These notes are for the course MENG 2012- Heat Transfer and the reference

for the notes is the recommended text Fundamentals of Heat and Mass Transfer-

F.P. Incropera and D.P. DeWitt

RADIATION

Heat transfer by conduction and convection requires the presence of a temperature

gradient in some form of matter. Heat transfer by thermal radiation requires no matter.

We associate thermal radiation with the rate at which energy is emitted by matter as a

result of its finite temperature. All forms of matter emit radiation.

Radiation cooling of a (a)Volumetric Surface phenomenon

heated solid The emission process

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

and liquids, radiation emitted from interior molecules is strongly absorbed by adjoining

molecules. Therefore emission from a solid or a liquid into an adjoining gas or a vacuum

is viewed as a surface phenomenon

This figure shows the complete electromagnetic spectrum.

(1) The short wavelength :- Gamma rays, X rays, and Ultraviolet radiation are of

interest to the high energy physicist and nuclear engineer.

(2) The long wavelength microwaves and radiowave are of interest to the electrical

engineer.

(3) The intermediate portion of the spectrum, which extends from approximately 0.1

to 100 μm includes portion of the UV and all the visible and infrared IR is termed

thermal radiation and is pertinent to heat transfer.

The intensity I, of radiation emitted by dA1 is defined as the rate dq at which radiant

energy is emitted by dA1 in a particular direction per unit solid angle and per unit area of

the projection of dA1 perpendicular to the direction r. The total emissive power is the

total rate at which a surface, at absolute temperature emits radiant energy per unit area of

that surface.

Fig. 12.7 – from the figure the solid angle subtended by area dAn is )(2r

dAnd . The

area dAn = r(sin θ) (d ) rd θ = r2

sin θ dθ d. The area used to define intensity is the

component of dA1 perpendicular to the direction of the radiation. From the next figure

this area is dA1 Cos θ

The spectral instensity is

I = dq/dA1 Cos θ. dω. d where dq/d = dq is the rate at which radiation of

wavelength leaves dA1 and passes through dAn. Rearranging equation:-

dq = I (dA1 Cos θ.) dω

I,e is defined as the rate at which radiant energy is emitted at the wavelength in the

(,) direction, per unit area of the emitting surface normal to this direction, per unit

solid angle about this direction, and per unit wavelength interval d about .

Radiosity accounts for all of the radiant energy leaving a surface. Radiosity is represented by the letter J. If the surface is both a diffuse reflector and

diffuse emitter, I is independent of θ and , i.e:- J = π I.

Blackbody Radiation

The black body is an ideal surface having the following properties.

(1) A black body absorbs all incident radiation, regardless of wave length and

direction.

(2) For a prescribed temperature and wavelength, no surface can emit more energy

than a blackbody.

(3) The blackbody is a diffuse emitter.

The black body serves as a standard against which the radiative properties of actual

surfaces may be compared.

Experiment shows that the total emissive power is a complicated function of temperature,

type of material, and surface condition.

The total rate at which a black surface emits radiant energy per unit area at the absolute

temperature, T, was found experimentally by Stefan and later shown theoretically by

Boltzmann and is given by the Stefan-Boltzmann Law as Eb = σT4 where Stefan

Boltzman constant σ is 5.670 x 10-8

W/m2. K

4

The blackbody is an ideal emitter, therefore it is convenient to choose the black body as a

reference. For real surfaces there is a radiative property known as emissivity (ε). It is

defined as the ratio of the radiation emitted by the surface to the radiation emitted by a

blackbody at the same temperature.

Surface absorption, reflection and transmission

Spectral irradiation, Gλ, is defined as the rate at which radiation of wavelength λ is

incident on a surface per unit area of the surface and per unit wavelength interval dλ

about λ. Total irradiation G (W/m2) encompasses all spectral contributions. In the most

general situation when irradiation interacts with a semitransparent medium, portions of

this radiation are reflected, absorbed and transmitted.

For a radiation balance G λ = G λ,ref + G λ,abs + G λ,tr

The absorptivity () is a property that determines the fraction of the irradiation absorbed

by a surface.

