UNIT-1 CONDUCTION

T.RAMESH AP/MECH
Dept. of Mechanical Engineering,
Kamaraj College of Engineering and Technology.
Virudhunagar
1
T.RAMESH AP/MECH , KCET
,VNR .

◦ Chapter 1: General Concepts of Heat
Conduction
◦ Chapter 2: One-Dimensional, Steady-State
Conduction
◦ Chapter 3: Two-Dimensional, Steady-State
Conduction
◦ Chapter 4 Transıent Heat Conductıon
2
,VNR .

Chapter 1
3
T.RAMESH AP/MECH , KCET ,VNR .

 Fourier’s law, 1-D form:
 Fourier’s law, general form:
- q” is the heat flux vector, which has three
components; in Cartesian coordinates:
(magnitude)
4
dx
dTkxq −=′′
Tkq ∇−=′′

kqjqiqq zyx
ˆˆˆ ′′+′′+′′=′′

222
zyx qqqq ′′+′′+′′=′′

,VNR .

 ∇T is the temperature gradient,
which is:
◦ a vector quantity that points in
direction of maximum temperature
increase
◦ always perpendicular to constant
temperature surfaces, or isotherms
(Cartesian)
(Cylindrical)
(Spherical)
5
k
z
T
j
y
T
i
x
T
T ˆˆˆ
∂
∂
+
∂
∂
+
∂
∂
=∇
k
z
T
j
T
r
i
r
T
T ˆˆ1ˆ
∂
∂
+
∂
∂
+
∂
∂
=∇
φ
k
T
r
j
T
r
i
r
T
T ˆ
sin
1ˆ1ˆ
φθθ ∂
∂
+
∂
∂
+
∂
∂
=∇
,VNR .

 k is the thermal conductivity of the
material undergoing conduction,
which is a tensor quantity in the
most general case:
◦ most materials are homogeneous,
isotropic, and their structure is time-
independent; hence:
which is a scalar and usually assumed to
be a constant if evaluated at the average
temperature of the material
6
),,,,( Ttzyxkk

=
),(Tkk =
,VNR .

 Total heat rate (q) is found by
integrating the heat flux over the
appropriate area:
 k and ∇ T must be known in order to
calculate q” from Fourier’s law
◦ k is usually obtained from material
property tables to find ∇T, another
equation is required; this additional
equation is derived by applying the
conservation of energy principle to a
differential control volume undergoing
conduction heat transfer; this yields the
general Heat Diffusion (Conduction)
Equation
7
∫ ⋅′′=
A
Adqq

,VNR .

 For a homogeneous, isotropic solid
material undergoing heat
conduction:
 This is a second-order, partial
differential equation (PDE); its
solution yields the temperature field,
T(x,y,z,t), within a given solid
material
8
t
T
cq
z
T
k
zy
T
k
yx
T
k
x ∂
∂
=+





∂
∂
∂
∂
+





∂
∂
∂
∂
+





∂
∂
∂
∂
ρ
,VNR .

 For constant thermal conductivity (k):
 For k = constant, steady-state
conditions, and no internal heat
generation
◦ this is known as Laplace’s equation,
which appears in other branches of
engineering science (e.g., fluids,
electrostatics, and solid mechanics)
9
y)diffusivit(thermal
:where
,
1
2
2
2
2
2
2
c
k
t
T
k
q
z
T
y
T
x
T
ρ
α
α
=
∂
∂
=+
∂
∂
+
∂
∂
+
∂
∂ 
00 2
2
2
2
2
2
2
=∇=
∂
∂
+
∂
∂
+
∂
∂
T
z
T
y
T
x
T
or,
:)0( =q
,VNR .

 Boundary Conditions: known conditions
at solution domain boundaries
 Initial Condition: known condition at t =
0
 Number of boundary conditions
required to solve the heat diffusion
equation is equal to the number of
spatial dimensions multiplied by two
 There is only one initial condition, which
takes the form
◦ where Ti may be a constant or a function of
x,y, and z
10
iTzyxT =)0,,,(
,VNR .

