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HEAT TRANSFERby
CONDUCTION
Prepared by,
Kumargaurao D PunaseAssistant Professor
Dept of Chemical Engg.COES, UPES
Dehradun
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First mechanism - molecularinteraction (e.g. gas)
Greater motion of molecule
at higher energy level(temperature) imparts energyto adjacent molecules atlower energy levels
Second mechanism by free
electrons (e.g. solid)The mechanism of heat conduction
in different phases of substance
Fouriers law of Heat Conduction
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Steady state heat conduction
One directional heat flow
Isotropic and homogeneous material i.e. thermal conductivityhas a constant value in all the directions
Constant temperature gradient and a linear temperature profile
No internal heat generation
Assumptions of Fouriers law: Contd .
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Summary: Fouriers Law
It is phenomenological, i.e. based on experimental evidence
Is a vector expression indicating that the heat flux is normal toan isotherm, in the direction of decreasing temperature
Applies to all states of matter
Defines the thermal conductivity, ie.
)/( xT
qk x
Contd .
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Chapter 2 6
Thermal Properties of Matter
Fouriers law for heat conduction:
The proportionality constant is a transport property , known asthermal conductivity k (units W/m.K)
the thermal conductivity of a material can be defined as therate of heat transfer through a unit thickness of the material
per unit area per unit temperature difference. The thermal
conductivity of a material is a measure of the ability of thematerial to conduct heat.
Usually assumed to be isotropic (independent of the direction
of transfer): k x=ky=kz=k
dxdT
k q x
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7
Thermal Conductivity: Solids
Solid comprised of free electrons and atoms bound in lattice
Thermal energy transported through Migration of free electrons, k e Lattice vibrational waves, k l
l e k k k )(y,resistivitelectrical1
eek where
Contd .
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Thermal Conductivity: Gases
Intermolecular spacing is much larger Molecular motion is random Thermal energy transport less effective than in solids; thermal
conductivity is lower
Kinetic theory of gases:
cnk Where, n the number of particles per unit volume,the mean molecular speed and
the mean free path (average distance travelled before a collision) c
Contd .
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Thermal Conductivity: Liquids
Physical mechanisms controlling thermal conductivity not well
understood in the liquid state
Generally k decreases with increasing temperature (exceptions
glycerine and water)
k decreases with increasing molecular weight.
3/1
3/4
M Ac
k p
The parameter A depends on the temperature of the liquid;
The quantity (Ac p) is constant for all liquids
Contd .
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Thermal Conductivity: Insulators Low temperature insulation (cork, rock wool, glass wool and thermocole)
are used when the enclosures is at a temperature lower than the ambienttemperature and it is desired to prevent the enclosure from gaining heat.
High temperature insulation (asbestos, diatomaceous earth, magnesia
etc.) are used when the enclosures is at a temperature higher than theambient temperature and it is desired to prevent the enclosure fromloosing heat.
Super insulators include powders, fibres or multi-layer materials that
have been evacuated of all air
The low conductivity of insulating materials is due primarily to air that iscontained in the pores rather than the low conductivity of the solidsubstance.
Contd .
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Variation of Thermal Conductivity with temperatureand pressure
For most materials, the dependence of thermal conductivityon temperature is almost linear
T k k 10Where, k 0 is the thermal conductivity at 0 0C and
Is a constant value depending upon the material; may be positive ornegative depending on whether k increases or decreases withtemperature.
Thermal conductivity is very weakly dependent on pressurefor solids and liquids and essentially independent of pressurefor gases at standard atmospheric pressure
Contd .
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The variation of the thermal conductivity of the various solids,
liquids and gases with the temperature
Contd .
Adapted from Heat and MassTransfer A Practical Approach,Y.A. Cengel, Third Edition,McGraw Hill 2007.
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The range of thermal conductivity of various materials at room
temperature
Contd .
Adapted from Heat and MassTransfer A Practical Approach,Y.A. Cengel, Third Edition,McGraw Hill 2007.
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Thermal Diffusivity
The product C p , which is frequently encountered in heattransfer analysis, is called the heat capacity of a material.
Both the specific heat C p and the heat capacity C p representthe heat storage capability of a material.
