Transient and Steady Heat Conduction in Arbitrary Bodies with Arbitrary Boundary and Initial Conditions

1968 ◽  
Vol 90 (1) ◽  
pp. 103-108 ◽  
Author(s):  
E. M. Sparrow ◽  
A. Haji-Sheikh
Keyword(s):  
Heat Conduction ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Arbitrary Boundary ◽  
Closed Form Solutions ◽  
Thermal Boundary Conditions ◽  
Transient Problems ◽  
Conceptual Changes ◽  
Spatially Varying ◽  
Steady Heat Conduction

A method of analysis is described which yields closed-form solutions for two-dimensional heat conduction problems for bodies of arbitrary shape. Three-dimensional problems can also be treated without basic conceptual changes. The method accommodates rather general thermal boundary conditions including arbitrary spatial variations in surface temperature or in surface heat flux, or a convective (or linearized radiative) exchange with a fluid having spatially varying temperature and heat transfer coefficient. For transient problems, the initial temperature may be arbitrarily distributed. Once the solution method has been developed, its practical realization is rather direct, being facilitated by the use of widely available computer routines. A numerical example to illustrate the method is worked out.

Heat Conduction in Multilayered Rectangular Domains

2007 ◽  
Vol 129 (4) ◽  
pp. 440-451 ◽  
Author(s):  
James Geer ◽  
Anand Desai ◽  
Bahgat Sammakia
Keyword(s):  
Thermal Conductivity ◽  
Heat Conduction ◽  
Heat Generation ◽  
Analytical Study ◽  
Three Dimensional ◽  
Thermal Boundary Conditions ◽  
Wide Range ◽  
Potential Applications ◽  
Steady State Heat ◽  
Spatially Varying

This paper presents the results of an analytical study of steady state heat conduction in multiple rectangular domains. Any finite number of domains that are equally sized (in plane) may be considered in the current analysis. The thermal conductivity and thickness of these domains may be different. The entire geometry composed of these connected domains is considered as adiabatic on the lateral surfaces and can be subjected to a wide range of thermal boundary conditions at the top and bottom. For example, the bottom of the stack may be adiabatic, while the top of the stack may be exposed to a uniform heat transfer coefficient. Spatially varying heat generation rates can be applied in each of the domains. The solutions are found to be in agreement with known solutions for simpler geometries. The analytical solution presented here is very general in that it takes into account the interface resistances between the layers. One application of this analytical study relates to the thermal management of three-dimensional stacks of computer devices and interconnect layers. The devices would have spatially nonuniform power dissipation within them, and the interconnect layers would have a significantly lower thermal conductivity than the devices. Interfacial defects, such as delamination or air voids, between the devices and the interconnect layers may be included in the model. Another possible application is to the study of hot spots in a chip stack with nonuniform heat generation. Many other potential applications may also be simulated.

Heat Conduction in Three Dimensional Stacked Packages

2005 ◽  
Author(s):  
Anand Desai ◽  
James Geer ◽  
Bahgat Sammakia
Keyword(s):  
Heat Conduction ◽  
Thermal Management ◽  
Heat Generation ◽  
Analytical Study ◽  
Three Dimensional ◽  
The Other ◽  
Heat Generation Rate ◽  
Potential Applications ◽  
Steady State Heat ◽  
Spatially Varying

This paper presents the results of an analytical study of steady state heat conduction in multiple rectangular domains. Any finite number of such domains may be considered in the current study. The thermal conductivity and thickness of these domains may be different. The entire geometry composed of these connected domains is considered as adiabatic on the lateral surfaces and can be subjected to uniform convective cooling at one end. The other end of the geometry may be adiabatic and a specified, spatially varying heat generation rate can be applied in each of the domains. The solutions are found to be in agreement with known solutions for simpler geometries. The analytical solution presented here is very general in that it takes into account the interface resistances between the layers. One application of this analytical study relates to the thermal management of a 3-D stack of devices and interconnect layers. Another possible application is to the study of hotspots in a chip stack with non uniform heat generation. Many other potential applications may also be simulated.

