The document discusses heat transfer via fins. It defines fins as protrusions that increase the surface area in contact with a fluid to facilitate heat transfer. It presents the governing differential equation for one-dimensional heat transfer through a fin and describes the boundary conditions and solutions for different fin types, including infinitely long fins, short fins with insulated tips, and fins with convection at the tip. It also provides equations for calculating the heat transfer rate for different fin configurations.

1. 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
)(
∞
−T
s
T
)(
∞
−= T
s
ThAQ
conv
Q
dxx
Q
x
Q +
+
=
dx
dT
Ak
x
Q −= dx
dx
Td
Ak
dx
dT
Akdx
dx
dT
Ak
dx
d
dx
dT
Ak
dxx
Q 2
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
conv
Q
dxx
Q
x
Q +
+
=
( )∞
−+−−=− TTdxPhdx
dx
Td
Ak
dx
dT
Ak
dx
dT
Ak )(2
2
( )∞
−= TT
s
Ah
conv
Q ( )∞
−= TTdxPh
conv
Q )( dxpermeterAs )(=
( )∞
−= TTdxPhdx
dx
Td
Ak )(2
2
( )∞
−= TT
KA
Ph
dx
Td
2
2
( )∞
−= TTm
dx
Td 2
2
2
KA
Ph
m =
KA
Ph
m =2
0)(2)(
2
2
=− xm
dx
xd
θ
θ ∞−= TTxlet )(θ
dx
dT
dx
xd
=
)(θ
mxmx
eCeCx 21)( += −
θ
mxCmxCx sinhcosh)( 21 +=θ

3. 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
L
T
0)(2)(
2
2
=− tm
dx
td
θ
θ
mxmx
eCeCt 21)( += −
θ
b
TTatx == 0
∞
==∞= T
L
TTatx
∞−= TT
b bθ
0=
L
θ
1C
b
=θ
b
TTatx == 0 ∞−= TT
b bθ
∞
==∞= T
L
TTatx 0=
L
θ
∞= 20 C 02 =C
mx
e
b
x −
= θθ )(
mx
e
b
x −
=
θ
θ )(
mx
e
T
b
T
TT
−
=
∞
−
∞
−
0
)(
0 =
−=
=
−=
xdx
xd
Ak
xdx
dT
AkQ
θ
)( bmAkQ θ−−=
mx
e
b
x −
=
θ
θ )(
mx
b ex −
=θθ )(
mx
b em
dx
xd −
−= θ
θ )(bmAkQ θ=
)( ∞−= TT
KA
hP
AkQ b
)( ∞−= TThPKAQ b

4. 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,
b
TTatx == 0 ∞−= TT
b bθ
0==
dx
d
atLx
θ
( )∞
−= TTm
dx
Td 2
2
2
KA
Ph
m =
KA
Ph
m =2
0)(2)(
2
2
=− xm
dx
xd
θ
θ ∞−= TTxlet )(θ
dx
dT
dx
xd
=
)(θ
mxCmxCx sinhcosh)( 21 +=θ
b
TTatx == 0 ∞−= TT
b bθ
1C
b
=θ
( ) ( )mxmCmxm
bdx
td
coshsinh
)(
2+= θ
θ
( ) ( )mLmCmLm
b
sinhcosh0 2+= θ
( )
( )mL
mL
bC
cosh
sinh
2
θ−
=
( )
( )
mx
mL
mL
bmx
b
t sinh
cosh
sinh
cosh)(







−
+=
θ
θθ
( )
( )
mx
mL
mL
mx
b
t
sinh
cosh
sinh
cosh
)(






−=
θ
θ
( ) ( )
( )mL
mxmLmLmx
b
t
cosh
sinhsinhcoshcosh)( −
=
θ
θ
( )
( )mL
xLm
b
t
cosh
cosh)( −
=
θ
θ
( )
( )mL
xLm
T
b
T
TT
cosh
cosh −
=
∞
−
∞
−

5. 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
T
b
T
TT
cosh
cosh −
=
∞
−
∞
−
0
)(
0 =
−=
=
−=
xdx
xd
Ak
xdx
dT
AkQ
θ
( )
( )[ ]
0
sinh
cosh
=








−−−=
x
xLmm
mL
bAkQ
θ
( )
( )mL
xLm
b
x
cosh
cosh)( −
=
θ
θ
( )
( ) 




