Extended Surfaces

Chapter ThreeSection 3.6

Lecture 6

Nature and Rationale

Nature and Rationale of Extended Surfaces• An extended surface (also know 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.

– Why is heat transfer by conduction in the x-direction not, in fact, one- dimensional?

– If heat is transferred from the surface to the fluid by convection, what surface condition is dictated by the conservation of energy requirement?

Nature and Rationale (cont.)

– What is the actual functional dependence of the temperature distribution in the solid?– If the temperature distribution is assumed to be one-dimensional, that is, T=T(x) , how should the value of T be interpreted for any x location?

– How does vary with x ?cond,xq

– When may the assumption of one-dimensional conduction be viewed as an excellent approximation? The thin-fin approximation.

• 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). They are particularly beneficial when is small,

as for a gas and natural convection.h

• Some typical fin configurations:

Straight fins of (a) uniform and (b) non-uniform cross sections; (c) annularfin, and (d) pin fin of non-uniform cross section.

Heat Transfer from Extended SurfacesExtended Surface to Increase Heat Transfer

Rate

Heat Transfer from Extended Surfaces

Extended Surface to Increase Heat Transfer

Rate

Three ways to increase q

Increase h

Reduce

Increase A

)( TThAq s

T

Heat Transfer from Extended Surfaces

Applications of Fins

Heat Transfer from Extended Surfaces

A General Conduction Analysis

Heat Transfer from Extended Surfaces

Applying Conservation of energy to the

differential element:

From Fourier’s Law

convdxxx dqqq

dx

dTkAq cx

Heat Transfer from Extended Surfaces

From Taylor Series Expansion:

Heat of Convection

dxdx

dqqq x

xdxx

dxdx

dTA

dx

dk

dx

dTkAq ccdxx )(

)]([ TTAhddq sconv

Heat Transfer from Extended Surfaces

Heat Transfer Eqn:

Or

0)()( TTdx

dA

k

h

dx

dTA

dx

d sc

0))(1

()1

(2

2

TTdx

dA

k

h

Adx

dT

dx

dA

Adx

Td s

c

c

c

Fins of Uniform Section Area

0)(2

2

TTkA

hP

dx

Td

c

Fins of Uniform Section AreaHeat Transfer Eqn:

Where dAs/dx = P (fin perimeter), dAC/dx =

0

Define Excess Temperature

0)(2

2

TTkA

hP

dx

Td

c

TxTx )()(

Fins of Uniform Section Area

Define:

Heat Transfer Eqn.:

General Solution

ckA

hPm 2

022

2

m

dx

d

mxmx eCeCx 21)(

Fins of Uniform Section Area

Need Two b.c.’s to evaluate C1 and C2.

Fins of Uniform Section Area

b.c.-1:

b.c.-2:

Or b.c.-2

From b.c.-1:

From b.c.-2:

bb TT )0(

Lxcc dx

dTkATLThA

])([

Lxdx

dkLh

)(

21 CCb

)()( 1221mLmLmLmL eCeCkmeCeCh

Fins of Uniform Section AreaSolving for C1 and C2:

The hyperbolic function definitions (Page

1014):

mLmkhmL

xLmmkhxLm

b sinh)/(cosh

)(sinh)/()(cosh

)(2

1cosh

)(2

1sinh

xx

xx

eex

eex

Fins of Uniform Section Area

Overall Heat Transfer Rate:

Above equation can be obtained from

energy conservation of the fin.

mLmkhmL

mLmkhmLhpkAcqq bbf sinh)/(cosh

cosh)/(sinh

Fins of Uniform Section Area

If the fin tip is adiabatic (second tip b.c.)

0Lxdx

d

mL

xLm

b cosh

)(cosh

mLhPkAq bcf tanh

Fins of Uniform Section Area

If the fin tip is at constant T (Third tip b.c.)

LL )(

mL

xLmmxbL

b sinh

)(sinhsinh)(

mL

mLhPkAq bL

bcf sinh

/cosh

Fins of Uniform Section Area

If the fin is very long (Fourth tip b.c.)

