Short Notes: Heat Transfer Through Fins | Short Notes for Mechanical Engineering PDF Download

 Page 1


Heat Transfer Through Fins 
Heat Transfer from Extended Surface (Fin)
A fin is a surface that extends from an object to increase the rate of heat transfer 
to or from the environment by increase convection. Adding a fin to an object 
increases the surface area and can sometimes be an economical solution to heat 
transfer problems.
Finned surfaces are commonly used in practice to enhance heat transfer. In the 
analysis of the fins, we consider steady operation with no heat generation in the fin. 
We also assume that the convection heat transfer coefficient h to be constant and 
uniform over the entire surface of the fin.
Page 2


Heat Transfer Through Fins 
Heat Transfer from Extended Surface (Fin)
A fin is a surface that extends from an object to increase the rate of heat transfer 
to or from the environment by increase convection. Adding a fin to an object 
increases the surface area and can sometimes be an economical solution to heat 
transfer problems.
Finned surfaces are commonly used in practice to enhance heat transfer. In the 
analysis of the fins, we consider steady operation with no heat generation in the fin. 
We also assume that the convection heat transfer coefficient h to be constant and 
uniform over the entire surface of the fin.
• The rate of heat transfer from a solid surface to atmosphere is given by Q = 
M A T where, h and AT are not controllable.
• So, to increase the value of Q surface area should be increased. The extended 
surface which increases the rate of heat transfer is known as fin.
Generalized Equation for Fin Rectangular fin
d l e _ _ 1 d .it de h d i.
d x 1 A( dx dx kA( dx
Where Ac and As are cross-sectional and surface area, 
And Q(x)=t(x)-ta
Perimeter
Rectangular fin diagram
Heat balance equation if Ac constant and As °° P(x) linear
d 't
d x '
hp
T7c
( t ~ t a)= 0
0
• General equation of 2n d order 0 = Ciem x + C 2e'm x
• Heat dissipation can take place on the basis of three cases
Case 1: Heat Dissipation from an Infinitely Long Fin (I — ? »)
• In such a case, the temperature at the end of Fin approaches to surrounding 
fluid temperature ta as shown in figure. The boundary conditions are given 
below
• At x=0, t=to
• 6 = to - ta = 6o
• At x = 1 — ? °°
• t = tg, 6 = 0
0 = 00 e'm x
Page 3


Heat Transfer Through Fins 
Heat Transfer from Extended Surface (Fin)
A fin is a surface that extends from an object to increase the rate of heat transfer 
to or from the environment by increase convection. Adding a fin to an object 
increases the surface area and can sometimes be an economical solution to heat 
transfer problems.
Finned surfaces are commonly used in practice to enhance heat transfer. In the 
analysis of the fins, we consider steady operation with no heat generation in the fin. 
We also assume that the convection heat transfer coefficient h to be constant and 
uniform over the entire surface of the fin.
• The rate of heat transfer from a solid surface to atmosphere is given by Q = 
M A T where, h and AT are not controllable.
• So, to increase the value of Q surface area should be increased. The extended 
surface which increases the rate of heat transfer is known as fin.
Generalized Equation for Fin Rectangular fin
d l e _ _ 1 d .it de h d i.
d x 1 A( dx dx kA( dx
Where Ac and As are cross-sectional and surface area, 
And Q(x)=t(x)-ta
Perimeter
Rectangular fin diagram
Heat balance equation if Ac constant and As °° P(x) linear
d 't
d x '
hp
T7c
( t ~ t a)= 0
0
• General equation of 2n d order 0 = Ciem x + C 2e'm x
• Heat dissipation can take place on the basis of three cases
Case 1: Heat Dissipation from an Infinitely Long Fin (I — ? »)
• In such a case, the temperature at the end of Fin approaches to surrounding 
fluid temperature ta as shown in figure. The boundary conditions are given 
below
• At x=0, t=to
• 6 = to - ta = 6o
• At x = 1 — ? °°
• t = tg, 6 = 0
0 = 00 e'm x
• Heat transfer by conduction at base
Case 2: Heat Dissipation from a Fin Insulated at the End Tip
• Practically, the heat loss from the long and thin film tip is negligible, thus the 
end of the tip can be, considered as insulated.
• At x=0, t=t0 and 9 = t0 - ta = 90
at
. _ . . d9 .
x = /. O = 0 i.e. — = 0 
c h c
9 _ r — r c _ cosh m(l— x) 
ft r 0 - r a cosh m l 
Qm =yjPhk A( (r0- r a)tanh ml
Case 3: Heat Dissipation from a Fin loosing Heat at the End Tip
• The boundary conditions are given below.
• At x=0, t = to and 9 = 9o
• At X=7, Q conduction = Q convection
dt
dx
= h As ( t - r )
9= ft
cosh m(/ — x)+ ¦ — ^ sin h tn(l-x) 
mk
cosh ml + — sinh ml 
mk
QJbl= y jhPkAt 9t
tanh m l+— 
mk
0 h
H----- tanh m l
mk
Page 4


