sign in sign up
<<
Home , Emissivity, Flux, Radiosity (radiometry)

AME 60634 Int. Trans. : Overview • Radiation - Emission – 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 of the matter – emission corresponds to 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 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. or quanta); in other cases, radiation behaves as an electromagnetic wave – radiation is characterized by a λ and 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 – thermal radiation is confined to the , 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 – 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 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 (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 (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 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:

• The spectral heat (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 – 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 – 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 – 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 : k =1.3805 ×10−23 J/K λ [exp(hco λkT) −1] 8 speed of (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. B. Go σT

€ AME 60634 Int. Heat Trans. Radiation: Black Body

D. B. Go AME 60634 Int. Heat Trans. 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 2 is Idif = 70 W/m -sr. What is the total irradiation on the earth’s surface?

SCHEMATIC:

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

D. B. Go AME 60634 Int. Heat Trans. 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 – , ε

• Non-Ideal Irradiation – absorptivity, α – reflectivity, ρ – transmissivity, τ

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

Iλ,e (λ,θ,φ,T) ε λ,θ,φ,T = λ,θ ( ) Iλ,b (λ,T) – Spectral, Hemispherical Emissivity (directional average) π 2π 2 ∫ ∫ Iλ,e (λ,θ,φ)cosθ sinθ dθ dφ € Eλ (λ,T) 0 0 ελ (λ,T) = = Eλ,b (λ,T) π Iλ,b (λ,T)

– Total, Directional Emissivity (spectral average)

Ie (θ,φ,T) εθ (θ,φ,T) = Ib (T) D. B. Go

€ AME 60634 Int. Heat Trans. Radiation: Emissivity • Emissivity – Total, Hemispherical Emissivity (directional average) π ∞ 2π 2 ∫ ∫ ∫ Iλ,e (λ,θ,φ)cosθ sinθdθdφdλ E(T) 0 0 0 ε(T) = = 4 E b (T) σT

which can be simplified to ∞ € ∫ ελ (λ,T)E λ,b (λ,T)dλ ε(T) = 0 σT4 – to a reasonable approximation, the total, hemispherical emissivity is equal to the total, normal emissivity

€ ε = εn

€

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

• Representative temperature variations

D. B. Go AME 60634 Int. Heat Trans. Radiation: Absorption/Reflection/Transmission

• 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

Gλ = Gλ,abs + Gλ,ref + Gλ,tr

• An opaque material only has a surface response – there is no € transmission (volumetric effect)

Gλ = Gλ,abs + Gλ,ref

• 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 D. B. Go AME 60634 Int. Heat Trans. Radiation: Absorptivity • Spectral, Directional Absorptivity – assuming negligible temperature dependence

Iλ,i,abs(λ,θ,φ) α λ,θ (λ,θ,φ) = Iλ,i (λ)

• Spectral, Hemispherical Absorptivity (directional average)

π € 2π 2 ∫ ∫ α λ,θ (λ,θ,φ)Iλ,i (λ,θ,φ)cosθ sinθdθdφ Gλ,abs(λ) 0 0 α λ (λ) = = π 2π 2 Gλ (λ) ∫ ∫ Iλ,i (λ,θ,φ)cosθ sinθdθdφ 0 0

• Total, Hemispherical Absorptivity € ∞

α λ (λ)Gλ (λ)dλ G ∫ α = abs = 0 G ∞ ∫ Gλ (λ)dλ 0 D. B. Go

€ AME 60634 Int. Heat Trans. Radiation: Reflectivity • Spectral, Directional Reflectivity – assuming negligible temperature dependence

Iλ,i,ref (λ,θ,φ) ρλ,θ (λ,θ,φ) = Iλ,i (λ)

• Spectral, Hemispherical Reflectivity (spectral average)

π € 2π 2 ∫ ∫ ρλ,θ (λ,θ,φ)Iλ,i (λ,θ,φ)cosθ sinθdθdφ Gλ,ref (λ) 0 0 ρλ (λ) = = π 2π 2 Gλ (λ) ∫ ∫ Iλ,i (λ,θ,φ)cosθ sinθdθdφ 0 0 • Total, Hemispherical Reflectivity ∞ specular – polished diffuse – rough surfaces € ρ λ G λ dλ surfaces G ∫ λ ( ) λ ( ) ρ = ref = 0 G ∞ ∫ Gλ (λ)dλ 0

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

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

Gλ,tr (λ) τ λ (λ) = Gλ (λ) • Total, Hemispherical Transmissivity ∞

τ λ (λ)Gλ (λ)dλ G ∫ € τ = tr = 0 G ∞ ∫ Gλ (λ)dλ 0

• Representative spectral variations €

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

• Semi-Transparent Materials

and 1 α λ + ρλ + τ λ =1 α + ρ + τ =

• Opaque Materials € €

α λ + ρλ =1 and α + ρ =1

€ €

D. B. Go AME 60634 Int. Heat Trans. Radiation: Kirchhoff’s Law

• Kirchhoff’s Law – spectral, directional surface properties are equal

ελ,θ = α λ,θ

• €Kirchhoff’s Law (spectral) – spectral, hemispherical surface properties are equal – for diffuse surfaces or diffuse irradiation

ελ = α λ

• Kirchhoff’s Law (blackbodies) – total, hemispherical properties are equal € – when the irradiation is from a blackbody at the same temperature as the emitting surface ε = α

