Two-dimensional Transient Heat Conduction Research Articles

A boundary element modelling for two-dimensional transient heat conduction

In this paper a boundary element formulation for heat conduction problems are presented. The different types of boundary conditions of conduction, convection and radiation are included in the integral equations. The transient fundamental solution is used and by considering small time limits the integration over the time domain are calculated analytically. In the absence of a radiation boundary condition a set of linear system of equations would be developed. While, in the presence of radiation boundary conditions, a set of nonlinear system of equations is resulted. Also presented in this paper a multi-region boundary element analysis for heat conduction problems composed of different materials having different thermal properties. The compatibility and equilibrium are used to link the systems of equations developed for each region. Examples are provided to check the accuracy of the method and the implementation.

Two-Dimensional Transient Heat Conduction Analysis Using Spreadsheets

A method for using spreadsheets to perform a numerical analysis of one- and two-dimensional transient heat conduction is described. The numerical method is based on the explicit nodal equations and is described in terms of the thermal resistance and capacitance formulation. The spreadsheet setup uses an iteration macro and can use either a single-layer spreadsheet or multiple-layer linked spreadsheets. The method allows the use of various boundary conditions, including a specified surface temperature, insulated surfaces, a convective heal transfer coefficient distribution on any surface, and an equivalent radiation heat transfer coefficient. A variable grid can be used that allows a finer grid in areas of high gradients. Thermal properties may be heterogeneous and temperature dependent. The results are highly visual and can be easily plotted in two- and three-dimensional graphs.

SOLUTION CONTOUR-BASED ADAPTIVE REGRIDDIIMG SCHEME FOR TRANSIENT DIFFUSION PROBLEMS

An adaptive regridding scheme within the framework of irregular triangles and finite differences is presented. The new grid's nodal points are selected on solution contours, Boundary adaptation is accomplished by using a pseudo-triangle gradient level indicator. The technique is demonstrated by adoptively solving the linear two-dimensional transient heat conduction equation on both rectangular and circular domains. Results indicate that the adaptive regridding scheme works as accurately as a fine grid, especially in regions where a steep change of solution occurs, but at a significantly lower computational cost. Apart from promising to be a general method, the scheme is simple in that it does not require the evaluation of weight functions, the solution of a set of differential equations, or the construction of a solution monitor surface to obtain new nodes.

Heat transfer in tissues radiated by a 432 MHz directional antenna

Heat transfer in tissues during electromagnetic heating obtained by a 432 MHz directional antenna for therapeutic purposes is considered. The two-dimensional transient heat conduction equation, which takes into account the electromagnetic heating and the cooling owing to blood circulation is solved by finite-difference means. Solutions in cases of inhomogeneous tissues and of tissues containing large blood vessels are given. Also, the analytical solution of the corresponding one-dimensional steady-state problem in the principal axis of the electromagnetic field is derived. This solution is in good agreement with the two-dimensional numerical solution and may therefore be employed for making quick calculations.

Hybrid Laplace transform/finite-element method for two-dimensional transient heat conduction

A powerful method of analysis, the combined use of the Laplace transform and the finite-element method, is applicable to the problem of time-dependent heat flow systems. The present method removes the time terms using the Laplace transform and then solves the associated equation with the finite-element technique. The associated temperature transform is inverted by the method of Dubner and Abate for obtaining the required results. The present results are compared in tables to the analytical solutions and other numerical data, and it is found that the present solutions are stable and convergent to them. There exists no time step; thus, the present method is a useful tool in solving lengthy problems or in other engineering applications.

Application of the Integral Method to Two-Dimensional Transient Heat Conduction Problems

Application of the Integral Method to Two-Dimensional Transient Heat Conduction Problems

Discussion: “Stability and Oscillation Characteristics of Finite-Element, Finite-Difference, and Weighted-Residuals Methods for Transient Two-Dimensional Heat Conduction in Solids” (Yalamanchilli, R. V. S., and Chu, S.-C., 1973, ASME J. Heat Transfer, 95, pp. 235–239)

Discussion: “Stability and Oscillation Characteristics of Finite-Element, Finite-Difference, and Weighted-Residuals Methods for Transient Two-Dimensional Heat Conduction in Solids” (Yalamanchilli, R. V. S., and Chu, S.-C., 1973, ASME J. Heat Transfer, 95, pp. 235–239)

Heat conduction model for the calculation of the volume of molten metal in electric discharges

The craters obtained by electric sparks in electro-discharge machining are numerically calculated by use of a thermal model. The computational procedure is based on the exact solution of the related two-dimensional transient heat conduction problem in a finite or semi-infinite body subjected to a stepwise or time-dependent uniform heat flux over a small circular part of its upper surface. Optimal geometrical conditions, relative to a fast convergence and negligible influence of the boundaries, are demonstrated. By superposition of rectangular heat flux pulses over circular parts with different diameters, the influence of the channel widening can be investigated. It is shown that a theoretical model should include the possibility of accounting for the plasma channel widening in order to obtain acceptable agreement with experimental data or to predict the optimal operating conditions.