Heat Transfer Through Fins 
Heat Transfer from Extended Surface (Fin)
A fin is a surface that extends from an object to increase the rate of heat transfer 
to or from the environment by increase convection. Adding a fin to an object 
increases the surface area and can sometimes be an economical solution to heat 
transfer problems.
Finned surfaces are commonly used in practice to enhance heat transfer. In the 
analysis of the fins, we consider steady operation with no heat generation in the fin. 
We also assume that the convection heat transfer coefficient h to be constant and 
uniform over the entire surface of the fin.
• The rate of heat transfer from a solid surface to atmosphere is given by Q = 
M A T where, h and AT are not controllable.
• So, to increase the value of Q surface area should be increased. The extended 
surface which increases the rate of heat transfer is known as fin.
Generalized Equation for Fin Rectangular fin
d l e _ _ 1 d .it de h d i.
d x 1 A( dx dx kA( dx
Where Ac and As are cross-sectional and surface area, 
And Q(x)=t(x)-ta
Perimeter
Rectangular fin diagram
Heat balance equation if Ac constant and As °° P(x) linear
d 't
d x '
hp
T7c
( t ~ t a)= 0
0
• General equation of 2n d order 0 = Ciem x + C 2e'm x
• Heat dissipation can take place on the basis of three cases
Case 1: Heat Dissipation from an Infinitely Long Fin (I — ? »)
• In such a case, the temperature at the end of Fin approaches to surrounding 
fluid temperature ta as shown in figure. The boundary conditions are given 
below
• At x=0, t=to
• 6 = to - ta = 6o
• At x = 1 — ? °°
• t = tg, 6 = 0
0 = 00 e'm x
• Heat transfer by conduction at base
Case 2: Heat Dissipation from a Fin Insulated at the End Tip
• Practically, the heat loss from the long and thin film tip is negligible, thus the 
end of the tip can be, considered as insulated.
• At x=0, t=t0 and 9 = t0 - ta = 90
at
. _ . . d9 .
x = /. O = 0 i.e. — = 0 
c h c
9 _ r — r c _ cosh m(l— x) 
ft r 0 - r a cosh m l 
Qm =yjPhk A( (r0- r a)tanh ml
Case 3: Heat Dissipation from a Fin loosing Heat at the End Tip
• The boundary conditions are given below.
• At x=0, t = to and 9 = 9o
• At X=7, Q conduction = Q convection
dt
dx
= h As ( t - r )
9= ft
cosh m(/ — x)+ ¦ — ^ sin h tn(l-x) 
mk
cosh ml + — sinh ml 
mk
QJbl= y jhPkAt 9t
tanh m l+— 
mk
0 h
H----- tanh m l
mk
x = 6
Fin Efficiency:
Fin efficiency is given by
Actual heat rate from fin Q 
Maximum heat transfer rate Q„^
• If / — ? « > (infinite length of fin),
¦yjljPkA, fl0 1 
T]~ h{Pl + b6)9c “ Tv hP
• If fin is with insulated tip,
O r JhPki tan h ml 
r i - — -------------------
hPW0
• If finite length of fin,
e .J h K T '
tan h ml - \-----
mk
1 + • — tan h ml 
mk
h(Pl+ b6)60
Note: The following must be noted for a proper fin selection:
• The longer the fin, the larger the heat transfer area and thus the higher the rat 
e of heat transfer from the fin.
• The larger the fin, the bigger the mass, the higher the price, and larger the fluid 
friction.
• The fin efficiency decreases with increasing fin length because of the decreas 
e in fin temperature with length.
Fin Effectiveness:
The performance of fins is judged on the basis of the enhancement in heat transfer 
relative to the no-fin case, and expressed in terms of the fin effectiveness:
£ fh , — "
Qtk
fa
H (T „ - Ts )
heat transfer rate from the fin 
heat transfer rate from the surface area of A b
(< 1 fin acts as insulation
= 1
>1
fin does not affect heat transfer 
fin enhances heat transfer
For a sufficiently long fin of uniform cross-section Ac, the temperature at the tip of 
the fin will approach the environment temperature, T°°. By writing energy balance 
and solving the differential equation, one finds
Page 5


