15. Radiation Introduction

7/28/2019 15. Radiation Introduction

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15. RADIATION HEAT TRANSFER

Heat conduction and convection - always a fluid which

transfers the heat (gas, liquid, solid) – motion of atoms ormolecules

Heat conduction and convection is not possible in a vacuum

In most practical applications all three modes occurconcurrently at varying degrees



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A hot object in a vacuum

chamber looses heat by

radiation only

Unlike conduction and convection, heat

transfer by radiation can occur between two

bodies, even when they are separated by

a medium colder than both of them

c o n v

e c t i o

n

r a d i a t i o n

What will be a final equilibrium temperature of the body

surface?

Can you write an energy balance equation between the body andsurrounding air and the hot source (fire)?



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Theoretical foundation of radiation was establishedby Maxwell

Electromagnetic wave motion or electromagnetic radiation

Electromagnetic waves travel at the speed of light c in a

vacuum

Electromagnetic waves are characterized by their frequency f

or wavelength λ: λ=c/f

c=co /n co light speed in a vacuumn 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 energy

In 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 to

heat transfer –

Thermal radiationλ from 0,1μm to 100 μm

As a result of energy transition

in molecules, atoms andelectrons.

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 vary

with wavelength and direction

• Monochromatic or s pectral : 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 (zář ivost):

Radiation emitted from a surface

• Spectral emissive power λ E :

λ E

per unit area per unit wavelength,

= rate of emitted radiation

mW/m2μ

• Total emissive power E :

, E = Integration of λ E over all values of λ2

W/m :

( ) ( )∫∞

=

0

, λλ d T E T E

∫∞

=0

λλd E E

λ

λ

10.1Fig.



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• Irradiation: Radiation energy incident on a surface

• Spectral irradiation λG :

λG per unit area per unit wavelength,

= rate of radiation energy incident upon a surfacemμW/m

2−

• Total irradiationG :

G = integration of λG over all values of λ :

( ) ( )∫∞

=

0

, λλ d T G T G

• Radiosity: The sum of emitted and reflected radiation

• Spectral radiosity λ J :

λ J = rate of radiation leaving a surface per unit area per

unit wavelength, mμW/m2−



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In the above definitions, summation in all directions is

implied although the term hemispherical is not used

• Total radiosity J :

( ) ( )∫∞

=

0

, λλ d T J T J

J = integration of λ J 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

λb E = spectral emissive power of a blackbody:

( )1)/exp(

,2

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−

−

T C

C T E b

λ

λλλ C 1 and C2 are constants

Blackbody Radiation

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



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Planck's Law

Blackbody Radiation

2879,6 T λmax =

Maximum emitted energy at

specific temperatures given by

Wien law:

Note - by qualitative judgment -

energy emitted in visible range

for 2000 K – tungsten wire in

a light bulb.

Thermal radiation 0,1 to 100 μm



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Stefan-Boltzmann Law

Based on:

• Experimental data by Stefan (1879)

• Theoretical derivation by Boltzmann (1884)4T E b σ =

b E = total blackbody emissive power (all wavelengthsand all directions), [W/m2]

428-K W/m105.67 − is the Stefan-Boltzmann

constant

It can also be arrived at using Planck's law

Stefan-Boltzmann law



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( ) ( )

4

02

51

0b λb

T σ

d λ1 )T / λC ( exp

λC

dx λ ,T E T E

=

∫ =−

=

=∫=

∞ −

∞

• Stefan-Boltzmann law gives the total radiation emitted from

a 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 E b,0- λ

∫=−

λ

0

b λ λb,0 T)d λ( E (T) E ,λ



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Define a dimensionless quantity f λ(T):

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λ

0 b, λ

λσ T

(T)d λ E (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 ⇒ f λ1 = 0,000321

λ2T=0, 76.2500=1900 ⇒ f λ2 = 0,053035

f λ2 - f λ1 = 0,0527

Only about 5% of radiation is emitted

in the visible range. The remaining

95% 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

G

G

G G E

J

10.2Fig.

rG

tG

Irradiation incident on a real surface can

be absorbed, reflected and transmitted.

