Chapter 5
Numerical Methods in Heat Conduction
580
Starting with an energy balance on a disk volume element, the onedimensional transient explicit
finite difference equation for a general interior node for
Tz
in a cylinder whose side surface is insulated
for the case of constant thermal conductivity with uniform heat generation is to be obtained.
t
(,)
Analysis
We consider transient onedimensional heat conduction in the axial
z
direction in an insulated
cylindrical rod of constant crosssectional area
A
with constant heat generation
and constant
conductivity
k
with a mesh size of
0
g
&
z
Δ
in the
z
direction.
Noting that the volume element of a general
interior node
m
involves heat conduction from two sides and the volume of the element is
z
A
V
Δ
=
element
,
the transient explicit finite difference formulation for an interior node can be expressed as
t
T
T
xC
A
x
A
g
x
T
T
kA
x
T
T
kA
i
m
i
m
i
m
i
m
i
m
i
m
Δ
−
Δ
=
Δ
+
Δ
−
Δ
−
+
+
−
1
0
1
1
+
ρ
&
•
•
m
1
m
m
+1
Insulation
Disk
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This note was uploaded on 01/22/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.
 Fall '10
 Dr.DanielArenas
 Thermodynamics, Energy, Mass, Heat

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