Chapter 1: Basic Concepts - Chapter Notes, Heat Transfer, Engineering Notes | EduRev

: Chapter 1: Basic Concepts - Chapter Notes, Heat Transfer, Engineering Notes | EduRev

 Page 1


BOOKCOMP, Inc. — John Wiley & Sons / Page 1 / 2nd Proofs /HeatTransferHandbook / Bejan
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[First Page]
[1], (1)
Lines: 0 to 95
———
11.25609pt PgVar
———
Normal Page
* PgEnds: PageBreak
[1], (1)
CHAPTER 1
Basic Concepts
ALLAND.KRAUS
UniversityofAkron
Akron,Ohio
1.1 Heat transfer fundamentals
1.1.1 Introduction
1.1.2 Conduction heat transfer
One-dimensional conduction
One-dimensional conduction with internal heat generation
1.1.3 Spreading resistance
1.1.4 Interface–contact resistance
1.1.5 Lumped-capacity heating and cooling
1.1.6 Convective heat transfer
Heat transfer coef?cient
Dimensionless parameters
Natural convection
Forced convection
1.1.7 Phase-change heat transfer
1.1.8 Finned surfaces
1.1.9 Flow resistance
1.1.10 Radiative heat transfer
1.2 Coordinate systems
1.2.1 Rectangular (Cartesian) coordinate system
1.2.2 Cylindrical coordinate system
1.2.3 Spherical coordinate system
1.2.4 General curvilinear coordinates
1.3 Continuity equation
1.4 Momentum and the momentum theorem
1.5 Conservation of energy
1.6 Dimensional analysis
1.6.1 Friction loss in pipe ?ow
1.6.2 Summary of dimensionless groups
1.7 Units
1.7.1 SI system (Syst` eme International d’Unit´ es)
1.7.2 English engineering system (U.S. customary system)
1.7.3 Conversion factors
Nomenclature
References
1
Page 2


BOOKCOMP, Inc. — John Wiley & Sons / Page 1 / 2nd Proofs /HeatTransferHandbook / Bejan
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[First Page]
[1], (1)
Lines: 0 to 95
———
11.25609pt PgVar
———
Normal Page
* PgEnds: PageBreak
[1], (1)
CHAPTER 1
Basic Concepts
ALLAND.KRAUS
UniversityofAkron
Akron,Ohio
1.1 Heat transfer fundamentals
1.1.1 Introduction
1.1.2 Conduction heat transfer
One-dimensional conduction
One-dimensional conduction with internal heat generation
1.1.3 Spreading resistance
1.1.4 Interface–contact resistance
1.1.5 Lumped-capacity heating and cooling
1.1.6 Convective heat transfer
Heat transfer coef?cient
Dimensionless parameters
Natural convection
Forced convection
1.1.7 Phase-change heat transfer
1.1.8 Finned surfaces
1.1.9 Flow resistance
1.1.10 Radiative heat transfer
1.2 Coordinate systems
1.2.1 Rectangular (Cartesian) coordinate system
1.2.2 Cylindrical coordinate system
1.2.3 Spherical coordinate system
1.2.4 General curvilinear coordinates
1.3 Continuity equation
1.4 Momentum and the momentum theorem
1.5 Conservation of energy
1.6 Dimensional analysis
1.6.1 Friction loss in pipe ?ow
1.6.2 Summary of dimensionless groups
