Int.$HeatTrans.$ Radiation: Overviesst/teaching/AME60634/lectures/AME60634_F13...surface consists of two components that may be approximated as being diffusely distributed with the

AME 60634 Int. Heat Trans.

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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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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)

€

λ =cν

€

c = co = 2.998 ×108 m/sin a vacuum


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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


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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.

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

€

dω =dAn

r2


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•  Solid Angle Radiation: Directional Considerations

€

dAn = r2 sinθdθdφ

€

dω =dAn

r2= sinθdθdω

€

ωhemi = sinθdθdφ0

π2

∫0

2π

∫ = 2π sr

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


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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:

–  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

€

Iλ,e λ,θ,φ( ) =dq

dA1 cosθ( )dωdλ[W/m2-sr-µm]


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•  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

€

dqλ =dqdλ

= Iλ,e λ,θ,φ( )cosθdA1dω

€

d " " q λ =dq

dλdA1= Iλ,e λ,θ,φ( )cosθdω = Iλ,e λ,θ,φ( )cosθ sinθdθdφ

€

" " q λ = Eλ λ( ) = Iλ,e λ,θ,φ( )cosθ sinθdθdφ0

π2

∫0

2π

∫ Wm2 ⋅µm)

* +

,

- .


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•  The total heat flux from the surface due to radiation is emission over all wavelengths and directions è 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

€

" " q = E = Iλ,e λ,θ,φ( )cosθ sinθdθdφdλ0

π2

∫0

2π

∫ = Eλ λ( )dλ0

∞

∫ 0

∞

∫ Wm2

)

* + ,

- .

€

" " q = E = Iλ,e λ( )dλ cosθ sinθdθdφ0

π2

∫0

2π

∫ = π Iλ,e λ( )dλ0

∞

∫ = πIe 0

∞

∫ Wm2

)

* + ,

- .


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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)


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•  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 è total irradiative power

Radiation: Irradiation Heat Flux

€

" " q λ = Gλ λ( ) = Iλ,i λ,θ,φ( )cosθ sinθdθdφ0

π2

∫0

2π

∫ Wm2 ⋅µm)

* +

,

- .

€

" " q = G = Iλ,i λ,θ,φ( )cosθ sinθdθdφdλ0

π2

∫0

2π

∫ = Gλ λ( )dλ0

∞

∫ 0

∞

∫ Wm2

)

* + ,

- .


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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

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

" " q λ = Jλ λ( ) = Iλ,e +r λ,θ,φ( )cosθ sinθdθdφ0

π2

∫0

2π

∫ Wm2 ⋅µm)

* +

,

- .

€

" " q = J = Iλ,e +r λ,θ,φ( )cosθ sinθdθdφdλ0

π2

∫0

2π

∫ = Jλ λ( )dλ0

∞

∫ 0

∞

∫ Wm2

)

* + ,

- .


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•  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 è ideal emitter

–  a black body absorbs all incident radiation (no reflection) è ideal absorber

–  a black body is defined as a diffuse emitter


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Radiation: Black Body •  Planck Distribution

–  the spectral emission intensity of a black body •  determined theoretically and confirmed experimentally

–  spectral emissive power

!!qλ = Eλ λ( ) = Iλ,b λ,T( )cosθ sinθ dθ dφ0

π2

∫0

2π

∫ →

Eλ,b λ,T( ) = π Iλ,b λ,T( ) = C1

λ 5 exp C2 λT( )−1%& '( W

m2 ⋅µm%

&*

'

(+

€

Iλ,b λ,T( ) =2hco

2

λ5 exp hco λkT( ) −1[ ]

€

Planck constant : h = 6.6256 ×10−34 J ⋅ sBoltzmann constant : k =1.3805 ×10−23 J/Kspeed of light (vacuum) : co = 2.998 ×108 m/s

€

C1 = 2πhco2 = 3.742 ×108 W ⋅µm4/m2

C1 = hco k =1.439 ×104 µm/K


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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

€

λmax =C3

TC3 = 2897.8 µm ⋅K


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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 (λ1 < λ < λ2) is €

" " q = Eb = Eλ,b λ,T( )dλ0

∞

∫ = πIλ,b λ,T( )dλ0

∞

∫ =σT4 Wm2

(

) * +

, - Stefan-Boltzmann Law

€

Stefan - Boltzmann constant : σ = 5.670 ×10−8 W/m2 −K4

€

Eb λ1 < λ < λ2( )Eb

= F λ1−λ2( ) =

Eλ,b λ,T( )dλλ1

λ2

∫

σT 4

€

this can be rewritten as

€

F λ1−λ2( ) = F 0−λ2( ) − F 0−λ1( ) =Eλ,b λ,T( )dλ

0

λ2

∫ − Eλ,b λ,T( )dλ0

λ1

∫σT 4

the following function is tabulated

€

F 0−λ( ) =Eλ,b λ,T( )dλ

0

λ

∫σT 4


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Radiation: Black Body


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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?