 Specified surface temperature, e.g.,
 Specified surface heat flux, e.g.,
 Specified convection (h, T∞ given),
e.g.,
 Specified radiation (ε, Tsur given), e.g.,
11
0),,,0( TtzyT =
0
0
q
x
T
k
x
′′=
∂
∂
−
=
[ ]),,,0(
0
tzyTTh
x
T
k
x
−=
∂
∂
− ∞
=
[ ]),,,0(44
0
tzyTT
x
T
k sur
x
−=
∂
∂
−
=
εσ
,VNR .

 Choose a coordinate system that best fits
the problem geometry.
 Identify the independent variables
(x,y,z,t), e,g, is it a S-S problem? Is
conduction 1-D, 2-D, or 3-D? Justify
assumptions.
 Determine if k can be treated as constant
and if
 Write the general heat conduction
equation using the chosen coordinates.
 Reduce equation to simplest form based
upon assumptions.
 Write boundary conditions and initial
condition (if applicable).
 Obtain a general solution for T(x,y,z,t) by
some method; if impossible, resort to
numerical methods.
12
.0=q
,VNR .

 Solve for the constants in the
general solution by applying the
boundary conditions and initial
condition to obtain a particular
solution.
 Check solution for correctness (e.g.,
at boundaries or limits such as x =
0, t = 0, t → ∞ , etc.)
 Calculate heat flux or total heat rate
using Fourier’s law, if required.
 Optional: rearrange solution into a
nondimensional form
13
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 GIVEN: Rectangular copper bar of
dimensions L x W x H is insulated on
the bottom and initially at Ti
throughout . Suddenly, the ends are
subjected and maintained at
temperatures T1 and T2 , respectively,
and the other three sides are
exposed to forced convection with
known h, T∞.
 FIND: Governing heat equation,
BCs, and initial condition
14
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Chapter 2
15
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 Plane Wall
◦ if k = constant, general heat diffusion
equation reduces to
◦ separating variables and integrating
yields
◦ where T(x) is the general solution;
◦ C1 and C2 are integration constants that
are determined from boundary
conditions
16
002
2
=





=
dx
dT
dx
d
dx
Td
or
211 )( CxCxTC
dx
dT
+== thenand
x
L
,VNR .

 Plane Wall, cont.
◦ suppose the boundary conditions are
◦ integration constants are then found to
be
◦ the particular solution for the temperature
distribution in the plane wall is now
17
21 )(and)0( ss TLxTTxT ====
12
12
1 s
ss
TC
L
TT
C =
−
= and
112 )()( sss T
L
x
TTxT +−=
,VNR .

 Plane wall, cont.
◦ The conduction heat rate is found from
Fourier’s law:
◦ If k were not constant, e.g., k = k(T), the
analysis would yield
 note that the temperature distribution would
be nonlinear, in general
18
( )211 ss TT
L
kA
kAC
dx
dT
kAq −=−=−=
∫ +′′= CxqdTTk )(
,VNR .

 Electric Circuit Analogy
◦ heat rate in plane wall can be written as
◦ in electrical circuits we have Ohm’s law:
◦ analogy:
19
constantmaterial
differenceetemperatur
=
−
=
kAL
TT
q ss
/
)( 21
R
V
i
∆
=
)resistance(electric)resistance(thermal
(voltage)re)(temperatu
(current)rate)(heat
R
kA
L
VT
iq
↔
↔
↔
,VNR .

 Cylindrical Wall
 Spherical Wall
20
k
rr
Rt
π2
)/ln( 12
=
k
rr
Rt
π4
/1/1 21 −
=
l
r1
r2
r2
r1
,VNR .

 Convection
 Radiation
21
)resistancethermale(convectivconvt
s
s
R
hA
hA
TT
TThAq
,
1
/1
)(
=
−
=−= ∞
∞
)resistancethermal(radiative
where
radt
r
ssr
r
s
sr
R
Ah
TTTTh
Ah
TT
TTAhq
,
22
1
))((
/1
)(
=
++=
−
=−=
∞∞
∞
∞
εσ
,VNR .

