advanced heat transfer lecture1

8/13/2019 Advanced Heat Transfer Lecture1

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Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)

Lecture-1. Governing Laws for Thermal Radiation

Contents of the lecture

1.1 Heat Transfer Mechanisms

1.6 Geometrical Considerations

1.7 Governing Laws for Thermal Radiation1.8 Blackbody Radiation in a Wavelength Interval

1.11 Blackbody Emission into a Medium Other than Vacuum

1.10 Historical Note Origin of Quantum Mechanics

1.12 Summary

1.2 Electromagnetic Radiation


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What is heat transfer?

Heat transfer (or heat) is energy in transit due toa temperature difference

HEAT TRANSFER MODES


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The convention (in this lecture series) is

Heat transfer rate Q& in W (J/s)

Amount of heat (energy) Q in J

Heat flux q& in W/m2


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Radiation which is given off by a body

because of its temperature is called

thermal radiation

A body of a temperature larger than 0 K

emits thermal radiation


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RELEVANCE OF THERMAL RADIATION

4

2

4

1

21

21

TTQ

TTQ

TTQ

radiation

convection

conduction

&

&

&

When no medium is present radiation is the only

mode of heat transfer


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ELECTROMAGNETIC WAVES

Classical theory

Quantum theory

vhEphoton = sJ1063.634 = h


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SPEED, FREQUENCY and WAVELENGTH

For any wave:

=w

Determined

by the medium

Determined by

the source

For electromagnetic waves:

=cc=3108 m/s ( in vacuum)


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SPEED, FREQUENCY and WAVELENGTH

For a medium other than vacuum:

medium

mediumn

cc =

The frequency stays the same so,

medium

medium

n

=


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COMMON UNITS FOR WAVELENGTH

1 micrometer = 10-6 m

1 nanometer = 10-9 m

1 angstrom = 10-10 m


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Example 1.1 (Calculate energy of photons)

Frequency

(Hz)

Photon

energy in J

Energy inelectron

volts

Number ofphotons in a

joule of energy

Short radio

waves

=1076.6310-27 4.110-8 1.51026

Visible light

waves=1015

6.6310-19

4.1 1.51018

X-rays

=1018 6.6310-16 4.1103 1.51015

Gamma

rays

=1020

6.6310-14 4.1105 1.51013


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THERMAL RADIATION


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1.6 Geometrical Considerations

1.6.1 Normal to a Surface Element


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1.6.2 Solid Angle


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Example 1.2

Derive formula for calculating the length of an arc andthe circumference of a circle.

dRds =

( ) ==2

1

12

RdRs

Plane anglein radiance

radiansinanglePlaneRadiusarcanofLength =

2circletheofnceCircumfere =R


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Derive formula for calculating the area of a sphere

dRdA = 2

( ) ==2

1

12

22

RdRA

The solid angle

in steradians

( ) steradiansinangleSolidRadius

spheretheofpartaofArea

2

=

=

How to calculate the solid angle?


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2R

dAd s=

( ) ( ) ddRdRdRdAs == sinsin2


ddd = sin


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Now we can complete the integration since we know

how to calculate the solid angle:

===

ddRdRA2

1

2

1

2

1

sin22

( ) ( ) ==2

1cos122 R

( ) )cos(cos 21122 =R

22 2)01(2e)(hemispherArea RR ==

Solid angle for a hemisphere is 2

Solid angle for a sphere is 4


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1.6.3 Area and Projected Area

cos=dAdAP


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1.6.4 Radiation Intensity and Irradiation

msrm

Wi

)AreaProjected(inintensityspectraltheis 2

'

indicates direction

srm

Wi

)AreaProjected(inintensitytotaltheis

2

'

=0

'' dii


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Irradiation

=directionsall

' cos),,( dig&

= =

==

2

0

2/

0

' sincos),,( ddig&

for isotropic incoming radiation

==2/

0

' )2()2sin(2

1

dig&

[ ] '2/0

' )2cos(2

1

ii ==


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1.7.1 Black Body RadiationReal surfaces (bodies)

gggg &&&& ++=

reflectivity

absorptivity

transmissivity

1.7 Governing Laws for Thermal Radiation


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BLACK BODY RADIATION

Definition of a black body

A black body is defined as an ideal body that all

incident radiation pass into it and internally absorbs

all the incident radiation.

This is true for radiation of all wavelengths and for all angles

of incidence


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BLACK BODY RADIATION

Properties:

Black body is a perfect emitterIn a black body enclosure radiation is isotropic

Black body is a perfect emitter in each direction

Black body is a perfect emitter at any wavelength

Total radiation of a black body into vacuum is a

function of temperature only


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===hemisphere

b

hemisphere

bbb ididie''' coscos &

The angular distribution of radiation intensity

emitted by a black body


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1.7.2 Plancks Radiation Law

1

1),(),(

/5

1'

2

==

TCbb e

CTiTe

&

216

1 mW107418.3 =

C

Km01438769.1 2

2

= C


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Plancks Radiation Law


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Plancks Radiation Law


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( ) 1

1),(/2

5

1

5

=TC

eT

C

T

Te b

&


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See Example 1.4 of the lecture notes to understand

the meaning of:

