Stefan Boltzmann Law

11 Key excerpts on "Stefan Boltzmann Law"

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  Heat Transfer 2

  Radiative Transfer

  • Michel Ledoux, Abdelkhalak El Hami(Authors)
  • 2021(Publication Date)
  • Wiley-ISTE

    (Publisher)

  pdV”. The sign of the calculated work may be significant. Here, in terms of equality of two calculations of energy quantity, we can discard it.

  And by identification we find:

  [2.26]

  which is the starting point for many scientific publications. It is therefore important to know this expression when exploring the literature.

  NOTE.– Readers familiar with statistical thermodynamics will have recognized in the term the energy ratio of the photon to the Boltzmann term . This is not surprising, since Planck’s rationale was based exactly on a statistical calculation (Boltzmann was a pupil of Planck who contributed to this theory).

  2.3. Stefan–Boltzmann law

  2.3.1. Establishing the law

  In the subject of radiation, the main laws have been expressed on an experimental basis, before being integrated in a synthetic theoretical vision based on Planck’s theory.

  The Stefan–Boltzmann law was posed experimentally through the work of Joseph Stefan in 1879 using Tyndall‘s experimental data. The theoretical foundations were posed in the context of thermodynamics by one of Stefan’s doctoral students, Ludwig Boltzmann (1844–1906), in 1884.

  We will establish it here as a result of the luminance data, or more precisely the emittance from a black body surface.

  The total emittance of a black body in all directions and across the whole (infinite) wavelength spectrum is calculated by integrating the total monochromatic emittance of λ = 0 to λ → ∞

  [2.27]

  [2.28]

  This integral is transformed easily by carrying out a change of variable:

  [2.29]

  [2.30]

  and:

  [2.31]

  Posing

  [2.32]

  We obtain

  [2.33]

  σ is a constant equal to:

  [2.34]

  The constant σ

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  The Fundamentals of Radiation Thermometers

  • Peter Coates, David Lowe(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  For example, by taking logs and differentiating, we may show that the uncertainties in the spectral radiance, the temperature and the wavelength may be related by Δ L λ L λ = c 2 λT Δ T T + c 2 λT -5 Δ λ λ The combination c 2 /λT occurs so frequently in radiation thermometry analysis that I have termed it ‘the Planck parameter’, denoted by the symbol ‘ p ’. It was at one time common to approximate the variation of spectral radiance with temperature by a power law, that is, L λ ( λ, T ) ∝ T n It is relatively simple to show that for small changes the exponent is in fact the Planck parameter p . As p is itself a strong function of temperature, the power law approximation is only valid over a very limited range, and its use is discouraged in favour of the Wien equation. Finally, we note that if the spectral exitance of a black-body is integrated over all wavelengths, the total exitance is obtained as a function of temperature, that is, the Stefan-Boltzmann law. Making the substitution x = hν kT and using the mathematical relationship Z ∞ 0 x 3 e x -1 dx = 4 15 we obtain M ( T ) = 2 π 5 k 4 15 c 2 h 3 T 4 enabling the Stefan-Boltzmann constant to be evaluated in terms of other fun-damental constants 2 . Its value, to an accuracy adequate for most purposes, is 5 . 67 × 10 -8 W m -2 K -4 , which must make it the easiest fundamental constant to remember! 2 The value of c is fixed by the S.I. definition of the metre and the definition of the second. The stated intention of the Bureau International des Poids et Mesures (BIPM) is to specify the kelvin by fixing the value of the Boltzmann constant with zero uncertainty. Also, the kilogram is to be specified by similarly fixing the Planck constant. The Stefan-Boltzmann constant will therefore also be fixed with zero uncertainty.

