Chapter 2
Heat Conduction Equation
229
We consider a thin ring shaped volume element of width
Δ
z
and thickness
Δ
r
in a cylinder. The
density of the cylinder is
ρ
and the specific heat is
C.
In general, an
energy balance
on this ring element
during a small time interval
Δ
t
can be expressed as
t
E
Q
Q
Q
Q
z
z
z
r
r
r
Δ
Δ
=
−
+
−
Δ
+
Δ
+
element
)
(
)
(
&
&
&
&
But the change in the energy content of the element can be expressed as
Δ
Δ
Δ
ΔΔ
Δ
E
m
C
T
T
C
r
r
z
T
T
t
tt t
=
EE
tt
element
=−
−
=
−
++
+
()
(
)
ρ
π
2
Substituting,
(
&&
)(
)( )
QQ
Cr
rz
TT
t
rr
r
zz
z
−+
−=
−
+
Δ
Δ
ρπ
2
Δ
z
r+
Δ
r
r
Dividing the equation above by
2
rr z
Δ
Δ
gives
−
−
−
−
=
−
1
2
1
2
ππ
+
r z
r
r r
z
C
t
r
z
Δ
Δ
r
2
Noting that the heat transfer surface areas of the element for heat conduction in the
r
and
z
directions are
Ar
z
==
2
Δ
and
,
Δ
respectively, and taking the limit as
Δ
Δ
Δ
t
,
and
→
0 yields
t
T
C
z
T
k
z
T
k
r
r
T
kr
r
r
∂
∂
ρ
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Dr.DanielArenas
 Thermodynamics, Energy, Mass, Heat, Heat Transfer, Thermal conductivity, ΔZ

Click to edit the document details