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

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 Radiation

1.8 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


What is heat transfer?

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

HEAT TRANSFER MODES


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


Radiation which is given off by a bodybecause of its temperature is called thermal radiation

A body of a temperature larger than 0 K emits thermal radiation


A scene from “Silence of the lambs”

taken with an ordinary camera

taken with an infrared camera

A photograph of a car

The number plate has been wiped out


RELEVANCE OF THERMAL RADIATION

42

41

21

21

TTQ

TTQ

TTQ

radiation

convection

conduction

When no medium is present radiation is the onlymode of heat transfer


ELECTROMAGNETIC WAVES

Classical theory

Quantum theory

vhEphoton sJ 1063.6 34 h


SPEED, FREQUENCY and WAVELENGTH

For any wave:

w

Determinedby the medium

Determined by the source

For electromagnetic waves:

cc=3·108 m/s ( in vacuum)


SPEED, FREQUENCY and WAVELENGTH

For a medium other than vacuum:

mediummedium n

cc

The frequency stays the same so,

mediummedium n


COMMON UNITS FOR WAVELENGTH

1 micrometer = 10-6 m

1 nanometer = 10-9 m

1 angstrom = 10-10 m


Example 1.1 (Calculate energy of photons)

Frequency (Hz)

Photon energy in J

Energy inelectron

volts

Number of photons in a

joule of energy

Short radio wavesν=107

6.63·10-27 4.1·10-8 1.5·1026

Visible light wavesν=1015

6.63·10-19 4.1 1.5·1018

X-rays ν=1018 6.63·10-16 4.1·103 1.5·1015

Gamma rays

ν=1020

6.63·10-14 4.1·105 1.5·1013


THERMAL RADIATION


1.6 Geometrical Considerations

1.6.1 Normal to a Surface Element


1.6.2 Solid Angle


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

radiansin angle PlaneRadiusarcan ofLength

2circle theof nceCircumfere R


Derive formula for calculating the area of a sphere

dRdA 2

2

1

1222

RdRA

The solid anglein steradians

steradiansin angle SolidRadius

sphere theofpart a of Area2

How to calculate the solid angle?


2R

dAd s

ddRdRdRdAs sinsin 2


ddd sin


Now we can complete the integration since we knowhow to calculate the solid angle:

ddRdRA2

1

2

1

2

1

sin22

2

1cos12

2

R

)cos(cos 21122 R

22 2)01(2e)(hemispher Area RR

Solid angle for a hemisphere is 2

Solid angle for a sphere is 4


1.6.3 Area and Projected Area

cosdAdAP


1.6.4 Radiation Intensity and Irradiation

msrm

Wi

)Area Projected(inintensity spectral theis

2'

indicates direction

srm

Wi

)Area Projected(inintensity total theis

2'

0

'' dii


Irradiation

directions all

' 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


For isotropic radiation

' ig

'ig

An important integral in radiation

hemisphere

ddd

2

0

2/

0

sincoscos


1.7.1 Black Body Radiation

Real surfaces (bodies)

gggg

reflectivity

absorptivity

transmissivity

1.7 Governing Laws for Thermal Radiation


BLACK BODY RADIATION

Definition of a black body

A black body is defined as an ideal body that allincident radiation pass into it and internally absorbsall the incident radiation.

This is true for radiation of all wavelengths and for all anglesof incidence


BLACK BODY RADIATION

Properties:

Black body is a perfect emitter

In 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


hemisphere

b

hemisphere

bbb ididie ''' coscos

The angular distribution of radiation intensity emitted by a black body


1.7.2 Planck’s Radiation Law

1

1),(),( /5

1'

2 TCbb e

CTiTe

2161 m W107418.3 C

Km 01438769.1 22 C


Planck’s Radiation Law


1

1),(/2

51

5

TCeT

C

T

Te b


See Example 1.4 of the lecture notes to understand

the meaning of:

Frequency distribution

Cumulative frequency distribution

Relative cumulative frequency distribution


Height per

class (cm)

Number of students

-Frequency

Class mark

(cm)

153-159

160-166

167-173

174-180

181-187

188-194

195-201

202-208

4

12

18

25

33

22

11

5

TOTAL 130

156

163

170

177

184

191

198

205

Example 1.4


Histogram and frequency polygon of heights of 130 students

Example 1.4


Q

P

dxxfArea 130)51122332518124()(

class theof width theis 7cmΔ

(130) students ofnumber totaltheArea

Example 1.4


Height (cm) Number of students

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 cm

Less than 201 cm

Less than 208 cm

0

4

16

34

59

92

114

125

130

Cumulative distribution(less than the upper class boundary)

Example 1.4


Students smaller than 174 cm

174

0

)(1

)18124(1

18124174cm) than F(less dxxf

The relative cumulative distribution

0

174

0

)(

)(

130

18124)cm 174 than less(

dxxf

dxxf

F

Example 1.4


Cumulative distributionExample 1.4


1.7.3 Wien’s Displacement Law

We are looking for a wavelength that maximizes the Planck’s function for a given temperature

1/51/5

1 11

1),( 2

2

TCTCb eC

e

CTe

1/

61 1)5( 2 TCb eC

d

ed

0)1(1)1( 22/2/51 22

T

Cee

C TCTC

TCe

CT

/

2

21

1

5


2

2/

1( )

5 1 C T

Cf T T

e


Wien’s Law

Kμm 2,898C3max T


1.7.4 Stefan-Boltzmann Law

0

?),( dTee bb

2

04 31 1

4/520 11

b C T

C T Ce d d

C ee

T

C

2

0 3

1 15d

e

4442

1

15TT

C

Ceb

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


1.8 Blackbody Radiation in a Wavelength Interval

2

1

2

1

21),(

1

),(

),(

4

0

_

dTe

TdTe

dTe

F TT


2

1

21),(

14_

dTe

TF bTT

TTbb FFdTedTeT 12

2 1

_0_0

0 04

),(),(1


2 1

1 2 2 1_ 0_ 0_5 50 0

( , ) ( , )1( ) ( )

T T

b bT T T T

e T e TF d T d T F F

T T


1.9 Blackbody Emission into a Medium Other than Vacuum

21

21 /2 nCchC mm nC

k

chC m

m /22

n

ccm

nm

1

1),(

/51

2

TCb e

CTe

),(),( 3 TenTe bmmb


),(),( 3 TenTe bmbm

n- refractive index

Planck’s function in vacuum


Stefan-Boltzmann Law

42 Tnebm

Wien’s Displacement Law

n

CTn

3max,


1.10 Historical Note – Origin of Quantum Mechanics


1

1),(

/5

Tbb e

aTe

The challenge was inderiving a and b constantsfrom the first principle


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 ofaction (h) somehow into the classical theory continued fora number of years, and they cost me a great deal of efforts”


In 1905 Albert Einstein made an assumptionthe energy of a light was concentrated intolocalized bundles – later called photons

hEPlanck, the originator of the h constant, did not acceptat once Einstein’s photons. In 1913 Planck wrote aboutEinstein “that he sometimes have missed the target in hisspeculations, as for example in his theory of light quanta,cannot really be held against him”In 1918 – Planck received a Nobel prize “for his discoveryof energy quanta”

In 1921 – Einstein received his Nobel prize “for his service totheoretical physics and specially for discovery of the law of photoelectric effect”


1.12 Summary

Students should understand:

The concepts of radiation intensity and emissive power

The radiation laws for black-body radiation

Planck’s law

Wien’s law

Stefan-Boltzmann law

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

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