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Purdue ME 606: Introduction

Table of Contents

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Fundamentals of Radiation

Definition: Energy transport and exchange by electromagnetic waves.

Variables:

c = speed of light = c = λν

λ = wavelength of light (microns)

ν = frequency (Hz, 1/s)

c0 = speed of light in vacuum = 2.9979 * 108m/s

c = \frac{c_0}{n}, n = index of refraction

another form is:

\nu = \frac{\omega}{2 \pi} = \frac{c}{\lambda} = c \eta, \eta = \frac{1}{\lambda} = wavenumber, number of waves in one unit of length.

Other Relevant Constants

Planck’s constant: h = 6.6262×10 − 34 Js

Boltzmann’s constant: kb = 1.3806×10 − 23 J/K

Stefan-Boltzmann constant: σ = 5.6696×10 − 8 W/m 2 K 4

Electrical permittivity of vacuum: ε0 = 8.8542×10 − 12 C 2 /Nm 2

Magnetic permittivity of vacuum: μ0 = 4π×10 − 7 Ns 2 /C 2

A Few Relevant Trigonometric Identities

sin(2a) = 2cos(a)sin(a)

cos(2a) = cos2a − sin2a = 1 − 2sin2a = 2cos2a − 1

Blackbody Radiation

  1. Perfect absorber of all wavelengths and directions
  2. Perfect emitter, emits all the radiation that is theoretically possible, at all wavelengths and directions
  3. Its intensity is uniform, meaning there is no specular (angular) dependence on emission or absorption

Spectrum

Energy of a single wave is quantized and \epsilon = h \nu = h \frac{c}{\lambda} , where h is Planck’s constant. The reduced Planck’s constant is \hbar = \frac{h}{2 \pi}.

Origins of Radiation

  • Emitted radiation originates from top 1-10 μm of a surface.
  • Both electrons and phonons can emit (and absorb) photons

Thermal Radiation

  • Results from changes in energy levels of atoms and molecules due soley to the temperature of a system.
  • No radiation is transmitted at T = 0 K.
  • Radiation represents the 2nd basic heat transfer mechanism
    • Conduction occurs by microscopic 1-way processes transmitting energy in opposing directions.
    • Radiation occurs by long range interactions where heat is carried by EM waves

Physics of Radiation

The spectral emmisive power of a black body Ebλ depends on temperature and is governed by Planck’s law

E_{b\lambda} = \frac{c_1}{n^2\lambda^5[\exp(\frac{c_2}{\lambda T})-1]}

where n is the refractive index, c1 = 3.7418 * 10 − 16 W/m 2 is the first radiation constant and c2 = 1.4388 * 10 − 2 mK. The spectral (wavelength) dependence of radiation can be explored at this site.

The location of the maximum thermal radiation intensity is given by Wien’s displacement law

nλmaxT = 2898 μmK

The fraction of emitted radiation between 0 and λT F0→λT is

F_{0 \rightarrow \lambda T} = \frac{1}{\sigma} \int_0^{\lambda T} \frac{E_{b \lambda}}{T^5} d(\lambda T) = \frac{E_b(0 \rightarrow \lambda T)}{\sigma T^4}= \frac{E_b(0 \rightarrow \lambda T)}{E_b}

Blackbody Radiation

  1. Consider a cavity with a small aperture as pictured above
    • Entering rays are reflected internally and absorbed independently.
    • The cavity is a blackbody absorber independent of inner wall properties
  2. Close the slot
    • The radiation inside the cavity G(Tw) fills it uniformly
  3. Introduce a small black disk at temperature Td. Assume the disk is small enough that it does not affect G(Tw).
    • The disk absorbs AdG(Tw)
    • The disk emits AdEb(Td)
    • In equilibrium
AdG(Tw) = AdEb(Td) (l st law)
Tw = Td (2 nd law)
G = Eb(Tw)

Last modified on 15 Oct, 2008