 Since the surface areas of cylinders
and spheres increase with r, there
exist competing heat transfer effects
with the addition of insulation under
convective boundary conditions (see
Example 3.4)
 A critical radius (rcr) exists for radial
systems, where:
◦ adding insulation up to this radius will
increase heat transfer
◦ adding insulation beyond this radius will
decrease heat transfer
 For cylindrical systems,rcr = kins/h
 For spherical systems, rcr = 2kins/h
22
,VNR .

 Thermal contact resistance exists
at solid-solid interfaces due to
surface roughness, creating gaps
of air or other material:
23
K/W)(mareaunitperresistancethermal
areacontactapparentwhere
2
,
,
,
=′′
=
′′
=
−
=
ct
c
c
ct
c
BA
ct
R
A
A
R
qA
TT
R
A
B
q
,VNR .

 R”t,c is usually experimentally
measured and depends upon
◦ thermal conductivity of solids A and B
◦ surface finish & cleanliness
◦ contact pressure
◦ gap material
◦ temperature at contact plane
 See Tables 3.1, 3.2 for typical
values
24
,VNR .

 Given: two, 1cm thick plates of
milled, cold-rolled steel, 3.18µm
roughness, clean, in air under 1 MPa
contact pressure
 Find: Thermal circuit and compare
thermal resistances
25
,VNR .

 Thermal energy can be generated
within a material due to
conversion from some other
energy form:
◦ Electrical
◦ Nuclear
◦ Chemical
 Governing heat diffusion equation
if k = constant:
26
systemsCartesianfor
where 2
2
2
2
0/
dx
Td
T
kqT
=∇
=+∇ 
,VNR .

 Consider plane wall exposed to
convection where Ts>T∞:
 How could you enhance q ?
◦ increase h
◦ decrease T∞
◦ increase As (attach fins)
27
,VNR .

 x = longitudinal direction of fin
 L = fin length (base to tip)
 Lc = fin length corrected for tip area
 W = fin width (parallel to base)
 t = fin thickness at base
 Af = fin surface area exposed to
fluid
 Ac = fin cross-sectional area, normal
to heat flow
 Ap = fin (side) profile area
 P = fin perimeter that encompasses
Ac
 D = pin fin diameter
 Tb = temperature at base of fin
28
,VNR .

 If L >> t and k/L >> h, then the
temperature gradient in the
longitudinal direction (x) is much
greater than that in the transverse
direction (y); therefore
 Another way of viewing fin heat
transfer is to imagine 1-D
conduction with a negative heat
generation rate along its length
due to convection
29
)conductionD-(1iqq x
ˆ≅′′

,VNR .

 Fin Effectiveness
 Fin Efficiency
◦ for a straight fin of uniform cross-section:
◦ where Lc = L + t / 2 (corrected fin
length)
30
)(, ∞−
=
=
TThA
q
bbc
f
f
finw/oareabasefromHT
finsinglefromHT
ε
)(max ∞−
==
=
TThA
q
q
q
T
bf
ff
b
f
atwerefinentireifHT
finsinglefromHT
η
c
c
f
mL
mL )tanh(
=η
,VNR .

 Calculate corrected fin length, Lc
 Calculate profile area, Ap
 Evaluate parameter
 Determine fin efficiency ηf from
Figure 3.18, 3.19, or Table 3.5
 Calculate maximum heat transfer
rate from fin:
 Calculate actual heat rate:
31
( )finsrrectangulafor2//2/3
cpc mLkAhL =
)(max, ∞−= TThAq bff
max,fff qq η=
,,, 3
1
,2
1
,, LtALtAtLA parptripcrecp ===
,VNR .

 Analysis:
 “Optimal” design results:
32
constantwithSet == p
f
A
dL
dq
0
profilerrectangulaannular,for
profileparabolicconcavefor1.7536
profiletriangularfor1.3094
profilerrectangulafor1.0035
2/
3
/
12
2/3
+
=
=
=
=
rr
kAhL pc
,VNR .

