Civil Engineering Reference
In-Depth Information
q
x
(
x
a
)
Insulated
a
q
x
(
x
b
)
b
x
(a)
A
d
q
x
d
x
q
x
q
x
d
x
d
x
(b)
Figure 5.8
Insulated body in one-dimensional heat
conduction.
only. Surfaces of the body normal to the
x
axis are assumed to be perfectly
insulated, so that no heat loss occurs through these surfaces. Figure 5.8b shows
the control volume of differential length d
x
of the body, which is assumed to be
of constant cross-sectional area and uniform material properties. The principle of
conservation of energy is applied to obtain the governing equation as follows:
E
in
+
E
out
(5.53)
Equation 5.53 states that the energy entering the control volume plus energy gen-
erated internally by any heat source present must equal the increase in internal
energy plus the energy leaving the control volume. For the volume of Fig-
ure 5.8b, during a time interval d
t
, Equation 5.53 is expressed as
E
generated
=
E
increase
+
q
x
+
∂
d
x
A
d
t
q
x
∂
q
x
A
d
t
+
QA
d
x
d
t
=
U
+
(5.54)
x
where
q
x
=
heat flux across boundary (W/m
2
, Btu/hr-ft
2
);
Q
=
internal heat generation rate (W/m
3
, Btu/hr-ft
3
);
U
=
internal energy (W, Btu).
The last term on the right side of Equation 5.54 is a two-term Taylor series
expansion of
q
x
(
x
,
t
) evaluated at
x
+
d
x
. Note the use of partial differentiation,
since for now, we assume that the dependent variables vary with time as well as
spatial position.
The heat flux is expressed in terms of the temperature gradient via Fourier's
law of heat conduction:
k
x
∂
T
q
x
=−
(5.55)
∂
x
