PROBLEM 4.73
KNOWN:
Electrical heating elements with known dissipation rate embedded in a ceramic plate of
known thermal conductivity; lower surface is insulated, while upper surface is exposed to a convection
rocess.
p
FIND:
(a) Temperature distribution within the plate using prescribed grid spacing, (b) Sketch isotherms
to illustrate temperature distribution, (c) Heat loss by convection from exposed surface (compare with
element dissipation rate), (d) Advantage, if any, in not setting
Δ
x =
Δ
y, (e) Effect of grid size and
convection coefficient on the temperature field.
SCHEMATIC:
ASSUMPTIONS:
(1) Steadystate, twodimensional conduction in ceramic plate, (2) Constant
properties, (3) No internal generation, except for Node 7 (or Node 15 for part (e)), (4) Heating element
pproximates a line source of negligible wire diameter.
a
ANALYSIS:
(a) The prescribed grid for the symmetry element shown above consists of 12 nodal points.
Nodes 13 are points on a surface experiencing convection; nodes 46 and 812 are interior nodes.
Node
7 is a special case of the interior node having a generation term; because of symmetry,
= 25 W/m.
The finitedifference equations are derived as follows:
ht
q
′
Continued.
..
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′
PROBLEM 4.73 (Cont.)
Surface Node 2
.
From an energy balance on the prescribed control volume with
Δ
x/
Δ
y = 3,
= 0;
in
out
EE
−=
±±
abc
qqqq
′′′
+++
()
32
52
12
2
TT
yT T
y
kh
x
T
T
kk
x
2x
y
∞
0
−
−
Δ−
Δ
+Δ
−
+
=
ΔΔ
Δ
.
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 Spring '10
 LEE,J.S.
 Thermodynamics, Heat, Heat Transfer, TI, grid size, Heating Element, prescribed control volume

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