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University of Ottawa, CHG 2314, B. Kruczek
1
CHG 2314
Heat Transfer
Part 2c
TwoDimensional SteadyState Conduction
 Analytical solution
 Graphical method and shape factors
 Dimensionless conduction rate
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2
Multidimensional effects

Surface area for conduction changes in the direction of heat flow in a Cartesian
system
z
Long prismatic solid with two
surfaces insulated and the other
two maintained at different
temperatures
z
“End effects” – thick wall oven

Inherently multidimensional systems
z
Temperature gradients exist in more than one direction
University of Ottawa, CHG 2314, B. Kruczek
3
The heat equation and methods of solution
( )
22
0
qx
,y
TT
k
xy
∂∂
+
+=
±

Methods of solution
z
Exact/Analytical: Separation of Variables (Section 4.2)
¾
Limited to simple geometries and boundary conditions
z
Approximate/Graphical: Flux plotting (Sections 4.S.1  supplemental material)
¾
Requires zero heat generation

We will limit our discussion to twodimensional conduction.
z
For constant thermal conductivity, 2D heat diffusion equation in Cartesian
coordinates becomes:
z
Approximate/Numerical: Finite Difference, Finite Element or Boundary
Element Method (Sections 4.4 and 4.5) – not required
¾
Of limited value for quantitative considerations but a quick aid to establishing physical
insights
¾
Most useful approach and adaptable to any level of complexity
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4
Exact/Analytical solutions

Analytical solutions can be obtained for a limited number of multiD problems
z
Consider twodimensional conduction in a thin rectangular plate with the edges
maintained at constant temperatures as shown:
z
Governing differential equation:
z
Boundary conditions:
0
2
2
2
2
=
∂
∂
+
∂
∂
y
x
θθ
1
21
where:
TT
θ
−
=
−
()
( )
(
)
1
,
0
,
0
0
,
0
,
0
=
=
=
=
W
x
and
y
L
x
and
y
University of Ottawa, CHG 2314, B. Kruczek
5
Exact solutions
z
The analytical solution is obtained by using the
method of separation of variables
z
The temperature profile and heat flow can then be plotted:
•
The lines at
θ
= 0.1, 0.25, 0.5, 0.75 are isotherms,
as are the edges at
θ
= 0 and
θ
= 1
•
The arrows are heat flow lines, which are
perpendicular
to the isotherms
()
( )
∑
∞
=
+
−
=
1
1
sinh
sinh
sin
1
2
,
n
n
L
W
n
L
y
n
L
x
n
n
y
x
π
ππ
θ
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6
Graphical method

The objective of the graphical method is
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This note was uploaded on 05/30/2011 for the course CHG 2314 taught by Professor Kruz. during the Spring '10 term at University of Ottawa.
 Spring '10
 kruz.

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