1.7 Units
1.7.1 SI system (Syst` eme International d’Unit´ es)
1.7.2 English engineering system (U.S. customary system)
1.7.3 Conversion factors
Nomenclature
References
1
BOOKCOMP, Inc. — John Wiley & Sons / Page 2 / 2nd Proofs /HeatTransferHandbook / Bejan
2 BASICCONCEPTS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[2], (2)
Lines: 95 to 139
———
6.28009pt PgVar
———
Normal Page
PgEnds: T
E
X
[2], (2)
1.1 HEAT TRANSFER FUNDAMENTALS
1.1.1 Introduction
Practitioners of the thermal arts and sciences generally deal with four basic thermal
transport modes: conduction, convection, phase change, and radiation. The process
by which heat diffuses through a solid or a stationary ?uid is termedheatconduction.
Situations in which heat transfer from a wetted surface is assisted by the motion of
the ?uid give rise to heat convection, and when the ?uid undergoes a liquid–solid
or liquid–vapor state transformation at or very near the wetted surface, attention is
focused on thisphase-changeheattransfer. The exchange of heat between surfaces,
or between a surface and a surrounding ?uid, by long-wavelength electromagnetic
radiation is termedthermalheatradiation.
It is our intent in this section to describe brie?y these modes of heat transfer, with
emphasis on an important parameter known as thethermalresistance to heat transfer.
Simple examples are given for illustration; detailed descriptions of the same topics
are presented in specialized chapters.
1.1.2 Conduction Heat Transfer
One-Dimensional Conduction Thermal diffusion through solids is governed
byFourier’slaw, which in one-dimensional form is expressible as
q=-kA
dT
dx
(W) (1.1)
where q is the heat current, k the thermal conductivity of the medium, A the cross-
sectional area for heat ?ow, and dT/dx the temperature gradient, which, because it
is negative, requires insertion of the minus sign in eq. (1.1) to assure a positive heat
?ow q. The temperature difference resulting from the steady-state diffusion of heat
is thus related to the thermal conductivity of the material, the cross-sectional area A,
and the path length L (Fig. 1.1), according to
(T
1
- T
2
)
cd
= q
L
kA
(K) (1.2)
The form of eq. (1.2), where k and A are presumed constant, suggests that in a way
that is analogous to Ohm’s law governing electrical current ?ow through a resistance,
it is possible to de?ne a conduction thermal resistance as
R
cd
=
T
1
- T
2
q
=
L
kA
(K/W) (1.3)
One-Dimensional Conduction with Internal Heat Generation Situations
in which a solid experiences internal heat generation, such as that produced by the
?ow of an electric current, give rise to more complex governing equations and require
Page 3


