Post on 26-Aug-2020
Heat & Mass Transfer
Week_05
Instructor: Mr. Adnan Qamar
Mechanical Engineering Department
1
One Dimensional Steady State Heat Conduction
Logarithmic Mean Area for the Hollow Cylinder and Plane Wall
Let us consider a cylinder and a slab both made of the same material of uniform
thermal conductivity (k)
2
Logarithmic Mean Area for the Hollow Cylinder and Plane Wall
3
Logarithmic Mean Area for the Hollow Cylinder and Plane Wall
4
Logarithmic Mean Area for the Hollow Cylinder and Plane Wall
5
One Dimensional Steady State Heat Conduction
Logarithmic Mean Area for the Hollow Sphere and Plane Wall
6
Logarithmic Mean Area for the Hollow Sphere and Plane Wall
7
One Dimensional Steady State Heat Conduction
Variable Thermal Conductivity
➢ Effects of Various Parameters on the Thermal Conductivity of Solids
8
Effects of Various Parameters on the Thermal Conductivity of Solids
9
One Dimensional Steady State Heat Conduction
Plane Wall (Slab) with variable Thermal Conductivity (kT)
10
Plane Wall (Slab) with variable Thermal Conductivity (kT)
11
Plane Wall (Slab) with variable Thermal Conductivity (kT)
12
Plane Wall (Slab) with variable Thermal Conductivity (kT)
13
Plane Wall (Slab) with variable Thermal Conductivity (kT)
14
One Dimensional Steady State Heat Conduction
Hollow Cylinder with variable Thermal Conductivity (kT)
15
Hollow Cylinder with variable Thermal Conductivity (kT)
v
16
Hollow Cylinder with variable Thermal Conductivity (kT)
17
One Dimensional Steady State Heat Conduction
Spherical System with variable Thermal Conductivity (kT)
18
Spherical System with variable Thermal Conductivity (kT)
19
Spherical System with variable Thermal Conductivity (kT)
20
Spherical System with variable Thermal Conductivity (kT)
21
One Dimensional Steady State Heat Conduction
Heat Conduction Through A Composite Wall
22
Heat Conduction Through A Composite Wall
23
Heat Conduction Through A Composite Wall
24
Heat Conduction Through A Composite Wall
25
One Dimensional Steady State Heat Conduction
Thermal Contact Resistance
26
Thermal Contact Resistance
27
One Dimensional Steady State Heat Conduction
Overall Heat Transfer Coefficient (U)
28
Overall Heat Transfer Coefficient (U)
29
Overall Heat Transfer Coefficient (U)
30
Overall Heat Transfer Coefficient (U)
Then compare equation (15) & (16);
31
One Dimensional Steady State Heat Conduction
Series and Parallel One Dimensional Heat Transfer Through A Composite Wall
32
Series and Parallel One Dimensional Heat Transfer Through A Composite Wall
33
Series and Parallel One Dimensional Heat Transfer Through A Composite Wall
34
Series and Parallel One Dimensional Heat Transfer Through A Composite Wall
35
Overall Heat Transfer Through a Plane Wall
Consider a plane wall with thickness “dx” as shown in Fig;
Fig: Overall heat transfer through a plane wall
36
Overall Heat Transfer Through a Plane Wall
Convection Heat Transfer 1:
Conduction Heat Transfer:
Convection Heat Transfer 2:
Cross multiplying and adding equations 1,2 &3 we will have;
37
)1(1
)( 11
1
11111
conv
convR
TT
Ah
TTTTAhQ
)2()( 312121
cond
condR
TT
kA
dx
TT
dx
TTkAQ
)3(1
)( 22
2
22222
conv
convR
TT
Ah
TTTTAhQ
21 convcondconv QQQQ
Overall Heat Transfer Through a Plane Wall
Where ΣR=1/UA, and U= overall heat transfer coefficient. It is the combined
effect of conduction and convection.
38
)4()(11 21
21
21
21
TTUA
R
TT
AhkA
dx
Ah
TTQ
222111
21
11 TTTTTT
AhQ
KA
dxQ
AhQ
21
21
)11
( TTAhKA
dx
AhQ
Overall Heat Transfer Through a Composite Wall
Consider a composite wall with thickness “dx1 & dx2” as shown in Fig;
Fig: Overall heat transfer through a composite wall
39
Overall Heat Transfer Through a Composite Wall
Convection Heat Transfer 1:
Conduction Heat Transfer 1:
Conduction Heat Transfer 2:
40
)1(1
)( 11
1
11111
conv
convR
TT
Ah
TTTTAhQ
)2()(
1
21
1
1
21
1
2111
cond
condR
TT
Ak
dx
TT
dx
TTAkQ
)3()(
2
32
2
2
32
2
3222
cond
condR
TT
Ak
dx
TT
dx
TTAkQ
Overall Heat Transfer Through a Composite Wall
Convection Heat Transfer 2:
Cross multiplying and adding equations 1,2, 3& 4, we will have;
41
)4(1
)(2
23
2
232322
conv
convR
TT
Ah
TTTTAhQ
2211 convcondcondconv QQQQQ
23322111
22
2
1
1
1
1...