But C p expresses it per unit mass whereas C p expresses it perunit volume, as can be noticed from their units J/kgC andJ/m 3C, respectively.
Thermal diffusivity is the ratio of the thermal conductivity to
the heat capacity:
pck
storedHeatconductedHeat
(m 2/s)
Contd .
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Note that the thermal conductivity k represents how well amaterial conducts heat, and the heat capacity C p represents
how much energy a material stores per unit volume.
Therefore, the thermal diffusivity of a material can be viewed asthe ratio of the heat conducted through the material to the heatstored per unit volume.
A material that has a high thermal conductivity or a low heatcapacity will obviously have a large thermal diffusivity. Thelarger the thermal diffusivity, the faster the propagation of heat
into the medium.
A small value of thermal diffusivity means that heat is mostlyabsorbed by the material and a small amount of heat will beconducted further.
Contd .
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Introduction to development of 3D equation
d
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Contd .
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
C d
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Contd .
C d
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Contd .
C td
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Contd .
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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The Conduction Rate Equation
In reality we must account for heat transfer in three dimensions Temperature is a scalar field T(x,y,z) Heat flux is a vector quantity. In Cartesian coordinates:
z y x qqq k jiq
for isotropic medium z T k q
yT k q
xT k q z y x ,,
T k z
T
y
T
x
T
k
k jiqWhere three dimensional del operator in Cartesian coordinates:
z y x k ji
C td
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Energy Conservation Equation st
st g out in E dt
dE E E E
dz z dy ydx xout qqq E z y xin qqq E
Contd .
Contd
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E=mcpT=( V)cpT= (dxdydz)c pT
dy
dx
qx qx+dx
x
y
q KA T
x K dydz
T
x
q q dq q q
xdx
x x x
x dx x x x x
( )
dx
dy
dz
Contd .
Contd
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where from Fouriers law
z
T dxdyk
z
T kAq
yT
dxdz k yT
kAq
x
T dydz k
x
T kAq
z z
y y
x x
)(
)(
)(
dz z
qqq
dy y
qqq
dx
x
qqq
z
z dz z
y ydy y
x xdx x
Energy in
Energy out
Contd .
Contd
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Thermal energy generation due to an energy source : Manifestation of energy conversion process (between thermal
energy and chemical/electrical/nuclear energy)
Positive (source) if thermal energy is generatedNegative (sink) if thermal energy is consumed
)(dV dz dydxqq E g
Energy storage term Represents the rate of change of thermal energy stored in the
matter in the absence of phase change.
)( dz dydxt T
c E p st
t T c p /is the time rate of change of the sensible (thermal) energy ofthe medium per unit volume (W/m 3)
q is the rate at which energy is generated per unit volume of themedium (W/m 3)
Contd .
Contd
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dxdydz t T
C dxdydz qqqqqqq pdz z dy ydx x z y x
dxdydz t
T C dxdydz qdz
z
qdy
y
qdx
x
q p
z y x
z
T kdxdyq
yT kdxdz q
xT
kdydz q
z
y
x
st st
g out in E dt dE
E E E Contd .
Contd
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dxdydz t
T C dxdydz qdxdydz
z
T k
z dxdydz
y
T k
ydxdydz
x
T k
x P
t T C q
z T k
z yT k
y xT k
x P
Heat (Diffusion) Equation: at any point in the medium the rate of
energy transfer by conduction in a unit volume plus thevolumetric rate of thermal energy must equal to the rate ofchange of thermal energy stored within the volume.
Contd .
Contd
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dx x x qqdx
x
T k
x
t T
k q
z T
yT
xT
1
2
2
2
2
2
2
Net conduction heat flux into the controlled volume
If the thermal conductivity is constant.
Where = k /( Cp) is the thermal diffusivity
t T
C q z T
k z y
T k
y xT
k x P
Contd .
Contd
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0
q z T
k z y
T k
y xT
k x
0)( dxdT
k dxd
Under steady-state condition, there can be no change in theamount of energy storage.
If the heat transfer is one-dimensional and there is noenergy generation, the above equation reduces to
Under steady-state, one-dimensional conditions with noenergy generation, the heat flux is a constant in the
direction of transfer.
