Heat Transfer Chapter 12

7/27/2019 Heat Transfer Chapter 12

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Chapter 12:

Fundamentals of ThermalRadiation

Yoav PelesDepartment of Mechanical, Aerospace and Nuclear Engineering

Rensselaer Polytechnic Institute


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Objectives

Classify electromagnetic radiation, and identify thermal radiation,

Understand the idealizedblackbody, and calculate the total andspectral blackbody emissive power,

Calculate the fraction of radiation emitted in a specifiedwavelength band using theblackbody radiation functions,

Understand the concept ofradiation intensity,

Develop a clear understanding of the properties emissivity,absorptivity, relflectivity, and transmissivity on spectral,directional, and total basis,

Apply Kirchhoffs law to determine the absorptivity of a surfacewhen its emissivity is known,

Model the atmospheric radiationby the use of an effective sky

temperature, and appreciate the importance of greenhouse effect.


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Introduction

Unlike conduction and convection, radiation does notrequire the presence of a material medium to take place.

Electromagnetic waves orelectromagnetic radiation represent the energy emitted by matter as a result of the

changes in the electronic configurations of the atoms ormolecules.

Electromagnetic waves are characterized by theirfrequency orwavelength

c the speed of propagation of a wave in that medium.

c

= (12-1)

c = c0/n ; c0 (vacuum) = 3 x 108 (m/s)

n =


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Thermal Radiation

Electromagnetic radiation covers a widerange of wavelengths.

Ofparticular interest in the study of heattransfer is the thermal radiation.

Temperature is a measure of the strength

of these activities at the microscopic level.

Thermal radiation is defined as thespectrum that extends from about 0.1 to

100 m. Radiation is a volumetric phenomenon.

However, frequently it is more convenient

to treat it as a surface phenomenon.


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Blackbody Radiation

A body at a thermodynamic (or absolute) temperatureabove zero emits radiation in all directions over a wide

range of wavelengths.

The amount of radiation energy emitted from a surface ata given wavelength depends on:

the material of the body and the condition of its surface,

the surface temperature.

A blackbody the maximum amount of radiation thatcan be emitted by a surface at a given temperature.

At a specified temperature and wavelength, no surfacecan emit more energy than a blackbody.


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A blackbody absorbs allincident radiation, regardless

of wavelength and direction.

A blackbody emits radiation energy uniformly in all

directionsper unit area normal to direction of

emission.

Radiation IntensityI(,T)


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The radiation energy emitted by a blackbody per unit

time and per unit surface area (StefanBoltzmann law)

= 5.67 x 10-8 W/m2K4.

Examples of approximate blackbody:

snow,

white paint,

( ) ( )4 2 W/mbE T T= (12-3)

();

blackbody!


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a large cavity with a small opening.

((diffuse) radiative properties

blackbody ())

---- blackbody


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( - - )

- - -


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The spectral blackbody emissive power (Planck 1901)

( )( )

( )

( )( )

2

2 8 4 2

1 0

4

2 0

2

,

4

0

1

52

W/m m

2 3.74177 10 W m m

/ 1.43878 10 m K

( ) (W/m )

,exp 1

bb

b

E

C hc

C hc k

E dT T

CE T

C T

= =

= =

= =

=

(12-4)

The variation of the spectral blackbody emissive power

with wavelength is plotted in Fig. 129.


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at any specified

temperature a maximum

exists,

at any wavelength, the

amount of emitted radiation

increases with increasingtemperature,

as temperature increases,

the curves shift to theshorter wavelength,

the radiation emitted by the

sun (5780 K) is in thevisible spectrum.

( ) ( )max power 2897.8 m KT = (12-5)

Wiens displacement law :


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The radiation energy emitted by a

blackbody per unit area over awavelength band from =0 to =1is determined from

( ) ( ) ( )1

1

2,0

0, W/mb bE T E T d

= (12-7)

( )( )

( )

( )0 0

4

0

, ,; 1 or 2

,

n n

n

b b

b

E T d E T df T n

TE T d

= = =

(12-8)

The values offare listed in Table 122.

