NASA 5P-164

THERMAL
RADIATION
HEAT
_O TRANSFER
cOo
O
v

NASA SP-164

THERMAL
RADIATION
HEAT
TRANSFER
Volume I

The Blackbody, Electromagnetic Theory,

and Material Properties

Robert Siegel and John R. Howell

Lewis Research Center

Cleveland, Ohio

Scientific and Technical In[ormation Division

OFFICE OF TECHNOLOGY UTILIZATION 1968
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Washington, D.C.
 PREFACE

Several years ago it was realized that thermal radiation was becoming
of increasing importance in aerospace research and design. This im-
portance arose from several areas: high temperatures associated with
increased engine efficiencies, high-velocity flight which is accompanied
by elevated temperatures from frictional heating, and the operation of
devices beyond the Earth's atmosphere where convection vanishes and
radiation becomes the only external mode of heat transfer. As a result,
a course in thermal radiation was initiated at the NASA Lewis Research
Center as part of an internal advanced study program.
The course was divided into three main sections. The first dealt with
the radiation properties of opaque materials including a discussion of
the blackbody, electromagnetic theory, and measured properties. The
second discussed radiation exchange in enclosures both with and with-
out convection and conduction. The third section treated radiation in
partially transmitting materials-chiefly gases.
When the course was originated, there was not available any single
radiation textbook that covered the desired span of material. As a result
the authors began writing a set of notes; the present publication is an
outgrowth of the notes dealing with the first of the three main sections.
During the past few years, a few radiation textbooks have appeared in
the literature; hence, the need for a single reference has been partially
satisfied. The objectives here are more extensive than the content of a
standard textbook intended for a one-semester course. Many parts of
the present discussion have been made quite detailed so that they will
serve as a source of reference for some of the more subtle points in
radiation theory. The detailed treatment has resulted in some of the sec-
tions being rather long, but the intent was to be thorough rather than to
try to conserve space. The sections have been subdivided so that specific
portions can be located for easy reference.
This volume is divided into five chapters. The introduction discusses
the conditions where thermal radiation is of importance and indicates
some of the inherent differences and complexities of radiation problems
as compared with convection and conduction.

Chapter 2 deals with the blackbody, which is defined as a perfect

absorber. It is important to understand the behavior of a blackbody before
considering real materials, as the blackbody provides an ideal perform-
ance with which real material performance can be compared. First the
blackbody is discussed qualitatively with its properties being deduced

Ul
 THERMAL RADIATION HEAT TRANSFER

from the original definition of a perfect absorber. A quantitative elabora-

tion, including a numerical tabulation, then provides the blackbody
emission as a function of wavelength and temperature.
The third chapter is completely devoted to the definitions of emis-
sivity, absorptivity, and reflectivity. These properties are used to com-
pare the radiative performance of real materials with the ideal (blackbody)
behavior. A functional notation has been introduced that includes prime
superscripts to denote directional quantities and by which ambiguities
in the various hemispherical and directional quantities are avoided. An
extensive examination of the property definitions is made in order to
demonstrate when it is valid to use various reciprocity relations and
equalities, such as Kirchhoff's laws relating emissivity and absorptivity.
The restrictions on these relations are summarized in tables for con-
venient reference.
The use of classical electromagnetic theory for the prediction of
radiative properties is the subject of chapter 4. The electromagnetic
theory discussed deals with ideal surfaces and hence does not account
for the many factors (e.g., contamination and roughness) that influence
the behavior of real surfaces. In spite of this shortcoming, the theory
does provide a valuable basis for many observed trends and serves to
relate optical and electrical properties to radiative properties.
The final chapter illustrates the radiative performance of real materials
by showing a number of examples of property variations with wavelength
and temperature.
Each chapter contains numerical examples to acquaint the reader with
the use of the analytical relations. It is hoped that these examples will
help bridge the gap between theory and practical application.

iv
 CONTENTS
CHAPTER PAGE

1 INTRODUCTION ....................................................... 1
1.1 IMPORTANCE OF THERMAL RADIATION ....................... 1
1.2 SYMBOLS ..................................................................... 3
1.3 COMPLEXITIES INHERENT IN RADIATION PROBLEMS... 3
1.4 WAVE AGAINST QUANTUM MODEL .............................. 5
1.5 ELECTROMAGNETIC SPECTRUM .................................... 6

RADIATION FROM A BLACKBODY ........................ 9

2.1 SYMBOLS ..................................................................... 9
2.2 DEFINITION OF A BLACKBODY .................................... 11
2.3 PROPERTIES OF A BLACKBODY .................................... 11
2.3.1 Perfect Emitter ........................................................... 11
2.3.2 Radiation Isotropy in a Black Enclosure ........................... 12
2.3.3 Perfect Emitter in Each Direction ................................... 13
2.3.4 Perfect Emitter at Every Wavelength .............................. 13
2.3.5 Total Radiation a Function Only of Temperature ............... 13
2.4 EMISSIVE CHARACTERISTICS OF A BLACKBODY ............ 15
2.4.1 Definition of Blackbody Radiation Intensity ....................... 15
2.4.2 Angular Independence of Intensity ................................. 16
2.4.3 Blackbody Emissive Power-Definition and Cosine Law
Dependence ............................................................ 18
2.4.4 Hemispherical Spectral Emissive Power of a Blackbody ...... 19
2.4.5 Spectral Emissive Power Through a Finite Solid Angle ......... 20
2.4.6 Spectral Distribution of Emissive Power ........................... 20
2.4.7 Approximations for Spectral Distribution .......................... 25
2.4.7.1 Wien's formula ...................................................... 26
2.4.7.2 Rayleigh-Jeans formula ............................................ 26
2.4.8 Wien's Displacement Law ............................................. 26
2.4.9 Total Intensity and Emissive Power ................................. 27
2.4.10 Behavior of Maximum Intensity With Temperature ............ 29
2.4.11 Blackbody Radiation in a Wavelength Interval .................. 29
2.4.12 Blackbody Emission in a Medium Other Than a Vacuum ...... 35
2.5 EXPERIMENTAL PRODUCTION OF A BLACKBODY ......... 36
2.6 SUMMARY OF BLACKBODY PROPERTIES ....................... 37
2.7 HISTORICAL DEVELOPMENT ......................................... 43
REFERENCES ........................................................................ 45

3 DEFINITIONS OF PROPERTIES FOR NON-

BLACK SURFACES .................................................. 47
3.1 INTRODUCTION ............................................................ 47
3.1.1 Nomenclature ............................................................ 52
3.1.2 Notation ..................................................................... 53
 THERMAL RADIATION HEAT TRANSFER

PAGE
CHAPTER
3.2 SYMBOLS ..................................................................... 54
3.3 EMISSIVITY .................................................................. 55
3.3.1 Directional Spectral Emissivity _'_ ( _, fl, O, TA ) .................... 55
3.3.2 Averaged Emissivities ................................................... 57
3.3.2.1 Directional total emissivity e'(fl, 0, TA) ........................ 57
3.3.2.2 Hemispherical spectral emissivity e_()t, TA) .................. 59
3.3.2.3 Hemispherical total emissivity ¢(TA) ........................... 59
3.4 ABSORPTIVITY .............................................................. 64
3.4.1 Directional Spectral Absorptivity a'x(X, fl, O, TA).................. 64
3.4.2 Kirchhoff's "Law ........... ................................................ 65
3.4.3 Directional Total Absorptivity a'(g, 0, TA) ........................ 66
3.4.4 Kirchhoff's Law for Directional Total Properties .................. 67
3.4.5 Hemispherical Spectral Absorptivity ct_(),, TA).................... 67
3.4.6 Hemispherical Total Absorptivity a(TA) ........................... 68
3.4.7 Summary of Kirchhoff's Law Relations ........................... 71
3.5 REFLECTIVITY ............................................................... 72
3.5.1 Spectral Reflectivities ................................................... 72
3.5.1.1 Bidirectional spectral reflectivity p_(_,, fir, 0r, 8, 0) ......... 72
3.5.1.2 Reciprocity for bidirectional spectral reflectivity ........... 73
3.5.1.3 Directional spectral reflectivities ................................ 74
3.5.1.4 Reciprocity for directional spectral reflectivity ............ 75
3.5.1.5 Hemispherical spectral reflectivity px(),) ..................... 76
3.5.1.6 Limiting cases for spectral surfaces ........................... 77
3.5.1.6.1 Diffusely reflecting surfaces ................................ 77
3.5.1.6.2 Specularly reflecting surfaces .............................. 78
3.5.2 Total Reflectivities ...................................................... 80
3.5.2.1 Bidirectional total reflectivity p"([Jr, Or, f3, O) ................. 80
3.5.2.2 Reciprocity ............................................................ 81
3.5.2.3 Directional total reflectivity p'. ................................ 81
3.5.2.4 Reciprocity ............................................................ 82
3.5.2.5 Hemispherical total reflectivity p ............................... 82
3.5.3 Summary of Restrictions on Reciprocity Relations Between
Reflectivities ............................................................ 83
3.6 RELATIONS AMONG REFLECTIVITY, ABSORPTIVITY,
AND EMISSIVITY ........................................................ 84
3.7 CONCLUDING REMARKS ................................................ 88
REFERENCE ......................................................................... 88

4 PREDICTION OF RADIATIVE PROPERTIES BY

CLASSICAL ELECTROMAGNETIC THEORY... 89
4.1 INTRODUCTION ............................................................ 89
4.2 SYMBOLS ..................................................................... 90
4.3 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETIC
THEORY ..................................................................... 91
4.4 RADIATIVE WAVE PROPAGATION ................................. 92
4.4.1 Propagation in Perfect Dielectric Media .......................... 93
4.4.2 Propagation in Isotropic Media of Finite Conductivity ......... 98
4.4.3 Energy of an Electromagnetic Wave ................................ 100

vi
 CONTENTS

CHAPTER PAGE

4.5 LAWS OF REFLECTION AND REFRACTION .................... 101

4.5.1 Incidence and Reflection of a Wave From Dielectric or Trans-
parent Media (K Negligible Compared to, n) .................... 107
4.5.2 Incidence on an Absorbing Medium ................................ 109
4.6 APPLICATION OF ELECTROMAGNETIC THEORY RELA-
TIONS TO RADIATIVE PROPERTY PREDICTIONS ........ 110
4.6.1 Radiative Properties of Dielectrics (K---_0) ........................ 111
4.6.1.1 Reflectivity ............................................................ 111
4.6.1.2 Emissivity ............................................................ 113
4.6.2 Radiative Properties of the Metals ................................. 116
4.6.2.l Reflectivity and emissivity relations using optical
constants ........................................................... 116
4.6.2.2 Relation between emissive and electrical properties ...... 122
4.6.3 Summary of Prediction Equations ................................... 128
4.7 EXTENSIONS OF THE THEORY OF RADIATIVE PROP-
ERTIES ....................................................................... 130
REFERENCES ........................................................................ 130

5 RADIATIVE PROPERTIES OF REAL MATE.

RIALS ........................................................................... 133
5.1 INTRODUCTION ............................................................ 133
5.2 SYMBOLS ..................................................................... 133
5.3 RADIATIVE PROPERTIES OF METALS ........................... 134
5.3.1 Directional Variations ................................................... 135
5.3.2 Effect of Wavelength ................................................... 136
5.3.3 Effect of Surface Temperature ....................................... 137
5.3.4 Effect of Surface Roughness .......................................... 138
5.3.5 Effect of Surface Impurities .......................................... 141
5.4 RADIATIVE PROPERTIES OF OPAQUE NONMETALS ...... 146
5.4.1 Spectral Measurements ................................................ 148
5.4.2 Variation of Total Properties With Temperature .................. 150
5.4.3 Effect of Surface Roughness .......................................... 152
5.4.4 Semiconductors ........................................................... 155
5.5 SPECIAL SURFACES ...................................................... 156
5.5.1 Modification of Spectral Characteristics ........................... 157
5.5.2 Modification of Directional Characteristics ........................ 165
5.6 CONCLUDING REMARKS ................................................ 167
REFERENCES ........................................................................ 168

APPENDIX ........................................................................ 17l

INDEX .................................................................................. 185

vii
 Chapter 1. Introduction

All substances continuously emit electromagnetic radiation by virtue

of the molecular and atomic agitation associated with the internal energy
of the material. In the equilibrium state, this internal energy is in direct
proportion to the temperature of the substance. The emitted radiant
energy can range from radio waves, which can have wavelengths of miles,
to cosmic rays with wavelengths of less than 10 -I° centimeter (cm). In
this volume, only radiation that is detected as heat or light will be con-
sidered; this is termed thermal radiation, and it occupies an intermediate
wavelength range. This range is defined explicitly in section 1.5.
Although radiant energy constantly surrounds us, we are not very
aware of it because our bodies are able to detect only portions of it di-
rectly. Other portions require detection by use of some intermediate
instrumentation. Our eyes are sensitive direct detectors of light, being
able to form images of objects, but are relatively insensitive to heat
(infrared) radiation. Our skin is a direct detector for heat radiation but
not a good one. The skin is not aware of images of warm or cool surfaces
around us unless the heat radiation is large. We require indirect means
such as infrared-sensitive film in a camera to form images using heat
radiation.
Before discussing the nature of thermal radiation in detail, it is well
to consider why thermal radiation is so important in our modern
technology.

1.i IMPORTANCE OF THERMAL RADIATION

One of the factors that causes some of the important applications of

thermal radiation to arise is the dependence of radiant emission on tem-
perature. For conduction and convection the transfer of energy between
two locations depends on the temperature difference of the locations to
approximately the first power2 The transfer of energy by thermal radia-
tion, however, depends on the differences of the individual absolute tem-
peratures of the bodies each raised to a power in the range of about 4 or 5.
From this basic difference between radiation and the convection and
conduction energy exchange mechanisms, it is evident that the impor-
tance of radiation becomes intensified at high absolute temperature
levels. Consequently, radiation contributes substantially to the heat

For free convection or when variable property effects are included, the power of the temperature difference may be-
come larger than unity but usually in convection and conduction does not approach 2.
2 THERMAL RADIATION HEAT TRANSFER

transfer in furnaces and combustion chambers and in the energy emis-

sion from a nuclear explosion. The laws of radiation govern the tempera-
ture distribution within the Sun and the radiant emission from the Sun
or from a source duplicating the Sun in a solar simulator. Some devices
for space applications are designed to operate at high temperature levels
in order to achieve high thermal efficiency. Hence, radiation must often
be considered when calculating thermal effects in devices such as a
rocket nozzle, a nuclear powerplant, or a gaseous core nuclear rocket.
A second distinguishing feature of radiative transfer is that no medium
need be present between two locations in order for radiant interchange
to occur. The radiative energy will pass perfectly through a vacuum.
This is in contrast to convection and conduction where a physical
medium must be present to carry the energy with the convective flow
or to transport it by means of thermal conduction. When no medium
is present, radiation becomes the only significant mode of heat transfer.
Some common instances are the heat leakage through the evacuated
walls of a Dewar flask or thermos bottle, or the heat dissipation from
the filament of a vacuum tube. A more recent application is the radiation
used to reject waste heat from a powerplant operating in space.
Radiation can be of importance in some instances even though the
temperature levels are not elevated and other modes of heat transfer
are present. The following example is quoted from a Cleveland news-
paper published in the spring of 1964. A florist "noted the recurrence
of a phenomenon he has observed for two seasons since using plastic
coverings over [flower] flats. Water collecting in the plastic has formed
ice a quarter-inch thick [at night] when the official [temperature] reading
was well above freezing. 'I'd like an answer to that, I supposed you
couldn't get ice without freezing temperatures.' " The florist's oversight
was in considering only the convection to the air and omitting the night-
time radiation loss occurring between the water covered surface and the
very cold heat sink of outer space.
Another similar illustration is the discomfort that a person experiences
in a room where cold surfaces are present. Cold window surfaces, for
example, have a chilling effect as the body radiates directly to them
without receiving compensating energy from them. Covering the windows
with a shade or drape will greatly decrease the bodily discomfort.
Finally, we note that the thermal radiation that we shall examine is
in the wavelength region that gives mankind heat, light, photosynthesis,
and all their attendant benefits. This in itself is strong justification for
studying thermal radiation. Our existence depends on the solar radiant
energy incident upon the Earth. Understanding the interaction of this
radiation with the atmosphere and surface of the Earth can provide
additional benefits in its utilization.
 INTRODUCTION 3

1.2 SYMBOLS

C speed of electromagnetic radiation propagation in medium

Co speed of electromagnetic radiation propagation in vacuum
k thermal conductivity
n index of refraction, Co/C
qc energy per unit area per unit time resulting from heat con-
duction
qr radiant energy per unit area per unit time arriving at surface
element
q$ radiant energy per unit area per unit time arriving from unit
surface element
qv radiant energy per unit area per unit time arriving from unit
volume element
S surface area
T temperature
V volume
x, y, z coordinates in Cartesian system
arbitrary direction
wavelength in vacuum
b' frequency

1.3 COMPLEXITIES INHERENT IN RADIATION PROBLEMS

First let us discuss some of the mathematical complexities that arise

from the basic nature of radiation exchange. In conduction and convec-
tion heat transfer, energy is transported through a physical medium.
The energy transferred into and from an infinitesimal volume element
of solid or fluid depends on the temperature gradients and physical
properties in the immediate vicinity of the element. For example, for
the relatively simple case of heat conduction in amaterial (no convection)
with temperature distribution T(x, y, z) and constant thermal conduc-
tivity k, the heat conduction is obtained by locally applying the following
Fourier conduction law:

OT
(l-l)
qc -- k c?_
In{
direction

For an elemental cube within a solid as shown in figure 1-1(a), a con-

sideration of the net heat flow in and out of all the faces using the terms
in the figure yields the Laplace equation governing the heat conduction
within the material
 THERMAL RADIATION HEAT TRANSFER

02T F 02T ' _2T

ax 2 -0-_ + -_z2 = 0 (1-2)

The terms in this equation depend only on local temperature derivatives

in the material.
A similar although more complex analysis can be made for the con-
vection process, again demonstrating that the heat balance depends
only on the conditions in the immediate vicinity of the location being
considered.

_,_i -[c_T c_2T.

-K/_-y
+_-y2
_) dxdz
l J' /c_T

dz
../._dz_ ..... T_y dz
1/ Solid material
f,f

(a)

_ volume of differential
material
,_ ,-Radiating /

I i

Radiating V
differential
surface /- -- ..........

element--" ,1""'_" qs _'"___ f

(b)

(a) Heat conduction terms for volume element in solid.

(b) Radiation terms for enclosure filled with radiating material.

FIGURE 1-1.-Comparison of types of terms for conduction and radiation heat balances.

In radiation, energy is transmitted between separated elements

without the need of a medium between the elements. Consider a heated
enclosure of surface S and volume V filled with radiating material (such
as gas or glass) as shown in figure 1-1(b). If qs is the radiant energy flux
(energy per unit area and per unit time) arriving at dA from an element
on the surface dS of the enclosure and qv arrives at dA from an element of
the medium dV, then the total radiation arriving per unit area at dA is
 INTRODUCTION 5

qr=fsqsdS+fvqvdV (1-3)

These types of terms lead to heat balances in the form of integral equa-
tions which are generally not as familiar to the engineer as differential
equations. When radiation is combined with conduction and/or con-
vection, the presence of both integral and differential terms having dif-
ferent powers of temperature leads to nonlinear integrodifferential
equations. These are, in general, extremely difficult to solve.

In addition to the mathematical difficulties, there is a second com-

plexity inherent in radiation problems. This is in accurately specifying
the physical property values to be inserted into the equations. The
difficulty in specifying accurate property values arises because the
properties for solids depend on many variables such as: surface rough-
ness and degree of polish, purity of material, thickness of a coating such
as paint on a surface (for a thin coating the underlying material may
have an effect), temperature, wavelength of radiation, and angle at which
radiation leaves the surface. Unfortunately, many measurements have
been reported where all the pertinent surface conditions have not been
precisely defined.

1.4 WAVE AGAINST QUANTUM MODEL

The theory of radiant energy propagation can be considered from two

viewpoints-classical electromagnetic wave theory and quantum me-
chanics. The quantum-mechanical view of the interaction of radiation
and matter yields, in most cases, equations that are remarkably similar
to the classical results. With a few exceptions, thermal radiation may
therefore be viewed as a phenomenon based on the classical concept
of the transport of energy by electromagnetic waves. These exceptions,
however, include some of the most important effects common to radiative
transfer studies, such as the spectral distribution of the energy emitted
from a body and the radiative properties of gases. These can only be
explained and derived on the basis of quantum effects in which the
energy is assumed to be carried by discrete particles (photons). The
"true" nature of electromagnetic energy (i.e., waves or quanta) is not
known, nor is it generally important to the engineer. Throughout the
present work, the wave theory will generally be adhered to because it
has the greatest utility in engineering calculations and also generally
produces the same formal equations as the quantum theory. Occasional
reference will be made to phenomena where quantum arguments must
be invoked.
 THERMAL RADIATION HEAT TRANSFER

1.5 ELECTROMAGNETIC SPECTRUM

Within the framework of the wave theory, electromagnetic radiation

follows the laws governing transverse waves that oscillate in a direction

perpendicular to the direction of travel. The speed of propagation for

electromagnetic radiation is the same as for light, light after all is simply

I
Cosmic
rays

10-4 - 10-8 --
1022

Gamma
10-2 - rays
10-6 --
102
0 I

i00 - 10-4
1018 X-rays

102 - 10-2 T visib,e

1016 Ultraviolet (-0.4 - 0.7 gmI

E
_t.104 - _ 100 -- 10-4 --
l Red _ NAear infrared
_- 1014
>_,
-= Thermal (_0.7 - 25 gm)

radiation Far in_rared

_106 -- _ 102 -- 10-2
(25 - 1000 tim)
_ 1012 i0-31--
! !

