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Chapter (3)
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By the end of today’s lecture, you should be able to:
Learn how to deal with heat generation problems.
Learn how to treat the extended surfaces (fins)
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Consider situations for which thermal energy is being Generated, as in
chemical reactions, electrical energy, nuclear energy,
:Assumptions
• The energy generation is uniform per unit volume
• Steady state.
• One dimension
• Constant thermal conductivity
.constq
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sT
x
x=0 x=Lx=L
Tmax = T00
dx
dT
q
sT
Th,Th,
t
TcqTk p
From the general equation:
Apply the previous assumptions:
0 qTk
02
2
qdx
Tdk
1Cxk
q
dx
dT
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2
2CxCx
k
qT
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sT
x
x=0 x=Lx=L
Tmax = T00
dx
dT
q
sT
Th,Th,
From the boundary conditions:
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2
2CxCx
k
qT
At X = L T = Ts
At X = -L T = Ts
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2
2CLCL
k
qTs
21
2
2CLCL
k
qTs
From the above equations C1 = 0
2
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Lk
qTC s
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sT
x
x=0 x=Lx=L
Tmax = T00
dx
dT
q
sT
Th,Th,
)(2
22 xLk
qTT s
For maximum temperature:
For symmetric heat generation,
the maximum temperature at x = 0
0dx
dTIn this case:
1Cxk
q
dx
dT
01 Cxk
qX = 0
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sT
x
x=0 x=Lx=L
Tmax = T00
dx
dT
q
sT
Th,Th,
2
max02
Lk
qTTT sx
Relationship between surface temperature
and environmental temperature
Heat balance:
convgen qVqq
)(2)2*( TThALAq s
Lh
qTTs
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From the general equation:
01
k
q
dr
dTr
dr
d
r
srrr TTanddr
dTr )(0 0
0
q
r0 L0dx
dT
Th,
sT
sTr
r
k
rqrT
2
0
22
0 14
)(
By using the boundary conditions:
The temperature distribution is:
By the same method, the relationship between
surface temperature and environmental temperature is:h
rqTTs
2
0
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Fins are extended surfaces that are utilized in the removal of heat
from a body
One can increase heat transfer by increasing the heat transfer
coefficient or increasing the surface area
Finned surfaces are manufactured by extruding, welding, or
wrapping a thin metal sheet on a surface
Fins are often seen in electrical appliance such as in computer
power supply cooling or substation transformers and are also used
for engine cooling such as car radiators
Example of heat sink mounted
on a CPU microchip
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Many designs are possible, where Fin designs are only limited by imagination
:(a) Straight fin of uniform cross-section
(b) Straight fin non-uniform cross-section
(c) Annular fin
(d) Pin fin
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How fins work
Fins enhance heat transfer from a surface by exposing a larger
surface area to convection and radiation
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Variation of Temperature along a Fin
0 2 4 6 8 10 12 14290
300
310
320
330
340
350
360
370375
290
T x( )
150 x
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Assumptions:
• Steady state.
• One dimension
• Constant thermal conductivity
• No heat generation
•Uniform cross section area
T∞, = temp. of surrounding
Tb = fin temp. at base
A = cross-sectional area of fin
P = perimeter of fin
L = fin length
Tb
T
Tb
x
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For the differential Element shown;
The heat balance equation is:
dxat x
element theFrom
Convection
heat of Rate
dxat x
element theFrom
Conduction
heat of Rate
at x
element theInto
Conduction
heat of Rate
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Substitute in the equation;
)(2
2
TTkA
hP
dx
Td
Put,
kA
hpmandTT
2
02
2
2
mdx
d
TTpdxhqconv )(dx
dTkAqx dx
dx
dqqq xdxx
convdxxx qqq
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The general solution to this differential
equation is:
Integrate the equation,
mxmx eCeCx 21)(
02
2
2
mdx
d
From the boundary conditions:
At X = 0 T = Tb bb TT
1st boundary condition:
Boundary condition at fin base
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CASE I. Infinitely Long Fin (Tfin tip =T ∞)
Assume a very long fin, L approaches infinity. Thus, the
temperature at the end of the fin should be practically equal to
the temperature of the surrounding fluid. Then;
At X = L T = T∞
0 TT
2nd boundary condition:
At X = L different conditions are possible
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Determination of C1 and C2 leads to,
Heat transfer:
mxmx eCeCx 21)(Substitute in the previous equation:
mxex b
)( x
kA
hp
fbfx eTTTT
,,)(
OR;
fbc TThPkAq ,
xkA
hp
fb
fxe
TT
TT
,
,)(
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fbc TThPkAq ,
For Very long fin:
Where:
P is the perimeter,
Ac is the cross-sectional area of the
fin, and
x is the distance from the fin base.
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The fin has a finite length, L, and the end is insulated, which means that;
At X = L
Or:
Heat transfer
CASE II. Negligible Heat Loss from the Fin Tip (Insulated fin tip,
Qfin tip = 0)
mL
xLm
TT
TT
fb
fx
b cosh
cosh
,
,)(
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The fin has a finite length, L, and losses heat by convection from its end
At X = L
CASE III. Convection (or Combined Convection and Radiation)
from Fin Tip
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CASE III. Convection (or Combined Convection and Radiation)
from Fin Tip
A practical way of accounting for the heat loss from the fin tip is to
replace the fin length L in the relation for the insulated tip case by a
Defined length as:
Corrected length
P
ALL c
c
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CASE III. Convection (or Combined Convection and Radiation)
from Fin Tip
Corrected length
Using the proper relations for Ac and p, the corrected lengths for
rectangular and cylindrical fins are easily determined to be:
2finr rectangula,
tLLc
4fin lcylindrica,
DLLc
Where:
t is the thickness of the rectangular fins and
D is the diameter of the cylindrical fins.
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The maximum heat loss through convection will be
when the driving force (i.e. temperature difference
between the base and fluid) remains the same at all
points along the fin.
To determine, how an actual fin design compares to
this theoretical fin, we calculate the fin efficiency.
A definition of the fin efficiency is therefore,
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For example, for Case II, (insulated tip):
Fin efficiency relations are developed for fins of various profiles and
are plotted charts as shown in the following charts
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Fins are used to enhance heat transfer, and the use of fins on a surface
cannot be recommended unless the enhancement in heat transfer justifies
the added cost and complexity associated with the fins. In fact, there is
no assurance that adding fins on a surface will enhance heat transfer. 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 expressed in
terms of the fin effectiveness fin is defined as:
fin < 1 , fin = 1 , fin > 1
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When determining the rate of heat transfer from a finned
surface, we must consider the unfinned portion of the surface as
well as the fins. Therefore, the rate of heat transfer for a surface
containing n fins can be expressed as:
