Radiative Properties • When radiation strikes a surface, a portion of it is

reflected, and the rest enters the surface. • Of the portion that enters the surface, some are

absorbed by the material, and the remaining radiation is transmitted through.

• The ratio of reflected energy to the incident energy is called reflectivity, ρ.

• Transmissivity (τ) is defined as the fraction of the incident energy that is transmitted through the object.

• Absorptivity (α) is defined as the fraction of the incident energy that is absorbed by the object.

• The three radiative properties all have values between zero and 1. • Furthermore, since the reflected, transmitted, and absorbed radiation must

add up to equal the incident energy, the following can be said about the three properties:

Emissivity

• A black body is an ideal emitter. • The energy emitted by any real surface is less than the

energy emitted by a black body at the same temperature. • At a defined temperature, a black body has the highest

monochromatic emissive power at all wavelengths. • The ratio of the emissive power of real body Eto the

blackbody emissive power bat the same temperature is the hemispherical emissivity of the surface.

bEE

The Emission Process

• For gases and semitransparent solids, emission is a volumetric phenomenon.

• In most solids and liquids the radiation emitted from interior molecules is strongly absorbed by adjoining molecules.

• Only the surface molecules can emit radiation.

Spherical Geometry

dA

Vectors in Spherical Geometry

Zenith Angle :

zimuthal Angle : (,r)

Hemispherical Black Surface Emission

4TEI bb

Black body Emissive Intensity

Real Surface emission

The radiation emitted by a real surface is spatially distributed:

),,( TII real Directional Emissive Intensity:

Directional Emissivity:

bITIT ),,(),,(

Directional Emissivity

Planck Radiation Law

• The primary law governing blackbody radiation is the Planck Radiation Law.

• This law governs the intensity of radiation emitted by unit surface area into a fixed direction (solid angle) from the blackbody as a function of wavelength for a fixed temperature.

• The Planck Law can be expressed through the following equation.

1

12, 5

2

kThc

e

hcTE

h = 6.625 X 10-27 erg-sec (Planck Constant)

K = 1.38 X 10-16 erg/K (Boltzmann Constant)

C = Speed of light in vacuum

Real Surface Monochromatic emission

The radiation emitted by a real surface is spatially distributed:

),,,(, TII real Directional MonochromaticEmissive Intensity:

Directional Monochromatic Emissivity:

,

),,(),,(bI

TIT

Monochromatic emissive power E

• All surfaces emit radiation in many wavelengths and some, including black bodies, over all wavelengths.

• The monochromatic emissive power is defined by:• dE = emissive power in the wave band in the infinitesimal

wave band between and d

dTEdE ,

The monochromatic emissive power of a blackbody is given by:

1

12, 5

2

kThc

e

hcTE

Shifting Peak Nature of Radiation

Wein’s Displacement Law:• At any given wavelength, the black body monochromatic

emissive power increases with temperature.• The wavelength max at which is a maximum decreases as

the temperature increases.• The wavelength at which the monochromatic emissive

power is a maximum is found by setting the derivative of previous Equation with respect to

de

hcd

dTdE kT

hc

1

12

,5

2

mKT 8.2897max

Wien law for three different stars

Stefan-Boltzmann Law

• The maximum emissive power at a given temperature is the black body emissive power (Eb).

• Integrating this over all wavelengths gives Eb.

05

2

0 1

12,

de

hcdTEkThc

444

42

15

2 TT

khc

hcTEb

The total (hemispherical) energy emitted by a body, regardless of the wavelengths, is given by:

4ATQemitted

• where ε is the emissivity of the body,• A is the surface area, • T is the temperature, and • σ is the Stefan-Boltzmann constant, equal to 5.67×10-8 W/m2K4. • Emissivity is a material property, ranging from 0 to 1, which

measures how much energy a surface can emit with respect to an ideal emitter (ε = 1) at the same temperature

The total (hemispherical emissive power is, then, given by

Here, can be interpreted as either the emissivity of a body,

which is wavelength independent, i.e., is constant, or as the

average emissivity of a surface at that temperature.

A surface whose properties are independent of the wavelength is known as a gray surface.

The emissive power of a real surface is given by

00

)( dEdEE b

Define total (hemispherical) emissivity, at a defined temperature

0

0

0

0

)(

dE

dE

dE

dE

b

b

b

Absorptivity , Reflectivity and Transmissivity

• Consider a semi-transparent sheet that receives incident radiant energy flux, also known as irradiation, G .

• Let dG represent the irradiation in the waveband to d.

• Part of it may be absorbed, part of it reflected at the surface, and the rest transmitted through the sheet.

• We define monochromatic properties,

Monochromatic Absorptivity :dGdG

Monochromatic reflectivity :dGdG

Monochromatic Transmissivity :dGdG

Total Absorptivity :GGd

0

Total reflectivity :GG

d

0

Total Transmissivity :GGd

0

Conservation of Irradiation

The total Irradiation = GGGG

GG

GG

GG 1

1

3..