1. TTTTTTTT
AhQ
AK
dxQ
AK
dxQ
AhQ
21
22
2
1
1
1
)11
( TTAhAk
dx
Ak
dx
AhQ
Overall Heat Transfer Through a Composite Wall
Where ΣR=1/UA, and U= overall heat transfer coefficient. It is the combined
effect of conduction and convection.
42
)5(11
21
22
2
1
1
1
21
R
TT
AhAk
dx
Ak
dx
Ah
TTQ
)6()( 21 TTUAQ
Overall Heat Transfer Through a Hollow Cylinder
Consider a hollow cylinder with length “L” as shown in Fig;
Fig: Overall heat transfer through a hollow cylinder
43
Overall Heat Transfer Through a Hollow Cylinder
Convection Heat Transfer 1:
Conduction Heat Transfer:
Convection Heat Transfer 2:
44
)1(1
)(1
11
11
1111111
conv
convR
TT
Ah
TTTTAhQ
)2(
ln.2
1)( 21
1
2
1221
cond
condR
TT
r
r
KL
TTTTkAQ
)3(1
)(2
22
22
2222222
conv
convR
TT
Ah
TTTTAhQ
Overall Heat Transfer Through a Hollow Cylinder
Cross multiplying and adding equations 1,2 & 3, we will have;
45
QQQQ convcondconv 21
222111
221
2
11
1.ln.
2
1.
1.
TTTTTT
AhQ
r
r
KLQ
AhQ
21
221
2
11
1ln.
2
11.
TT
Ahr
r
kLAhQ
)4(1
ln.2
1121
221
2
11
21
R
TT
Ahr
r
KLAh
TTQ
)5()( 21 TTUAQ
Overall Heat Transfer Through a Hollow Cylinder
Where ΣR=1/UA, and U= overall heat transfer coefficient. It is the combined
effect of conduction and convection.
Note: If heat is moving inside the fluid, then take area A1 and if the heat is
moving outside the fluid then take A2.
46
Overall Heat Transfer Through a Composite Hollow Cylinder
Consider a composite hollow cylinder with length “L” as shown in Fig;
Fig: Overall heat transfer through a composite hollow cylinder
47
Overall Heat Transfer Through a Composite Hollow Cylinder
Convection Heat Transfer 1:
Conduction Heat Transfer 1:
Conduction Heat Transfer 2:
48
)1(1
)(1
11
11
1111111
conv
convR
TT
Ah
TTTTAhQ
)2(
ln2
1)(
1
21
1
2
1
212111
cond
condR
TT
r
r
Lk
TTTTAkQ
)3(
ln2
1)(
2
32
2
3
2
323222
cond
condR
TT
r
r
Lk
TTTTAkQ
Overall Heat Transfer Through a Composite Hollow Cylinder
Convection Heat Transfer 2:
Cross multiplying and adding equations 1,2,3 & 4, we will have;
49
)4(1
)(2
23
32
2323322
conv
convR
TT
Ah
TTTTAhQ
QQQQQ convcondcondconv 2211
23322111
222
3
21
2
111
1.ln.
2
1.ln.
2
1.
1.
TTTTTTTT
AhQ
r
r
LkQ
r
r
LkQ
AhQ
Overall Heat Transfer Through a Composite Hollow Cylinder
Where ΣR=1/UA, and U= overall heat transfer coefficient. It is the combinedeffect of conduction and convection.
Note: If heat is moving inside the fluid, then take area A1 and if the heat ismoving outside the fluid then take A3.
50
21
322
3
21
2
111
1ln
2
1ln
2
11
TT
Ahr
r
LKr
r
LKAhQ
)5(
2
1ln
2
1ln
2
1
2.
1
322
3
21
2
111
21
Lrhr
r
LKr
r
LKLrh
TTQ
)6()( 21 TTUAQ
Overall Heat Transfer Through a Hollow Sphere
Consider a composite hollow sphere with radius “r1 & r2” as shown in Fig;
Fig: Overall heat transfer through a hollow sphere
51
Overall Heat Transfer Through a Hollow Sphere
Convection Heat Transfer 1:
Conduction Heat Transfer:
Convection Heat Transfer 2:
52
)1(1
)(1
11
11
1111111
conv
convR
TT
Ah
TTTTAhQ
)2(
]11
[4
1)( 21
21
2121
conv
condR
TT
rrk
TTTTkAQ
)3(1
)( 22
22
2222222
conv
convR
TT
Ah
TTTTAhQ
Overall Heat Transfer Through a Hollow Sphere
Cross multiplying and adding equations 1,2 & 3, we will have;
53
)4(1
)11
(4
1121
222111
21
R
TT
AhrrkAh
TTQ
222111
222111
1)
11(
4
11
TTTTTT
AhQ
rrkQ
AhQ
21
222111
1)
11(
4
11
TT
AhrrkAhQ
)5()( 21 TTUAQ
Overall Heat Transfer Through a Hollow Sphere
Where ΣR=1/UA, and U= overall heat transfer coefficient. It is the combined
effect of conduction and convection.
