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AME 60634 Int. Heat Trans. D. B. Go 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

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AME 60634 Int. Heat Trans. D. B. Go 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) in a vacuum

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AME 60634 Int. Heat Trans. D. B. Go 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 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

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AME 60634 Int. Heat Trans. D. B. Go 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, dA n, and propagating in a particular direction ( θ, ϕ ) is quantified in terms of a differential solid angle associated with the direction, dω. dA n unit element of surface on a hypothetical sphere and normal to the ( θ, ϕ ) direction

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AME 60634 Int. Heat Trans. D. B. Go Solid Angle Radiation: Directional Considerations the solid angle ω has units of steradians (sr) the solid angle ω hemi associated with a complete hemisphere

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Spectral Intensity Spectral Intensity, I λ,e –a quantity used to specify the radiant heat flux (W/m 2 ) within a unit solid angle about a prescribed direction (W/m 2 -sr) and within a unit wavelength interval about a prescribed wavelength (W/m 2 -sr-μm) –associated with emission from a surface element dA 1 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 ( dA 1 cosθ ) stems from the existence of surfaces for which, to a good approximation, I λ,e is independent of direction. Such surfaces are termed diffuse, and the radiation is said to be isotropic. the projected area is how dA 1 appears along θ, ϕ [W/m 2 -sr-μm]

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AME 60634 Int. Heat Trans. D. B. Go 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

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AME 60634 Int. Heat Trans. D. B. Go 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 Radiation: Heat Flux

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Irradiation Irradiation –electromagnetic waves incident on a surface is called irradiation –irradiation can be either absorbed or reflected Spectral Intensity, I λ,i –a quantity used to specify the incident radiant heat flux (W/m 2 ) within a unit solid angle about the direction of incidence (W/m 2 - sr) and within a unit wavelength interval about a prescribed wavelength (W/m 2 -sr-μm) and the projected area of the receiving surface ( dA 1 cosθ )

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AME 60634 Int. Heat Trans. D. B. Go 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 Radiation: Irradiation Heat Flux

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Radiosity Radiosity –for opaque surfaces –accounts for all radiation leaving a surface emission reflection Spectral Intensity, I λ,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

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AME 60634 Int. Heat Trans. D. B. Go 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 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 ideal emitter –a black body absorbs all incident radiation (no reflection) ideal absorber –a black body is defined as a diffuse emitter

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Black Body Planck Distribution –the spectral emission intensity of a black body determined theoretically and confirmed experimentally –spectral emissive power

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AME 60634 Int. Heat Trans. D. B. Go 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, E λ,b, occurs at λ max which is determined by Wiens displacement law

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Black Body Stefan-Boltzmann 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 ( λ 1 < λ < λ 2 ) is Stefan-Boltzmann Law this can be rewritten as the following function is tabulated

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Black Body

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AME 60634 Int. Heat Trans. D. B. Go Example: Radiation According to its directional distribution, solar radiation incident on the earths 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/m 2 and the total intensity of the diffuse radiation is I dif = 70 W/m 2 -sr. What is the total irradiation on the earths surface?

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AME 60634 Int. Heat Trans. D. B. Go 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). (a)Calculate the band emission fractions for the visible region for each light source. (b)Calculate the wavelength corresponding to the maximum spectral intensity for each light source.

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Surface Properties 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

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Emissivity Emissivity –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)

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Emissivity Emissivity –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

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Emissivity Representative spectral variations Representative temperature variations

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Absorption/Reflection/Transmission Three responses of semi-transparent medium to irradiation, G λ –absorption within medium, G λ,abs –reflection from the medium, G λ,ref –transmission through the medium, G λ,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

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Absorptivity Spectral, Directional Absorptivity –assuming negligible temperature dependence Spectral, Hemispherical Absorptivity (directional average) Total, Hemispherical Absorptivity

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Reflectivity Spectral, Directional Reflectivity –assuming negligible temperature dependence Spectral, Hemispherical Reflectivity (spectral average) Total, Hemispherical Reflectivity diffuse – rough surfaces specular – polished surfaces

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Reflectivity Representative spectral variations