Contd .
Contd
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Boundary and Initial Conditions Heat equation is a differential equation:
Second order in spatial coordinates: Need 2 boundary conditions First order in time: Need 1 initial condition
Boundary Conditions
1) B.C. of first kind (Dirichlet condition):
x
T(x,t)
Ts
Constant Surface Temperature At x=0, T(x,t)=T(0,t)=T s
x=0
Contd .
Contd
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Boundary and Initial Conditions2) B.C. of second kind (Neumann condition): Constant heat flux at the
surface
1. Finite heat flux
xT
xT
k q x
s slopeconst0
"
0slope00
"
xT
xT
k q x
s
xT(x,t)
x
T(x,t)
qx = q s
2. Adiabatic surface qx=0
Contd .
Contd .
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Boundary and Initial Conditions3) B.C. of third kind: When convective heat transfer occurs
at the surface
)],0([0
t T T h x
T k
x
T(x,t)
T(0,t)
x
hT ,
= q conduction q convection
Contd .
Contd .
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Contd .
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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Cylindrical coordinates
Contd .
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Cylindrical coordinates
z T
k z qT
r k
qr T
k r q ";";"
)1
(" Z T
k T
r j
r T
ik T k q
When the del operator is expressed in cylindrical coordinates,the general form of the heat flux vector , and hence the FouriersLaw, is
Contd .
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Heat Conduction Equation
In cylindrical coordinates:
t T
cq z T
k z
T k
r r T
kr r r p
211
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Spherical coordinates
Contd .
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Spherical coordinates
T
r k
qT
r k
qr T
k r q ";sin";"
)sin11
("
T r
k T
r j
r T
ik T k q
When the del operator is expressed in spherical coordinates,the general form of the heat flux vector , and hence the FouriersLaw, is
Contd .
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Heat Conduction Equation
In spherical coordinates:
t T
cqT
k r
T k
r r T
kr r r
p
sinsin
1
sin
11222
22
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Energy balance for the wall
rate ofheat transferinto the wall
rate ofheat transferout of the wall
rate of changeof the energyof the wall
- =
dt dE
QQ wall out in
0dt
dE wall
consQ wall cond ,
steady operation; since there is no change in the
temperature of the wall with time at any point
The rate of heat transfer through the wall is constant
If there is no heat generation
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FOURIERS LAW OF HEAT CONDUCTION
wall cond Q , dx
dT kAQ wall cond , (W)
and A constant, then
dxdT constant also
Temperature through the wall varies linearly withx. Temperature distribution in the wall understeady conditions is a straight line.
2
1,0
T
T T wall cond
L
x kAdT dxQ
LT T
kAQ wall cond 21
,
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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THERMAL RESISTANCE
wall wall cond R
T T Q 21,
kA L
Rwall
(W)
(0C / W)
Depends on the geometry andthe thermal properties of themedium
e RV V I 21 A L R ee
e R21 V V e
Electrical resistance
Voltage differenceacross the resistance
Electricalconductivity
Contd .
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NEWTONS LAW OF COOLING FORCONVECTION HEAT TRANSFER RATE
)( T T hAQ S S conv
conv
S conv R
T T Q
S
conv
hA
R1
conv R
h
Convection resistance of surface
(W)
(0C / W)
Convection heat transfercoefficient
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
Contd .
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RADIATION
rad
surr S surr S S rad rad R
T T T T AhQ
)(
S rad rad Ah
R1
surr s surr s surr S S
rad rad T T T T T T A
Qh 22)(
rad convcombined hhh
)( 44 surr S S rad T T AQ
Contd .
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The thermal resistance network for heat transfer through a plane wallsubjected to convection on both sides and the electrical analogy
THERMAL RESISTANCE NETWORK
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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ONE DIMENSIONAL STEADY HEATFLOW
Rate of
heat convection
from the wall
Rate of
heat convection
into the wall
Rate of
heat conduction
through the wall
= =
)()( 22221
111 T T Ah LT T
kAT T AhQ
Ah
T T
kA L
T T
Ah
T T Q
2
2221
1
11
/1//1
2,
2221
1,
11
convwall conv RT T
RT T
RT T
Q
adding the numerators and denominators
total RT T
Q 21
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Total Thermal Resistance
total RT T
Q 21
Ah Ak L
Ak L
Ah R
R R R R R
total
convwall wall convtotal
22
2
1
1
1
2,2,1,1,
11
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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Thermal Contact Resistance
gapcontact QQQ
erfacec T AhQ int
erfacec T
AQh
int
/(W/m 2 0C)
(m 2 0C/ W) AQ
T
h R erface
cc
/
1 int
hC: thermal contact conductance
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
Contd .
