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RADIATION HEAT TRANSFER

Heat conduction and convection - always a fluid which transfers the heat (gas, liquid, solid) – motion of atoms or moleculesHeat conduction and convection is not possible in a vacuumIn most practical applications all three modes occur concurrently at varying degrees

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A hot object in a vacuumchamber looses heat byradiation only

Unlike conduction and convection, heattransfer by radiation can occur between two bodies, even when they are separated by a medium colder than both of them

convection

radiation

What will be a final equilibrium temperature of the body surface? Can you write an energy balance equation between the body and surrounding air and the hot source (fire)?

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Theoretical foundation of radiation was establishedby MaxwellElectromagnetic wave motion or electromagnetic radiationElectromagnetic waves travel at the speed of light c in a vacuumElectromagnetic waves are characterized by their frequency for wavelength λ: λ=c/fc=co/n co light speed in a vacuum

n refraction index of a medium (n=1 for air and most gases, n=1,5 for glass, 1,33 for water)

In all material medium, there is attenuation of the energyIn a vacuum there is no attenuation of the energy

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Electromagnetic radiation covers a wide range of wavelengths

Radiation that is related toheat transfer –Thermal radiationλ from 0,1μm to 100 μmAs a result of energy transition in molecules, atoms and electrons.

Thermal radiation is emitted by all matter whose temperature is above absolute zero.

Everything around us emits (and absorbs) radiation.

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• Thermal radiation includes entire visible (0,4 to 0,76 μm)and infrared light and a portion of ultraviolet radiation.

• Bodies start emit visible radiation at 800K (red hot) and tungsten wire in the lightbulb at 2000K (white hot) to emit a significant amount of radiation in the visible range.

• Bodies at room temperature emit radiation in infrared range 0,7 to 100 μm.

• Sun (primary light source) emits solar radiation –0,3 to 3 μm – almost half is visible, remaining is ultraviolet and infrared.

• Body that emits radiation in the visible range is called light source.

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Spectral and Directional Distribution

Radiation characteristics varywith wavelength and direction

• Monochromatic or spectral: Characteristics at a given λ• Total: Integrated values over all wavelengths• Directional: At a given direction

• Diffuse radiation: Uniform in all directions • Hemispherical: Integrated values over all directions

The assumption of diffuse radiation will be made throughout

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Emissive Power E, Irradiation G and Radiosity J

• Emissive Power (zářivost):Radiation emitted from a surface

• Spectral emissive power λE :

λEper unit area per unit wavelength,

= rate of emitted radiationmW/m2μ

• Total emissive power E:,E = Integration of λE over all values of λ 2W/m :

( ) ( )∫∞

=

0, λλλ dTETE

∫∞

=0

λλdEE

λ

λE

10.1 Fig.

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• Irradiation: Radiation energy incident on a surface• Spectral irradiation λG :

λGper unit area per unit wavelength,

= rate of radiation energy incident upon a surfacemμW/m2−

• Total irradiationG:G = integration of λG over all values of λ :

( ) ( )∫∞

=

0 , λλλ dTGTG

• Radiosity: The sum of emitted and reflected radiation • Spectral radiosity λJ :

λJ = rate of radiation leaving a surface per unit area perunit wavelength, mμW/m2−

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In the above definitions, summation in all directions isimplied although the term hemispherical is not used

• Total radiosity J:

( ) ( )∫∞

=

0 , λλλ dTJTJ

J = integration of λJ over all values of λ :

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Characteristics of blackbody:(1) It absorbs all radiation incident upon it(2) It emits the maximum energy at a given temperature

and wavelength(3) Its emission is diffuse

Planck's Law λbE = spectral emissive power of a blackbody:

( )1)/exp(

,2

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−=

−

TCCTEb λλλλ C1 and C2 are constants

Blackbody Radiation

Blackbody: An ideal radiation surface used as standard for describing radiation of real surfaces

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Planck's Law

Blackbody Radiation

2879,6Tλmax =

Maximum emitted energy atspecific temperatures given byWien law:

Note - by qualitative judgment -energy emitted in visible range for 2000 K – tungsten wire ina light bulb.

Thermal radiation 0,1 to 100 μm

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Stefan-Boltzmann LawBased on: • Experimental data by Stefan (1879)• Theoretical derivation by Boltzmann (1884)

4TEb σ=

bE = total blackbody emissive power (all wavelengths and all directions), [W/m2]

428- KW/m105.67 −×=σ is the Stefan-Boltzmannconstant

It can also be arrived at using Planck's law

Stefan-Boltzmann law

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( ) ( )

4

0 2

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0 bλb

Tσ

dλ1)T/λC(exp

λC

dxλ,TETE

=

∫ =−

=

=∫=

∞ −

∞

• Stefan-Boltzmann law gives the total radiation emitted froma black body at all wavelengths from λ=0 to λ=∞.

• Often an interest in radiation over some wavelength band –light bulb – how much is emitted in the visible range?

• We use a procedure to determine Eb,0-λ

∫=−

λ

0bλλb,0 T)dλ(E(T)E ,λ

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Define a dimensionless quantity fλ(T):

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λ0 b,λ

λ σT(T)dλE

(T)f ∫=

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Want to know how energy is emitted in the visible range 0,40 to 0,76 μm.

λ1T=0,40.2500=1000 ⇒ fλ1 = 0,000321

λ2T=0, 76.2500=1900 ⇒ fλ2 = 0,053035

fλ2 - fλ1 = 0,0527

Only about 5% of radiation is emittedin the visible range. The remaining95% is in the infrared region in theform of heat.

