AME 60634 Int. Heat Trans. Radiation: Overview • Radiation - Emission – thermal radiation is the emission of electromagnetic waves when matter is at an absolute temperature greater than 0 K – emission is due to the oscillations and transitions of the many electrons that comprise the matter • the oscillations and transitions are sustained by the thermal energy of the matter – emission corresponds to heat transfer from the matter and hence to a reduction in the thermal energy stored in the matter • Radiation - Absorption – radiation may also be absorbed by matter – absorption results in heat transfer to the matter and hence to an increase in the thermal energy stored in the matter D. B. Go AME 60634 Int. Heat Trans. Radiation: Overview • Emission – emission from a gas or semi- transparent solid or liquid is a volumetric phenomenon – emission from an opaque solid or liquid is a surface phenomenon • emission originates from atoms & molecules within 1 µm of the surface • Dual Nature – in some cases, the physical manifestations of radiation may be explained by viewing it as particles (A.K.A. photons or quanta); in other cases, radiation behaves as an electromagnetic wave – radiation is characterized by a wavelength λ and frequency ν which are related through the speed at which radiation propagates in the medium of interest (solid, liquid, gas, vacuum) c in a vacuum λ = c = c = 2.998 ×108 m/s ν o D. B. Go € € AME 60634 Int. Heat Trans. Radiation: Spectral Considerations • Electromagnetic Spectrum – the range of all possible radiation frequencies – thermal radiation is confined to the infrared, visible, and ultraviolet regions of the spectrum 0.1< λ <100 µm € • Spectral Distribution – radiation emitted by an opaque surface varies with wavelength – spectral distribution describes the radiation over all wavelengths – monochromatic/spectral components are associated with particular wavelengths D. B. Go AME 60634 Int. Heat Trans. Radiation: Directional Considerations • Emission – Radiation emitted by a surface will be in all directions associated with a hypothetical hemisphere about the surface and is characterized by a directional distribution – Direction may be represented in a spherical coordinate system characterized by the zenith or polar angle θ and the azimuthal angle ϕ. - The amount of radiation emitted from a surface, dAn, and propagating in a particular direction (θ,ϕ) is quantified in terms of a differential solid angle associated with the direction, dω. dA dω = n r2 dAn è unit element of surface on a hypothetical sphere and D. B. Go normal to the (θ,ϕ) direction € AME 60634 Int. Heat Trans. Radiation: Directional Considerations • Solid Angle dA dA = r2 sinθdθdφ dω = n = sinθdθdω n r2 • the solid angle ω has units of steradians (sr) • the solid angle ω associated with a complete hemisphere € hemi € π 2π 2 ωhemi = ∫ ∫ sinθdθdφ = 2π sr 0 0 D. B. Go € AME 60634 Int. Heat Trans. Radiation: Spectral Intensity • Spectral Intensity, Iλ,e – a quantity used to specify the radiant heat flux (W/m2) within a unit solid angle about a prescribed direction (W/m2-sr) and within a unit wavelength interval about a prescribed wavelength (W/m2-sr-µm) – associated with emission from a surface element dA1 in the solid angle dω about θ, ϕ and the wavelength interval dλ about λ and is defined as: dq 2 Iλ,e (λ,θ,φ) = [W/m -sr-µm] (dA1 cosθ)dωdλ – the rational for defining the radiation flux in terms of the projected area (dA1cosθ) stems from the existence of surfaces for which, to a good approximation, Iλ,e is independent€ of direction. Such surfaces are termed diffuse, and the radiation is said to be isotropic. • the projected area is how dA1 appears along θ, ϕ D. B. Go AME 60634 Int. Heat Trans. Radiation: Heat Flux • The spectral heat rate (heat rate per unit wavelength of radiation) associated with emission dq dqλ = = Iλ,e (λ,θ,φ)cosθdA1dω dλ • The spectral heat flux (heat flux per unit wavelength of radiation) associated€ with emission dq dqλ" " = = Iλ,e (λ,θ,φ)cosθdω = Iλ,e (λ,θ,φ)cosθ sinθdθdφ dλdA1 • The integration of the spectral heat flux is called the spectral € emissive power – spectral emission (heat flux) over all possible directions π 2π 2 ) W , q E I , , cos sin d d λ"" = λ (λ) = ∫ ∫ λ,e (λ θ φ) θ θ θ φ + 2 . 0 0 * m ⋅ µm- D. B. Go € AME 60634 Int. Heat Trans. Radiation: Heat Flux • The total heat flux from the surface due to radiation is emission over all wavelengths and directions è total emissive power π ∞ 2π 2 ∞ ) W , q E I , , cos sin d d d = E d "" = = ∫ ∫ ∫ λ,e (λ θ φ) θ θ θ φ λ ∫ λ (λ) λ + 2 . 