This is a document that shows information about physical design strategies for an engineering part of our course

2. Cartesian
Cartesian
z
T
k
q
y
T
k
q
x
T
k
q z
y
x











 ;
;
Source
z
T
y
T
x
T
k
z
T
v
y
T
v
x
T
v
C
t
T
C z
y
x
p
p 
































2
2
2
2
2
2


Similarity between LUMPED ANALYSIS and RECTANGULAR FINS

3. 1D PLANAR heat conduction and convective boundary
conditions (pin-fin in a heat exchanger)
Consider a thin fin extending from a wall at temperature T0 into an ambient at
temperature T∞. The rod is long enough so that it is reasonable to assume its
free end is adiabatic.
Cross-sectional area of the rod is A, thermal conductivity is K and perimeter is
P. Get the temperature distribution within the rod.

4. )
(x
f
T 
At first glance, it appears that we can quickly simplify the governing equation and
make use of the two boundary conditions at x=0 and x=L and solve the problem
Source
z
T
y
T
x
T
k
z
T
v
y
T
v
x
T
v
C
t
T
C z
y
x
p
p 
































2
2
2
2
2
2


Governing equation for constant K
Objective

5. Governing Equation
We can represent the heat loss through the
Fin as a negative energy source:
Convective heat flux through the fin surface x
Area of the fin surface/Volume
AL
pL
T
T
h
Q
)
( 



Therefore, the governing equation becomes
L
x
A
p
T
T
h
x
T
k 




 
0
,
)
(
2
2
kA
hp
m 
2


 
T
T 0
2
2
2






m
x

6. Solving Diffy Q’s is often a “guessing game”

7. 1 2
( ) mx mx
T x C e C e T


  
BC 1:
BC 2:
0
0
x
T T


Therefore, the general solution is given by:
Infinitely long fin

9. The insulated tip fin
0
cosh( )
( ) ( )
cosh( )
mL mx
T x T T T
mL
 

  
Using:
)
cosh(
)
sinh(
);
sinh(
)
cosh( x
x
dx
d
x
x
dx
d


2nd bc gives using:
Show that all the conductive heat transfer at x=0 is equal to the total
convective heat loss across the fin surface
1st BC gives:

10. The insulated tip fin
0
cosh( )
( ) ( )
cosh( )
mL mx
T x T T T
mL
 

  
Using:
)
cosh(
)
sinh(
);
sinh(
)
cosh( x
x
dx
d
x
x
dx
d


2nd bc gives using:
Show that all the conductive heat transfer at x=0 is equal to the total
convective heat loss across the fin surface
1st BC gives:

11. Fin efficiency
The fin efficiency is defined as the ratio of the energy transferred through a real
fin to that transferred through an ideal fin.
An ideal fin is thought to be one made of a perfect or infinite conductor material.
A perfect conductor has an infinite thermal conductivity so that the entire fin is at
the base material temperature.
For an adiabatic fin