An Iterative Boundary Integral Numerical Solution for General Steady Heat Conduction Problems

1981 ◽  
Vol 103 (1) ◽  
pp. 26-31 ◽  
Author(s):  
M. S. Khader ◽  
M. C. Hanna
Keyword(s):  
Boundary Conditions ◽  
Present Method ◽  
Three Dimensional ◽  
Computational Procedure ◽  
Boundary Integral ◽  
Temperature Dependent ◽  
Convective Boundary ◽  
Arbitrary Boundary ◽  
Steady Heat Conduction ◽  
Steady Heat

An iterative boundary integral numerical method for solving the steady conduction of heat is developed. The method is general for two- and three-dimensional regions with arbitrary boundary shapes. The development is generalized to include the first, second, and third kind of boundary conditions and also radiative boundary and temperature-space dependent convective coefficient cases. With Kirchhoff’s transformation, cases of temperature-dependent thermal conductivity with general boundary conditions are also accounted for by the present method. A variety of problems are analyzed with this method and their solutions are compared to those obtained analytically. A comparison between the present method and the finite difference predictions is also investigated for a case of mixed temperature and convective boundary conditions. Moreover, two-dimensional regions with three kinds of boundary conditions and irregular-shaped boundaries are used to illustrate the versatility of the technique as a computational procedure.

scholarly journals TOPOLOGY OF STEADY HEAT CONDUCTION IN A SOLID SLAB SUBJECT TO A NONUNIFORM BOUNDARY CONDITION: THE CARSLAW–JAEGER SOLUTION REVISITED

2012 ◽  
Vol 53 (4) ◽  
pp. 308-320 ◽  
Author(s):  
R. G. KASIMOVA ◽  
YU. V. OBNOSOV
Keyword(s):  
Heat Conduction ◽  
Complex Potential ◽  
Singular Integrals ◽  
Thermal Protection ◽  
Closed Form Solutions ◽  
Heat Flows ◽  
Model Flow ◽  
Temperature Boundary ◽  
Steady Heat Conduction ◽  
Steady Heat

AbstractTemperature distributions recorded by thermocouples in a solid body (slab) subject to surface heating are used in a mathematical model of two-dimensional heat conduction. The corresponding Dirichlet problem for a holomorphic function (complex potential), involving temperature and a heat stream function, is solved in a strip. The Zhukovskii function is reconstructed through singular integrals, involving an auxiliary complex variable. The complex potential is mapped onto an auxiliary half-plane. The flow net (orthogonal isotherms and heat lines) of heat conduction is compared with the known Carslaw–Jaeger solution and shows a puzzling topology of three regimes of energy fluxes for temperature boundary conditions common in passive thermal insulation. The simplest regime is realized if cooling of a shaded zone is mild and heat flows in a slightly distorted “resistor model” flow tube. The second regime emerges when cooling is stronger and two disconnected separatrices demarcate the back-flow of heat from a relatively hot segment of the slab surface to the atmosphere through relatively cold parts of this surface. The third topological regime is characterized by a single separatrix with a critical point inside the slab, where the thermal gradient is nil. In this regime the back-suction of heat into the atmosphere is most intensive. The closed-form solutions obtained can be used in assessment of efficiency of thermal protection of buildings.

A Numerical Study of Three-Dimensional Roll Cells within Rigid Boundaries

1979 ◽  
Vol 101 (2) ◽  
pp. 233-237 ◽  
Author(s):  
A. M. C. Chan ◽  
S. Banerjee
Keyword(s):  
Natural Convection ◽  
Initial Conditions ◽  
Numerical Study ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
Convection Roll ◽  
Marker And Cell Method

Three-dimensional natural convection, roll cells within rigid enclosures have been studied with a previously developed numerical technique based on the marker and cell method. Given identical initial conditions, the velocity and temperatures fields are found to be sensitive to the thermal boundary conditions on the side walls and the aspect ratio of the enclosure. Two-dimensional results are also obtained and compared with the corresponding three-dimensional results. The two-dimensional calculations do not agree well with the three dimensional ones, especially for enclosures having aspect ratios less than unity. This indicates that care must be taken in analyzing natural convection problems of this type with two-dimensional methods.

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