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Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 1
FINNED SURFACES
Introduction:
Convection: Heat transfer between a solid surface and a moving fluid is governed by the
Newton’s cooling law: where Ts is the surface temperature and T� is the
fluid temperature. Therefore, to increase the convective heat transfer, one can
• Increase the temperature difference between the surface and the fluid.
• Increase the convection coefficient h. This can be accomplished by increasing the fluid
flow over the surface since h is a function of the flow velocity and the higher the velocity,
the higher the h. Example: a cooling fan.
• Increase the contact surface area A. Example: a heat sink with fins.
Many times, when the first option is not in our control and the second option (i.e. increasing h) is
already stretched to its limit, we are left with the only alternative of increasing the effective
surface area by using fins or extended surfaces. Fins are protrusions from the base surface into
the cooling fluid, so that the extra surface of the protrusions is also in contact with the fluid.
Most of you have encountered cooling fins on air-cooled engines (motorcycles, portable
generators, etc.), electronic equipment (CPUs), automobile radiators, air conditioning equipment
(condensers) and elsewhere.
Temperature distribution and Heat transfer in fins (rectangular or circular)
4
Governing differential equation:
Let us take small element dx on a rectangular fin with uniform cross sectional area A
)(∞
−TsT
)(∞
−= TsThAQ
convQ
dxxQ
xQ +
+=
dx
dTAk
xQ −= dx
dx
TdAk
dx
dTAkdx
dx
dTAk
dx
d
dx
dTAk
dxxQ
2
2
−−=
−+−=+
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 2
This equation is called, one dimensional fin equation for fins of uniform cross section(second
order, linear, ordinary equation)
The solution is,
Where C1 and C2 are constants
C1 and C2 are constant are determined by the application of the two boundary condition, one
specified for the fin tip.
x= 0 at T=Tb and the other boundary condition there are several possibities
1. Infinitely long fin
2. Fin insulated at its end (i. negligible heat loss from the end of the fin)
3. Fin loosing heat from its end by convection
4. Fin with specified temperature at its end
convQ
dxxQ
xQ +
+=
( )∞
−+−−=− TTdxPhdxdx
TdAk
dx
dTAk
dx
dTAk )(
2
2
( )∞
−= TTsAh
convQ ( )
∞−= TTdxPh
convQ )( dxpermeterAs )(=
( )∞
−= TTdxPhdxdx
TdAk )(
2
2
( )∞
−= TTKA
Ph
dx
Td2
2
( )∞
−= TTmdx
Td 22
2
KA
Phm =
KA
Phm =2
0)(2)(2
2
=− xmdx
xdθ
θ ∞−= TTxlet )(θ
dx
dT
dx
xd=
)(θ
mxmx eCeCx 21)( += −θ
mxCmxCx sinhcosh)( 21 +=θ
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 3
Infinitely long fin
Assuming that fin tip is insulated
Boundary conditions,
Heat transfer rate
(Refer HMTDB Page No-50)
∞= T
LT
0)(2)(2
2
=− tmdx
tdθ
θ
mxmx eCeCt 21)( += −θ
bTTatx == 0
∞==∞= T
LTTatx
∞−= TTb bθ
0=L
θ
1Cb=θ
bTTatx == 0 ∞−= TT
b bθ
∞==∞= T
LTTatx 0=
Lθ
∞= 20 C 02 =C
mxeb
x −= θθ )(
mxe
b
x −=θθ )(
mxeT
bT
TT−=
∞−∞
−
0
)(
0 =−=
=−=
xdx
xdAk
xdx
dTAkQ
θ
)( bmAkQ θ−−=
mxe
b
x −=θθ )(
mx
