Advanced Heat Transfer Lecture1 by HC120520153636

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```									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
1.10 Historical Note – Origin of Quantum Mechanics
1.11 Blackbody Emission into a Medium Other than Vacuum
1.12 Summary

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 to
a 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

Amount of heat (energy) Q in J

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

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 body
because of its temperature is called

A body of a temperature larger than 0 K

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
A scene from “Silence of the lambs”

A photograph of a car
taken with                                                                                   taken with
an                                                                                          an
ordinary                                                                                     infrared
camera                                                                                       camera

The number plate has been wiped out
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)


Qconduction  T1  T2

Q            T T
convection                  1           2

4    4

When no medium is present radiation is the only
mode of heat transfer

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

Quantum theory
E photon  h  v                     h  6.63 10             34
J s
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   
Determined                                            Determined by
by the medium                                         the source

For electromagnetic waves:

c   
c=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:

c
c medium 
n medium
The frequency stays the same so,


 medium 
nmedium

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)
Energy in       Number of
Frequency        Photon
electron      photons in a
(Hz)      energy in J                    joule of energy
volts
waves       6.63·10-27       4.1·10-8       1.5·1026
ν=107
Visible light
waves       6.63·10-19          4.1         1.5·1018
ν=1015
X-rays
ν=1018      6.63·10-16        4.1·103       1.5·1015

Gamma
rays                   6.63·10-14                  4.1·105                   1.5·1013
ν=1020
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)
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 and
the circumference of a circle.

ds  R  d                                               Plane angle
2
s  R   d  R   2  1 
1

Circumfere of the circle  R  2
nce
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Derive formula for calculating the area of a sphere

dA  R  d         2
The solid angle
2
A  R   d  R  2  1 
2                       2

1

Area of a part of the sphere 
2

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)
How to calculate the solid angle?

dAs
d  2
R

dAs  R  d   R  sin   d   R 2  sin   d  d

d  sin   d  d

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Now we can complete the integration since we know
how to calculate the solid angle:
2                    2 2
A  R 2   d  R 2    sin  d  d 
1                  1 1

 R   2  1    cos   
2                                          2
1

 R   2  1   (cos 1  cos 2 )
2

Area (hemispher e)  R  2  (1  0)  2  R      2                                              2

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

dAP  dA  cos

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

indicates direction

W
i  is the spectral intensity in 2
'

m (Projected Area )  sr  m

W
i  is the total intensity in 2
'

m (Projected Area )  sr

i   i  d
'           '

0
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)

g                  i ( , ,  )  cos  d
'

all directions

2  / 2
g          i ( , ,  )  cos  sin   d  d 
'

0 0

 /2
1
      i   sin(2 )  d (2 ) 
g            '

2           0

    i   cos(2 ) 0    i
1       '                 /2     '

2
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)

g     i
                         '

g   i
                            '

  2  / 2


hemisphere
cos  d 

  cos  sin   d  d  
0      0

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1.7 Governing Laws for Thermal Radiation
Real surfaces (bodies)
g    g    g   g
                    

reflectivity
absorptivity

transmissivity

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

Definition of a black body

A black body is defined as an ideal body that all
incident radiation pass into it and internally absorbs
This is true for radiation of all wavelengths and for all angles
of incidence
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)

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)
The angular distribution of radiation intensity
emitted by a black body

eb            ib  cos  d  ib                        cos  d    ib
'                 '                                            '

hemisphere                                    hemisphere
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)

C1                    1
eb ( , T )    ib ( , T ) 
                                    '
     C 2 / T
 e 5
1
16
C1  3.7418 10                                       W m            2

2
C2  1.438769 10                                            mK

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)

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
eb ( , T )
                C1            1
            C / T
T 5
  T  e 2  1
5

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)
Example 1.4
Height per               Number of                    Class mark
class (cm)                 students                      (cm)
-Frequency
153-159                      4                             156
160-166                     12                             163
167-173                     18                             170
174-180                     25                             177
181-187                     33                             184
188-194                     22                             191
195-201                     11                             198
202-208                      5                             205
TOTAL 130
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Example 1.4