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  An Introduction to the Atomic and Radiation Physics of Plasmas

  • G. J. Tallents(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  We show later that the total thermal radiation energy integrated over all frequencies varies as (k B T ) 4 . 84 Radiation Emission in Plasmas ћw (eV) Relative energy density 100 eV 150 eV 200 eV Figure 4.1 The relative energy density per unit photon energy per unit volume for thermal Planckian radiation as a function of photon energy. Three sample radiation temperatures (100 eV, 150 eV and 200 eV) as indicated are plotted. The Planck radiation energy density is referred to as black-body radiation because of early studies of the absorption and emission of infra-red radiation by Gustav Kirchhoff (1824–1887) and others. Kirchhoff found that a black object is an equally good absorber and emitter of infra-red radiation and coined the term black body. A black object at slightly higher temperature than room temperature (for example, an old fashioned stove) can emit a Planckian spectrum of radiation with a peak of emission in the infra-red and little radiation in the visible. It is a clear sign of over-heating if a black stove starts to emit in the visible (usually red) spectral range. We discuss in Section 4.3 Kirchhoff’s law that emission and absorption are proportional. 4.1.3 The Planck Radiation Flux and Intensity The Planck black-body energy density expression given by Equation 4.7 has radi- ation uniformly distributed in angle. At the walls of the black-body cavity or if an object is placed inside the cavity, the intensity and flux of the radiation, which are measures of the power per unit area, are important. For angular variations of intensity and flux, we use units of steradian defined as the solid angle subtended by an area of a sphere of unit radius at the centre of the sphere. For example, if the area extends to all the surface of the sphere, that is to all angles, this solid angle is 4π steradian. The intensity of radiation < I p (ω) > represented by the radiation power per unit area integrated over all angles (4π steradian) associated with the Planck radiation

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  Radiative Heat Exchange in the Atmosphere

  • K. Ya. Kondrat'Yev(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  THERMAL RADIATION 13 Introducing ψ(ξ) = I J>d| , we finally obtain ρ = Ψ(ξ 2 )-Ψ(ξ 1 ). (1.31) The function Ψ(ξ) is shown in Fig. 2. 3. The Stefan-BoUzmann law. By integrating the expression (1.27) *φ* I'UU 0-9 0-8 0-7 0-6 0-5 0-4 0-3 0-2 0-1 ^~~~ ^ 0000000 ^ ^r s / / / / S 1 I —-I I I ! 0-5 1-0 1-5 2-0 2-5 3-0 3-5 4-0 FIG. 2. The function Ψ(ξ). for all wavelengths between 0 and oo we obtain the total intensity of a perfect black body E E -r * « -^ z dx The last integral equals π 4 /15. Hence 15c 2 Ä 3 π ' where a = 2π 5 & 15c% 3 (1.32) Since the intensity of a perfect black surface is independent of direction (Lambert's law) then, according to (1.2), we obtain the following expression for the radiative flux of a perfectly black sur-face: Β = πΕ = σΤ (1.33) The constant a is usually defined as a = 5.75 X 10 5 erg/cm 2 sec deg 4 , or in other units, a = 0.826 X 10 10 cal/cm 2 min deg 4 . However, 14 RADIATIVE HEAT EXCHANGE IN THE ATMOSPHERE it should be pointed out that in the latest measurements 14 this con-stant was found to be a = 5.669 X 10 12 W/cm 2 deg 4 , which in other units is a = 0.816 X 10-10 cal/cm 2 min deg 4 . Table 1 of the Appendix gives the values of aT* for a = 0.814 x 10 10 compiled from the data of Elovskikh. 4. Wieris displacement law. It can be shown by differentiation of Planck's formula (1.27) with respect to λ, and subsequent determina-tion of the wavelength X m which corresponds to the maximum of the function Ε λ (Τ), (for example, Ref. 81) that the following relations exist: X m T = a, (1.34) where a = 0.2897 cm deg if the wavelength is given in cm: E ltm = cT*, (1.35) where c = -1.301 x l O -» W/cmV· deg 5 . The relation (1.34) is called Wien's displacement law. An examina-tion of this relation shows that it gives the displacement in wavelength of the intensity maximum of the perfect black body as a function of the temperature of the latter.