b ex −=θθ )(
mx
b emdx
xd −−= θθ )(bmAkQ θ=
)( ∞−= TTKA
hPAkQ b
)( ∞−= TThPKAQ b
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 4
Fin of finite length with insulated tip (short fin with end insulated)
Boundary conditions,
bTTatx == 0 ∞−= TT
b bθ
0==dx
datLx
θ
( )∞
−= TTmdx
Td 22
2
KA
Phm =
KA
Phm =2
0)(2)(2
2
=− xmdx
xdθ
θ ∞−= TTxlet )(θ
dx
dT
dx
xd=
)(θ
mxCmxCx sinhcosh)( 21 +=θ
bTTatx == 0 ∞−= TT
b bθ
1Cb=θ
( ) ( )mxmCmxmbdx
tdcoshsinh
)(2+= θ
θ
( ) ( )mLmCmLmb
sinhcosh0 2+= θ
( )( )mL
mLbCcosh
sinh
2
θ−=
( )( )
mxmL
mLbmx
bt sinh
cosh
sinhcosh)(
−+=
θθθ
( )( )
mxmL
mLmx
b
tsinh
cosh
sinhcosh
)(
−=
θθ
( ) ( )( )mL
mxmLmLmx
b
t
cosh
sinhsinhcoshcosh)( −=
θθ
( )( )mL
xLm
b
t
cosh
cosh)( −=
θθ
( )( )mL
xLm
TbT
TT
cosh
cosh −=
∞−∞
−
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 5
(Refer HMTDB Page No 50)
Heat transfer rate
(Refer HMTDB Page No 50)
Heat transfer can also be find out by,
(Refer the method of last derivation)
( )( )mL
xLm
TbT
TT
cosh
cosh −=
∞−∞
−
0
)(
0 =−=
=−=
xdx
xdAk
xdx
dTAkQ
θ
( )( )[ ]
0
sinhcosh
=
−−−=
x
xLmmmL
bAkQθ
( )( )mL
xLm
b
x
cosh
cosh)( −=
θθ
( )( )
−=
mL
xLmx b
cosh
cosh)( θθ
( )( )[ ] )10(sinh
cosh
)(−−= xLmm
mL
b
dx
xd θθ
( )( )[ ]xLmm
mL
b
dx
xd−−= sinh
cosh
)( θθ( )mL
mL
bAmkQ sinhcosh
θ=
( ) mLTbTAmkQ tanh∞−=
( ) mLTbThPKAQ tanh∞−=
∫ ∞−=L
o
TxTdxPhQ ))(()(
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 6
Fins with convection at the tip
Boundary conditions
Solution for the second order differential equation
0)(22
)(2=− xm
dx
xdθ
θ
0)( θθ =
∞−= T
bTx 0=xat
0)()(
=+ xLh
dx
xdk θ
θLxat =
)(sin2
)(cos1
)( xmhCxmhCx +=θ
)(sin2
)(cos1
)( xmhCxmhCx +=θ0
)( θθ =∞
−= TbTx 0=xat
010+=Cθ
)(sin2
)(cos0
)( xmhCxmhx +=θθ
0)()(
=+ xLh
dx
xdk θ
θ Lxat =
)(cos2
)(sin0
)(xmhmCxmhm
dx
xd+=θ
θ
LxxA
Lh
Lxdx
xdkA
==
=− )(
)(θ
θ
[ ] [ ] 0)(sin)(cos)(cos)(sin 2020 =+++ LmhCLmhLhlmhmCLmhmk θθ
0)()(
=+ xLh
dx
xdk θ
θLxat =
[ ] [ ] 0)(sin)(cos)(cos)(sin 2020 =+++ LmhChLmhhLmhmCkLmhmk LLθθ
[ ] [ ] 0)(sin)(cos)(cos)(sin 2200 =+++ LmhChLmhmCkLmhhLmhmk LLθθ
+−=
+ )(sin)(cos)(cos)(sin 20 Lmh
km
Lh
LmhCLmhkm
Lh
Lmhθ
10C=θ
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 7
)(sin)(cos
)(cos)(sin0
2
Lmhkm
Lh
Lmh
Lmhkm
Lh
Lmh
C
+
+
−=
θ
)(sin
)(sin)(cos
)(cos)(sin
)(cos0
)( 0 xmh
Lmhkm
Lh
Lmh
Lmhkm
Lh
Lmh
xmhx
+
+
−= θθθ
)(sin
)(sin)(cos
)(cos)(sin
)(cos
0
)(xmh
Lmhkm
Lh
Lmh
Lmhkm
Lh
Lmh
xmhx
+
+
−=θθ
)(sin)(cos
)(sin)(cos)(sin)(sin)(cos)(cos
0
)(
Lmhkm
Lh
Lmh
xmhLmhkm
Lh
LmhLmhkm
Lh
Lmhxmhx
+
+−
+
=θθ
)(sin)(cos
)(sin)(cos)(sin)(sin)(cos)(sin)(cos)(cos
0
)(
Lmhkm
Lh
Lmh
xmhLmhkm
Lh
xmhLmhxmhLmhkm
Lh
Lmhxmhx
+
+−
+
=θθ
[ ]
)(sin)(cos
)(sin)(cos)(cos)(sin)(sin)(sin)(cos)(cos
0
)(
Lmhkm
Lh
mLh
xmhLmhkm
Lh
xmhLmhkm
Lh
xmhLmhxmhLmhx
+
−+−
=θθ
[ ] [ ]
)(sin)(cos
)(sin)(cos)(
0
)(
Lmhkm
Lh
mLh
xLmhkm
Lh
xLmh
TbT
TxTx
+
−+−
=∞−∞−
=θθ
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 8
Heat transfer rate
Dividing )(cos Lmh in the equation
(HMTDB Page No 50)
0
)(
0 =−=
=−=
xdx
xdAk
xdx
dTAkQ
θ
[ ] [ ]
+
−+−
=
)(sin)(cos
)(sin)(cos
)( 0
mLhkm
Lh
Lmh
xLmhkm
Lh
xLmh
x θθ
[ ] [ ]
)(sin)(cos
)1)((cosh)1)((sin)(
0
Lmhkm
