Chapter 5
Numerical Methods in Heat Conduction
559E
The top and bottom surfaces of a Vgrooved long solid bar are maintained at specified temperatures
while the left and right surfaces are insulated. The temperature at the middle of the insulated surface is to
be determined.
Assumptions 1
Heat transfer through the bar is given to be steady and twodimensional.
2
There is no heat
generation within the bar.
3
Thermal properties are constant.
Analysis
The nodal spacing is given to be
Δ
x
=
Δ
y
=
l
=1 ft, and the general finite difference form of an
interior node for steady twodimensional heat conduction with no heat generation is expressed as
0
4
0
4
node
bottom
right
top
left
2
node
node
bottom
right
top
left
=
−
+
+
+
→
=
+
−
+
+
+
T
T
T
T
T
k
l
g
T
T
T
T
T
&
There is symmetry about the vertical plane passing through the center. Therefore,
T
1
=
T
9
,
T
2
=
T
10,
T
3
=
T
11,
T
4
=
T
7
, and
T
5
=
T
8
. Therefore, there
are only 6 unknown nodal temperatures, and thus we need only 6
equations to determine them uniquely. Also, we can replace the symmetry lines by insulation and utilize
the mirrorimage concept when writing the finite difference equations for the interior nodes.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Dr.DanielArenas
 Thermodynamics, Mass, Heat, Heat Transfer, Thermal conductivity

Click to edit the document details