A Class of Moving Boundary Problems with a Source Term. Application of a Reciprocal Transformation

We consider a new Stefan-type problem for the classical heat equation with a latent heat and phase-change temperature depending of the variable time. We prove the equivalence of this Stefan problem with a class of boundary value problems for the nonlinear canonical evolution equation involving a source term with two free boundaries. This equivalence is obtained by applying a reduction to a Burgers equation and a reciprocal-type transformations. Moreover, for a particular case, we obtain a unique explicit solution for the two different problems

Nonlinear Stefan Problem for one-phase generalized heat equation with heat flux and convective boundary condition

In this article we consider a mathematical model of an initial stage of closure electrical contact that involves a metallic vaporization after instantaneous exploding of contact due to arc ignition with power $P_0$ on fixed face $z=0$ and heat transfer in material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component with heat flux and convective boundary conditions prescribed at the known free boundary $z = \alpha (t)$. The temperature field in the liquid region of such kind of material can be modelled by Stefan problem for the generalized heat equation. The method of solution is based on similarity variable, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we have to determine temperature solution for the liquid phase and location of melting interface. Existence and uniqueness of the solution is proved by using the fixed point Banach theorem. The solution for two cases of thermal coefficients, in particular, constant and linear thermal conductivity are represented, existence and uniqueness for each type of solution is proved

Microwave heating of fluid/solid layers : a study of hydrodynamic stability and melting front propagation

In this work we study the effects of externally induced heating on the dynamics of fluid layers, and materials composed of two phases separated by a thermally driven moving front. One novel aspect of our study is in the nature of the external source, which is provided by the action of microwaves acting on dielectric materials. The main challenge is to model and solve systems of differential equations, which couple fluid dynamical motions (the Navie- Stokes equations for nonisothermal flows) and electromagnetic wave propagation (governed by Maxwell\u27s equations). When an electromagnetic wave impinges on a material, energy is generated within the material due to dipolar and ohmic heating. The electrical and thermal properties of the material dictate the dynamics of the heating process, as well as steady state temperature profiles. Such forms of heating have received little attention in studies of hydrodynamic instabilities of non-isothermal flows, such as the classical Benard problem, for instance. The novel feature, which allows possibilities for fluid management and control, is the non-local coupling between the electromagnetic field and the temperature distribution within the fluid. In the first part of the thesis, we consider hydrodynamic instabilities of such systems with particular emphasis on conditions for onset of convection. This is achieved by solving the linear stability equations in order to identify parameter values, which produce instability. The analysis and subsequent numerical solutions are carried out both for materials with constant dielectric attributes (in such cases the electric field equations decouple and they can be solved in closed form), and materials with temperature dependent Conductivities, dielectric permittivities and dielectric loss factors. In the latter case we incorporate known data for water or ethanol into our numerical solutions. Our solutions provide a complete picture of onset conditions as a function of input power levels and microwave frequency (or equivalently fluid layer thickness). In addition, in the case of water, the flow is found to be more stable for constant attributes as compared with temperature dependent attributes; that is, a higher power is required to set the fluid layer into convective motions in the latter case. We have also established that onset is obtained at power levels well below those needed to cause thermal runaway and consequently boiling of the water layer, for instance. Our results also identify different parameter ranges, which can produce convection cells of different sizes with the same power input. Such results are directly related to the micro-wave radiation, which provides the heating, and in particular the distribution of the electric field within the fluid layer. Several interesting experiments are suggested by the theoretical predictions. The second problem studied is concerned with the use of microwave radiation in the processing of materials, which contain two phases separated by a moving front, which forms and propagates due to a jump in temperature flux across the interface separating the phases. The problem is an extension of the classical Stefan problem with the propagation caused by temperature gradients induced by the electromagnetic radiation. \u27We have modeled and solved the problem of two phases separated by a planar interface and in the absence of fluid motion if melting is involved. The boundary conditions are those of convective cooling at the top surface and either a heat sink (to maintain a frozen state for ice, for example) or a perfectly electro magnetically reflective bounding surface at the bottom. Known data modeling a water-ice system have been used, but the methods are the same for other materials. We have addressed the cases of constant and temperature obtained by solving a coupled system of nonlinear differential equations leading to an eigenvalue problem for the interfacial position. In addition, a time-dependent code was developed in order to study transient motions towards steady-state, starting from initial configurations of a thin water layer on the ice, for example. Our results indicate that for a given power level there can be two stable steady-state positions for the melting front as well as an unstable one. Existence of multiple states is a consequence of electromagnetic wave resonances within the material and their global effects on the thermal distribution. Such behavior leads to a theoretical framework in efforts to control the position of phase separation interfaces in the processing of materials

Convection in the Melt

A physical problem involving the melting/freezing of a phase-change material (PCM) is the applied setting of this research. The development of models that couple the partial differential equations for energy transport and fluid motion with phases of differing densities is a primary goal of the research. In Chapter 2, a general framework is developed for the formulation of conservation laws that admit interfaces. A notion of weak solution is developed and its relation with classical and other weak formulations is discussed. Conditions that hold across various kinds of interfaces are also developed. The formulation is examined for the conservation of mass, momentum and energy in Chapter 3. In Chapter 4, a numerical method for the solution of conservation law equations is given. The method uses a Crank-Nicolson time discretization and solves the implicit equations with a Newton/Approximate Factorization technique. The method captures interfaces and is consistent with the control volume weak formulations of Chapter 2. The numerical solution converges to the distributional solution of the conservation law. In Chapter 5, three applications of the theory are developed and numerical computations are presented. First, a one dimensional problem is studied involving conservation of mass. momentum and energy in a phase-change material with a liquid density larger than that of the solid. The second application is a suction problem in two dimensions. The bulk movement of a liquid and void are simulated with and without the effects of surface tension. The third application is to a three-dimensional simulation of the heating of a cylindrical canister of PCM in 1-g and 0-g. For this simulation the Marangoni stress is the important driving force on the flow

Numerical Formulation for the Prediction of Solid/Liquid Change of a Binary Alloy

A computational model is presented for the prediction of solid/liquid phase change energy transport including the influence of free convection fluid flow in the liquid phase region. The computational model considers the velocity components of all non-liquid phase change material control volumes to be zero but fully solves the coupled mass-momentum problem within the liquid region. The thermal energy model includes the entire domain and uses an enthalpy like model and a recently developed method for handling the phase change interface nonlinearity. Convergence studies are performed and comparisons made with experimental data for two different problem specifications. The convergence studies indicate that grid independence was achieved and the comparison with experimental data indicates excellent quantitative prediction of the melt fraction evolution. Qualitative data is also provided in the form of velocity vector diagrams and isotherm plots for selected times in the evolution of both problems. The computational costs incurred are quite low by comparison with previous efforts on solving these problems

Numerical simulation of one-dimensional heat transfer in composite bodies with phase change

A numerical simulation was developed to investigate the one dimensional heat transfer occurring in a system composed of a layered aircraft blade having an ice deposit on its surface. The finite difference representation of the heat conduction equations was done using the Crank-Nicolson implicit finite difference formulation. The simulation considers uniform or time dependent heat sources, from heaters which can be either point sources or of finite thickness. For the ice water phase change, a numerical method which approximates the latent heat effect by a large heat capacity over a small temperature interval was applied. The simulation describes the temperature profiles within the various layers of the de-icer pad, as well as the movement of the ice water interface. The simulation could also be used to predict the one dimensional temperature profiles in any composite slab having different boundary conditions