BOOKCOMP, Inc. — John Wiley & Sons / Page 1 / 2nd Proofs /HeatTransferHandbook / Bejan
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[First Page]
[1], (1)
Lines: 0 to 95
———
11.25609pt PgVar
———
Normal Page
* PgEnds: PageBreak
[1], (1)
CHAPTER 1
Basic Concepts
ALLAND.KRAUS
UniversityofAkron
Akron,Ohio
1.1 Heat transfer fundamentals
1.1.1 Introduction
1.1.2 Conduction heat transfer
One-dimensional conduction
One-dimensional conduction with internal heat generation
1.1.3 Spreading resistance
1.1.4 Interface–contact resistance
1.1.5 Lumped-capacity heating and cooling
1.1.6 Convective heat transfer
Heat transfer coef?cient
Dimensionless parameters
Natural convection
Forced convection
1.1.7 Phase-change heat transfer
1.1.8 Finned surfaces
1.1.9 Flow resistance
1.1.10 Radiative heat transfer
1.2 Coordinate systems
1.2.1 Rectangular (Cartesian) coordinate system
1.2.2 Cylindrical coordinate system
1.2.3 Spherical coordinate system
1.2.4 General curvilinear coordinates
1.3 Continuity equation
1.4 Momentum and the momentum theorem
1.5 Conservation of energy
1.6 Dimensional analysis
1.6.1 Friction loss in pipe ?ow
1.6.2 Summary of dimensionless groups
1.7 Units
1.7.1 SI system (Syst` eme International d’Unit´ es)
1.7.2 English engineering system (U.S. customary system)
1.7.3 Conversion factors
Nomenclature
References
1
BOOKCOMP, Inc. — John Wiley & Sons / Page 2 / 2nd Proofs /HeatTransferHandbook / Bejan
2 BASICCONCEPTS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[2], (2)
Lines: 95 to 139
———
6.28009pt PgVar
———
Normal Page
PgEnds: T
E
X
[2], (2)
1.1 HEAT TRANSFER FUNDAMENTALS
1.1.1 Introduction
Practitioners of the thermal arts and sciences generally deal with four basic thermal
transport modes: conduction, convection, phase change, and radiation. The process
by which heat diffuses through a solid or a stationary ?uid is termedheatconduction.
Situations in which heat transfer from a wetted surface is assisted by the motion of
the ?uid give rise to heat convection, and when the ?uid undergoes a liquid–solid
or liquid–vapor state transformation at or very near the wetted surface, attention is
focused on thisphase-changeheattransfer. The exchange of heat between surfaces,
or between a surface and a surrounding ?uid, by long-wavelength electromagnetic
radiation is termedthermalheatradiation.
It is our intent in this section to describe brie?y these modes of heat transfer, with
emphasis on an important parameter known as thethermalresistance to heat transfer.
Simple examples are given for illustration; detailed descriptions of the same topics
are presented in specialized chapters.
1.1.2 Conduction Heat Transfer
One-Dimensional Conduction Thermal diffusion through solids is governed
byFourier’slaw, which in one-dimensional form is expressible as
q=-kA
dT
dx
(W) (1.1)
where q is the heat current, k the thermal conductivity of the medium, A the cross-
sectional area for heat ?ow, and dT/dx the temperature gradient, which, because it
is negative, requires insertion of the minus sign in eq. (1.1) to assure a positive heat
?ow q. The temperature difference resulting from the steady-state diffusion of heat
is thus related to the thermal conductivity of the material, the cross-sectional area A,
and the path length L (Fig. 1.1), according to
(T
1
- T
2
)
cd
= q
L
kA
(K) (1.2)
The form of eq. (1.2), where k and A are presumed constant, suggests that in a way
that is analogous to Ohm’s law governing electrical current ?ow through a resistance,
it is possible to de?ne a conduction thermal resistance as
R
cd
=
T
1
- T
2
q
=
L
kA
(K/W) (1.3)
One-Dimensional Conduction with Internal Heat Generation Situations
in which a solid experiences internal heat generation, such as that produced by the
?ow of an electric current, give rise to more complex governing equations and require
BOOKCOMP, Inc. — John Wiley & Sons / Page 3 / 2nd Proofs /HeatTransferHandbook / Bejan
HEATTRANSFERFUNDAMENTALS 3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[3], (3)
Lines: 139 to 173
———
-2.4009pt PgVar
———
Normal Page
PgEnds: T
E
X
[3], (3)
T
x
Figure1.1 Heat transfer by conduction through a slab.
greater care in obtaining the appropriate temperature differences. The axial temper-
ature variation in the slim, internally heated conductor shown in Fig. 1.2 is found to
equal
T = T
o
+ q
g
L
2
2k

