Isotherm Migration Method Research Articles

Numerical solutions for a Stefan problem

The initial version of a Stefan problem is the melting of a semi-infinite sheet of ice. This problem is described by a parabolic partial differential equation along with two bou ndary conditions on the moving boundary which are used to determine the boundary itself and complete the solution of the differenti al equation. In this paper firstly, we use variable space grid method, boundary immobilisation method and isotherm migration method to get rid of the trouble of the Stefan problem. Then, collocation finite element method based on cubic B-spline bases function s is applied to model problem. The numerical schemes of finite element methods provide a good numerical approximation for the model problem. The numerical results show that the present results are in good agreement with the exact ones.

The isotherm migration method in spherical coordinates with a moving heat source

An isotherm migration method is presented to track the isotherms at the surface of a semi-infinite workpiece that is being heated by a point source with a linear motion. The governing heat conduction equation is first transformed into a one-dimensional isotherm migration equation in spherical coordinates, which expresses the isotherm velocities as a function of the isotherm positions and their temperature derivatives. A finite difference formulation of this equation is then derived together with the corresponding boundary conditions. It is shown through a number of simulations that this system converges to the analytical solution when the temperature mesh is refined. The additional presence of a phase change results in a Stefan problem that can be solved approximately with some minor modifications to the method. The resulting system of equations can be used as a simple but accurate thermal model for some laser-material interaction processes such as laser heat treatments and laser cladding.

Multigrid Methods for Incompressible Heat Flow Problems with an Unknown Interface

Finite-volume/multigrid methods are presented for solving incompressible heat flow problems with an unknown melt/solid interface, mainly in solidification applications, using primitive variables on collocated grids. The methods are implemented based on a multiblock and multilevel approach, allowing the treatment of a complicated geometry. The inner iterations are based on the SIMPLE scheme, in which the momentum interpolation is used to prevent velocity/pressure decoupling. The outer iterations are set up for interface update through the isotherm migration method. V-cycle and full multigrid (FMG) methods are tested for both two- and three-dimensional problems and are compared with a global Newton's method and a single-grid method. The effects of Prandtl and Rayleigh numbers on the performance of the schemes are also illustrated. Among these approaches, FMG has proven to be superior on performance and efficient for large problems. Sample calculations are also conducted for horizontal Bridgman crystal growth, and the performance is compared with that of traditional single-grid methods.

Simultaneous drying and pyrolysis of solid fuel particles

Drying and devolatilization are studied at combustion temperatures. The surface temperature of particles at the end of drying can significantly exceed the temperature when devolatilization starts, implying that drying and pyrolysis may partly overlap. Devolatilization is controlled by heat transfer, when the particle size is large. The critical particle size at which heat transfer dominates chemical kinetics is discussed. A model for calculating the intrinsic rate of generation of volatiles in the regime of heat transfer control is presented. A novel isotherm migration method is used for the computation of simultaneous drying and pyrolysis inside a fuel particle. It applies to the study of heat transfer in a one-dimensional geometry with moving phase-change boundaries, internal fluid flow and mass generation, including steep temperature and density profiles, as frequently encountered in combustion.

Adaptive finite element analysis of axisymmetric freezing

The paper describes an adaptive finite element analysis of the transient axisymmetric freezing process. Adaptivity schemes are applied to both space and time tessellations. Error equidistribution adjusts the nodal positions and Taylor series analysis of time truncation error selects the time step. In implicit time integration, the location of the freezing front and the nodal temperatures at the next time is solved by full Newton all at once. The method appears to be accurate; in cases for which closed-form solutions are available, it agrees well with them. It also avoids the problem found in the Modified Isotherm Migration Method where the freezing front tends to retrogress when solid just forms on the outer surface cooled by convection.

A note on the use of the isotherm migration method

In this note we present some simple modifications of the isotherm migration method (IMM) so that a broader class of conduction problems (than the usual Stefan problems) can be effectively tackled by its use. Some numerical illustrations, based on finite differences, are presented which show that, for heat flow problems with arbitrary conditions, an accurate numerical process can be based on the IMM formulation.

