HEAT CONDUCTION THROUGH FINSFINS

Prabal TalukdarPrabal TalukdarAssociate Professor

Department of Mechanical EngineeringDepartment of Mechanical EngineeringIIT Delhi

E-mail: prabal@mech.iitd.ac.inp

Mech/IITD

IntroductionIntroduction)TT(hAQ ssconv ∞−=&

There are two ways to increase the rate of heat transfer• to increase the convection heat transfer coefficient h o c e se e co vec o e s e coe c e• to increase the surface area As

• Increasing h may require the installation of a pump or fan, • Or replacing the existing one with a larger onep g g g• The alternative is to increase the surface area by attaching to the surface extended surfaces called fins• made of highly conductive materials such as aluminum• made of highly conductive materials such as aluminum

Heat sinks

Pin fins

The thin plate fins of a car radiator greatly increase the rate of heat transfer to the air

Fins from nature

Longitudinal fins

Rectangular Trapezoidal

Triangular Concave parabolic

Convex parabolic

Radial fins:

Rectangular profile Triangular profile Hyperbolic profile

Pins:

g p Triangular profile

(a)Cylindrical (b)conical (c) concave parabolic (d) convex parabolic

Radial fin coffee cup

Fin Equation

dxdTkAQ cx −=&dx

dxdxQdQQ x

xdxx&

&& +=+

( )∞−= TThdAQd sconv&

convdxxx QdQQ &&& += +

Energy Balance:

)TT(hdAdxdxQdQ s

xx ∞−++=

&&

dx

0)TT(dx

dAkh

dxdTA

dxd s

c =−−⎟⎠⎞

⎜⎝⎛

∞

0)TT(dx

dAkh

A1

dxdT

dxdA

A1

dxTd s

c

c

c2

2=−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+ ∞

2 ⎞⎛⎞⎛

Fins with uniform cross sectional area

0)(112

2

=−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+ ∞TT

dxdA

kh

AdxdT

dxdA

AdxTd s

c

c

c

0dA 0=dx

dA c

PxAs = Pd

dA s =dx

0)(2

2

=−⎟⎟⎠

⎞⎜⎜⎝

⎛− ∞TT

kAhP

dxTd

⎟⎠

⎜⎝ kAdx c

Excess temperature θ

∞−≡ TxTx )()(θ

2θd 022 =− θθ m

dxd

(a) Rectangular Fin (b) Pin fin

General solution and Boundary Conditions

022

θθd 022 =− θm

dx

ckAhPm =2

c

∞−≡ TxTx )()(θ

The general solution is of the form

mxmx eCeCx −+= 21)(θ

g

Convection from tip

q denotes Q&q denotes Q

BCs

T)0(T[ ] Lxcc |

dxdTkAT)L(ThA =∞ −=−

bb TT)0(T)0(T

θ≡−=θ=

∞

∞ dx

Lx|dxdk)L(h =θ

−=θ

)xL(msinh)mk/h()xL(mcosh −+−=

θmLsinh)mk/h(mLcoshb +

=θ

Heat transfer from the fin surface:Heat transfer from the fin surface:

0xc0xcbf |dxdkA|

dxdTkAQQ ==

θ−=−== &&

mLcosh)mk/h(mLsinhhPkAQ bcf+

θ=&

dxdx

mLsinh)mk/h(mLcoshhPkAQ bcf +

θ

dA)(hdA]T)(T[hQ θ∫∫•

Another way of finding Q

finAfinAfin dA)x(hdA]T)x(T[hQfinfin

θ∫=−∫= ∞

Insulated tipInsulated tip

BC 1: BC 2:

General Sol: mxmx eCeCx −+= 21)(θ

bb TT θθ ≡−= ∞)0( 0| == Lxddθ

C bb TT θθ ∞)0( | = Lxdx

L )(hθmL

xLm

b cosh)(cosh −

=θθ

00 || == −=−== xcxcbf dxdkA

dxdTkAqq θ

mLhPkAq bcf tanhθ=

Prescribed temperaturePrescribed temperatureThis is a condition when the temperature at the tip is known p p(for example, measured by a sensor)

( )mL

xLmmxbL

sinh)(sinhsinh −+

=θθ

θθ

mLb sinhθ

( )L

mLhPkAq bLbcf i h

cosh θθθ −=

mLq bcf sinh

Infinitely Long Fin (Tfin tip = T∞)y g ( fin tip ∞)