_ I0-2'-
108 - 104 -- _ 100
Rada!,
10 lo 10-I
..C
television,
i00 -- and radio

108 --
1010 -- 106 -- 102
1o
I- T
Shortwave
broadcasting
102-
1012 -- i0_ _ 1o4 -
106- 103 --
Longwave
broadcasting

1014 -- i01[
(al
104-

(a) Type of radiation.

FIGURE 1-2.-Spectrum of
 INTRODUCTION 7

the special case of electromagnetic radiation in a small region of the

spectrum. In vacuum the speed of propagation is co = 2.998 × 10 _° centi-
meters per second (cm/sec) or 186 000 miles per second (mi/sec). The
speed c in a medium is less than co and for dielectrics is commonly
given in terms of the index of refraction n = co/c, where n is greater than
unity. 2 For glass n is about 1.5, while for gases n is very close to 1.

I
Conversionof matter
to radiant ener_

Decelerationof
Radioactive
high-energy
disintegration particles

IT Electron
bombardment IT
Synchrotron
radiation

._L_ ronic transi-

T.._.tions in gases
l
Vibration-rotation transitions in gases,
T molecularvibrations in solidsand liquids,
Rotationtransitions and boundelectro9 transitions in solids
in gasesand lattice
vibrations in solids

t
Amplifiedoscillations
in electronic circuits

(b)

(b) Production mechanism.

electromagnetic radiation.
8 THERMAL RADIATION HEAT TRANSFER

The types of electromagnetic radiation can be classified according to

their wavelength h (or frequency v where c =xv). Common units for
wavelength measurement are the micron (/xm) where 1/xm = 10 -4 cm or
10 -s meter (m) and the angstrom (_) where 1 /_= 10 -'° m. Hence,
104/_ = 1 ttm. A chart of the radiation spectrum is shown in figure 1-2.
A set of conversion factors for basic units in radiative transfer is given in
tables I to III in the appendix.
The region of interest here includes a portion of the long wave fringe
of the ultraviolet, the visible light region which extends from wave-
lengths of approximately 0.4 to 0.7/zm, and the infrared region which
extends from beyond the red end of the visible spectrum to about _ = 1000
/zm. The infrared region is sometimes divided into the near infrared,
extending from the visible region to about _ = 25 ttm, and the far infrared
composed of the longer wavelength portion of the infrared spectrum.
The column at the far right in figure 1-2 indicates the various mech-
anisms by which electromagnetic radiation is produced. Some of the
descriptions are from a quantum-mechanical viewpoint in which elec-
trons or molecules in a state of agitation undergo transitions from one
energy state to another. These transitions result in a radiative energy
release. The transitions may occur spontaneously, or they may be ini-
tiated by the presence of a radiation field.
In this chapter we have discussed the importance of thermal radia-
tion, the dil_culties inherent in radiation problems, and the wavelength
region occupied by thermal radiation within the electromagnetic spec-
trum. In the next chapter, the radiative behavior of the ideal radiating
surface, termed a black surface, will be examined. Using the behavior
of this ideal as a standard for comparison, the behavior of radiative
energy for conditions of interest to the engineer will be discussed in
succeeding chapters.

t For metals the index of refraction is a complex quantity of which n is only the real part. In this case n can be less than
unity which at first glance might convey the impression that the propagation speed in metals is greater than co. This is not
the case; the imaginary part of the complex index must also be considered and this part is greater than unity. A detailed
discussion is given in chapter 4.
 Chapter 2. Radiation from a Blackbody
Before discussing the idealized concept of the blackbody, let us
examine a few aspects of the interaction of incident radiant energy with
matter. The idea we are concerned with is that the interaction at the
surface of a body is not the result of only a surface property but depends
as well on the bulk material beneath the surface.
When radiation is incident on a homogeneous body, some of the radia-
tion is reflected and the remainder penetrates into the body. The radiation
may then be absorbed as it travels through the medium. If the material
thickness required to substantially absorb the radiation is large com-
pared with the thickness dimension of the body or if the material is
transparent, then most of the radiation will be transmitted entirely
through the body and will emerge with its nature unchanged. If, on the
other hand, the material is a strong internal absorber the radiation that
is not reflected from the body will be converted into internal energy
within a very thin layer adjacent to the surface. A very careful distinction
must be made between the ability of a material to let radiation pass
through its surface and its ability to internally absorb the radiation after
it has passed into the body. For example, a highly polished metal will
generally reflect all but a small portion of the incident radiation, but the
radiation passing into the body will be strongly absorbed and converted
to internal energy within a very short distance within the material. Thus
the metal has strong internal absorption ability, although it is a poor
absorber for the incident beam since most of the incident beam is re-
flected. Nonmetals may exhibitthe opposite tendency. Nonmetals may
allow a substantial portion of the incident beam to pass into the material,
but a larger thickness will be required than in the case of a metal to
internally absorb the radiation and convert it into internal energy.
When all the radiation that passes into the body is absorbed internally,
the body is called opaque.
If metals in the form of very fine particles are deposited on a sub-
surface, the result is a surface of low reflectivity. This combined with
the high internal absorption of the metal causes this type of surface to
be a good absorber. This is the basis for formation of the metallic
"blacks" such as platinum or gold black.

9
295-763 0L-68--2
10 THERMAL RADIATION HEAT TRANSFER

2.1 SYMBOLS

surface area
constants in Planck's spectral energy distribution (see table
IV of the appendix)
C3 constant in Wien's displacement law (see table IV of the
appendix)
c speed of light in medium other than a vacuum
Co speed of light in vacuum
E energy emitted per unit time
e emissive power
Fo-_ fraction of total blackbody intensity or emissive power lying
in spectral region 0- k
h Planck's constant
i radiant intensity
k Boltzmann constant
n refractive index
q rate of energy
r radius
T absolute temperature
azimuthal, or cone angle (measured from normal of surface)
the quantity C2/hT
19 wave number
0 circumferential angle
K extinction coefficient for electromagnetic radiation
wavelength in vacuum
hm wavelength in medium other than a vacuum
frequency
or Stefan-Bohzmann constant (eq. (2-22))
O) solid angle
t

Superscript:

directional heat flow quantity

Subscripts:

b blackbody
max corresponding to maximum energy
n normal direction
p projected
s sphere
19 wave number dependent
h spectrally (wavelength) dependent
h_-K_ in wavelength span h_ to ks
hT evaluated at hT
t, frequency dependent
 RADIATION
FROM A BLACKBODY 11

2.2 DEFINITION OF A BLACKBODY

A blackbody is defined as an ideal body that allows all the incident

radiation to pass into it (no reflected energy) and absorbs internally all
the incident radiation (no transmitted energy). This is true of radiation
for all wavelengths and for all angles of incidence. Hence the blackbody
is a perfect absorber of incident radiation. All other qualitative aspects
of blackbody behavior can be derived from this definition.
The concept of a blackbody is basic to the study of radiative energy
transfer. As a perfect absorber, it serves as a standard with which real
absorbers can be compared. As will be seen, the blackbody also emits
a maximum energy and hence serves as an ideal standard of comparison
for a body emitting radiation. The radiative properties of the ideal
blackbody have been well established by use of quantum theory, and
have been verified by experiment.
Only a few surfaces such as carbon black, carborundum, platinum
black, and gold black approach the blackbody in their ability to absorb
radiant energy. The blackbody derives its name from the observation
that good absorbers of incident visible light do indeed appear black to
the eye. However, except for the visible region the eye is not a good
indicator of absorbing ability in the wavelength range of thermal radia-
tion. For example, a surface coated with white oil paint is a very good
absorber for infrared radiation emitted at room temperature, although
it is a poor absorber for the shorter wavelength region characteristic of
visible light.

2.3 PROPERTIES OF A BLACKBODY

Aside from being a perfect absorber of radiation, the blackbody has

other important properties, which will now be discussed.

2.3.1 Perfect Emitter

Consider a blackbody at a uniform temperature placed within a

perfectly insulated enclosure of arbitrary shape whose walls are also
composed of blackbodies at some uniform temperature initially different
from that of the enclosed blackbody (fig. 2-1). After a period of time, the
blackbody and the enclosure will attain a common uniform equilibrium
temperature. In this equilibrium condition, the blackbody must radiate
exactly as much energy as it absorbs. To prove this, consider what would
happen if the incoming and outgoing amounts of radiation were not
equal. Then the enclosed blackbody would either increase or decrease
in temperature. This would involve a net amount of heat transferred from
a cooler to a warmer body which is in violation of the second law of ther-
modynamics. It follows then that because the blackbody is by definition
12 THERMAL RADIATION HEAT TRANSFER

absorbing the maximum possible radiation from the enclosure at each

wavelength and from each direction, it must also be emitting the maxi-
mum total amount of radiation. This is made clear by considering any
less-than-perfect absorber, which must emit less energy than the
blackbody to remain in equilibrium.

at uniform temperature

.P Blackbody at uni-
form temperature

"-Surface element exchanging radiant

energy with the blackbody

FIGURE 2--1.- Enclosure geometry for derivation of blackbody properties.

2.3.2 Radiation Isotropy in a Black Enclosure

Now consider the isothermal enclosure with black walls and arbitrary
shape shown in figure 2-1, and move the blackbody to another position
and rotate it to another orientation. The blackbody must still be at the
same temperature because the whole enclosure remains isothermal.
Consequently, the blackbody must be emitting the same amount of
radiation as before. To be in equilibrium, the body must still be receiving
the same amount of radiation from the enclosure walls. Thus, the total
radiation received by the blackbody is independent of body orientation
or position throughout the enclosure; therefore, the radiation traveling
through any point within the enclosure is independent of position or
direction. This means that the black radiation filling the enclosure is
isotropic.
 RADIATION FROM A BLACKBODY 13

In addition to emitting the maximum possible total radiation, the

blackbody emits the maximum possible energy at each wavelength and
in each direction. This is shown by the following arguments:

2.3.3 Perfect Emitter in Each Direction

Consider an area element on the surface of the black isothermal en-

closure and an elemental blackbody within the enclosure. Some of the
radiation from the surface element strikes the elemental body and is at
an angle to the body surface. All this radiation, by definition, is absorbed.
In order to maintain thermal equilibrium and isotropic radiation through-
out the enclosure, the radiation emitted back into the incident direction
must equal that received. Since the body is absorbing the maximum
radiation from any direction, it must be emitting the maximum in any
direction. Furthermore since the black radiation filling the enclosure is
isotropic, the radiation received or emitted in any direction by the
enclosed black surface, per unit projected area normal to that direction,
must be the same.

2.3.4 Perfect Emitter at Every Wavelength

Consider a blackbody inside an enclosure with the whole system in

thermal equilibrium. The enclosure boundary is specified as being of a
very special type-it emits and absorbs radiation only in the small
wavelength interval d_l around _1. The blackbody, being a perfect
absorber, absorbs all the incident radiation in this wavelength interval.
To maintain the thermal equilibrium of the enclosure, the blackbody
must reemit radiation in this same wavelength interval; the radiation can
then be absorbed by the enclosure boundary which only absorbs in this
particular wavelength interval. Since the blackbody is absorbing a
maximum of the radiation in d_, it must be emitting a maximum in
dM. A second enclosure can now be specified that only emits and ab-
sorbs in the interval dX2 around ;t2. The blackbody must then emit a
maximum at the wavelength ;,2. In this manner the blackbody is shown
to be a perfect_emitter at each wavelength. The special nature of the
enclosure assumed in this discussion is of no significance relative to
the blackbody, because the emissive properties of a body depend only
on the nature of the body and are independent of the enclosure.

2.3.5 Total Radiation a Function Only of Temperature

If the enclosure temperature is altered, the enclosed blackbody

temperature must adjust and become equal to the new enclosure temper-
ature (i.e., the complete isolated system must tend toward thermal
14 THERMAL RADIATION HEAT TRANSFER

equilibrium). The system will again be isothermal, and the absorbed and
emitted energy of the blackbody will again be equal to each other al-
though the magnitude differs from the value for the previous enclosure
temperature. Since by definition the body absorbs (and hence emits)
the maximum amount corresponding to this temperature, the character-
istics of the surroundings do not affect the emissive behavior of the
blackbody. Hence, the total radiant energy emitted by a blackbody
is a function only of its temperature.
Further, the second law of thermodynamics forbids net energy transfer
from a cooler to a hotter surface without doing work on the system. If the
radiant energy emitted by a blackbody increased with decreasing
temperature, we could easily build a device to violate this law. Consider,
for example, the infinite parallel black plates shown in figure 2-2. The
upper plate is held at temperature/'1, which is higher than the tempera-

, I01.E2j
!

! TI>T2 ./
I E

J i

I Q2 : E2 - E1

FIGURE 2--2.--Device violating second law of thermodynamics.

ture T2 of the lower plate. If the emission of energy decreased with

increasing temperature, then the energy emitted per unit time by plate 2,
E2, is larger than that emitted by plate 1, El. Because the plates are
black, each absorbs all energy emitted by the other. To maintain the
temperature of the plates, an amount of energy Q_ =E2-E1 must be
extracted from plate 1 per unit time and an equal amount added to
plate 2. Thus, we are transferring net energy from the colder to the
warmer plate without doing external work. Experience as embodied in
the second law of thermodynamics says that this cannot be done. There-
fore, the radiant energy emitted by a blackbody must increase with
temperature.
From these arguments, the total radiant energy emitted by a black-
body is expected to be proportional only to a monotonically increasing
function of temperature.
 RADIATION FROM A BLACKBODY 15

2.4 EMISSIVE CHARACTERISTICS OF A BLACKBODY

2.4.1 Definition of Blackbody Radiation Intensity 3 .

Consider an elemental surface area d// surrounded by a hemisphere

of radius r as shown in figure 2-3. A hemisphere has a surface area of
2err 2 and subtends a solid angle of 2¢r steradians (sr) about a point at

FIGURE 2-3.--Spectral emission intensity from black surface.

the center of its base. Hence, by considering a hemisphere of unit

radius, the solid angle about the center of the base can be regarded di-
rectly as the area on the unit hemisphere. Direction is measured by the
angles 0 and/3 as shown in figure 2-3, where the angle/3 is measured
from the direction normal to the surface. The angular position for 0= 0
is arbitrary.
The radiation emitted in any direction will be defined in terms of
the intensity. There are two types of intensities: the spectral intensity
refers to radiation in an interval dX around a single wavelength, while

The system of units and definition of terms used here have been made as self-consistent as possible to avoid confusion.

This is not true for all areas of radiation, where separate interests and needs have caused a gi'eat ,_ariety of inconsistent

systems of units and definitions to be used. A good example of this was provided to the authors by Dr. Fred Nicodemus,

who sent a data sheet used in the field of ophthalmology to define units of luminance. Enough comment is probably pro-

vided by the following equality taken from the data sheet:

1 nit = 3.14 apostilb = 104 Bougie-Hectom_tre-Carr_ = 2.919 foot-lambert.

16 THERMAL RADIATION HEAT TRANSFER

the total intensity refers to the combined radiation including all wave-
lengths. The spectral intensity of a blackbody will be given by i_b(_).
The subscripts denote, respectively, that one wavelength is being con-
sidered and that the properties are for a blackbody. The prime denotes
that radiation in a single direction is being considered. The notation is
explained in detail in chapter 3, section 3.1.2. The spectral intensity is
the energy emitted per unit time per unit small wavelength interval
around the wavelength _, per unit elemental projected surface area
normal to the (/3, 0) direction and into a unit elemental solid angle cen-
tered around the direction (/3, 0). As will be shown in section 2.4.2 the
blackbody intensity defined in this way (i.e., on the basis of projected
area) is independent of direction: hence, the symbol for blackbody
intensity is not modified by any (/3, 0) designation. The total intensity
i_ is defined analogously to i_b, except that it includes the radiation/'or
all wavelengths; hence, the subscript _ and the functional dependence
(_) do not appear. The spectral and total intensities are related by the
integral over all wavelengths

tb = J_=0 i'_b(_)dh (2--1)

2.4.2 Angular Independence of Intensity

The angular independence of the blackbody intensity can be shown

by considering a spherical isothermal blackbody enclosure of radius r
with a blackbody element dA at its center, as shown in figure 2-4(a).
Once again, the enclosure and the central elemental body are in thermal
equilibrium. Thus, all radiation in transit throughout the enclosure
must be isotropic. Consider radiation in a wavelength interval d_ about
that is emitted by an element dAs on the enclosure surface and travels
toward the central element dA (fig. 2-4(b)). The emitted energy in this
direction per unit solid angle and time is i_b,n (_)dAsd_. The normal
spectral intensity of a blackbody is used because the energy is emitted
normal to the black wall element dAs of the spherical enclosure. The
amount of energy per unit time that impinges upon dA depends on the
solid angle that dA occupies when viewed from the location of dAs.
This solid angle is the projected area of dA normal to the (/3, 0) direction
divided by r 2. The projected area of dA is

dAp= d_l cos/3 (2-2)

Then the energy absorbed by dA is

t °J dA cos/3
(2-3)
Q_b(_.,/3, O) = t_b" ,(X)dA,d& r2
 RADIATION FROM A BLACKBODY 17

/-Spherical
/ black enclosure

dAs- I dA s

/ /
LdA /dg

(a) (b) (c)

(a) Black element dA within black (b) Energy transfer (c) Energy transfer

spherical enclosure, from dA= to d//p. from d,4p to d.4=.

FIGURE 2--4. -- Energy exchange between element of enclosure surface and element within
enclosure.

The energy emitted by dA in the 03, 0) direction and incident on dAB

(fig. 2-4(c)) must be equal to that absorbed from dAB, or equilibrium
would be disturbed; hence,

i_b(x,/_, o)aA, r-_ dX= Q;_(X,/3, o) =i'_,.(h)dd, dAc°s--------_

dh (2--4)
r2

Then, by virtue of equation (2-2),

i_b(_t, 1_, O)=i_b, .(X) _ function of/8, 0 (2-5)

Equation (2-5) shows that the intensity of radiation from a blackbody,

as defined here on the basis of projected area, is independent of the
direction of emission. Neither the subscript n nor the 03, 0) notation is
really needed for complete description of the black intensity. Since
the blackbody is always a perfect absorber and emitter, these properties
of the blackbody are independent of its surroundings. Hence, these
results are independent of both the assumptions used in the derivation
]8 THERMAL RADIATION HEAT TRANSFER

of a spherical enclosure and thermodynamic equilibrium with the

surroundings. 4

2.4.3 Blackbody Emissive Power--Dofinition and Cosine Law Dependence

The intensity has been defined on the basis of projected area. It is

useful also to define a quantity which gives the energy emitted in a given
direction per unit of actual (unprojected) surface area. This is defined as
e_.b(h, /3, 0) which is the energy emitted by a black surface per unit time
within a unit small wavelength interval centered around the wavelength
h, per unit elemental surface area and into a unit elemental solid angle
dto centered around the direction (,8, 8). The energy in the wavelength
interval dh centered about h emitted per unit time in any direction
Q_(h,/3, 8) can then be expressed in the two forms

Q_,o(h,/3, 8) = e_o(h,/3, O)dAdtodh = i[b(h) dA cos/3 dtodh

Consequently, there exists the relation

e'xa ( h, /3, O) = i'xo( h ) cos/3 = e'xb( h, /3) (2-6)

It is evident from the i'Xb(h ) COS /3 term in equation (2--6) that exb(h, /3, 8)

does not depend on 0 and hence can be expressed as e'xb(h, /3). The
quantity e'xb(h, /3) is called the directional spectral emissive power for a
black surface. In the case of some nonblack surfaces, there will be a
dependence of eL on angle O.
Equation (2--6) is known as Lambert's cosine law, and surfaces having
a directional emissive power that follows this relation are known as
"diffuse" or "cosine law" surfaces. A blackbody, because it is always
a diffuse surface, serves as a standard for comparison with the direc-
tional properties of real surfaces that do not, in general, follow the
cosine law.

