Lecture 9 Conduction with Internal Heat Generation
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Oct 30, 2023
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14
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9/19/23
1
MECE 4364 Heat Transfer
Prof. Dong Liu
Department of Mechanical Engineering
University of Houston
1
Lecture 7
1
Last Lecture
¨
Heat conduction equation in a cylindrical wall
¨
The solutions
¨
Thermal resistance
¨
Thermal network
2
࠵?
࠵?࠵?
࠵?
࠵?࠵?
࠵?࠵?
= 0
࠵?
!
= −࠵?࠵?
࠵?࠵?
࠵?࠵?
=
2࠵?࠵?࠵?
࠵?
"
− ࠵?
#
࠵?࠵?
࠵?
#
/࠵?
"
࠵?
࠵?
=
$
!
%$
"
&’
!
!
/!
"
࠵?࠵?
!
!
"
+
࠵?
#
r
1
r
2
T
1
T
2
࠵?
)*&
=
࠵?
"
− ࠵?
#
࠵?
!
=
࠵?࠵?
࠵?
#
/࠵?
"
2࠵?࠵?࠵?
࠵?
!
=
࠵?
+,"
− ࠵?
+,#
࠵?
-.-/&
࠵?
-.-/&
= ࠵?
).’0,"
+
࠵?
)*&,"
+ ࠵?
)*&,#
+
࠵?
)*&,1
+ ࠵?
).’0,1
2
9/19/23
2
Example 1
¨
Steam at
࠵?
+,"
= 320℃
flows in a cast iron pipe (
࠵? = 80 ࠵?/࠵?࠵?)
whose inner and outer
diameters are
࠵?
"
= 5 ࠵?࠵?
and
࠵?
#
= 5.5 ࠵?࠵?.
The pipe is covered with 3-cm-thick glass wool
insulation with
࠵? = 0.05
2
34
.
Heat is lost to the surroundings at
࠵?
+,#
= 5℃
by natural convection
and radiation, with a combined heat transfer coefficient of
ℎ
#
= 18
2
3
"
4
.
Taking the heat
transfer coefficient inside the pipe to be
ℎ
"
= 60
2
3
"
4
.
Determine the rate of heat loss from the
steam per unit length of the pipe.
Also determine the temperature drops across the pipe shell
and the insulation.
3
3
Critical Radius of Insulation
¨
Adding more insulation to a plane wall or to the attic always decreases heat
transfer.
The thicker the insulation, the lower the heat transfer rate
¤
This is expected, since the heat transfer area A is constant, and adding insulation
always increases the thermal resistance of the wall without increasing the
convection resistance
¨
However, this may not be the case for a cylindrical pipe
4
¨
Consider a cylindrical pipe of outer radius r
1
whose outer
surface temperature T
1
is maintained constant
¨
The pipe is now insulated with a material whose thermal
conductivity is k and outer radius is r
2
(
the larger r
2
, the thicker
the insulation
)
¨
Heat transfer from the insulated pipe to the surrounding air is
¨
What if we want to minimize the heat loss?
࠵?
!
=
࠵?
"
− ࠵?
+
࠵?
5’6
+ ࠵?
).’0
=
࠵?
"
− ࠵?
+
࠵?࠵?
࠵?
#
/࠵?
"
2࠵?࠵?࠵?
+
1
ℎ
2࠵?࠵?
#
࠵?
4
9/19/23
3
Critical Radius of Insulation
¨
The variation of q
r
with the outer radius of the
insulation r
2
shows a maximum at which
¨
The
critical radius of insulation
is
¨
Should we worry about r
cr
when insulating hot-
water pipes?
¤
r
cr
is the largest when k is large and h is small
¤
For
࠵?
!"#
= 0.05
$
!%
and
ℎ
!&’
= 5
$
!
"
%
5
࠵?
!
=
࠵?
"
− ࠵?
+
࠵?
5’6
+ ࠵?
).’0
=
࠵?
"
− ࠵?
+
࠵?࠵?
࠵?
#
/࠵?
"
2࠵?࠵?࠵?
+
1
ℎ
2࠵?࠵?
#
࠵?
࠵?࠵?
!
࠵?࠵?
#
= 0
࠵?
)!,)*&5’78!
=
࠵?
ℎ
࠵?
࠵?
࠵?
!"#
࠵?
(")*
5
Example 2
6
6
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4
Example 2
7
࠵?
࠵?
