Heat Generation and Conduction through Composite Walls

​
wall with uniform generation
A
B
​
heat generation rate (kW/
3
m
)
4
contact resistance (
2
m
K/W)
0.05
Temperature profiles are calculated for steady-state conduction through a series of three walls with different thermal conductivity. There is also a contact resistance between each pair of walls. Heat is generated at a constant rate in either wall
A
or
B
(select using buttons). Set the rate of heat generation and the contact resistance between the walls with sliders. The distance between the walls is exaggerated so that the temperature profile is more easily seen. The left side of wall
A
is insulated, and heat is transferred from the right side of wall
C
to the flowing air, which is at 20 °C.

Details

The heat flux
q''
from the wall with generation is uniform and directed toward the right:
q''=Q
L
i
,
where
q''
is in
W/
2
m
,
L
i
is the thickness of the wall with generation
i=(A,B)
(m) and
Q
is the volumetric heat generation rate (
W/
3
m
).
A thermal circuit is considered from the right side of the wall with generation to the air with forced convection. If
A
is the wall with generation, then the heat flux can be written as:
q''=
T
s,1
-
T
∞
1/h
=
T
s,2
-
T
s,1
L
C
/
k
C
=
T
s,3
-
T
s,2
″
R
tc
=
T
s,4
-
T
s,3
L
B
/
k
B
=
T
s,5
-
T
s,4
″
R
tc
,
where
″
R
tc
is contact resistance (
2
m
K/W
),
T
s,1
is the right surface of wall
C
and
T
s,5
is the right side of wall
A
(°C). However, if wall
B
is the wall with generation, then the thermal circuit can only be used to solve up to
T
s,3
, the right side of wall
B
.
The temperature profile for heat generation in either wall
A
or
B
is calculated using:
2

T

2
x
=-
Q
k
i
,
where
k
i
is the thermal conductivity of the wall with generation.
The boundary conditions are:
T
x
=0
,
T(
L
i
)=
T
s,rightsideofwall
;
the left side (
L
i
=0
) is well insulated so the flux is zero. The temperature of the right side of the wall with generation is determined from the thermal circuit.

References

[1] T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, Introduction to Heat Transfer, 6th ed., Hoboken: John Wiley and Sons, 2011.

Permanent Citation

Adam J. Johnston, Rachael L. Baumann, John L. Falconer
​
​"Heat Generation and Conduction through Composite Walls"​
​http://demonstrations.wolfram.com/HeatGenerationAndConductionThroughCompositeWalls/​
​Wolfram Demonstrations Project​
​Published: October 4, 2016