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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

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) in a vacuum

Electromagnetic Spectrum the range of all possible radiation frequencies thermal radiation is confined to the infrared, visible, and ultraviolet regions of the spectrum 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

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. dAn  unit element of surface on a hypothetical sphere and normal to the (θ,ϕ) direction

Solid Angle the solid angle ω has units of steradians (sr) the solid angle ωhemi associated with a complete hemisphere

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: 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 [W/m2-sr-μm]

Radiation: Heat Flux The spectral heat rate (heat rate per unit wavelength of radiation) associated with emission The spectral heat flux (heat flux per unit wavelength of radiation) associated with emission The integration of the spectral heat flux is called the spectral emissive power spectral emission (heat flux) over all possible directions

Radiation: Heat Flux The total heat flux from the surface due to radiation is emission over all wavelengths and directions  total emissive power If the emission is the same in all directions, then the surface is diffuse and the emission is isotropic

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)

The integration of the spectral heat flux is called the spectral irradiation spectral irradiation (heat flux) over all possible directions The total heat flux to the surface due to irradiation over all wavelengths and directions  total irradiative power

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 The total heat flux from the surface due to irradiation over all wavelengths and directions  total radiosity

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  ideal emitter a black body absorbs all incident radiation (no reflection)  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 emission from a black body

the spectral emission intensity of a black body determined theoretically and confirmed experimentally spectral emissive power

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

the total emissive power of a black body is found by integrating the Planck distribution the fraction of the total emissive power within a wavelength band (λ1 < λ < λ2) is Stefan-Boltzmann Law this can be rewritten as the following function is tabulated

Example: Radiation According to its directional distribution, solar radiation incident on the earth’s surface consists of two components that may be approximated as being diffusely distributed with the angle of the sun θ. Consider clear sky conditions with incident radiation at an angle of 30° with a total heat flux (if the radiation were angled normal to the surface) of 1000 W/m2 and the total intensity of the diffuse radiation is Idif = 70 W/m2-sr. What is the total irradiation on the earth’s surface?

Example: Radiation The human eye, as well as the light-sensitive chemicals on color photographic film, respond differently to lighting sources with different spectral distributions. Daylight lighting corresponds to the spectral distribution of a solar disk (approximated as a blackbody at 5800 K) and incandescent lighting from the usual household lamp (approximated as a blackbody at 2900 K). Calculate the band emission fractions for the visible region for each light source. Calculate the wavelength corresponding to the maximum spectral intensity for each light source.

Real surfaces do not behave like ideal black bodies non-ideal surfaces are characterized by factors (< 1) which are the ratio of the non-ideal performance to the ideal black body performance these factors can be a function of wavelength (spectral dependence) and direction (angular dependence) Non-Ideal Radiation Factor emissivity, ε Non-Ideal Irradiation absorptivity, α reflectivity, ρ transmissivity, τ

characterizes the emission of a real body to the ideal emission of a black body and can be defined in three manners a function of wavelength (spectral dependence) and direction (angular dependence) a function of wavelength (spectral dependence) averaged over all directions a function of direction (angular dependence) averaged over all wavelengths Spectral, Directional Emissivity Spectral, Hemispherical Emissivity (directional average) Total, Directional Emissivity (spectral average)

Total, Hemispherical Emissivity (directional average) to a reasonable approximation, the total, hemispherical emissivity is equal to the total, normal emissivity which can be simplified to

Representative spectral variations Representative temperature variations

Three responses of semi-transparent medium to irradiation, Gλ absorption within medium, Gλ,abs reflection from the medium, Gλ,ref transmission through the medium, Gλ,tr Total irradiation balance An opaque material only has a surface response – there is no transmission (volumetric effect) The semi-transparency or opaqueness of a medium is governed by both the nature of the material and the wavelength of the incident radiation the color of an opaque material is based on the spectral dependence of reflection in the visible spectrum

Spectral, Directional Absorptivity assuming negligible temperature dependence Spectral, Hemispherical Absorptivity (directional average) Total, Hemispherical Absorptivity

Spectral, Directional Reflectivity assuming negligible temperature dependence Spectral, Hemispherical Reflectivity (spectral average) Total, Hemispherical Reflectivity specular – polished surfaces diffuse – rough surfaces

Representative spectral variations

Spectral, Hemispherical Reflectivity assuming negligible temperature dependence Total, Hemispherical Transmissivity Representative spectral variations

