Wilson Braz
MANE 6630
September 30, 2009
Homework 3
Multi-Dimensional Steady State Heat Conduction:
Analytical Methods
1
Consider a steady state heat conduction in a square plate (x 2 [0; 1]; y 2 [0; 1] in meters)
with constant thermal conductivity. The following boundary conditions are prescribed:
At x = 0,
T 0, y   0
at x = 1,
T 1, y   sin y 
1
at y = 0
1
if 0  x 
2 x
2
T x,0  
1


2
1

x

if  x  1
2
and at y = 1
T x,1  0
Use the method of separation of variables to obtain a solution for this problem. Evaluate
the resulting series and plot your results.
SOLUTION:
We can us the Laplace equation to obtain the temperature distribution in the square
plate.
 2T  2T

0
x 2 y 2
This problem contains non-homogeneous boundary conditions. Since the Laplace
equation is linear, we can use the principal of superposition to obtain a solution to
the problem.
T x, y   T1 x, y   T2 x, y 
Using the method of separation of variables:
T1 x, y   X x  Y  y 
Substitute into Laplace equation:
2 X
 2Y
Y 2 X 2 0
x
y
---------------------------1  2 X 1  2Y


X x 2 Y y 2
---------------------------Introduce the separation constant  .
2 X
 X x   0
x 2
 2Y
 Y  y   0
y 2
---------------------------Let:
  2
The solution to the above equations become:
X x  Asin x  B cos x and
Y x  C sinh y  D cosh y
---------------------------Applying the boundary conditions to the general rectangular element yields:
nx
X n x   An sin
W
ny
Yn  x   Bn sin
H
Where H is the height in the y direction and W is the width in the x direction.
The solution is the linear combination of all eigenfunctions.

T x, y    Tn x, y    X n x Yn  y 
n 1

T x, y    C n sin
n 1
nx
ny
sinh
W
H
Next, we apply the general solution and the principal of superposition to the problem
given:

T1 x, y    C n sinh nx sin ny
n 1
Where
1
cn1 
 T 1, y sin ny dy
0
1
sinh n  sin ny  dy
1

2
 sin y   sin ny dy
0
1
sinh n  sin ny  dy
.
2
0
0
Since the given boundary condition is sin y  , the only the first Fourier coefficient is
non-zero, therefore T1 becomes:
T1 x, y   C1 sinh x  sin y
T2 is given as:

T2 x, y    C n sin nx sinh n 1  y 
n 1
Where
1
C n2 
 T x,0sin nx dx
0
1
sinh n  sin nx  dx
2
0
0.5

1
 2 x sin nx dx
0
0.5
sinh n   sin nx  dx
2
0

 21  x   sin nx dx
0.5
1
sinh n   sin nx  dx
2
0.5
The following graphic was created by inputting the previous equation into and plotting
on a 3D plot in MAPLE:
2
The temperature inside a circular cylinder (radius = 1, height = 1, in meters) is
independent of the angular coordinate  (axisymmetric system). The following boundary
conditions are prescribed:
At z = 0
T (r, 0) = 0
At z = 1
T (r, 1) = (1 - r)3
At its side surface, r = 1
T (1, z) = 0
Also, note that because of symmetry, the temperature has an extremum at r = 0. Use
the method of separation of variables to obtain a solution for this problem. Evaluate the
resulting series and plot your results.
SOLUTION:
The governing heat equation for the problem given is as follows:
 2T  0
1      2T
0
r  
r r  r  z 2
---------------------------Let:
T r, z   Rr  Z z 
T RZ
R

Z
 ZR'
r
r
r
 2T
 2 RZ

Z
 ZR' '
r 2
r 2
 2T
 RZ ' '
z 2
---------------------------1
ZR' ZR' ' RZ ' '  0
r
1 R' R' '
Z ''


  2
r R R
Z
r
R' 2 R' '
Z ''
r
 r 2
  2
R
R
Z
Using separation of variables:
r 2 R' 'rR' 2 r 2 R  0  R  A1 J 0 n r   A2Y0 n r  (Bessel equation)
Z ' ' 2 Z  0  Z  B1 sinh  1  z   B2 coshz 
After applying the prescribed boundary conditions, A2 and B2 = 0. The temperature
distribution becomes:

T r , z    C n J 0 n z sinh n z 
n 1
where:
1
 r 1  r  J  r dr
3
0
Cn 
n
0
sinh n  r J 0 n r  dr
2
and  n are the eigenvalues of J 0 n   0 .
3
Consider the problem of steady state heat conduction inside a cylinder (radius = 0.1,
height = 1, in meters), of constant thermal conductivity k = 1 W/mK and subjected to a
process of constant internal heat generation at a rate of g = 105 W/m3. Assume all the
external surfaces of the cylinder are maintained at zero temperature. Use the method of
separation of variables to obtain a solution for this problem. Evaluate the resulting series
and plot your results.
SOLUTION:
The governing heat equation for the problem given is as follows:
k 2T  g  0
1      2T
g
r   2  
r r  r  z
k
Boundary conditions:
a bcdef ghi j klmnopq rs t uv wx yz ABCDEFGHIJKLMNOPQRSTUVWXYZ