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Transmissivity Spectral, Hemispherical Reflectivity –assuming negligible temperature dependence Total, Hemispherical Transmissivity Representative spectral variations

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Irradiation Balance Semi-Transparent Materials Opaque Materials and

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Kirchhoffs Law Kirchhoffs Law –spectral, directional surface properties are equal Kirchhoffs Law (spectral) –spectral, hemispherical surface properties are equal –for diffuse surfaces or diffuse irradiation Kirchhoffs Law (blackbodies) –total, hemispherical properties are equal –when the irradiation is from a blackbody at the same temperature as the emitting surface

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Kirchhoffs Law Kirchhoffs Law (spectral) –true if irradiation is diffuse –true if surface is diffuse Kirchhoffs Law (blackbody) –true if irradiation is from a blackbody at the same temperature as the emitting surface –true if the surface is gray ? ?

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Gray Surfaces Gray Surface –a surface where α λ and ε λ are independent of λ over the spectral regions of the irradiation and emission Gray approximation only valid for:

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Example The spectral, hemispherical emissivity absorptivity of an opaque surface is shown below. (a)What is the solar absorptivity? (b)If Kirchhoffs Law (spectral) is assumed and the surface temperature is 340 K, what is the total hemispherical emissivity?

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AME 60634 Int. Heat Trans. D. B. Go 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.

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Exchange Between Surfaces 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.

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AME 60634 Int. Heat Trans. D. B. Go Radiation: View Factor (Shape Factor) View Factor, F ij –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 dA i to the differential area dA j –the rate of radiosity (emission + reflection) intercepted by dA j –The view factor is the ratio of the intercepted radiosity to the total radiosity the view factor is based entirely on geometry

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AME 60634 Int. Heat Trans. D. B. Go Radiation: View Factor Relations Reciprocity Summation –from conservation of radiation (energy), for an enclosure

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AME 60634 Int. Heat Trans. D. B. Go Radiation: View Factors 2-D Geometries

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AME 60634 Int. Heat Trans. D. B. Go Radiation: View Factors 3-D Geometries

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Blackbody Radiation Exchange 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 A i )

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Gray Radiation Exchange 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

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Gray Radiation Exchange 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

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Gray Radiation Exchange The equivalent circuit for a radiation network consists of two resistances –resistance at the surface –resistances between all bodies

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Gray Radiation Exchange Methodology of an enclosure analysis –apply the following equation for each surface where the net radiation heat rate q i is known –apply the following equation for each remaining surface where the temperature T i (and thus E bi ) is known –determine all the view factors –solve the system of N equations for the unknown radiosities J 1, J 2, …, J N –apply the following equation to determine the radiation heat rate q i for each surface of known T i and T i for each surface of known q i

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Gray Radiation Exchange Special Case –enclosure with an opening (aperture) of area A i through which the interior surface exchange radiation with large surroundings at temperature T sur T sur AiAi Treat the aperture as a virtual blackbody surface with area A i, T i = T sur and

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Two Surface Enclosures 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

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Two Surface Enclosures Special Cases

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Reradiating Surface Reradiating Surface –idealization for which G R = J R hence q R = 0 and J R = E b,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

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Reradiating Surface The temperature of the reradiating surface T R may be determined from knowledge of its radiosity J R. With q R = 0 a radiation balance on the surface yields

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AME 60634 Int. Heat Trans. D. B. Go Radiation: Multimode Effects In an enclosure with conduction and convection heat transfer to/from one or more surface, the foregoing treatments of the radiation exchange may be combined with surface energy balances to determine thermal conditions Consider a general surface condition for which there is external heat addition (e.g., electrically) as well as conduction, convection and radiation appropriate analysis for N-surface, two-surface, etc. enclosure

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AME 60634 Int. Heat Trans. D. B. Go Example: Radiation Exchange A cylindrical furnace for heat treating materials in a spacecraft environment has a 90- mm diameter and an overall length of 180 mm. Heating elements in the 135 mm long section maintain a refractory lining at 800 °C and ε = 0.8. the other linings are insulated but made of the same material. The surroundings are at 23 °C. Determine the power required to maintain the furnace operating conditions.

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