Ch 2 - Steady state 1-D, Heat conduction.pdf

2021-8 2
Objectives
 Understand the conditions under which a heat transfer problem can be approximated as being one-
dimensional.
 Obtain the differential equation of heat conduction in various coordinate systems, and simplify it for
steady one-dimensional case.
 Identify the thermal conditions on surfaces, and express them mathematically as boundary and initial
conditions.
 Solve one-dimensional heat conduction problems and obtain the temperature distributions within a
medium and the heat flux.
 Analyze one-dimensional heat conduction in solids that involve heat generation.
 Evaluate heat conduction in solids with temperature-dependent thermal conductivity.

Part A: One-Dimensional, Steady-State Conduction
without Thermal Energy Generation

2021-8 4
Introduction To Heat Conduction
• Steady implies no change with time at any point within
the medium
• Transient implies variation with time or time dependence
• In the special case of variation with time but not with
position, the temperature of the medium changes
uniformly with time. Such heat transfer systems are called
lumped systems.
• Heat transfer problems are also classified as being:
 one-dimensional
 two dimensional
 three-dimensional
 Steady versus Transient Heat Transfer

2021-8 5
Introduction To Heat Conduction
 Simplest Case: One-Dimensional, Steady-State Conduction with No Thermal Energy Generation.
Common Geometries:
The Plane Wall: Described in rectangular (x) coordinate. Area perpendicular to direction of heat transfer is
constant (independent of x).
 rectangular T(x, y, z, t)
The Tube Wall: Radial conduction through tube wall (cylindrical).
 cylindrical T(r, , z, t)
The Spherical Shell: Radial conduction through shell wall.
 spherical T(r, , , t).

2021-8 6
Heat Conduction in a Plane Wall
For one-dimension (1-D)

2021-8 7
Heat Generation
• Examples:
 electrical energy being converted to heat at a rate of I2R,
 fuel elements of nuclear reactors,
 exothermic chemical reactions.
• Heat generation is a volumetric phenomenon.
• The rate of heat generation units : W/m3 or Btu/h·ft3.
• The rate of heat generation in a medium may vary with time
as well as position within the medium.
 In addition, there may occur change in the amount
of the internal thermal energy stored by the
material in the control volume;

2021-8 8

2021-8 9
Sub: the values of C1 and C2 in Eqn. 1
The temperature distribution
1
1
2 )
( T
L
x
T
T
Tx 


 General equation
Integrating twice
T(x) = C1 x + C2 ----------(1)
Boundary conditions B.C
0
2
2

dx
T
d
1
C
dx
dT

L
T
T
C
T
T
L
x
at
)
( 1
2
1
2






1
2
1
0 T
C
T
T
x
at 





2021-8 10
 Heat Flux and Heat Rate q:
q 

)
(W/m
C
-k
dx
dT
k
- 2
1




q
(W)
C
A
-k
dx
dT
A
k
- 1


q
th
R
T
kA
L
T
T
T
T
L
kA
q






)
(
)
( 2
1
2
1


kA
L
Rth
Where Thermal Resistances (K/W) in a plane wall

2021-8 11
 Thermal Resistance Concept
Analogy between thermal and electrical
resistance concepts.
rate of heat transfer  electric current
thermal resistance  electrical resistance
temperature difference  voltage difference
• Conduction resistance of the wall: Thermal resistance of the
wall against heat conduction.
• Thermal resistance of a medium depends on the geometry and
the thermal properties of the medium.
Electrical resistance

2021-8 12
 Convection resistance of the surface: Thermal resistance of the
surface against heat convection.
• When the convection heat transfer coefficient is very large (h → ), the
convection resistance becomes zero and Ts  T.
• That is, the surface offers no resistance to convection, and thus it does
not slow down the heat transfer process.
• This situation is approached in practice at surfaces where boiling and
condensation occur.
Schematic for convection
resistance at a surface.

2021-8 13
 Radiation resistance of the surface: Thermal resistance of the surface
against radiation.
Schematic for convection
and radiation resistances at
a surface.
• Radiation heat transfer coefficient hrad
• Ts and Tsurr must be in K in the evaluation of hrad
Combined heat transfer coefficient

2021-8 14
 Thermal circuit for plane wall with adjoining fluids:
Schematic for convection and radiation resistances at a
surface.