 −
=
mL
xLm
x b
cosh
cosh
)( θθ
( )
( )[ ] )10(sinh
cosh
)(
−−= xLmm
mL
b
dx
xd θθ
( )
( )[ ]xLmm
mL
b
dx
xd
−−= sinh
cosh
)( θθ
( )
mL
mL
bAmkQ sinh
cosh
θ
=
( ) mLT
b
TAmkQ tanh
∞
−=
( ) mLT
b
ThPKAQ tanh
∞
−=
∫ ∞
−=
L
o
TxTdxPhQ ))(()(

6. 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)(2
2
)(2
=− xm
dx
xd
θ
θ
0
)( θθ =
∞
−= T
b
Tx 0=xat
0)(
)(
=+ x
L
h
dx
xd
k θ
θ
Lxat =
)(sin
2
)(cos
1
)( xmhCxmhCx +=θ
)(sin
2
)(cos
1
)( xmhCxmhCx +=θ
0
)( θθ =
∞
−= T
b
Tx 0=xat
0
10
+=Cθ
)(sin
2
)(cos
0
)( xmhCxmhx +=θθ
0)(
)(
=+ x
L
h
dx
xd
k θ
θ Lxat =
)(cos
2
)(sin
0
)(
xmhmCxmhm
dx
xd
+=θ
θ
Lx
xA
L
h
Lxdx
xd
kA
=
=
=
− )(
)(
θ
θ
[ ] [ ] 0)(sin)(cos)(cos)(sin 2020 =+++ LmhCLmh
L
hlmhmCLmhmk θθ
0)(
)(
=+ x
L
h
dx
xd
k θ
θ
Lxat =
[ ] [ ] 0)(sin)(cos)(cos)(sin 2020 =+++ LmhChLmhhLmhmCkLmhmk LLθθ
[ ] [ ] 0)(sin)(cos)(cos)(sin 2200 =+++ LmhChLmhmCkLmhhLmhmk LLθθ








+−=








+ )(sin)(cos)(cos)(sin 20 Lmh
km
L
h
LmhCLmh
km
L
h
Lmhθ
10
C=θ

7. Heat and mass transfer Fins
Yashawantha K M, Dept. of Marine Engineering, SIT, Mangaluru Page 7
)(sin)(cos
)(cos)(sin
0
2
Lmh
km
L
h
Lmh
Lmh
km
L
h
Lmh
C
+








+
−=
θ
)(sin
)(sin)(cos
)(cos)(sin
)(cos
0
)( 0 xmh
Lmh
km
L
h
Lmh
Lmh
km
L
h
Lmh
xmhx














+
+
−= θθθ
)(sin
)(sin)(cos
)(cos)(sin
)(cos
0
)(
xmh
Lmh
km
L
h
Lmh
Lmh
km
L
h
Lmh
xmh
x














+
+
−=
θ
θ
)(sin)(cos
)(sin)(cos)(sin)(sin)(cos)(cos
0
)(
Lmh
km
L
h
Lmh
xmhLmh
km
L
h
LmhLmh
km
L
h
Lmhxmh
x
+








+−








+
=
θ
θ
)(sin)(cos
)(sin)(cos)(sin)(sin)(cos)(sin)(cos)(cos
0
)(
Lmh
km
L
h
Lmh
xmhLmh
km
L
h
xmhLmhxmhLmh
km
L
h
Lmhxmh
x
+








+−








+
=
θ
θ
[ ]
)(sin)(cos
)(sin)(cos)(cos)(sin)(sin)(sin)(cos)(cos
0
)(
Lmh
km
L
h
mLh
xmhLmh
km
L
h
xmhLmh
km
L
h
xmhLmhxmhLmh
x
+








−+−
=
θ
θ
[ ] [ ]
)(sin)(cos
)(sin)(cos)(
0
)(
Lmh
km
L
h
mLh
xLmh
km
L
h
xLmh
T
b
T
TxTx
+
−+−
=
∞
−
∞
−
=
θ
θ

8. 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
xd
Ak
xdx
dT
AkQ
θ
[ ] [ ]














