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Lecture-1. Governing Laws for Thermal Radiation Contents of the lecture 1.1 Heat Transfer Mechanisms 1.2 Electromagnetic Radiation 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 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” 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) RELEVANCE OF THERMAL RADIATION Qconduction T1 T2 Q T T convection 1 2 Qradiation T1 T2 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 Short radio 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) 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 and the circumference of a circle. ds R d Plane angle in radiance 2 s R d R 2 1 1 Length of an arc Radius Plane angle in radians 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 in steradians 2 A R d R 2 1 2 2 1 Area of a part of the sphere Radius Solid angle in steradians 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) 1.6.4 Radiation Intensity and Irradiation 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) Irradiation g i ( , , ) cos d ' all directions 2 / 2 g i ( , , ) cos sin d d ' 0 0 for isotropic incoming radiation /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) For isotropic radiation g i ' g i ' An important integral in radiation 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 1.7.1 Black Body 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) 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 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) The angular distribution of radiation intensity emitted by a black body eb ib cos d ib cos d ib ' ' ' hemisphere hemisphere Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws) 1.7.2 Planck’s Radiation Law C1 1 eb ( , T ) ib ( , T ) ' C 2 / T e 5 1 16 C1 3.7418 10 W m 2 2 C2 1.438769 10 mK 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) eb ( , 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 eb ( , T ) C1 e5 C 2 / T 1 1 C1 e 5 C2 / T 1 1 deb 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 eb ( , 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 F1T _ 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 F1T _ 2T e ( , T ) d T 1 1 2 1 4 b e ( , T ) d eb ( , 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 eb ( , T ) eb ( , T ) F1T _ 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 eb ( , 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 emb (m , T ) n eb ( , T ) 3 Advanced Heat Transfer - Prof. Dr.-Ing. R. Weber - Winter 2005/2006 - Lecture 1 (Governing Laws) ebm (m , T ) n eb ( , 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 eb ( , 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 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)