2021-8 15
 Temperature drop
• Once Q is evaluated, the surface temperature T1 can be determined from
A
h
kA
L
A
h
Rtotal
2
1
1
1



total
R
T
T
q
)
,
( 2
,
1 
 

k
L
R cond
th 

 ,
h
R conv
th
1
,



)
/
( W
K
Rth  W
K
m
Rth /
2



• Thermal Resistance for Unit Surface Area:
Units:

2021-8 16
 Multilayer Plane Walls

2021-8 17
 Multilayer Plane Walls



 



th
th
overall
R
T
T
R
T
q
)
( ,
, 4
1
         
 
A
h
A
k
L
A
k
L
A
k
L
A
h
T
T
q
C
C
B
B
A
A
x
4
1
4
1
1
1 /
/
/
/
/
,
,








     
....
/
/
/
,
,
,








A
k
L
T
T
A
k
L
T
T
A
h
T
T
q
B
B
A
A
s
s
x
3
2
2
1
1
1
1
1
Overall Heat Transfer Coefficient (U) :


 th
total R
UA
R
1
         
 
 
4
1 1
1
1
1
h
k
L
k
L
k
L
h
A
R
U
C
C
B
B
A
A
tot /
/
/
/
/ 





)
( 4
,
1
, 
 


 T
T
UA
T
UA
q overall

2021-8 18
 Series – Parallel Composite Wall

2021-8 19
 Series – Parallel Composite Wall
2
1
4
1
q
q
R
R
R
T
T
R
T
q
H
equv
E
th
total 








H
equv
E
total
R
T
T
R
T
T
R
T
T
q 4
3
3
2
2
1 





where
G
F
G
F
G
F
equv
R
R
R
.
R
R
R
R




1
1
1
H
F
E
G
F
G
total
R
R
R
T
T
R
R
R
q
q





 4
1
1
H
G
E
G
F
F
total
R
R
R
T
T
R
R
R
q
q





 4
1
2

2021-8 20
 Example 2.1

2021-8 21
 Example 2.1 (Cont.)

2021-8 22
 Example 2.1 (Cont.)
• Alternative thermal resistance network for Example
• We could also solve this problem by going to the other extreme and
assuming the surfaces parallel to the x-direction are adiabatic. The
thermal resistance network in this case will be as shown in Figure.

2021-8 23
 Thermal Contact Resistance
 


A
K
L
R
A
K
L
R
B
B
c
th
A
A
th ,
Where

c
th
R , thermal contact resistance (K/W)
W
K
m
q
T
T
R
x
B
A
c
th
2
, 






W
K
A
R
R c
th
c
th 


 ,
,
The value of thermal contact resistance depends on:
• surface roughness,
• material properties,
• temperature and pressure at the interface
• type of fluid trapped at the interface.

2021-8 24
 Thermal contact resistance is significant and can
even dominate the heat transfer for good heat
conductors such as metals, but can be disregarded for
poor heat conductors such as insulations.
hc thermal contact conductance
 The thermal contact resistance can be minimized
by applying
• a thermal grease such as silicon oil
• a better conducting gas such as helium or hydrogen
• a soft metallic foil such as tin, silver, copper, nickel,
or aluminum
(m2.K/W)
(b) interfacial fluid
(a) Vacuum interface
2.75
1.05
0.720
0.525
0.265
Air
Helium
Hydrogen
Silicone oil
Glycerin
10,000 kN/m2
0.7-4.0
0.1-0.5
0.2-0.4
0.2-0.4
100 kN/m2
6-25
1-10
1.5-3.5
1.5-5.0
Contact pressure
Stainless steel
Copper
Magnesium
Aluminum
Table 2.1 Thermal contact resistance for (a) metallic interfaces under
vacuum conditions and (b) aluminum - aluminum interface (10  m
surface roughness, 105 N/m2) with different interfacial fluids.
h
t
R 

,



2021-8 25
Heat Conduction in a Hollow Cylinder
 Fourier’s law of heat conduction for heat transfer through the cylindrical
layer can be expressed as
• (for steady-state, one-dimensional without heat generation,
constant thermal conductivity)
L
r
A
L
r
A
o
o
i
i


2
2


rL
A 
2

Conduction resistance of the cylinder layer

2021-8 26
 The thermal resistance network for a cylindrical shell subjected to
convection from both the inner and the outer sides.