4It should be noted that some exceptions do exist for most of the blackbody "laws" presented in this chapter. The
exceptions are of minor importance in almost any practical engineering situation but need to be considered when ex-
tremely rapid transients are present in a radiative transfer process. If the transient period is of the order of the time scale
of whatever process is governing the emission of radiation from the body in question, then the emission properties of
the body may lag the absorption properties. In such a case, the concepts of temperature used in the derivations of the
blackbody laws no longer hold rigorously. The treatment of such problems is outside the scope of this work.
 RADIATION FROM A BLACKBODY 19

2.4.4 Hemispherical Spectral Emissive Power of a Blackbody

In calculations of total radiant energy rejection by a surface, there is

needed the spectral emissive power integrated over all solid angles of a
hemispherical envelope placed over a black surface. This quantity is
called the hemispherical spectral emissive power of a black surface exb(h).
It is the energy leaving a black surface per unit time per unit area and per
unit wavelength interval around h. Figure 2-5 shows the elemental area
dd at the center of the base of a unit hemisphere. By definition, a solid

FIGUIIE 2-5.-Unit hemisphere used to obtain relation between blackbody intensity and
hemispherical emissive power.

angle anywhere above dA is equal to the intercepted area on the unit

hemisphere. An element of this hemispherical area is given by

dxo = sin/3 d/3dO

Hence, the spectral emission from dA per unit time and unit surface area
passing through the element on the hemispherical area is given by

e'_b(h, /3) sin/3 d/3dO

By virtue of equation (2--6), this is equal to

e_b(h,/3)dto = i_b(h) cos/3 sin/3 d[3dO (2-7)

To obtain the blackbody emission passing through the entire hemi-

20 THERMAL RADIATION HEAT TRANSFER

sphere, equation (2-7) is integrated over all solid angles to give

" cos/3 sin/3 d/3dO (2-8a)

f021r _7r/2

O1"

exb(X) = 2_'i _,b(X) fo' sin/3 d(sin/3) = rri_b (X) (2-8b)

where the prime notation is absent in the designation of the hemispheri-

cal quantity. Also from equation (2-6), when the emission is normal to
the surface (/3 = 0) so that cos/3 = 1,

e'xb, n( _k ) ---- i_b (h.)

and, from equation (2-8b),

exb ()_) = 1re,b,, (_) (2--9)

Hence, purely from the geometry involved, this simple relation is found:
The blackbody hemispherical emissive power is zr times the directional
emissive power normal to the surface or lr times the intensity. This
relation will prove to be very Useful in relating directional and hemi-
spherical quantities in following chapters.

2.4.5 Spectral Emissive Power Through a Finite Solid Angle

Sometimes the emission through only part of the hemispherical solid

angle enclosing an area element may be desired. The emission through
the solid angle extending from/3_ to/32 and 01 to 02 is found by modifying
the limits of integration in equation (2-8a)

exb (h, /3, --/3z, 01 -- 02) = i _b(h) r| o2rt_ cos/3 sin _ dE dO

jo, J_,

= i,xb(h ) (sin"/32- sin" [3,) (02 -- 0,) (2-10)

2

2.4.6 Spectral Distribution of Emissive Power

Some of the blackbody characteristics that have been discussed are:

The blackbody is defined as a perfect absorber and is also a perfect
 RADIATION FROM A BLACKBODY 21

emitter. Its spectral intensity and therefore its spectral emissive power
are only functions of the temperature of the blackbody. The emitted
blackbody spectral energy follows Lambert's cosine law.

All these blackbody properties have been demonstrated by thermo-

dynamic arguments. However, a very important fundamental property
of the blackbody remains to be presented. This is the formula that
gives the magnitude of the emitted energy at each of the wavelengths
that comprise the radiation spectrum. This relation cannot be obtained
from purely thermodynamic arguments. Indeed, the search for this
formula led Planck to investigation and hypothesis that became the
foundation of quantum theory. The derivation of the spectral distribution
is beyond the scope of interest of the present discussion. Therefore
the results will be presented here without derivation. The interested
reader may consult various standard physics texts (refs. 1 to 3) for the
complete development.

It has been shown by the quantum arguments of Planck (ref. 4) and

verified experimentally that for a blackbody the spectral distributions of
hemispherical emissive power and radiant intensity in a vacuum are
given as a function of absolute temperature and wavelength by

2¢rC1
exb ( )t ) --_ 7ri 'xo( _t) = )t 5 ( eCz/xr_ 1) (2-1 la)

This is known as Planck's spectral distribution of emissive power. As

will be shown later, for radiation into a medium where the speed of light
is not close to co, equation (2-11a) must be modified by including index
of refraction multiplying factors (see section 2.4.12). For most engineer-
ing work, the radiant emission is into air or other gases with an index of
refraction so close to unity that equation (2-11a) is applicable. The
values of the constants C1 and Cz are given in table IV of the appendix
in two common systems of units. These constants are equal to C1 = hc2o
and C._=hco/k where h is Planck's constant and k is the Boltzmann
constant.5, 6 Equation (2-11a) is of great importance as it provides quan-
titative results for the radiation from a blackbody.

EXAMPLE 2-1: A plane black surface is radiating at a temperature

of 1500 ° F. What is the directional spectral emissive power of the
blackbody at an an_e of 60 ° from the normal and at a wavelength of
6#m?

h = 6.625 x 10 -t_ (erg) (sec) and k = 1.380 × 10 - _6 erg/OK.

t In some literature, the constant CI is defined as 2whcto.
22 THERMAL RADIATION HEAT TRANSFER

From equation (2-11a),

i_(6/zm)= 2 x 0.1889 x l0 s (Btu)(/-_m)4/(hr)(sq ft)

65(p_m)._ ( e2589s/6 × 1960 _ 1 ) (sr)

Btu
= 606 (hr) (sq ft) (/xm) (sr)"

From equation (2-6) the directional emissive power is

Btu
e_,b(6/xm, 60 °) = 606 cos 60 ° = 303
(hr) (sq it) (/xm) (sr)

Alternate forms of equation (2-11a) are sometimes employed where

frequency or wave number is used rather than wavelength. The use of
frequency has an advantage when radiation travels from one medium
into another, as in this instance the frequency remains constant while
the wavelength changes because of the change in propagation velocity.
To make the transformation of equation (2-11a) to frequency, note
that in vacuum X=Co/V, and hence dk=-(Co/p2)dv. Then the hemi-
spherical emissive power in the wavelength interval dk becomes

2¢rCidX - 27rCw'_dv
ex°(h)d)_= hS(eC_/xT- 1)-- cao(eC_'/CoT--l) - e,o(v)dp (2-11b)

The quantity e_b(v) is the emissive power per unit frequency about v.
The wave number _/= 1/4 is the number of waves per unit length.
Then

and

exb(h)dh= 27rC_'o3dTI _-- e,b('o)d'o (2-11c)

( e% "_/v- 1 )

The quantity e_b(_) is the emissive power per unit wave number about _.
To understand better the implications of equation (2-11a), it has been
plotted in figure 2-6. Here the hemispherical spectral emissive power
is given as a function of wavelength for several different values of the
 RADIATION FROM A BLACKBODY 23_

108

108

I07

107

106 -- $ Blackbody

"_ _ temperature,

105
"/'_000 127781
1o' i . ,
104

\\\i
103

N 1 000 15551_" -_ ,_

values,exbO_max,
T)

loll

2 4 6 8 10 12
Violet Wavelength,
),,
pm
lion
(0.4 - O.7pm)

FIGURE 2--6.- Hemispherical spectral emissive power of blackbody for several different
temperatures.

absolute temperature. One characteristic that is quite evident is that

the energy emitted at all wavelengths increases as the temperature
increases. It was shown in section 2.3.5, and it is known from common
experience that the total (including all wavelengths) radiated energy
must increase with temperature; the curves show that this is also true
24 THERMAL RADIATION HEAT TRANSFER

for the energy at each wavelength. Another characteristic is that the

peak spectral emissive power shifts toward a smaller wavelength as
the temperature is increased. A cross plot of figure 2-6 giving energy
as a function of temperature for fixed wavelengths shows that the energy
emitted at the shorter wavelength end of the spectrum increases more
rapidly with temperature than the energy at the long wavelengths.
The position of the range of wavelengths included in the visible spec-
trum is included in figure 2-6. For a body at 1000 ° R only a very small
amount of energy would be in the visible region and would not be suf-
ficient to be detected by eye. Since the curves at the lower tempera-
tures slope downward from the red toward the violet end of the spectrum,
as the temperature is raised the red light becomes visible first. 7 Higher
temperatures make visible additional wavelengths of the visible light
range, and at a sufficiently high temperature the light emitted becomes
white, representing radiation composed of a mixture of all the visible
wavelengths.
For the filament of an incandescent lamp to operate efficiently, the
temperature must be high, otherwise too much of the electrical energy
would be dissipated as radiation in the infrared region rather than in
the visible range. Most tungsten filament lamps operate at about 5400 ° R,
and thus do give off a large fraction of their energy in the infrared, but
their filament vaporization rate limits the temperature to near this value.
The Sun emits a spectrum quite similar to that of a blackbody at a tem-
perature of about 10 000 ° R, and an appreciable amount of energy re-
lease is in what we sense as the visible region. This may be because
evolution has caused the eye to be most sensitive in the spectral region
of greatest energy. If the eye were sensitive in other regions (for ex-
ample, the infrared so that we could see thermal images in the "dark"),
our definition of the "visible region of the spectrum" would change. If
man finds life in other solar systems, where the Sun has an effective
temperature different from ours, it will be interesting to discover what
wavelength range encompasses the "visible spectrum" if the beings
there possess sight.
Equation (2-11a) can be placed in a more convenient form that elimi-
nates the need for providing a separate curve for each value of T. This
is done by dividing by the fifth power of temperature to give

exb(h, T) rri_b(,k, T) 21rC,

(2-12)
T_ T5 (hT)5(eC_/xr- 1)

This equation gives the quantity exb(h, T)/T 5 in terms of the single vari-

rThis occurs at the so-called Draper point of 977 ° F (ref. 5), at which red light first becomes visible from a heated
object in darkened surroundings.
 RADIATION FROM A BLACKBODY 25

I I I I I I I I I I

-- Planck's law
..... Wien's distribution
----- Rayleigh-Jeans distribution
14 O00x10-15 240x10-15 },maxT) l III
kT, (pm)(°R) 2606 5216 7394111069 4] _I
.kT, (pm)(°K) 1448- -2898-4108-I 6 149 2_ 2211--
Percent 1
12 000-- 200 ,er.issivei i
powerbeiow I I
'1
'kT i I
_- IOOOO- '_
A

_ 160
! / ]1 ' " I

!/
_4
E .c:
_ 120
I/ : I
_2 6000 -- ,_
(Xl

i/ II

:
I

I
' t !
_ 4000--
_
_
80
Ii/ : i

: _" I I
201111-
40
- i/]
, !!
II
V
0-- 0
.2 •4 •6 .8 1 2 4 6xlO4
Wavelength-temperature product, ;_T, (pm)(°R)

Illll I I I J I IIII I
.6 •8 1 2 4 6 8 10 20
Wavelength-temperature product, kT, (pm)(°K)

FIGURE 2-7•--Spectral distribution of blackbody hemispherical emissive power.

able XT. A plot of this relation is given in figure 2-7 and replaces the
multiple curves in figure 2-6. A compilation of values is presented in
table V of the appendix.

EXAMPLE 2--2: For a blackbody at 1500 ° R what is the spectral hemi-

spherical emissive power at a wavelength of 2 /xm? Use table V of the
appendix.
The value of XT is 3000 (/xm)(°R). From table V, at this AT, e_/T 5
= 87.047 × 10 -15 Btu/(hr)(sq ft)(/xm)(°R) 5. Then e_b(2/_m) = 87.047 × 10 -15
(1500) 5 = 661 Btu/(hr)(sq ft)(/xm).

2•4.7 Approximations for Spectral Distribution

Planck's spectral distribution gives the maximum (blackbody) inten-

sity of radiation which any body can emit at a given wavelength for a

295-763 0L-68--3
26 THERMAL RADIATION HEAT TRANSFER

given temperature. This intensity serves as an optimum standard with

which real surface performance can be compared. In chapter 3, the
methods of comparison will be defined. Planck's distribution also pro-
vides a means to evaluate the maximum radiative performance that can
be attained for any radiating device.
Some approximate forms of Planck's distribution are occasionally
useful because of their simplicity. Care must be taken to use them only
in the range where their accuracy is acceptable.
2.4.7.1 Wien'sformula.-If the term ec_/xr is much larger than 1, equa-
tion (2-12) reduces to

i'_b( h, T) _ 2C1
(2-13)
Ta ( hT)5e%lar

which is known as Wien's formula. It is accurate to within 1 percent for

hT less than 5400(/xm)(°R).
2.4.7.2 Rayleigh-'Jeans formula.- Another approximation is found by
taking the denominator of equation (2-12) and expanding it in a series
to give

C2+1 fC2N
2 1 (C2 3 (2-14)

For hT much larger than C2, this series can be approximated by the single
term C2/AT, and equation (2-12) becomes

i_b(A, T) _ 2C, 1
(2-15)
T5 C2 (AT) 4

This is known as the Rayleigh-Jeans formula and is accurate to within

1 percent for AT greater than 14 × 105 (gm)(°R). This is well outside the
range generally encountered in thermal radiation problems, since a
blackbody emits over 99.9 percent of its energy at AT values below this.
The formula has utility for long-wave radiation of other classifications,
such as radio waves.
A comparison of these approximate formulae with the Planck distri-
bution is shown in figure 2-7.

2.4.8 Wien's Displacement Law

Another quantity of interest with regard to the blackbody emissive

spectrum is the wavelength Amax at which the emitted energy is a max-
imum for a given temperature. This maximum shifts toward shorter
 RADIATION FROM A BLACKBODY 27

wavelengths as the temperature is increased, as shown by the dotted

line in figure 2-6. The value of hmaxT can be found at the peak of the
distribution curve given in figure 2-7. Alternately it can be found ana-
lytically by differentiating Planck's distribution from equation (2-12)
and setting the left side equal to zero. This gives the transcendental
equation

m x =?tX
1 -- e - Cz/>'max T) (2-16)

The solution to this equation is of the form

_-maxT = C3 (2-17)

which is one form of Wien's displacement law. Values of the constant

C3 are given in table IV of the appendix. Equation (2-17) indicates that
the peak emissive power and intensity shift to a shorter wavelength at
a higher temperature in inverse proportion to T.

EXAMPLE 2-3: For a blackbody to radiate its maximum energy at

the center of the visible spectrum what would its temperature have to be?
Figure 1-2 shows the visible spectrum spans the range 0.4 to 0.7/xm,
and the center of the range is at 0.55 gm. From equation (2-17)

T- C3 5216 (tsm)(°R)= 9480 ° R

_max 0.55 gm

This is close to the effective surface temperature of the Sun.

2.4.9 Total Intensity and Emissive Power

The previous discussion has provided the energy per unit wavelength
interval that a blackbody radiates at each wavelength. It will now be
shown how the total intensity of radiation, which includes the radiation
for all wavelengths, can be determined. The result is a surprisingly
simple relation.
The energy emitted over the small wavelength interval dh is given by
i'xb(h)dh. Integrating the spectral intensity over all wavelengths from
h= 0 to h= _ gives the total intensity

t f _¢

ib= Jo i_b(X)dX (2-18)

This integral may be evaluated by substitution of Planck's distribution

28 THERMAL RADIATION HEAT TRANSFER

from equation (2-12) and a transformation of variables in terms of

_= C,,/_T. Equation (2-18) then becomes

.,
tb= f( 2C1
_5(eC2/_T_ 1) d)_

=f" (C.,_ _ V " ___ _ _

L (e_',/_'T,--1)

-- 2CIT4
_ fo (e¢--_.3 1) dE (2-19)

From a table of integrals (ref. 6), this can be evaluated as

.t
2Ci T 4 rr4
lb = (2--20)
c4 15

Defining a new constant results in

., = --
tb O" T4 (2-21)
7/"

where the constant is

2C,¢r _ Btu
o'= _15C-------7
= 0.1712 × 10 -s (hr)(sq ft)(°R 4) (2-22)

The hemispherical total emissive power of a surface is then

eb = fo exb()t)da= f: 7"gtab()t)dh= o-T 4 (2-23)

which is known as the Stefan-Boltzmann law, where cr is the Stefan-

Boltzmann constant. The value of tr as determined experimentally
differs slightly from that calculated by equation (2-22). This is indicated
in table IV of the appendix.

EXAMPLE 2-4: The beam emitted normal to a blackbody surface is

found to have a total radiation per unit solid angle and per unit surface
area of 3000 Btu/(hr)(sq ft)(sr). What is the surface temperature?
The hemispherical total emissive power is related to the total emis-
sive power in the normal direction by eb = n-e_, ,. Hence, from equation
 RADIATION FROM A BLACKBODY 29

(2-23) T= (rre;, nlo') 1/4= (3000¢r/0.173 × 10-8) 1/4= 1528 ° R. The experi-
mental value of the Stefan-Boltzmann constant has been used.

EXAMPLE 2--5: A black surface is radiating with a hemispherical total

emissive power of 2000 Btu/(hr)(sq ft). What is the surface temperature?
At what wavelength is its maximum spectral emissive power?
From the Stefan-Bohzmann law, the temperature of the blackbody is
T= (edor)'/2= (2000/0.173 × 10-s) 1/4= 1037 ° R. Then from Wien's dis-
placement law, )kma x _--- C3/.T -_- 5216/1037 = 5.04 gm.

2.4.10 Behavior of Maximum Intensity with Temperature

The spectral intensity of a black surface i_,b(h) was shown in equation

(2-5) to be independent of the angle of emission. Integrating over all
wavelengths of course did not change this angular independence.
The intensity of a surface is what the eye interprets as "brightness."
The Sun, which radiates with a spectral distribution of intensity similar
to that of a blackbody at 10 000 ° R, appears equally bright across its
surface to the unaided eye. The Sun thus gives qualitative experimental
verification that the intensity is indeed invariant with direction of emis-
sion, because the radiation reaching us from the center of the solar disk
was emitted normal to the surface, while that from the edge was emitted
at nearly 90 ° to the normal.
The intensity at a given wavelength is found from Planck's spectral
distribution. It is interesting to note that substitution of Wien's displace-
ment law (eq. (2-17)) into equation (2-12) gives

•, [ 2C1 ] (2-24)
thmaxb = T5 LC_(e¢7_ - 1)J

This shows that the maximum intensity increases as temperature to the

fifth power. Indeed, because ixb/T 5 is a function only of ;,T as shown by
equation (2-12), it is evident that if the blackbody temperature is
changed from T_ to /'2 and at the same time the wavelengths h_ and h,,
are chosen such that h_T_ = h2T2, the value of i_b/1_ remains unchanged.
Therefore, the intensity at h2 for temperature /'2 increases as tempera-
ture to the fifth power from the value at h_ for temperature T_. This is the
general statement of Wien's law.

2.4.11 Blackbody Radiation in a Wavelength Interval

The Stefan-Boltzmann law shows that the hemispherical total emissive

power of a blackbody is given by
30 THERMAL RADIATION HEAT TRANSFER

eb : fo_ e_(X)dX = ofT 4

It is often desirable in calculations of radiative exchange to determine

the fraction of the total emissive power that is emitted in a given wave-

r Distribution for
L

blackbody temperature T

,w /r Band emission
E
_o _'_'_'_ "._.
"_ , for interval XI - X_
I

._'_, _"_-_ F_ f:_,_';_,_.. /- Emission under

_'_ _ _.',"._t_;'.._.,_// enf_Jre curve
i-
_o

9.,.-_'_._._-i_,_.;', - , . _:,,_:._--- =...,

E

XI X2 -----,,-
Wavelength, X

FIGURE 2--8.--Emitted energy in wavelength band.

length band as illustrated by figure 2-8. This fraction is designated by

F_,__= and is given by the ratio

1 4 f'_e_b(h)dX (2-25)
o.T
Fx,__,- f: e_,(X)dX

The last integral in equation (2-25) can be expressed by two integrals

each beginning at X = 0

F_,__,= _-_ r rx, e_>(X)dX-Jofx, ex.(X)dX ] =Fo-x,-Fo-x,

1 [Jo
(2-26)

The fraction of the emissive power for any wavelength band can there-
fore be found by having available the values of Fo-x as a function X.
The Fo-xl function is illustrated by figure 2-9(a) where it would equal
the crosshatched area divided by the total area (shaded) under the curve.
 RADIATION FROM A BLACKBODY 31

_',,_
:.-_ _._,,,.,:_ _:..

Wavelength,
).

I,--
<-

Wavelength-temperature
product._T

(a) In terms of curve for specific temperature. Entire area under curve, oft _.
(b) In terms of universal curve. Entire area under curve, o'.

FIGURE 2-9.-Physical representation of F factor, where F0-_, or F0-_,r is ratio of cross-

hatched to shaded area.

For a blackbody , because of the simple manner in which the hemi-

spherical emissive power is related to the intensity (eq. (2-8b)) the Fx,_x_
function also gives the fraction of the intensity which lies in the wave-
length interval )_1--)_. Since e_o depends on T, the application of equa-
tion (2-26) would require that F0-x be tabulated for each T. There is
32 THERMAL RADIATION HEAT TRANSFER

no need to have this complexity, however, as it is possible to arrange

the F function in terms of only the single variable AT, figure 2-9(b). In
this way a universal set of F values is obtained that can apply for all
temperatures and wavelengths. The universal form is found by re-
writing equation (2-26) as

LJO Ts d(_T)--jo _ d(_T) =Fo-x,r--Fo-x,r

(2-27)

As shown by equation (2-12), exb/T 5 is only a function of AT so that the

integrands in equation (2-27) are only dependent on the AT variable.
The F0-xT values are given in table V of the appendix and a plot of
F0-xT as a function of _,T is shown in figure 2-10.