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Thermal contact resistance is inverse of thermalcontact conduction,
Depends on
Surface roughness, Material properties,
Temperature and pressure at interface, Type of fluid trapped at interface
Contd .
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Effect of metalliccoatings on thermalcontact conductance
For soft metals withsmoot surfaces at highpressures
Thermal contactconductance
Thermal contactresistance
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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Overall heat transfer coefficient
q1 = convection
q2 = conduction
q3 = convection
q = q 1 = q2 = q3
The overall heat transfer bycombined conduction andconvection is frequently expressed interms of an overall heat transfercoefficient U
overall T UAq
?U
Contd
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)( T T hAq wconv
hAT T
q wconv /1
hA Rconv /1
Convection Boundary condition
Conduction
)( 12 T T xkA
qcond
Ak x
Rn
ncond
kA xT T
qcond /21
Contd
Contd
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)()()( 222111 B A T T AhT T xkA
T T Ahq
AhkA x Ah
T T q B A
21 /1//1
R Ahk xhU
11/1//1
1
21
Overall heat transfer coefficient
Contd
21
111hk
xhU
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Thermal resistance networkthrough a two-layer plane
T UAQ
total RUA
1
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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THERMAL RESISTANCE NETWORKS
)11
)((21
212
21
1
2121 R R
T T R
T T R
T T QQQ
total RT T
Q 21
21
111 R R Rtotal
21
21
R R R R Rtotal
Resistances are parallel
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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COMBINED SERIES-PARALLEL ARRANGEMENT
total RT T Q 1
convconvtotal R R R R R R
R R R R 321
21312
11
11 Ak
L R
22
22 Ak
L R
33
33 Ak
L R 3
1hA Rconv
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
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Problem:
Two slabs, each 100mm thick and made of materials with thermalconductivities of 16 and 200 W/m 0C, are placed in contact whichis not perfect. Due to roughness of surfaces, only 40% of area is incontact and air fills 0.02 mm thick gap in the remaining area.If the extreme surfaces of the arrangement are at temperaturesof 250 0C and 30 0C, determine the heat flow through thecomposite system, the contact resistance and temperature dropin contact.
Take thermal conductivity of air as 0.032 W/m 0C and assume thathalf of the contact is due to either metal.
1 D Steady Heat Transfer in the Plane Wall
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For an infinite flat plate (y, z
infinite),find the steady statetemperature profile with thetemperature at x=0 remainsat T 1 and the temperature atx=L remains at T 2
X=0 X=L
T1
T2
1-D, Steady Heat Transfer in the Plane Wall
The Governing Differential Equation
t T
C q z T
k z y
T k
y xT
k x P
=0 =0 =0 =0
T is not function of y and z
H t E tiContd .
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022
xT ---
Heat Equation :
0
d dT
k dx dx
022
dxT d k
Contd .
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Integrate eq.
Integrate again,
T=C1.x+C2
Apply the boundary conditions:
T1=C2 --- T2=C1.L+C2 ---
1C dxdT
B. C. T=T1 at x=0B. C. T=T2 at x=L
Contd .
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Putting, into
L x
T T T T
L x
T T T T
T x L
T T T
LT T C T LC T
12
1
121
112
121112
).(
.
.
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HEAT CONDUCTION IN CYLINDERS AND SPHERES
Steady-state heat conduction
Heat is lost from a hot-water
pipe to the air outside in theradial direction.
Heat transfer from a longpipe is one dimensional
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,McGraw Hill 2007.