SCHEMATIC:


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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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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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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)

€

ελ,θ λ,θ,φ,T( ) =Iλ,e λ,θ,φ,T( )Iλ,b λ,T( )

ελ λ,T( ) =Eλ λ,T( )Eλ,b λ,T( )

=Iλ,e λ,θ,φ( )cosθ sinθ dθ dφ

0

π2

∫0

2π

∫π Iλ,b λ,T( )

€

εθ θ,φ,T( ) =Ie θ,φ,T( )Ib T( )


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Radiation: Emissivity •  Emissivity

–  Total, Hemispherical Emissivity (directional average)

–  to a reasonable approximation, the total, hemispherical emissivity is equal to the total, normal emissivity

€

ε T( ) =E T( )Eb T( )

=Iλ,e λ,θ,φ( )cosθ sinθdθdφdλ

0

π2

∫0

2π

∫0

∞

∫σT4

which can be simplified to

€

ε T( ) =ελ λ,T( )Eλ,b λ,T( )dλ

0

∞

∫σT4

€

ε = εn


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Radiation: Emissivity •  Representative spectral variations

•  Representative temperature variations


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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

•  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

€

Gλ =Gλ,abs +Gλ,ref +Gλ,tr

€

Gλ =Gλ,abs +Gλ,ref


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Radiation: Absorptivity •  Spectral, Directional Absorptivity

–  assuming negligible temperature dependence

•  Spectral, Hemispherical Absorptivity (directional average)

•  Total, Hemispherical Absorptivity

€

αλ,θ λ,θ,φ( ) =Iλ,i,abs λ,θ,φ( )

Iλ,i λ( )

€

αλ λ( ) =Gλ,abs λ( )Gλ λ( )

=αλ,θ λ,θ,φ( )Iλ,i λ,θ,φ( )cosθ sinθdθdφ

0

π2

∫0

2π

∫

Iλ,i λ,θ,φ( )cosθ sinθdθdφ0

π2

∫0

2π

∫

€

α =Gabs

G=

αλ λ( )Gλ λ( )dλ0

∞

∫

Gλ λ( )dλ0

∞

∫


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Radiation: Reflectivity •  Spectral, Directional Reflectivity


•  Spectral, Hemispherical Reflectivity (spectral average)

•  Total, Hemispherical Reflectivity

€

ρλ,θ λ,θ,φ( ) =Iλ,i,ref λ,θ,φ( )

Iλ,i λ( )

€

ρλ λ( ) =Gλ,ref λ( )Gλ λ( )

=ρλ,θ λ,θ,φ( )Iλ,i λ,θ,φ( )cosθ sinθdθdφ

0

π2

∫0

2π

∫


π2

∫0

2π

∫

€

ρ =Gref

G=

ρλ λ( )Gλ λ( )dλ0

∞

∫

Gλ λ( )dλ0

∞

∫

diffuse – rough surfaces

specular – polished surfaces


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Radiation: Reflectivity •  Representative spectral variations


D. B. Go

Radiation: Transmissivity •  Spectral, Hemispherical Reflectivity


•  Total, Hemispherical Transmissivity

•  Representative spectral variations

€

τλ λ( ) =Gλ,tr λ( )Gλ λ( )

€

τ =Gtr

G=

τλ λ( )Gλ λ( )dλ0

∞

∫

Gλ λ( )dλ0

∞

∫


D. B. Go

Radiation: Irradiation Balance •  Semi-Transparent Materials

•  Opaque Materials €

αλ + ρλ + τλ =1

€

α + ρ + τ =1and

€

αλ + ρλ =1

€

α + ρ =1and


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Radiation: Kirchhoff’s Law •  Kirchhoff’s Law

–  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

€

ελ,θ =αλ,θ

€

ελ =αλ

€

ε =α


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Radiation: Kirchhoff’s Law •  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

€

ελ =αλ

€

ελ =ελ,θ λ,θ,φ( )cosθ sinθdθdφ

0

π2

∫0

2π

∫

cosθ sinθdθdφ0

π2

∫0

2π

∫

€

αλ =αλ,θ λ,θ,φ( )Iλ,i λ,θ,φ( )cosθ sinθdθdφ

0

π2

∫0

2π

∫


π2

∫0

2π

∫

?