x
L
-

x
L

2

where T
o
is the edge temperature. When the two ends are cooled to an identical
temperature, and when the volumetric heat generation rate q
g
(W/m
3
) is uniform
throughout, the peak temperature is developed at the center of the solid and is given
by
T
max
= T
o
+ q
g
L
2
8k
(K) (1.4)
Alternatively, because q
g
is the volumetric heat generation q
g
= q/LWd, the
center–edge temperature difference can be expressed as
T
max
- T
o
= q
L
2
8kLWd
= q
L
8kA
(1.5)
where the cross-sectional area A is the product of the width W and the thicknessd.
An examination of eq. (1.5) reveals that the thermal resistance of a conductor with a
distributed heat input is only one-fourth that of a structure in which all of the heat is
generated at the center.
Page 4


BOOKCOMP, Inc. — John Wiley & Sons / Page 1 / 2nd Proofs /HeatTransferHandbook / Bejan
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[First Page]
[1], (1)
Lines: 0 to 95
———
11.25609pt PgVar
———
Normal Page
* PgEnds: PageBreak
[1], (1)
CHAPTER 1
Basic Concepts
ALLAND.KRAUS
UniversityofAkron
Akron,Ohio
1.1 Heat transfer fundamentals
1.1.1 Introduction
1.1.2 Conduction heat transfer
One-dimensional conduction
One-dimensional conduction with internal heat generation
1.1.3 Spreading resistance
1.1.4 Interface–contact resistance
1.1.5 Lumped-capacity heating and cooling
1.1.6 Convective heat transfer
Heat transfer coef?cient
Dimensionless parameters
Natural convection
Forced convection
1.1.7 Phase-change heat transfer
1.1.8 Finned surfaces
1.1.9 Flow resistance
1.1.10 Radiative heat transfer
1.2 Coordinate systems
1.2.1 Rectangular (Cartesian) coordinate system
1.2.2 Cylindrical coordinate system
1.2.3 Spherical coordinate system
1.2.4 General curvilinear coordinates
1.3 Continuity equation
1.4 Momentum and the momentum theorem
1.5 Conservation of energy
1.6 Dimensional analysis
1.6.1 Friction loss in pipe ?ow
1.6.2 Summary of dimensionless groups
1.7 Units
1.7.1 SI system (Syst` eme International d’Unit´ es)
1.7.2 English engineering system (U.S. customary system)
1.7.3 Conversion factors
Nomenclature
References
1
BOOKCOMP, Inc. — John Wiley & Sons / Page 2 / 2nd Proofs /HeatTransferHandbook / Bejan
2 BASICCONCEPTS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[2], (2)
Lines: 95 to 139
———
6.28009pt PgVar
———
Normal Page
PgEnds: T
E
X
[2], (2)
1.1 HEAT TRANSFER FUNDAMENTALS
1.1.1 Introduction
Practitioners of the thermal arts and sciences generally deal with four basic thermal
transport modes: conduction, convection, phase change, and radiation. The process
by which heat diffuses through a solid or a stationary ?uid is termedheatconduction.
Situations in which heat transfer from a wetted surface is assisted by the motion of
the ?uid give rise to heat convection, and when the ?uid undergoes a liquid–solid
or liquid–vapor state transformation at or very near the wetted surface, attention is
focused on thisphase-changeheattransfer. The exchange of heat between surfaces,
or between a surface and a surrounding ?uid, by long-wavelength electromagnetic
radiation is termedthermalheatradiation.
It is our intent in this section to describe brie?y these modes of heat transfer, with
emphasis on an important parameter known as thethermalresistance to heat transfer.
Simple examples are given for illustration; detailed descriptions of the same topics
are presented in specialized chapters.
1.1.2 Conduction Heat Transfer
One-Dimensional Conduction Thermal diffusion through solids is governed
byFourier’slaw, which in one-dimensional form is expressible as
q=-kA
dT
dx
(W) (1.1)
where q is the heat current, k the thermal conductivity of the medium, A the cross-
sectional area for heat ?ow, and dT/dx the temperature gradient, which, because it
is negative, requires insertion of the minus sign in eq. (1.1) to assure a positive heat
?ow q. The temperature difference resulting from the steady-state diffusion of heat
is thus related to the thermal conductivity of the material, the cross-sectional area A,
and the path length L (Fig. 1.1), according to
(T
1
- T
2
)
cd
= q
L
kA
(K) (1.2)
The form of eq. (1.2), where k and A are presumed constant, suggests that in a way
that is analogous to Ohm’s law governing electrical current ?ow through a resistance,
it is possible to de?ne a conduction thermal resistance as
R
cd
=
T
1
- T
2
q
=
L
kA
(K/W) (1.3)
One-Dimensional Conduction with Internal Heat Generation Situations
in which a solid experiences internal heat generation, such as that produced by the
?ow of an electric current, give rise to more complex governing equations and require
BOOKCOMP, Inc. — John Wiley & Sons / Page 3 / 2nd Proofs /HeatTransferHandbook / Bejan
HEATTRANSFERFUNDAMENTALS 3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[3], (3)
Lines: 139 to 173
———
-2.4009pt PgVar
———
Normal Page
PgEnds: T
E
X
[3], (3)
T
x
Figure1.1 Heat transfer by conduction through a slab.
greater care in obtaining the appropriate temperature differences. The axial temper-
ature variation in the slim, internally heated conductor shown in Fig. 1.2 is found to
equal
T = T
o
+ q
g
L
2
2k

