Kevin D. Cole

Heat conduction using Green's functions

Introduction to Greens Functions Heat Flux and Temperature Differential Energy Equation Boundary and Initial Conditions Integral Energy Equation Dirac Delta Function Steady Heat Conduction in One Dimension GF in the Infinite One-Dimensional Body Temperature in an Infinite One-Dimensional Body Two Interpretations of Greens Functions Temperature in Semi-Infinite Bodies Flat Plates Properties Common to Transient Greens Functions Heterogeneous Bodies Anisotropic Bodies Transformations Non-Fourier Heat Conduction Numbering System in Heat Conduction Geometry and Boundary Condition Numbering System Boundary Condition Modifiers Initial Temperature Distribution Interface Descriptors Numbering System for g(x, t) Examples of Numbering System Advantages of Numbering System Derivation of the Greens Function Solution Equation Derivation of the One-Dimensional Greens Function Solution Equation General Form of the Greens Function Solution Equation Alternative Greens Function Solution Equation Fin Term m2T Steady Heat Conduction Moving Solids Methods for Obtaining Greens Functions Method of Images Laplace Transform Method Method Of Separation of Variables Product Solution for Transient GF Method of Eigenfunction Expansions Steady Greens Functions Improvement of Convergence and Intrinsic Verification Identifying Convergence Problems Strategies to Improve Series Convergence Intrinsic Verification Rectangular Coordinates One-Dimensional Greens Functions Solution Equation Semi-Infinite One-Dimensional Bodies Flat Plates: Small-Cotime Greens Functions Flat Plates: Large-Cotime Greens Functions Flat Plates: The Nonhomogeneous Boundary Two-Dimensional Rectangular Bodies Two-Dimensional Semi-Infinite Bodies Steady State Cylindrical Coordinates Relations for Radial Heat Flow Infinite Body Separation of Variables for Radial Heat Flow Long Solid Cylinder Hollow Cylinder Infinite Body with a Circular Hole Thin Shells, T = T (phi, t) Limiting Cases for 2D and 3D Geometries Cylinders with T = T (r, z, t ) Disk Heat Source on a Semi-Infinite Body Bodies with T = T (r, phi, t ) Steady State Radial Heat Flow in Spherical Coordinates Greens Function Equation for Radial Spherical Heat Flow Infinite Body Separation of Variables for Radial Heat Flow in Spheres Temperature in Solid Spheres Temperature in Hollow Spheres Temperature in an Infinite Region Outside a Spherical Cavity Steady State Steady-Periodic Heat Conduction Steady-Periodic Relations One-Dimensional GF One-Dimensional Temperature Layered Bodies Two- and Three-Dimensional Cartesian Bodies Two-Dimensional Bodies in Cylindrical Coordinates Cylinder with T = T (r, phi, z,omega) Galerkin-Based Greens Functions and Solutions Greens Functions and Greens Function Solution Method Alternative form of the Greens Function Solution Basis Functions and Simple Matrix Operations Fins and Fin Effect Conclusions Applications of the Galerkin-Based Greens Functions Basis Functions in some Complex Geometries Heterogeneous Solids Steady-State Conduction Fluid Flow in Ducts Conclusion Unsteady Surface Element Method Duhamels Theorem and Greens Function Method Unsteady Surface Element Formulations Approximate Analytical Solution (Single Element) Examples Problems References Appendices Index

Solutions of the heat conduction equation in multilayers for photothermal deflection experiments

Exact expressions are presented for the deflection of a laser beam passing parallel to and above the surface of a sample heated by a periodically modulated axisymmetric laser beam. The sample may consist of any number of planar films on a thick substrate. These exact expressions are derived from a local Green’s function treatment of the heat conduction equation, and contain an exact analytical treatment of the absorption of energy in the multilayered system from the heating laser. The method is based on calculation of the normal component of the heat fluxes across the layer boundaries, from which either the beam deflections or the temperature anywhere in space can be easily found. A central part of the calculation is a tridiagonal matrix equation for the N+1 normal boundary fluxes, where N equals the number of films in the sample, with the beam deflections given as simple functions of the normal heat flux through the top surface of the sample. Even though any layer or layers in the sample (including the s...

Conjugate heat transfer from a small heated strip

Abstract This study addresses the electronic cooling problem from the perspective of scaling laws applied to a simple conjugate heat transfer geometry, steady shear flow over a heated strip on a flat plate. The results are reported in a compact form in terms of a modified Nusselt number and a combined parameter ( solk f k s )Pe 1 3 . Numerical results are reported that apply to a wide range of values for fluid flow, thermal conductivity, and thickness of the flat plate, and the results include simple design correlations. A new dimensionless parameter is suggested for determining when the fluid axial heat conduction can be neglected.

Green's functions, temperature and heat flux in the rectangle

Abstract Steady heat conduction in the rectangle is treated with the method of Greens functions. Single-sum series for the Greens functions are reported in terms of exponentials which have better numerical properties than hyperbolic functions. Series expressions for temperature and heat flux caused by spatially uniform effects are presented. The numerical convergence of these series is improved, in some cases by a factor of 1000, by replacing slowly converging portions of the series with fully summed forms. This work is motivated by high-accuracy verification of finite-difference and finite-element codes.

