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In-Depth Information
time interval t is simply
R
b
a
tf dx. We now have the following mathematical
expression for our basic energy balance:
t.q.a; t/
q.b; t//
C
Z
b
a
tf
dx
D
Z
b
a
%
@e
@t
t
dx
:
(7.21)
The
mathematical
similarity with the mass conservation equation (7.14) is striking;
only a factor % differs due to a slightly different definition of q. Using integration by
parts “backward” on the left-hand side transforms this difference to an integral over
Œa; b:
Z
b
a
@x
dx
C
Z
b
a
tf dx
D
Z
b
a
@q
%
@e
@t
t dx :
Collecting the integrals yields
%
@e
@x
f
dx
D
0:
Z
b
@t
C
@q
a
For this equation to hold for an arbitrary interval Œa; b, the integrand must vanish, a
requirement that leads to the PDE
%
@e
@t
C
@q
@x
f
D
0:
(7.22)
We have two unknown functions, e.x; t/ and q.x; t/, and thus we need an addi-
tional equation. Observe also that the
temperature
, which is the quantity we want to
compute, does not yet enter our model. However, we will relate both e and q to the
temperature T .
In thermodynamics there is a class of relations called
equations of state
.One
such relation reads e
D
e.T; V /, relating internal energy e to temperature T and
density %
D
1=V (V is referred to as volume). By the chain rule, we have
ˇ
ˇ
ˇ
ˇ
V
ˇ
ˇ
ˇ
ˇ
T
@e
@t
D
@e
@T
@t
C
@e
@V
@t
:
@T
@V
The first coefficient .@e=@T /
V
is called the
specific heat capacity at constant
volume
, denoted by c
v
,
ˇ
ˇ
ˇ
ˇ
V
c
v
@e
@T
:
The heat capacity will in general vary with T and V (
D
1=%), but taking it as a
constant is reasonable in many applications. We will do so here. One can find tables
in the literature with the constant values of c
v
for many substances.
The other term, .@e=@V /
T
, can, by the second law of thermodynamics, be shown
to be proportional to the derivative of T with respect to V (or %). The term models
temperature effects due to compressibility and volume expansion. These effects are
often small and can be neglected. We will therefore set .@e=@V /
T
D
0.