A dimensionless quantityfcalled the blackbody radiation

function is defined:


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Table 12-2 Blackbody Radiation Functionsf( ) ( )

1

1

0

4,bE T df T

T

= (12-8)

( ) ( ) ( )1 2 2 1

f T f T f T =

(12-9)


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Radiation Intensity

The direction of radiation passing

through a point is best described

in spherical coordinates in termsof the zenith angle and theazimuth angle .

Radiation intensity is used to describe how the emitted

radiation varies with the zenith and azimuth angles (

).

A differentially small surface in space dAn, through which

this radiation passes, subtends a solid angle dwhen

viewed from a point on dA.

dAn


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The differential solid angle dsubtended by a differential

area dSon a sphere of radius rcan be expressed as

2(sr : steradisin a ) n

dSd d d

r = (12-11)

edQ


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Note: d r !

2sindSd d d

r =


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( ) 2W

,cos cos sin m sr

e ee

dQ dQI

dA d dA d d

=

Radiation Intensity ()the rate at which radiation energy is emitted in the(, ) directionper unit area normal to this direction

andper unit solid angle about this direction.

(12-13)

( )

( )

, cos ( )

, cos sin

ee

e

dQdE I d

dA

I d d

=

=

The radiation fluxis the emissive powerE:

(12-14)


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The emissive power from the surface into the hemisphere

surrounding it can be determined by

For a diffusely emittingsurface, the intensity of the

emitted radiation is independent of direction and thus

Ie = constant:

( ) ( )2 / 2 2

0 0, cos sin W m

hemisphere

e

E dE

I d d

= =

=

=

(12-15)

2 / 2

0 0cos sin

hemisphere

e e

dE

d

E

d II

= =

=

= =

(12-16)


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For a blackbody, which is a diffuse emitter, Eq. 12

16 can be expressed as

whereEb = T4 is the blackbody emissive power.Therefore, the intensity of the radiation emitted by

a blackbody at absolute temperature Tis

b bE I=(12-17)

( )

( )

42W m - sr bb

E TI T

= = (12-18)

Not a function of and


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Intensity of incident radiation

Ii(,) the rate at which radiation energy dG is incidentfrom the (,) directionper unit area of the

receiving surface normal to this direction andper

unit solid angle about this direction.


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The radiation flux incident on a surface from all

directions is called irradiation G

When the incident radiation is diffuse (Ii = constant):

( )

( )

2 / 2

0 0

2

, cos sin

Units: W m

ihemisphere

G dG I d d

= =

= =

(12-19)

iG I= (12-20)


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Radiosity (J) the rate at which radiation energy leaves

a unit area of a surface in all directions:

( ) ( )2 / 2

2

0 0, cos sin W me rJ I d d

+= =

= (12-21)


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For a surface that is both a diffuse emitterand a diffuse

reflector,Ie+rf(,) = , Ie+r:

( )

( )

2 / 2

0 0

2 / 2

0 0

2

, cos sin

= cos sin

=

Units: W m

e r

e r

e r

J I d d

I d d

I

+= =

+ = =

+

= (12-21)

2( W m )e rJ I += (12-22)


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Spectral Quantities the variation of radiation with wavelength.


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Thespectral intensity for emitted radiationI,e(,,)

for example, is simply the total radiation intensityI(,) per unit wavelength interval about .

Then thespectral emissive powerbecomes

( ), 2W

, ,cos m sr m

ee

dQI

dA d d

=

(12-23)

( )2 / 2

,0 0

, , cos sineE I d d

= =

= (12-24)


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The spectral intensity of radiation emitted by a

blackbody at a thermodynamic temperature Tat awavelength has been determined by Max Planck, and

is expressed as

Then the spectral blackbody emissive poweris ()

( )( )

( )2

20

5

0

2, W/m sr mexp 1

bhcI T

hc kT

=

(12-28)

( ) ( ), ,b bT I T =(12-29)

b bE I= (12-17)

Radiation Intensity


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2 = 400

1 = 550

A1 = 3 cm2

T1 = 600 K

A2 = 5 cm2


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Radiative Properties

Many materials encountered in practice, such as metals,wood, and bricks, are opaque () to thermalradiation, and radiation is considered to be a surface

phenomenon for such materials.

In these materials thermal radiation is emitted or

absorbed within the first few microns of the surface.