Light bulb.

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Radiation of Real Surfaces

Objective: Develop a methodology for determining radiation heat exchange between real surfaces.

• Surface radiation properties

• The graybody

• Kirchhoff's law

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Absorptivity a, Reflectivity r, Transmissivity t

Gα

Gτ

GρG E

J

10.2 Fig.

rG

tG

Irradiation incident on a real surface can be absorbed, reflected and transmitted.

Remind: radiosity J (total radiation leavingthe surface) is a sum of emitted E and reflected rG radiation.

a = total absorptivity = fraction absorbedr = total reflectivity = fraction reflected t = total transmissivity = fraction transmitted

GGtrGaG =++

1=++ tra

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Similarly

1=++ λλλ tra

aλ = spectral absorptivityrλ = spectral reflectivitytλ = spectral transmissivity

Opaque material: 0== λtt

Simplification: 1=+ ra

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Emissivity (emisivita, poměrná zářivost)

Total emissivity ε(T):Ratio of emissive power of a surface to that of a blackbody at the same temperature:

( ) ( )TETET

b

)(=ε

Spectral emissivity λε :Ratio of the spectral emissive power of a surface to that of a blackbody at the same temperature:

( ) ( )( )TE

TETb ,

,,λλλε

λ

λλ =

λ

λE

10.3 Fig.

blackbodysurface real

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Kirchhoff's Law

It is much easier to determine emissivity ε than absorptivity a.By experiments. But how we can determine absorptivity?

Kirchhoff’s law says that under certain conditions:

( ) ( )TT αε =Total ( ) ( )TT ,, λαλε λλ =Spectral

Kirchhoff’s law is used to determine aλ(λ,T) from experimentaldata on ελ(λ,T)

Equality of emissivity and absorptivityQuite different physical quantitiesJust numerical equality

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Graybody ApproximationThe graybody concept is introduced to simplify the analysis of radiation exchange between bodies

Graybody: An ideal surface for which thespectral emissivity ελ is independent of λ

λ

λE

10.3 Fig.

blackbodysurface realgray body

λ

λE

10.3 Fig.

blackbody

body gray

approx. 0,75 Eb

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Thus: == )(),( TT ελελ constant independent of λ

It follows from Kirchhoff's Law that

( ) ( ) graybodya forTT αε =

NOTE:(1) Radiation properties ε, a and r are assigned single

values instead of a spectrum of values(2) Data on ε give r and a for opaque surface.

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Radiation Exchange Between Black Surfaces

Two black surfaces withareas 1S and 2S attemperatures 1T and 2T

Objective: Determine the netheat transfer 21Q −& between the two surfaces

21 TT >

1T2T

1S

2S

1E2E

12Q&

1

2

Important factors:• Configuration• Surface area• Surface temperature • Radiation properties (for gray body) • Surrounding surfaces • Space medium

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The View Factor(1) Definition and use:

• It is a geometric factor

• Also known as shape factor and configuration factor

The view factor is the fraction of radiation energyleaving surface S1 which is intercepted by S2

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1Q& = rate of radiation energy leaving surface 1, = S1E1

2Q& = rate of radiation energy leaving surface 2, = S2E2

21Q −& = net radiation energy exchanged between 1 and 221−F = fraction of radiation energy leaving 1 and

reaching 2

12−F = fraction of radiation energy leaving 2 and reaching 1

1T2T

1S

2S

1E2E

21Q −&

1

2

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For black surfaces:

The net energy exchanged between the surfaces 1 and 2:

21-2212-112-1 bb EFSEFSQ −=& (a)

Radiation that leaves the surface 1: 111 bESQ =&

1121 bESF −and is intercepted by the surface 2:

222 bESQ =&Radiation that leaves the surface 2:

2212 bESF −and is intercepted by the surface 1:

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If 21 TT = then 21 bb EE = and 12Q& = 0.

( ) ( )44212-11212-112-1 TTFSEEFSQ bb −=−= σ&

122211 −− = FSFS (b)

Combine (a) and (b) and use Stefan-Boltzmann law :4TEb σ=

Reciprocal rule (vztah recoprocity)

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(2) Rules: • Reciprocal rule can be generalized

ijjjii FSFS −− =

• Additive rule: Conservation of energy - see the figure.

( ) 3-12-132-1 FFF +=+

Multiply by 1S

3-112-113)21 FSFSFS +=+−(1

Use the reciprocal rule

1-331-221-3)+(232 )( FSFSFSS +=+

10.5 Fig.

1

23

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• Enclosure or summation rule: All energy leaving one surface must be received by some or all other surfaces

1 1131211 =++++ nFFFF K

niFn

jji ,,3,2,1 1=

1=K=∑

• Conclusion: iiF = 0 for a plane or convex surface and

iiF ≠ 0 for a concave surface

(3) Determination of view factors:

• Simple configurations: By physical reasoning:

n

4

32

1

6.10 Fig.

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21A 2A

1

112 =F

Apply the reciprocal rule

211122 −− = FSFS

21212112 /)/( SSFSSF == −−

• Other methods:• Surface integration method: Can involve tedious

double integrals• View factor algebra method: Known factors are used

in a superposition scheme together with the three view factor rules to construct factors for other configurations

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View factor for parallel rectangles

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View factor for perpendicular rectangleswith a common side