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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)

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BB.. SS.. AA.. CCoolllleeggee ooff EEnngggg.. AAnndd TTeecchhnnoollooggyy

MMaatthhuurraa,, UUttttaarr PPrraaddeesshh,, ((IInnddiiaa))

SSuuppppoorrtteedd BByy::

PPuurrvvii BBhhoooosshhaann

In This Chapter We Cover the Following Topics

Art. Content Page

3.1 Extended Surfaces 2

3.2 Fins of Uniform Cross-Sectional Area 3

3.3 Fin Performance (Effectiveness) 6

3.4 Limitation of an Extended Surface 10

3.5 Rectangular Fin of Minimum Weight 10

3.6 Generalized Equation for Fins 12

3.7 Fin of Minimum Weight 14

3.8 Fin Arrangement 15

3.9 Cylindrical Fins 15

References:

1- J. P. Holman, Heat Transfer, 9th Edn, MaGraw-Hill, New York, 2002.

2- James R. Welty, Charles E. Wicks, Robert E. Wilson, Gregory L. Rorrer

Fundamentals of Momentum, Heat, and Mass Transfer, 5th Edn, John Wiley & Sons,

Inc., 2008.

3- F. Kreith and M. S. Bohn, Principal of Heat Transfer, 5th Edn, PWS Publishing Co.,

Boston, 1997.

4- P. K. Nag, Heat and Mass Transfer, 2nd Edn, MaGraw-Hill, New Delhi 2005.

Please welcome for any correction or misprint in the entire manuscript and your

valuable suggestions kindly mail us [email protected].




2 Chapter 3: Extended Surfaces (Fins)

3.1 EXTENDED SURFACES

Convection heat transfer between a surface (at Tw) and the fluid surrounding it (at T∞) is

given by

where h is the heat transfer coefficient and A is the surface area of heat transfer. For

gases h (= kf/δt) is low, since the thermal conductivity kf of a gas film is low.

For heat transfer from a hot gas to a liquid through a wall hgas << hliquid. To compensate

for low heat transfer coefficient, surface area A on the gas side may be extended for a

given Q. Such an extended surface is termed as fin.

Let us consider the plane wall of Diagram 3.1. If Tw is fixed, there are three ways in

which the heat transfer rate may be increased.

Diagram 3.1 Combined conduction and convection in a bar

1. The convection coefficient h could be increased by increasing the fluid velocity

or/and the fluid temperature T∞ could be reduced. However, increasing h even to

the maximum possible value is often insufficient to obtain the desired heat

transfer rate or the costs related to blower or pump power required to increase h

may be prohibitive.

2. The second option of reducing T∞ is often impractical.

3. The heat transfer rate may be increased by increasing the surface area across

which convection occurs. This may be done by using fins that extend from the

wall into the surrounding fluid (Diagram 3.2). The thermal conductivity of the fin

material has a strong effect on the temperature distribution along the fin and

thus the degree to which the heat transfer rate is enhanced.

Diagram 3.2 Use of fins to enhance heat transfer from a plane wall: Bare surface and Finned surface

Diagram 3.3 shows different fin configurations. A straight fin is any extended surface

that is attached to a plane wall. It may be of uniform cross-sectional area, or its cross-

sectional area may vary with the distance x from the wall. An annular fin is one that is

circumferentially attached to a cylinder. A pin fin or spine is an extended surface of

circular cross-sections. Pin fins may also be of uniform or non-uniform cross-section.

Fluid

0

L




3 Heat and Mass Transfer By Brij Bhooshan

Diagram 3.3 Fin configurations (a) Straight fin of uniform cross-section, (b) straight fin of nonuniform cross-

section, (c) pin fin

3.2 FINS OF UNIFORM CROSS-SECTIONAL AREA

Let us first consider the simplest case of straight and pin fins of uniform cross-section

(Diagram 3.4). Each fin is attached to the base surface of T0 and extends into a fluid of

temperature T∞. The perimeter of the fin, P, which is uniform is 2(z + t) or, P ≈ 2z, if the

thickness of the fin is small (t <<< z). If the fin is very thin and its length z is long. It can

be assumed that there is no radial temperature variation and heat gets conducted

axially along the length. This heat is then dissipated to the surroundings by convection.

The problem thus reduces to axial heat conduction along the fin with distributed heat

sink from the sides. It is thus treated as one-dimensional heat conduction.

Diagram 3.4 Conduction and convection in a straight fin or a thin rod

Let us consider a small volume element at a distance x from the base or root of the fin of

thickness dx. The rate at which heat enters the element is Qx and the heat leaving the

element is Qx+dx. In that small distance dx, let dQConv be the heat transferred by

convection. Then by energy balance.

