Conduction Heat Transfer

Wednesday, June 1, 2011
Conduction heat transfer phenomena are found throughout virtually all of the physical world and the industrial domain. The analytical description of this heat transfer mode is one of the best understood. Some of the bases of understanding of conduction date back to early history. It was recognized that by invoking certain relatively minor simplifications, mathematical solutions resulted directly. Some of these were very easily formulated. What transpired over the years was a very vigorous development of applications to a broad range of processes. Perhaps no single work better summarizes the wealth of these studies than does the book by Carslaw and Jaeger (1959). They gave solutions to a broad range of problems, from topics related to the cooling of the Earth to the current-carrying capacities of wires. The general analyses given there have been applied to a range of modern-day problems, from laser heating to temperature-control systems.

Today conduction heat transfer is still an active area of research and application. A great deal of interest has developed in recent years in topics like contact resistance, where a temperature difference develops between two solids that do not have perfect contact with each other. Additional issues of current interest include non-Fourier conduction, where the processes occur so fast that the equation described below does not apply. Also, the problems related to transport in miniaturized systems are garnering a great deal of interest. Increased interest has also been directed to ways of handling composite materials, where the ability to conduct heat is very directional.

Much of the work in conduction analysis is now accomplished by use of sophisticated computer codes. These tools have given the heat transfer analyst the capability of solving problems in nonhomogenous media, with very complicated geometries, and with very involved boundary conditions. It is still important to understand analytical ways of determining the performance of conducting systems. At the minimum these can be used as calibrations for numerical codes.

Fourier’s Law

The basis of conduction heat transfer is Fourier’s Law . This law involves the idea that the heat flux is proportional to the temperature gradient in any direction n .Thermal conductivity, k , a property of materials that is temperature dependent, is the constant of proportionality.

Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 In many systems the area A is a function of the distance in the direction n. One important extension is that this can be combined with the first law of thermodynamics to yield the heat conduction equation For constant thermal conductivity, this is given as

In this equation, a is the thermal diffusivity and is the internal heat generation per unit volume. Some problems, typically steady-state, one-dimensional formulations where only the heat flux is desired, can be solved simply from Equation (1). Most conduction analyses are performed with Equation (2). In the latter, a more general approach, the temperature distribution is found from this equation and appropriate boundary conditions. Then the heat flux, if desired, is found at any location using Equation (1) . Normally, it is the temperature distribution that is of most importance. For example, it may be desirable to know through analysis if a material will reach some critical temperature, like its melting point. Less frequently the heat flux is desired.

While there are times when it is simply desired to understand what the temperature response of a structure is, the engineer is often faced with a need to increase or decrease heat transfer to some specific level. Examination of the thermal conductivity of materials gives some insight into the range of possibilities that exist through simple conduction. Of the more common engineering materials, pure copper exhibits one of the higher abilities to conduct heat with a thermal conductivity approaching 400 W/m^2 K.

Aluminum, also considered to be a good conductor, has a thermal conductivity a little over half that of copper. To increase the heat transfer above values possible through simple conduction, more-involved designs are necessary that incorporate a variety of other heat transfer modes like convection and phase change. Decreasing the heat transfer is accomplished with the use of insulations.

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