Note: If heat is moving inside the fluid, then take area A1 and if the heat is
moving outside the fluid then take A3.
54
Overall Heat Transfer Through a Composite Hollow Sphere
Consider a composite hollow sphere with radius “r1 & r2” as shown in Fig;
Fig: Overall heat transfer through a composite hollow sphere
55
Overall Heat Transfer Through a Hollow Sphere
Convection Heat Transfer 1:
Conduction Heat Transfer 1:
Conduction Heat Transfer 2:
56
)1(1
)(1
11
11
1111111
conv
convR
TT
Ah
TTTTAhQ
)2(
]11
[4
1)( 21
211
21211
conv
condR
TT
rrk
TTTTAkQ
)3(
]11
[4
1)(
2
2
32
322
32322
cond
condR
TT
rrk
TTTTAkQ
Overall Heat Transfer Through a Composite Hollow Sphere
Convection Heat Transfer 2:
Cross multiplying and adding equations 1,2,3 & 4, we will have;
57
)4(1
)( 23
32
2323322
conv
convR
TT
Ah
TTTTAhQ
QQQQQ convcondcondconv 2211
23322111
3232221111
1.
11
4
1.
11
4
1.
1.
TTTTTTTT
AhQ
rrkQ
rrkQ
AhQ
21
3232221111
1]
11[
4
1]
11[
4
11
TT
AhrrkrrkAhQ
Overall Heat Transfer Through a Composite Hollow Sphere
Where ΣR=1/UA, and U= overall heat transfer coefficient. It is the combined
effect of conduction and convection.
Note: If heat is moving inside the fluid, then take area A1 and if the heat is
moving outside the fluid then take A3.
58
)5(1
)11
(4
1)
11(
4
1121
2232221111
21
R
TT
AhrrKrrKAh
TTQ
)6()( 21 TTUAQ
One Dimensional Steady State Heat Conduction
Critical Radius of Insulation
59
Critical Radius of Insulation
60
Critical Radius of Insulation
61
Critical Radius of Insulation
62
Critical Radius of Insulation
63
Critical Radius of Insulation
64
Critical Radius of Insulation
65
One Dimensional Steady State Heat Conduction
Critical Radius of Insulation For Cylinder
66
Conduction Heat Transfer-Class Problems
Example 5.1:
67
Conduction Heat Transfer-Class Problems
68
Conduction Heat Transfer-Class Problems
Under Steady state, heat flux is constant throughout the wall and each layer
69
Conduction Heat Transfer-Class Problems
70
Conduction Heat Transfer-Class Problems
71
Conduction Heat Transfer-Class Problems
Example 5.2:
72
Conduction Heat Transfer-Class Problems
73
Conduction Heat Transfer-Class Problems
74
Conduction Heat Transfer-Class Problems
75
Conduction Heat Transfer-Class Problems
Example 5.3:
76
Conduction Heat Transfer-Class Problems
77
Conduction Heat Transfer-Class Problems
78
Conduction Heat Transfer-Class Problems
79
Conduction Heat Transfer-Class Problems
80
Conduction Heat Transfer-Class Problems
Example 5.4:
81
Conduction Heat Transfer-Class Problems
82
Conduction Heat Transfer-Class Problems
83
Conduction Heat Transfer-Class Problems
84
Conduction Heat Transfer-Class Problems
Example 5.5:
85
Conduction Heat Transfer-Class Problems
86
Conduction Heat Transfer-Class Problems
87
Conduction Heat Transfer-Class Problems
88
Conduction Heat Transfer-Class Problems
89
Conduction Heat Transfer-Class Problems
Example 5.6:
90
Conduction Heat Transfer-Class Problems
91
Conduction Heat Transfer-Class Problems
92
Conduction Heat Transfer-Class Problems
93
Conduction Heat Transfer-Class Problems
94
Conduction Heat Transfer-Class Problems
95
Conduction Heat Transfer-Class Problems
96
Conduction Heat Transfer-Class Problems
Example 5.7:
97
Conduction Heat Transfer-Class Problems
98
Conduction Heat Transfer-Class Problems
99
Conduction Heat Transfer-Class Problems
100
Conduction Heat Transfer-Class Problems
Example 5.8:
101
Conduction Heat Transfer-Class Problems
102
Conduction Heat Transfer-Class Problems
103
Conduction Heat Transfer-Class Problems
104