Energy in left face = Energy out right face + Energy lost by convection

Let excess temperature at any section θ = T T∞

A

(a)

(b)

(c)





Let m2 = hP/kA, then we have

Using D operation

Either (D – m) = 0

or (D + m) = 0

where C1 and C2 are constants.

The general solution for temperature distribution is

with boundary conditions

(1) When x = 0, T = T0, θ = θ0 = T0 − T∞

(2) When x = L, T = TL, θ = θL = TL − T∞

The other boundary condition depends on the physical situation. Several cases may be

considered:

CASE 1 The fin is very long, and the temperature at the end of the fin is essentially

that of the surrounding fluid.

CASE 2 The fin is of finite length and loses heat by convection from its end.

CASE 3 The end of the fin is insulated so that dT/dx = 0 at x = L.

Case 1: The fin is very long, and the temperature at the end of the fin is essentially that

of the surrounding fluid. The boundary conditions are at θ = θ0 at x = 0; and θ = 0 at x =

∞. Then using Eqn. (3.7 and 3.8) we have

After solving we get C1 = 0 and C2 = θ0. Thus we have

Case 2 The fin is of finite length and loses heat by convection from its end. At tip

From Eqn. 3.8 on differentiation and substitution,





Eqns. (3.7) and (3.11) the constants C1 and C2 are given to be

The temperature distribution, Eq. (3.6), becomes

At x = L,

Heat transfer from the fin base

It may be noted that conservation of energy demands that the rate at which heat is

transferred by convection from the fin must be equal to the rate at which heat is

conducted through the base of the fin.

Case 3: The fin is thin and long enough so that the heat loss from the tip is negligible. All

the heat Q0 is convected out along the length and no heat is dissipated from the tip

surface.

Since C1 + C2 = θ0; then C1 + C2e2mL = θ0

The temperature distribution, Eq. (3.6), becomes





This is the temperature distribution along the fin.

At the tip, x = L,

It gives the temperature of the fin at its tip (TL).

The rate of heat transfer

It is found from Eqs. (3.17) and (3.18), that as L increases, Q (i.e. tanh mL) increases

rapidly at first and then the rate slowly decreases and becomes asymptotic at mL = 3,

which indicates that any further increase in length will not substantially increase the

rate of heat transfer. Also, as L increases, θL decreases.

Diagram 3.5 Schematic represents of four boundary conditions at the tip of a fin

Case 4:

If the temperature is given at the fin tip TL, then θL = TL − T∞ , and the resulting

expressions for temperature distribution

and heat transfer are

3.3 FIN PERFORMANCE (EFFECTIVENESS)

For a plane surface area, the thermal convection resistance is equal to (1/hA). Addition

of fin to a plane increases the surface area for convection. However, the thermal

resistance reduced for convection. However, the thermal resistance for conduction, over

that portion of the original surface at which the fins added, increases. Hence addition of

fins will not always increase the rate of heat transfer. The effectiveness of fin is

expressed as

(a) Convection heat

transfer at the tip

(b) Adiabatic tip

(c) Fixed tip temp.

(d) Infinite fin





where A is the cross-sectional area of the fin.

In any design, εf should be as large as possible, and in general, the use of fins is rarely

justified unless εf ≥ 2.

For any one of the four tip conditions given earlier, the effectiveness for a fin of uniform

cross-section may be obtained by dividing the appropriate expression for Q0 by hAθ0.

For the infinite fin (Case 1) is

For Case 3 with negligible tip loss

It is observed that fin effectiveness is enhanced by

1. If k is large, hence fins are usually made of aluminium. Which has a higher

thermal conductivity, and they are of low cost and density.

2. If heat transfer coefficient h is small, therefore fins are usually used to increases

the rate of heat transfer to the gas medium and they are less effective when the

medium is liquid (hliq >> hgas, hnatural convection << hforce convection).

3. If δ is small. Therefore, large numbers of thin and closely spaced fins are more

effective than fewer and thickness fins.

Increasing the ratio of the perimeter to the cross-sectional area of the fin, P/A.

Fin performance may also be quantified in terms of a thermal resistance. Treating the

difference between the base and fluid temperatures as the driving potential, a fin

resistance may be defined as

This result is very useful in the sense that a finned surface can be represented by a

thermal circuit.

Dividing Eq. (3.24) into the expression for thermal resistance due to convection at the

exposed base

Putting in Eq. (3.21), then we have

Hence, the fin effectiveness may be interpreted as a ratio of thermal resistances, and to

increase εf it is necessary to reduce the conductivity/convection resistance of the fin Rt, f.

If the fin is to enhance heat transfer, its resistance must not exceed that of the exposed

surface, Rt, b.