.2 •4 • .6 .8 1 2 4 6 810x104
Wavelength-temperature product, ),T, (pm)(°R)

I I I I I I llll I I I i D
.1 .2 .4 .6 .8 1 2 4 6xlO 4
Wavelencjth temperature product, XT, (pm)(°K)

FIGURE 2--10.-Fractional blackbody emissive power in range 0 to kT.

 RADIATION FROM A BLACKBODY 33

For calculations involving desired accuracy of greater than 1 percent,

it should be noted that the values of F0-_T in table V were computed
using the constants C_ and C2 of table IV of the appendix. The value of
o', the Stefan-Boltzmann constant, which corresponds to these C_ and
C2 values is the calculated value shown in table IV. This calculated o"
should be used to determine the total energy quantities that are to be
multiplied by the F factors to obtain the energy in a wavelength interval.
Use of the experimental value for o" introduces an error of slightly less
than 1 percent because of the inconsistency of the experimental cr with
the one used in equation (2-27) to obtain table V.
The tabulated Fo-_r values are also available in expanded form for
use where even greater accuracy is required. The tables of Pivovonsky
and Nagel (ref. 7), for example, tabulate values for every AT interval of
10 (/xm)(°K) over a very wide range of AT.
Polynomial approximation expressions for Fo-_ are also available
and are included in the appendix.
The compilation of values of F0-_T has a number of uses as illustrated
in the following examples.

EXAMPLE 2-6: A blackbody is radiating at a temperature of 5000 ° R.

An experimenter wishes to measure the total radiant emission by use
of a radiation detector. This detector absorbs all radiation in the _, range
0.8 to 5/_m, but detects no energy outside that range. What percentage
correction will the experimenter have to apply to his energy measure-
ment? If the sensitivity of the detector could be extended in range by
0.5 /xm at only one end of the sensitive range, which end should be
extended?
Taking )t_T= 0.8 x 5000 = 4000 (/xm)(°R) and _.2T_--- 5 )<5000 :- 25 000
(/_m)(°R) results in the fraction of energy outside the sensitive range
being

F0-_: + F_:__ = Fo-_: + ( 1 -- F0-_:) = (0.1050 + 1 -- 0.9621 ) = 0.1429

or a correction of 14.3 percent of the total incident energy. Extending

the sensitive range to the longer wavelength side of the measurement
interval adds little accuracy because of the small slope of the curve
of F against AT in that region, so extending to shorter wavelengths would
provide the greatest increase in detected energy.

EXAMPLE 2-7: The experimenter of the previous example has designed

a radiant energy detector which can only be made sensitive over any
1-/xm range of wavelength. He wants to measure the total emissive power
of two blackbodies, one at 5000 ° and the other at 10 000 ° R. He plans to
adjust his 1-gm interval to give a 0.5-/xm sensitive band on each side of
34 THERMAL RADIATION HEAT TRANSFER

the peak blackbody emissive power. For which blackbody should he

expect to detect the greatest percentage of the total emissive power?
What will the percentage be in each case?
Wien's displacement law tells us that the peak emissive power will
occur at kmax = (5216/T) /zm in each case. Because for the higher
temperature a wavelength interval of 1 /zm will give a wider spread of
_,T values around the peak of kmaxT on the normalized blackbody curve
(fig. 2-7), the measurement should be more accurate for the I0 000 ° R
case. For the 10 000 ° R blackbody, hm_, = 0.5216/zm, and hiT= (0.5216
--0.5000) x 10000=216 (/zm)(°R). Similarly, h2T= 1.0216 × 10000
= 10 216 (/.tm)(°R). The percentage of detected emissive power is then

100(F0-1o 2_6- F0-zls) : 100 x (0.708-- 0) = 70.8 percent

A similar calculation for the 5000 ° R blackbody shows that 51.7 percent
of the emissive power is detected.
Some commonly used values of F0-xr are given in table 2-I. It is
interesting to note that exactly one-fourth of the total emissive power

TABLE 2-I. -- FRACTION OF BLACKBODY

EMISSION CONTAINED IN THE RANGE
O-_T

hT

(/zm) (°R) (/zm) (°K)

o.ol
26O6 1448

5 216 = _,max_r 2 898

7 394 4108
ll 069 6 149
4180O 23 220

lies in the wavelength range below the peak of the Planck spectral
distribution at any temperature. This relation appears to have no simple
physical explanation and must be put down alongside those other
phenomena such as gravitational attraction and the Stefan-Bohzmann
fourth power law in which nature provides us with a simple law to de-
scribe an apparently complex event.
 RADIATION FROM A BLACKBODY 35

2.4.12 Blackbody Emission in a Medium Other Than ct Vacuum

The previous expressions for blackbody emission have been for

emission into a vacuum. When emission is considered from a location
within a large volume of medium other than a vacuum, the quantities
C_ and C2 appearing in Planck's energy distribution equation (eq. (2-11a))
should be replaced by the quantities

C'1 = hc _ (2-28a)

C'2 = hc/ k (2-28b)

so that

2rrC',
exrnb( Am)dkm --- ASm(eC,i_mr_ 1 ) dAm (2-29)

where k is the Boltzmann constant, h is Planck's constant, c is the speed

of propagation of light in the medium considered, and )tin is the wave-
length in the medium.
Since the speed c depends on the medium under consideration, it is
better to define C_ and C2 in terms of Co, the speed of light in vacuum, so
that C1 and C2 are then truly constants. For a dielectric, the speed in the
medium is given by c= coin where n is the index of refraction. Planck's
distribution for the energy in a wavelength interval dam becomes (note
that Am is the wavelength in the medium)

2 7rc2h 2 _-c2oh
e_mb( km ) -- A5 ( ech/kXm T_ 1) dam = n2A 5m
( eC oh/nk_,mT _ 1) dam

2rrCi
dam (2-30)
n2kSm( eCz/nxm T- l )

If n can be considered independent of wavelength, then dA_=d(-hn)

1 dA and
n

2_C1n 2
ex mO(A) dam = As ( e C2/)tT__ 1 ) dh (2-31 )

In equations (2-30) and (2-31), Cl= hcZo and C2 =hco/k which are the
values of C, and C2 presented in table IV of the appendix. The A is the
36 THERMAL RADIATION HEAT TRANSFER

wavelength in a vacuum, while hm is the wavelength in the medium. Note

that the refractive index cancels out of the exponential term when h is
used. Equation (2-31) gives the emissive power for a wavelength interval
in the medium in terms of the corresponding wavelength interval in
vacuum where h = nhm.
The integration of equation (2-31) over all wavelengths follows directly
from equation (2-19) when n is constant. This yields the Stefan-Boltz-
mann law for hemispherical total emissive power in a medium of refrac-
tive index n

eb, m = n2o'T 4 (2-32)

The emission within glass (n - 1.5) can thus be 2.25 times that from a
surface into air.
Finally, Wien's displacement law by similar arguments becomes

hmaxT: nhmax, roT= C3 (2-33)

where hmax is the wavelength at peak emission into a vacuum and

hmax, m is the wavelength at peak emission into a medium.
For metals, it is shown in chapter 4 that the simple index of refraction
must be replaced by the complex index of refraction n--iK. In deter-
mining the speed of propagation c in terms of Co in metals, the simple
refractive index n in equations (2-30) to (2-33) must be replaced by the
modulus of the complex refractive index, In--iKI = (n" + K") _/', remem-
bering that the restriction of wavelength independence has been imposed.
These refinements will not be carried in succeeding sections because
their applicability to engineering radiation problems is small. A notable
exception is the work of Gardon and others (refs. 8 to 10) dealing with
radiation effects in molten glass.

2.5 EXPERIMENTAL PRODUCTION OF A BLACKBODY

When making experimental measurements of ihe radiative properties

of real materials, it is desirable to have a black surface for reference so
that a direct comparison can be made between the real surface and the
ideal (black) surface. Since perfectly black surfaces do not exist in
nature, a special technique is utilized to provide a very close approxima-
tion to a black area. Figure 2-11 shows a metal cylinder that has been
hollowed out to form a cavity with a small opening. If an incident beam
passes into the cavity as shown, it strikes the cavity wall and part is
absorbed with the remainder being reflected. The reflected portion
 RADIATION
FROM A BLACKBODY 37

F-Insulation
_-Polishec!
surface
/-Heater \
\
/-Copper
ylinder/-Reflected

Highly
absorbing
I
surfaceJ
Incident //
beam-J/
Black
areaj"

FIGURE2--11.-Cavity used to produce blackbody area.

strikes other parts of the wall and is again partially absorbed. It is

evident that, if the opening to the cavity is very small, very little of the
original incident beam will manage to escape back out through the
opening. Thus by making the opening sufficiently small, the opening
area approaches the behavior of a black surface because essentially
all the radiation passing in through it is absorbed. To help keep the
cavity at a uniform temperature so that the internal radiation will all
be in thermal equilibrium, the cavity shown in figure 2-11 is machined
from a copper cylinder and surrounded by insulation. By heating the
cavity, a source of black radiation is obtained at the opening since, as
previously discussed in section 2.3.1, a perfectly absorbing surface is
also perfectly emitting. The polished surface at the front of the cavity
aids in shielding the opening from stray radiation from the surroundings.
The attainment of isothermal conditions in such a cavity (often
referred to as a "hohlraum") is a difficult but necessary condition in
the accurate experimental determination of radiative properties.

2.6 SUMMARY OF BLACKBODY PROPERTIES

It has been shown in this chapter that the ideal blackbody possesses
38 THERMAL RADIATION HEAT TRANSFER

TABLE 2-II. - BLACKBODY

Symbol Name Definition

Emission in any direction per unit of pro-

"Xb(_,,T) Spectral intensity
jected area normal to that direction, and per
unit time, wavelength interval about X, and
solid angle

Total intensity Emission, Including all wavelengths, in any

i_(T)
direction per unit of projected area normal
to that direction, and per unit time and
solid angle

Emission per unit solid angle in direction 13

e_b(X,P,T) Directional spectral
emissive power per unit surface area, wavelength interval,
and time

eb(l_,T) Directional total Emission, including all wavelengths, in di-

emissive power rection I_ per unit surface area, solid angle,
and time
 RADIATION FROM A BLACKBODY 39

RADIATION QUANTITIES

Geometry Formula

2C I

oT 4

i_b cos [B

11X °T4cos
4O THERMAL RADIATION HEAT TRANSFER

TABLE 2-II.--BLACKBODY

Symbol Name Definition

_b()'._1- I_,el- _. 1) Finite solid angle spec- Emission in solid angle 131< __< I_,
tral emissive power B1 < B < 82 per unit surface area, wave-
length interval, and time

Finite solid angle total Emission, including all wavelengths, in

eb(l_1 - 132,81- 82,T)
emissive power solid angle 131< 13< _2, O1 < B <_82 per unit
surface area an-dtime - -

exb(X1 -X2,1_1 - _Z,61- e2,T] Finite solid angle band Emission in solid angle 131_<13< 132,
emlssive power e 1 < e < e2 and wavelengt'h-ban-d _'1 - )_2
per-un_ surface area and time

%(x, T) Hemispherical spectral Emission into hemispherical solid angle

emissive power per unit surface area, wavelength inter-
val, and time
 RADIATION FROM A BLACKBODY 41

RADIATION QUANTITIES-- Continued

Geometry Formula

(sin2 I_ - sin 2 l_l_

sin _- sinz fill

°14T 'e2 "Sl)" _'n2 _ Fo-ll)2

- sin2 Pl) (Fo'x2 -

_2

_ri_,b

295--763 0L-68.---4
42 THERMAL RADIATION HEAT TRANSFER

TABLE 2-1I.-- BLACKBODY

Symbol Name Definition

Hemispherical band Emission in wavelength band hI - h2 into

emissive power hemispherical solid angle per unit surface
area and time

ebITi Hemispherical total Emission, including all wavelengths, into

emissive power hemispherical solid angle per unit surface
area and time

certain fundamental properties that make it a standard with which real

radiating bodies can be compared. These properties, listed here for
convenience, are the following:
(1) The blackbody is the best possible absorber and emitter of radiant
energy at any wavelength and in any direction.
(2) The total radiant intensity and hemispherical total emissive power
of a blackbody are given by the Stefan-Bohzmann law:

Iri_ = eb = o'T 4

(3) The blackbody directional spectral and total emissive power

follow Lambert's cosine law:

e_b()t,/3) : eXb, n()t) COS/3

e_ (/3) = eb, , cos/3

(4) The spectral distribution of intensity of a blackbody is given by

P|anck's distribution:
 RADIATION FROM A BLACKBODY 43

_ADIATION QUANTITIES-- Continued

Geometry Formula

aT4

2C1
i_b(k ) =
_5 (eC'/_r-- 1)

(5) The wavelength at which the maximum spectral intensity of

radiation for a blackbody occurs is given by Wien's displacement law:

Because of the many definitions introduced in this chapter, it is

convenient to summarize the quantities in tabular form. This has been
done in table 2-II. The formulas for the quantities are given in terms
of either the spectral intensity i_(M, which is computed from Planck's
law, or the surface temperature T.

2.7 HISTORICAL DEVELOPMENT

The derivation of the approximate spectral distributions of Wien and of

Rayleigb and Jeans, the Stefan-Boltzmann law, and Wien's displacement
44 THERMAL RADIATION HEAT TRANSFER

law are all seen to be logical consequences of the spectral distribution of

intensity as derived by Max Planck. However, it is interesting to note
that all of these relations were formulated prior to publication of Planck's
work in 1901 and were originally derived through fairly complex thermo-
dynamic arguments.
Joseph Stefan (ref. 11) proposed in 1879, after study of some experi-
mental results, that emissive power was related to the fourth power of
the absolute temperature of a radiating body. Ludwig Edward Boltzmann
(ref. 12) was able to derive the same relation in 1884 by analyzing a
Carnot cycle in which radiation pressure was assumed to act as the pres-
sure of the working fluid.
Wilhelm Carl Werner Otto Fritz Franz (Willy) Wien (ref. 13) derived
the displacement law in 1891 by consideration of a piston moving within
a mirrored cylinder. He found that the spectral energy density in an
isothermal enclosure and the spectral emissive power of a blackbody
are both directly proportional to the fifth power of the absolute tempera-
ture when "corresponding wavelengths" are chosen. The relation pre-
sented in section 2.4.8 (eq. (2-17)) is more often cited as Wien's dis-
placement law, but is actually a consequence of the previous sentence.
Wien (ref. 14) also derived his spectral distribution of intensity through
thermodynamic argument plus assumptions concerning the absorption
and emission processes.
Lord Rayleigh (1900) and Sir James Jeans (1905) based their spectral
distribution on the assumption that the classical idea of equipartition
of energy was valid (refs. 15 and 16).
The fact that measurements and some theoretical considerations s
indicated Wien's expression for the spectral distribution to be invalid
at high temperatures and/or large wavelengths led Planck to an investiga-
tion of harmonic oscillators which were assumed to be the emitters and
absorbers of radiant energy. Various further assumptions as to the
average energy of the oscillators led Planck to derive both the Wien and
the Rayleigh-Jeans distributions. Planck finally found an empirical
equation which fit the measured energy distributions over the entire
spectrum. In determining what modifications to the theory would allow
derivation of this empirical equation, he was led to the assumptions
which form the basis of the quantum theory. As we have seen, his equa-
tion leads directly to all the results derived previously by Wien, Stefan,
Boltzmann, Rayleigh, and Jeans.
For an interesting and informative comprehensive review of the
history of the field of thermal radiation, the article by Barr (ref. 17) is
recommended.

s It was felt that as temperature approaches large values, the intensity of a blackbody should not approach a finite
limit. Examination of Wien's formula (eq. (2-13)) shows that this condition is not met. Planek's distribution law (eq. (2-11)),
however, does satisfy the condition.
 RADIATION FROM A BLACKBODY 45

REFERENCES

1. RICHTMYER, F. K.; AND KENNARD, E. H.: Introduction to Modern Physics. Fourth ed.,
McGraw-Hill Book Co., Inc., 1947.
2. TER HAAR, D.: Elements of Statistical Mechanics. Seeond ed., Holt, Rinehart and
Winston, Ine., 1960.
3. TRIBUS, MYRON: Thermostatics and Thermodynamics; an Introduction to Energy,
Information and States of Matter, with Engineering Applications. D. Van Nostrand
Co., Inc., 1961.
4. PLANCK, MAX: Distribution of Energy in the Spectrum. Ann. d. Physik, vol. 4, no. 3,
Mar. 1901, pp. 553-563.
5. DRAeER, JOHN W.: On the Production of Light by Heat. Phil. Mag, Set. 3, vol. 30,
1847, pp. 345-360.
6. DWIGHT, HERBERT B.: Tables of Integrals and Other Mathematical Data. Fourth ed.,
Macmillan Book Co., 1961, p. 231.
7. PIVOVONSKY, MARK; AND NAGEL, MAX R.: Tables of Blackbody Radiation Functions.
Macmillan Book Co., 1961.
8. GARDON, ROBERT: The Emissivity of Transparent Materials. Am. Ceramic Soc. J.,
vol. 39, no. 8, Aug. 1956, pp. 278-287.
9. GARDON, ROBERT: A Review of Radiant Heat Transfer in Glass. Am. Ceramic Soc. J.,
vol. 44, no. 7, July 1961, pp. 305-312.
0. KELLETT, B. S.: The Steady Flow of Heat Through Hot Glass. Opt. Soc. Am. J., vol. 42,
no. 5, May 1952, pp. 339-343.
11. STEFAN, JOSEPH: Ueber die Beziehung zwischen der W_mestrahlung und der Tem-
peratur. Sitz. ber. Akad. Wiss. Wien., vol. 79, pt. II, 1879, pp. 391-428.
12. BOLTZMANN, LUDWIG: Ableitung des Stefan'schen Gesetzes, Betreffend die Ab-
h_ingigkeit der Wiirmestrahlung yon der Temperatur aus der Electromagnetischen
Lichttheorie. Ann. d. Physik, Ser. 2, vol. 22, 1884, pp. 291-294.
13. WIEN, WILLY: Temperatur und Entropie der Strahlung. Ann. d. Physik, Set. 2, vol. 52,
1894, pp. i32-165.
14. WIEN, WILLY: Ueber die Energievertheilung im Emissionsspectrum eines Schwarzen
KSrpers. Ann. d. Physik, Ser. 3, vol. 58, 1896, pp. 662-669.
15. LORD RAYLEIGH: The Law of Complete Radiation. Phil. Mag., vol. 49, June 1900,
pp. 539-540.
16. JEANS, SIR JAMES: On the Partition of Energy Between Matter and the Ether. Phil.
Mag., vol. 10, 1905, pp. 91-97.
17. BARR, E. SCOTT: Historical Survey of theEarly Development of the Infrared Spectral
Region. Am. J. Phys., vol. 28, no. 1, Jan. 1960, pp. 42-54.
 Chapter 3. Definitions of Properties for Nonblack Surfaces

3.1 INTRODUCTION

In chapter 2 the radiative behavior of a blackbody was presented in

detail. The ideal behavior of the blackbody serves as a standard with
which the lberformance of real radiating bodies can be compared. The
radiative behavior of a real body depends on many factors such as com-
position, surface finish, temperature, wavelength of the radiation, angle
at which radiation is either being emitted or intercepted by the surface,
and the spectral distribution of the radiation incident on the surface.
Various emissive, absorptive, and reflective properties, both unaveraged
and averaged, are used to describe the radiative behavior of real materi-
als relative to blackbody behavior.
The definitions of radiative properties of opaque materials are given
in this chapter. To make them of greatest value, the definitions are
presented rigorously and in detail. Since the definitions are numerous,
the reader should not expect to read the present chapter in as complete
detail as the discussion on the blackbody in chapter 2. Rather, some of
the alternate forms of defining the same quantity can be briefly scanned
to obtain an overall view of what information is available, and the chap-
ter then used as a reference source. The sections have been subdivided
and made fairly independent to facilitate use for reference purposes.
The rigorous examination of radiative property definitions arises
from the need to properly interpret available property data for use in
heat-transfer computations. A limited amount of data in the literature
provides detailed directional and spectral measurements. Because of the
difficulties in making these detailed measurements, most of the tabulated
property values are averaged quantities. An averaged radiative perform-
ance has been measured for all directions, all wavelengths, or both. A
clear understanding of the averages involved can be obtained from the
definitions of this chapter. The definitions also reveal relations between
various averaged properties in the form of equalities or reciprocity rela-
tions. This enables the researcher or engineer to make maximum use of
the available property information. Thus, for example, absorptivity data
can be obtained from measured emissivity data if certain restrictions are
observed. These restrictions have often been misunderstood, resulting in
confusion or inaccuracy in applying measured properties.
By detailed examination of the derivation of property definitions, the
restrictions on the property relations are demonstrated. As an aid to

47
48 THERMAL RADIATION HEAT TRANSFER

understanding these definitions, figure 3-1 provides a schematic repre-

sentation of the types of directional properties. The various parts of this
figure will be referred to as each definition is introduced to help provide
a physical interpretation of the quantities being discussed. Further,

(a)

(a) Directional emissivity _'_, 0, TA).

