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Chapter 5
Numerical Methods in Heat Conduction
578
Starting with an energy balance on a volume element, the twodimensional transient
explicit
finite
difference equation for a general interior node in rectangular coordinates for
for the case of
constant thermal conductivity and no heat generation is to be obtained.
Txyt
(,,)
Analysis
(See Figure 549 in the text). We consider a rectangular region in which heat conduction is
significant in the
x
and
y
directions, and consider a unit depth of
Δ
z
= 1 in the
z
direction. There is no heat
generation in the medium, and the thermal conductivity
k
of the medium is constant. Now we divide the
xy
plane of the region into a
rectangular mesh
of nodal points which are spaced
Δ
x
and
Δ
y
apart in the
x
and
y
directions, respectively, and consider a general interior node (
m
,
n
) whose coordinates are
x
m x
=Δ
and
. Noting that the volume element centered about the general interior node (
m
,
n
)
involves heat
conduction from four sides (right, left, top, and bottom) and expressing them at previous time step
i
,
the
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Energy, Mass, Heat

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