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- 1. Radiation Heat Transfer P M V Subbarao Professor Mechanical Engineering Department IIT Delhi A means for basic life on earth……..
- 2. Radiative Properties • When radiation strikes a surface, a portion of it is reflected, and the rest enters the surface. • Of the portion that enters the surface, some are absorbed by the material, and the remaining radiation is transmitted through. • The ratio of reflected energy to the incident energy is called reflectivity, ρ. • Transmissivity (τ) is defined as the fraction of the incident energy that is transmitted through the object. • Absorptivity (α) is defined as the fraction of the incident energy that is absorbed by the object. • The three radiative properties all have values between zero and 1. • Furthermore, since the reflected, transmitted, and absorbed radiation must add up to equal the incident energy, the following can be said about the three properties: ∀ α + τ +ρ = 1
- 3. Emissivity • A black body is an ideal emitter. • The energy emitted by any real surface is less than the energy emitted by a black body at the same temperature. • At a defined temperature, a black body has the highest monochromatic emissive power at all wavelengths. • The ratio of the emissive power of real body E to the blackbody emissive power Εb at the same temperature is the hemispherical emissivity of the surface. bE E =ε
- 4. The Emission Process • For gases and semitransparent solids, emission is a volumetric phenomenon. • In most solids and liquids the radiation emitted from interior molecules is strongly absorbed by adjoining molecules. • Only the surface molecules can emit radiation.
- 5. Spherical Geometry θ φ dA
- 6. Vectors in Spherical Geometry Zenith Angle : θ Αzimuthal Angle : φ (φ, θ,r)
- 7. Hemispherical Black Surface Emission π σ π 4 TE I b b == Black body Emissive Intensity
- 8. Real Surface emission The radiation emitted by a real surface is spatially distributed: ),,( TIIreal ϕθ=Directional Emissive Intensity: Directional Emissivity: bI TI T ),,( ),,( ϕθ ϕθε =
- 9. Directional Emissivity
- 10. Planck Radiation Law • The primary law governing blackbody radiation is the Planck Radiation Law. • This law governs the intensity of radiation emitted by unit surface area into a fixed direction (solid angle) from the blackbody as a function of wavelength for a fixed temperature. • The Planck Law can be expressed through the following equation. ( ) 1 12 , 5 2 − = kT hc e hc TE λ λ λ h = 6.625 X 10-27 erg-sec (Planck Constant) K = 1.38 X 10-16 erg/K (Boltzmann Constant) C = Speed of light in vacuum
- 11. Real Surface Monochromatic emission The radiation emitted by a real surface is spatially distributed: ),,,(, TIIreal ϕθλλ =Directional MonochromaticEmissive Intensity: Directional Monochromatic Emissivity: λ ϕλθ ϕλθε , ),,( ),,( bI TI T =
- 12. Monochromatic emissive power Eλ • All surfaces emit radiation in many wavelengths and some, including black bodies, over all wavelengths. • The monochromatic emissive power is defined by: • dE = emissive power in the wave band in the infinitesimal wave band between λ and λ+dλ. ( ) λλ dTEdE ,= The monochromatic emissive power of a blackbody is given by: ( ) 1 12 , 5 2 − = kT hc e hc TE λ λ λ
- 13. Shifting Peak Nature of Radiation
- 14. Wein’s Displacement Law: • At any given wavelength, the black body monochromatic emissive power increases with temperature. • The wavelength λmax at which is a maximum decreases as the temperature increases. • The wavelength at which the monochromatic emissive power is a maximum is found by setting the derivative of previous Equation with respect to λ. ( ) λ λ λ λ λ d e hc d d TdE kT hc −= 1 12 , 5 2 mKT µλ 8.2897max =
- 15. Wien law for three different stars
- 16. Stefan-Boltzmann Law • The maximum emissive power at a given temperature is the black body emissive power (Eb). • Integrating this over all wavelengths gives Eb. ( ) ∫∫ ∞∞ − = 0 5 2 0 1 12 , λ λ λλ λ d e hc dTE kT hc ( ) 44 4 42 15 2 TT k hc hc TEb σ π = =
- 17. The total (hemispherical) energy emitted by a body, regardless of the wavelengths, is given by: 4 ATQemitted εσ= • where ε is the emissivity of the body, • A is the surface area, • T is the temperature, and • σ is the Stefan-Boltzmann constant, equal to 5.67×10-8 W/m2 K4 . • Emissivity is a material property, ranging from 0 to 1, which measures how much energy a surface can emit with respect to an ideal emitter (ε = 1) at the same temperature