(b) Hemispherical emissivity _(TA).

FIGURE 3-1.-Pictorial description of directional and hemispherical radiation properties.

 DEFINITIONS FOR NONBLACK SURFACES 49

table 3-I lists each of the properties, its symbolic notation, and the
equation number of its definition. The notation is described in section
3.1.2.

(c) Directional absorptivity a' (_, 0, TA).

(d) Hemispherical absorptivity a(T,O.

FIGUttE3-1.-- Continued.
50 THERMAL RADIATION HEAT TRANSFER

Measured properties of real materials are given in chapter 5 to demon-

strate the practical use of the relations derived here.

(e)

(f)
i r • , i i

(e) Bidirectional reflectivity p"(fl,-, 0,., _, O, TA).

(f) Directional-hemispherical reflectivity p' (/3, O, TA).

FIGURE 3-1.-- Continued.

 DEFINITIONS FOR NONBLACK SURFACES 51

i_-(13
r, Or, TA)

(g)

(g) Hemispherical-directional reflectivity p ' (_r, Or, TA).

(h) Hemispherical reflectivity p(Ta).

FIGURE 3--1. -- Concluded.

52 THERMAL RADIATION HEAT TRANSFER

TABLE 3--I.--SUMMARY OF SURFACE PROPERTY DEFINITIONS

I
Quantity Defining Descriptive
equation figure
I Symbol

Emissivity

Directional spectral ................................... (E L .............. 3-2 3-1(a)

Directional total ........................................ (E I .............. 3-3 3-1(a)
Hemispherical spectral .............................. 3-5 3-1(b)
Hemispherical total .................................... (E ............... 3-6 3-l(b)

Absorptivity

Directional spectral .................................... 3-10a 3-1(c)

Directional total ......................................... t ............. 3-14 3-1(c)
Hemispherical spectral ............................... 3-16 3-1(d)
Hemispherical total .................................... Ot .............. 3-18 3-1(d)

Reflectivity

Bidirectional spectral .................................. p_ ............. 3-20 3-1(e)

Directional-hemispherical spectral ................. p_(/3, /_)...... 3-24 3-1(f)
Hemispherical-directional spectral ................. p_(/3,, er) ..... 3-26 3-1(g)
Hemispherical spectral ............................... p_ .............
3-29 3-1(h)
Bidirectional total ...................................... p" .............
3-39 3-](e)
Directional-hemispherical total ..................... p'(_, e)...... 3-41a 3-t(f)
Hemispherical-directional total ..................... p'_3r, _r) .... 3-41b 3-1(g)
Hemispherical total .................................... p ..............
3-43 3-1(h)

3.1.1 Nomenclature

A number of suggestions have been made in an effort to standardize

the nomenclature of radiation. One controversy centers around the end-
ing "ivity" for the various radiative properties of materials. The National
Bureau of Standards is attempting to standardize nomenclature, and in
their publications reserve this ending for the properties of an optically
smooth substance with an uncontaminated surface (emissivity, reflec-
tivity, etc.), while assigning the "ance" ending (i.e., cmittance, reflect-
ance, etc.), to measured properties where there is a need to specify
surface conditions.
 DEFINITIONS FOR NONBLACK SURFACES 53

It is the practice in most fields of science to assign the "ivity" ending

to intensive properties of materials, such as in the cases of electrical
resistivity, thermal conductivity, or diffusivity. The "ance" ending is
reserved, however, for extensive properties of materials as in electrical
resistance or conductance. Use of the term "emittance" as defined in the
previous paragraph does not follow this convention, since "emittance"
would still be an intensive property as long as opaque materials are
considered. Further, it seems cumbersome to define two terms for the
same concept, using one term to differentiate the one very special case
of the perfectly prepared pure substance.
For these reasons, the "ivity" ending will be used throughout this
book for the radiative properties of opaque materials whether for ideal
uncontaminated surfaces or for properties with some given surface
condition. The "ance" ending can then be reserved for an extensive
property such as the emittance of a layer of water where the emittance
would vary with thickness. The derived relations of course apply regard-
less of the nomenclature adopted.
It must be noted that the "ance" ending is often found in the literature
dealing with the experimental determination of surface properties. The
term "emittance" is also used in some references to describe what we
have called emissive power.

3.1.2 Notation

Because of the many independent variables that must be specified

for radiative properties, a concise but accurate notation is necessary.
The notation to be used here is an extension of that introduced in the
preceding chapter. A functional notation is used to explicitly give the
variables upon which a quantity depends. For example _(_, fl, 0, TA)
shows that ¢_ depends on the four variables noted. The prime denotes a
directional quantity, and the _, subscript specifies that the quantity
is spectral. Certain quantities depend upon two directions (four angles);
these will be given a double prime. A hemispherical quantity will not
have a prime, and a total quantity will not have a _ subscript. A quantity
that is directional in nature, that is, it is evaluated on a "per unit solid
angle" basis, will always have a prime even if in a specific case its
numerical value is independent of direction; the independence of direc-
tion is denoted by the absence of (/3, 0) in the functional notation. Simi-
larly a spectral quantity will always have a _, subscript even when in
specific cases the numerical value does not vary with wavelength; such
a specific case would not have a _ in the functional notation.
Additional notation is needed for the energy rate Q for a finite area
in order to keep consistent mathematical forms for energy balances.
54 THERMAL RADIATION HEAT TRANSFER

Thus, d2Q'x denotes as before a directional-spectral quantity, but the

second differential is needed to denote that the energy is of differential
order in both wavelength and solid angle. Thus, dQ' and dQ_ are dif-
ferential quantities with respect to solid angle and wavelength, respec-
tively. If a differential area is involved, the order of the derivative is
correspondingly increased.
This notation may appear somewhat redundant, but the usefulness will
become clear in dealing with certain special cases, such as gray and
diffuse bodies. In addition, the shorthand value of referring to ¢_ in the
text rather than writing out the term "directional spectral emissivity"
should be apparent. A study of table 3-I will help clarify the notation
system being used.
The three main sections of the chapter each deal with a different
property, that is, emissivity, absorptivity, and reflectivity. In each of
these sections the most basic unaveraged property is presented first;
for example, in the first section, the directional spectral emissivity is
presented. Then the averaged quantities are obtained by integration.
The section on absorptivity also contains forms of Kirchhoff's law relat-
ing absorptivity to emissivity. The section on reflectivity includes the
reciprocity relations.

3.2 SYMBOLS

A surface area
C a coefficient
e radiative emissive power
F fraction of blackbody total emissive power
i radiation intensity
Q energy rate; energy per unit time
q energy flux; energy per unit area per unit time
t" distance between emitting and absorbing elements
T absolute temperature
O_ absorptivity
cone angle measured from normal of surface
0 circumferential angle
E emissivity
)t wavelength
P reflectivity
or Stefan-Boltzmann constant, table IV of the appendix
60 solid angle

integration over solid angle of entire enclosing hemisphere

 DEFINITIONS FOR NONBLACK SURFACES 55

Subscripts:

A of surface A
a absorbed
b blackbody
d diffuse
e emitted or emitting
i incident
p projected
r reflected
s specular
h spectrally dependent

Superscripts:

' directional
" bidirectional

3,3 EMISSIVITY

The emissivity is a measure of how well a body can radiate energy

as compared with a blackbody. The emitting ability can depend on
factors such as body temperature, the particular Wavelength being
considered for the emitted energy, and angle at which the energy is
being emitted. The emissivity is usually measured experimentally at a
direction normal to the surface and as a function of wavelength. In
calculating energy loss by a body, the emission into all directions is
required, and for such a calculation an emissivity is needed that is
averaged over all directions and wavelengths. For radiant interchange
between surfaces, emissivities averaged over wavelength but not
direction might be needed; in other cases, when spectral effects become
large, spectral values averaged only over direction are used. Thus,
various avergged emissivities may be required by the analyst, and they
must often be obtained from available measured values.
In this section, the basic derivation of the directional spectral emis-
sivity is given. This emissivity is then averaged in turn with respect to
wavelength, direction, and then wavelength and direction simultaneously.
Values averaged with respect to wavelength are termed "total"
quantities; averages with respect to direction are termed "hemispheri-
cal" quantities. This convention will be adhered to throughout this
publication.

3.3.1 Directional Spectral Emissivity ¢_(X,/3, 0, TA)

Consider the geometry for emitted radiation shown in figure 3-l(a).

As discussed in chapter 2 the radiation intensity is the energy per unit
56 THERMAL RADIATION HEAT TRANSFER

time emitted in direction (/3, 0) per unit of the projected area dAp normal
to this direction, per unit solid angle and per unit wavelength band. In
some texts the intensity has been defined relative to the actual surface
area rather than the projected value. By basing the intensity on the pro-
jected area as is done here, there is the advantage that for a black surface
the intensity has the same value for all directions. Unlike the intensity
from a blackbody, the emission from a real body does depend on direction
and hence the (/3, 0) designation is included in the notation for intensity.
The energyleaving a real surface dA of temperature TA, per unit time
in the wavelength interval dh and within the solid angle doJ, is then
given by

d3Q'x(h, [3, O, Ta)= i_,(h,/3, 0, Ta)dA cos/3 dhdto=e'x(h, /3, O, TA)d/ldhd¢o

(3-1a)

For a blackbody the intensity is independent of direction and was

designated in chapter 2 by i'ab()t). The TA notation is introduced here to
clarify when properties are temperature dependent so that the blackbody
intensity is designated as i_(h, TA). The energy leaving a black area
element per unit time within dh and d¢o is

d3O'xb ( M /3, TA)= i '_b( h, T A) dA cos/3 dhdto = e _ ( h, /3, T a ) dA dh dto

(3-1b)

The emissivity is then defined as the ratio of the emissive ability of the
real surface to that of a blackbody; this provides the definition

3 t
d Ox(h,/3, O, TA)
Directional spectral emissivity =- e_(h, /3, O, TA) -- 3 ,
d'Qxo(h, [3, TA)

i'_(X, /3, O, TA) e'x(h, /3, O, TA)

(3-2)
i'xb(A, TA) e_(A, /3, TA)

This is the most fundamental emissivity, because it includes the de-

pendence on wavelength, direction, and surface temperature.

EXAMPLE 3--]: At 60 ° from the normal, a surface heated to 1500 ° R

has a directional spectral emissivity of 0.70 at a wavelength of 5/xm.
The emissivity is isotropic with respect to the angle 0. What is the
spectral intensity in this direction?
From table V of the appendix, for a blackbody at hTA of 7500 (/xm)(°R),
 DEFINITIONS FOR NONBLACK SURFACES 57

exb(h, TA)/T_ = 163.5 x 10 -_5 Btu/(hr)(sq ft)(ftm)(°R). Then

i_(5 ftm, 60 °, 1500 ° R)= E_ (5/xm, 60 °, 1500 ° R)i_b (5 ftm, 1500 ° R)

--
-- Eh' (5/.tm, 60 °, 1500 ° R) e_ (5 gm, 1500 ° R)
7r

163.5 × 10 -t'_
=0.70× (1500) 5 = 276 Btu/(hr)(sq ft)(/xm)(sr)
7/"

3.3.2 Averaged Emissivities

From the directional spectral emissivity as given in equation (3-2),

an averaged emissivity can now be derived by proceeding along one of
two approaches: averaging over all wavelengths or averaging over all
directions.
3.3.2.1 Directional total emissivity e'(13, 0, TA).-Looking first at an
average over all wavelengths, the radiation emitted into direction (/3, 0),
including the contributions from all wavelengths, is found by inte_ating
the directional spectral emissive power to give the directional total
emissive power (as in chapter 2 the term "total" denotes that radiation
from all wavelengths is included)

e'([3, O, TA)= fo_ e'_(h, _, O, TA)dh

Similarly from table 2-II the directional total emissive power for a
blackbody is given by

e_(fl, TA)= fo_ e_(h, fl, TA)dh=_rT] _'c°s fl

The directional total emissivity is the ratio of e'(/3, 0, TA) for the real
surface to e_(/3, TA) emitted by a blackbody at the same temperature;
that is,

e'(fl, 0, TA)
Directional total emissivity=--E'(fl, O, T_)- e,b({j ' TA)

fo e_,(h, _, O, TA)dX
(3-3a)

295-763 OL-68--5
58 THERMAL RADIATION HEAT TRANSFER

The e'_(h, fl, O, TA) in the numerator can be replaced in terms of

_,(h, f_, 0, TA) by using equation (3-2) to give

Directional total emissivity (in terms of directional spectral emissiv-

ity)_ e'(fl, 0, T_)

e_(h,/3, 0, TA)e'_(h, fl, TA)dh

o-T_
-- COS
q'f

_;(x, t_, o, T_)i'_(X, T_)dX

(3-3b)
o-T_

Thus if the wavelength dependence of _(h, fl, 0, TA) is known, the

• ' (/3, O, TA) is obtained as an integrated average weighted by the black-
body emissive power. The e_,(h, /3, 0, TA) must be known with good
accuracy in the region where e_(h, /3, Ta) is large, so that the inte-
grand of equation (3-3b) will be accurate where it has large values.

EXAMPLE 3-2: At 1000 ° R the E_(h, /3, O, TA) can be approximated

by 0.8 in the range h = 0 to 5/xm and 0.4 for h > 5 ttm. What is the value
of e' (/3, 0, TA )?
From equation (3-3b),

fO _c
E_,(h, /3, O, TA)e_,b(h,
t t
B, TA)dh

C (fl, O, TA)=" o'T,_

-- COS /3
77

Apply the following relation obtained from table 2-II:

e_(h, TA) cos/3

e'_(h, fl, TA) =
T$"

This yields

e'(/3, 0. T.)= STA--

0.8
,_L [e_(X,T] T.)] d(XTA)

+ fs:A [? exb( T] TA).] d(hTA)

 DEFINITIONS FOR NONBLACK SURFACES 59

From equation (2-27),

_' (/3, 0, T_) = 0.8 Fo-50oo + 0.4 Fso00-_ = 0.8(0.223) + 0.4(0.777) = 0.490

Since 77.7 percent of the emitted blackbody energy at 1000 ° R is in the

region for h > 5 pm, the result is weighted heavily toward the 0.4 emis-
sivity value.
3.3.2.2 Hemispherical spectral emissivity _(h, TA).-Now return to
equation (3-2) and consider the average obtained by integrating the
directional spectral quantities over all directions of a hemispherical
envelope covering the surface (fig. 3-1(b)). The spectral radiation
emitted by a unit surface area into all directions of the hemisphere is
termed the hemispherical spectral emissive power and is found by in-
tegrating the spectral energy per unit solid angle over all solid angles.
This is analogous to equation (2-8a) for a blackbody and is given by

ex()t, TA)= ft a i_(h, fl, O, TA) cos fl dw

f
The notation
Jo d¢o signifies integration over the hemispherical solid
angle. Here, i_ (h,/3, 0, TA) cannot in general be removed from under
the integral sign as was done for a blackbody. By using equation (3-2)
this can be written as

ex(h, TA) =i_b(h, TA) ft_ _(h, fl, O, TA) cos/3doJ (3-4a)

For a blackbody the hemispherical spectral emissive power is from

equation (2-8b)

e_a(h, T_):TTi_b()t , TA) (3-4b)

The ratio of actual to blackbody emission from the surface (eq. (3-4a)
divided by eq. (3-4b)) provides the following definition:

Hemispherical spectral emissivity (in terms of directional spectral

emissivity -= _(_,, TA)

_ e_()t, TA) =-_ ¢_(h,/3, O, TA) cos /3 doJ (3-5)

e_b ( )t, T,t )

3.3.2.3 Hemispherical total emissivity ¢(TA).-To derive the hemi-

spherical total emissivity, consider that from a unit area the spectral
60 THERMAL RADIATION HEAT TRANSFER

emissive power in any direction is derived from equation (3-2)as

_(_, [3, O, TA)i'_(_, TA) cos /3. This is integrated over all k and co to
give the hemispherical total emissive power. Dividing by o'T_, which is
the hemispherical total emissive power for a blackbody, results in the
following emissivity:

Hemispherical total emissivity (in terms of directional spectral emis-

sivity - _(Ta)

eb (TA) o"T_

(3-6a)
o-T_

By using equation (3-3b) this can be placed in a second form

Hemispherical total emissivity (in terms of directional total emissiv-

ity) =- _(TA)

= _ _' (fl, 0, TA ) cos/3 dto (3-6b)

If the order of the integrations is interchanged in equation (3-6a), there

results

e(TA) --
F i_a(_, T,) [L ¢_(x,/_, 0, TA) cos p&o] dX
o'-T_

Equation (3-5) is then utilized to obtain a third form

Hemispherical total emissivity (in terms of hemispherical spectral

emissivity) -= E (TA)

7r J: e_(A, TA)i'_()t, TA)C_

o'T 4 (3-6c)

Substituting equation (3-4b) gives

 DEFINITIONS FOR NONBLACK SURFACES 61

f: EX(_, TA)e_b(_, TA)d_

E(TA) : o.T4 (3--6d)

To interpret equation (3-6d) physically, look at figure 3-2. In figure

3-2(a) is shown the emissivity _ for a surface temperature TA. The solid
curve in figure 3-2(b) is the hemispherical spectral emissive power for a
blackbody at TA. The area under the solid curve is o'T4 which is the
denominator of equation (3-6d) and is equal to the radiation emitted per
unit area by a black surface including all wavelengths and directions.

/
_<

'r=

Wavelength, X

(a) Measured emissivity values.

(b) Interpretation of emissivity as ratio of actual emissive power to blackbody emissive
power.

FIGURE 3-2.-Physical interpretation of hemispherical spectra] and total emissivities.

62 THERMAL RADIATION HEAT TRANSFER

The dashed curve in figure 3-2(b) is the product ex(h, TA)eXb(h, TA) and
the area under this curve is the integral in the numerator of equation
(3-6d) which is the emission from the real surface. Hence ¢(TA) is the
ratio of the area under the dashed curve to that under the solid curve.
From a slightly different viewpoint, at each h the quantity ex is the or-
dinate of the dashed curve divided by the ordinate of the solid curve.
As shown in figure 3-2, for hi the hemispherical spectral emissivity is
ex(Xl, TA) =b/a.

EXAMPLE 3--3: A surface at 1800 ° R is isotropic in the sense that e'

is independent of 0, but depends on 13 as shown in figure 3-3. What is
the hemispherical total emissivity and the hemispherical total emissive
power?

1.0 B
A
t_
f ¢ '(fi. 1800° R)

_-0.85cosI_ _"_._
._>
T,
E
B
co
$
.,_
.2 _'_"
._o

-- I I [ I [ [
0 l0 20 30 40 50 60 70 80 , 90
Anglefromnormal,13,deg

FIGURE 3-3.-Directional total emissivity at 1800 ° R for example 3-3.

The e'_, 1800 ° R) can be approximated in this case quite well by

the function 0.85 cos /3 (dashed line). Then from equation (3-6b) the
hemispherical total emissivity is

_ 1 fz_ f,/2 (cos3/3_1"_

e(TA) --_J0=oJ_=o 0.85 sin/3 cos 2/3d/3dO =- 1.70 _/Io = 0.57

The hemispherical total emissive power is then

e(TA) = e(TA)crT 4= 0.57 × 0.173 x 10 -s × ]8004= 10 300 Btu/(hr)(sq ft)

Generally the e'(/3, TA) will not be well approximated by a convenient

analytical function, and the integration must be carried out numerically.
 DEFINITIONS FOR NONBLACK SURFACES 63

t-

"-
_ ._

:F I
0 2 4 6 8
Wavele_th, _, _tm

FIGURE 3--4.-Hemispherical spectral emissivity for example 3-4. Surface temperature TA,
2000 ° R.

EXAMPLE 3--4: The ex(h, TA) for a surface at T_=2000 ° R can be

approximated as shown in figure 3-4. What is the hemispherical total
emissivity and the hemispherical total emissive power of the surface?
From equation (3-6d)

e(TA) =_-T_1 f o° ¢_(h, Ta)e_(h, TA)dh=-_ lfo: 0.1 e_'(_t'TA)

T'-_ TAdh

+lfo'J2 6 0"4 e_(X,_ T,O "a

_,r dh+lj_cr 0.2 e_b(h'T5 TA) Tadh

This yields

0.1 f4°°°eXb t _ _ 0.4/'12°°°exb

e(Ta)='_ - Jo 7'-_
5_(^Ta''+-'_'-- J4ooo -_d(krA)

+o.2
o" J,2 ooo _ d(hT_)

where the quantity exblT 5 is a function of hTA. From equation (2-27)

this can be written as

¢(TA) = 0.1Fo-4ooo + 0.4(Fo-,2 ooo- Fo-4ooo) -4-0.2(1- Fo-,2 ooo)

=- 0.3Fo-4ooo+ 0.2Fo-1_ ooo+ 0.2

=- 0.3(0.1051) -4-0.2(0.7877) -4-0.2 = 0.3260

The hemispherical total emissive power is

e(TA) = e(TA)orT 4= 0.326x 0.173 × 10-s(2000) 4= 9020 Btu/(hr)(sq ft).