hLmh
xLmmkm
hxLmhm
dx
xd
L
L
+
−−+−−= θ
θ
( )
+
+
∞−=
)(tan1
)(tan
Lmhkm
h
km
hmLh
TbTAk
LPhQ
L
L
[ ] [ ]
0
)(sin)(cos
)1)((cosh)1)((sin
0
=+
−−+−−−=
x
Lmhkm
Lh
Lmh
xLmmkm
Lh
xLmhm
AkQ θ
[ ] [ ]
)(sin)(cos
)(cosh)(sin
0
Lmhkm
hLmh
mLmkm
hmLhm
AkQL
L
+
−−−= θ
+
+
=
)(sin)(cos
)(cosh)(sin
0
Lmhkm
hLmh
Lmkm
hmLh
AmkQL
L
θ
( )
+
+
∞−=
)(sin)(cos
)(cosh)(sin
Lmhkm
hLmh
Lmkm
hmLh
TbTAk
LPhQ
L
L
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 9
Fin of finite length with specified temperature at its end
Temperature distribution
Boundary conditions
0)(22
)(2=− xm
dx
xdθ
θ
0== xatbTT
LxatLTT ==
∞−= T
bT
bθ
∞−= T
LT
Lθ
)(sin2
)(cos1
)( xmhCxmhCx +=θ
)(sin2
)(cos1
)( xmhCxmhCx +=θ
0== xatbTT
∞−= T
bT
bθ
1C
b=θ
)(sin2
)(cos1
)( xmhCxmhCx +=θ
LxatLTT ==
∞−= T
LT
Lθ
)(sin2
)(cos mLhCLmhbL
+=θθ
)(sin
)(cos
2 mLh
LmhbLCθθ −
=
)(sin)(sin
)(cos)(cos)( xmh
mLh
LmhbLxmh
bx
−+=
θθθθ
[ ])(sin
(sin)(cos)(sin)(cos)(
mLh
xmhLmhbL
mLhxmhbx
θθθθ
−+=
)(sin
)(sin)(cos)(sin)(sin)(cos)(
mLh
xmhLmhb
xmhL
mLhxmhbx
θθθθ
−+=
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 10
Heat transfer
(HMTDB Page No 50)
Fin efficiency
The fin efficiency is defined as the ratio of the energy transferred through a real fin to
that transferred through an ideal fin. An ideal fin is thought to be one made of a perfect or
infinite conductor material. A perfect conductor has an infinite thermal conductivity so that the
entire fin is at the base material temperature.
For long fin
(HMTDB Page No 50)
)(sin
)(sin)(sin)(
mLh
xmhL
xLmhbx
θθθ
+−=
∫ ∞−=
L
o
TxTdxPhQ ))(()(
dxxL
o
PhQ )(θ∫=
dxmLh
xmhL
xLmhb
L
o
PhQ
+−∫=
)(sin
)(sin)(sin θθ
Lom
xmhL
m
xLmhb
mLh
PhQ
+
−−= )
)(cos))(cos
)(sin
θθ
[ ])1)(cos()cos1()(sin
−+−−= LmhL
mLhbmLhm
PhQ θθ
[ ])1)(cos()1(cos)(sin
−+−= LmhL
mLhbmLhm
PhQ θθ
[ ])1(cos)()(sin
−+= mLhLbmLhm
PhQ θθ
PkAhmLh
mLh
LT
LTT
bTQ
)(sin
)1(cos)()(
−
∞−+
∞−=
∞−= T
bT
bθ
∞−= T
LT
Lθ
idealQ
realQ
=η)(
)(
∞−
∞−
=T
bTLPh
TbThPkA
mL
1=
kA
hPm =
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 11
For Short fin (tip is insulated)
(HMTDB Page No 50)
Fin Effectiveness
How effective a fin can enhance heat transfer is characterized by the fin effectiveness
which is as the ratio of fin heat transfer and the heat transfer without the fin. For an adiabatic fin:
for long fin
For Short fin (tip is insulated)
idealQ
realQ
=η)(
)tanh()(
∞−∞
−=
TbTLPh
mLTbThPkA
mL
mL)tanh(=
)(f
ε
finwithoutQ
finwithQ
f=ε
)(
)(
∞−
∞−
=T
bTAh
TbThPkA
hA
Pk=
)(
)tanh()(
∞−∞
−=
TbTAh
mLTbThPkA
)tanh(mLhA
Pk==
finwithoutQ
finwithQ
f=ε
kA
hPm =
kA
hPm =
Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 12
REFERENCES
1. Heat transfer-A basic approach, Ozisik, Tata McGraw Hill, 2002
2. Heat transfer , J P Holman, Tata McGraw Hill, 2002, 9th edition
3. Principles of heat transfer, Kreith Thomas Learning, 2001
4. Heat and Mass Transfer Data Book, C.P Kothandarman , S Subramanyan, new age
international publishers ,2010, 7th edition
5. Fundamental of Heat and Mass transfer, M Thirumaleshwar,Pearson,2013
6. Pradeep Dutta, “HEAT AND MASS TRANSFER”, Web based course material under the
NPTEL, Phase 1, 2006.