Frequency distribution

Cumulative frequency distribution

Relative cumulative frequency distribution

E l 1 4


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156

163

170

177

184

191

198205

4

12

18

25

33

22

115

TOTAL 130

153-159

160-166

167-173

174-180

181-187

188-194

195-201202-208

Class mark

(cm)

Number of

students-Frequency

Height per

class (cm)

Example 1.4


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Histogram and frequency polygon of heights of 130 students

Example 1.4

149 156 163 170 177 184 191 198 205 2120

5

10

15

20

25

30

35

f(x)

QP

Nu

mberofstudentsperheight

Height (cm)

Example 1 4


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=+++++++==Q

P

dxxfArea 130)51122332518124()(

classtheofwidththeis7cm=

(130)studentsofnumbertotaltheArea

Example 1.4

Example 1 4


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0

4

16

34

59

92

114125

130

Less than 153 cm

Less than 160 cm

Less than 167 cm

Less than 174 cm

Less than 181 cm

Less than 188 cm

Less than 195 cmLess than 201 cm

Less than 208 cm

Number of studentsHeight (cm)

Cumulative distribution

(less than the upper class boundary)

Example 1.4


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Example 1 4


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Cumulative distributionExample 1.4

150 160 170 180 190 200 210

0

20

40

60

80

100

120

F(x)Cu

mulativeFreq

uency(No.of

Students)

Height (cm)

0.0

0.2

0.4

0.6

0.8

1.0

F

(x)RelativeCumulativeFre

quency


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1.7.3 Wiens Displacement Law

We are looking for a wavelength that maximizesthe Plancks function for a given temperature

( )1

/51/5

1 11

1),( 22

==

TCTCb eC

eCTe &

( ) += 1/

6

1

1)5(2 TCb

e

C

d

ed

&

( ) ( ) 0)1(1)1( 22/2/5

1 22 =+

T

Cee

C TCTC

( )TC

e

CT

=

/

2

21

1

5


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( )22

/

1

( ) 5 1 C T

C

f T Te =

0.000 0.002 0.004 0.006 0.008 0.010

-0.010

-0.005

0.000

0.005

0.010

(C3-Wien's constant)

max

T = 0.0028977756 mK

T in mK

f(

T)in

mK


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Wiens Law

Km2,898C3max ==T

1.7.4 Stefan-Boltzmann Law


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1.7.4 Stefan Boltzmann Law

==0

?),( dTee bb &&

( )204 3

1 1

4/520

11b C T

C T C

e d dC ee

= = & TC

= 2

0 3

1 15d

e

=

44

4

2

1

15TT

C

Ceb ==

&

8 2 45.67 10 W/(m K ) = Stefan-Boltzmann

constant

1 8 Bl kb d R di ti i W l th I t l


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1.8 Blackbody Radiation in a Wavelength Interval

=

=

2

1

2

1

21),(

1

),(

),(

4

0

_

dTe

TdTe

dTeF TT &

&

&

21


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=

=2

1

21),(

14_

dTe

T

F bTT &

TTbb FFdTedTeT 12

2 1

_0_0

0 0

4),(),(

1

=

= &&


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2 1

1 2 2 1_ 0_ 0_5 50 0

( , ) ( , )1( ) ( )

T T

b b

T T T T

e T e T F d T d T F F

T T

= =

& &

1 9 Bl kb d E i i i M di O h h V


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1.9 Blackbody Emission into a Medium Other than Vacuum

21

21 /2 nCchC mm == nCk

chC mm /22 =

=

n

ccm = nm

=

11),(

/5

1

2 =

TCbe

CTe

&

),(),( 3 TenTe bmmb && =


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Stefan-Boltzmann Law

42Tnebm = &

Wiens Displacement Law

n

CTn

3max, =

1 10 i i O i i f Q i


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1.10 Historical Note Origin of Quantum Mechanics

1 Th h ll i


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1

1),(

/5

=Tbb e

aTe

&

The challenge was in

deriving a and b constantsfrom the first principle


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Quantification of energy (Max Planck 1990)

vhmE =

m=1,2,3,... quantum number

Ten years later Planck wrote:

My futile attempts to fit the elementary quantum of

action (h) somehow into the classical theory continued for

a number of years, and they cost me a great deal of efforts

In 1905 Albert Einstein made an assumption


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the energy of a light was concentrated into

localized bundles later called photons

=hEPlanck, the originator of the h constant, did not acceptat once Einsteins photons. In 1913 Planck wrote about

Einstein that he sometimes have missed the target in his

speculations, as for example in his theory of light quanta,

cannot really be held against him

In 1918 Planck received a Nobel prize for his discovery

of energy quanta

In 1921 Einstein received his Nobel prize for his service to

theoretical physics and specially for discovery of the law of

photoelectric effect

1 12 Summary


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1.12 Summary

Students should understand:

The concepts of radiation intensity and emissive power

The radiation laws for black-body radiation

Plancks law

Wiens law

Stefan-Boltzmann law

advanced heat transfer lecture1

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