64 THERMAL RADIATION HEAT TRANSFER

3.4 ABSORPTIVITY

The absorptivity is defined as the fraction of the energy incident on

a body that is absorbed by the body. The incident radiation is the result
of the radiative conditions at the source of the incident energy. The
spectral distribution of the incident radiation is independent of the
temperature or physical nature of the absorbing element (unless the
radiation emitted from the surface is partially reflected back to the
surface). Compared with emissivity, additional complexities are intro-
duced into the absorptivity because the directional and spectral charac-
teristics of the incident radiation must now be accounted for.
Experimentally it is often easier to measure the emissivity than the
absorptivity; hence, it is desirable to have relations between these two
quantities so that measured values of one will allow the other to be cal-
culated. Such relations are developed in this section along with the
definitions of the absorptivity quantities.

3.4.1 Directional Spectral Absorptivity cry(A, /3, 0, TA)

Figure 3-1(c) illustrates the energy incident on a surface element

dA from the (/3, 0) direction. The line from dA in direction (/3, 0) passes
normally through an area element dAe on the surface of a hemisphere
of radius r placed over dA. The incident spectral intensity passing
through dAe is i_,._(h, /3, 0). This is the energy per unit area of the
hemisphere, per unit incident solid angle (the shaded solid angle in
fig. 3-1(c)), per unit time, and per unit wavelength interval. The energy
within the incident solid angle strikes the area dA of the absorbing
surface. The fraction of this incident energy that is absorbed is defined
as the directional spectral absorptivity a'_(h, /3, O, TA). In addition to
depending on the wavelength and direction of the incident radiation, the
spectral absorptivity is a function of the absorbing surface temperature.
The energy per unit time incident from direction (fl, 0) in the wavelength
interval dh is

d3Q'_. _( h, fl, O) = i'_, i( h, fl, O) dAe--dA cos/3 d_ (3-7)

r2

where dA cos /3/r 2 is the solid angle subtended by dA when viewed

from dA_. Note that

dd c°s /3 dA_ =r2-_j-2_cosfl dA = d(o cos fl dA (3-8)

 DEFINITIONS FOR NONBLACK SURFACES 65

where d_ is the solid angle subtended by dAe when viewed from dA

located at the center of the base of the hemisphere (fig. 3-1(c)). Equation
(3-8) will be used in many of the derivations that follow. Equation (3-7)
can then be written as

3 t .t
d Q_, _(h,/3, 0) = t_, i(h, fl, 0) d_ cos fl d_qdh (3-9)

The amount of the incident energy d3Q'_, i that is absorbed is desig-

3 t
nated as d Q_, a- Then the ratio is formed

Directional spectral absorptivity =- a'_ ( k, fl, O, TA ) -- d3Q'_" a( h, _, O, TA )

d 3Ox,t i(h, fl, 0)

d3Q'x, a(h, fl, O, TA)

(3-10a)
tx, i(h, _, 0) dA cosfl doodk

If the incident energy is from black surroundings at uniform temperature

Tb, then there is the special case

O_x(h, fl, O, TA) = d3Qx, a (h, fl, O, TA)

(3-lOb)
i_b, i(h, Tb ) dA cos fl doJdh

3.4.2 Kirchhoff's Law

This law is concerned with the relation between the emitting and
absorbing abilities of a body. The law can have various conditions im-
posed on it depending on whether spectral, total, directional, or hemi-
spherical quantities are being considered. From equations (3-1) and
(3-2) the energy emitted per unit time by an element dA in a wavelength
interval dh and solid angle tho is

d3Q'_, e= i_(k, fl, O, TA ) d_ cos fl dtodh

=_x(h,[J,O, TA)ixb(h, Ta) dAcos_doJdh (3-11)

If the element d.4 at temperature TA is assumed to be placed in an

isothermal black enclosure also at temperature TA, then the intensity
of the energy incident on dA from the direction (/3, 0) (recalling the
isotropy of intensity in a black enclosure) will be i_,b(_,, TA). To maintain
the isotropy of the radiation within the black enclosure, the absorbed and
emitted energies given by equations (3-10b) and (3-11) must be equal.
66 THERMAL RADIATION HEAT TRANSFER

Equating these gives

Cx(;t,/3, O, TA) = cr_,(_,,/3, 0, TA) (3-12)

This equality is a fixed relation between the properties of the material

and holds without restriction. This is the most general form of Kirchhoff's
law. 9

3.4.3 Directional Total Absorptivity ct' (/3, 0, TA)

The directional total absorptivity is the ratio of the energy including

all wavelengths that is absorbed from a given direction to the energy
incident from that direction. The total energy incident from the given
direction is obtained by integrating the spectral incident energy (eq.
(3--9)) over all wavelengths to obtain

O)dh (3-13a)
d2Q_(/3, 0) =cos/3dAdCOfo i_,. i(h,/3,

The radiation absorbed is determined by integrating equation (3-10a)

over all wavelengths, that is,

d2Q'(/3, O, TA) =cos /3dAdto a_(h,/3, O, TA)i'x.i(M/3, O)dh

(3-13b)

The following ratio is then formed:

Directional total absorptivity--a'(/3, 0, TA)= dZQ_(/3' O, TA)

d2Q[ (/3, O)

ff a_,(X,/3, 0, TA)i_,i(X, t3, O)dX

(3-14a)
fo i'X. ,()t, 0, O)d)t

By use of Kirchhoff's law (eq. (3-12)) an alternate form of equation

(3-14a) is

9As will be discussed in chapter 4 in connection with rad/ation properties of electrical conductors, radiation is polarized
in the sense of having two wave components vibrating at right angles to each other and to the propagation direction
For the special case of black radiation the two components of polarization are equal. To be strictly accurate, equation
(3-12) holds only for each component of polarization; and for equation (3-12) to be valid as written for all incident energy,
the incident radiation must be polarized into equal components.
 DEFINITIONS FOR NONBLACK SURFACES 67

¢_(h, fl, O, Ta)i'_. _(h, /3, O)dh

(3-14b)
f[ i'_.i(x, /3, O)dX

3.4.4 Kirchhoff's Law for Directional Total Properties

The general form of Kirchhoff's law (eq. 3-12) shows that c_ and o_
are equal. It is now of interest to examine this equality for the directional
total quantities. This can be accomplished by comparing a special case
of equation (3-14b) with equation (3-3b). If in equation (3-14b) the
incident radiation has a spectral distribution proportional to that of a
blackbody at TA, then i_,.i(h, /3, 0)=C(/3, O)i'_a(h, TA) and equation
(3-14b) becomes

[o:_d(x,/3, o, TA)i_b(h, TA)dX

0((/3, O, TA)=' --_'([3, O, TA)
fo_ i'xb(h, TA)dh (=cr-_)

Hence when _ and od are dependent on wavelength, a'(/3, 0, Ta)

= _' (/3, 0, TA) only when the incident radiation meets the restriction
i'_. i(h,/3, O) = C(/3, O)i'_,(h, Ta) where C is independent of wavelength.
There is another important case when the relation a'(/3, 0, TA)
=4'(/3, 0, TA) is valid. If the directional emission from a surface has
the same wavelength dependence as a blackbody, i_(h, /3, 0, T.4)
:C(/3, 0)_/_(h, TA), then the E_ is independent of h. From equations
(3-3b) and (3-14b) if _(/3, 0, TA)and hence _x_(/3, 0, T_) do not depend
on X, then, for the direction (fl, 0), _, c_,, _', and a' are all equal. A sur-
face exhibiting such behavior is termed a directional gray surface.

3.4.5 Hemispherical Spectral Absorptivity o_x(_, TA)

The hemispherical spectral absorptivity is the fraction of the spectral

energy that is absorbed from the spectral energy incident from all di-
rections over a surrounding hemisphere (fig. 3-1(d)). The spectral energy
from an element d,4, on the hemisphere that is intercepted by a surface
element dA is given by equation (3-9). The incident energy on dA from
all directions of the hemisphere is then given by the integral

dZQx. i = dAdh fa i'x, i(X,/3, 0) cos/3 dto (3-15a)

68 THERMAL RADIATION HEAT TRANSFER

The amount absorbed is found by integrating equation (3-10a) over the

hemisphere

d2Qx. a=dAdh f_, a'_(h, [3, O, TA)i'_._(M [3, O) cos [3dto

(3-15b)

The ratio of these quantities gives

Hemispherical spectral absorptivity =- ax ( h, TA)

dZQx. a ft_ a_(h, [3, O, TA)i'_.i(h, [3, O) cos [3dto

(3-16a)
d2Qx, i i'_, i(h, [3, 0) cos [3 dto

or by using Kirchhoff's law

e_(X, [3, O, TA)i'x, i(X,/3, 0) cos [3 dto

(3-16b)

f_ i'_, i(h, [3, O) cos/3 dto

The hemispherical spectral absorptivity and emissivity can now be

compared by looking at equations (3-16b) and (3-5). It is found that for
the general case, where a_, and e_ are functions of h, [3, 0, and TA,
ax(h, TA)=e_(h, TA) only if i'_.i(h) is independent of[3 and O, that is,
if the incident spectral intensity is uniform over all directions. If this is
so, the i[,i can be canceled in equation (3-16b) and the denominator
becomes 7r which then compares with equation (3-5).
For the case t_(h, TA)=E_(h, TA), that is, the directional spectral
properties are independent of angle, then the hemispherical spectral
properties are related by ax(h, TA) = ex (h, TA) for any angular variation
of incident intensity. Such a surface is termed a diffuse spectral surface.

3.4.6 Hemispherical Total Absorptivity ot(TA)

The hemispherical total absorptivity represents the fraction of energy

absorbed that is incident from all directions of the enclosing hemi-
sphere and for all wavelengths as shown in figure 3-1(d). The total
incident energy" that is intercepted by a surface element dA is deter-
mined by integrating equation (3-9) over all h and all ([3, 0) of the hemi-
sphere which results in
 DEFINITIONS
FORNONBLACK
SURFACES 69

Similarly by integrating equation (3-]0a), the total amount of energy

absorbed is equal to

a_(x,/_, 0, T,)i'_.,(X, _, O)dx] cos/_ d(,,

(3-17b)

The ratio of absorbed to incident energy provides the definition

Hemispherical total absorptivity (in terms of directional spectral ab-

sorptivity or emissivity) = a ( TA )

_ dQ_(TA)_ I,_ If: a_(h, fl, O, TA)i[, i(h, fl, O)dh] cos fl dto

foif: (3-18a)

or from Kirchhofl's law

¢_(h, fl, O, TA)i'_,i(?t, fl, ,O)dh cos fldto

a(TA) ="
fo [ff i_. ,(h, fl, O)dh] cos fldt0

(3-18b)

Equation (3-18b) can be compared with equation (3-6a) to determine

under what conditions the hemispherical total absorptivity and emis-
sivity are equal. It is recalled in equation (3-6a) that

crT]= f,., [fo :¢i'_(_t, TA)dh] cos fl dto

The comparison reveals that for the general case when ¢_ and a;, vary
with both wavelength and angle, then a(T_)= ¢(TA) only when the in-
cident intensity is independent of the incident angle and has the same
spectral form as that emitted by a blackbody with temperature equal to
the surface temperature TA, that is, only when
70 THERMAL RADIATION HEAT TRANSFER

i'_, i()t, _, O)= Ci'_(h, Ta)

where C is a constant. Some more restrictive cases are listed in table 3-II.

TABLE 3--II.--SUMMARY O1r KIBCHHOFF'S LAW RELATIONS BETWEEN ABSORPTIVITY AND

EMISSIVITY

Equality Restrictions
Type of quantity

Directional spectral ........ None

=_(x, t_, 0, TA)

Directional total ............. a'_, 0, T_) Incident radiation must have a spec-

=E'_, 0, T_) tral distribution proportional to that

of a blackbody at TA
i_,,_(X, _, 0)=C(/3, O)i'_(X, TA);
or a_,(/3, 0, TA) = _,(/3, 0, TA) are inde-
pendent of wavelength (directional-
gray surface)

Incident radiation must be independ-

Hemispherical spectral ....
ent of angle
i_,, _O,)= C(_);
or a_,(X, TA)= el(X, TA) do not depend
on angle (diffuse-spectral surface)

a(TA) = E(T_) Incident radiation must be independ-

Hemispherical total ........
ent of angle and have a spectral distri-
bution proportional to that of a black-
body at TA
i'_, _(_t)= Ci'_(X, TA);
or incident radiation independent of
angle and
a[(_, 0, TA)=_[_, 0, TA) are inde-
pendent of g (directional-gray sur-
face);
or incident radiation from each direc-
tion has spectral distribution propor-
tional to that of a blackbody at TA and

a'_(It, TA)= C_(_t, TA) are independent

of angle (diffuse-spectral surface);
or a'_(TA)=¢_ (TA) are independent of
wavelength and angle (diffuse-gray
surface)
 DEFINITIONS FOR NONBLACK SURFACES 71

Substituting equation (3-14a) into equation (3-18a) gives the fol-

lowing alternate forms:

Hemispherical total absorptivity (in terms of directional total absorp-

tivity) = a(TA)

=ft_ [ff i_,i(h, [3, 0)d_t] a'([3, 0, TA) cos [3dto

(3-18c)
fo[f; i'x.i(X, [3, O)dX ]
cos/3dt_

or

a'(/3, 0, Ta)i_(/3, O) cos ,8dto

(3-18d)
fta i_(fl, O) cos/3 dto

where i_(/3, O) is the incident total intensity from direction (/3, 0).
Changing the order of integration in equation (3-18a) and then sub-
stituting equation (3-16a) give

Hemispherical total absorptivity (in terms of hemispherical spectral

absorptivity) =- c_(TA)

d)t
_fo_[ax(h, Ta)fa i_.i(h, fl, O)cos_dto]
(3-18e)
fo [ft_ i'x.i(h, /3, 0) cos _dco] dh

or

otx (X, TA ) dZQx. i

a(TA) = f°_
(3-18f)
ff d2Qx.

where d2Qx. _ is the spectral energy incident from all directions that is
intercepted by the surface element dA.

3.4.7 Summary of Kirchhoff's Law Relations

The restrictions on application of Kirchhoff's law are summarized in

table 3-II.
72 THERMAL RADIATION HEAT TRANSFER

3.5 REFLECTIVITY

The reflective properties of a surface are more complicated to specify

than either the emissivity or absorptivity. This is because the reflected
energy depends not only on the angle at which the incident energy im-
pinges on the surface, but additionally on the direction being con-
sidered for the reflected energy. Some of the pertinent reflectivity quan-
tities will now be defined.

3.5.1 Spectral Reflectivities

3.5.1.1 Bidirectional spectral reflectivity p_ (h,/3r, 0r,/3, 0).-Consider

incident spectral radiation on a surface from direction (/3, 0) as shown
in figure 3-1(e). Part of this energy is reflected into the (/3_, 0r) direction
and provides a part of the reflected intensity in the (/3r, 0r) direction.
The subscript r will always denote quantities evaluated after reflection.
The entire magnitude of the i_. r(h,/3r, 0r) is the result of summing the
reflected intensities produced by the incident intensities i'x,i(h, /3, 0)
from all incident directions (/3, 0) of the hemisphere surrounding the sur-
face element. The contribution to i_, r(h,/3r, 0r) produced by the inci-
dent energy from only one (/3, 0) will be designated as iX, _(h,/3_, 0r,/3, 0)
and it depends on both the incidence and reflection angles.
The energy from direction (/3, 0) intercepted by d/l per unit area and
wavelength is from equation (3-9),

daQ'x, i(X,/3, 0)
dA dX = i_, i(X, /3, O) cos/3 dto (3-19)

The bidirectional spectral reflectivity is a ratio expressing the contribu-

tion that i_, i (},,/3, 0) cos/3 dxo makes to the reflected spectral intensity
in the (/3r, 0r) direction.

Bidirectional spectral reflectivity = p'_(h, /3r, Or,/3, 0)

= i_, ,(x,/3r, Or, /3, o) (3-20)

i_,. i(X,/3, 0) cos /3 dxo

Although the reflectivity is a function of surface temperature, the TA

notation modifying p will be omitted at present for simplicity. The ratio
in equation (3-20) is a reflected intensity divided by the intercepted
intensity arriving within solid angle d¢o. Having cos/3 tho in the denom-
inator means that when p"_(h,/3r, Or, /3, O)i'_. i(h, /3, O) cos /3 do_ is in-
tegrated over all incidence angles to provide the refected intensity
i_,. r(h, fir, 0r), this reflected intensity will be properly weighted by the
 DEFINITIONS FOR NONBLACK SURFACES 73

amount of energy intercepted from each direction. Since i" is one

differential order smaller than i', the dm in the denominator prevents
p_(k, fir, Or, fl, 0) from being a differential quantity. For a diffuse reflec-
tion the incident energy from (/3, 0) contributes equally to the reflected
intensity for all (fir, 0r). It will be shown that the form of equation
(3-20) leads to some convenient reciprocity relations.
3.5.1.2 Reciprocity for bidirectional spectral reflectivity.- It is generally
true that p_(k,/3r, 0r, fl, 0) is symmetric with regard to reflection and
incidence angles, that is, p_ for energy incident at (/3, 0) and reflected at
_r, 0r) is equal to p_ for energy incident at (/3r, Or) and reflected at (fl, 0).
This is demonstrated by considering a nonblack element dA2 located
within an isothermal black enclosure as shown in figure 3-5. For the
isothermal condition, the net energy exchange between black elements

FIGURE 3-5.--Enclosure used to prove reciprocity of bidirectional spectral reflectivity.

d//_ and dAa must be zero. This energy exchange is by two possible
paths. The first is the direct exchange along the dashed line. This direct
exchange between black elements is uninfluenced by the presence of
d,42 and hence is zero as it would be in a black isothermal enclosure
without d,42. If the net exchange along this path is zero and net exchange
including all paths between d,4_ and dz43 is zero, then net exchange
along the remaining path having reflection from dd2 must also be zero.
295-7630L-68--6
74 THERMAL RADIATION HE_T TRANSFER

We can now write the following for the energy traveling along the
reflected path:

d4O,,
v_. ,-2-3 = d4Q_, 3-2-, (3-21a)

The energy reflected from dAz that reaches dA3 is

d 4_" _-- "" r( )k, /3r, 0r, /3, 0) COS /3 r d_42d,43 cos/33 dk
t_k, I--2--3 Lk, r 2

or, using equation (3-19),

dl41 co$ /31

it __ .
d 4 Qx,,_2_3-px( _, /3r, Or,/3, O)i'_. ,(k, T) cos/3 COS /3r

X d/_2 dz43 COS /33 dk (3-21b)

r._

Similarly,

dl43 cos/33
d413,, ,, ., _.
v_.3-2-, =P_(_,/3, 0,/3r, Or)t_,3( , T) cos /3r _ cos/3

X dA2 d_/i cos/31 dk (3-21c)

r2

Substituting equations (3-21b) and (3-21c) into equation (3-21a) gives

....
P_(k,/3r, Or,/3, O) tx, ,(X, T)-p,k(_k,
-- " /3, O, /3r, Or)tk,
"' 3(k, T)

or, because i_.l(k, T)=i_,.3(k, T)=i'_b(k, T), we find the following

reciprocity relation for p'_:

p_(X,/3r, or,/3, O)=p_(X,/3, o,/3r, Or) (3-22)

3.5.1.3 Directional spectral reflectivities.-If i"x,r is multiplied by

dk cos /3rdAdoJr and integrated over the hemisphere for all/3r and 0r,
the energy per unit time is obtained that is reflected into the entire
hemisphere as the result of an incident intensity from one direction

?
3 t ( .it
d Qx, r(k, /3, O)= dkdA Jr_ tk, r(k, /3r, Or, /3, O) COS /3rdoJr
 DEFINITIONS FOR NONBLACK SURFACES 75

By use of equation (3-20) this is equal to

d3Q_, r()t, fl, 0) = i_, _(_,,fl, 0) COSfldtod_dAfp'_Ot , or, 0)

j,_

The directional-hemispherical spectral reflectivity is then defined as

the energy refected into all solid angles divided by the incident energy
from one direction (fig. 3-1(f)). This gives equation (3-23) divided by
the incident energy from equation (3-19)

Directional-hemispherical spectral reflectivity (in terms of bidirectional

spectral reflectivity) =- p_(_, _, 0)

d3Q_, r(_., _,
--d3Q_,i(_., fl, 0)
0) - f,., p_(_t,_r, Or,_,O) COS/3r dtor (3-24)

Equation (3-24) defines how much of the radiant energy incident from
one direction will be reflected into all directions. Another directional
reflectivity is useful when one is concerned with the reflected intensity
into one direction resulting from incident radiation coming from all
directions. It is called the hemispherical-directional spectral reflectivity
(fig. 3-1(g)). The reflected intensity into the _r, 0r) direction is found
by integrating equation (3-20) over all incident directions

i_,, r(_,, fir, 0r) = fa P_(_' fir, 0r, _, O)i'_, _(_,, fl, 0) cos_ dto (3-25)

The hemispherical-directional spectral reflectivity is then defined as the

reflected intensity in the _r, 0r) direction divided by the integrated
average incident intensity

Hemispherical-directional spectral reflectivity (in terms of bidirectional

spectral reflectivity) =- p_(_,, _r, 0r)

