radiation lecture notes
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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
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