Fins or Extended Surfaces

FINSWEB.TEX

ECE 309 M.M. Yovanovich

Fins or Extended Surfaces

Fins or extended surfaces are used to increase the heat transfer rate from surfaces which are convectively cooled by gases air under natural or forced convection. The characteristics of ns: a they are metallic, b they having di erent shapes, c the n length is much larger than the thickness or diameter, d there is perfect or imperfect contact at the base, e the n tip is adiabatic or it is cooled, f the temperature distribution is one-dimensional because Bi 0:2.

Derivation of Fin Equation

The derivation of the n equation is based on a heat balance over the boundaries of a di erential volume dV = Ax dx where Ax is the variable conduction area. The heat conduction rate into the volume through the boundary located at x according to Fourier's Law of Conduction is:

Q_ x

=

,k

Ax

dx dx

where x = T x , Tf is the local temperature excess. The heat conduction
rate out of the control volume through the boundary located at x + dx is

Q_ x+dx = Q_ x + ddxQ_ x dx + higher order terms of Taylor series expansion The heat loss rate from the surface of the n by convective cooling according to Newton's Law of cooling is

Q_ loss = h P x x dx

where h is the uniform heat transfer coe cient and P x is the local n perimeter. For steady-state and in the absence of thermal sources or sinks, the energy balance over the boundaries of the control volume, ie:

Q_ x = Q_ x+dx + Q_ loss

leads to the following energy balance:

d dx

Q_ x

dx

+

h

P

x

x

dx

=

0

General Fin Equation

Assuming the thermal conductivity to be constant, and after some manipulations the general n equation is obtained:

d2 dx2

+

1 Ax

dAx dx

dx dx

,

h k

P x Ax

x

=

0;

0xL

where L is the length of the n. The above equation is second-order with variable coe cients. It requires two boundary conditions: i at the n base x = 0 and ii at the n tip x = L.

Boundary Conditions

At the n base x = 0 there are two possible conditions: a perfect contact

where T 0 = Tb which requires that 0 = b = Tb , Tf and Tb is the base

temperature, or there is imperfect contact at the base in which case we have

qb = hc Tb , T 0 = ,k dT=dx where hc is the contact conductance. The

selection of the imperfect contact case leads to the boundary condition of the

third kind:

d0 = ,hc , 0

dx k b

At the n tip x = L there are three possible conditions: a adiabatic insulated

tip where dT=dx = d=dx = 0, b perfect contact with the uid where T L =

Tf, therefore L = 0, and c convective cooling at the n tip such that

qtip = he T L , Tf = ,k dT L=dx where he is the convective coe cient. The

third case will be considered here because it leads to the general n solution.

Therefore at the n tip we take:

dL dx

=

, hke

L

Fin Equation for Constant Cross-Sections

For constant cross-section ns: Ax = A and P x = P , therefore the general n equation becomes:

d2 dx2

,

m2

=

0;

0xL

where the n parameter is de ned as

m2

=

hP kA

2

and its units are m,2. The hyperbolic form of the solution of the previous second-order di erential equation is chosen:

= C1 cosh mx + C2 sinh mx

The temperature gradient is

d dx

=

m

C1

sinh

mx

+

m

C2

cosh

mx

Dimensionless Fin Parameters

We introduce the following three dimensionless n parameters which account for heat transfer through the base, the n tip and the n sides:

Bic = hkcL;

Bie = hkeL;

mL

=

q

hP kA

L;

Bi = hkte

0:2

where te is some e ective thickness of the n cross-section. The e ective thickness of a rectangular cross-section: 2t by w where w is the width and w 2t is te = t. This relation is consistent with the de nition:

te

=

A P

=

cross

, sectional
perimeter

area

For a circular n of diameter d, te = A=P = =4d2=d = d=4; for a n of square cross-section where A = 4w2, te = A=P = 4w2=8w = w=2.

Constants of Integration

After some algebraic manipulations the constants of integration for the two boundary conditions of the third kind give:

C

=

"
1

+

mL

,1

K

1b

Bic

and

C = ,

"
1

+

mL

,1

K

2

b

Bic

The n function is de ned as

= mmLL t+anBhime tLan+h mBiLe

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Fin Heat Transfer Rate
The relation for the heat transfer rate through the n can be obtained by the application of Fourier's Law of Conduction at the n base:
Q_ n = ,kAbddx0 = ,k A m C2 W
Fin Resistance
The n resistance is de ned as:
R n = TbQ_,nTf = "1 + mBLic  hphP kA i,1
The general solution and corresponding relations can be used for any constant cross-section n which has contact resistance and end cooling. The special cases which are frequently presented in heat transfer texts arise from the previous general solution and results.
Special Cases of the General Solution Perfect Contact at the Fin Base and End Cooling For this case we put hc  109 or Bic  109. This leads to 0 = b or T 0 = Tb.
The n resistance becomes:
R n = phP1kA
where the n function = mL; Bie. When Bie = 0, = tanh mL, and
when Bie = 1, = coth mL. Also when mL  2:65, the numerical values of tanh mL and coth mL are within 1 of 1, and therefore  1 for all values of
Bie.
Perfect Contact at Fin Base and Adiabatic Fin Tip
For this case we put Bic = 109 and Bie = 0. The n function becomes: = tanh mL and the n resistance relation becomes:
R n = phP kA1tanh mL
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In nitely Long Fin With Perfect Contact at Fin Base

For this case we put Bic = 109 and take mL  2:65, and the

reduces to

R n = phP1 kA

Criterion for In nitely Long Fins

The criterion for in nitely long ns is

L



2:

s
65

k

A

in nitely long

hP

n resistance

Temperature Distributions for the Special Cases

Perfect Contact at the Fin Base and Tip Cooling

xb = cosh mx , sin mx;
where is de ned above.

0xL

Perfect Contact at Fin Base and Fin Tip

xb = sinhsimnhLm,L x; 0  x  L

Perfect Contact at Fin Base and Adiabatic Fin Tip

xb = coschomshLm,L x; 0  x  L In nitely Long Fin With Perfect Contact at Base

xb = e,mx;

0  x  L  2:65skA
hP

Fin E ciency

The n e ciency is de ned for ns with perfect base contact and adiabatic tip

as:

Q_ n

 = Q_ ideal 1

5

where the ideal n heat transfer rate is de ned as Q_ ideal = Z L h P dx
0

which becomes

Q_ ideal = h P L b

when x = b which corresponds to ns whose thermal conductivity approaches in nitely large values.

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