P;(k, _r, Or, _, O)i_.i()t, _, 0) cos fld¢o

(3-26)
l fa i'_._(X,B, O) cos _ da,

3.5.1.4 Reciprocity for directional spectral reflectivity.-A reciprocity

relation can also be found for p_ in the following manner. When the
incident intensity is uniform over all incident directions, equation (3-26)
reduces to
76 THERMAL RADIATION HEAT TRANSFER

Hemispherical-directional spectral reflectivity (for uniform incident

intensity) = p_(A, fir, Or)

= ft_ p'_(h,/3r, 0r,/3, 0) cos/3 d_o (3-27)

By comparing equations (3-24) and (3-27) and noting equation (3-22),

the reciprocal relation for p_ results (restricted to uniform incident
intensity)

pi(x,/3, 0) = pi(x,/3r, 0r) (3-28)

where (/3r, 0r) and (/3, 0) are the same angles. This means that the
reflectivity of a material irradiated at a given angle of incidence (/3, 0) as
measured by the energy collected over the entire hemisphere of reflec-
tion is equal to the reflectivity for uniform irradiation from the hemi-
sphere as measured by collecting the energy at a single angle of reflection
(/3r, Or) when (/3r, Or) is the same angle as (/3, 0). This relation is employed
in the design of "hemispherical reflectometers" for measuring radiative
properties (ref. 1).
3.5.l.S Hemispherical spectral reflectivity p_(h).-If the incident
spectral radiation arrives from all angles over the hemisphere (fig. 3-1(h)),
then all the radiation intercepted by the area element dA of the surface
d2Qx, _ is given by equation (3-15a) as

d"Qx, i( h ) = dhdA f,_ i'x, i(h,/3, 0)cos/3 dto

The amount of d2Qx, i that is reflected is, by integration of equation

(3-24),

dZQx, r(h) = ff,t pa(h,/3,

t O) d 3Qx,t _(h,/3, O)

dhdA (_ p_(h,/3, O)i'x, _(h,/3, O) cos/3 dto

jt.a

The fraction of d2Qx, i(h) that is reflected provides the definition

Hemispherical spectral reflectivity (in terms of directional-hemispherical

spectral reflectivity) _ px(h)
 DEFINITIONS FOR NONBLACK SURFACES 77

_ d2Qx, r(X) dhdA f

d2Qx, i(X) = d2Qx, i(X) J,, p](x,/3, o)i'x, i(x,/3, o) cos/3 do (3-29)

3.5.1.6 Limiting cases for spectral surfaces.-Two important limiting

cases of spectrally refecting surfaces will be discussed in this section.
3.5.1.6.1 Diffusely reflecting surfaces: For a diffuse surface the incident
energy from direction (/3, 0) that is reflected produces a reflected in-
tensity that is uniform over all (/3r, Or) directions, but the amount of
energy reflected may vary as a function of incident angle. 1° When viewing
_a diffuse surface element irradiated by an incident beam, the element
will appear equally bright from all viewing directions. The bidirectional
spectral reflectivity is then independent of (/3r, Or), and equation (3-24)
simplifies to

p_.,
' d(
h ,/3, O)
= "
pl()t, /3, O) f cos/3r dto¢

Carrying out the integration gives for a diffuse surface

p_, d(h, /3, O)= 7rp_(h, /3, t9) (3-30)

so that for any incidence angle the directional-hemispherical spectral

reflectivity is equal to zr times the bidirectional spectral reflectivity.
This is because p;,, d accounts for the reflected energy into all (/3r, 0r)
directions, while p[ accounts for the reflected intensity into only one
direction. This is analogous to the relation between blackbody hemi-
spherical emissive power and intensity, e_(h) = 7r/_()t).
Equation (3-25) provides the intensity in the (/3r, Or) direction when
the incident radiation is distributed over (/3, 0) values. If the surface is
diffuse, and if the bidirectional reflectivity is independent of incidence
angle, and if the incident intensity is uniform for all incident angles,
equation (3-25) reduces to

•, ., )f ,, .,
tX, r(h)=p_(h)tx, i(h Jt_ cos/3dto=Trpx(h)t_,,_h) (3-31a)

By using equation (3-30), which applies for the diffuse surface,

•t
tX, r(]t)=pX, t d( _ .t
)tX, t(h) (3-31b)

t* It itt often tacitly assumed that diffuse reflectivities are independent of angle of incidence (fl, #), but this is not a

necessary condition for the diffuse definition.

78 THERMAL RADIATION HEAT TRANSFER

so that the refected intensity in any direction for this case is simply the
hemispherical-directional reflectivity (which has been assumed independ-
ent of incidence angle) times the incident intensity. For the assumed
uniform irradiation, the spectral energy per unit time intercepted by
the surface element dA from all angular directions in the hemisphere is

d_Q_, i( X ) = 7ri'x, i( X) dAd_

so that

, d2Qx i(X)
iL _(X)= p_,_(X) ....... (3-31c)
7r dAdh

3.5.1.6.2'Specularly refecting surfaces: Mirror-like, or specular, sur-

faces obey well-known laws of reflection. The perfect specular reflector
and the perfect diffuse surface provide two relatively simple special
cases that can be used for the calculation of heat exchange in enclosures.
For an incident beam from a single direction, a specular reflector, by
definition, obeys a definite relation between incident and reflected angles.
The reflected beam is at the same angle from the surface normal as the
incident beam and is in the same plane as that formed by the incident
beam and normal. Hence,

/3r =/3, 0r = 0 + 7r (3-32)

and at all other angles, the bidirectional spectral reflectivity of a specular

surface is zero. We can write

p_(X,/3, O,/3r, Or)specular -_- p_(_-,/3, O,/3r =/3, Or = 0-_- 7f') _ p_, s( X, /3, O)

(3-33)

and the bidirectional spectral reflectivity of a specular surface is con-

sidered to be only a function of the incident direction.
For the intensity of radiation reflected from a specular surface into
the solid angle around q3r, Or), equation (3-25) gives, for an arbitrary
directional distribution of incident intensity,

i_, r (_, /3r, Or) = fe_ P_" s()k'/3, O)i_, i(X, /3, O) cos /3 dto (3-34a)

The integrand of equation (3-34a) has a nonzero value only in the small
 DEFINITIONS FOR NONBLACK SURFACES 79

solid angle around the direction (/3, 0) because of the properties of

p_, s(X, r, 0). Equation (3-34a) can then be written as

t_, -" "x,fl,

" r(_,, _r, 0r) = px,,( 0)tx,"' i(X, fl, 0) cos/_d¢o (3-34b)

Let us now consider for a moment the general equation for bidirec-
tional spectral reflectivity (eq. (3-20)). When written for a specular
surface, it becomes

i;. r(_, _r = _, er = o+ _) = p:, s(_', _, o)i'_, i(x, _, _) cos/_

(3-35)

This result is the intensity reflected into a solid angle around (fir, _r)
from a single beam incident at (fl=flr, 0= 0r--T r). The right side of
equation (3-35) is seen to be identical to the right side of equation (3-34b),
which gives the intensity reflected into the solid angle around (fir, 0r)
from distributed incident radiation. The point of this line of reasoning
is to demonstrate the following rather obvious fact: In examining the
radiation reflected from a specular surface into a given direction, only
that radiation incident at the (fl, 0) defined by equation (3-32) need be
considered as contributing to the reflected intensity regardless of the
directional distribution of incident energy.

From equations (3-26) and (3-34a), the hemispherical-directional

spectral reflectivity for uniform irradiation of a specular surface is
given by

foP_,s(X,_, o)i'_,i(x) cos_& i_, r(_',_r, 0r)

p'x,s( X,_,r, Or)--" (3-36a)

l fa i, i(X) cosfl d(° i_,,_(X)

Comparison with equation (3-34b) gives the relation between bidirec-

tional and hemispherical-directional spectral reflectivities for a specular
surface with uniform incident intensity as

pL,(x, _r, 0r)= p_, _(_,,_, 0) cos _ (3 -36b)

Use of the reciprocity relation (eq. 3-28) shows that the directional-
hemispherical reflectivity p_._(X, r, 0) for a single incident beam is

P;,s( _, [3, O) = PX, s (_" _r, Or) = p;, s (_k, _, O) COS_r doJr (3-36C)
80 THERMAL RADIATION HEAT TRANSFER

where the restrictions of equation (3-32) still apply and the incident
intensity is uniform.
The hemispherical spectral reflectivity of a uniformly irradiated
specular reflector is, from equation (3-29),

px, s(h) = 1 f,, pLs(h,[3,0) cos[3dto (3-37)

If p_,s is independent of incident angle, evaluation of the integral in

equation (3-37) gives

px.s (h) = p _, _(h) (3-38)

3.5.2 Total Reflectivities

The previous reflectivity definitions have dealt only with spectral

radiation; the definitions are now considered that include the contribu-
tions from all wavelengths.

3.5.2.1 Bidirectional total reflectivity p"q3r, Or, [3, 0).-The bidirec-

tional total reflectivity gives the contribution made by the total energy
incident from direction (/3, 0) to the reflected total intensity into the
direction ([3r, Or). By analogy with equation (3-20),

Bidirectional total reflectivity =- p"([3r, 0r, [3, 0)

°i/,, r(X, [3r, 0r, [3, O)dh

cos [3 dto fo _
tx. i(h,/3,
.t
O)dX

i'([3r, Or, [3, O) (3-39a)

t i ([3, O) cos [3 dto

As an alternate form, the reflected energy is given by integrating equa-

tion (3-20) over all wavelengths

i r_Pr,
"_ 0r,[3,0) = cos [3 dto fo _ p_'(h,[3r, Or,[3,0)tx, "' i(X,[3,0) dx

so that equation (3-39a) can also be written as

 DEFINITIONS FOR NONBLACK SURFACES 81

Bidirectional total reflectivity (in terms of bidirectional spectral reflec-

tivity) = p"(/3r, 0r,/3, 0)

fo _¢
o'S(x,/3r, or,/3, o)t_. ,(x,/3, O)dX
• t

i_(/3, O) (3-39b)

where i_(/3, O) = f0 _
tx, i(X,/3,
• t
O)dh.

3.5.2.2Reciprocity.-Rewriting equation (3-39b) for the case of

energy incident from direction _r, Or) and reflected into direction (/3, O)
gives

fo_p_(h,/3, 0,/3r, Or)i_,,,(h,/3r, Or)dk

p"(/3, O, _, Or)-- (3-39c)
i;(13r, Or)

Comparison of equations (3-39b) and (3-39c) shows that

p"(/3, O, /3r, Or)= p"(/3r, Or,/3, O) (3-40)

if the spectral distribution of incident intensity is the same for all

directions or in a less restrictive sense if i'x, _h, /3, O)= Ci'x, i(h, /3r, Or).
3.5.2.3 Directional total reflectivity p'.-The directional-hemispherical
total reflectivity is the fraction of the total energy incident from a single
direction that is reflected into all angular directions. The spectral
energy from a given direction that is intercepted by the surface is
i'x, i(h,/3, 0) cos/3 dtodhdet. The portion of this energy that is reflected is
p'_(k, r, O)i'x, t(k,/3, 0) cos/3 d_dhd/l. If these quantities are integrated
over all wavelengths to provide total values, the following definition is
formed:

Directional-hemispherical total reflectivity (in terms of directional-

hemispherical spectral reflectivity) = p'(/3, 0)

_ d2Q;(/3, O)
d2Q _(fl , O)

fo_ p_,(k,/3, O)i_,i(X,/3, O)dk

(3-41a)
fo_ i[, ,(x,/3, O)dX
82 THERMAL
RADIATION
HEAT TRANSFER

Another directional total reflectivity specifies the fraction of radiation

reflected into a given (/3r, 0r) direction when there is uniform irradiation.
The total radiation intensity reflected into the (/3r, 0r) direction when
the incident intensity is uniform for all directions is

i_-(/3r, 0r) = fo i_,r(X, fir, 0r)dh= f[ i'_,,(h)p[(k, /3r, Or)dh

where p_(h,/3r, Or) was discussed in connection with equation (3-27).

Then the reflectivity can be defined as the reflected intensity divided
by the incident intensity:

Hemispherical-directional total reflectivity (for uniform irradia-

tion) _ p'(/3r, Or)

fO _¢
p_,(x, &, or) t_, _(x)dX
• t

(3--41b)

3.5.2.4 Reciprocity. _- Equations (3--41a) and (3-41b) are now compared,

bearing in mind that the latter is restricted to uniform incident intensity.
With this restriction, from equation (3-28) p_,(X, /3, 0)=p_,(_,,/3r, 0r),
it is als%found that

p '(flr, Or) = p'(/3, O) (3--42)

where (/3r, 0r) and (/3, 0) are the same angles when there is afixed spectral
distribution of the incident radiation such that

i'_, ,Or,/3, O)= Ci'_, i(X)

3.5.2.5 Hemispherical total reflectivity p.- If the incident total radiation

arrives from all angles over the hemisphere, the total radiation inter-
cepted by a unit area at the surface is given by equation (3-17a). The
amount of this radiation that is reflected is

dQr= dA f,., p' (_, O)i; (fl, O) cos fl do_

The ratio of these two quantities is then the hemispherical total reflec-
tivity, which is the fraction of all the incident energy that is reflected
including all directions of reflection; that is,
 DEFINITIONS FOR NONBLACK SURFACES 83

Hemispherical total reflectivity (in terms of directional-hemispherical

total reflectivity) ----p

(3-43a)
- dQr = d/tf,,
dQ---_dQ--] P' (_' o)_,., (_, 0) cos # d_,

Another form is found by using d2Ox, _h) which is the incident hemispher-
ical spectral energy intercepted by the surface. The amount of this that
is reflected is p,(h)d2Qx,_ where px(h) is the hemispherical spectral
reflectivity from equation (3-29). Then integrating yields

Hemispherical total reflectivity (in terms of hemispherical spectral

reflectivity) =- p

_px (X) dZQx, i(k)

(3-43b)
dQi

3.5.3 Summary of Restrictions on Reciprocity Relations Between

Reflectivities

In table 3-III, a summary is presented of the restrictive conditions

TABLE 3-III.- SUMMARY OF RECIPROCITY RELATIONS BETWEEN REFLECTIVITIES

Type of quantity Equality Restrictions

A. Bidirectional spectral pi(X, p, e, .8,, 0,) None

(eq. (3-22)) -p_(X,

-- It
#r, or, .8, O)

B. Directional spectral p;,(x,f, o)= p_(X,_,, Or) p._(X, f,, Or) is for uniform
(eq. (3-28)) where f = .8, incident intensity
and 0 = Or or p'_ (k ) independent of
r, 0, fir, and 0r

i'_,i(x, [3, o)=CiLi(x, _T, o_)

C. Bidirectional total ¢'(.8, o, .8,, Or) or p;(f, O, fir, Or)
(eq. (3-40)) = p"(.8,, Or,.8, 0)
independent of wavelength

D. Directional total p'q3, e)=p'q3,., o,.) One restriction from both

(eq. (3-42)) where f = f, B and C
and O = Or
84 THERMAL RADIATION HEAT TRANSFER

necessary for application of the various reciprocity relations for reflec-

tivities.

3.6 RELATIONS AMONG REFLECTIVITY, ABSORPTIVITY, AND EMISSIVITY

From the definitions of absorptivity and reflectivity as fractions of

incident energy absorbed or reflected, it is evident that for an opaque
body (no radiation transmitted through the body) some simple relations
exist between these surface properties. By using Kirchhoff's law (see
section 3.4.7) and taking note of the restrictions involved, further
relations can be found in certain cases between the emissivity and the
reflectivity.
Because the spectral energy per unit time d3Q'x, i incident upon d,4
of an opaque body from a solid angle dto is either absorbed or reflected,
it is evident that

d 3 Qx,,(x,/3,
t
0) = d3Q_.,,_(X,/3, 0, Ta) + dZQ_,,,.(X,/3, 0, Ta)

or

' d3Q_, ,.(h, fl, 0, Ta)

d Qx, a( h,/3, 0, TA) _ -- 1 (3-44)
3 t
dZQ_, i(h, fl, O) d Qx, ,(h, fl, 0)

Since the energy is incident from the direction (/3, 0), the two energy
ratios of equation (3-44) are the directional spectral absorptivity (eq.
(3-10a)) and the directional-hemispherical spectral reflectivity (eq.
(3-24)), respectively. Substituting gives

at(x,/3, o, T_) +p;,(x,/3, o, T_) = 1 (3-45)

Kirchhoff's law (eq. (3-12)) can then be applied without restriction to

yield

E_(h,/3, 0, TA) --I--p_(h,/3, 0, TA) --_- 1 (3-46)

When the total energy arriving at de/ from a given direction is con-
sidered, equation (3--44) becomes

2 t
d 2 qa(/3,
t
0, Ta) 4 dQr(fl, O, Ta) 1 (3-47)
2 t
d Q_(/3, o) d 2 0_(/3,
t
0)

Substituting equation (3-14a) and (3-41a) for the energy ratios results in
 DEFINITIONS FOR NONBLACK SURFACES 85

_'_, o, T_)+p'(B, O, T_)=I (3-48)

The absorptivity is the directional total value, and the reflectivity is

the directional-hemispherical total value.
KirchhofPs law for directional total properties (section 3.4.4) can then
be applied to give

_'(/3, 0, TA)+F(/3, 0, TA)= 1 (3-49)

under the restrictions that the incident radiation obeys the relation
i_, dh,/3, 0)= C(fl, 0)i_n(h, TA) or the surface is directional gray.
If the incident spectral energy is assumed to be arriving at dA from
all directions over the hemisphere, equation (3-44) gives

d2Qx, a(h, TA) d2Qx, rOt, TA)

_- = 1 (3-50)
d2Qx, _(h) d2Q_, i(h)

Equation (3-50) can then be written as

a_(_,, Ta)+ 0x(_-, TA)= 1 (3-51)

where the radiative properties are hemispherical spectral values from

equations (3-16) and (3-29). Substitution of the hemispherical spectral
emissivity ex(h, TA) for ax(h, Ta) in this relation is valid only if the
intensity of incident radiation is independent of incident angle, that is,
it is uniform over all incident directions, or if the ax and _x do not
depend on angle (see section 3.4.7). Under these restrictions, equation
(3-51) becomes

Ex(k, TA)+ px(h, TA)=- 1 (3--52)

If the incident energy on det is summed over all wavelengths and

directions, equation (3-44) becomes

dQa(TA) F dQr(TA) _ 1 (3-53)

dQ_ dQ_

The energy ratios are now the hemispherical total values of absorptivity
and reflectivity (eqs. (3-18) and (3-43a), respectively), and equation
(3-53) becomes

a(TA) +p(TA) = 1 (3--54)

86 THERMAL RADIATION HEAT TRANSFER

Again, certain restrictions apply if _(T.4) is substituted for a(Ta) to

obtain

((T_) +p(T4) = 1 (3-55)

The principal restrictions on the validity of this relation are that the
incident spectral intensity is proportional to the emitted spectral in-
tensity of a blackbody at TA and the incident intensity is uniform over
all incident angles; that is, i_._(X)=Ci_b(X, TA). Other special cases
where the substitution a(T_)= _(TA) can be made are listed in section
3.4.7.
When the body is not opaque so that some radiation is transmitted
entirely through it, a transmitted fraction must be introduced. This
topic is more properly discussed in connection with radiation in ab-
sorbing media.

EXAMPLE 3--5: Radiation from the Sun is incident on a surface in

orbit above the Earth's atmosphere. The surface is at 1800 ° R, and the
directional total emissivity is given in figure 3-3. If the incident energy
is at an angle 25 ° from the normal to the surface, what is the reflected
energy flux?
From figure 3-3 E'(25 °, 1800 ° R)=0.8. The spectrum of radiation
from the Sun is similar to that of a blackbody. Section 3.4.7 shows that
a' (25 °, 1800 ° R) = C(25 °, 1800 ° R)= 0.8, only when the incident spec-
trum is proportional to that emitted by a blackbody at T.4= 1800 ° R.
This is not the case here since the Sun acts like a blackbody at 10 000 ° R.
Hence, a' # 0.8, and without a' we cannot determine p': the emissivity
data given are insufficient to work the problem.

.8

._ .4

.2
to
E

Z
I I I I
4 6 8 10
Wavelength,
k IJm

FIGURE 3-6.--Directional spectral emissivity in normal direction for example 3-6.