x
L
-

x
L

2

where T
o
is the edge temperature. When the two ends are cooled to an identical
temperature, and when the volumetric heat generation rate q
g
(W/m
3
) is uniform
throughout, the peak temperature is developed at the center of the solid and is given
by
T
max
= T
o
+ q
g
L
2
8k
(K) (1.4)
Alternatively, because q
g
is the volumetric heat generation q
g
= q/LWd, the
center–edge temperature difference can be expressed as
T
max
- T
o
= q
L
2
8kLWd
= q
L
8kA
(1.5)
where the cross-sectional area A is the product of the width W and the thicknessd.
An examination of eq. (1.5) reveals that the thermal resistance of a conductor with a
distributed heat input is only one-fourth that of a structure in which all of the heat is
generated at the center.
BOOKCOMP, Inc. — John Wiley & Sons / Page 4 / 2nd Proofs /HeatTransferHandbook / Bejan
4 BASICCONCEPTS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[4], (4)
Lines: 173 to 196
———
-1.48895pt PgVar
———
Long Page
PgEnds: T
E
X
[4], (4)
x
1
x
2
T
max
T
o
Peak temperature
Edge temperature
x
L
Figure1.2 Temperature variation in an internally heated conductor.
1.1.3 Spreading Resistance
In con?gurations where there is a discrete heat source on the surface of a conducting
medium, provision must be made for the lateral spreading of the heat generated
in successive “layers” in the conducting medium below the source. The additional
resistance associated with this lateral ?ow of heat is called thespreadingresistance.
According to Yovanovich and Antonetti (1988), the spreading resistance for a small
heat source on a thick conductor or heat spreader (required to be three to ?ve times
thicker than the square root of the heat source area) can be expressed as
R
sp
=
1- 1.410+ 0.344
3
+ 0.043
5
+ 0.034
7
4ka
(K/W) (1.6)
where  is the ratio of the heat source area to the substrate area, k the thermal
conductivity of the conductor, and a the square root of the area of the heat source.
For relatively thin conducting layers on thicker substrates, such as encountered
in the cooling of microcircuits, eq. (1.6) cannot provide an acceptable prediction of
R
sp
. Instead, use can be made of the numerical results plotted in Fig. 1.3 to obtain the
requisite value of the spreading resistance.
1.1.4 Interface–Contact Resistance
Heat transfer across the interface between two solids is generally accompanied by
a measurable temperature difference, which can be ascribed to a contact or inter-
face thermal resistance. For perfectly adhering solids, geometrical differences in the
crystal structure (lattice mismatch) can impede the ?ow of phonons and electrons
Page 5


