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(c) Controlled mass (in-)flow of the substance,
k
@
u
@n
D
Q.t/;
Q.t/ is prescribed.
(7.42)
u
@n
@
Note:
is the derivative in the outward normal direction to the boundary,
@
u
@n
D
@
u
@x
u
@n
@
D
@
u
@x
at x
D
a and
at x
D
b.
i.e.,
Heat Conduction
The governing PDE for heat conduction in a material at rest reads
k
@
u
@x
C
f.x;t/:
c
v
%
@
u
@t
D
@
(7.43)
@x
The parameters are the following:
-
u
.x; t /, the temperature
-
c
v
.x/, the heat capacity, at constant volume, of the material
-
%.x/m the density of the material
-
k.x/, the heat conduction coefficient of the material
-
f.x;t/, the supply of heat per unit time and volume
The set of relevant boundary conditions in heat conduction problems is listed next.
(a) Controlled temperature:
u
D
U
0
.t /;
U
0
.t / is prescribed.
(7.44)
(b) Insulated boundary (no heat flow):
k
@
u
@n
D
0:
(7.45)
(c) Controlled heat flow:
@
u
@n
D
Q.t/;
Q.t/ is prescribed.
(7.46)
(d) Cooling law:
k
@
u
@n
D
h
T
.
u
u
s
/;
h
T
;
u
s
.t / are prescribed,
(7.47)
where h
T
is a heat transfer coefficient, modeling the heat exchange with the
surrounding medium, which has temperature
u
s
.t /.