Iλ,i λ,θ,φ( ) = Iλ,i λ( )ελ,θ λ,θ,φ( ) = ελ; αλ,θ λ,θ,φ( ) =αλ

€

α =αλGλ λ( )dλ

0

∞

∫G

€

ε =ελEλ,b λ,T( )dλ

0

∞

∫Eb T( )

€

ε =α?

€

Gλ λ( ) = Eλ,b λ,T( )→G = Eb T( )

€

ελ ≠ f λ( ); αλ ≠ f λ( )


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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


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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

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

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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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

–  The view factor is the ratio of the intercepted radiosity to the total radiosity

dqi→ j = Ii cosθidAidω j→i = Jicosθi cosθ j

πR2dAidAj

€

Fij =qi→ j

AiJi=1Ai

cosθi cosθ j

πR2dA j

A j

∫ dAiAi

∫

the view factor is based entirely on geometry


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Radiation: View Factor Relations •  Reciprocity

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

€

Fij =1Ai

cosθi cosθ j

πR2dA j

A j

∫ dAiAi

∫ Fji =1Aj

cosθ j cosθiπR2

dAiAi

∫ dAjAj

∫

€

AiFij = A jFji

€

AiFijj=1

N

∑ =1


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Radiation: View Factors •  2-D Geometries


D. B. Go

Radiation: View Factors •  3-D Geometries


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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

€

Ji = Ebi =σTi4

€

qij = qi→ j − q j→i

qij = AiFij Ebi − A jFjiEbj

qij = AiFijσ Ti4 −Tj

4( )

€

qi = AiFijσ Ti4 −Tj

4( )j=1

N

∑

(heat loss from Ai)


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Radiation: Gray Radiation Exchange •  General assumption for opaque, diffuse, gray surfaces

•  Equivalent expressions for the net radiation (heat) transfer from surface i €

εi =α i =1− ρi

qi = Ai Ji −Gi( ) ⇒ Fig. (b)

qi = Ai Ei −αiGi( )⇒ Fig. (c)

thus for gray bodies the resistance at the surface is

and the driving potential is

€

Rrad ,surface =1−εi( )εiAi

€

Ebi − Ji

qi =Ebi − Ji

1−εi( ) εiAi ⇒ Fig. (d)


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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

€

Rrad ,space =1

AiFij

€

Ji − J j

€

qi = AiFij Ji − J j( )j=1

N

∑ =Ji − J j( )AiFij( )

−1j=1

N

∑

€

qi =Ebi − Ji1−εi( ) εiAi

=Ji − J j( )AiFij( )

−1j=1

N

∑Radiation energy balance on surface i :

net energy leaving = energy exchange with other surfaces


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


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 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

€

qi =Ji − J j( )AiFij( )

−1j=1

N

∑

€

Ebi − Ji1−εi( ) εiAi

=Ji − J j( )AiFij( )

−1j=1

N

∑

€

qi =Ebi − Ji1−εi( ) εiAi


D. B. Go

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


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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

q1 = −q2 =σ T1

4 −T24( )

1−ε1( )ε1A1

+1

A1F12+1−ε2( )ε2A2


D. B. Go

Radiation: Two Surface Enclosures •  Special Cases


D. B. Go

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

€

q1 = −q2 =σ T1

4 −T14( )

1−ε1( )ε1A1

+1

A1F12 + 1 A1F1R( ) + 1 A2F2R( )[ ]−1+1−ε2( )ε2A2


D. B. Go

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

€

J1 − JR1 A1F1R( )

=JR − J21 A2F2R( )

TR =JRσ

$

% &

'

( )

14


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

€

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

€

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


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.

Documents

Int.$HeatTrans.$ Radiation: Overviesst/teaching/AME60634/lectures/AME60634_F13...surface consists of two components that may be approximated as being diffusely distributed with the