- 18. The total (hemispherical emissive power is, then, given by Here, ε can be interpreted as either the emissivity of a body, which is wavelength independent, i.e., ελ is constant, or as the average emissivity of a surface at that temperature. A surface whose properties are independent of the wavelength is known as a gray surface. The emissive power of a real surface is given by ∫∫ ∞∞ == 00 )( λλελ λλ dEdEE b Define total (hemispherical) emissivity, at a defined temperature ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ == 0 0 0 0 )( λ λλε λ λ ε λ λ λ λ dE dE dE dE b b b
- 19. Absorptivity α, Reflectivity ρ, and Transmissivity τ • Consider a semi-transparent sheet that receives incident radiant energy flux, also known as irradiation, G . • Let dG represent the irradiation in the waveband λ to λ + dλ. • Part of it may be absorbed, part of it reflected at the surface, and the rest transmitted through the sheet. • We define monochromatic properties,
- 20. Monochromatic Absorptivity : dG dGα λα = Monochromatic reflectivity : dG dGρ λρ = Monochromatic Transmissivity : dG dGτ λτ = Total Absorptivity : G G d α λ λαα == ∫ ∞ 0 Total reflectivity : G G d ρ λ λρρ == ∫ ∞ 0 Total Transmissivity : G G d τ λ λττ == ∫ ∞
- 21. Conservation of Irradiation The total Irradiation = τρα GGGG ++= G G G G G G τρα ++=1 1=++ τρα
- 22. 3.. Blackbody Radiation • The characteristics of a blackbody are : • It is a perfect emitter. • At any prescribed temperature it has the highest monochromatic emissive power at all wave lengths. • A blackbody absorbs all the incident energy and there fore α = αλ = 1. • It is non reflective body (τ=0). • It is opaque (τ = 0). • It is a diffuse emitter
- 23. Radiative Heat Transfer Consider the heat transfer between two surfaces, as shown in Figure. What is the rate of heat transfer into Surface B? To find this, we will first look at the emission from A to B. Surface A emits radiation as described in 4 , AAAemittedA TAQ σε= This radiation is emitted in all directions, and only a fraction of it will actually strike Surface B. This fraction is called the shape factor, F.
- 24. The amount of radiation striking Surface B is therefore: 4 , AAABAinceidentB TAFQ σε→= The only portion of the incident radiation contributing to heating Surface B is the absorbed portion, given by the absorptivity αB: 4 , AAABABabsorbedB TAFQ σεα →= Above equation is the amount of radiation gained by Surface B from Surface A. To find the net heat transfer rate at B, we must now subtract the amount of radiation emitted by B: 4 , BBBemittedB TAQ σε=
- 25. The net radiative heat transfer (gain) rate at Surface B is emittedBabsorbedBB QQQ ,, −= 44 BBBAAABABB TATAFQ σεσεα −= →
- 26. Shape Factors • Shape factor, F, is a geometrical factor which is determined by the shapes and relative locations of two surfaces. • Figure illustrates this for a simple case of cylindrical source and planar surface. • Both the cylinder and the plate are infinite in length. • In this case, it is easy to see that the shape factor is reduced as the distance between the source and plane increases. • The shape factor for this simple geometry is simply the cone angle (θ) divided by 2π
- 27. • Shape factors for other simple geometries can be calculated using basic theory of geometry. • For more complicated geometries, the following two rules must be applied to find shape factors based on simple geometries. • The first is the summation rule. 122211 →→ = FAFA • This rule says that the shape factor from a surface (1) to another (2) can be expressed as a sum of the shape factors from (1) to (2a), and (1) to (2b). • The second rule is the reciprocity rule, which relates the shape factors from (1) to (2) and that from (2) to (1) as follows: ba FFF 212121 →→→ +=
- 28. Thus, if the shape factor from (1) to (2) is known, then the shape factor from (2) to (1) can be found by: 12 1 2 21 →→ = F A A F If surface (2) totally encloses the surface 1: 121 =→F
- 29. Geometric Concepts in Radiation • Solid Angle: 2 r dA d n =ω sind d dω θ θ ϕ= Emissive intensity ( ) SrW/m ddS dq I n e 2 , ω ϕθ = Monochromatic Emissive intensity ( ) mSrW/m dddS dq I n e µ λω ϕθλλ 2 , ,, =
- 30. Total emissive power dS ndS cosndS dS θ= n e e all directions dS E I d dS ω= ∫ 2 2 , 0 0 0 ( , , )cos sine eq E I d d d π π λ λ θ ϕ θ θ θ ϕ λ ∞ = = ∫ ∫ ∫

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