BOOKCOMP, Inc. — John Wiley & Sons / Page 1 / 2nd Proofs /HeatTransferHandbook / Bejan
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[First Page]
[1], (1)
Lines: 0 to 95
———
11.25609pt PgVar
———
Normal Page
* PgEnds: PageBreak
[1], (1)
CHAPTER 1
Basic Concepts
ALLAND.KRAUS
UniversityofAkron
Akron,Ohio
1.1 Heat transfer fundamentals
1.1.1 Introduction
1.1.2 Conduction heat transfer
One-dimensional conduction
One-dimensional conduction with internal heat generation
1.1.3 Spreading resistance
1.1.4 Interface–contact resistance
1.1.5 Lumped-capacity heating and cooling
1.1.6 Convective heat transfer
Heat transfer coef?cient
Dimensionless parameters
Natural convection
Forced convection
1.1.7 Phase-change heat transfer
1.1.8 Finned surfaces
1.1.9 Flow resistance
1.1.10 Radiative heat transfer
1.2 Coordinate systems
1.2.1 Rectangular (Cartesian) coordinate system
1.2.2 Cylindrical coordinate system
1.2.3 Spherical coordinate system
1.2.4 General curvilinear coordinates
1.3 Continuity equation
1.4 Momentum and the momentum theorem
1.5 Conservation of energy
1.6 Dimensional analysis
1.6.1 Friction loss in pipe ?ow
1.6.2 Summary of dimensionless groups
1.7 Units
1.7.1 SI system (Syst` eme International d’Unit´ es)
1.7.2 English engineering system (U.S. customary system)
1.7.3 Conversion factors
Nomenclature
References
1
BOOKCOMP, Inc. — John Wiley & Sons / Page 2 / 2nd Proofs /HeatTransferHandbook / Bejan
2 BASICCONCEPTS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[2], (2)
Lines: 95 to 139
———
6.28009pt PgVar
———
Normal Page
PgEnds: T
E
X
[2], (2)
1.1 HEAT TRANSFER FUNDAMENTALS
1.1.1 Introduction
Practitioners of the thermal arts and sciences generally deal with four basic thermal
transport modes: conduction, convection, phase change, and radiation. The process
by which heat diffuses through a solid or a stationary ?uid is termedheatconduction.
Situations in which heat transfer from a wetted surface is assisted by the motion of
the ?uid give rise to heat convection, and when the ?uid undergoes a liquid–solid
or liquid–vapor state transformation at or very near the wetted surface, attention is
focused on thisphase-changeheattransfer. The exchange of heat between surfaces,
or between a surface and a surrounding ?uid, by long-wavelength electromagnetic
radiation is termedthermalheatradiation.
It is our intent in this section to describe brie?y these modes of heat transfer, with
emphasis on an important parameter known as thethermalresistance to heat transfer.
Simple examples are given for illustration; detailed descriptions of the same topics
are presented in specialized chapters.
1.1.2 Conduction Heat Transfer
One-Dimensional Conduction Thermal diffusion through solids is governed
byFourier’slaw, which in one-dimensional form is expressible as
q=-kA
dT
dx
(W) (1.1)
where q is the heat current, k the thermal conductivity of the medium, A the cross-
sectional area for heat ?ow, and dT/dx the temperature gradient, which, because it
is negative, requires insertion of the minus sign in eq. (1.1) to assure a positive heat
?ow q. The temperature difference resulting from the steady-state diffusion of heat
is thus related to the thermal conductivity of the material, the cross-sectional area A,
and the path length L (Fig. 1.1), according to
(T
1
- T
2
)
cd
= q
L
kA
(K) (1.2)
The form of eq. (1.2), where k and A are presumed constant, suggests that in a way
that is analogous to Ohm’s law governing electrical current ?ow through a resistance,
it is possible to de?ne a conduction thermal resistance as
R
cd
=
T
1
- T
2
q
=
L
kA
(K/W) (1.3)
One-Dimensional Conduction with Internal Heat Generation Situations
in which a solid experiences internal heat generation, such as that produced by the
?ow of an electric current, give rise to more complex governing equations and require
BOOKCOMP, Inc. — John Wiley & Sons / Page 3 / 2nd Proofs /HeatTransferHandbook / Bejan
HEATTRANSFERFUNDAMENTALS 3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[3], (3)
Lines: 139 to 173
———
-2.4009pt PgVar
———
Normal Page
PgEnds: T
E
X
[3], (3)
T
x
Figure1.1 Heat transfer by conduction through a slab.
greater care in obtaining the appropriate temperature differences. The axial temper-
ature variation in the slim, internally heated conductor shown in Fig. 1.2 is found to
equal
T = T
o
+ q
g
L
2
2k