Some materials like glass and waterexhibit differentbehavior at different wavelengths:

Visible spectrum () semi-transparent,

Infrared spectrum () opaque.


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Emissivity () Emissivity of a surface the ratio of the radiation

emitted by the surface at a given temperature to theradiation emitted by a blackbody at the same temperature.

The emissivity of a surface is denoted by , and it variesbetween zero and one, 0 1.

The emissivity of real surfaces varies with: the temperature of the surface,

the wavelength, and

the direction of the emitted radiation.

Spectral directional emissivity the most elementalemissivity of a surface at a given temperature.

S t l di ti l i i it


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Spectral directional emissivity

The subscripts and are used to designate spectraland directionalquantities, respectively.

The total directional emissivity (intensities integrated

over all wavelengths)

The spectral hemispherical emissivity

( ) ( )( ),

, , , ,, , ,,

e

b

TTI T

=

(12-30)

( )( )

( )

, ,, ,

e

b

I TT

I T

= (12-31)

( )( )

( )

,,

,b

E TT

E T

= (12-32)


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The total hemispherical emissivity

SinceEb(T) = T4 the total hemispherical emissivity

can also be expressed as

To perform this integration, we need to know thevariation of spectral emissivity with wavelength at the

specified temperature.

( ) ( )( )b

E TT

E T = (12-33)

( )( )

( )

( ) ( )0

4

, ,b

b

T E T d E TT

E T T

= = (12-34)


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Gray and Diffuse Surfaces

Diffuse surface a surface which properties are independentofdirection.


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Gray surface surface properties are independentof wavelength.


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emissivity wavelength


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emissivity temperature


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Average emissivity ()

Absorptivity Reflectivity and Transmissivity


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Absorptivity, Reflectivity, and Transmissivity

When radiation strikes a surface, part of it: is absorbed (absorptivity, ),

is reflected (reflectivity, ),

and the remaining part, if any, is transmitted (transmissivity, ).


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Absorptivity ():

Reflectivity ():

Transmissivity ():

Absorbed radiation

Incident radiation

absG

G= = (12-37)

Reflected radiationIncident radiation

refGG

= = (12-38)

Transmitted radiation

Incident radiation

trG

G= = (12-39)

Th fi t l f th d i i th t th f


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The first law of thermodynamics requires that the sum ofthe absorbed, reflected, and transmitted radiation be equalto the incident radiation.

Dividing each term of this relation by G yields

For opaque surfaces, = 0, and thus

These definitions are fortotal hemisphericalproperties.

abs ref tr G G G G+ + = (12-40)

1 + + = (12-41)

1 + = (12-42)

Lik i i it th ti l b


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Like emissivity, these properties can also be

defined for a specific wavelength and/ordirection.

Spectral directional absorptivity

Spectral directional reflectivity

( )( )

( ),

,

,

, ,, , , ,

abs

i

I

I

= (12-43)

( )( )

( ),

,

,

, ,, ,

, ,

ref

i

I

I

= (12-43)

S t l h i h i l b ti it


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Spectral hemispherical absorptivity

Spectral hemispherical reflectivity

Spectral hemispherical transmissivity

( )( )

( ),absG

G

= (12-44)

( )

( )

( ),refG

G

= (12-44)

( )( )

( ),trG

G

= (12-44)

The average absorptivity reflectivity and transmissivity


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The average absorptivity, reflectivity, and transmissivity

of a surface can also be defined in terms of their spectral

counterparts as

The reflectivity () differs somewhat from the other

properties in that it is bidirectionalin nature.

0 0 0

0 0 0

, ,G d G d G d

G d G d G d

= = =

(12-46)

Direction of incident radiation ()Direction of reflection ()


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For simplicity, surfaces are assumed to reflect

in a perfectlyspecularordiffuse manner.

Kirchhoffs Law (1860)


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Kirchhoff s Law (1860) Consider a small body of surface areaAs, emissivity ,

and absorptivity at temperature Tcontained in a large

isothermal enclosure at the same temperature.

A large isothermal enclosure forms a blackbody cavity


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A large isothermal enclosure forms a blackbody cavityregardless of the radiative properties of the enclosuresurface.(enclosure small body radiation blackbodyradiation)

The body in the enclosure is too small to interfere withthe blackbody nature of the cavity.