A LONG CYLINDERICAL PIPE
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STEADY STATE OPERATION
dr dT
kAQ cy l cond ,
Fouriers law of conduction
cyl cond Q , constant
2
1
2
1
, T
T T
r
r r
cyl cond
kdT dr A
Q
rL A 2
)/ln(2 1221, r r
T T Lk Q cyl cond
cyl cyl cond
R
T T Q 21,
Lk
r r Rcy l
2
)/ln( 12
There are flow of gases on bothComposite cylindrical wall
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sides of the wall, which remainat temp T i & To. Looking for(1) temperature profile(2) heat flux requited to
maintain steady state
r1 r2
r3
Ti
T1 T2
T3 To
p y
))(2(
)()/ln(
2
)()/ln(
2))(2(
33
3223
2
2112
1
11
oo
ii
T T Lr h
T T r r Lk
T T r r
Lk T T Lr hQ
Contd .
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65
oi
oi
Lhr Lk r r
Lk r r
Lhr
T T Q
32
23
1
12
1 21
2)/ln(
2)/ln(
21
ResistanceTotalDifferenceeTemperatur OverallQ
I fi it C li dFor steady radial flow of heat through the
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Infinite Cylindery g
wall of a hollow cylinder. The inner surfaceis maintained at temperature T 1, whileouter surface is kept at temperature T 2.
Find the temperature profile across thewall and the heat flux required to maintainsteady state.T1
T2
r1
r2
T is not function of and z
t T
cq z T
k z
T k
r r T
kr r r p
2
11
Governing equation in cylindrical coordinate=0 =0 =0 =0
The GDE isContd .
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The GDE is
0)(1
r
T r
r r k
21
1
1
ln
0)(
cr cT
r
c
r
T
cr
T r
r
T r
r
---
Contd .
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68
Apply boundary conditions
2212
2111
ln
ln
cr cT
cr cT
B.C. r=r 1, T=T1 r=r 2, T=T2
Contd .
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)/ln(
)/ln()ln(ln
21
211
21121121
r r T T c
r r cr r cT T
222122122
2221212
2221
212
lnlnlnlnln)()ln(ln
ln.)/ln(
r T r T r T r T ccr T T r r T
cr r r
T T T
Contd .
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70
)ln)(ln()ln)(ln(
)ln)(ln()ln(ln)ln(ln
lnlnlnlnlnlnln
221212
22121221
2221221221
21
r r T T r r T T
r r T T r r T r r T
r T r T r T r T r r r
T T T
)/ln()/ln(
21
2
21
2
r r r r
T T T T
The variation of temperature within the cylinder is determined to be
STEADY STATE HEAT
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Ti
To ri
ro
dr dT
kAQ
24 r A
dr
dT r k Q 24
oi
o
i
T
T
r
r dT
r
dr
k
Q24
io
o
i
i
o
o
i
T T
r
r
T
T
r
r T
r k Q
dT r dr
k Q 1
44 2
STEADY STATE HEATCONDUCTION FOR
SPHERES
TContd .
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RT
Q
oi
oi
r r
T T k Q
11)(4
k r r R oi
4
11
k r r
r r R
oi
io
sph 4OR
Contd .
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k r r r r
R sph21
12
4
sph sphcond R
T T Q 21,
including convection
22
221
12
41
4 hr k r r r r
Rtotal
total RT T
Q 1
Temperature distribution inside the spherical shell
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p p
t T cqT k
r T k
r r T kr
r r p
sinsin1
sin11 222
22
=0 =0 =0 =0
01 2
2
r
T kr
r r
02
r T
r r
21
r C
dr dT
21)( C
r
C r T
1)(C
Contd .
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22
1222
21
1111
)(
)(
C r C
T T r T
C r
T T r T
12
11222
21
12
211
r r T r T r
C
T T
r r
r r C
12
112221
12
21)(r r
T r T r T T
r r r r r
r T
Applying BoundaryConditions
The constantsobtained are as:
The variation of temperature within the spherical shell isdetermined to be
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CRITICAL RADIUS OFINSULATION
)2(1
2)/ln(
2
12
11
Lr h Lk r r
T T R RT T
Qconvins
CYLINDER
However, as r 2 A 0 convection heat transfer resistance of surface
=>Q
As r2 thickness of wall heat resistance in the wall => Q
The heat transfer from the pipe may increase or decrease,Contd .
