Page 158 - Mechanical Engineers' Handbook (Volume 4)

1 Conduction Heat Transfer 147

Writing an energy balance for a three-dimensional body and utilizing Fourier’s law of
heat conduction yields an expression for the transient diffusion occurring within a body or
substance:
k
T T T T
x x y k y z k z ˙q c p x t

This expression, usually referred to as the heat diffusion equation or heat equation, provides
a basis for most types heat conduction analyses. Specialized cases of this equation can be
used to solve many steady-state or transient problems. Some of these specialized cases are
Thermal conductivity is a constant

2
2
2
T T T ˙ q c T
p

x 2 y 2 z 2 k k t
Steady-state with heat generation
k
T k T k T ˙q 0
x x y y z z
Steady-state , one-dimensional heat transfer with no heat sink (i.e., a fin)

T ˙ q
x x k 0
One-dimensional heat transfer with no internal heat generation

T c T
p
x x k t
In the following sections, the heat diffusion equation will be utilized for several specific
cases. However, in general, for a three-dimensional body of constant thermal properties
without heat generation under steady-state heat conduction the temperature field satisfies the
expression

2
T 0
1.1 Thermal Conductivity

The ability of a substance to transfer heat through conduction can be represented by the
constant of proportionality, k, referred to as the thermal conductivity. Figure 1 illustrates the
characteristics of the thermal conductivity as a function of temperature for several solids,
liquids, and gases. As shown, the thermal conductivity of solids is higher than liquids, and
liquids higher than gases. Metals typically have higher thermal conductivities than nonmetals,
with pure metals having thermal conductivities that decrease with increasing temperature,
while the thermal conductivity of nonmetallic solids generally increases with increasing
temperature and density. The addition of other metals to create alloys, or the presence of
impurities, usually decreases the thermal conductivity of a pure metal.
In general, the thermal conductivity of liquids decreases with increasing temperature.
Alternatively, the thermal conductivity of gases and vapors, while lower, increases with
increasing temperature and decreases with increasing molecular weight. The thermal con-
ductivities of a number of commonly used metals and nonmetals are tabulated in Tables 1