PROBLEM 4.7
KNOWN:
Boundary conditions on four sides of a square plate.
FIND:
Expressions for shape factors associated with the
maximum
and
average
top surface
temperatures.
Values of these shape factors.
The maximum and average temperatures for
specified conditions.
SCHEMATIC:
0
0
x
W
y
W
T
1
T
1
T
1
′′
s
q
ASSUMPTIONS:
(1) Twodimensional, steadystate conduction, (2) Constant properties.
ANALYSIS:
We must first find the temperature distribution as in Problem 4.5.
Problem 4.5
differs from the problem solved in Section 4.2 only in the boundary condition at the top surface.
Defining
θ
= T – T
∞
, the differential equation and boundary conditions are
22
θθ
+
= 0
xy
∂∂
s
yW
θ
θ
(0,y)
0
θ
(L,y)
0
θ
(x,0)
0
k
q
y
=
∂
===
=
∂
(1a,b,c,d)
The solution is identical to that in Section 4.2 through Equation (4.11),
n
n=1
n
π
xn
π
y
θ
=
C sin
sinh
L
∞
∑
L
(2)
To determine C
n
, we now apply the top surface boundary condition, Equation (1d).
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 Spring '10
 LEE,J.S.
 Thermodynamics, Heat Transfer, Trigraph, CN, 0 L, shape factors

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