Heat Transfer Through Fins 
Heat Transfer from Extended Surface (Fin)
A fin is a surface that extends from an object to increase the rate of heat transfer 
to or from the environment by increase convection. Adding a fin to an object 
increases the surface area and can sometimes be an economical solution to heat 
transfer problems.
Finned surfaces are commonly used in practice to enhance heat transfer. In the 
analysis of the fins, we consider steady operation with no heat generation in the fin. 
We also assume that the convection heat transfer coefficient h to be constant and 
uniform over the entire surface of the fin.
• The rate of heat transfer from a solid surface to atmosphere is given by Q = 
M A T where, h and AT are not controllable.
• So, to increase the value of Q surface area should be increased. The extended 
surface which increases the rate of heat transfer is known as fin.
Generalized Equation for Fin Rectangular fin
d l e _ _ 1 d .it de h d i.
d x 1 A( dx dx kA( dx
Where Ac and As are cross-sectional and surface area, 
And Q(x)=t(x)-ta
Perimeter
Rectangular fin diagram
Heat balance equation if Ac constant and As °° P(x) linear
d 't
d x '
hp
T7c
( t ~ t a)= 0
0
• General equation of 2n d order 0 = Ciem x + C 2e'm x
• Heat dissipation can take place on the basis of three cases
Case 1: Heat Dissipation from an Infinitely Long Fin (I — ? »)
• In such a case, the temperature at the end of Fin approaches to surrounding 
fluid temperature ta as shown in figure. The boundary conditions are given 
below
• At x=0, t=to
• 6 = to - ta = 6o
• At x = 1 — ? °°
• t = tg, 6 = 0
0 = 00 e'm x
• Heat transfer by conduction at base
Case 2: Heat Dissipation from a Fin Insulated at the End Tip
• Practically, the heat loss from the long and thin film tip is negligible, thus the 
end of the tip can be, considered as insulated.
• At x=0, t=t0 and 9 = t0 - ta = 90
at
. _ . . d9 .
x = /. O = 0 i.e. — = 0 
c h c
9 _ r — r c _ cosh m(l— x) 
ft r 0 - r a cosh m l 
Qm =yjPhk A( (r0- r a)tanh ml
Case 3: Heat Dissipation from a Fin loosing Heat at the End Tip
• The boundary conditions are given below.
• At x=0, t = to and 9 = 9o
• At X=7, Q conduction = Q convection
dt
dx
= h As ( t - r )
9= ft
cosh m(/ — x)+ ¦ — ^ sin h tn(l-x) 
mk
cosh ml + — sinh ml 
mk
QJbl= y jhPkAt 9t
tanh m l+— 
mk
0 h
H----- tanh m l
mk
x = 6
Fin Efficiency:
Fin efficiency is given by
Actual heat rate from fin Q 
Maximum heat transfer rate Q„^
• If / — ? « > (infinite length of fin),
¦yjljPkA, fl0 1 
T]~ h{Pl + b6)9c “ Tv hP
• If fin is with insulated tip,
O r JhPki tan h ml 
r i - — -------------------
hPW0
• If finite length of fin,
e .J h K T '
tan h ml - \-----
mk
1 + • — tan h ml 
mk
h(Pl+ b6)60
Note: The following must be noted for a proper fin selection:
• The longer the fin, the larger the heat transfer area and thus the higher the rat 
e of heat transfer from the fin.
• The larger the fin, the bigger the mass, the higher the price, and larger the fluid 
friction.
• The fin efficiency decreases with increasing fin length because of the decreas 
e in fin temperature with length.
Fin Effectiveness:
The performance of fins is judged on the basis of the enhancement in heat transfer 
relative to the no-fin case, and expressed in terms of the fin effectiveness:
£ fh , — "
Qtk
fa
H (T „ - Ts )
heat transfer rate from the fin 
heat transfer rate from the surface area of A b
(< 1 fin acts as insulation
= 1
>1
fin does not affect heat transfer 
fin enhances heat transfer
For a sufficiently long fin of uniform cross-section Ac, the temperature at the tip of 
the fin will approach the environment temperature, T°°. By writing energy balance 
and solving the differential equation, one finds
n*)-Tx
T ,-T m
where Ac is the cross-
sectional area, x is the distance from the base, and p is perimeter. 
The effectiveness becomes:
To increase fins effectiveness, one can conclude:
• The thermal conductivity of the fin material must be as high as possible
• The ratio of perimeter to the cross-sectional area p/Ac should be as high as 
possible
• The use of fin is most effective in applications that involve low convection 
heat transfer coefficient, i.e. natural convection.