a body from a moving heat source with given dependences of the thermophysical parameters on temperature.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 19–22, 1987.