Heat and Mass Transfer II(ME-315)

Chapter 3

Heat Transfer from Extended Surfaces: Heat transfer from surfaces with uniform cross-sections

Solutions based on various boundary conditions,

Fin efficiency curves

Applications and design.

The rate of heat transfer from a surface at a temperature Ts to the surrounding medium at T∞ is given by Newton’s law of cooling as

If Ts and T∞ are fixed how can we increase the heat transfer ?

Increase A or h

Need for Fins

Need for Fins

We can increase the surface area by attaching to the surface an extended surfaces called fins made of highly conductive materials such as aluminum. Finned surfaces are manufactured by extruding, welding, or wrapping a thin metal sheet on a surface.

Fins enhance heat transfer from a surface by exposing a larger surface area to convection and radiation.

Need for Fins

Chapter 3

The thin plate fins of a car radiator greatly increase the rate of heat transfer to the air

Fin Configurations

Assumptions

1. The heat conduction in the fin is steady and one-dimensional.2. There is no energy generation in the fin.3. The convective environment is characterized by a uniform and constant heat transfer coefficient and temperature.4. The fin has a constant thermal conductivity.5. The contact between the base of the fin and the primary surface is perfect.6. The fin has a constant base temperature.

Fin Equation

General Equation

From Fourier’s law

where Ac is the cross-sectional area, which may vary with x . Since the conduction heat rate at x+dx may be expressed as

General Equation

General Equation

This result provides a general form of the energy equation for an extended surface. Its solution for appropriate boundary conditions provides the temperature distribution

Fins with uniform cross sectionFor the prescribed fins, Ac is a constant and As = Px, where As is the surface area measured from the base to x and P is the fin perimeter. Accordingly,

dAc /dx = 0 and dAs /dx = P,

Reduces to

dAc /dx = 0 and dAs /dx = P, Since

To simplify the equation

Its general solution is of the form

To evaluate the constants C1 and C2 of we need to specify appropriate boundary conditions. Like temperature at the base of the fin (x= 0)

Longitudinal Convecting Fins1. Rectangular Fin

1. Constant base temperature and convecting tip: Applying an energy balance to a control surface about this tip (x=L)

Incorporating both boundary conditions in general solution we get

BC1 @ x=0

BC2 @ x=L

Above equation will yield temperature profile as shown in figure . Note that the magnitude of the temperature gradient decreases with increasing x. This trend is a consequence of the reduction in the conduction heat transfer qx(x) with increasing x due to continuous convection losses from the fin surface.

Heat transfer from FinWe can calculate total heat transfer either considering conduction from the base or convection from total surface of fin both will yield same result

Conduction:

Convection:

where Af is the total, including the tip, fin surface area

Introducing expression of θ in both equation will yield same following result:

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Case: Convective heat loss from the fin tip is negligible( adiabatic case).

From general solution we get

Fin heat transfer rate

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As discussed earlier heat transfer rate can be written as:

Case: when temperature at the fin tip is known

From general solution we get

Fin heat transfer rate

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As discussed earlier heat transfer rate can be written as:

Infinitely Long Fin (Tfin tip=T) For a sufficiently long fin the temperature at the fin tip

approaches the ambient temperature Boundary condition: (L→∞)=T(L)-T∞=0

Infinitely Long Fin (Tfin tip=T) For a sufficiently long fin the temperature at the fin tip

approaches the ambient temperature Boundary condition: ϴ(L→∞)=T(L)-T∞=0

The temperature distribution:

heat transfer from the entire fin

/( ) cx hp kAmx

b

T x T e eT T

--¥

¥

- = =-

( )0

c c bx

dTQ kA hpkA T Tdx ¥

==- = -

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Corrected Length concept

A practical way of accounting for the heat loss from the fin tip is to replace the fin length L in the relation for the insulated tip case by a corrected length defined as

Fin Efficiency To maximize the heat transfer from a fin the

temperature of the fin should be uniform (maximized) at the base value of Tb

In reality, the temperature drops along the fin, and thus the heat transfer from the fin is less

To account for the effect we define a fin efficiency

or

,max

finfin

fin

QQ

h = =

Actual heat transfer rate from the fin

Ideal heat transfer rate from the finif the entire fin were at base temperature

,max ( ) fin fin fin fin fin bQ Q hA T Th h ¥= = -

Fin Efficiency

For constant cross section of very long fins:

For constant cross section with adiabatic tip:

( )( ),

,max

1 1 fin c b clong fin

fin fin b

Q hpkA T T kAQ hA T T L hp mL

h ¥

¥

-= = = =-

( )( ),

,max

tanh

tanh

fin c badiabatic fin

fin fin b

Q hpkA T T mLQ hA T T

mLmL

h ¥

¥

-= = -

=

Afin = P*L

Fin Effectiveness The performance of the fins is judged on the basis of

the enhancement in heat transfer relative to the no-fin case.

The performance of fins is expressed in terms of the fin effectiveness

defined as

( )

fin finfin

no fin b b

Q QQ hA T T

e¥

= = =- Heat transfer

rate from the surface of area

Ab

Heat transfer rate from the fin of base

area Ab

Fin effectiveness

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Efficiency

Efficiency of straight fins (rectangular, triangular, and parabolic profiles).

Efficiency of annular fins of rectangular profile

Efficiency

Fin Effectiveness

An effectiveness of fin ε=1 indicates that the addition of fins to the surface does not affect heat transfer at all. That is, heat conducted to the fin through the base area Ab is equal to the heat transferred from the same area Ab to the surrounding medium.

An effectiveness of fin ε< 1 indicates that the fin actuallyacts as insulation, slowing down the heat transfer from the surface. This situation can occur when fins made of low thermal conductivity materials are used. An effectiveness of fin ε > 1 indicates that fins are enhancing heat transfer from the surface, as they should. However, the use of fins cannot be justifiedunless fin is sufficiently larger than 1.

Fin Effectiveness

Both the fin efficiency and fin effectiveness are related to the performance of the fin, but they are different quantities. However, they are related to each other by

Efficiency of Common Fin Shapes

Efficiency of Common Fin Shapes

Effect of Fins on Heat Transfer from Steam Pipes

Steam in a heating system flows through tubes whose outer diameter is D1= 3 cm and whose walls are maintained at a temperature of 120°C. Circularaluminum fins (k = 180 W/m · °C) of outer diameter D2 = 6 cm and constant thickness t = 2 mm are attached to the tube, as shown in Fig.

Effect of Fins on Heat Transfer from Steam Pipes

The space between the fins is 3 mm, and thus there are 200 fins per meter length of the tube. Heat is transferred to the surrounding air at T = 25°C, with a combined heat transfer coefficient of h= 60 W/m2 · °C. Determine the increase in heat transfer from the tube per meter of its length as a result of adding fins.

SOLUTION Circular aluminum fins are to be attached to the tubes of a heating system. The increase in heat transfer from the tubes per unit length as a result of adding fins is to be determined.

Effect of Fins on Heat Transfer from Steam Pipes

Assumptions 1 Steady operating conditions exist. 2 The heat transfer coefficient is uniform over the entire fin surfaces. 3 Thermal conductivity is constant.4 Heat transfer by radiation is negligible.Properties The thermal conductivity of the fins is given to be k = 180 W/m · °C.Analysis In the case of no fins, heat transfer from the tube per meter of its length is determined from Newton’s law of cooling to be

Effect of Fins on Heat Transfer from Steam Pipes

Efficiency can be found directly from the efficiency chart if we know the corrected length

=

Effect of Fins on Heat Transfer from Steam Pipes

Effect of Fins on Heat Transfer from Steam Pipes