 Fin heat rate:
 Define fin thermal resistance:
 Single fin thermal circuit:
33
ff
b
bfffff
hA
TT
TThAqq
η
ηη
/1
)(max,
∞
∞
−
=
−==
ff
ft
hA
R
η
1
, =
,VNR .

 Total heat transfer =
heat transfer from N fins +
heat transfer from exposed base
Thermal circuit:
◦ Where
34
( )bffb
bbbffbft
AANh
hAhANqNqq
+=
+=+=
ηθ
θθη
b
convb
ff
ft
bc
ct
ct
hA
R
hAN
R
NA
R
R
11
,,
,
"
,
, === ,,
η
,VNR .

 Overall thermal resistance:
35
)(,
,,1
1
)(
)(
)(,
/1
11
1
cot
b
t
bft
bcctff
f
t
f
co
tco
cot
R
TT
q
ANAA
ARhAC
CA
NA
hA
R
∞−
=
+=
′′+=






−−=
=
then
array)ofareasurface(total
where
η
η
η
η
,VNR .

 Given: Annular array of 10
aluminum fins, spaced 4mm apart C-
C, with inner and outer radii of 1.35
and 2.6 cm, and thickness of 1 mm.
Temperature difference between
base and ambient air is 180°C with a
convection coefficient of 125 W/m2
-
K. Contact resistance of 2.75x10-4
m2
-K/W exists at base.
 Find: a) Total heat rate w/o and
with fins
b) Effect of R”t,c on heat rate
36
,VNR .

Chapter 3
37
,VNR .

 Heat Diffusion Equation reduces to:
 Solving the HDE for 2-D, S-S heat
conduction by exact analysis is
impossible for all but the most
simple geometries with simple
boundary conditions.
38
cartesian)D,-(2
or
equation)s(Laplace'
0
0
2
2
2
2
2
=
∂
∂
+
∂
∂
=∇
y
T
x
T
T
,VNR .

 Analytical Methods
◦ Separation of variables (see section 4.2)
◦ Laplace transform
◦ Similarity technique
◦ Conformal mapping
 Graphical Methods
◦ Plot isotherms & heat flux lines
 Numerical Methods
◦ Finite-difference method (FDM)
◦ Finite-element method (FEM)
39
,VNR .

 The heat rate in some 2-D
geometries that contain two
isothermal boundaries (T1, T2) with k
= constant can be expressed as
◦ where S = conduction shape factor
(see Table 4.1)
 Define 2-D thermal resistance:
40
)( 21 TTSkq −=
Sk
R Dcondt
1
)2(, =−
,VNR .

 Practical applications:
◦ Heat loss from underground spherical
tanks: Case 1
◦ Heat loss from underground pipes and
cables: Case 2, Case 4
◦ Heat loss from an edge or corner of an
object: Case 8, Case 9
◦ Heat loss from electronic components
mounted on a thick substrate: Case 10
41
,VNR .

42
TransıentTransıent
HeatHeat
ConductıonConductıon
Chapter 4
,VNR .

43
Objectives
• Assess when the spatial variation of
temperature is negligible, and temperature
varies nearly uniformly with time, making the
simplified lumped system analysis applicable
• Obtain analytical solutions for transient one-
dimensional conduction problems in
rectangular, cylindrical, and spherical
geometries using the method of separation of
variables, and understand why a one-term
solution is usually a reasonable approximation
• Solve the transient conduction problem in large
mediums using the similarity variable, and
predict the variation of temperature with time
and distance from the exposed surface
• Construct solutions for multi-dimensional
transient conduction problems using the product
solution approach
,VNR .

44
LUMPED SYSTEM ANALYSIS
Interior temperature of
some bodies remains
essentially uniform at all
times during a heat
transfer process.
The temperature of such
bodies can be taken to be
a function of time only,
T(t).
Heat transfer analysis that
utilizes this idealization is
known as lumped system
analysis.
A small copper ball can be
modeled as a lumped
system, but a roast beef
cannot.
,VNR .

45
Integrating
with
T = Ti at t
= 0
T = T(t) at
t = t
The geometry and
parameters
involved in the
lumped system
analysis.
time
const
ant
,VNR .

