HEAT CONDUCTION THROUGH
FINS
Prabal Talukdar
Associate Professor
Department of Mechanical Engineering
IIT Delhi
E-mail: prabal@mech.iitd.ac.in
p
Mech/IITD
Introduction
&
Q
conv = hA s (Ts − T∞ )
There are two ways to increase the rate of heat transfer
• too increase
c e se thee convection
co vec o heat
e transfer
s e coefficient
coe c e h
• to increase the surface area As
• Increasing h may require the installation of a pump or fan,
• Or replacing
p
g the existingg one with a larger
g one
• 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
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
Triangular
Convex parabolic
Trapezoidal
Concave parabolic
Radial fins:
Rectangular
g
p
profile
Triangular profile
Hyperbolic profile
Pins:
Radial fin coffee cup
(a)Cylindrical (b)conical (c) concave parabolic (d) convex parabolic
Fin Equation
dT
Q& x = − kA c
dx
d Q& x
&
&
Q x + dx = Q x +
dx
dx
&
dQ
conv = hdA s (T − T∞ )
Energy Balance:
&
dQ
x dx + hdA ( T − T )
&
= Qx +
s
∞
dx
dT ⎞
h dA s
(T − T∞ ) = 0
⎟−
dx ⎠
k dx
Q& x = Q& x + dx + d Q& conv
d ⎛
⎜A
dx ⎝
d 2T
dx
2
c
1 dA c ⎛ dT ⎞ ⎛ 1 h dA s ⎞
⎟ (T − T∞ ) = 0
+
⎜
⎟ − ⎜⎜
A c dx ⎝ dx ⎠ ⎝ A c k dx ⎟⎠
Fins with uniform cross sectional area
1 dA c ⎛ dT ⎞
d 2T
+
⎜
⎟−
2
dx
A c dx ⎝ dx ⎠
dA c
=0
dx
dA s
A s = Px
= P
d
dx
⎛ 1 h dA s ⎞
⎜⎜
⎟⎟ ( T − T ∞ ) = 0
⎝ A c k dx ⎠
d 2 T ⎛ hP ⎞
⎟⎟ ( T − T ∞ ) = 0
− ⎜⎜
2
dx
⎝ kA c ⎠
Excess temperature θ
θ ( x ) ≡ T ( x ) − T∞
d 2θ
2
−
m
θ =0
2
dx
(a) Rectangular Fin (b) Pin fin
General solution and
Boundary Conditions
d 2θ
2
−
m
θ =0
2
dx
hP
m2 =
kA c
θ ( x ) ≡ T ( x ) − T∞
The g
general solution is of the form
θ ( x ) = C 1 e mx + C 2 e − mx
Convection from tip
q denotes
BCs
T (0 ) = T∞
θ (0 ) = Tb − T∞ ≡ θ b
hA c [T ( L ) − T∞ ] = − kA c
h θ( L ) = − k
θ
cosh m ( L − x ) + ( h / mk ) sinh m ( L − x )
=
θb
cosh mL + ( h / mk ) sinh mL
dθ
|x =L
dx
dT
|x = L
dx
&
Q
Heat transfer from the fin surface:
dθ
& =Q
& = − kA dT |
Q
=
−
kA
|x =0
f
b
c
x =0
c
dx
dx
& =
Q
f
hPkA c θ b
sinh mL + ( h / mk ) cosh mL
cosh mL + ( h / mk ) sinh mL
Another way of finding Q
•
Q fin = ∫ A fin h [ T ( x ) − T∞ ]dA fin = ∫ A fin h θ ( x ) dA fin
Insulated tip
General Sol:
BC
C 1:
θ ( x ) = C 1 e mx + C 2 e − mx
θ ( 0 ) = Tb − T∞ ≡ θ b
θ
cosh
h m( L − x)
=
θb
cosh mL
dT
dθ
q f = q b = − kA c
| x = 0 = − kA c
|x=0
dx
dx
qf =
hPkA c θ b tanh mL
BC 2:
dθ
|x= L = 0
d
dx
Prescribed temperature
This is a condition when the temperature
p
at the tip
p is known
(for example, measured by a sensor)
(θ θ )sinh mx + sinh m ( L − x )
θ
= L b
θb
sinh mL
qf =
hPkA c θ b
cosh mL − (θ L θ b )
sinh
i h mL
L
Infinitely
y Long
g Fin ((Tfin tip
θ ( L ) = T ( L ) − T∞ = 0
Boundary condition at the fin tip:
The general solution is of the form
θ ( x ) = C 1 e mx + C 2 e − mx
Possible when C1 → 0
θ( x ) = C 2 e − mx
Apply boundary condition at base and find T