x
L
-

x
L

2

where T
o
is the edge temperature. When the two ends are cooled to an identical
temperature, and when the volumetric heat generation rate q
g
(W/m
3
) is uniform
throughout, the peak temperature is developed at the center of the solid and is given
by
T
max
= T
o
+ q
g
L
2
8k
(K) (1.4)
Alternatively, because q
g
is the volumetric heat generation q
g
= q/LWd, the
center–edge temperature difference can be expressed as
T
max
- T
o
= q
L
2
8kLWd
= q
L
8kA
(1.5)
where the cross-sectional area A is the product of the width W and the thicknessd.
An examination of eq. (1.5) reveals that the thermal resistance of a conductor with a
distributed heat input is only one-fourth that of a structure in which all of the heat is
generated at the center.
BOOKCOMP, Inc. — John Wiley & Sons / Page 4 / 2nd Proofs /HeatTransferHandbook / Bejan
4 BASICCONCEPTS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[4], (4)
Lines: 173 to 196
———
-1.48895pt PgVar
———
Long Page
PgEnds: T
E
X
[4], (4)
x
1
x
2
T
max
T
o
Peak temperature
Edge temperature
x
L
Figure1.2 Temperature variation in an internally heated conductor.
1.1.3 Spreading Resistance
In con?gurations where there is a discrete heat source on the surface of a conducting
medium, provision must be made for the lateral spreading of the heat generated
in successive “layers” in the conducting medium below the source. The additional
resistance associated with this lateral ?ow of heat is called thespreadingresistance.
According to Yovanovich and Antonetti (1988), the spreading resistance for a small
heat source on a thick conductor or heat spreader (required to be three to ?ve times
thicker than the square root of the heat source area) can be expressed as
R
sp
=
1- 1.410+ 0.344
3
+ 0.043
5
+ 0.034
7
4ka
(K/W) (1.6)
where  is the ratio of the heat source area to the substrate area, k the thermal
conductivity of the conductor, and a the square root of the area of the heat source.
For relatively thin conducting layers on thicker substrates, such as encountered
in the cooling of microcircuits, eq. (1.6) cannot provide an acceptable prediction of
R
sp
. Instead, use can be made of the numerical results plotted in Fig. 1.3 to obtain the
requisite value of the spreading resistance.
1.1.4 Interface–Contact Resistance
Heat transfer across the interface between two solids is generally accompanied by
a measurable temperature difference, which can be ascribed to a contact or inter-
face thermal resistance. For perfectly adhering solids, geometrical differences in the
crystal structure (lattice mismatch) can impede the ?ow of phonons and electrons
BOOKCOMP, Inc. — John Wiley & Sons / Page 5 / 2nd Proofs /HeatTransferHandbook / Bejan
HEATTRANSFERFUNDAMENTALS 5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
[5], (5)
Lines: 196 to 214
———
7.93205pt PgVar
———
Long Page
* PgEnds: Eject
[5], (5)
Figure1.3 Thermal resistance for a circular discrete heat source on a two-layer conducting
medium. (From Yovanovich and Antonetti, 1988.)
across the interface, but this resistance is generally negligible in engineering design.
However, when dealing with real interfaces, the asperities present on each of the sur-
faces (Fig. 1.4) limit actual contact between the two solids to a very small fraction
of the apparent interface area. The ?ow of heat across the gap between two solids in
nominal contact is by solid conduction in areas of actual contact and ?uid conduction
across the “open” spaces. Radiation across the gap can be important in a vacuum
environment or when surface temperatures are high. The heat transferred across an
interface can be found by adding the effects of solid-to-solid conduction and conduc-
tion through the ?uid and recognizing that solid-to-solid conduction in the contact
zones involves heat ?owing sequentially through the two solids. With the total con-
tact conductance h
co
, taken as the sum of solid-to-solid conductance h
c
and the gap
conductance h
g
,
h
co
= h
c
+ h
g
(W/m
2
· K) (1.7a)
the contact resistance based on the apparent contact area A
a
may be de?ned as
R
co
=
1
h
co
A
a
(K/W) (1.7b)
In eq. (1.7a), h
c
is given by (Yovanovich and Antonetti, 1988)
h
c
= 1.25k
s
m
s

P
H

0.95
(1.8a)
Read More
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!