Boundary condition at the fin tip: 0T)L(T)L( =−=θ ∞ as ∞→L

mx2

mx1 eCeC)x( −+=θ

The general solution is of the form

Possible when C1 → 0

mxC)( −θ

Apply boundary condition at base and find T

mx2eC)x( =θ

hP

ckAhPx

b e)TT(T)x(T−

∞∞ −+=

dT•)TT(hPkA|

dxdTkAQ bc0xclongfin ∞= −=−=

Corrected fin lengthg

PALL c

c +=Corrected fin length:

Multiplying the relation above by the perimeter gives

Pcg

the perimeter gives Acorrected = Afin (lateral) + Atip

tLL2

LL gularfintanrec,c +=

DLL lfincylindricac +=

Corrected fin length Lc is defined such that heat transfer from a fin of

4lfincylindrica,c

g clength Lc with insulated tip is equal to heat transfer from the actual fin of length L with convection at the fin tip.

Fin Efficiencyy== •

•

finfin

Q

Qη

Actual heat transfer rate from the finIdeal heat transfer rate from the fin

In the limiting case of zero thermal resistance or infinite thermal cond cti it (k ) the temperat re

max,finQ if the entire fin were at base temperature

infinite thermal conductivity (k →∞ ), the temperature of the fin will be uniform at the base value of Tb.

)(••

TThAQQ )(max, ∞−== TThAQQ bfinfinfinfinfin ηη

kATThPkAQcbcfin 11)(==

−== ∞

•

η

mLmLTThPkAQ

mLhpLTThAQ

f

bfinfin

longfin

tanhtanh)(

)(max,

−

==−

==

•

∞•η

mLmL

TThAmLTThPkA

Q

Q

bfin

bc

fin

finipinsulatedt

tanh)(

tanh)(

max,

=−

==∞

∞•η

Fin Effectiveness===

•

•

•

)( TThAQQ finfin

finεHeat transfer rate fromthe fin of base area AbHeat transfer rate from− ∞ )( TThAQ bb

nofinHeat transfer rate fromthe surface of area Ab

h kAQ•

)(

cbb

bc

nofin

finlongfin hA

kPTThA

TThPkA

Q

Q=

−−

==∞

∞• )(

)(ε

1. k should be as high as possible, (copper, aluminum, iron). Aluminum is preferred: low cost and weight, resistance to corrosion.p g ,

2. p/Ac should be as high as possible. (Thin plate fins and slender pin fins)

3. Most effective in applications where h is low. (Use of fins justified if when the medium is gas and heat transfer is by natural convection).

Fin Effectiveness••

QQ finfinHeat transfer rate fromth fi f b A=

−==

∞• )( TThA

Q

Q

Q

bb

fin

nofin

finfinε the fin of base area Ab

Heat transfer rate fromthe surface of area Ab

Does not affect the heat transfer at all.1=finεFin act as insulation (if low k material is used)

Enhancing heat transfer (use of fins justified if εfin>2)1

1

>

<

fin

fin

ε

ε

Overall Fin EfficiencyWhen determining the rate of heat transfer from a finned surface, we must consider the

y,

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

)()(,

∞∞

•••

−+−=

+=

TThATThA

QQQ

bfinfinbunfin

finunfinfintotal

η

We can also define an overall effectiveness for a finned surface as the

))(( ∞+ −= TTAAh bfinfinunfin η

effectiveness for a finned surface as the ratio of the total heat transfer from the finned surface to the heat transfer from thesame surface if there were no fins

•

)())((

,

,,

∞

∞•

•

−

−+==

TThATTAAh

Q

Q

bnofin

bfinfinunfin

nofintotal

fintotaloverallfin

ηε

Efficiency of circular, rectangular, and triangular fins on a plain surface of width w (from Gardner, Ref 6).

Efficiency of circular fins of length L and constant thickness t (fromGardner, Ref. 6).

Proper length of a fin“a” means “m”

Fi i h i l d b li fil i l• Fins with triangular and parabolic profiles contain less materialand are more efficient requiring minimum weight

• An important consideration is the selection of the proper fin length L. Increasing the length of the fin beyond a certain value cannot be justified unless the added benefits outweigh the addedcannot be justified unless the added benefits outweigh the added cost.

Th ffi i f t fi d i ti i b 90 t•The efficiency of most fins used in practice is above 90 percent