Blackbody Radiation

• The characteristics of a blackbody are :• It is a perfect emitter.• At any prescribed temperature it has the highest

monochromatic emissive power at all wave lengths.• A blackbody absorbs all the incident energy and there fore

• It is non reflective body (• It is opaque (= 0). • It is a diffuse emitter

Radiative Heat Transfer Consider the heat transfer between two surfaces, as shown in Figure. What is the rate of heat transfer into Surface B? To find this, we will first look at the emission from A to B. Surface A emits radiation as described in

4, AAAemittedA TAQ

This radiation is emitted in all directions, and only a fraction of it will actually strike Surface B. This fraction is called the shape factor, F.

The amount of radiation striking Surface B is therefore:

4, AAABAinceidentB TAFQ

The only portion of the incident radiation contributing to heating Surface B is the absorbed portion, given by the absorptivity αB:

4, AAABABabsorbedB TAFQ

Above equation is the amount of radiation gained by Surface B from Surface A. To find the net heat transfer rate at B, we must now subtract the amount of radiation emitted by B:

4, BBBemittedB TAQ

The net radiative heat transfer (gain) rate at Surface B is

emittedBabsorbedBB QQQ ,,

44BBBAAABABB TATAFQ

Shape Factors • Shape factor, F, is a geometrical

factor which is determined by the shapes and relative locations of two surfaces.

• Figure illustrates this for a simple case of cylindrical source and planar surface.

• Both the cylinder and the plate are infinite in length.

• In this case, it is easy to see that the shape factor is reduced as the distance between the source and plane increases.

• The shape factor for this simple geometry is simply the cone angle (θ) divided by 2π

• Shape factors for other simple geometries can be calculated using basic theory of geometry.

• For more complicated geometries, the following two rules must be applied to find shape factors based on simple geometries.

• The first is the summation rule.

122211 FAFA

• This rule says that the shape factor from a surface (1) to another (2) can be expressed as a sum of the shape factors from (1) to (2a), and (1) to (2b).

• The second rule is the reciprocity rule, which relates the shape factors from (1) to (2) and that from (2) to (1) as follows:

ba FFF 212121

Thus, if the shape factor from (1) to (2) is known, then the shape factor from (2) to (1) can be found by:

121

221 F

AAF

If surface (2) totally encloses the surface 1:

121 F

Geometric Concepts in Radiation

• Solid Angle:

2rdAd n

sind d d

Emissive intensity SrW/mddS

dqIn

e2 ,

Monochromatic Emissive intensity

mSrW/mdddS

dqIn

e

2

, ,,

Total emissive power dS

ndS

cosndS dS

ne e

alldirections

dSE I ddS

2 2

,0 0 0

( , , ) cos sine eq E I d d d

Radiation Heat Transfer• e = emissive power• G = total irradiation• J = total radiosity

1

In general:

Opaque material:

1

= absorptivity = reflectivity = transmissivity

= emissivity

e

T1

T2

T3

Energy

e

Ideal EmitterSchematicT3> T2> T1

Radiation Heat Transfer

• Black Body– absorptivity = – emissivity = – ideal emissive power = eb

4Teb 1

f

f

4Tegray

bgray ee

• Gray Body– absorptivity < 1– emissivity < 1– emissive

power<1

e Gray Body

Black Body

Energy

Real Body

black

gray

ee

f

Schematic

Radiation Heat Transfer

1 j

ijF 1...... 111211 nj FFFF

jijiji FAFA Thermal Equilibrium

View Factor: Fij = fraction of radiation from surface i intercepted by surface j.

1 2

Radiation Heat Transfer

1be1J 2JRJ

11

11

A

22

21

ARFA 11

1

121

1FA

RFA 22

1

No net heat flux wall

Analog circuit

12Q11212 AqQ

Find:

2be

X

1 2

R

R

Radiation Heat Transfer

1be1J 2J

2be

RJ11

11

A

22

21

ARFA 11

1

121

1FA

RFA 22

1

RR FAFA 2211

11

1be1J 2J

2be

11

11

A

22

21

A

121

1FA

Radiation Heat Transfer

1be1J 2J

2be

11

11

A

22

21

ARR FAFA 2211

11

121

1FA

1be 1J 2J 2be

11

11

A

22

21

A

RR FAFA

FA

2211

121 111

1

RR FAFA

FAFA

2211

121121 111

Radiation Heat Transfer

1be 1J 2J 2be

11

11

A

22

21

A

121

1FA

RR FAFA

FAFA

2211

121121 111

22

2

12111

1 111

AFAA

1be 2be

22

2

12111

1121 111

1

AFAA

A

F

1be 2be

121

1FA

42

41112 TTAQ 12F

21112 bb eeAQ 12F

121tan FAceConduc

Radiation Heat Transfer

RR FAFA

FAFA

2211

121121 111

22

2

12111

1121 111

1

AFAA

A

F

Radiation Heat TransferConsider the radiation heat transfer between two infinite parallel plates.

1 2

netq ,12

2be1be1J 2J

22

21

A

11

11

A

121

1FA

12112 FF 21 AA

22

2

12111

1

21112 111

AFAA

eeAq bb

111

21

2112

bb eeq

1

Special Case

Radiation Heat TransferConsider the radiation heat transfer between a small object and infinite surroundings.

1

S

netSq ,1 bSe1be1J SJ

SS

S

A 1

11

11

A

SFA 11

1

11 SF 01 SAA

SS

bSbS

AFAA

eeAq

2

2

1111

1

111 111

bSbS eeq 111

10

1 SS

Special Case

44111 SS TTq