Civil Engineering Reference
In-Depth Information
Convection
q
h
q
in
q
x
q
m
Q
,
U
d
q
x
d
x
d
q
m
d
x
q
x
q
m
q
x
d
x
q
m
d
x
q
out
d
x
(a)
(b)
Figure 7.15
(a) One-dimensional conduction with convection and mass transport. (b) Control volume for
energy balance.
one-dimensional case. Figure 7.15a is essentially Figure 7.2a with a major phys-
ical difference. The volume shown in Figure 7.15a represents a flowing fluid (as
in a pipe, for example) and heat is transported as a result of the flow. The heat
flux associated with mass transport is denoted
q
m
, as indicated in the figure. The
additional flux term arising from mass transport is given by
(7.61)
q
m
=˙
mcT
(W or Btu/hr)
where
m
is mass flow rate (kg/hr or slug/hr),
c
is the specific heat of the fluid
(W-hr/(kg-
◦
C) or Btu/(slug-
◦
F)), and
T
(
x
)
is the temperature of the fluid (
◦
C or
◦
F). A control volume of length d
x
of the flow is shown in Figure 7.15b, where
the flux terms have been expressed as two-term Taylor series as in past deri-
vations. Applying the principle of conservation of energy (in analogy with
Equation 7.1),
˙
q
x
+
d
x
A
d
t
d
q
x
d
x
q
x
A
d
t
+
q
m
d
t
+
QA
d
x
d
t
=
U
+
q
m
+
d
x
d
t
d
q
m
d
x
(7.62)
+
+
q
h
P
d
x
d
t
Considering steady-state conditions,
U
0
, using Equations 5.51 and 7.2 and
=
simplifying yields
k
x
d
d
x
d
T
d
x
d
q
m
d
x
+
hP
A
(7.63)
+
Q
=
(
T
−
T
a
)
where all terms are as previously defined. Substituting for
q
m
into Equation 7.63,
we obtain
k
x
˙
T
d
d
x
d
T
d
x
d
d
x
mc
A
hP
A
(7.64)
+
Q
=
+
(
T
−
T
a
)
which for constant material properties and constant mass flow rate (steady state)
becomes
d
2
T
d
x
2
mc
A
˙
d
T
d
x
+
hP
A
(7.65)
k
x
+
Q
=
(
T
−
T
a
)
