07. Extended Surface Fins

7/28/2019 07. Extended Surface Fins

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7. EXTENDED SURFACES - FINS

The Function of Fins

Increase heat transfer rate for a fixed surfacetemperature, or

Lower surface temperature for a fixed heat transfer rate

Newton's law of cooling

)T(TaSQsss =&

Rewrite (3.63)

s

s

s aS

QTT &+=

sQ& Extending Ss for a fixed Tsincreases

Extending surface at fixed lowerssQ&

sT


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A surface is extended by adding fins Examples of fins:

Thin rods on the condenser in back of refrigerator.

Honeycomb surface of a car radiator. Corrugated surface of a motorcycle engine. Coolers of PC boards.

Types of Fins

finstraight

areaconstant(a) finpin(c)

finstraight

areavariable(b) finannular(d)

3.12Fig.


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Fin terminology and types

Fin base (pata ebra) Fin tip (konec ebra)

Heat Transfer and Temperature

Distribution in Fins

Heat flow through a fin isaxial and lateral (2-D)

Note temperature profile

Temperature distribution is 2-D

x

Th,

T

Tr

3.13Fig.

T,


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The Fin Approximation

Neglect temperature variation in the lateral direction r

and assume uniform temperature at any section:

T T(x)

Criterion for justifying this approximation:Biot number =Bi= /


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Conservation of energy for an element for steady stateand no energy generation:

outin EE && =

Energy in by conduction =x

Q&

Energy out by conduction = /dx)dxQ(dQ&&

+

The Fin Heat Equation


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Energy out by convection =conv

Qd&

Neglect radiation:

0Qddxxd

Qd

conv

x

=+&

&

(c)

(a)Q&=

(b)convx

xoutQddx

dxQdQE &&

& ++=

(a) and (b) into outin EE && = )(b

CsdA

cdq

xq dxdx

dqq xx+

dS=P.dx

convQd&

P


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0Pdx)T(Tdx

dx

dTA

dx

d=

(f)

A = cross-sectional conduction area

P.dx= dS= surface area of element through whichheat is convected

is a surface average heat transfer coefficient

(d) and (e) into 0Qddxxd

Qd

convx

=+&

&

(e))P.dxT(TQd conv =&

Fourier's and Newtons laws:

dx

dTAQx =& (d)


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TTOr assuming constant and replacing TwithT

Assume: constant

0)T(TA

P

dx

Td2

2=

Heat equation for fins

0)T(T

A

P

dx

)(T-Td2

2

=


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Boundary Conditions

Two B.C. are needed


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Determination of Fin Heat Transfer Rate oQ&

Conservation of energy applied to a fin at steady state:

(1) Conduction at the base: Fourier's law( )

0xodx

TTdAQ =

=&

Two methods to determine oQ&

= conduction at the base =

convection at the surfaceoQ&


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(2) Convection at the fin surface: Newton's law

= o P.dx]T[T(x)Q&

Introducing a new parameterfin parameter m

LA

PmL =

0)T(Tmdx

)T-(Td2

2

2

=

Rewrite Heat equation for fins 0)T(TA

P

dx

)(T-Td2

2

=


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` Heat equation is valid for:(1) Steady state

(2) Constant

(3) No energy generation

(4) No radiation

(5)Bi


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mx

2

mx

1eCeCTT(x)

+=

C1

and Care integration constants.

They depend on boundary conditions.

Solution to the equation: 0)T(Tmdx

)T-(Td22

2

=

Two boundary conditions: at the base and at the tip


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Consider 3 cases of constant area fins

(i) Specified temperature at the base, semi-infinite fin

(ii) Finite fin, specified temperature at the base,

insulated tip(iii) Finite fin, specified temperature at the base, heat

transfer at the tip


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The base is at temperature oT Ambient temperature is T

Case (i): Semi-infinite fin with specified

temperature at the base

Remind the solution: mx2

mx

1eCeCTT(x)

+=

B.C. are: T(0) = ,To (a)

= TTTT(0)0

(b)T() = ,T 0T)T( =

0 x

CTh,

Th,

oT cA3.16Fig.

T,

T,

P

A


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mx)exp(TT

TT(x)

o

=

=

mx))exp(T(TTT(x)o

=


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Fin heat transfer rateo

Q&

( ))TAm(T

dx

TTdAQ 00xo =

=

=&

( )= TTPAQ 0o&

Obviously,

no dependence on the fin length

( )= TTA

Q0

o&


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Case (ii): Finite length fin with specified

temperature at the base and insulated tip

Same as Case (i) except

the tip is insulated.

mx

2

mx

1eCeCTT(x)

+=General solution:

0=

=

==

LLx

x

TSQ& cA

C

th0

oT

Th,

xTh,

3.17Fig.