46
The temperature of a lumped
system approaches the
environment temperature as time
gets larger.
• This equation enables us
to determine the
temperature T(t) of a body
at time t, or alternatively,
the time t required for the
temperature to reach a
specified value T(t).
• The temperature of a
body approaches the
ambient temperature T∞
exponentially.
• The temperature of the
body changes rapidly at
the beginning, but rather
slowly later on. A large
value of b indicates that
the body approaches the
environment temperature
in a short time
,VNR .

47
Heat transfer to or from
a body reaches its
maximum value when
the body reaches the
environment
temperature.
The rate of convection
heat transfer between the
body and its environment
at time t
The total amount of heat transfer
between the body and the
surrounding medium over the time
interval t = 0 to t
The maximum heat transfer
between the body and its
surroundings
,VNR .

48
Criteria for Lumped System Analysis
Lumped system
analysis is applicable
if
When Bi ≤ 0.1, the
temperatures within the body
relative to the surroundings
(i.e., T −T∞) remain within 5
percent of each other.
Characterist
ic length
Biot number
,VNR .

49
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50
Small bodies with
high thermal
conductivities and
low convection
coefficients are
most likely to
satisfy the criterion
for lumped system
analysis.
Analogy between heat
transfer to a solid and
passenger traffic to an
island.
When the convection
coefficient h is high and k is
low, large temperature
differences oc
cur between the inner and
outer regions of a large solid.
,VNR .

51
TRANSIENT HEAT CONDUCTION IN LARGE
PLANE WALLS, LONG CYLINDERS, AND
SPHERES WITH SPATIAL EFFECTS
We will consider the variation of
temperature with time and position in one-
dimensional problems such as those
associated with a large plane wall, a long
cylinder, and a sphere.
Schematic of
the simple
geometries in
which heat
transfer is
Transient temperature
profiles in a plane wall
exposed to convection
from its surfaces for Ti
>T∞.
,VNR .

52
Nondimensionalized One-Dimensional Transient
Conduction Problem
,VNR .

53
Nondimensionalization
reduces the number of
independent variables
in one-dimensional
transient conduction
problems from 8 to 3,
offering great
convenience in the
presentation of results.
,VNR .

54
Exact Solution of One-Dimensional Transient
Conduction Problem
,VNR .

55
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56
The analytical solutions of
problems typically involve
infinite series, and thus the
evaluation of an infinite
number of terms to
determine the temperature at
a specified location and time.
,VNR .

57
Approximate Analytical and Graphical Solutions
Solution with one-term approximation
The terms in the series solutions converge rapidly
with increasing time, and for τ > 0.2, keeping the first
term and neglecting all the remaining terms in the
series results in an error under 2 percent.
,VNR .

58
,VNR .

59
(a) Midplane temperature
Transient temperature and heat transfer charts
(Heisler and Grober charts) for a plane wall of
thickness 2L initially at a uniform temperature
Ti subjected to convection from both sides to an
environment at temperature T∞ with a
convection coefficient of h. T.RAMESH AP/MECH , KCET
,VNR .

60
(b) Temperature distribution
,VNR .

61
(c) Heat transfer
,VNR .

62
The dimensionless temperatures anywhere in a plane
wall, cylinder, and sphere are related to the center
temperature by
The specified surface temperature corresponds to the case of
convection to an environment at T∞ with with a convection
coefficient h that is infinite.
,VNR .

63
The fraction of total heat
transfer Q/Qmax up to a
specified time t is determined
using the Gröber charts.
,VNR .

64
• The Fourier number is
a measure of heat
conducted through a
body relative to heat
stored.
• A large value of the
Fourier number
indicates faster
propagation of heat
through a body.
Fourier number at time
t can be viewed as the
ratio of the rate of heat
conducted to the rate
of heat stored at that
time.
The physical significance of the Fourier number
,VNR .

