Information Technology Reference
In-Depth Information
%
@e
@t
Crq
f
D
0:
The thermodynamic relation used to introduce T in the first term is as explained
in the one-dimensional case. In addition, we need a three-dimensional version of
Fourier's law:
q D
k
r
T:
Performing the same substitutions and eliminations as in the one-dimensional case,
we get the three-dimensional PDE for T.x;y;
z
;t/:
@T
@t
Dr
.k
r
T/
C
f:
%c
v
(7.26)
The term
r
.k
r
T/ arises from inserting
q D
k
r
T in
rq
. Observe that we
do not assume that k is constant here. If k is constant, we can write
r
.k
r
T/
D
k
rr
T
D
k
r
2
T . Equation (7.26) then takes a form similar to that of (7.4):
@T
@t
D
r
2
T
C
f;
(7.27)
where f
D
f=.%c
v
/ and
D
k=.%c
v
/, since we have divided by %c
v
.
Boundary Conditions
There are three common types of boundary conditions in heat transfer applications.
The simplest condition is to prescribe T at a part of the boundary. Such a condition
requires some external device at the boundary to control the temperature. Another
common condition models insulated boundaries, i.e., boundaries effectively covered
by some isolating material such that the heat cannot escape. This means that there
is no heat flow out of the boundary:
q n D
0, which, by Fourier's law, becomes
k
r
T
n D
0, often written as
k
@
@n
D
0. For a one-dimensional problem posed
on Œa; b, the outward unit normal vector points in the positive x-direction on x
D
a
and in negative x-direction on x
D
b. In both cases, @T =@x
D
0 is the relevant
condition.
A third boundary condition, modeling heat exchange with the surroundings, is
often used:
q n D
h
T
.T
T
s
/;
or, equivalently,
k
@T
@n
D
h
T
.T
T
s
/:
(7.28)
This law, commonly referred to as
Newton's law of cooling
, states that the net heat
flow out of the boundary is proportional to the difference between the tempera-
ture in the body (T ) and the temperature in the surroundings (T
s
). The constant of
proportionality is referred to as the
heat transfer coefficient
, and it often requires
