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Chapter 2
Heat Conduction Equation
222
We consider a thin cylindrical shell element of thickness
Δ
r
in a long cylinder (see Fig. 215 in the
text). The density of the cylinder is
ρ
, the specific heat is
C,
and the length is
L.
The area of the cylinder
normal to the direction of heat transfer at any location
is
Ar
L
=
2
π
where
r
is the value of the radius at that
location
.
Note that the heat transfer area
A
depends on
r
in this case
,
and thus it varies with location.
An
energy balance
on this thin cylindrical shell element of thickness
Δ
r
during a small time interval
Δ
t
can be
expressed as
&&
&
QQ
G
E
t
rr
r
−+
=
+Δ
Δ
Δ
element
element
where
Δ
Δ
ΔΔ
EE
E
m
C
T
T
C
A
r
T
tt
t
tt t
element
=−
=
Δ
T
−
=
−
++
+
()
ρ
r
A
g
V
g
G
Δ
=
=
&
&
&
element
element
Substituting,
t
T
T
r
CA
r
A
g
Q
Q
t
t
t
r
r
r
Δ
−
Δ
ρ
=
Δ
+
−
Δ
+
Δ
+
&
&
&
where
=
2
L
.
Dividing the equation above by
A
Δ
r
gives
−
−
+=
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Mass, Heat

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