7
Example 2
8
8
9/19/23
5
Example 3
9
9
Example 3
10
10
9/19/23
6
Example 3
11
11
Conduction with Heat Generation
(Section 3.4)
12
12
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9/19/23
7
Heat Generation in A Solid
¨
Many practical heat transfer applications involve the conversion of some form of
energy into thermal energy in the medium
¤
Such mediums are said to involve internal heat generation, which manifests itself as
a rise in temperature throughout the medium
¤
Some examples of heat generation are resistance heating in wires, exothermic
chemical reactions in a solid, and nuclear reactions in nuclear fuel rods where
electrical, chemical, and nuclear energies are converted to heat
13
¨
Heat generation is usually expressed
per unit
volume
of the medium, whose unit is W/m
3
¨
Example
¤
Heat generation in an electrical wire of outer
radius r
0
and length L
̇࠵? =
̇
࠵?
+*’,*-*./)&.
࠵?
=
࠵?
0
࠵?
*
࠵?࠵?
1
0
࠵?
13
1D Steady-State Heat Generation
A large plane wall of thickness 2L experiences a uniform heat generation,
determine the temperature variation within the wall
¨
Begin with the general heat conduction equation
¨
Assumptions
¤
1D
¤
Steady state
¤
Constant properties
¨
Integrating twice gives the general solution
14
࠵?
0
࠵?
࠵?࠵?
0
+
̇࠵?
࠵?
= 0
࠵?
0
࠵?
࠵?࠵?
0
+
࠵?
0
࠵?
࠵?࠵?
0
+
࠵?
0
࠵?
࠵?࠵?
0
+
̇࠵?
࠵?
=
1
࠵?
࠵?࠵?
࠵?࠵?
x = -L
x = L
T
T
1
T
2
!
q
࠵?
࠵?
= −
̇࠵?
2࠵?
࠵?
0
+ ࠵?
2
࠵? + ࠵?
0
Making a concrete slab
14
9/19/23
8
¨
Consider the boundary conditions
࠵? = −࠵?
:
࠵?
−࠵?
= ࠵?
"
= −
̇:
#;
࠵?
#
− ࠵?
"
࠵? + ࠵?
#
࠵? = ࠵?
:
࠵?
࠵?
= ࠵?
#
= −
̇:
#;
࠵?
#
+ ࠵?
"
࠵? + ࠵?
#
¨
Solving for C
1
and C
2
¨
The temperature field is
1D Steady-State Heat Generation
15
࠵?
࠵?
=
̇࠵?
2࠵?
࠵?
0
− ࠵?
0
−
࠵?
2
− ࠵?
0
2࠵?
࠵? +
࠵?
2
+ ࠵?
0
2
Parabolic profile
x = -L
x = L
T
T
1
T
2
!
q
࠵?
"
=
࠵?
#
− ࠵?
"
2࠵?
࠵?
#
=
̇࠵?
2࠵?
࠵?
#
+
࠵?
"
+ ࠵?
#
2
(what if
̇࠵? = 0
?)
15
¨
Special case
¤
࠵?
2
= ࠵?
0
= ࠵?
3
n
Maximum temperature
n
At x = 0
n
There is no heat transfer through this plane
n
Heat transfer at other locations?
n
It may be represented by an adiabatic surface
n
Share the same solution
1D Steady-State Heat Generation
16
࠵?
࠵?
=
̇࠵?
2࠵?
࠵?
0
− ࠵?
0
+
࠵?
0
− ࠵?
2
2࠵?
࠵? +
࠵?
0
+ ࠵?
2
2
࠵?
࠵?
=
̇࠵?
2࠵?
࠵?
0
− ࠵?
0
+ ࠵?
3
࠵?
࠵? = 0
=
̇
࠵?࠵?
0
2࠵?
+ ࠵?
3
࠵?࠵?
࠵?࠵?
#41
= 0
16
9/19/23
9
¨
Heat flux
¨
Special case
¤
࠵?
2
= ࠵?
0
= ࠵?
3
1D Steady-State Heat Generation
17
࠵?
55
࠵?
= −࠵?
࠵?࠵?
࠵?࠵?
=
̇࠵?࠵? − ࠵?
࠵?
0
− ࠵?
2
2࠵?
Linear heat flux profile
࠵?
࠵?
=
̇࠵?
2࠵?
࠵?
0
− ࠵?
0
+
࠵?
0
− ࠵?
2
2࠵?
࠵? +
࠵?
0
+ ࠵?
2
2
࠵?
55
࠵?
= −࠵?
࠵?࠵?
࠵?࠵?
=
̇࠵?࠵?
17
¨
A solid wall of thickness L is subject to the following
boundary conditions
¨
What is the surface temperature at x = L?