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  Turbulent Flows and Heat Transfer

  • Chia-Ch'iao Lin(Author)
  • 2015(Publication Date)
  • Princeton University Press

    (Publisher)

  SECTION I m i m ENGINEERING CALCULATIONS OF RADIANT HEAT EXCHANGE HOYT C. HOTTEL 1,1. Radiating Characteristics of Surfaces. Knowledge on the reader's part of the general nature of thermal radiation and the laws of black body radiation will be assumed in this section (see this volume, Sec. H). Fig. I,la summarizes, for numerical use, the radiating charac- teristics of a black body in the form of a plot of monochromatic emissive power 1 TFbx divided by the fifth power of the absolute temperature versus the wavelength-temperature product. The curve may be visualized as an intensity-wavelength distribution at I 0 absolute. An extra scale along the top permits a determination of the fraction of the spectral energy found below a given wavelength λ. The area under the curve is directly the Stefan-Boltzmann constant σ[0.1713 X IO -8 BTU/ft 2 hr ( 0 R) 4 ; 5.67 X 10~ 5 ergs/cm 2 sec ( 0 K) 4 ; 4.88 X IO -8 kg-cal/m 2 hr ( 0 K) 4 ; 1.00 X IO -8 CHU/ft 2 hr ( 0 K) 4 ], for use in the relation W h = σΤ 1 (1-1) where TT b is the total emissive power, throughout a solid angle of 2τ steradians, of a black body or "perfect" radiator. In evaluating radiant heat transfer between surfaces, one could con- sider monochromatic radiation exchange and integrate throughout the spectrum; certain advantages would appear. For most engineering pur- poses, however, it is simpler to formulate total radiation exchange, ex- pressing it in terms of the 4th power temperature law strictly applicable only to the black body or perfect radiator, and to let the more-or-less weak residual temperature function be taken care of by the variable total emissivity, absorptivity, or transmissivity of the pertinent bodies.

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  Fly by Night Physics

  How Physicists Use the Backs of Envelopes

  • Anthony Zee, A. Zee(Authors)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  So your unspoken question may be widespread. † The Boltzmann constant k , which (as was emphasized in chapter IV.1) is merely a conver-sion factor between energy units and the markings on some tubes containing mercury known as “degrees,” has been set to 1. 146 Chapter IV.3 This “sophisticated” dimensional analysis captures an essential piece of physics: The radiation is explosive! As the black hole radiates energy, M goes down, and T H goes up, and thus the black hole radiates faster. The radiative mass loss accelerates. Certainly not something you want to see in the kitchen: An object that gets hotter as it loses energy. If you so wish, we could easily write T H in everyday unnatural units by restoring c and . Way back when, we derived Kepler’s law without breaking a sweat by noticing that GM has dimension of L 3 / T 2 . Since temperature has the dimension of energy, and since [ ] = ET , we form / GM with dimen-sion of ET 3 / L 3 . Hence, the Hawking temperature, which, as has already been remarked, has the dimension of energy, is given by T H ∼ c 3 GM (2) Very gratifying to see that, indeed, with = 0 and quantum effects turned off, T H = 0 and the black hole does not radiate. Physics is consistent! In fact, the overall numerical constant in (2), which turns out to be 1 8 π , could be determined, with a touch of sophistication, ∗ in a couple lines of algebra. 4 A man of deep integrity Jacob Bekenstein, a student of John Wheeler’s, was the first to realize that black holes have entropy. I will let Wheeler tell the story in his trademark style: 5 One afternoon in 1970, . . . I told [Bekenstein] of the concern I always feel when a hot cup of tea exchanges heat energy with a cold cup of tea. By allowing that transfer of heat . . . I increase (the universe’s) microscopic disorder, its information loss, its entropy. “The conse-quences of my crime, Jacob, echo down to the end of time,” I noted.

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  Atomic Physics: 8th Edition

  • Max Born(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  For light quanta, as we have shown above (p. 224), the second subsidiary condition (constancy of number of particles) drops out. Thus we find in the usual way or

  The Bose-Einstein law of distribution for light quanta therefore runs (if we drop the index s)

  this gives for the energy density

  This is just Planck’s radiation formula, if we put β = 1/kT. The justification for this last step is given by thermodynamics; according to Boltzmann, S = k log W is to be regarded as the entropy, and it can then be shown (see Appendix XXXV , p. 459) that from the equation TdS = dQ we can infer that β = 1/kT (dQ is the increment of the heat content, or, at constant volume, of the energy content of the light quantum gas). From the Bose-Einstein statistics, therefore, Planck’s radiation law can be deduced in a way to which no objection can be taken.

  5. Einstein’s Theory of Gas Degeneration.

  After the brilliant success of the Bose-Einstein statistics with the fight quantum gas, it was a natural suggestion to try it in the kinetic theory of gases also, as a substitute for the Boltzmann statistics. The investigation, which was undertaken by Einstein (1925), is based on the hypothesis that the molecules of a gas are, like light quanta, indistinguishable from each other.