,T

,T

P

( )0dx

TTdLx =

=

Tip B.C.


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About hyperbolic functions:

xx

xxxxxx

ee

ee

coshx

sinhxtghx;

2

eecoshx;

2

eesinhx

+

==

+=

=

Two B.C. - fin base (specified temperature) and tipzero heat transfer rate give C1 and C2,


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mL

xLm

TT

TxT

o cosh

)-(cosh)(=

=

Solution:

ml

mL)TAm(TQ 0ocosh

sinh=&

( ))(tgh

0

mLTTPA

Qo =

&

Similarly heat transfer rate

(tepeln tok):

( )0xo

dx

TTdAQ =

=&

Tip temperature (x=L):mLTT

TT LL

cosh

1

0

=


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T0,014.(TTT 0L =

Compare with the semi-infinite fin: ( )= TTPA

Q0

o&

( )1=

TTPA

Q

0

o&

014,0

0

=

TT

TT LL


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Case (iii): Finite length fin with specified

temperature at the base and heattransfer at the tip

( )( ) mLmmL

xLmmxLm

TT

TxT

L

L

sinh/cosh

)(sinh/)(cosh)(

0

++=

cA

C

th0

oT

Th,

xTh,

3.17Fig.

,T

,T

P

,T


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Corrected LengthLc

Fins with insulated tips have simpler solutions thanfins with convection at the tip

Simplified model: Assume insulated tip andcompensate by increasing the length by Lc

The corrected length Lcis cc LLL

The correction incrementLc = t/2 fin half width0,0625

t

Error negligible if


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Fin Effectiveness (efektivnost) f

Fin Efficiency (innost) f

Fin performance is described by two parameters:

FinEffectiveness fenhancement due to fin addition.

: Measures heat transfer

A

Ratio of heat transfered by the fin and heat transfered

from the original areaA of the fin ( )= TTAQ obase

Defined as

( ) baseo

o

o

f Q

Q

TTA

Q

&

&&

=

=

ml

mL)TAm(TQ 0ocosh

sinh=&


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f must be >2, otherwise dont use the fin.

( )( ) A

P

TTA

TTAP

o

of =

=

For a semi-infinite fin:

How to increase fin effectivness:

high conductivity material

high perimeter to area ratio P/A thin fin and close

each to other use fin where heat transfer coefficient is low


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fS = total fin surface area

Introduce the above into equation for f

FinEfficiency fwith the maximum heat that the fin

can transfer. Defined as

: Compares heat transfer from a fin

o,max

of Q

Q

&

&

=

the base temperature

= heat transfer from fin if its entire surface is ato,maxQ&

( )= TTSQ ofo,max&

( )

=

TTS

Q

of

of

&


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Total Fin Surface Efficiency (innost) tot

( )==

TTS

Q

Q

Q

oTOT

TOT

TOT,max

TOTTOT

&

&

&

When the entire finned surface is at the

base temperatureTOT,maxQ

&

TOTSTotal finned surface (including surface between

individual fins) ofTOT SSS +=

fS

oS

TOTQ& Total heat transfer rate from the total finned

surface of an actual finned surface

( ) ( )+= TTSTTSQ

oooffTOT&


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( ) ( )

= TT

S

SSQ of

TOT

fTOTTOT 11&

After some rearrangement

Taking the definition of the total efficiency

( )( )

=

= TTSQTTS

Q oTOTTOTTOT

oTOT

TOTTOT

&&

( fTOTfTOT

S

S = 11

we come to the definition of a total efficiency


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Practical procedure how to calculate heat transfer rate

from a finned surface

fS

oS

1. Specify efficiency of an individual fin f using graphs

ofTOT SS +=

2. Calculate total surface of fins and

free space between fins

3. Calculate total efficiencyT OT

( )fTOT

fTOT

SS = 11

5. Calculate actual heat transfer rate

TOT,maxTOTTOT QQ&& =

4. Calculate maximum heat transfer rate

)T(TSQ oTOTTOT,max =&


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Fin efficiency for three types of straight fins

Graphs of fin efficiency


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Fin efficiency for annular fins

Graphs of fin efficiency

07. Extended Surface Fins

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