The reflectivity () is a property that determines the fraction of the incident radiation

reflected by a surface.

The Gray Surface

A gray surface may be defined as one for which and are independent of over

the spectral regions of the irradiation and the surface emission. It is a surface for which

= ie. ( = )

View Factor

The view factor Fij is defined as the fraction of the radiation leaving surface i which is

intercepted by surface j.

dqi - j = Ii Cos i dAi dωj-i ie:- The heat transfer from i to j is the

Intensity (Ii) The projected Area The solid angle at j

of surface I X Ai as seen from j X looking from i

to the direction r

dj-i = projected area of j in the direction to R (from )(2r

dAnd )

R2

= Cos j dAj

R2

dqi-j = Ii Cos i Cosj dAi dAj

R2

but Ji = Ii

dqi-j = Ji Cos i Cosj dAi dAj

R2

q i-j = Ji ∫Ai ∫Aj Cos i Cosj dAi dAj

R2

where J is uniform over the surface Ai. From the definition of view factor.

Fij = q i-j that is Radiation leaving Ai transferred to Aj

AiJi total radiation leaving Area Ai

Fij = (1/Ai) ∫Ai ∫Aj Cos i Cosj dAi dAj

R2

Similarly the view factor F j i will be

Fji = (1/Ai) ∫Ai ∫Aj Cos i Cosj dAi dAj

R2

Ai Fij = Aj Fji this is the reciprocity relation

Another important view factor relation pertains to the surface of an enclosure Fig. 13.2.

From the definition of the view factor, the summation rule

N

jijF

1

1, may be applied to

each of the N surfaces in the enclosure.

An example is the two surface enclosure shown below.

Since all the radiation leaving the surface (1) must reach (2) then F12 = 1. The same is

not true for surface 2, since (2) sees itself. Reciprocity relation

A2 F21 = A1 F12

2

1

2

112

2

121 1)(

A

A

A

AF

A

AF

summation: F11 + F12 = 1 but F11 = 0 F12 = 1

F21 + F22 = 1

F22 = 1 – F21 = 1 – ( )2

1

A

A

The summation rule may be applied to each of the N surfaces in the enclosure. This rule

follows from the conservation requirement that all radiation leaving surface i must be

intercepted by the enclosure surface.

The term Fii represents the fraction of the radiation that leaves surface i and is directly

intercepted by i.

i.e. the surfaces sees itself and Fii is not zero.

Figures 13.4, 13.5, 13.6 shows the view factor solutions for more complicate geometries.

Blackbody Radiation Exchange

Surfaces that approximate as blackbodies show no reflection. Energy only leaves as a

result of emission, and all incident radiation is absorbed.

qi-j = (Ai Ji) Fij ie the radiation transfer between surfaces i j is the total

radiation emission of surface i which is (Ai Ji) x View factor. For blackbodies radiosity

equal emissive power (Ji = Ei)

then

q i - j = Ai Fij Ei

Similarly q j - i = Aj Fji Ej

The net radiative exchange between the two surfaces is then

qij = qi-j – qj-i which gives

qij = Ai Fij (Ti4 – Tj

4) using Stefan Boltzman law

qi =

N

jjiiji TTFA

1

44

Radiation Exchange Between Diffuse,

Gray Surfaces in an Enclosure

Ref. Fig. 13.9. The term qi is the net rate at which radiation leaves surface i. Also it is

the rate at which energy would have to be transferred to the surface by other means to

maintain it at a constant temperature.

