Chapter 5
Numerical Methods in Heat Conduction
57
We consider three consecutive nodes
n
1,
n
, and
n
+1 in a plain wall. Using Eq. 56, the first derivative
of temperature
dT dx
/
at the midpoints
n
 1/2 and
n
+ 1/2 of the sections surrounding the node
n
can be
expressed as
x
T
T
dx
dT
x
T
T
dx
dT
n
n
n
n
n
n
Δ
−
≅
Δ
−
≅
+
+
−
−
1
2
1
1
2
1
and
Noting that second derivative is simply the derivative of the
first derivative, the second derivative of temperature at node
n
can be expressed as
n+
1
Δ
x
Δ
x
n
1
n
T
n
1
T
n
T
n+
1
T
(
x
)
x
2
1
1
1
1
2
1
2
1
2
2
2
x
T
T
T
x
x
T
T
x
T
T
x
dx
dT
dx
dT
dx
T
d
n
n
n
n
n
n
n
n
n
n
Δ
+
−
=
Δ
Δ
−
−
Δ
−
=
Δ
−
≅
+
−
−
+
−
+
which is the
finite difference representation
of the
second derivative
at a general internal node
n
. Note that
the second derivative of temperature at a node
n
is expressed in terms of the temperatures at node
n
and its
two neighboring nodes
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This note was uploaded on 01/19/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.
 Fall '10
 Dr.DanielArenas
 Thermodynamics, Mass, Heat

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