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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
What is heat transfer?
Heat transfer (or heat) is energy in transit due toa temperature difference
HEAT TRANSFER MODES
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
ELECTROMAGNETIC WAVES
Classical theory
Quantum theory
vhEphoton sJ 1063.6 34 h
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)
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)
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
SPEED, FREQUENCY and WAVELENGTH
For a medium other than vacuum:
mediummedium n
cc
The frequency stays the same so,
mediummedium n
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
COMMON UNITS FOR WAVELENGTH
1 micrometer = 10-6 m
1 nanometer = 10-9 m
1 angstrom = 10-10 m
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
THERMAL RADIATION
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1.6 Geometrical Considerations
1.6.1 Normal to a Surface Element
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1.6.2 Solid Angle
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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?
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
How to calculate the solid angle?
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
2R
dAd s
ddRdRdRdAs sinsin 2
How to calculate the solid angle?
ddd sin
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1.6.3 Area and Projected Area
cosdAdAP
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
For isotropic radiation
' ig
'ig
An important integral in radiation
hemisphere
ddd
2
0
2/
0
sincoscos
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1.7.1 Black Body Radiation
Real surfaces (bodies)
gggg
reflectivity
absorptivity
transmissivity
1.7 Governing Laws for Thermal Radiation
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
hemisphere
b
hemisphere
bbb ididie ''' coscos
The angular distribution of radiation intensity emitted by a black body
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Planck’s Radiation Law
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Planck’s Radiation Law
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1
1),(/2
51
5
TCeT
C
T
Te b
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
See Example 1.4 of the lecture notes to understand
the meaning of:
Frequency distribution
Cumulative frequency distribution
Relative cumulative frequency distribution
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Histogram and frequency polygon of heights of 130 students
Example 1.4
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Q
P
dxxfArea 130)51122332518124()(
class theof width theis 7cmΔ
(130) students ofnumber totaltheArea
Example 1.4
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Cumulative distributionExample 1.4
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
2
2/
1( )
5 1 C T
Cf T T
e
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Wien’s Law
Kμm 2,898C3max T
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1.8 Blackbody Radiation in a Wavelength Interval
2
1
2
1
21),(
1
),(
),(
4
0
_
dTe
TdTe
dTe
F TT
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
2
1
21),(
14_
dTe
TF bTT
TTbb FFdTedTeT 12
2 1
_0_0
0 04
),(),(1
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
),(),( 3 TenTe bmbm
n- refractive index
Planck’s function in vacuum
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Stefan-Boltzmann Law
42 Tnebm
Wien’s Displacement Law
n
CTn
3max,
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1.10 Historical Note – Origin of Quantum Mechanics
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1
1),(
/5
Tbb e
aTe
The challenge was inderiving a and b constantsfrom the first principle
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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”
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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”
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
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