Steady-Periodic Green's Functions and Thermal- Measurement Applications in Rectangular Coordinates

Methods of thermal property measurements based on steady-periodic heating are indirect techniques, in which the thermal properties are deduced from a systematic comparison between experimental data and heat-transfer theory. In this paper heat-transfer theory is presented for a variety of two-dimensional geometries applicable to steady-periodic thermal-property techniques. The method of Green’s functions is used to systematically treat rectangles, slabs (two dimensional), and semi-infinite bodies. Several boundary conditions are treated, including convection and boundaries containing a thin, highconductivity film. The family of solutions presented here provides an opportunity for verification of numerical results by the use of distinct, but similar, geometries. A second opportunity for verification arises from alternate forms of the Green’s function, from which alternate series expressions may be constructed for the same unique temperature solution. Numerical examples are given to demonstrate both verification techniques for the steady-periodic response to a heated strip. DOI: 10.1115/1.2194040

Fast-converging steady-state heat conduction in a rectangular parallelepiped

Abstract A Greens function approach for precisely computing the temperature and the three components of the heat flux in a rectangular parallelepiped is presented. Each face of the parallelepiped may have a different, but spatially uniform, boundary condition. Uniform volume energy generation is also treated. Three types of boundary conditions are included: type 1, a specified temperature; type 2, a specified flux; or type 3, a specified convection boundary condition. A general form of the Greens function covering all three types of boundary conditions is given. An algorithm is presented to obtain the temperature and flux at high accuracy with a minimal number of calculations for points in the interior as well as on any of the faces. Heat flux on type 1 boundaries, impossible to evaluate with traditional Fourier series, is found by factoring out lower-dimensional solutions. A numerical example is given. This research and resulting computer program was part of a code verification project for Sandia National Laboratories.

Thermal characterization of thin films by photothermally induced laser beam deflection

Abstract The photothermal deflection technique, also known as the “mirage effect”, is a powerful non-destructive means of evaluating the thermal properties of both bulk materials and thin films. In this experiment, the sample is heated by a modulated laser beam, and the deflection of a second laser beam passing through the heated region above the sample (parallel to the surface of the sample) is measured as a function of distance between the probe and heating beams. The temperature profile above the sample and hence the beam deflections, are determined by the sample properties. These measurements can be analyzed to determine the thermal diffusivity of the sample. For multilayered samples it is often possible to find the thermal constants for a single layer if the properties of the other materials in the sample are previously measured or otherwise known. We have non-linear regression software for the analysis for samples that may have any number of layers, based on a Greens function formalism with no approximations, for the calculation of beam deflections. We have applied this technique to a number of bulk and thin film systems, from which representative results are presented.

Steady-periodic heating of a cylinder

Steady periodic heating is an important experimental technique for measurement of thermal properties. In these methods the thermal properties are deduced from a systematic comparison between the data (such as temperature) and a detailed thermal model. This paper addresses steady-periodic heat transfer on cylindrical geometries with application to thermal-property measurements. The method of Greens functions is used to provide a comprehensive collection of exact analytical expressions for temperature in cylinders. Five kinds of boundary conditions are treated for one-, two-, and three-dimensional geometries. For some geometries an alternate form of the Greens function is given, which can be used for improvement of series convergence and for checking purposes to produce highly accurate numerical values. Numerical examples are given.

Microchannel Heat Transfer with Slip Flow and Wall Effects

Analysis is presented for conjugate heat transfer in a parallel-plate microchannel. Axial conduction in the fluid and in the adjacent wall is included. The fluid is a constant property gas with a slip-flow velocity distribution. The microchannel is heated by a small region on the channel wall. The analytic solution is given in the form of integrals by the method of Green’s functions. Quadrature is used to obtain numerical results for the temperature and heat transfer coefficient on the heated region for various Peclet number, Knudsen number, and wall materials. A region downstream of the heater is also explored. These results have application in the optimal design of small-scale heat transfer devices for biomedical applications, electronic cooling, and advanced fuel cells.

Design of Experiments for Thermal Characterization of Metallic Foam

Metallic foams are being investigated for possible use in the thermal protection systems of reusable launch vehicles. As a result, the performance of these materials needs to be characterized over a wide range of temperatures and pressures. In this paper a radiation/conduction model is presented for heat transfer in metallic foams. Candidates for the optimal transient experiment to determine the intrinsic properties of the model are found by two methods. First, an optimality criterion is used to nd an experiment to nd all of the parameters using one heating event. Second, a pair of heating events is used to determine the parameters in which one heating event is optimal for nding the parameters related to conduction, while the other heating event is optimal for nding the parameters associated with radiation. Simulated data containing random noise was analyzed to determine the parameters using both methods. In all cases the parameter estimates could be improved by analyzing a larger data record than suggested by the optimality criterion.