Efficiency

The thermal performance of a fin is also measured by a parameter called fin efficiency

(ηf). The maximum driving potential for convection is the temperature difference

between the base (x = 0) and the fluid, θ0 = T0 − T. Hence, the maximum rate at which





a fin could dissipate energy is the rate that would exist, if the entire fin surface were at

the base temperature, i.e. if the thermal conductivity of the fin is infinity.

where Af is the total surface area of the fin.

Actual heat transfer from a fin.

For a long thin fin with insulated tip

The fin efficiency is then

For a rectangular fin (from Diagram 3.4)

For a pin rod

where d is the diameter of pin.

For a rectangular fin is long, width and thin

The heat loss from the tip of fin can be taken in to account approximately by increasing

by t/2 and assuming that the tip insulated. This approximation keeps the surface area

from which heat is lost the same as the real cases, and fin efficiency will be





where the corrected fin length Lc = L + t/2.

Kreith and Bohn have, however, recommended Lc = L + A/P.

The error of this approximation is less than 8% when

For fins of non-uniform cross-section, the analysis dependent of graph for finding the

value of rc, Ap and other dimensions. It is necessary to calculate the maximum heat

transfer rate.

For rectangular fin of uniform cross-sectional area and the triangular and parabolic fins

of non-uniform cross- sectional area

for annular fins

In practice, a finned heat transfer surface is composed of the fin surface and the

unfinned surface. The total heat transfer will be

where A is the total area of fin and unfinned surface.

Application 1: The total efficiency for a finned surface may be defined as the ratio of

the total heat transfer of the combined area of the surface and fins to the heat which

would be transferred if this total area were maintained at the root temperature T0.

Show that this efficiency can be calculated from ηt = 1 − Af / A(1− ηt) where ηt = total

efficiency, Af = surface area of all fins, A = total heat transfer area, ηf = fin efficiency.

Solution: Fin efficiency,

Actual heat transfer from finned surface = ηfhAf (T0 − T∞)

Actual heat transfer from un finned surface which are at the root temperature:

h(A−Af)(T0 − T∞)

Actual total heat transfer = h(A −Af ) )(T0 − T∞) + ηfhAf (T0 − T∞)

By the definition of total efficiency,





3.4 LIMITATION OF AN EXTENDED SURFACE

The installation of fins on a heat transferring surface increases the heat transfer area

but it is not necessary that the rate of heat transfer would increase.

For long fins the rate of heat loss from the fin is given by

When h/mk = 1, or h = mk, then

which is equal to the heat loss from the primary surface with no extended surface. Thus

when h = mk, an extended surface will not increase the heat transfer rate from the

primary surface whatever be the length of the extended surface.

For h/mk > 1, Q < hAθ0 and hence adding a secondary surface reduces the heat transfer

and the added surface will act as an insulation.

For h/mk < 1, Q > hAθ0 and the extended surface will increase the heat transfer. The

heat transfer would be more effective when h/k is low for a given geometry.

3.5 RECTANGULAR FIN OF MINIMUM WEIGHT

Suppose the weight of one fin = b × L × z × ρ

where ρ is the density of fin material.

Consider A1 = b × L be the area of fin cross-section normal to z.

Diagram 3.6 Fin of rectangular cross-section

The length z is fixed at a given dimension, whereas the two dimensions b and L are to be

changed so as give maximum heat flow for a given area A1 as shown in Diagram 3.6.

We have

If the tip loss is neglected, then

For a given A1, Q1, will be maximum, when





Put

This is the condition for the maximum heat flow for a given weight of fin, giving the

optimum ratio of fin height to half the fin thickness.





Fin effectiveness

This equation makes it possible to determine the heat flow increase through the wall as

a result of the addition of fins.

3.6 GENERALIZED EQUATION FOR FINS

Let us consider an extended surface of arbitrary shape (Diagram 3.7).

Diagram 3.7 General one dimensional fin

Assuming heat to be flowing only in the longitudinal direction (x), an energy balance for

an elemental area of thickness dx at a distance x from the base is given as

Since both A and P are functions of x,

Dividing throughout by kA and putting T(x) T = θ(x),

This result provides a general form of the energy equation for an extended surface.

Now let us consider its application to the case of a triangular fin as shown in Diagram

(3.8).

Diagram 3.8 Triangular fin

The perimeter P ≈ 2z.

The cross-sectional area A = zbx/L. Substituting these values in Eq. (3.43), we obtain

L

P





It is a modified form of the Bessel equation, which in its general form for any value of n

is

The general solution of above equation is

where B and C are integral constants, and In and Kn are n order Bessel functions of the

first and second kind respectively.

