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HEAT CONDUCTION THROUGH FINSFINS
Prabal TalukdarPrabal TalukdarAssociate Professor
Department of Mechanical EngineeringDepartment of Mechanical EngineeringIIT Delhi
E-mail: [email protected]
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
Pin fins
The thin plate fins of a car radiator greatly increase the rate of heat transfer to the air
Fins from nature
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).
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