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0/ 2 dr Qd
The heat transfer from the pipe may increase or decrease,depending on which effect dominates.
T 1 ,T
2 , k, L, h, r
i are constant terms, therefore:
The variation of Q with the outerradius of the insulation r 2 is plotted:
The value of r 2 at which Q reachesa maximum is determined from therequirement that
Adapted from Heat and Mass Transfer A PracticalApproach, Y.A. Cengel, Third Edition, McGraw Hill2007.
Contd .
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222
2
21
2
2
21
2
2
11.
1ln
1
)(2
1ln
1)(2
0
r hkr
r hr r
k
T T L
dr hr r
r k
T T Ld
dr dQ
o
o
oi
oi
o
c
o
o
h
k r
r hkr
r hkr
,2
222
222
11
011
Critical radius of insulation for cylinderContd .
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hk
r cylinder cr ,Thermal conductivity
External convection heattransfer coefficient
y
Critical radius of insulation for shpere
hk
r spherecr 2
,
CHOSING INSULATION THICKNESSContd .
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cr
cr
cr
r r
r r
r r
2
2
2
Q max
Before insulation check for critical radius
CHOSING INSULATION THICKNESS
Contd .
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Contd .
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Optimum thickness of insulation
Conduction with Generation
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Conduction with GenerationThermal energy may be generated or consumed due to
conversion from some other energy form.If thermal energy is generated in the material at the expense ofsome other energy form, we have a source: is +ve Deceleration and absorption of neutrons in a nuclear reactor
Exothermic reactions Conversion of electrical to thermal energy:
V R I
V
E q
e g 2 where I is the current, R e the electrical
resistance, V the volume of the medium
If thermal energy is consumed we have a sink : is -ve
Endothermic reactions
q
q
The Plane Wall
0 L
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Consider one-dimensional,steady-state conduction in aplane wall of constant k, withuniform generation, andasymmetric surface conditions:
Heat conduction equation
02
2
k
q
dx
T d
The Governing Differential Equation
t
T C q
z
T k
z y
T k
y x
T k
x P
=0 =0 =0
2 qTd Contd .
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02 k q
dxT d
222
2
dxk qT d
k q
dxT d
1c xdxk
qdT
Boundary Conditions:
2,1, )(,)0( s s T LT T T
General Solution:
212
2C xC x
k
qT
Temperature ProfileContd .
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p
1,1,2,2 )(2)( s s s T L x
T T x Lxk
q xT
Profile is parabolic.Heat flux not independent of x
0 L
1,2 sT C Lk
q
L
T T C s s
2
)( 1,2,1
1,1,2, )(2)( s s s T x
L
T T x L
k q
xT
OR
Symmetrical DistributionContd .
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When both surfaces are maintained at a commontemperature, T s,1= Ts,2 = Ts 0 L
1,1,2, )(2)( s s s T x
L
T T x L
k q
xT
sT x x Lk q
xT 2
)(
Maximum temperature within a symmetric systemContd .
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0 L
sT x x L
k
q xT
2)(
0dxdT
The maximum temperature at anyposition x can be obtained as:
02
2 x L
k
q
dx
dT
02k q or02 x L
2
L x
At x=L/2, T = Tmax
Contd .
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, max
sT x x Lk q xT 2)(
sT L L
Lk
qT
222(max)
Putting x=L/2,
sT Lk q
T 28
(max)
Heat transfer then occurs towards both surfaces and for
Contd .
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Heat transfer then occurs towards both surfaces, and foreach surface it is given by:
L x xdxdT kAQ
or0
q
ALQ
x Lk q
kAQ L x x
2
22 or0
Heat transfer for both surfaces,
q ALQ
=(volume of conducting medium)(heat generating capacity)
Heat conducted to the wall surface is finally dissipated to the
Contd .