¨
What is the surface temperature at x = 0?
Example 1
18
࠵?
࠵?
=
̇࠵?
2࠵?
࠵?
0
− ࠵?
0
+ ࠵?
6
+
̇
࠵?࠵?
ℎ
࠵?
࠵? = ࠵?
= ࠵?
6
+
̇
࠵?࠵?
ℎ
= ࠵?
3
࠵?
࠵?
= −
̇࠵?
2࠵?
࠵?
0
+ ࠵?
2
࠵? + ࠵?
0
Standard approach
࠵?
࠵?࠵?
࠵?࠵?
࠵?࠵?
= −
̇࠵?
࠵?
࠵?
࠵? = 0
=
̇࠵?࠵?
0
2࠵?
+ ࠵?
6
+
̇
࠵?࠵?
ℎ
D
࠵?࠵?
࠵?࠵?
<=>
= 0
D
−࠵?
࠵?࠵?
࠵?࠵?
<=?
= ℎ
࠵?
࠵? = ࠵?
− ࠵?
+
18
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10
Example 1
19
A different approach to find T
s
:
࠵?
3
= ࠵?
6
+
̇
࠵?࠵?
ℎ
−࠵?
.7’8
55
࠵? +
̇࠵?
࠵?࠵?
= 0
For the solid wall, energy balance yields
−
̇
࠵?
79/
+
̇
࠵?
+
= 0
ℎ࠵?
࠵?
3
− ࠵?
6
=
̇࠵?
࠵?࠵?
19
Example 2
¨
A composite plane wall consists of materials A and B.
¨
A has uniform heat generation
̇࠵? = 1.5×10
:
࠵?/࠵?
;
. The inner surface of A is
insulated
¨
The outer surface of B (
࠵?
<
= 150 ࠵?/࠵?࠵?
) is cooled by a water stream
࠵?
6
=
30℃, ℎ = 1000 ࠵?/࠵?
0
࠵?
.
¤
Sketch the steady-state temperature distribution
¤
Determine T
0
and T
2
20
20
9/19/23
11
Example 2
¨
Sketch the steady-state
temperature distribution
21
¨
A: conduction with heat generation
¤
BCs:
¤
Parabolic temperature profile
¨
B: conduction without heat generation
¤
BCs:
¤
Linear temperature profile
21
Example 2
22
22
9/19/23
12
Example 2
23
!
" = 0
=
%&
!
̇
2)
+ !
"
23
Heat Generation in A Cylinder
¨
A long homogeneous resistance wire of radius
࠵?
>
= 0.2 ࠵?࠵?࠵?ℎ
in and thermal
conductivity
࠵? = 7.8
@-A
B D- ℉
is being used to boil water at atmospheric pressure
by the passage of electric current. Heat is generated in the wire uniformly as
a result of resistance heating at a rate of
̇࠵? = 2400 ࠵?࠵?࠵?/ℎ K ࠵?࠵?
1
. If the outer
surface temperature of the wire is measured to be
࠵?
6
= 226℉
, obtain a
relation for the temperature distribution, and determine the temperature at
the centerline of the wire when steady operating conditions are reached.
24
This heat transfer problem is similar to what we have solved,
except that
࠵? = ࠵?
࠵?
Boundary conditions
1
࠵?
࠵?
࠵?࠵?
࠵?
࠵?࠵?
࠵?࠵?
+
̇࠵?
࠵?
= 0
࠵?
࠵?
>
= ࠵?
6
= 226℉
D
࠵?࠵?
࠵?࠵?
!=>
= 0
24
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13
Heat Generation in A Cylinder
¨
Integrating once gives
¨
Applying the surface temperature BC
25
1
࠵?
࠵?
࠵?࠵?
࠵?
࠵?࠵?
࠵?࠵?
+
̇࠵?
࠵?
= 0
࠵?
࠵?
>
= ࠵?
6
= 226℉
D
࠵?࠵?
࠵?࠵?
!=>
= 0
࠵?
࠵?࠵?
࠵?࠵?
= −
̇࠵?
2࠵?
࠵?
#
+ ࠵?
"
࠵?
2
= 0
࠵?࠵?
࠵?࠵?
= −
̇࠵?
2࠵?
࠵?
࠵? = −
̇࠵?
4࠵?
࠵?
#
+ ࠵?
#
࠵?
6
= −
̇࠵?
4࠵?
࠵?
>
#
+ ࠵?
#
࠵?
#
= ࠵?
6
+ −
̇࠵?
4࠵?
࠵?
>
#
࠵?
࠵?