2021-8 27
 Multilayered Cylinders

2021-8 28
 Multilayered Cylinders
4
4
3
4
2
3
1
2
1
1
4
,
1
,
2
1
2
)
/
(
ln
2
)
/
(
ln
2
)
/
(
ln
2
1
h
L
r
L
k
r
r
L
k
r
r
L
k
r
r
h
L
r
T
T
q
C
B
A
r









 

)
(
)
( 4
,
1
,
4
,
1
,
4
,
1
,











 T
T
A
U
T
T
A
U
R
T
T
q o
o
i
i
tot
r
4
4
1
3
4
1
2
3
1
1
2
1
1
1
1
ln
ln
ln
1
1
h
r
r
r
r
k
r
r
r
k
r
r
r
k
r
h
U
C
B
A





 



 1
4
4
3
3
2
2
1
1 )
( th
R
A
U
A
U
A
U
A
U
• To determine the temperature at any radius , q = constant
4
4
4
,
4
4
3
4
3
2
3
3
2
1
2
2
1
1
1
1
1
,
2
1
2
)
/
ln(
2
)
/
ln(
2
)
/
ln(
2
1
h
L
r
T
T
Lk
r
r
T
T
Lk
r
r
T
T
Lk
r
r
T
T
h
L
r
T
T
q
c
B
A
r






 










2021-8 29
 Example 2.2

2021-8 30
 Example 2.2 (cont.)

2021-8 31
Heat Conduction in a Hollow Sphere
 The thermal resistance network for a spherical shell subjected to
convection from both the inner and the outer sides.
th
o
i
i
o
o
i
i
o
o
i
i
o
R
T
k
r
r
r
r
T
T
r
r
T
T
k
r
r
q










4
)
(
)
(
4
Thermal Resistance
)
1
1
(
4
1
4
,
o
i
i
o
i
o
sphere
th
r
r
k
k
r
r
r
r
R 






2021-8 32
 Multilayered Sphere
)
4
(
1
)
4
(
1
2
4
2
,
2
1
1
,
r
h
R
r
h
R
o
conv
i
conv




3
4
3
3
4
3
,
2
3
2
2
3
2
,
1
2
1
1
2
1
,
4
,
4
,
4 k
r
r
r
r
R
k
r
r
r
r
R
k
r
r
r
r
R sph
sph
sph









2
4
2
1
4
4
r
A
r
A
o
i




o
i
total
A
h
k
r
r
r
r
k
r
r
r
r
k
r
r
r
r
A
h
R
2
3
4
3
3
4
2
3
2
2
3
1
2
1
1
2
1
1
4
4
4
1














th
R
T
q

2021-8 33
 Example 2.3.

2021-8 34
 Example 2.3. (cont.)

2021-8 35

2021-8 36

2021-8 37
Heat Conduction
 Example 2.4.
A company used a storage tank consists of a cylindrical section that has length and inner diameter of L=1.8 m
and Di = 1000 mm, respectively, and two hemispherical end sections. The tank is constructed from 20 mm-
thick glass (Pyrex, k = 1.4 W/ m K) and is exposed to ambient air for which the temperature is 27 C and the
convection coefficient is 10 W/ m 2 K. The tank is used to store heated oil, which maintains at a temperature
of 150 C and heat transfer coefficient is 120 W/m2.K. Radiation effects may be neglected. Sketch the
thermal circuit and Determine:
a)The electrical power that must be supplied to a heater submerged in the oil
b) The temperatures of outer surface of cylinder side only (TS,o).
If the price of electricity is $0.08/kWh, Determine the annual cost of heat loss per .
Known: Geometry of an oil storage tank. Temperatures of stored oil and environmental conditions.

