Problem
Consider a 5-m-high, 8-m-long, and 0.22-m-thick wall whose representative cross section is as given in the figure. The thermal conductivities of various materials used, in W/m·K, are kA = kF =2,kB =8, kc = 20, kD = 15, and kE =35. The left and right surfaces of the wall are maintained at uniform temperatures of 300°C and 100°C, respectively. Assuming heat transfer through the wall to be one-dimensional, determine (a) the rate of heat transfer through the wall; (b) the temperature at the point where the sections B, D, and E meet; and (c) the temperature drop across the section F. Disregard any contact resistances at the interfaces.
Step-by-step solution
Step 1 of 13
a)Draw the cross section of
thick plane wall shown as follows:
Comment Step 2 of 13
Draw the thermal resistance network due to conduction as shown below.
Comment Step 3 of 13
Calculate the thermal resistance in wall
due to conduction, by using the equation as shown below.
Here, the thickness of the wall
is 
, thermal conductivity of wall 
is 
and the area normal to the direction of heat flow through 
is 
.Substitute
,
and 
.
Comments (8) Step 4 of 13
Calculate the thermal resistance in wall
due to conduction, by using the equation as shown below.
Here, the thickness of the wall
is 
, thermal conductivity of wall 
is 
and the area normal to the direction of heat flow through 
is 
.Substitute
,
and 
.
Comment Step 5 of 13
Calculate the thermal resistance in wall
due to conduction, by using the equation as shown below.
Here, the thickness of the wall
is
, thermal conductivity of wall 
is 
and the area normal to the direction of heat flow through 
is 
.Substitute
, 
and 
.
Comment Step 6 of 13
Calculate the thermal resistance in wall
due to conduction, by using the equation as follows:
Here, the thickness of the wall
is 
, thermal conductivity of wall 
is 
and the area normal to the direction of heat flow through 
is 
.Substitute
,
and 
.
Calculate the thermal resistance in wall
due to conduction, by using the equation as follows:
Here, the thickness of the wall
is 
, thermal conductivity of wall 
is 
and the area normal to the direction of heat flow through 
is 
.Substitute
, 
and
.
Comment Step 7 of 13
Calculate the thermal resistance in wall
due to conduction, by using the equation as shown below.
Here, the thickness of the wall
is 
, thermal conductivity of the wall 
is 
and the area normal to the direction of heat flow through 
is 
.Substitute
,
and 
.
Comments (5) Step 8 of 13
The three resistances
,
and 
in the middle are parallel.Calculate the equivalent resistance of
, 
and 
, by using the equation as shown below.
Substitute the known values in the above equation.
Comment Step 9 of 13
The two resistances
and
in the middle are parallel.Calculate the equivalent resistance of
and 
by using the equation as shown below.
Substitute the known values in the above equation.
Comment Step 10 of 13
Calculate the total thermal resistance.
Substitute the known values in the above equation.
Comment Step 11 of 13
Calculate the rate of heat transfer by using the equation as shown below.
Here, the temperature of the left surface is
and the temperature of the right surface is
.Substitute
,
and 
.
Calculate the rate of heat transfer through entire wall
.
Therefore, the rate of heat transfer through the wall is
.b)Calculate the total thermal resistance between left surface and the point where the sections B, D, and E meet.
Substitute the known values in the above equation.
Comments (2) Step 12 of 13
Calculate the temperature at the point where the sections B, D, and E meet.
Substitute
, 
and
.
Therefore, the temperature at B, D, and E interface is
.Comment Step 13 of 13
c)Calculate the temperature drop across the section F by using the equation as shown below.
Substitute
and
.
Therefore, the temperature drop across the section F is
.Comment