  The calculations run exactly as in the light quantum case, except that here a second subsidiary condition appears, on account of the conservation of the number of particles:

  The determination of the probability of a definite distribution n1 , n2 , . . . follows the same lines as before. The calculation of the most probable distribution leads now, owing to the presence of the second subsidiary condition, to the equation

  or, on dropping the suffix s,

  where again β = 1/kT (see Appendix XXXV , p. 459). Here the number g

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  Heat Transfer

  A Systematic Learning Approach

  • Naseem Uddin(Author)
  • 2024(Publication Date)
  • CRC Press

    (Publisher)

  have ζ 3 [ exp (ζ) − 1 ] = ζ 3 [ e − ζ + e − 2 ζ + e − 3 ζ + ⋯ ] Integrating with up till 6th term we have ∫ 0 ∞ ζ 3 [ exp (ζ) − 1 ] d ζ = 6.486741204 and the total emissive power of a blackbody. is E b = c 1 T 4 c 2 4 (6.486741204) = 5.67 410 × 10 − 8 T 4 E b = σ T 4 The constant σ = 5.67410 × 10 − 8 W / m 2 K 4 is called Stefan-Boltzmann constant. In US customary units, the value of this constant is 1.71344 ×10 -9 Btu ⋅ hr -1 ⋅ft -2 ⋅ ∘ R -4. The total intensity of the emitted radiation from blackbody is I b = E b / π Pioneers of Heat Transfer Josef Stefan (1835–1893) was professor at the University of Vienna and in 1879, he inferred from some of his experiments that blackbody emission is proportional to temperature to the fourth power. Previously, Ludwig Erhard Boltzmann (1844–1906) in 1889 derived the fourth-power law purely from thermodynamic relations. Close to our planet the biggest source of thermal radiation is our star - the sun. Solar energy coming to Earth varies over the year as the Earth orbait around sun is elliptical. The maximum irradiation received from sun is about 1412 W/m 2 in early January, and a minimum of about 1321 W/m 2 happens in early July. Example 13.1 Solar energy received by the earth's atmosphere, also called the solar constant is measured to be around 1367 W/m 2. Estimate the solar constant on Jupiter? The distance from earth to sun is 147.48 million km, and distance from sun to Jupiter is 740.85 million km. Solution The radiation from sun surface spreading spherically and reaching Earth is q s u n − e a r t h = Q A s = σ T s u n 4 4 π R s e 2 where R s e is the distance from sun to earth. The radiation from sun surface spreading spherically and reaching Jupiter is q s u n − J u p i t e r = Q A s = σ T s u n 4 4 π R s j 2 where R s j is the distance from sun to Jupiter. Taking the ratio of two equations we. have q s u n − J u p i t e r q s u n − e a r t h = R s e 2 R s j 2 = (R s e R s j) 2 According to give data we

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  Optical Coherence and Quantum Optics

  • Leonard Mandel, Emil Wolf(Authors)
  • 1995(Publication Date)
  • Cambridge University Press

    (Publisher)

  (13.1-15) !) Alternatively, since each photon of frequency a> carries an energy ha>, we can 13.1 Blackbody radiation 665 write a corresponding equation for the energy density u((o)d(o within the fre- quency interval dco, u(co)dco = h(o the distribution u{a>) is proportional to a> 2 , which is characteristic of Rayleigh's law, but it reaches a maximum for a frequency a> of order 3k B T/h, and then falls back towards zero. The position of the maximum is given by Wien's displacement law. Finally, by integrating over all frequencies, we arrive at the total energy density u of the electromagnetic field at temperature T, Jo - 1 ) -dco Y x 3 Jo e* - 1 dx (k B T) 4 where is the Riemann zeta function. It can be shown that £(4) = TT 4 /90. The fourth power law dependence of u on temperature T given by Eq. (13.1-17a) is known as the Stefan-Boltzmann law. 1 2 3 4 Wavelength (microns) Fig. 13.1 The distribution of radiant energy with wavelength according to Planck's law at three different temperatures. (Reproduced from Halliday and Resnick, 1970, p. 759.) 666 Radiation from thermal equilibrium sources In a completely analogous manner, we may integrate