2021-8 38
Heat Conduction
R4
qcyl
½ qspher
FIND: a) Heater power required , b) (TS,o) for cylinder side only
c) the annual cost of heat loss per year and the fraction of the hot oil energy cost of this company that is due to the heat loss
from the tank.
SCHEMATIC:
R4 R5
R6
R1 R2 R3
qsphe
qcyl
T∞,i
T∞,o

2021-8 39
Heat Conduction
sphere
cyl
hemi
cyl q
q
q
2
q
q 



3
2
1
o
,
i
,
cyl
,
th
o
,
i
,
cyl
R
R
R
T
T
R
T
T
q











o
o
i
o
i
i
o
,
i
,
cyl
h
L
r
2
1
r
/
r
ln
kL
2
1
h
L
r
2
1
T
T
q









10
x
2
x
52
.
0
x
2
1
5
.
0
/
52
.
0
ln
2
x
4
.
1
x
2
1
120
x
2
x
5
.
0
x
2
1
27
150
qcyl







Watt
1
.
6522
01886
.
0
123
0153
.
0
10
x
229
.
2
10
x
326
.
1
27
150
q
3
3
cyl 







6
5
4
o
,
i
,
sphere
,
th
o
,
i
,
sphere
R
R
R
T
T
R
T
T
q












2021-8 40
Heat Conduction
o
2
o
o
i
i
o
i
2
i
o
,
i
,
sphere
h
r
4
1
r
kr
4
r
r
h
r
4
1
T
T
q










10
)
52
.
0
(
4
1
52
.
0
x
5
.
0
x
4
.
1
x
4
5
.
0
52
.
0
120
)
5
.
0
(
4
1
27
150
q
2
2
sphere








Watt
8
.
3483
03645
.
0
123
0294
.
0
10
x
372
.
4
10
x
652
..
2
27
150
q
3
3
cyl 







The electrical power
= 10.0059 kW
The amount and cost of heat loss per year are
Cost of energy = (Amount of energy) (Unit cost)=
= (87651.68kWh) ($0.08/kWh) = $7012.13
Watt
9
.
10005
8
.
3483
1
.
6522
qtotal 


yr
/
kWh
68
.
87651
365
x
yr
/
h
24
kWx
9
.
005
..
10
time
x
q
Q 



2021-8 41
Heat Conduction
 Summary of Thermal Resistances
Thermal
Resistance
Equation for Heat
Flow
Geometry
Plane Wall
Long Hollow
cylinder
Hollow sphere
Convection surface
L
)
T
T
(
kA
q 2
1 
 KA
L
)
r
/
r
ln(
)
T
T
(
L
k
q
i
o
o
i 


2
kL
r
r i
o

2
)
/
ln(
i
o
o
i
i
o
r
r
)
T
T
(
k
r
r
q




4
k
r
r
r
r
i
o
i
o

4

)
T
T
(
hA
q s 

 hA
1

2021-8 42
Critical Radius of Insulation
 Critical Radius of Insulation
• Adding more insulation to a wall or to the attic always
decreases heat transfer since the heat transfer area is
constant, and adding insulation always increases the
thermal resistance of the wall without increasing the
convection resistance.
• In a cylindrical pipe or a spherical shell, the additional
insulation increases the conduction resistance of the
insulation layer but decreases the convection resistance
of the surface because of the increase in the outer
surface area for convection.
• The heat transfer from the pipe may increase or
decrease, depending on which effect dominates.

2021-8 43
Critical Radius of Insulation
 The critical radius of insulation for a cylindrical body
 The critical radius of insulation for a spherical body
• Insulating hot-water pipes or hot-water tanks
• Electric wires
• The rate of heat transfer from the cylinder increases with the addition of
insulation for r2 < rcr, reaches a maximum when r2 = rcr, and starts to
decrease for r2 > rcr.
• Thus, insulating the pipe may actually increase the rate of heat transfer
from the pipe instead of decreasing it when r2 < rcr.

Part B: One-Dimensional, Steady-State Conduction
with Thermal Energy Generation

2021-8 45
Heat Generation in a Solid
Heat Generation
• Examples:
 electrical energy being converted to heat,
 fuel elements of nuclear reactors,
 exothermic chemical reactions.
• Heat generation is a volumetric phenomenon.
• The rate of heat generation units : W/m3 or Btu/h·ft3.
• The rate of heat generation in a medium may vary with time
as well as position within the medium.

