Chapter 3

Extended Surfaces / Fins

Chapter 3

Extended Surfaces (Fins)

An extended surface (also known as a combined conduction-convection system or a fin) is a solid within which heat transfer by conduction is assumed to be one dimensional, while heat is also transferred by convection (and/or radiation) from the surface in a direction transverse to that of conduction

36

Chapter 3

Heat Transfer from Extended Surfaces

• Extended surfaces may exist in many situations but are commonly used as fins to enhance heat transfer by increasing the surface area available for convection (and/or radiation).

37

Chapter 3

Typical Fin Configurations

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Chapter 3

Fins of Uniform Cross-Sectional AreaAssuming one-dimensional, steady-state conduction in an extended surface of constant conductivity and uniform cross-sectional area with negligible generation and radiation, the fin equation is of the form:

02

2

)( TTkA

hP

dx

Td

c

where p is the fin perimeter

39

TTbb

TxT )(

Define:

(3.6.1)

Chapter 3

Boundary Conditions

• At the base T = Tb or (0)=b

• At the tip:Case A: Convection heat transfer

Case B: Adiabatic tip

Case C: Prescribed temperature, (L)=L

Case D: Infinite fin, T(L)=T or (L)=

Case A

40

Chapter 3

Solutions of Differential Equation

)( TThPkAq bcf

41

(3.6.2)

Chapter 3

True or False?

• Heat is transferred from hot water flowing through a tube to air flowing over the tube. To enhance the heat transfer rate the fins should be installed on the tube interior surface (the hot water side)

• Fins are particularly beneficial when h is small (typical for a gas or when only natural convection exists).

• Ideally the fin material should have a large thermal conductivity to minimize temperature variations from its base to its tip.

Chapter 3

Selection of fin material (Example 3.9)

kCu>kAl>kSS

(1)

(2)

(3)

SS

Al

Cu

Chapter 3

Example: Problem 3.116

Assessment of cooling scheme for gas turbine blade. (a) Determine whether the blade temperature is less than the maximum

allowable value (1050 °C) for the prescribed operating conditions (b) Evaluate blade cooling rate.

Assume that convective heat losses from the surface are negligible, i.e. adiabatic tip condition.

42

Chapter 3

Fin Performance

• Fin effectiveness: Ratio of the fin heat transfer rate qf to the heat transfer rate that would exist without the fin

bbc

ff hA

q

,

f should be as large as possible (at least >2)

• For a very long (infinite) fin (Case D boundary condition):

21 /

cf hA

kP

where b=Tb-T, and Ac,b is the fin cross-sectional area at the base

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(3.6.3)

(3.6.4)

Chapter 3

Fin Performance

ft

b

ft

bf RR

TTq

,,

• Fin heat transfer rate:

where Rt,f is the fin resistance

Can express fin effectiveness as a ratio of thermal resistances:

ft

btf R

R

,

, where Rt,b is the resistance due to convection of the exposed base (=1/hAc,b)

44

Chapter 3

Fin Performance• Fin efficiency: The ratio of the actual heat transfer rate from the fin to the

maximum rate at which a fin could dissipate energy

bf

fff hA

q

q

q

max

See Table 3.5 and Figures 3.18 and 3.19 for the efficiencies of common fin shapes

ffft hA

R

1,

We can use the efficiency to calculate the fin resistance

45

(3.6.5)

(3.6.6)

Chapter 3

Fin Arrays• Define the overall efficiency, o as

)( ft

fo A

NA 11

where N is the number of fins in the array, Af the surface area of each fin and At the total surface area.

• We can then calculate the heat rate for the fin array

btot hAq

• Thermal resistance of the fin array

toot hA

R

1,

)( ft

fo A

NA 11

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bt

tto hA

q

q

q

max

(3.6.7)

(3.6.8)

(3.6.9)

Chapter 3

Fin Manufacturing

Care must be exercised to ensure that the thermal contact resistance does not adversely influence the overall fin performance

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Chapter 3

ExampleAs more components are placed on a single integrated circuit (chip), the amount of heat dissipated increases. The maximum allowable chip operating temperature, is approximately 75°C. Suggest ways to maximize heat dissipation.

Top view Side view

Air, T=20°C

48

Chapter 3

Fins in Heat Exchangers

• Widely used to achieve large heat rates per unit volume, particularly when one or both fluids is a gas.

• Characterized by large heat transfer surface areas per unit volume (>700 m2/m3), small flow passages, and laminar flow.

Chapter 3

Fin (extended surface) effects

• Fins reduce the resistance to convection heat transfer, by increasing surface area.

• The expression for the overall heat transfer coefficient includes overall surface efficiency, or temperature efficiency, o, of the finned surface, which depends on the type of fin (see also Ch. 3.6.5)

hoho

hfconduction

co

cf

co hAA

RR

A

R

hAUA )()()()(

",

",

111

(11.5)

where c is for cold and h for hot fluids respectively