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a s T T hAq AL2
Heat conducted to the wall surface is finally dissipated to thesurrounding atmosphere at temperature T a
Lh
qT T a s 2
OR
Substituting this value of wall temperature, the temperature
distribution in term of surrounding atmospheric temperature T a
x x Lk
q L
hq
T xT a 22)(
Contd
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Note that at the plane of symmetry:
0q" 00
0
x xdx
dT
Equivalent to adiabatic surface
Cylinder with uniform heat generation
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Solid Cylinder Temperature distribution within a solid cylinder
224)( r Rk qT r T s
R
Ts
h,Ta
Temperature distribution is parabolic andthe maximum temperature T max occurring at the centre (r=0) of the rod isgiven by
2max 4
Rk
qT T s
The temperature distribution in term of surrounding atmospherictemperature T a
2242
r Rk
q R
hq
T T a
Hollow Cylinder with temperatures specified at the inside andt id f
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outside surfaces
Temperature distribution within a thickness of hollow cylinder
2
1
121
2221
2211
ln
ln
44)(
r r
r r
r r k
qT T r r
k q
T r T
Sphere with uniform heat generation
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Temperature distribution within a solid cylinder
226)( r Rk qT r T s R
Ts
h,Ta
Temperature distribution is parabolic and themaximum temperature T max occurring at thecentre (r=0) of the rod is given by
2max 6
Rk
qT T s
The temperature distribution in term of surrounding atmospherictemperature T a
2263
r Rk
q R
hq
T T a
Heat Transfer from Extended Surfaces
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An extended surface (also know as a combined conduction-convectionsystem or a fin) is a solid within which heat transfer by conduction isassumed to be one dimensional , while heat is also transferred by convection (and/or radiation ) from the surface in a direction transverse to that ofconduction.
Anatomy of A STRIP FIN
io n
Contd .
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thickness
x
x
Direction of Heat Conduction
D i r e c t i o n o
f H e a t C o n v e c t i o
Basic Geometric Features of Fins
Contd .
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profilePROFILE AREA
cross-section
CROSS-SECTION AREA
Basic Geometric Features of Fins
GARDNER-MURRAY ANALYSIS : ASSUMPTIONSContd .
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Steady state one dimensional conduction Model.No Heat sources or sinks within the fin .Thermal conductivity is constant and uniform in all directions.Heat transfer coefficient is constant and uniform over fin faces.Surrounding temperature is constant and uniform.Base temperature is constant and uniform over fin base.Fin width much smaller than fin height or length.No bond resistance between fin base and prime surface.
Heat flow off fin proportional to temperature excess.
Why Fins are needed?
Contd .
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Temperature gradient dT/dx ,
Surface temperature, T, Are expressed such that T is a function of x only.
Newtons law of cooling
Two ways to increase the rate of heat transfer: increasing the heat transfer coefficient , increase the surface area fins
conv s sQ hA T T
Extended surfaces may exist in many situations but are commonly used as
Contd .
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fins to enhance heat transfer by increasing the surface area available forconvection (and/or radiation).
Some typical fin configurations:
Straight fins of (a) uniform and (b) non-uniform cross sections; (c) annularfin, and (d) pin fin of non-uniform cross section.
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Innovative Fin Designs
Steady One-dimensional Conduction through Fins
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Steady One dimensional Conduction through Fins
Adapted from Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition,
McGraw Hill 2007.
Contd .
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Energy Balance on Volume Element (fin)Contd .
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Energy Balance on Volume Element (fin)
rate of heatconduction intothe element at x
rate of heatconduction fromthe element at
x+x
rate of heatconvection fromthe element
+=
))(( T T x P hQ S conv
conv x xcond xcond QQQ ,,
0)(,, T T xhP QQ xcond x xcond
Expressing the surface area, A s, in terms of width, x, times the perimeter P
Contd .
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0 x
0)( T T hP dxdQ cond
0)(,, T T hP x
QQ xcond x xcond
Taking limit x tends to zero and using the definition of derivative:
dxdT
kAQ ccond
0)(
TThPdT
kAd
Contd .
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0)(
T T hP dx
kAdx c
At constant A C and kSolution is;
C kAhp
m 2 T T
0)(22
T T kAhP
dxT d
c
Putting following constants as:
2d
Contd .
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022
mdxd
mxmx eC eC x 21)(
Equation (A) is a linear, homogeneous, second-order differentialequation with constant coefficients.
The general solution of Eq. (A) is
(A)
Boundary Conditions
Contd .