2021-8 46
Heat Generation in a Solid
e
gen
gen R
I
E
q 2

 
)
/
( 3
2
m
W
Volume
q
V
R
I
V
E
q
gen
e
gen





Where Re = electrical resistance 
c
e
A
L
R 

where ;
 = resistivity of the metal ( .m) in data book table A.14;
L = length of the wire or bar (m);
Ac = Cross-Sectional area of the wire or bar (m2 )
bar
gular
rec
for
t
b
A
rod
or
wire
for
d
A
c
c
tan
4
/
2

 
t = thickness of bar and b is width of bar
 Electrical Energy
The source may be uniformly distributed, as in the conversion from electrical to thermal energy (Ohmic heating):

2021-8 47
Heat Generation in Plane Wall
q1
q2
0

dx
dT
s
s
s T
T
T 
 2
1
Twice integration
1
C
k
x
q
dx
dT




Steady state
, B.C: at x=0 → dT/dx=0 , 0
1 
C
2
2
2
C
k
x
q
T 



s
s
s T
T
T
T
L
x 



 2
,
1
,
k
L
q
T
C s
2
2
2




Second integration
Temperature distribution
    )
1
(
2
2 2
2
2
2
2
L
x
k
L
q
T
x
L
k
q
T
T s
s
x 







a) Symmetrical boundary conditions
b) Adiabatic surface at mid-plane
q
L

2021-8 48
• maximum temperature exist at the midplane
max
0 T
T
T
x o 



k
L
q
T
T
T s
o
2
2
max





• To find surface temperature
conv
gen q
q 
)
)(
2
( 

 T
T
A
h
volume
x
q s

)
)(
2
(
)
2
( 

 T
T
A
h
LA
q s

h
L
q
T
Ts


 
• The rate of heat transfer from each surface
2
1
2
q
q
volume
x
q
R
I
qgen 


 
AL
q
k
L
q
kA
dx
dT
kA
q
L
x









)
(
1
AL
q
k
L
q
kA
dx
dT
kA
q
L
x







)
(
2
volume
x
q
L
A
q
q
q
qgen

 



 )
2
(
2
1

2021-8 49
c) Asymmetrical boundary conditions for Ts,1> Ts,2
xm
0

 m
x
x
dx
dT
q2
q1
qgen
L
Find
• The temperature distribution equation
• Value and position (location) of Tmax, xm
• The rate of heat transfer at each surface of the end q1 and q2
k
q
dx
T
d 


2
2
, Twice integration
1
C
k
x
q
dx
dT




  2
1
2
2
C
x
C
k
x
q
T x 




BC: at 1
2
1
0 ,
, s
s T
C
T
T
x 




at 


 2
,
s
T
T
L
x
1
1
2
2
2
,
, s
s T
L
C
k
L
q
T 




L
T
T
k
L
q
C
s
s 1
2
1
2
,
, 




-------(1)

2021-8 50
Sub. The values of C1 and C2 in Eqn. 1
• The temperature distribution
  1
1
2
2
2
2
,
,
, )
( s
s
s
x T
k
L
q
T
T
L
x
k
x
q
T 







• The position of maximum temperature at x= xm
0



dx
dT
x
x
at m
q
kC
x
C
k
x
q
m
m

 1
1
0 




• Sub: in Eqn. (2)
max
T
T 

• The rate of heat transfer
1
0
1 kAC
dx
dT
kA
q
x



)
k
L
q
(- 1
2 C
kA
dx
dT
kA
q
L
x







volume
x
q
q
q
q
q gen
total 



 2
1
xm
0

 m
x
x
dx
dT
q2
q1
qgen
L

2021-8 51
 Example 2.5.
Plane wall of material A with internal heat generation is insulated
on one side and bounded by a second wall of material B, which is
without heat generation and is subjected to convection cooling.
Find:
1) Sketch of steady-state temperature distribution in the composite.
2) Inner T1 and outer surface To temperatures of the composite.
Sol:
qgen
A
gen L
A
q
A
of
volume
x
q
q 
 