65
TRANSIENT HEAT CONDUCTION IN
SEMI-INFINITE SOLIDS
Schematic of a semi-infinite body.
Semi-infinite solid: An
idealized body that has a
single plane surface and
extends to infinity in all
directions.
The earth can be considered
to be a semi-infinite medium
in determining the variation of
temperature near its surface.
A thick wall can be modeled
as a semi-infinite medium if
all we are interested in is the
variation of temperature in
the region near one of the
surfaces, and the other
surface is too far to have any
impact on the region of
interest during the time of
observation.
For short periods of time, most
bodies can be modeled as semi-
infinite solids since heat does not
have sufficient time to penetrate
deep into the body. T.RAMESH AP/MECH , KCET
,VNR .

66
Transformation of
variables in the
derivatives of the
heat conduction
equation by the
use of chain rule.
Analytical solution for the case of constant temperature Ts on the surface
error
function
complemen
tary error
function T.RAMESH AP/MECH , KCET
,VNR .

67
Error function is a
standard mathematical
function, just like the sine
and cosine functions,
whose value varies
between 0 and 1.
,VNR .

68
Analytical
solutions for
different boundary
conditions on the
surface
,VNR .

69
Dimensionless
temperature
distribution for
in a semi-infinite solid
whose surface is
maintained at a
constant
temperature Ts.
,VNR .

70
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71
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72
Variation of temperature with position and time in a
semi-infinite solid initially at temperature Ti subjected
to convection to an environment at T∞ with a
convection heat transfer coefficient of h.
,VNR .

73
TRANSIENT HEAT CONDUCTION IN
MULTIDIMENSIONAL SYSTEMS
• Using a superposition approach called the product solution, the
transient temperature charts and solutions can be used to
construct solutions for the two-dimensional and three-
dimensional transient heat conduction problems encountered in
geometries such as a short cylinder, a long rectangular bar, a
rectangular prism or a semi-infinite rectangular bar, provided that
all surfaces of the solid are subjected to convection to the same
fluid at temperature T∞, with the same heat transfer coefficient h,
and the body involves no heat generation.
• The solution in such multidimensional geometries can be
expressed as the product of the solutions for the one-dimensional
geometries whose intersection is the multidimensional geometry.
The temperature in a
short cylinder exposed
to convection from all
surfaces varies in both
the radial and axial
directions, and thus
heat is transferred in
both directions.
,VNR .

74
A short cylinder of
radius ro and height a
is the intersection of a
long cylinder of radius
ro and a plane wall of
thickness a.
The solution for a multidimensional geometry is the product of the
solutions of the one-dimensional geometries whose intersection is
the multidimensional body.
The solution for the two-dimensional short cylinder of height a and
radius ro is equal to the product of the nondimensionalized
solutions for the one-dimensional plane wall of thickness a and the
long cylinder of radius ro.
,VNR .

75
A long solid bar of rectangular
profile a × b is the intersection of
two plane walls of thicknesses a
and b.
,VNR .

76
The transient heat transfer for a two-dimensional
geometry formed by the intersection of two one-
dimensional geometries 1 and 2 is
Transient heat transfer for a three-dimensional
body formed by the intersection of three one-
dimensional bodies 1, 2, and 3 is
,VNR .

77
Multidimensional solutions expressed as
products of one-dimensional solutions for bodies
that are initially at a uniform temperature Ti and
exposed to convection from all surfaces to a
medium at T∞
,VNR .

78
Multidimensional solutions expressed as products of one-
dimensional solutions for bodies that are initially at a uniform
temperature Ti and exposed to convection from all surfaces to a
medium at T∞
,VNR .

79
 Lumped System Analysis
◦ Criteria for Lumped System Analysis
◦ Some Remarks on Heat Transfer in Lumped
Systems
 Transient Heat Conduction in Large Plane
Walls, Long Cylinders, and Spheres with
Spatial Effects
◦ Nondimensionalized One-Dimensional
Transient Conduction Problem
◦ Exact Solution of One-Dimensional Transient
Conduction Problem
◦ Approximate Analytical and Graphical
Solutions
 Transient Heat Conduction in Semi-Infinite
Solids
◦ Contact of Two Semi-Infinite Solids
 Transient Heat Conduction in
Multidimensional Systems
,VNR .

UNIT-1 CONDUCTION

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UNIT-1 CONDUCTION