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  In Celebration Of K C Hines

  • Bruce H J Mckellar, Ken Amos(Authors)
  • 2009(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 8 The Boltzmann Equation in Fluorescent Lamp Theory Graeme Lister Osram Sylvania, CRSL, 71 Cherry Hill Dr., Beverly, MA 01915, USA E-mail: [email protected] Ken Hines was a wonderful mentor in my formative years as a physicist, and a great friend and companion throughout the rest of his life. He introduced me to the Fokker-Planck equation, which formed the basis of my Masters thesis, and showed me how to use it to derive the Boltzmann equation. Many years later, when my research took me into the field of gas discharge lighting, I was to re-discover the Boltzmann equation and apply it to fluorescent lamp modelling. Herein I discuss the important role the electron energy distribution function plays in understanding the physics of fluorescent lamps, and I describe some of the important insights gained from interpreting the Boltzmann equation. 8.1. Preamble The day I first met Ken remains fixed in my memory as if it were yesterday. I had finished my summer job at Defence Standards Laboratories one Fri-day, and I appeared at his office door the next Monday, ready to start my postgraduate studies. Ken was a tall man, with a waistline that belied the enormous appetite I would later observe in our meals together. He had a clear, deep, resonant voice and his first question to me was “’Graeme, have you had a holiday?” When I replied in the negative, he said “Well, go away and have one, you’re no damn use to me without having had a holiday!” So I spent a couple of weeks down on Victoria’s Mornington Peninsula, before returning to Melbourne University suitably refreshed, and ready to begin my career in physics. I set to work understanding the Fokker-Planck equation, and from Ken, I learned to use it to derive the Boltzmann equa-tion. After completing my Masters Thesis, I headed for the world of fully 129 130 G. Lister ionised plasmas, and the “moments” of the Boltzmann equation at Flinders University in South Australia, under the tutelage of Roger Hosking.

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  Using SI Units in Astronomy

  • Richard Dodd(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  m −2 . m −1 . sr −1 (9.12) Examples of log–log blackbody curves in frequency space for various astronom- ical objects are given in Figure 9.1 from the very low temperature (T = 2.73 K) of the cosmic background radiation to the very high temperature (T = 3.4 × 10 9 K) near the centre of a 25 M  star. 8 10 12 14 16 18 20 log(ν) Hz –20 –10 0 10 log(Bν) W.m –2 . sr –1 . Hz T = 3.4 x 10 9 K T = 1.6 x 10 7 K T = 5779 K T = 290 K T = 2.73 K a b c d e Figure 9.1. Blackbody curves computed using Equation (9.11). Curve ‘a’ is for a blackbody at the same temperature as the cosmic background radiation, curve ‘b’ for the surface of the Earth, curve ‘c’ for the surface of the Sun, curve ‘d’ for the centre of the Sun and curve ‘e’ for the centre of a 25 M  star. 9.4 Blackbody radiation 153 9.4.1 Wein’s displacement law Wein’s displacement law gives the wavelength λ max , or frequency ν max , of the maximum value of the specific intensity of a Planck curve. The wavelength or frequency is a function solely of the temperature, but is different in the sense that λ max  = c/ν max due to the different forms of the Planck function in wavelength and frequency space. Values for λ max and ν max may be found by differentiating the appropriate expression for specific intensity with respect to either λ or ν and setting the result equal to zero. For λ max the expression is: λ max = b T (9.13) where T is the blackbody temperature in kelvin and the Wein displacement constant b = 0.002 897 8 K . m. For ν max the expression is: ν max = 5.88 × 10 10 T (9.14) Example: compute λ max and ν max for a blackbody of temperature 2.726 K λ max = 0.002 897 8 2.726 = 1.063 × 10 −3 m = 1.063 mm (9.15) ν max = 5.88 × 10 10 × 2.726 = 1.603 × 10 11 Hz (9.16) 9.4.2 Stefan–Boltzmann law The Stefan–Boltzmann law states that the power P , radiated per unit surface area of a blackbody, is directly proportional to the fourth power of its absolute temperature, so that: P = σ T 4 (9.17) In SI units, P is measured in W.

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