C
x
x
T
T
x
X
q
h
T
T
K
L
T
T
h
K
L
T
T
R
R
T
T
L
q
q
o
B
B
B
B
conv
B
cond
A
105
1000
75000
30
1000
05
.
0
10
5
.
1
30
1000
1
30
05
.
0
10
5
.
1
1
1
6
2
2
6
2
2
1
1
,
1






















 




2021-8 52
qgen
C
x
q
x
h
k
L
T
T o
B
B
115
85
30
75000
1000
1
150
02
.
0
30
1
1 





















 
• The maximum temperature To
C
x
x
x
T
T
K
L
q
T
T
T
o
A
A
o
0
2
6
max
2
1
max
140
25
115
75
2
)
05
.
0
(
10
5
.
1
115
2











2021-8 53
Heat Generation in Radial Systems
Cylindrical (Tube) Wall Spherical Wall (Shell)
Solid Cylinder (Circular Rod) Solid Sphere

2021-8 54
Steady state and 1-D
with heat generation
 Solid Cylinder
0
1








k
q
dr
dT
r
dr
d
r

k
r
q
dr
dT
r
dr
d 








Tmax
Ts
By integration
1
2
2
C
k
r
q
dr
dT
r 







 
B.C at 0
0
0 1 



 C
dr
dT
r
k
r
q
dr
dT
2


 Second integration
2
2
4
C
k
r
q
T 



s
o T
T
r
r 


k
r
q
T
C s
4
2
0
2




At

2021-8 55
)
( 2
2
4
r
r
k
q
T
T o
s
r 



At max
T
T
r 

 0
k
r
q
T
T
T o
s
o
4
2




max
• To find Ts
conv
gen q
q 
)
)(
2
(
2


 T
T
L
r
h
L
r
x
q s
o
o 


h
r
q
T
T o
s
2


 
• The maximum temperature
Tmax
Ts

2021-8 56
 Hollow Cylinder
0
1








k
q
dr
dT
r
dr
d
r

k
r
q
dr
dT
r
dr
d 








1
2
2
C
k
r
q
dr
dT
r 







 
Second integration
By integration
2
1
2
4
C
r
C
k
r
q
T 


 ln

B.C: at
k
r
q
C
dr
dT
r
r o
o
2
0
2
1







0
T
T
r
r o 


at
i
i
T
T
r
r 


or

2021-8 57
• To find rate of heat transfer
conv
gen q
q 
r
r
k
r
q
r
r
k
q
T
T o
o
o
o ln
)
(
2
4
2
2
2 




















i
i
i
i
r
r r
C
k
r
q
k
dr
dT
r
k
q 1
2
2
2
2



)
( 2
2
2
2
2
2
2 i
o
i
o
i
i
r
r
r
r
q
r
r
k
q
k
r
q
k
q 
















 


)
(
)
( , 


 T
T
L
r
h
L
r
r
q i
s
i
i
o 
 2
2
2

)
(
)
(
, 



T
T
r
r
r
q
h
i
s
i
i
o
2
2
2


2021-8 58
 Solid Sphere
0
2








k
q
dr
dT
r
dr
d 
2
r
1
1
3
2
3
1
C
k
r
q
dr
dT
r




By integration

2021-8 59
at
)
( 2
2
6
r
r
k
q
T
T o
s
r 



max
T
T
r 

 0
k
r
q
T
T
T o
s
o
6
2
max




• To find Ts
conv
gen q
q 
)
)(
( 

 T
T
r
h
r
x
q s
o
o
2
3
4
3
4



h
r
q
T
T o
s
3


 

2021-8 60
 Example 2.6.
Known: Long rod experiencing uniform volumetric generation encapsulated by a circular sleeve exposed to convection.
Find: (a) Temperature at the interface between rod and sleeve and on the outer surface, (b) Temperature at center of rod.
Assumption: (1) One-dimensional radial conduction in rod and sleeve, (2) Steady-state conditions, (3) Uniform volumetric
generation in rod, (4) Negligible contact resistance between rod and sleeve.
Analysis: (a) Construct a thermal circuit for the sleeve,

2021-8 61

Ch 2 - Steady state 1-D, Heat conduction.pdf

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