On the process of rewetting a hot surface by a falling liquid film

A two-dimensional transient heat conduction model for rewetting a hot surface by a falling liquid film predicts that for stainless steel, Inconel or Zircaloy only a wall thickness of some 0.020 in. takes part in the rewetting process in steam at 100–1000 psia. The rewetting rate is nearly independent of heat flux and thermal conductivity, but increases with pressure and decreases with volumetric heat capacity.

Radial–Axial Transient Heat Conduction in a Region Bounded Internally by a Circular Cylinder of Finite Length and Appreciable Heat Capacity

The problem of two-dimensional transient heat conduction from a circular cylinder of finite length and appreciable heat capacity has been solved using a Laplace transformation with respect to time and a finite Fourier sine transformation with respect to the axial variable. A case of constant surface heat flux with the ends of the cylinder maintained at zero temperature is considered. The solution, valid for all values of time, is compared with that of Jaeger for the infinitely long cylinder. The results are of use in the evaluation of heat losses for the transient hot-wire method of determining the thermal conductivity of fluids.

Stability and Oscillation Characteristics of Finite-Element, Finite-Difference, and Weighted-Residuals Methods for Transient Two-dimensional Heat Conduction in Solids

The finite-element difference expression was derived by use of the variational principle and finite-element synthesis. Several ordinary finite-difference formulae for the La-placian term were considered. A particular finite-difference formula for the Laplacian term was chosen to bring the difference expressions of finite-element, finite-difference, and weighted-residuals (Galerkin) methods into the same format. The stability criteria were established for all three techniques by use of the general stability, von Neumann, and Dusinberre concepts. The oscillation characteristics were derived for all three techniques. The finite-element method is more conservative than the finite-difference method, but not so conservative as the Galerkin method in both stability and oscillation characteristics.

Formulation of the two-dimensional transient heat-conduction problem for the bladed disk of a gas turbine

A system of differential equations is derived for the transient heat conduction associated with a cooled gas turbine disk equipped with shrouded blades on a shank.

A Correlation in Displacement Velocity of Liquid Film Boundary formed on a Heated Vertical Surface in Emergency Cooling

Intense ebullition occurs at the liquid front heading a falling sheet of water over a heated vertical surface, as in the case of emergency cooling. It has been ascertained that the displacement velocity of this liquid front is nearly constant for the values of specified parameters such as wall temperature before cooling, flow rate of cooling liquid, and the physical properties of heated wall and cooling liquid. Observations were conducted on the liquid front velocity using a heated wall of stainless-steel and with tap water as cooling liquid. An analysis has been made on the basis of a two-dimensional transient heat conduction model, the result of which proves to be valid for explaining the liquid front behavior when account is taken of boiling heat transfer. The liquid front velocity is given by where N denotes a reference state. The above relation is applicable in the region of flow rate from 0.245 kg/m.sec to 2.45 kg/m.sec.

A Correlation in Displacement Velocity of Liquid Film Boundary formed on a Heated Vertical Surface in Emergency Cooling

Intense ebullition occurs at the liquid front heading a falling sheet of water over a heated vertical surface, as in the case of emergency cooling. It has been ascertained that the displace-ment velocity of this liquid front is nearly constant for the values of specified parameters such as wall temperature before cooling, flow rate of cooling liquid, and the physical properties of heated wall and cooling liquid. Observations were conducted on the liquid front velocity using a heated wall of stainless-steel and with tap water as cooling liquid. An analysis has been made on the basis of a two-dimensional transient heat conduction model, the result of which proves to be valid for explaining the liquid front behavior when account is taken of boiling heat transfer. The liquid front velocity is given bywhere N denotes a reference state. The above relation is applicable in the region of flow rate from 0.245 kg/m • sec to 2.45 kg/m • sec.

Two-dimensional transient ablation and heat conduction analysis for multimaterial thrust chamber walls.

Two-dimensional transient ablation and heat conduction analysis for multimaterial thrust chamber walls.

Flame spreading and ignition transients in solid grain propellants

An analysis of grain ignition and flame spreading in solid-propellant rocket engines was performed to calculate chamber pressure as a function of time during this phase. Assumed for the analysis were isothermal one-dimensional gas flow in the chamber, spatially uniform gas pressure, two-dimensional transient heat conduction in the grain, perfect gas properties, choked nozzle flow, and gas generation due only to grain surface deflagration. Also investigated is the case of a diaphragm stretched across the throat entrance which bursts with sufficient pressure build-up. Initial conditions include instantaneous gasless ignition of a section of the upstream end of the grain, uniform pressure and temperature in the gas, a uniformly distributed temperature in the unignited portion of the grain, and an axially varying gas velocity distribution. A quasi-analytic solution (analytic in each time step) for chamber pressure is coupled, through changes in the burning wall area, to a finite difference solution for grain temperature (using the stable Peaceman-Rachford alternating direction method). After the end of flame spreading, the pressure calculation alone is continued until steady-state conditions are approximated. A computer program was written to perform the calculating for varying input data.