Civil Engineering Reference
In-Depth Information
material properties. The convective heat transfer across the surface, denoted
q
h
,
represents the heat flow rate (heat flux) across the surface per unit
surface
area.
To apply the principle of conservation of energy to the control volume, we need
only add the convection term to Equation 5.54 to obtain
q
x
+
∂
d
x
A
d
t
q
x
∂
q
x
A
d
t
+
QA
d
x
d
t
=
U
+
+
q
h
P
d
x
d
t
(7.1)
x
where all terms are as previously defined except that
P
is the peripheral dimen-
sion of the differential element and
q
h
is the heat flux due to convection. The con-
vective heat flux is given by [2]
q
h
=
h
(
T
−
T
a
)
(7.2)
where
h
=
convection coefficient, W/(m
2
-
◦
C
)
,
Btu/(hr-ft
2
-
◦
F
)
=
temperature of surface of the body
T
a
=
ambient fluid temperature
Substituting for
q
h
and assuming steady-state conditions such that
U
T
=
0
,
Equation 7.1 becomes
A
d
q
x
QA
=
d
x
+
hP
(
T
−
T
a
)
(7.3)
which, via Fourier's law Equation 5.55, becomes
d
2
T
d
x
2
hP
A
k
x
+
Q
=
(
T
−
T
a
)
(7.4)
where we have assumed
k
x
to be constant.
While Equation 7.4 represents the one-dimensional formulation of conduc-
tion with convection, note that the temperature at any position
x
along the length
of the body is not truly constant, owing to convection. Nevertheless, if the cross-
sectional area is small relative to the length, the one-dimensional model can give
useful results if we recognize that the computed temperatures represent average
values over a cross section.
7.3.1 Finite Element Formulation
To develop the finite element equations, a two-node linear element for which
T
(
x
)
N
2
(
x
)
T
2
(7.5)
is used in conjunction with Galerkin's method. For Equation 7.4, the residual
equations (in analogy with Equation 5.61) are expressed as
x
2
=
N
1
(
x
)
T
1
+
k
x
T
a
)
N
i
(
x
)
A
d
x
d
2
T
d
x
2
hP
A
(7.6)
+
Q
−
(
T
−
=
0
i
=
1, 2
x
1