Therefore, the radiation incident on anypart of the surface of the small body isequal to the radiation emitted by ablackbody at temperature T.

G =Eb(T) = T4.

The radiation absorbed by the small body per


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The radiation absorbed by the small body perunit of its surface area is

The radiation emitted by the small body is

Considering that the small body is in thermal

equilibrium with the enclosure, the net rate ofheat transfer to the body must be zero.

Thus, we conclude that

4

absG G T = =

4

emitE T=

4 4

s sT A T =

(12-47)

( ) ( )T T = =

The restrictive conditions inherent in the derivation of Eq.


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q

12-47 should be remembered:

the surface irradiation correspond to emission from a blackbody

surface temperature = temperature of the source of irradiation,

steady state.

The derivation above can also be repeated for radiation at

a specified wavelength :

.

The form of Kirchhoffs law that involves no restrictionsis thespectral directionalform

(12-48)(T T =

( ) ( ), ,T T =

!

Green House Effect


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Green House Effect

() 0.3 m < l < 3.0 m (Solar Radiation)

() 0.9

Atmospheric and Solar Radiation


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Atmospheric and Solar Radiation

The sun ( ):

is a nearly sphericalbody.

diameter of D 1.39 x 109 (m),

mass of m 2 x 1030 (kg),

mean distance ofL=1.5 x 1011

(m) from the earth,emits radiation energy continuously at a rate of

Esun 3.8 x 1026 (W),

about 1.7 x 1017 (W) of this energy strikes the earth,

the temperature of the outer region of the sun isabout 5800 K.

The solar energy reaching the earths atmosphere is


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called the total solar irradiance Gs, whose value is

The total solar irradiance (the solar constant) represents

the rate at which solar energy is incident on a surfacenormal to the suns rays at the outer edge of the atmos-phere when the earth is at its mean distance from the

sun.

(12-49)21373 W/msG =

The value of the total solar irradiance (Gs) can be used toestimate the effective surface temperature of the sun


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estimate the effective surface temperature of the sun

from the requirement that

(12-50)( ) ( )2 2 44 4s sunL G r T =

(~ 5800 K)

The solar radiation undergoes considerable attenuationas it passes through the atmosphere as a result ofabsorption

and scattering.

The several dips on the spectral


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p p

distribution of radiation on the

earths surface are due to

absorptionby various gases:

oxygen (O2) atabout =0.76 m,

ozone (O3)

below 0.3 m almost completely, in the range 0.30.4 m considerably, some in the visible range,

water vapor(H2O) and carbon dioxide (CO2) in the infrared

region,

dust particles and other pollutants in the atmosphere at

various wavelengths.

Th l hi th th f i


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The solar energy reaching the earths surface is

weakened considerably by the atmosphere and to about950 W/m2 on a clear day and much less on cloudy or

smoggy days.

Practically all of the solar radiation reaching the earths

surface falls in the wavelength band from 0.3 to 2.5 m. Another mechanism that attenuates solar radiation as it

passes through the atmosphere isscatteringorreflectionby air molecules and other particles such as dust, smog,

and water droplets suspended in the atmosphere.

The solar energy incident on a surface on earth is

id d i f d d d ff


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considered to consist ofdirectand diffuseparts.

Direct solar radiation GD:

the part of solar radiation that reaches the earths surface

without being scattered or absorbed by the atmosphere.

Diffuse solar radiation Gd:

the scattered radiation is assumed to reach the earths

surface uniformly from all directions.

Then the totalsolar energy incident on the unit area of a


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horizontal surface on the ground is:

(12-49)

( )

2cos W/msolar D dG G G= +

Fluorescent Lamp


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A fluorescent lamp is a gas-discharge lamp that useselectricity to excite mercury vaporin argon orneon gas,

resulting in aplasma that produces short-wave ultraviolet

light. This light then causes aphosphor tofluoresce producing visible light.

(~1/4 incandescent lamps) Unlike incandescent lamps,

fluorescent lamps always

require aballast ( )

to regulate the flow of power

through the lamp.

Heat Transfer Chapter 12

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