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the composite. The outer surface is exposed to ambient air, which is at T ∞,o
and provides a convection coefficient of h o . Under steady-state conditions,
""
a uniform heat flux of q h is dissipated by the heater.
a. Sketch the equivalent thermal circuit of the system and express all
resistances in terms of relevant variables.
b. Obtain an expression that may be used to determine the heater temperature, T h .
c. Obtain an expression for the ratio of heat flows to the outer and inner
"
"
fluids, q /q i . How might the variables of the problem be adjusted to
o
minimize this ratio?
SOLUTION
a. See the sketch shown in Figure 2.11.
b. Performing an energy balance for the heater, E ˙ in = E ˙ out , it follows that
""
"
"
T h − T ∞,i
q h (2πr 2 ) = q i + q =
o
(h i 2πr 1 ) −1 + (ln(r 2 /r 1 )/2πk B ) + R ""
tc,B
T h − T ∞,o
+
(h o 2πr 3 ) −1 + (ln(r 3 /r 2 )/2πk A ) + R ""
tc,A
c. From the circuit,
"
(h i 2πr 1 ) −1 + (ln (r 2 /r 1 ) /2πk B ) + R ""
q o
(T h − T ∞,o )
tc,B
=
·
"
(h o 2πr 3 ) −1 + (ln (r 3 /r 2 ) /2πk A ) + R ""
q i
(T h − T ∞,i )
tc,A
2.2. Heat is generated at a rate q ˙ in a large slab of thickness 2L, as shown in
Figure 2.5a. The side surfaces lose heat by convection to a liquid at temperature T ∞ . Obtain the steady-state temperature distributions for the following
cases:
a. q ˙ = q ˙ o 1 − (x/L) 2 , with x measured from the centerplane.
b. q ˙ = a + b(T − T ∞ )
SOLUTION
a. q ˙ = q ˙ o 1 − (x/L) 2
( ) 2
d 2 T
q ˙ o
x
= −
1 −
dx 2
k
L
dT
q ˙ o
1 x 3
= −
x −
+ C 1
dx
k
3 L 2
4
q ˙ o 1
1 x
T = −
x 2 −
+ C 1 x + C 2
k 2
12 L 2
32
Analytical Heat Transfer
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the composite. The outer surface is exposed to ambient air, which is at T ∞,o
and provides a convection coefficient of h o . Under steady-state conditions,
""
a uniform heat flux of q h is dissipated by the heater.
a. Sketch the equivalent thermal circuit of the system and express all
resistances in terms of relevant variables.
b. Obtain an expression that may be used to determine the heater temperature, T h .
c. Obtain an expression for the ratio of heat flows to the outer and inner
"
"
fluids, q /q i . How might the variables of the problem be adjusted to
o
minimize this ratio?
SOLUTION
a. See the sketch shown in Figure 2.11.
b. Performing an energy balance for the heater, E ˙ in = E ˙ out , it follows that
""
"
"
T h − T ∞,i
q h (2πr 2 ) = q i + q =
o
(h i 2πr 1 ) −1 + (ln(r 2 /r 1 )/2πk B ) + R ""
tc,B
T h − T ∞,o
+
(h o 2πr 3 ) −1 + (ln(r 3 /r 2 )/2πk A ) + R ""
tc,A
c. From the circuit,
"
(h i 2πr 1 ) −1 + (ln (r 2 /r 1 ) /2πk B ) + R ""
q o
(T h − T ∞,o )
tc,B
=
·
"
(h o 2πr 3 ) −1 + (ln (r 3 /r 2 ) /2πk A ) + R ""
q i
(T h − T ∞,i )
tc,A
2.2. Heat is generated at a rate q ˙ in a large slab of thickness 2L, as shown in
Figure 2.5a. The side surfaces lose heat by convection to a liquid at temperature T ∞ . Obtain the steady-state temperature distributions for the following
cases:
a. q ˙ = q ˙ o 1 − (x/L) 2 , with x measured from the centerplane.
b. q ˙ = a + b(T − T ∞ )
SOLUTION
a. q ˙ = q ˙ o 1 − (x/L) 2
( ) 2
d 2 T
q ˙ o
x
= −
1 −
dx 2
k
L
dT
q ˙ o
1 x 3
= −
x −
+ C 1
dx
k
3 L 2
4
q ˙ o 1
1 x
T = −
x 2 −
+ C 1 x + C 2
k 2
12 L 2
32
Analytical Heat Transfer