Remind: radiosity J (total radiation leaving

the surface) is a sum of

emitted E and reflected rG radiation.

a = total absorptivity = fraction absorbed

r = total reflectivity = fraction reflected

t = total transmissivity = fraction transmitted

G G t rG aG =++

1=++ t r a



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Similarly

1=++ λ λ λ t r a

a λ

= spectral absorptivity

r λ = spectral reflectivity

t λ

= spectral transmissivity

Opaque material: 0== λ t t

Simplification: 1=+ r a



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Emissivity (emisivita, poměrná zářivost)

Total emissivity ε (T):

Ratio of emissive power of a

surface to that of a blackbodyat the same temperature:

( )

( )T E

T E T

b

)(=ε

Spectral emissivity λ :

Ratio of the spectral emissive power of a surface to that

of a blackbody at the same temperature:

( ))

( )T E

T E T

b,

,,

λ

λ λ ε

λ

λ λ =

λ

λ E

10.3Fig.

blackbody

surfacereal



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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:

( ) ( )T T α ε =Total

( ) ( )T T ,, λλ=Spectral

Kirchhoff’s law is used to determine a λ( λ ,T) from experimental

data on ε λ( λ ,T)

Equality of emissivity and absorptivity

Quite different physical quantities

Just numerical equality



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Graybody Approximation

The graybody concept is introduced to simplify theanalysis of radiation exchange between bodies

Graybody: An ideal surface for which the

spectral emissivity ελ is independent of λ

λ

λ E

10.3Fig.

blackbody

surfacereal

gray body

λ

λ E

10.3Fig.

blackbody

bodygray

approx. 0,75 E b



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Thus: =)(),( T T ελ constant independent of λ

It follows from Kirchhoff's Law that

( ) ( ) graybodya for T T =

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 with

areas 1 S and 2 S at

temperatures1

T and2

T

Objective: Determine the net

heat transfer 21Q −& between the two surfaces

21 T T >

1T

2T

1 S

2 S

1 E 2 E

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 energy

leaving surface S 1 which is intercepted by S 2



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1Q& = rate of radiation energy leaving surface 1, = S 1 E 1

2Q& = rate of radiation energy leaving surface 2, = S 2 E 2

21Q −& = net radiation energy exchanged between 1 and 2

21−F = fraction of radiation energy leaving 1 andreaching 2

12−F = fraction of radiation energy leaving 2 andreaching 1

1T

2T

1 S

2 S

1 E

2 E

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 E F S E F S Q −=& (a)

Radiation that leaves the surface 1: 111 b E S Q =&

1121 b E S F −and is intercepted by the surface 2:

222 b E S Q=&

Radiation that leaves the surface 2:

2212 b E S F −and is intercepted by the surface 1:



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If 21 T T = then 21 bb E E = and 12Q&

= 0.

( ) 44212-11212-112-1 T T F S E E F S Q bb −=−= σ &

122211 −− = F S S (b)

Combine (a) and (b) and use Stefan-Boltzmann law

:

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T E b σ =

Reciprocal rule (vztah recoprocity)



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(2) Rules:

• Reciprocal rule can be generalizedi j j j i i F S F S −− =

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

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

Multiply by 1 S

3-112-113)21 F S F S F S +=+−(1

Use the reciprocal rule

1-331-221-3)+(232 )( F S F S F S S +=+

10.5Fig.

1

2

3

12

3

i

n

(i=1 to n, j=1 to n)



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• Enclosure or summation rule: All energy leaving one

surface must be received by some or all other surfaces

11131211 =nF F F K

ni F n

j

j i ,,3,2,1 1=

1=

K

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

ii F ≠ 0 for a concave surface

(3) Determination of view factors:

• Simple configurations: By physical reasoning:

n

4

3

2

1

6.10Fig.



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2

1 A 2 A

1

112 =F

Apply the reciprocal rule

211122 −− = F S F S

21212112 /)/( S S F S S F == −−

• Other methods:

• Surface integration method: Can involve tediousdouble integrals

• View factor algebra method: Known factors are used

in a superposition scheme together with the threeview 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 rectangles

with a common side



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Examples of the View factor algebra method:

Determine F1-3 and F3-3

(i) Figure (a): F 1-2 is given in Jícha: P ř enos tepla a látky P ř íloha 2-3a, (př ípad 3 )

Apply summation rule to surface 1

1131211 =F F F

But 11 = 0

∴ 1213 1 F F −

1

2

3

Figure (a)

To determine F 33

apply summation rule to surface 3:

1333231 =F F F



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Symmetry, 3231 F

Thus, 3133 21F −

Reciprocal rule:1-333-11

F S F S =

∴ 3-1313-3 21 F S S F )(−=



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( ) 44212-11212-112-1 T T F S E E F S Q bb −=−= σ &

Just to remind the equation for radiation heat exchangebetween two black objects with a view factor F 1-2:

15. Radiation Introduction

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