Game Development Reference
In-Depth Information
The Heat Conduction Equation
To model a
transient,
or time-varying, heat conduction problem, a heat energy equation that
has a time derivative term must be developed. In order to come up with this equation, a heat
energy balance is evaluated for the system. As shown in Figure 12-2, the heat energy that enters
or leaves a section of material of width
Dx
is given by Fourier's law. The temperature derivative
at the exit plane is equal to the derivative at the inflow plane plus the second derivative (the
derivative of the derivative) of temperature with respect to
x
times
Dx
.
Figure 12-2.
A heat conduction energy balance
Assuming that the object is not generating its own heat, the heat energy accumulation in
the object over time is equal to the heat energy coming in minus the heat energy that is leaving
the object.
κ
⎛
⎞
2
∂
T
∂
T
∂
T
∂
T
⎟
⎜
cx
ρ
Δ
−+ +Δ
κ
x
⎟
⎜
(12.10)
⎟
⎜
p
⎜
⎟
2
∂
t
∂
x
⎝
∂
x
∂
x
⎠
The quantity
r
is the density of the material. The
specific heat capacity
,
c
p
, is a material
property that represents the amount of heat energy required to raise a unit mass of a material
by one degree in temperature. It has units of
J/kg-K
. The quantity,
r
Δ
x
, is the mass-per-unit
area of a slice of material of thickness
x
. Equation (12.10) can be cleaned up somewhat by
removing terms that cancel each other out. What remains is an equation that relates the time
rate of change of temperature at a given point to the second derivative of temperature with
respect to
x
at the point.
Δ
2
∂
T
∂
T
c
ρ
=
κ
(12.11)
p
2
∂
t
∂
x
Equation (12.11) is known as the
1-D heat conduction equation
. If the object were gener-
ating heat internally (a nuclear reactor, for example), the energy generation term would be
added to the right-hand side of the equation. Oftentimes the material properties are grouped
together in a modified version of the 1-D heat conduction equation.
2
∂
T
∂
T
=
K
(12.12)
2
∂
t
∂
x
