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.


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 (4.1.1).

Most conduction analyses are performed with Equation (4.1.2). 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 (4.1.1). 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/m2 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 insulation.

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