Heat & Mass Transfer Week 05 Instructor: Mr....
104
Heat & Mass Transfer Week_05 Instructor: Mr. Adnan Qamar Mechanical Engineering Department 1
Transcript of Heat & Mass Transfer Week 05 Instructor: Mr....
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)
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Logarithmic Mean Area for the Hollow Cylinder and Plane Wall
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Logarithmic Mean Area for the Hollow Cylinder and Plane Wall
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Logarithmic Mean Area for the Hollow Cylinder and Plane Wall
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One Dimensional Steady State Heat Conduction
Logarithmic Mean Area for the Hollow Sphere and Plane Wall
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Logarithmic Mean Area for the Hollow Sphere and Plane Wall
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One Dimensional Steady State Heat Conduction
Variable Thermal Conductivity
➢ Effects of Various Parameters on the Thermal Conductivity of Solids
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Effects of Various Parameters on the Thermal Conductivity of Solids
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One Dimensional Steady State Heat Conduction
Plane Wall (Slab) with variable Thermal Conductivity (kT)
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Plane Wall (Slab) with variable Thermal Conductivity (kT)
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Plane Wall (Slab) with variable Thermal Conductivity (kT)
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Plane Wall (Slab) with variable Thermal Conductivity (kT)
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Plane Wall (Slab) with variable Thermal Conductivity (kT)
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One Dimensional Steady State Heat Conduction
Hollow Cylinder with variable Thermal Conductivity (kT)
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Hollow Cylinder with variable Thermal Conductivity (kT)
v
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Hollow Cylinder with variable Thermal Conductivity (kT)
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One Dimensional Steady State Heat Conduction
Spherical System with variable Thermal Conductivity (kT)
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Spherical System with variable Thermal Conductivity (kT)
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Spherical System with variable Thermal Conductivity (kT)
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Spherical System with variable Thermal Conductivity (kT)
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One Dimensional Steady State Heat Conduction
Heat Conduction Through A Composite Wall
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Heat Conduction Through A Composite Wall
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Heat Conduction Through A Composite Wall
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Heat Conduction Through A Composite Wall
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One Dimensional Steady State Heat Conduction
Thermal Contact Resistance
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Thermal Contact Resistance
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One Dimensional Steady State Heat Conduction
Overall Heat Transfer Coefficient (U)
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Overall Heat Transfer Coefficient (U)
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Overall Heat Transfer Coefficient (U)
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Overall Heat Transfer Coefficient (U)
Then compare equation (15) & (16);
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One Dimensional Steady State Heat Conduction
Series and Parallel One Dimensional Heat Transfer Through A Composite Wall
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Series and Parallel One Dimensional Heat Transfer Through A Composite Wall
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Series and Parallel One Dimensional Heat Transfer Through A Composite Wall
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Series and Parallel One Dimensional Heat Transfer Through A Composite Wall
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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
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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;
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)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.
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)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
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Overall Heat Transfer Through a Composite Wall
Convection Heat Transfer 1:
Conduction Heat Transfer 1:
Conduction Heat Transfer 2:
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)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;
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)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.
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)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
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Overall Heat Transfer Through a Hollow Cylinder
Convection Heat Transfer 1:
Conduction Heat Transfer:
Convection Heat Transfer 2:
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)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;
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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.
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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
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Overall Heat Transfer Through a Composite Hollow Cylinder
Convection Heat Transfer 1:
Conduction Heat Transfer 1:
Conduction Heat Transfer 2:
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)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;
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)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.
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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
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Overall Heat Transfer Through a Hollow Sphere
Convection Heat Transfer 1:
Conduction Heat Transfer:
Convection Heat Transfer 2:
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)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;
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)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.
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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
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Overall Heat Transfer Through a Hollow Sphere
Convection Heat Transfer 1:
Conduction Heat Transfer 1:
Conduction Heat Transfer 2:
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)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;
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)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.
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)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
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Critical Radius of Insulation
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Critical Radius of Insulation
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Critical Radius of Insulation
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Critical Radius of Insulation
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Critical Radius of Insulation
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Critical Radius of Insulation
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One Dimensional Steady State Heat Conduction
Critical Radius of Insulation For Cylinder
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Conduction Heat Transfer-Class Problems
Example 5.1:
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
Under Steady state, heat flux is constant throughout the wall and each layer
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
Example 5.2:
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
Example 5.3:
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
Example 5.4:
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
Example 5.5:
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
88
Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
Example 5.6:
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
Example 5.7:
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
99
Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
Example 5.8:
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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Conduction Heat Transfer-Class Problems
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