It is the difference between the surface radiosity and irradiation.

i.e. qi = Ai (Ji – Gi) Radiative balance , Fig.13.9 (b)

But radiosity, Ji is the sum of emission Ei, and reflection i Gi which is the net radiative

transfer from the surface, i. That is Ji = Ei + i Gi . The net radiative transfer can also be

represented by the difference between surface emissive power and the absorbed

irradiation.

i.e. qi = Ai (Ei – αi Gi) Radiative balance , Fig. 13.9 (c)

but emissivity ε = bE

E and ρi = 1 – αi = 1 – εi

ie the reflected radiation is the difference between the total irradiation and what is

absorbed. For gray surfaces α = ε

Ji = εi Ebi + (1 – εi)Gi which gives Gi =i

ibii EJ

1

i

biiiiii

EJJAq

1 and

ii

ibii

A

JEq

/)1(

From the definition of view factor, the total rate at which radiation reaches surface, i,

from all surfaces including i, is

N

jjjjiii JAFGA

1

or from the reciprocity relation

N

j

jjiiii JFAGA1

n

jjijiii JFJAq

1

( )(11

n

j

jij

n

j

iijii JFJFAq

)(

1

n

jjiijii JJFAq

N

j iji

ji

iii

ibi

FA

JJ

A

JE

11)(/)1(

This expression represents a radiation balance for the radiosity node associated with

surface i. The rate of radiation transfer (current flow) to i through its surface resistance

must equal the rate of radiation transfer (current flow) from i to all other surfaces through

the corresponding geometrical resistances.

The Two Surface Enclosure

For such a system the net rate of radiation transfer from surface 1, q1, must equal the rate

of radiation transfer to surface 2, - q2. i.e q1 = -q2 = q1-2

From the network representation we see that the heat transfer from surface 1, q1 is the

difference between the ideal radiosity 1. bBodyBlackBodyBlack EieEJNote and the

actual radiosity J1 , divided by the surface resistance ie.

11

11A

Similarly for the radiation absorbed by surface 2, q2 .

The heat transfer across the space between surface 1 and 2 is given by the view factor

F12 x A1 x J1 = q12 . Hence using Stefan Boltzmann’s Law the net radiation exchange

between the surfaces may be expressed as

q12 = q1 = -q2 =

22

2

12111

1

42

41

111

AFAA

TT

Note: This result may be used for any two diffuse, gray surfaces that form an enclosure

e.g. Long (infinite) Concentric cylinders Table 13.3

2

1

2

1

2

1

2

2

r

r

r

r

A

A

also F11+ F12 =1, but F11 = 0 F12 = 1

2

1

2

2

121

1

1

4

2

4

1

22

2

12111

1

4

2

4

112

111111

A

A

F

ATT

AFAA

TTq

Substitute for 2

1

A

Aand F12

2

1

2

2

1

1

4

2

4

1

2

1

2

2

1

1

4

2

4

112

11111

1

r

r

ATT

r

r

ATTq

Important cases are shown in Table 13.3

Radiation Shields

Radiation shields constructed from low emissivity (high reflectivity) materials can be

used to reduce the net radiation transfer between two surfaces. The emissivity associated

with one side of the shield may differ from the other side and the radiosities will always

differ.

View Factor (A1 = A2)

F11 + F13 = 1 but F11 = 0 F13 = 1 and F33 + F32 = 1 but F33 = 0 F32 = 1

Hence

2,3

2,3

1,3

1,3

2

42

411

121111

)(

1

TTAq

The resistance becomes larger with smaller emissivity. Also q12 = q13 = q32

This can be extended to problems involving multiple radiation shields. In a special case

for which all the emissivities are equal with N shields.

012121

1q

Nq N

where 012q is the radiation transfer rate with no shields (i.e. N = 0)

Reradiating Surface

An ideal reradiating surface has zero net radiation transfer (qi = 0) since

qi = Ai (Ji – Gi) = 0 Gi = Ji = Ebi = Ti4

Therefore if the radiosity of a reradiating surface is known its temperature is readily

determined and is independent of the emissivity. With qR = 0 the net radiation transfer

from surface 1 must equal the net radiation transfer to surface 2. The network is analyzed

as a simple series parallel arrangement as in the simple diagram.

5

432

111

1R

RRR

RRTotal

Therefore, from the diagram

q1 = -q2 =

22

2

1

2211

121

11

1

21

1

11

11

A

FAFAFA

A

EE

RR

bb

Knowing J1 and J2 you can find JR from 011

22

2

11

1

R

R

R

R

FA

JJ

FA

JJand the temperature

from RR JT 4

R1

R3

R2

R4

R5