= ࠵?
6
+
̇࠵?
4࠵?
࠵?
>
#
− ࠵?
#
25
Heat Generation in A Two-Layer Cylinder
¨
This time the wire of radius
࠵?
"
is wrapped by a layer of
insulation material.
Determine
¤
The temperature at the center of the wire
࠵?
1
¤
The temperature at the interface of the wire and the
insulation layer
࠵?
&
¨
Solution
¤
The temperature in the wire can be formulated as
with
¤
The problem was already solved
26
1
࠵?
࠵?
࠵?࠵?
࠵?
࠵?࠵?
࠵?࠵?
+
̇࠵?
࠵?
= 0
࠵?
࠵?
"
= ࠵?
5
D
࠵?࠵?
࠵?࠵?
!=>
= 0
࠵?
࠵?
= ࠵?
5
+
̇࠵?
4࠵?
࠵?
"
#
− ࠵?
#
࠵?
&
is unknown
26
9/19/23
14
Heat Generation in A Two-Layer Cylinder
¤
Note that the insulation layer does not involve any heat
generation
¤
The boundary conditions are
¤
The solution was obtained from last lecture
¨
How to find
࠵?
&
?
¤
At the interface, heat flux in the wire and the insulation must
be the same
27
࠵?
࠵?
"
= ࠵?
5
࠵?
࠵?࠵?
࠵?
࠵?࠵?
࠵?࠵?
= 0
࠵?
࠵?
#
= ࠵?
6
࠵?
࠵?
=
$
#
%$
$
&’
!
!
/!
"
࠵?࠵?
!
!
"
+
࠵?
6
࠵?
&
is still unknown
−࠵?
F5!8
D
࠵?࠵?
࠵?࠵?
!
!
,F5!8
= −࠵?
5’6
D
࠵?࠵?
࠵?࠵?
!
!
,5’6
࠵?
5
=
̇:!
!
"
#;
%#&’
࠵?࠵?
!
"
!
!
+
࠵?
6
27
Heat Generation in A Solid Sphere
1D, steady-state conduction in a sphere of
constant
k
,
uniform generation
, and
convection cooling
:
General Solution:
T r
( )
=
−
!
q
6
k
⎛
⎝
⎜
⎞
⎠
⎟
r
2
−
C
1
r
+
C
2
Specific Solution:
What are the two BCs?
T r
o
(
)
=
T
s
dT
dr
r
=
0
=
0
1
r
2
d
dr
kr
2
dT
dr
⎛
⎝
⎜
⎞
⎠
⎟
+
!
q
=
0
centerline temperature/thermal
condition for
radial systems
is the
same as an insulated wall
T r
( )
=
!
qr
o
2
6
k
1
−
r
2
r
o
2
⎛
⎝
⎜
⎞
⎠
⎟
+
T
s
28
28
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C:/Users/DANIEL/Desktop/300L%202ND%20SEMESTER%20MATERIALS/Module2-chap2propertiesofpuresubstances-130703012604 phpap.
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V Draw
H Highlight
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40
MEC 451 - THERMODYNAMICS
Faculty of Mechanical Engineering, UITM
Supplementary Problems
The pressure in an automobile tire depends on the temperature of the air
in the tire. When the air temperature is 25°C, the pressure gage reads 210
kRa. If the volume of the tire is 0.025 m3, Cetermine the pressure rise in
the tire when the air temperature in the tire rises to 50°C. Also, determine
the amount of air that must be bled off to restore pressure to its original
value at this temperature. Assume the atmospheric pressure is 100 kPa.
[ 26 kPa, 0.007 kg]
1.
A 6 4 2:57 PM
Lucky CORER
144
Tum lock
314
%
8
1/2
24
6
1/4
23
T
K
F
pause
B
arrow_forward
To maximize production and minimize pumping costs, crude oil is heated to reduce its viscosity during
transportation from a production field.
(1) Consider a pipe-in-pipe configuration consisting of concentric steel tubes with an intervening insulating
material. The inner tube is used to transport warm crude oil through cold ocean water. The inner steel pipe (
ks = 40 W/m-K) has an inside diameter of Di, 1 = 150 mm and wall thickness t; = 20 mm while the outer
steel pipe has an inside diameter of D2 = 250 mm and wall thickness, = 1,. Determine the maximum
allowable crude oil temperature to ensure the polyurethane foam insulation (k, = 0.0425 W/m.K) between
the two pipes does not exceed its maximum service temperature of Tp, max = 70°C. The ocean water is at
Too, o
ho
T = -5°C and provides an external convection heat transfer coefficient of h = 500 W/m²K. The
convection coefficient associated with the flowing crude oil is h₁ = 450 W/m²K.