+
−+−
=
)(sin)(cos
)(sin)(cos
)( 0
mLh
km
L
h
Lmh
xLmh
km
L
h
xLmh
x θθ
[ ] [ ]
)(sin)(cos
)1)((cosh)1)((sin
)(
0
Lmh
km
h
Lmh
xLmm
km
h
xLmhm
dx
xd
L
L
+
−−+−−
= θ
θ
( )






+






+
∞
−=
)(tan1
)(tan
Lmh
km
h
km
h
mLh
T
b
TAk
L
PhQ
L
L
[ ] [ ]
0
)(sin)(cos
)1)((cosh)1)((sin
0
=
+
−−+−−
−=
x
Lmh
km
L
h
Lmh
xLmm
km
L
h
xLmhm
AkQ θ
[ ] [ ]
)(sin)(cos
)(cosh)(sin
0
Lmh
km
h
Lmh
mLm
km
h
mLhm
AkQ
L
L
+
−−
−= θ






+






+
=
)(sin)(cos
)(cosh)(sin
0
Lmh
km
h
Lmh
Lm
km
h
mLh
AmkQ
L
L
θ
( )






+






+
∞
−=
)(sin)(cos
)(cosh)(sin
Lmh
km
h
Lmh
Lm
km
h
mLh
T
b
TAk
L
PhQ
L
L

9. 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)(2
2
)(2
=− xm
dx
xd
θ
θ
0== xat
b
TT
Lxat
L
TT ==
∞
−= T
b
T
b
θ
∞
−= T
L
T
L
θ
)(sin
2
)(cos
1
)( xmhCxmhCx +=θ
)(sin
2
)(cos
1
)( xmhCxmhCx +=θ
0== xat
b
TT
∞
−= T
b
T
b
θ
1
C
b
=θ
)(sin
2
)(cos
1
)( xmhCxmhCx +=θ
Lxat
L
TT ==
∞
−= T
L
T
L
θ
)(sin
2
)(cos mLhCLmh
bL
+=θθ
)(sin
)(cos
2 mLh
Lmh
bLC
θθ −
=
)(sin
)(sin
)(cos
)(cos)( xmh
mLh
Lmh
bLxmh
b
x







 −
+=
θθ
θθ
[ ]
)(sin
(sin)(cos)(sin)(cos
)(
mLh
xmhLmh
bL
mLhxmh
bx
θθθ
θ
−+
=
)(sin
)(sin)(cos)(sin)(sin)(cos
)(
mLh
xmhLmh
b
xmh
L
mLhxmh
bx
θθθ
θ
−+
=

10. 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
xmh
L
xLmh
bx
θθ
θ
+−
=
∫ ∞
−=
L
o
TxTdxPhQ ))(()(
dxx
L
o
PhQ )(θ∫=
dx
mLh
xmh
L
xLmh
b
L
o
PhQ







 +−
∫=
)(sin
)(sin)(sin θθ
L
om
xmh
L
m
xLmh
b
mLh
Ph
Q








+
−−
= )
)(cos
)
)(cos
)(sin
θθ
[ ])1)(cos()cos1(
)(sin
−+−−= Lmh
L
mLh
bmLhm
Ph
Q θθ
[ ])1)(cos()1(cos
)(sin
−+−= Lmh
L
mLh
bmLhm
Ph
Q θθ
[ ])1(cos)(
)(sin
−+= mLh
LbmLhm
Ph
Q θθ
PkAh
mLh
mLh
L
T
L
TT
b
TQ
)(sin
)1(cos
)()(
−




∞
−+
∞
−=
∞
−= T
b
T
b
θ
∞
−= T
L
T
L
θ
ideal
Q
real
Q
=η
)(
)(
∞
−
∞
−
=
T
b
TLPh
T
b
ThPkA
mL
1
= kA
hP
m =

11. 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)
ideal
Q
real
Q
=η
)(
)tanh()(
∞
−
∞
−
=
T
b
TLPh
mLT
b
ThPkA
mL
mL)tanh(
=
)(
f
ε
finwithout
Q
finwith
Q
f
=ε
)(
)(
∞
−
∞
−
=
T
b
TAh
T
b
ThPkA
hA
Pk
=
)(
)tanh()(
∞
−
∞
−
=
T
b
TAh
mLT
b
ThPkA
)tanh(mL
hA
Pk
==
finwithout
Q
finwith
Q
f
=ε
kA
hP
m =
kA
hP
m =

12. 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.