D. B. Go € AME 60634 Int. Heat Trans. Radiation: Kirchhoff’s Law

• Kirchhoff’s Law (spectral) π π 2π 2 2π 2 α λ,θ,φ I λ,θ,φ cosθ sinθdθdφ ∫ ∫ ελ,θ (λ,θ,φ)cosθ sinθdθdφ ∫ ∫ λ,θ ( ) λ,i ( ) 0 0 ? 0 0 ε = α λ = π λ π 2π 2 2π 2 ελ = α λ I λ,θ,φ cosθ sinθdθdφ ∫ ∫ cosθ sinθdθdφ ∫ ∫ λ,i ( ) 0 0 0 0 I , , I – true if irradiation€ is diffuse λ,i (λ θ φ) = λ,i (λ) – true if surface is diffuse ελ,θ (λ,θ,φ) = ελ ; αλ,θ (λ,θ,φ) = αλ € € • Kirchhoff’s Law (blackbody) ∞ ∞ ε E λ,T dλ α G λ dλ ∫ λ λ,b ( ) ? ∫ λ λ ( ) ε = 0 α = 0 ε = α E b (T) G – true if irradiation is from a blackbody at the same temperature as the emitting surface Gλ (λ) = E λ,b (λ,T) → G = E b (T) € € €

– true if the surface is gray ελ ≠ f (λ); α λ ≠ f (λ) D. B. Go €

€ AME 60634 Int. Heat Trans. Radiation: Gray Surfaces

• Gray Surface

– a surface where αλ and ελ are independent of λ over the spectral regions of the irradiation and emission

Gray approximation only valid for:

λ1 < λ < λ4

€

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

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

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

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

D. B. Go AME 60634 Int. Heat Trans. Radiation: View Factor (Shape Factor)

• 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 cosθ cosθ dq = I cosθ dA dω = J i j dA dA i→ j i i i j→i i π R2 i j – The view factor is the ratio of the intercepted radiosity to the total radiosity q 1 cosθ cosθ F = i→ j = i j dA dA ij A J A ∫ ∫ πR2 j i i i i Ai A j the view factor is based entirely on geometry

D. B. Go € AME 60634 Int. Heat Trans. Radiation: View Factor Relations

• Reciprocity

1 cosθ cosθ 1 cosθi cosθ j F j i dA dA F = dA dA ji = 2 i j ij ∫ ∫ 2 j i A ∫ ∫ π R A πR j Aj Ai i Ai A j

AiFij = A j F ji € • Summation – from conservation of€ radiation (energy), for an enclosure

N

∑ AiFij =1 j=1

€

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

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

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

• For a blackbody there is no reflection (perfect absorber)

J = E = σT 4 i bi i • 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

qij = qi→ j − q j →i

qij = AiFij E bi − A j F ji E bj 4 4 qij = AiFijσ(Ti − Tj )

(heat loss from Ai) • Net radiation (heat) transfer from surface i due to exchange with all €( N) surfaces of an enclosure

N 4 4 qi = ∑ AiFijσ(Ti − Tj ) j=1 D. B. Go

€ AME 60634 Int. Heat Trans. Radiation: Gray Radiation Exchange

• General assumption for opaque, diffuse, gray surfaces

εi = α i =1− ρi

• Equivalent expressions for the net radiation (heat) transfer from surface i € qi = Ai (Ji −Gi ) ⇒ Fig. (b)

qi = Ai (Ei −αiGi ) ⇒ Fig. (c)

Ebi − Ji qi = ⇒ Fig. (d) (1−εi ) εi Ai

thus for gray bodies the resistance at the surface is

(1−εi ) Rrad,surface = εi Ai and the driving potential is

E bi − Ji €

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

Net radiation (heat) transfer from surface i due to exchange with all (N) surfaces of an enclosure

N N J − J q A F J J ( i j ) i = ∑ i ij ( i − j ) = ∑ −1 j=1 j=1 (AiFij )

thus for gray bodies the resistance between two bodies (space or geometrical resistance) € 1 Rrad,space = AiFij and the driving potential is J − J € i j N J − J E bi − Ji ( i j ) Radiation energy balance on surface i : qi = = ∑ −1 € 1−εi εi Ai ( ) j=1 (AiFij ) net energy leaving = energy exchange with other surfaces

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

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

D. B. Go AME 60634 Int. Heat Trans. Radiation: Gray Radiation Exchange • Methodology of an enclosure analysis

– apply the following equation for each surface where the net radiation heat rate qi is known N (Ji − J j ) qi = ∑ −1 j=1 (AiFij )

– apply the following equation for each remaining surface where the temperature Ti (and thus E ) is known € bi N E − J Ji − J j bi i = ( ) 1−ε ε A ∑ −1 ( i ) i i j=1 (AiFij ) – 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

E bi − Ji qi = (1−εi ) εi Ai

D. B. Go

€ AME 60634 Int. Heat Trans. Radiation: Gray Radiation Exchange • 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

εi = α i =1

€

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

4 4 σ T1 −T2 q = −q = ( ) 1 2 1−ε 1 1−ε ( 1) + + ( 2 ) ε1A1 A1F12 ε2 A2

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

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

• Reradiating Surface

– 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

4 4 σ(T1 − T1 ) q1 = −q2 = (1−ε1) 1 (1−ε2 ) + −1 + ε1A1 A F 1 A F 1 A F ε2 A2 D. B. Go 1 12 + [( 1 1R ) + ( 2 2R )]

€ AME 60634 Int. Heat Trans. Radiation: 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

J − J J − J 1 R = R 2 (1 A1F1R ) (1 A2F2R )

1 $ J ' 4 T = & R ) R % σ (

€

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

qi,ext = qi,rad + qi,conv + qi,cond

€

qi,rad → appropriate analysis for N-surface, two-surface, etc. enclosure

D. B. Go

€ AME 60634 Int. Heat Trans. 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.

D. B. Go