Game Development Reference
In-Depth Information
Fourier's Law
In general, heat conduction will occur in all three directions, but that general case is too compli-
cated for most game programming applications. The mathematics can be considerably simplified
if the heat conduction is assumed to be one-dimensional, acting only in the direction normal
to the surface of the object. For the analysis presented in this section, we will assume that heat
conduction occurs in the x-direction. According to a relation known as
Fourier's law
, the rate
of heat conduction at any point in the object is proportional to the temperature gradient at
that point.
dT
q
=−
κ
(12.9)
x
dx
The quantity
q
x
is the heat energy transfer rate (energy per unit time) per unit area.
In the SI system of units,
q
x
has units of
W/m
2
, where the
W
stands for Watts. The
thermal
conductivity
,
, is a material property that measures how well a material conducts heat. It has
SI units of
W/m-K
. A material with a higher thermal conductivity will have a higher energy
transfer rate than will a material with a lower thermal conductivity. Materials with low thermal
conductivities are called
insulating materials
, because they do a poor job of conducting heat.
Table 12-4 lists thermal conductivity and density values for some common solid materials.
The thermal conductivity and density of metals are affected somewhat by temperature. The
values shown in Table 12-4 are for a temperature of 293
K (or 20
o
C)
. The density shown for
wood is a representative value as different types of wood will have different densities.
κ
Table 12-4.
Typical Thermal Conductivity Values for Various Materials
Density (
kg/m
3
)
Material
Thermal Conductivity (
W/m-K
)
Water
0.6
1000
Glass
0.9
2600
Wood
0.4
780
Concrete
1.3
2200
Aluminum
237
2700
Copper
390
8960
Iron
80
7000
Stainless steel
16
7900
Plastic insulation
0.03
50
Fourier's law assumes a steady-state situation in which the temperature profile through
the object is a function of distance but is constant over time. This situation will not always be
the case. For example, consider a metal rod that is kept at room temperature. If a lit match is
held next to one end of the rod, heat will flow into the rod, and the temperature profile through
the rod will change over time.