0, LL

mx

b

e

bcf hPkAq

Fins of Uniform Section Area

Tip Condition(x=L) T distribution Fin Heat Transfer

A

B

C

D

b / fq

LxdxdkLh

)/()(

0/ Lx

dxd

LL )(

0)( L

mLmkhmL

mLmkhmLhpkAc b sinh)/(cosh

cosh)/(sinh

mL

xLm

cosh

)(cosh

mL

xLmmxbL

sinh

)(sinhsinh)(

mxe

mLmkhmL

xLmmkhxLm

sinh)/(cosh

)(sinh)/()(cosh

mLhPkA bc tanh

mL

mLhPkA bL

bc sinh

/cosh

bchPkA

Example 3.9 (pages 162-164)

A very long rod 5 mm in diameter has one end maintained at

100 ºC. The surface of the rod exposed to ambient air at 25

ºC with a convection heat transfer coefficient of 100 W/m2 K.

1. Determine the temperature distributions along rods

constructed from pure copper, 2024 aluminum alloy, and

type AISI 316 stainless steel. What are the corresponding

heat losses from the rods?

2. Estimate how long the rods must be for the assumption of

infinite length to yield accurate estimate of the heat loss.

Example 3.9

Known: A long circular rod exposed to ambient air

Find:

1. T distribution and heat loss when rod is fabricated from

copper an aluminm alloy, or stainless steel

2. How long rods must be to assume infinite length

Schematic:

Example 3.9

Assumptions:

1. Steady-state;

2. 1-D conduction in x direction;

3. Constant properties.

4. Negligible radiation;

5. Uniform heat transfer coefficient;

6. Infinite long rod

Properties:

Example 3.9

•   1/2

Example 3.9

Example 3.9

•  

Example 3.9

•  

Example 3.9

Comments:

The mL 2.65 was based on heat loss. If the requirement is

to predict the T distribution accurately, a larger mL value

(4.6) is needed. It was based on exp(-mL)<0.01.

Problem 3.126

3.126 Turbine blades mounted to a rotating disc in a gas turbine engine are exposed to a gas stream that is at T=1200 ºC and maintains a convection coefficient of

h=250 W/m2 K over the blade. The blades, which are fabricated from Inconel, k 20 W/m K, have a length of L=50 mm. The blade profile has a uniform cross-sectional area of A=6xl0-4m2 and a perimeter of P =110 mm. A proposed blade-cooling scheme, which involves routing air through the supporting disc, is able to maintain the base of each blade at a temperature of T = 300 ºC.

Problem 3.126

(a). If the maximum allowable blade temperature is 1050 ºC and the blade tip may be assumed to be adiabatic, is the proposed cooling scheme satisfactory?

(b) For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant?

Problem: Turbine Blade Cooling

Problem 3.126: Assessment of cooling scheme for gas turbine blade.Determination of whether blade temperatures are lessthan the maximum allowable value (1050°C) for prescribed operating conditions and evaluation of bladecooling rate.

Assumptions: (1) One-dimensional, steady-state conduction in blade, (2) Constant k, (3)Adiabatic blade tip, (4) Negligible radiation.

Analysis: Conditions in the blade are determined by Case B of Table 3.4.

(a) With the maximum temperature existing at x = L, Eq. 3.80 yields

Schematic:

1

coshb

T L - T

T - T mL

1/22 4 2250 W/m K 0.11m/20W/m K 6 10 m1/ 2cm hP/kA = 47.87 m-1

mL = 47.87 m-1 0.05 m = 2.39

From Table B.1 (or by calculation), Hence, cosh 5.51.mL

and, subject to the assumption of an adiabatic tip, the operating conditions are acceptable.

Eq. 3.81 and Table B.1 yield

Hence,

Comments: Radiation losses from the blade surface contribute to reducing the blade temperatures, but what is the effect of assuming an adiabatic tip condition? Calculatethe tip temperature allowing for convection from the gas.

o o o1200 C (300 1200) C/5.51 1037 CT L

(b) With 1/22 4 2 o1/2 250W/m K 0.11m 20W/m K 6 10 m 900 C 517Wc bM hPkA ,

tanh 517W 0.983 508Wfq M mL

508Wb fq q

Problem: Turbine Blade Cooling (cont.)

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