ABSTRACT
Whenever a temperature gradient exits in a solid, heat will fl ow from a hightemperature region to a low-temperature region. The basic governing heat conduction equations are obtained by considering a plate with a surface area A and a thickness Δx, as shown in Figure 4.1. One side is maintained at a temperature T1, and the other side at temperature T2. Experimental observation indicates that the rate of heat fl ow is directly proportional to the area and temperature difference, but inversely proportional to the plate thickness. The proportionality sign is replaced by an equal sign by introducing the constant k, as follows
1 2T TQ kA x
−
=
Δ (4.1)
where k is the thermal conductivity of the plate, and this property depends on the type of the plate material. Equation 4.1 is also called Fourier’s law. Fourier’s law can also be expressed in differential form in the direction of the normal coordinate:
d
d
T Q kA
n = −
(4.2)
Also, Fourier’s law can be expressed for multidimensional heat fl ux fl ow:
T T T Q k i j k
x y z
⎛ ⎞∂ ∂ ∂ ″ = − + +⎜ ⎟⎝ ∂ ∂ ∂ ⎠ (4.3)
An energy balance can be applied to a differential volume, dx dy dz, for conduction analysis in a Cartesian coordinate, as shown in Figure 4.2. The objective of this energy balance is to obtain the temperature distribution within the solid. The temperature distribution can be used to determine the heat fl ow at a certain surface, or to study the thermal stress.