Histogram and frequency polygon of heights of 130 students

35

30
Number of students per height

f(x)

25

20

15
P
10                                                                    Q

5

0
149       156   163   170   177    184    191      198    205        212
Height (cm)
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Example 1.4

Q

Area   f ( x)  dx    (4  12  18  25  33  22  11  5)   130
P

Δ  7cmis the width of the class

Area  the total number of students (130)

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Example 1.4
Cumulative distribution
(less than the upper class boundary)
Height (cm)         Number of students
Less than 153 cm               0
Less than 160 cm               4
Less than 167 cm              16
Less than 174 cm              34
Less than 181 cm              59
Less than 188 cm              92
Less than 195 cm             114
Less than 201 cm             125
Less than 208 cm             130
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Example 1.4

Students smaller than 174 cm

174
1                     1
F(less than174cm)  4  12  18     (4  12  18)  
                                                         f ( x)  dx
0

The relative cumulative distribution
174

4  12  18                                f ( x)  dx
F (less than 174 cm)                                           0

130      
 f ( x)  dx
0

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Example 1.4
Cumulative distribution
1.0
F(x) Cumulative Frequency (No. of Students)

120

F(x) Relative Cumulative Frequency
0.8
100

80                                                                                                 0.6

60
0.4
40

0.2
20

0                                                                                                 0.0
150           160          170          180          190           200          210
Height (cm)
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

eb ( , T ) 

C1
 e5
      C 2 / T
1
1
 C1    e       5
   C2 / T
1  1


deb
d
C1 C2 / T
 (5)  6  e


1
1                          

C1
  5
 (1)  e          C 2 / T
1        e
2       C 2 / T
C2
  (1)    0
T
2

  T    C2 / T
C2     1
5 1 e
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
C2     1
f (  T )         C2 /  T
  T 
5 1 e
0.010

0.005
max·T = 0.0028977756 m·K
(C3-Wien's constant)
f(·T) in m·K

0.000

-0.005

-0.010
0.000   0.002      0.004          0.006           0.008          0.010
·T in m·K
Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Wien’s Law

max T  C3  2,898 μm  K

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
1.7.4 Stefan-Boltzmann Law

eb   eb ( , T )  d  ?
      
0

C1                        T 4  C1    3 0
C2
eb                                 d               d                          
0  5   eC 2   /  T
 1          C2  e  1
4
 T
0
3               
 e 1  d  15


C1 
eb  4   T 4    T 4

C2 15

Stefan-Boltzmann
  5.67 10 W/(m  K )         8                2           4
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

 e ( , T )  d

1
2

4  

F1T _ 2T         1
       e ( , T )  d


T 1
 e ( , T )  d

Advanced Heat Transfer - Prof.0Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
2
1
4  b
F1T _ 2T           e ( , T )  d 

  T 1
1    2               1

4   b
       e ( , T )  d   eb ( , T )  d   F0 _ 2T  F0 _ 1T
                
 T  0
                  0                   


Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
T                        1T
1  2 eb ( , T )                eb ( , T )           
F1T _ 2T                    d (T )                  d (T )   F0 _ 2T  F0 _ 1T
 0
       T 5
0
T 5



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
C1                   1
eb ( , T ) 
                                           C2 / T
 e5
1
c                                                
cm                                        m 
n                                               n
h  cm
C1m  2  h  cm  C1 / n 2
2
C2 m              C2 / n
k

emb (m , T )  n  eb ( , T )
                                                  3

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
ebm (m , T )  n  eb ( , T )
                                                     3

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

ebm  n    T
                     2                    4

Wien’s Displacement Law

C3
max,n  T 
n

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)
a            1                   The challenge was in
eb ( , T ) 
                            b / T
 e
5
1             deriving a and b constants
from the first principle

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
Quantification of energy (Max Planck – 1990)

E  m h  v
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”

Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws)
In 1905 Albert Einstein made an assumption
the energy of a light was concentrated into
localized bundles – later called photons

E  h 
Planck, the originator of the h constant, did not accept
at once Einstein’s 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”
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