T ( x ) = T∞ + ( T b − T∞ ) e
•
Q longfin = − kA c
−x
dT
|x =0 =
dx
hP
kA c
hPkA
c
( T b − T∞ )
as
= T∞)
L→∞
Corrected fin length
g
Corrected fin length:
g
Lc = L +
Ac
P
Multiplying the relation above by
the perimeter gives
Acorrected = Afin (lateral) + Atip
t
Lc,rectangularfin = L +
2
L c,cylindrica lfin = L +
D
4
Corrected fin length
g Lc is defined such that heat transfer from a fin of
length Lc with insulated tip is equal to heat transfer from the actual
fin of length L with convection at the fin tip.
Fin Efficiency
y
•
η fin =
Actual heat transfer rate from the fin
Q fin
= Ideal heat transfer rate from the fin
•
Q fin , max
if the entire fin were at base
temperature
In the limiting case of zero thermal resistance or
infinite thermal cond
conductivity
cti it (k →∞ ),
) the temperat
temperature
re
of the fin will be uniform at the base value of Tb.
•
•
Q fin = η fin Q fin , max = η fin hA fin (Tb − T∞ )
•
η longfin =
Q fin
•
=
Q fin , max
hPkA c (Tb − T∞ ) 1
=
hA fin (Tb − T∞ )
L
•
η insulatedt ip =
Q fin
f
•
Q fin , max
=
kA c
1
=
hp
mL
hPkA c (Tb − T∞ ) tanh mL tanh mL
=
hA fin (Tb − T∞ )
mL
Fin Effectiveness
•
ε fin =
Q fin
•
•
=
Q nofin
Q fin
hA b (Tb − T∞ )
•
ε longfin =
Q fin
•
Q nofin
=
=
Heat transfer rate from
the fin of base area Ab
Heat transfer rate from
the surface of area Ab
hPkA
h
kA c (Tb − T∞ )
=
hA b (Tb − T∞ )
kP
hA c
1. k should be as high as possible, (copper, aluminum, iron).
Aluminum is p
preferred: low cost and weight,
g , resistance to corrosion.
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
•
•
Q fin
ε fin =
•
Q nofin
=
Q fin
hA b (Tb − T∞ )
=
Heat transfer rate from
th fifin off base
the
b
area Ab
Heat transfer rate from
the surface of area Ab
ε fin = 1
Does not affect the heat transfer at all.
ε fin < 1
Fin act as insulation (if low k material is used)
ε fin > 1
Enhancing heat transfer (use of fins justified if εfin>2)
Overall Fin Efficiency
y
When determining the rate of heat transfer
from a finned surface,, we must consider the
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
•
•
•
Q total , fin = Q unfin + Q fin
= hA unfin (Tb − T∞ ) + η fin hA fin (Tb − T∞ )
= h ( Aunfin +η fin A fin )( Tb − T∞ )
We can also define an overall
effectiveness for a finned surface as the
ratio of the total heat transfer from the
finned surface to the heat transfer from the
same surface if there were no fins
•
ε fin , overall =
Q total , fin
•
Q total , nofin
=
h ( Aunfin + η fin A fin )( Tb − T∞ )
hA nofin (Tb − T∞ )
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 (from
Gardner, Ref. 6).
Proper length of a fin
“a” means “m”
• Fins
Fi with
i h triangular
i
l andd parabolic
b li profiles
fil contain
i less
l
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 added
cost.
•The
Th efficiency
ffi i
off mostt fins
fi usedd in
i practice
ti is
i above
b
90 percentt