0 0 0 0 * m - • If the emission is the same in all directions, then the surface is € diffuse and the emission is isotropic π ∞ 2π 2 ∞ ) W , q E I d cos sin d d = I d = I "" = = ∫ λ,e (λ) λ ∫ ∫ θ θ θ φ π ∫ λ,e (λ) λ π e + 2 . 0 0 0 0 * m - € D. B. Go AME 60634 Int. Heat Trans. Radiation: Irradiation • Irradiation – electromagnetic waves incident on a surface is called irradiation – irradiation can be either absorbed or reflected • Spectral Intensity, Iλ,i – a quantity used to specify the incident radiant heat flux (W/m2) within a unit solid angle about the direction of incidence (W/m2- sr) and within a unit wavelength interval about a prescribed wavelength (W/m2-sr-µm) and the projected area of the receiving surface (dA1cosθ) D. B. Go AME 60634 Int. Heat Trans. Radiation: Irradiation Heat Flux • The integration of the spectral heat flux is called the spectral irradiation – spectral irradiation (heat flux) over all possible directions π 2π 2 ) W , q G I , , cos sin d d λ"" = λ (λ) = ∫ ∫ λ,i (λ θ φ) θ θ θ φ + 2 . 0 0 * m ⋅ µm- • The total heat flux to the surface due to irradiation over all wavelengths and directions è total irradiative power € π ∞ 2π 2 ∞ ) W , q G I , , cos sin d d d = G d "" = = ∫ ∫ ∫ λ,i (λ θ φ) θ θ θ φ λ ∫ λ (λ) λ + 2 . 0 0 0 0 * m - € D. B. Go AME 60634 Int. Heat Trans. Radiation: Radiosity • Radiosity – for opaque surfaces – accounts for all radiation leaving a surface • emission • reflection • Spectral Intensity, Iλ,e+r – a quantity used to specify emitted and reflected radiation intensity • The integration of the spectral heat flux is called the spectral radiosity – spectral emission+reflection (heat flux) over all possible directions π 2π 2 ) W , q J I , , cos sin d d λ"" = λ (λ) = ∫ ∫ λ,e+r (λ θ φ) θ θ θ φ + 2 . 0 0 * m ⋅ µm- • The total heat flux from the surface due to irradiation over all wavelengths and directions è total radiosity π € ∞ 2π 2 ∞ ) W , q J I , , cos sin d d d = J d "" = = ∫ ∫ ∫ λ,e+r (λ θ φ) θ θ θ φ λ ∫ λ (λ) λ + 2 . 0 0 0 0 * m - D. B. Go € AME 60634 Int. Heat Trans. Radiation: Black Body • Black Body – an idealization providing limits on radiation emission and absorption by matter – for a prescribed temperature and wavelength, no surface can emit more than a black body è ideal emitter – a black body absorbs all incident radiation (no reflection) è ideal absorber – a black body is defined as a diffuse emitter • Isothermal Cavity – Approximation of Black Body – after multiple reflections, virtually all radiation entering the cavity is absorbed – emission from the aperture is the maximum possible emission for the temperature of cavity and the emission is diffuse – cumulative effect of emission and reflection off the cavity wall is to provide diffuse irradiation corresponding to D. B. Go emission from a black body AME 60634 Int. Heat Trans. Radiation: Black Body • Planck Distribution – the spectral emission intensity of a black body • determined theoretically and confirmed experimentally −34 2hc 2 Planck constant : h = 6.6256 ×10 J ⋅ s I λ,T = o λ,b ( ) 5 Boltzmann constant : k =1.3805 ×10−23 J/K λ [exp(hco λkT) −1] 8 speed of light (vacuum) : co = 2.998 ×10 m/s – spectral emissive power π € 2π 2 q!λ! = Eλ (λ) = ∫ ∫ Iλ,b (λ,T)c€o sθ sinθ dθ dφ → 0 0 C % W ' E λ,T = π I λ,T = 1 λ,b ( ) λ,b ( ) 5 % ' * 2 + λ &exp(C2 λT) −1( &m ⋅µm( 2 8 4 2 C1 = 2πhco = 3.742 ×10 W ⋅ µm /m 4 C1 = hco k =1.439 ×10 µm/K D. B. Go € AME 60634 Int. Heat Trans. Radiation: Black Body • Planck Distribution – emitted radiation varies continuously with wavelength – at any wavelength, the magnitude of the emitted power increases with temperature – the spectral region where the emission is concentrated depends on temperature • comparatively more radiation at shorter wave lengths sun approximated by 5800 K black body The maximum emission power, Eλ,b, occurs at λmax which is determined by Wien’s displacement law C λ = 3 max T C3 = 2897.8 µm⋅ K € D. B. Go AME 60634 Int. Heat Trans. Radiation: Black Body • Stefan-Boltzmann Law – the total emissive power of a black body is found by integrating the Planck distribution ∞ ∞ ( W + q E E ,T d = I ,T d = T4 "" = b = ∫ λ,b (λ ) λ ∫ π λ,b (λ ) λ σ * 2 - Stefan-Boltzmann Law 0 0 ) m , Stefan - Boltzmann constant : σ = 5.670 ×10−8 W/m2 − K4 – the fraction of the total emissive power within a wavelength band € (λ < λ < λ ) is λ2 € € 1 2 ∫ E λ,b (λ,T)dλ E b (λ1 < λ < λ2 ) λ1 = F = (λ1 −λ2 ) 4 E b σT this can be rewritten as λ2 λ1 ∫ E λ,b (λ,T)dλ − ∫ E λ,b (λ,T)dλ F = F − F = 0 0 € (λ1 −λ2 ) (0−λ2 ) (0−λ1 ) σT 4 the following function is tabulated λ € ∫ E λ,b (λ,T)dλ 0 F(0−λ) = 4 D.