(2) It is proposed to enhance the performance of…
arrow_forward
a. A slab of thermal insulator is 100 cm² in cross section and 2 cm thick. Its
thermal conductivity is 2.4 x 10 -+ cal/(s cm C"). If the temperature difference
between opposite faces is 180 F', how much heat flows through the slab in one
day?
arrow_forward
A food product containing a water content of 85% is frozen. Estimate the heat type products at -10 °C when 82% of the water is in its frozen state. Hot product type dry is 2.5 kJ/(kg °C) is assumed to heat this type of water at -10 °C is equal to heat water type at 0 °C, and the heat of the types of ice follow function Cp ice = 0 0062 x T frozen + 2 0649.
a. Cp frozen products kJ/kg °C.
arrow_forward
Thermal Modeling
The figure shows a thermal system involving two compartments with one
containing a heater. The temperature of the compartment containing the
heater is T1, the temperature of the other compartment is T2 and the
temperature surrounding the compartment is T3, develop equations how
temperatures T1 and T2 will vary with time.
All the walls of the containers have the same resistance and negligible
capacity. The two containers have the same capacity C. Also R1=R2=R
Design the thermal modeling equation in chambers 1 and 2?
CT
CT,
C T; T
arrow_forward
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- dated metncpdf Thermodynamics An Engineering X E Module2-chap2propertiesofpure x O File C:/Users/DANIEL/Desktop/300L%202ND%20SEMESTER%20MATERIALS/Module2-chap2propertiesofpuresubstances-130703012604 phpap. ID Page view A Read aloud V Draw H Highlight O Erase 40 MEC 451 - THERMODYNAMICS Faculty of Mechanical Engineering, UITM Supplementary Problems The pressure in an automobile tire depends on the temperature of the air in the tire. When the air temperature is 25°C, the pressure gage reads 210 kRa. If the volume of the tire is 0.025 m3, Cetermine the pressure rise in the tire when the air temperature in the tire rises to 50°C. Also, determine the amount of air that must be bled off to restore pressure to its original value at this temperature. Assume the atmospheric pressure is 100 kPa. [ 26 kPa, 0.007 kg] 1. A 6 4 2:57 PM Lucky CORER 144 Tum lock 314 % 8 1/2 24 6 1/4 23 T K F pause Barrow_forwardTo maximize production and minimize pumping costs, crude oil is heated to reduce its viscosity during transportation from a production field. (1) Consider a pipe-in-pipe configuration consisting of concentric steel tubes with an intervening insulating material. The inner tube is used to transport warm crude oil through cold ocean water. The inner steel pipe ( ks = 40 W/m-K) has an inside diameter of Di, 1 = 150 mm and wall thickness t; = 20 mm while the outer steel pipe has an inside diameter of D2 = 250 mm and wall thickness, = 1,. Determine the maximum allowable crude oil temperature to ensure the polyurethane foam insulation (k, = 0.0425 W/m.K) between the two pipes does not exceed its maximum service temperature of Tp, max = 70°C. The ocean water is at Too, o ho T = -5°C and provides an external convection heat transfer coefficient of h = 500 W/m²K. The convection coefficient associated with the flowing crude oil is h₁ = 450 W/m²K. (2) It is proposed to enhance the performance of…arrow_forwarda. A slab of thermal insulator is 100 cm² in cross section and 2 cm thick. Its thermal conductivity is 2.4 x 10 -+ cal/(s cm C"). If the temperature difference between opposite faces is 180 F', how much heat flows through the slab in one day?arrow_forward
- A food product containing a water content of 85% is frozen. Estimate the heat type products at -10 °C when 82% of the water is in its frozen state. Hot product type dry is 2.5 kJ/(kg °C) is assumed to heat this type of water at -10 °C is equal to heat water type at 0 °C, and the heat of the types of ice follow function Cp ice = 0 0062 x T frozen + 2 0649. a. Cp frozen products kJ/kg °C.arrow_forwardThermal Modeling The figure shows a thermal system involving two compartments with one containing a heater. The temperature of the compartment containing the heater is T1, the temperature of the other compartment is T2 and the temperature surrounding the compartment is T3, develop equations how temperatures T1 and T2 will vary with time. All the walls of the containers have the same resistance and negligible capacity. The two containers have the same capacity C. Also R1=R2=R Design the thermal modeling equation in chambers 1 and 2? CT CT, C T; Tarrow_forward
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- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning
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