In order to make use of the solutions available for the standard form of Eq. (3.45), Eq.

(3.44) is converted into the same form of Eq. (3.45) as follows:

Put ξ2 = 2Lh/bk and multiplying Eq. (3.44) by x2, then we get

Now again put Γ = 2ξ√x, and x = Γ2/4ξ2, then

Now using Eq. (3.47)

The above equation is identical to the modified Bessel equation of zero order (n = 0) and

its general solution is

where I0 and K0 axe modified zero order Bessel functions of the first and second kind

respectively. Some typical values of I0(Γ) and K0(Γ) are taken from table. It is seen that

I0(0) = 1, while K0(0) = .

The constants of integration B and C are obtained by the utilizing boundary conditions;

At the root, θ = θ0, at x = L,

At the tip, θ = finite at x = 0

Since at x = 0, K0(0) approaches infinity, C = 0.

From the first boundary condition,

The heat flow rate Q from the fin is given by

We know that from the properties of Bessel functions that





So that for n = 0,

Diagram 3.9 Fin of triangular cross-section

If the heat flow into the triangular fin (Diagram 3.9) is optimised in order to determine

the best ratio of height L to base b, the following expression as shown by Eckeit and

Drake results.

The temperature excess at the tip of the fin is

The ratio of the thickness of the triangular fin to the thickness of the rectangular fin

with equal heat flow is 1.31, and the ratio of the cross-sectional areas is 1:1.44.

Therefore, the weight saved by using the triangular fin is 44%.

3.7 FIN OF MINIMUM WEIGHT

In such a fin, every part should be utilised to the same degree, and the specific rate of

heat flow, q should be constant throughout the fin. The heat flow lines are equally

spaced and parallel to the fin axis (Diagram 3.10).

Diagram 3.10 Fin with smallest weight

Since

L

L





or

Temperature decreases linearly along any flow line from the value T0 at the root of the

fin to that at the tip which approaches T of the surrounding fluid when becomes zero.

For a finite value of there will be a temperature discontinuity between the fin tip and

the surrounding fluid as shown in Diagram 3.10.

Let us consider a surface element of the fin at a distance x. The element is inclined to

the fin axis by the angle . The specific rate of heat flow from this element is

where

Thus the contour lines of the fin are circles which meet tangentially at the tip for the

smallest weight of a given heat flow. The difference of weight between a fin in the shape

of a circular arc and the fin of triangular shape is very small.

3.8 FIN ARRANGEMENT

The cross-sectional area A1 = (bL) necessary for a given heat flow in the example of a

rectangular fin is derived by combining Eqs. (3.57) and (3.58), and solving for A1.

This equation (3.55) shows that it is advantageous to make the fins as thin (b) or as

small (A1 = bL) as possible. To double the heat flow, the area of one fin (A1) or thickness

b must be increased eight times, whereas it is sufficient to use two fins of the original

size.

Equation (3.55) shows that A1 L/k. The mass of the fin is, therefore, proportional to

ρ/k. It may be seen that by using aluminium, instead of copper, a weight saving of 50%

can be achieved. Iron fins have a ten-fold weight, and stainless steel about 50-fold

weight.

3.9 CYLINDRICAL FINS

Fins which are arranged around tubes are called cylindrical fins system is shown in

Diagram 3.. Here again the treatment is substantially the same as for rectangular fin

except that the area must be allowed to vary with the radius.





Diagram 3.11 Element of annular fin

The area normal to the heat flux vector can be written as

A = 2 rb

and the periphery can be expressed as

P = 4 r

Let us consider an annular element of radius r and thickness dr (Diagram 3.11).

By making an energy balance

Suppose T T = θ, then

The equation is recognised as the Bessel's equation of zero order and its solution is

I0 = modified Bessel function, 1st kind, and K0 = modified Bessel function, 2nd kind,

zero order.

The constants B and C are evaluated by applying the two boundary conditions:

At r = r1, T = Tw, θ = Tw T,

At r = r2, dT/dr = 0, dθ/dr = 0, since b << (r2 r1).

Using above conditions, we get temperature distribution

I1(mr) and K1(mr) are Bessel function of order one and m = (2h/kb)1/2.

The rate of heat transfer is

The fin efficiency for a convective tip then

where the tip radius r2 is replaced by a corrected radius r2c = r2 + b/2.The results are

represented graphically.

Top view of a annular fin Annular fin of uniform thickness

Top view

Front view

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HHeeaatt aanndd MMaassss TTrraannssffeerr …brijrbedu.org/Brij Data/Brij HMT/SM/Chapter-3 Extended Surfaces...3.1 EXTENDED SURFACES ... Diagram 3.1 Combined conduction and convection