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Consider two plane walls in contact (called a composite wall)
as shown below. The individual walls are labeled 1 and 2 as are
each the thermal conductivity and thickness . Assume the wall boundaries convect heat to the environment on both
sides. Each side may have different convection coefficients and environmental temperature .
Figure 7.11: Composite Wall With Convection Boundary
Conditions

Each layer must satisfy the heat conduction equation
whose solution is a linear
function in . Consequently, we have the following solution for layers 1 and 2:
To evaluate the constants , , , and , the boundary conditions at points and and the interface conditions at must be satisfied.
 A)
 Convective boundary condition at : (note direction of ,
)
 B)
 Interface boundary condition between wall 1 and 2 at :

(7.24) 

(7.25) 
 C)
 Convective boundary condition at : (note change in direction,
)
Consequently we have four equations and four unknowns (, , , and ) as follows:
The above system of four equations can be solved for , , , and and the result substituted back into equation (7.22)
to obtain and . The heat flux in the direction is given by

(7.28) 
For 1D slab heat flow, heat can flow only in one direction (in
this case, the direction). Consequently, in the absence of heat sinks/sources in
a layer, the heat flux must remain a constant as it passes through
the convective air layer on the left, through each slab and finally
through the convective air layer on the right.
To simplify the solution for a composite wall (with
no internal heat source), we seek to develop a simplified relation
between the overall heat flux through the composite wall and the
given temperature gradient from one side of the composite wall to
the other:
where effective heat transfer coefficient of the composite wall,
effective thermal resistance of the composite
wall and, for the case of convection boundary conditions on each side
of the composite wall, the known temperature gradient from left to
right is given by
.
The solution of the ODE for heat transfer through a single layer
with no heat source requires that the temperature variation in the
layer is a linear function of : where and are constants of integration dependent on boundary conditions. If
the temperature on either side of a wall of thickness is and , then
. For the composite wall shown
below, we introduce the following notation. Define the temperature
at the left boundary to be , at the interface , and at the right boundary as shown in the figure below. At this point, , , and are unknown.
Figure 7.12: Composite Wall With Convective Boundary
Conditions

In the absence of a heat source within the body, the temperature
in each layer will be a linear function of so that we may write the following equations for the temperature in
each layer:
At (the interface), these equations already satisfy the interface
condition that
. Therefore, only the heat flux boundary condition
needs to be satisfied at the interface and the convective boundary
condition at the left and right boundaries of the composite wall.
From conservation of energy, the heat energy through a given area () must be constant as it enters on the left and leaves on the
right boundary (since we assumed there is no internal heat generation,
). Since heat flow is normal to wall, each layer has same normal
area (so area cancels out). Thus, the heat flux must remain a constant as it passes through the convective air layer
on the left, through each slab and finally through the convective
air layer on the right and we can write

(7.31) 
Equation (7.31) may be separated
into 4 equations by considering each heat flux term individually
to obtain:
Add these four equations, (1) through (4) to obtain

(7.33) 
or,

(7.34) 
The fractional term in (7.34)
may be defined as the effective heat transfer coefficient
:

(7.35) 
where n is the number of layers in the composite wall. We may also
define the effective thermal resistance by the reciprocal of :

(7.36) 
Consequently, the heat flux through the composite wall with convection boundary conditions
on both sides of the wall is given by
Note that thermal resistance terms (like or ) are additive similar to resistors in electrical theory.
The last result may be expanded to include various boundary conditions
on the left and right side of the composite wall. For example, for
a composite wall with 3 layers we obtain the following summary of
results:
Summary of Conduction Through Composite Walls



(7.38) 




where the heat flux in each layer is given by:
Note: the fist and last terms below represent heat flux through the
fluid layers where convection occurs (terms with ) and the 2
through 4
terms represent heat flux through the solid layers where
conduction occurs (terms with ).
Considering the definition of (7.38) for the various cases of different
boundary conditions we note that when there is convection on the
left and right, the terms and appear in . When there is convection only on the right, only appears, etc. For three solid layers, we have for each of the three layers. This suggests the following
simplified definition of :

(7.40) 
where
means to include the term only if there is convection on left () or right ()
The general solution procedure then consists of three steps:
 1.
 Evaluate effective thermal resistance using (7.40)
 2.
 Evaluate the heat flux for the composite wall using
where
.
 3.
 Evaluate the temperatures for each layer using (7.39)
working from left to right through the layers. can be obtained from the first equation in (7.39),
from second equation, etc.
Example 75
Consider a twolayer composite wall with 1D heat transfer through
the layers and free convection of air on either side with
. Assume the
thickness of each layer is cm. The temperature difference from left to right is
C. Find for the following situations:
 a)
 Material 1glass; Material 2glass
 b)
 Material 1copper; Material 2glass
 c)
 Material 1copper; Material 2teflon
Solution
 a)
 is first converted to metric:
 b)

 c)

Note that in case c), the introduction of teflon, which is a good
insulator with a relatively low coefficient of thermal conductivity
, yields a higher effective resistance and correspondingly lower heat flux, .
Example 76
Consider a two layer composite wall of copper and teflon as shown
below. The copper has a thickness of 10 cm but the thickness of
the teflon is to be determined. The temperature on the left boundary
is equal to C and on the right boundary C. Determine the thickness of the teflon layer so that the
heat flux is equal to
.
Given:
Find:
Solution
Example 77
Consider steadystate heat conduction through a cylindrical wall
with convection on both sides of the cylindrical wall. Find the
temperature of the wall.
The heat transfer equation in cylindrical coordinates is given
by

(7.42) 
In the absence of internal a heat source in the solid, the solution
provided above will always hold.
at A

(7.43) 
at B

(7.44) 
Substituting (7.42) into the previous
two boundary condition equations yields:



(7.45) 




Equations (7.45) may be solved
for and and substituted into (7.42) to
obtain the solution for the temperature distribution .
Example 78
Consider steadystate heat conduction through a cylindrical wall
with specified temperature on the boundaries of the cylindrical
wall. Find the temperature of the wall.
The solution of the heat flow equation in cylindrical coordinates
is given by
Applying the boundary conditions at the inner and outer radius
gives
Subtracting equation (2) from equation (1) gives:
or,
Substituting into equation (1) above gives
Substituting and into yields
or
The heat flux in the radial direction is given by:
or
Note that the heat flux is a function of radial position . This is necessary because the area through which the heat flows increases
as increases. The radial flow for time is given by
.
Note that is independent of (as it should be) since there is no internal heat source and thus
the heat flow must be the same at all radial positions, .
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