TECHNICAL NOTE UNCONDITIONALLY STABLE IMPLEMENTATION OF THE ISOTHERM MIGRATION METHOD

In this paper, we present a particularly efficient implementation of the isotherm migration method for solving melting and freeing problems in which the hopscotch difference scheme is utilized to overcome the usual time-step restriction imposed on explicit-type numerical integrators. The method is illustrated on a simple one-phase problem, and numerical results are presented.

Isotherm migration method applied to fusion problems with convective boundary conditions

A numerical technique based on the Isotherm Migration Method is presented for solving Moving Boundary Problems involving convective boundary conditions. The method is illustrated by solving sample problems in one dimension as well as in two dimensions. The comparison of present results with those of earlier authors shows an extremely good agreement.

Mathematical Modeling of Microwave Thawing by the Modified Isotherm Migration Method

ABSTRACTA mathematical model of microwave thawing of homogeneous food products is developed and solved numerically using the Modified Isotherm Migration Method. The model is used to predict thawing time and temperature profiles for microwave thawed meat cylinders at three frequencies (2450 MHz, 915 MHz, 300 MHz) and different power levels. Model and experimental results for thawing a lean beef cylinder heated at low microwave power using 2450 MHz frequency compare well. The advantage of using 915 or 300 MHz power over 2450 MHz power is shown by calculations. The results show that microwaves significantly accelerate the thawing rate. The mathematical model is explored as a tool for designing optimal microwave/convective heating protocols for rapidly thawing foods in desired temperature ranges.

An efficient approach to isotherm migration method in two dimensions

Analyse numerique par deux methodes l'une implicite et l'autre explicite du mouvement de l'interface liquide-solide au cours de la solidification d'un liquide dans un prisme rectangulaire de longueur definie

Moving boundary problems in simple shapes solved by isotherm migration

AbstractThe Modified Isotherm Migration Method (MIMM) is developed for solving one‐dimensional Stefan‐type problems of multifront propagation in slabs, cylinders and spheres with a boundary condition of the third kind (radiation‐type boundary condition). The method is illustrated for all three geometries with applications to the thawing of ice and to the freezing of a 50–50 dispersion of tetradecane in water, a system in which two freezing fronts propagate simultaneously.

Progressive freezing of composites analyzed by isotherm migration methods

AbstractThe Isotherm Migration Method of numerically analyzing conduction with phase change is modified to handle surface and external resistances, appearance, and propagation of successive fronts into slabs of composite material of one or more constituents finely dispersed in a matrix, different components transforming at different temperatures. Procedures are developed for selecting temperature intervals between isotherms, initially locating isotherms by means of analytical approximations to the earliest stages, and thereafter inserting and removing isotherms as appropriate, particularly when each component gives rise to a separate front.

Analysis of Propagation of Freezing and Thawing Fronts

ABSTRACTThe modified isotherm migration method (MIMM) is used to calculate the temperature profiles and freezing (thawing) times for symmetrically cooled (heated) slabs with meat‐like thermal properties. MIMM, which is reviewed briefly is not restricted to the usual idealizations such as zero surface resistance, uniform critical temperature distribution, and infinitely large sample. Dimensional analysis is used to identify the dimensionless groups controlling freezing and thawing front propogation in slabs, cylinders, and spheres having uniform thermal properties and constant freezing temperature. The MIMM results provide a basis for evaluating the qualitative origins and the quantitative extent of the success of Plank's (1913) old model. The freezing times predicted by the moving front model used here agree with those predicted by the empirical correlation found by Cleland and Earle for the Karlsruhe test substance.

On an implicit scheme for the isotherm migration method along orthogonal flow lines in two dimensions

Abstract In a previous paper the authors developed an explicit scheme for the numerical solution of the equations of isotherm migration along orthogonal flow lines in two space dimensions. This paper presents an implicit scheme. It is used to solve a two-dimensional one-phase Stefan problem and results are compared with those obtained by other methods.

Isotherm Migration Method in two dimensions

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.