Game Development Reference
In-Depth Information
The
thermal diffusivity
,
K
, has units of
m
2
/s
and is equal to the ratio of the thermal
conductivity divided by the specific heat capacity and density.
c
κ
K
=
(12.13)
ρ
p
The thermal diffusivity can be thought of as a measure of the ability of a material to transfer
heat by conduction. Values of thermal diffusivity for various materials are shown in Table 12-5.
As before, the values shown in the table are for a temperature of 293
K
. There are order of
magnitude differences of thermal diffusivity for different materials. Those with high diffusivities
are good conductors, and those with low diffusivities are good insulators.
Table 12-5.
Thermal Diffusivity Values
Thermal Diffusivity (
m
2
/s
)
Material
Water
1.4
e
- 7
Glass
4.3
e
- 7
Wood
2.14
e
- 6
Concrete
6.6
e
- 7
Aluminum
9.975
e
- 5
Copper
1.116
e
- 4
Iron
2.545
e
- 5
Stainless steel
4.50
e
- 6
Plastic insulation
3.0
e
- 7
Solving the Heat Conduction Equation
The 1-D heat conduction equation allows us to compute the temperature profile through a
solid object as a function of time. All we have to do now is to solve the equation. The heat
conduction equation is a partial differential equation, or PDE, in that there is both a time deriv-
ative and a spatial derivative in the equation. If you remember when differential equations
were discussed in Chapter 2, the text did not cover how to solve PDEs because it was a difficult
topic that was beyond the scope of this topic. But now we are faced with having to solve one.
All is not lost, because fortunately the 1-D heat conduction equation is simple enough that
there is an analytical solution to it. It involves a bit of mathematical trickery in performing what
is called a change of variables to make the PDE look like an ODE. To accomplish this conversion,
a variable
u
is declared that is a function of both
x
and
t
.
x
u
=
(12.14)
t