Semi-Transparent Materials Opaque Materials and and

spectral, directional surface properties are equal Kirchhoff’s Law (spectral) spectral, hemispherical surface properties are equal for diffuse surfaces or diffuse irradiation Kirchhoff’s Law (blackbodies) total, hemispherical properties are equal when the irradiation is from a blackbody at the same temperature as the emitting surface

Kirchhoff’s Law (spectral) true if irradiation is diffuse true if surface is diffuse Kirchhoff’s Law (blackbody) true if irradiation is from a blackbody at the same temperature as the emitting surface true if the surface is gray ? ?

a surface where αλ and ελ are independent of λ over the spectral regions of the irradiation and emission Gray approximation only valid for:

Radiation: Example The spectral, hemispherical emissivity absorptivity of an opaque surface is shown below. What is the solar absorptivity? If Kirchhoff’s Law (spectral) is assumed and the surface temperature is 340 K, what is the total hemispherical emissivity?

Radiation: Example A vertical flat plate, 2 m in height, is insulated on its edges and backside is suspended in atmospheric air at 300 K. The exposed surface is painted with a special diffuse coating having the prescribed absorptivity distribution and is irradiated by solar-simulation lamps that provide spectral irradiation characteristic of the solar spectrum. Under steady conditions the plate is at 400 K. (a) Find the plate absorptivity, emissivity, free convection coefficient, and irradiation. (b) Estimate the plate temperature if if the irradiation was doubled.

Overview Enclosures consist of two or more surfaces that envelop a region of space (typically gas-filled) and between which there is radiation transfer. Virtual, as well as real, surfaces may be introduced to form an enclosure. A nonparticipating medium within the enclosure neither emits, absorbs, nor scatters radiation and hence has no effect on radiation exchange between the surfaces. Each surface of the enclosure is assumed to be isothermal, opaque, diffuse and gray, and to be characterized by uniform radiosity and irradiation.

View Factor, Fij geometrical quantity corresponding to the fraction of the radiation leaving surface i that is intercepted by surface j General expression consider radiation from the differential area dAi to the differential area dAj the rate of radiosity (emission + reflection) intercepted by dAj The view factor is the ratio of the intercepted radiosity to the total radiosity the view factor is based entirely on geometry

Reciprocity Summation from conservation of radiation (energy), for an enclosure

2-D Geometries

3-D Geometries

For a blackbody there is no reflection (perfect absorber) Net radiation exchange (heat rate) between two “blackbodies” net rate at which radiation leaves surface i due to its interaction with j OR net rate at which surface j gains radiation due to its interaction with i Net radiation (heat) transfer from surface i due to exchange with all (N) surfaces of an enclosure (heat loss from Ai)

General assumption for opaque, diffuse, gray surfaces Equivalent expressions for the net radiation (heat) transfer from surface i thus for gray bodies the resistance at the surface is and the driving potential is

Net radiation (heat) transfer from surface i due to exchange with all (N) surfaces of an enclosure thus for gray bodies the resistance between two bodies (space or geometrical resistance) and the driving potential is Radiation energy balance on surface i : net energy leaving = energy exchange with other surfaces

The equivalent circuit for a radiation network consists of two resistances resistance at the surface resistances between all bodies

Methodology of an enclosure analysis apply the following equation for each surface where the net radiation heat rate qi is known apply the following equation for each remaining surface where the temperature Ti (and thus Ebi) is known determine all the view factors solve the system of N equations for the unknown radiosities J1, J2, …, JN apply the following equation to determine the radiation heat rate qi for each surface of known Ti and Ti for each surface of known qi

Special Case enclosure with an opening (aperture) of area Ai through which the interior surface exchange radiation with large surroundings at temperature Tsur Tsur Ai Treat the aperture as a virtual blackbody surface with area Ai, Ti = Tsur and

Simplest enclosure for which radiation exchange is exclusively between two surfaces and a single expression for the rate of radiation transfer may be inferred from a network representation of the exchange

Special Cases

idealization for which GR = JR hence qR = 0 and JR = Eb,R approximated by surfaces that are well insulated on one side and for which convection is negligible on the opposite (radiating) side Three-surface enclosure with a reradiating surface

The temperature of the reradiating surface TR may be determined from knowledge of its radiosity JR. With qR = 0 a radiation balance on the surface yields