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Several boundary conditions are typically employed: At the fin base
Specified temperature boundary condition, expressedas: (0)= b= T b-T
At the fin tip1. Specified temperature
2. Infinitely Long Fin3. Adiabatic tip
4. Convection (andcombined convection).
Contd .
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Base ( x = 0) condition 0 b bT T
Tip ( x = L) conditions A. :Conve ti |c on / x Lkd dx h L
B. :A / |diabati 0c x Ld dx
Fixed temper C. :ature L L D. (I >2.65): 0nfinite fin mL L Fin Heat Rate:
0| f f c x A sd
q kA h x dAdx
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Solutions of Differential EquationContd .
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)( T T hPkAq bc f
Fin Efficiency
To maximize the heat transfer from a fin the temperature
Fin Performance
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To maximize the heat transfer from a fin the temperatureof the fin should be uniform (maximized) at the base value
of T b In reality, the temperature drops along the fin , and thus
the heat transfer from the fin is less To account for the effect we define
a fin efficiency
or,max
fin fin
fin
Q
Q Actual heat transfer rate from the fin
Ideal heat transfer rate from the finif the entire fin were at base temperature
,max ( ) fin fin fin fin fin bQ Q hA T T
For constant cross section of very long fins:
Contd .
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For constant cross section of very long fins:
For constant cross section with adiabatic tip:
, ,max
1 1 fin c b clong fin
fin fin b
Q hpkA T T kAQ hA T T L hp mL
, ,max
tanh
tanh
fin c badiabatic fin
fin fin b
Q hpkA T T mL
Q hA T T
mLmL
Afin = P*L
Fin Effectiveness
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The performance of the fins is judged on the basis of the
enhancement in heat transfer relative to the no-fin case. The performance of fins is expressed
in terms of the fin effectiveness fin
defined as
fin fin
finno fin b b
Q Q
Q hA T T
Heat transfer ratefrom the surface
of area A b
Heat transfer ratefrom the fin of base
area A b
with , and / f ch k A P
Fin Arrays
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Representative arrays of(a) rectangular and(b) annular fins.
Total surface area:t f b A NA A
Number of fins Area of exposed base ( prime surface)
Define terms: A b: base area exposed to coolantAf : surface area of a single finAt: total area including base area and total finned surface,
N: total number of fins
( ) ( )t b f b b f f bq q Nq hA T T N hA T T
Overall fin efficiency for an array of fins: Contd .
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Overall surface efficiency and resistance :
,1b
t o
t o t
Rq hA
1 1 f o f t
NA
A
bt
t t o hA
q
max
OR
[( ) ]( ) [ (1 )]( )
[1 (1 )]( ) ( )
Define overall fin efficiency: 1 (1 )
t f f f b t f f b
f t f b O t bt
f O f
t
h A NA N A T T h A NA T T
NAhA T T hA T T A
NA
A
Total heat rate:
,
bt f f b b b o t b
t o
q N hA hA hA R
Contd .
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,,
,
,
1( ) where
Compare to heat transfer without fins
1( ) ( )( )
where is the base area (unexposed) for the finTo enhance heat transfer
Th
bt t O b t O
t O t O
b b b f b
b f
t O
T T q hA T T R
R hA
q hA T T h A NA T T hA
A A A
Oat is, to increase the effective area . t A
=Ab+NAb,f
Thermal Resistance ConceptContd .
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T1 T
Tb T2
T1 T Tb T2
L1 t
R1=L1/(k 1A)
Rb=t/(k bA)
)/(1, Ot Ot hA R
1 1
1 ,b t O
T T T T q R R R R
A=Ab+NAb,f
Topics to be referred by the students from Textbooks
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Topics to be referred by the students from Textbooks
Shape factor
Effect of variable conductivity
Logarithmic mean radius
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References:
Heat Transfer A Practical Approach, Y.A. Cengel, Second Edition, Tata McGraw Hill2003.
Heat and Mass Transfer A Practical Approach, Y.A. Cengel, Third Edition, TataMcGraw Hill 2007.
Heat Transfer , P. K. Nag, First Edition, Tata McGraw Hill 2002.
Heat and Mass Transfer , D.S. Kumar, Sixth Edition, S.K. Kataria & Sons 2004.