 DEFINITIONS FOR NONBLACK SURFACES 87

EXAMPLE 3-6: A surface at T_= 1000 ° R has a spectral emissivity in

the normal direction that can be approximated as shown in figure 3-6.
This surface is maintained at 1000 ° R by cooling water and is then
enclosed by a black hemisphere heated to Ti = 3000 ° R. What will the
reflected intensity be into the direction normal to the surface?
From equation (3-46)

p_(k, _=0 °, TA)----]--e_(X, ,8-----0 °, TA)

which is the reflectivity into the hemisphere for radiation arriving from
the normal direction. From reciprocity, for uniform incident intensity
over the hemisphere,

p_(x, B,=0o, T_)=p_(x,/3=0 o, T_)

Hence, the reflectivity into the normal direction resulting from the
incident radiation from the hemisphere is (by use of fig. 3-6)

p_(0 <_ k < 2, flr=0 °, Ts)=0.7

p_(2 _< h < 5, fir=0 °, Ts) = 0.2

p_(5 _<x _< _, _=0 °, T_)= 0.5

The incident intensity is i_. i(k, Ti)= i_(k, 3000 ° R). From the relation
preceding equation (3-41b), the reflected intensity is

i;(_= 0°) = fo i_,b(;t, T,)p_(k, 13_=0°, T_)d_,

¢r J0 L o'T_ , p_(_,, #r=0 °, TA)d(XTd ]

From equation (2-27) this becomes

i 'r(_ = 0°) = o'T____

(0.7F0-2T i + 0.2F2Ti-sz i+ 0.5sTi- =)
¢r

0.1712
-- (30)4[0.7(0.347)+0.2(0.869--0.347)
71"

+0.5(1 -- 0.869) J
= 18 200 Btu/(hr) (sq ft) (gm) (sr)
88 THERMAL RADIATION HEAT TRANSFER

3.7 CONCLUDING REMARKS

In this chapter, a precise system of nomenclature has been introduced,

and careful definitions of the radiative properties have been given. The
defining equations are summarized in table 3-I for convenience, along
with the symbols used here.
By using these definitions, it was possible to examine the restrictions
on the various forms of Kirchhoff's law relating emissivity to absorptivity.
These restrictions are sometimes a source of confusion, and it is hoped
that the summary given (table 3-II, section 3.4.7) will make clear the
conditions when o_ can be set equal to _. These restrictions are also
invoked when deriving the relation E+p= 1 from the general relation
+ p = 1 for opaque bodies.
The detailed definitions given made it possible to derive the reciprocal
relations for reflectivities and examine the restrictions involved. These
restrictions are listed in a convenient summary in table 3-III, section
3.5.3.

REFERENCE

1. BRANDENBERG, W. M.: The Reflectivity of Solids at Grazing Angles. Measurement

of Thermal Radiation Properties of Solids. Joseph C. Richmond, ed. NASA SP-31,
1963, pp. 75-82.
 Chapter 4. Prediction of Radiative Properties by
Classical Electromagnetic Theory

4.1 INTRODUCTION

James Clerk Maxwell, in 1864, published an article defining what is

generally conceded to be the crowning achievement of classical physics:
the relation between electrical and magnetic fields and the realization
that electromagnetic waves propagate with the speed of light, indicating
strongly that light itself is in the form of an electromagnetic wave (ref. 1).
Ahhough quantum effects have since been shown to be the controlling
phenomena in electromagnetic energy propagation, it is possible and
indeed necessary to describe many of the properties of light and radiant
heat by the classical wave approach.
It will be demonstrated in this chapter that the reflectivity, emissivity,
and absorptivity of materials can in certain cases be calculated from the
optical and electrical properties of the materials. The relations between
the radiative properties of a material and its optical and electrical
properties are found by considering the interaction that occurs when an
electromagnetic wave traveling through one medium is incident on the
surface of another medium.
The analysis will be based on the assumption that there is an ideal
interaction between the incident waves and the surface. Physically this
means that the results are for optically smooth, clean surfaces that
reflect in a specular fashion. The wave propagation and surface inter-
action will be investigated here in a somewhat simplified fashion by using
Maxwell's fundamental _quations relating electric and magnetic fields.
For ideal surface conditions, it is possible to perform more accurate
property computations by using theory that is more rigorous than the
wave analysis presented here. However, the labor involved is generally
not justified, because neither the simplified nor the more sophisticated
approach can account for the effects of surface preparation. The depar-
tures of real materials from the ideal materials assumed in the theory
are often responsible for introducing large variations of measured
property values from theoretical predictions. These departures are
caused by factors such as impurities, surface roughness, surface con-
tamination, and crystal structure modification by surface working.
Although in practice there can be large effects of surface condition,
the theory presented here does serve a number of useful purposes. It

295-763 0L-68--7 89
90 THERMAL RADIATION HEAT TRANSFER

provides an understanding of why there are basic differences in the

radiative properties of insulators and electrical conductors, and reveals
general trends that help unify the p'resentation of experimental data.
These trends are also useful when it is required for engineering calcula-
tions to extrapolate limited experimental data into another range. The
theory has utility in the theoretical understanding of the angular behavior
of the directional reflectivity, absorptivity, and emissivity. Since the
electromagnetic theory applies for pure substances with ideally smooth
surfaces, it provides a means by which one limit of attainable properties
can be computed; for example, the maximum reflectivity or minimum
emissivity of a metallic surface can be determined.
The derivation of radiative property relations from classical theory
is carried out in some detail in sections 4.3 through 4.5. The results are
then gathered and their use demonstrated in section 4.6. Those readers
interested only in the use of the results for property predictions are
invited to pass over the derivation portions to section 4.6.

4.2 SYMBOLS

Cl, C2 constants in Planck spectral energy distribution

c speed of electromagnetic wave
Co speed of electromagnetic wave in vacuum
E amplitude.of electric intensity wave
e emissive power
H amplitude of magnetic intensity wave
K dielectric constant, T/To
n refractive index
complex refractive index, n- iK
re electrical resistivity
s instantaneous rate of energy transport per unit area
S Poynting vector, eq. (4-24)
T absolute temperature
t time
X, y, z coordinates in Cartesian system referenced to interface
between media (fig. 4-1)
¢ ¢ t

x ,y ,z coordinates in Cartesian system referenced to wave

propagating in a medium (fig. 4-1)
angle measured from normal of surface; cone angle
3/ permittivity
emissivity
0 circumferential angle
K extinction coefficient
h wavelength
 PREDICTIONS
BY ELECTROMAGNETIC THEORY 91

/x magnetic permeability
frequency
P reflectivity
X angle of refraction
(D angular frequency

L integration over solid angle of entire enclosing hemisphere

Subscripts:

A property of body or surface A

b black
i incident
M maximum value
n normal
o in a vacuum
r reflected
s specular
t transmitted
x, y, z components in x, y, or z direction
x', y', z' components in x', y', or z' direction
spectral
1, 2 medium 1 or 2
3_ perpendicular component
H parallel componer_t

Superscript:

directional quantity

4.3 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETIC THEORY

Maxwell's equations can be used to describe the interaction of electric

and magnetic fields within any isotropic medium, including a vacuum,
under the condition of no accumulation of static charge. With these
restrictions the equations are, in mks units,

-: oE
v xH= -57+7 (4-I)

vxE=- oi7
/_-_-_ (4-2)

V. E= 0 (4-3)

V./t = 0 (4-4)
92 THERMAL RADIATION HEAT TRANSFER

TABLE 4-I.--QUANTITIES FOR USE IN ELECTROMAGNETIC EQUATIONS IN MKS UNITS

Symbol Quantity Units Value

c Speed of electromagnetic m/sec

wave propagation.
m/sec 2.9979 × 10 s
Co Speed of electromagnetic
wave propagation in
vacuum.
E Electric intensity ................. N/C (newtons/
coulomb)
H Magnetic intensity .............. C/(m) (sec)
K Dielectric constant, "//To ........
re Electrical resistivity ............. (ohm)(m),
(N)(mZ)(sec)/C 2
S Instantaneous rate of energy (N) (m)/(sec) (m 2)
transport per unit area.
X, y, Z Cartesian coordinate m

x', y',z' position.

T Electrical permittivity ........... CZ/(N) (m'-')

Electrical permittivity of C2/(N) (m 2) 1
To x 10 -9
vacuum. 4¢t × 8.9875

g Magnetic permeability .......... (N) (sec_)/C 2

Magnetic permeability of (N) (secZ)/C z 4_r × 10 -7

vacuum.

where/t and E are the magnetic and electric intensities, respectively, Y

is the permittivity, re is the electrical resistivity, and ix is the magnetic
permeability of the medium. The mks units for these quantities are shown
in table 4-I. Zero subscripts denote quantities evaluated in a vacuum.
The solutions to these equations will reveal how radiation waves travel
within a material and what the interaction is between the electric and
magnetic fields. By knowing how the waves move in each of two adjacent
media and applying coupling relations at the interface between the
media, the relations governing reflection and absorption will be
formulated.

4.4 RADIATIVE WAVE PROPAGATION

The derivation of radiative wave propagation for perfect dielectrics

will be considered in section 4.4.1, and then media of finite electrical
conductivity will be analyzed in section 4.4.2.
 PREDICTIONS BY ELECTROMAGNETIC THEORY 93

4.4.1 Propagation in Pedect Dielectric Media

For simplicity, the situation will first be considered where the medium
is a vacuum or other insulator having an electrical resistivity so large
that the last term in equation (4-1), E/re, can be neglected. With this
simplification, equations (4-1) and (4-2) can be written out in Cartesian
coordinates to provide two sets of three equations relating the x, y, and
z components of the electric and magnetic intensities, that is,

OH_ OHy _ T OEx (4-5a)

Oy Oz Ot

OH_ OH_ OEy

- T (4-5b)
Oz Ox Ot

OH u OHm. OE_
ax Oy = T-_ (4-5c)

OEz OEv OH_ (4-6a)

-Oy Oz _=- I_ Ot

OE_ OE_ OH_ (4-6b)

Oz Ox =---t_ ¢9t

OEy OEx._ OH_ (4-6c)

0---_- Oy tx Ot

From equations (4-3) and (4-4), we get

OEx _ __
OEy __, --=
OE_ 0 (4-7)
Ox " Oy Oz
and

OH_ 4- oily + OH_ = 0 (4-8)

Ox ¢9y Oz

The interaction of a wave of incident electromagnetic radiation with a

material will be considered. The coordinate system x, y, z will be fixed
to the material with the x direction normal to the surface. A second
coordinate system x', y', z' is fixed to the path of the incident wave
(fig. 4-1).
94 THERMAL RADIATION HEAT TRANSFER

y'

z' rL wave L

p,an[ /

FIgu]lz 4-1.-Definition of coordinate systems.

For simplicity, a plane wave of incident radiation is considered

that is propagating in the x' direction. From the definition of a plane
wave, this wave has all quantities concerned with it constant over any
y'-z' plane at any given time. Hence O/Oy'=O/Oz'=O. For these
conditions, equations (4-5) to (4-8) reduce to the following:

OEx,
0 = TOt (4-9a)

OHm, OE u,
ax' = y Ot (4-9b)

OH u,_ OE_,
Ox' _ Ot (4-9c)

OHx,
0 =--/.t Ot (4-10a)

OE_, OHu,
Ox "=-ix O----t- (4-lOb)

OE u, = OHm,
(4-10e)
Ox' ix Ot

0__,=E 0 (4-11)
Ox '
 PREDICTIONS BY ELECTROMAGNETIC THEORY 95

oHx,= 0 (4-12)
OX'

The H components are then eliminated by differentiating equations

(4-9b) and (4-9c) with respect to t and equations (4-10b) and (4-10c)
with respect to x' to obtain

02Hz , 02Ey,
OtOx' = y Ot 2 (4-13a)

02Hit,_ 02Ez ,
0--_x' y _ (4-13b)

O"E_,_ OZH__ (4-14a)

Ox '2 _ Ox'Ot

O2Ey'- 02H_' (4-14b)

Ox '_ ----I_ Ox'Ot

Equations (4-13a) and (4--14b) are then combined to eliminate H_,

and similarly (4-13b) and (4-14a) to eliminate Hy,. This provides the
following two equations:

0 2E u , 0 2E u, (4-15a)
IZY OtC-= Ox,-----_z

and

02E,, O2E z,
/xy -7 = 0x'2 (4--15b)

These wave equations govern the propagation of the y' and z' compo-
nents of the electric intensity in the x' direction. For simplicity in the
remainder of the derivation, it will be assumed that the electromagnetic
waves are polarized such that the vector g is contained only within the
x'-y' plane (see fig. 4-2).. Then Ez, and its derivatives are zero and
equation (4-15b) need not be considered. The vector/_ will have only
x' and y' components.
With regard to the x' components of/_ and H, from equations (4-9a),
(4-10a), (4-11), and (4-12), OEx, l_t = _Ex,/_x' = OHx,/Ot= OHx,/Ox' = O.
Hence, the electric and magnetic intensity components in the direction
of propagation are both steady and independent of the propagation
direction, x'. Consequently, the only time-varying component of/_ is
Ey, as governed by equation (4-15a). Since this component is normal
to x', the direction of propagation, the wave is a transverse wave.
96 THERMAL RADIATION HEAT TRANSFER

yt

XI

ZI

FIGURE 4-2.--Electric field wave polarized in x'-y' plane, traveling in x' direction with
companion magnetic field wave.

Equation (4-15a) is recognized as the wave equation that describes

the propagation of the wave component Ey, in the x' direction. The
general solution of this equation is

t ,+ t

where f and g are any differentiable functions. The f function provides

propagation in the positive x' direction, while the g function accounts for
propagation in the negative x' direction. Since the present discussion
deals with a wave moving in the positive direction, only the f function
will be present in the analysis.
To obtain the wave propagation speed, consider an observer moving
along with the wave; the observer will always be at a fixed value of Ey,.
The x' location of the observer must then vary with time such that the
argument of f, x'-- (t/X/-_gy) is also fixed. Hence, dx'/dt = 1/X/-_gT. The
relation

Ev,=f(x' _gy) (4-16b)

thus represents a wave with y' component Ey, propagating in the positive
x' direction with speed 1/V_gV. In free space, the propagation speed of
the wave is Co, the speed of electromagnetic radiation in vacuum, so that
there is the relation Co = _//]')/*LO')/O
-11

H Independent measurements of #to, y., and co validate the result. The fact that Maxwell's equations predict 'that

all electromagnetic radiation propagates in vacuum with speed co was considered convincing evidence that light is a

form of electromagnetic radiation, and was one of the early triumphs of the electromagnetic theory.
 PREDICTIONS BY ELECTROMAGNETIC THEORY 97

Accompanying the Ey, wave component is a companion wave compo-

nent of the magnetic field. If equation (4--9b) is differentiated with
respect to x' and equation (4-10c) with respect to t, the results can be
combined to yield

2H z , _ c92Hz,
/x_/ 0t 2 ¢3x,2 (4-17)

Equation (4-17) is the same wave equation as equation (4-15a). Hence,

the Hz, component of the magnetic field propagates along with Ey, as
shown in figure 4-2.
Any propagating waveform as designated by theffunction in equation
(4-16b) can be represented using Fourier series as a superposition
of waves, each wave having a different fixed wavelength. Let us then
consider only one such monochromatic wave, and note that any wave-
form could then be built up from a number of monochromatic compo-
nents. For convenience in later portions of the analysis, the wave com-
ponent will be given in complex form.
Suppose that at the origin (x '= 0) the waveform variation with time is

Ey, = Ev,u exp (itot)

A position on the wave that leaves the origin (x' = 0) at time tl arrives at
location x' after a time interval x'/c, where c is the wave speed in the
medium. A wave traveling in the positive x' direction is then given by

Ey'= E_'uexp [ito (t-_)]

or

Ey,=Ey,uexp [ioJ (t- X/_gyx')] (4--18a)

This is a solution to the governing wave equation (eq. 4-15a) as shown

by comparison with equation (4-16b). If desired, other forms of the
solution can be obtained by using the relations w = 2try = 2rrc/k = 27rco/)to,
where )t and _,o are the wavelengths in the medium and in a vacuum,
respectively.
The simple refractive index n is defined as the ratio of the wave
speed in vacuum Co to the speed in the medium c= 1/V_g T. Hence,
98 THERMAL RADIATION HEAT TRANSFER

rt =--= Co
C

and equation (4-18a) can be written as

Ey' = Ey'M exp [ico (t-_o x') ] (4-18b)

As shown by equation (4-18), the wave propagates with undiminished

amplitude through the medium. This is a consequence of the assumption
that the medium can be regarded as a perfect dielectric, that is, one with
zero conductivity. In many real materials the conductivity is significant
and the last term on the right in equation (4-1) cannot be neglected. As
will now be shown, the inclusion of this term will lead to an attenuation
of the wave.

4.4.2 Propagation in Isotropic Media of Finite Conductivity

For simplicity, a single plane wave is again considered as described by

equations (4-18). If an exponential attenuation with distance is introduced
(it will be shown by equations (4-21) to (4-23) that this obeys Maxwell's
equations), the wave takes the form

Eu;=Eu'M exp[ito (t--_oX')] exp (--_oKx' ) (4-19a)

where K is termed the extinction coefftcient for the medium. The attenua-
tion term indicates an absorption of the energy of the wave as it travels
through the medium. The present form of the attenuation exponent was
chosen so that the exponential terms could be combined into the relation

(4-19b)
Eu'-= Eu'M exp [ico [t-- (n-- iK) _o]}

Using complex number relations, equation (4-19b) can be written for

later reference as

' X'

(4-1%)
 PREDICTIONS BY ELECTROMAGNETIC THEORY 99

A comparison of equation (4-19b) with equation (4-18b) shows that the

simple refractive index n has been replaced by a complex term that will
be termed the complex refractive index ft. Thus,

fi = n- it¢ (4-20)

It remains to be shown that equation (4--19b) constitutes a solution

of the governing equations with the last term on the right of equation (4-1)
included. With this term retained, equation (4--15a) takes the form

"> 2

(4-21)
_Y Ot 2 OX '2 re Ot

The waveform of equation (4-19b) is substituted into equation (4-21)

and the following equality results:

c2gy = (n- iK) "-+ igkoco

2_rre (4-22a)

where _,o is the wavelength in a vacuum. Equation (4-22a) provides the

relation between the wavelength and the properties of the medium neces-
sary for the wave to satisfy Maxwell's equations. Equating the real and
imaginary parts of equation (4-22a) yields

n 2 -- K 2= _,/_')/c 2 (4-22b)

and

nK ---IXX°C° (4--22C)
47rre

These equations may be solved for the components of the complex

refractive index, n and K, in terms of it, y, Xo, Co, and re to yield

n ----f- 1+ 1+ (4-23a)

and

• ttyc2o Xo 2
(4-23b)
100 THERMAL RADIATION HEAT TRANSFER

In the solutions, positive signs were chosen in front of the square roots
since n and K must physically be positive real quantities.
Comparison of equation (4-18b), the solution to the wave equation for
dielectric media, with equation (4-19b), the solution of the wave equation
for conducting media, shows the solutions to be identical with one excep-
tion: The simple refractive index n appearing in the dielectric solution is
replaced for conductors by the complex refractive index (n-- iK). This is a
most important observation. It means that any general relations that we
derive for dielectrics will also hold for conductors provided that we
substitute the complex index (n-iK) for the simple refractive index n.
Extensive use will be made of this analogy in succeeding sections.

4.4.3 Energy of an Electromagnetic Wave

The instantaneous energy carried per unit time and per unit area by
an electromagnetic wave is given by the cross product of the electric
and magnetic intensity vectors. This product is called the Poynting
vector S where

S=ExH

and according to the properties of the cross product, S' is a vector

propagating at right angles to the/_ and/1 vectors in a direction defined
by the right-hand rule. For the plane wave under consideration as shown
in figure 4-2, the propagation is in the positive x' direction. The magni-
tude of S" is given for the plane wave by

I-SI= Eu,H_, (4-24)

If E u, is given by equation (4-19b), then equation (4-10c), which holds

for conductors as well as dielectrics, can be used to find Hz, as follows:

OHz,_ OEu, _-- ito (n -- iK)Eu, = -- i¢o-_Eu '

--tx Ot c3x' Co Co

Then noting the t dependence of E u, in equation (4-19b) and integrating

yield the following relation between electric and magnetic intensities:

Hz, = _ Eu, (4-25)

p.co

The constant of integration has been taken to be zero. It would corre-

spond to the presence of a steady magnetic intensity in addition to that
 PREDICTIONS BY ELECTROMAGNETIC THEORY I01

induced by E v, and is zero for the conditions of the present discussion.

When Hz, is substituted in equation (4-24), the magnitude of the
Poynting vector becomes
2
]5] =--Ey, (4-26)
/zco

Thus, the instantaneous energy per unit time and area carried by the
wave is proportional to the square of the amplitude of the electric
intensity.
Because 171 is a monochromatic property, it is seen by examination
of its definition to be proportional to the quantity we have called spectral
radiant intensity. For radiation passing through a medium, the exponen-
tial decay factor in the radiant intensity must then be, by virtue of equa-
tion (4-26), equal to the square of the decay term in Ey,. Thus, from
equation (4-19a) the intensity decay factor is exp (-2toKx'/co) or
exp (-- 4rrKx'/Xo).

4.5 LAWS OF REFLECTION AND REFRACTION

In the previous derivations, the wave nature of the propagating

radiation has been revealed and the characteristics of movement through
an isotropic medium have been found. The analysis provided a complex
refractive index which is related to the velocity of propagation and the
wave attenuation as it moves through a medium. Now the interaction of
the electromagnetic wave with the interface between two media will be
considered. This will provide laws of reflection and refraction in terms
of the complex refractive indices which are in turn related to the electric
and magnetic properties of the media by means of equation's (4_23).
For simplicity throughout this discussion a simple cosine _vave will
be utilized as obtained by retaining only the cosine term in equation
(4-19c). This wave is moving in the x' direction and strikes the interface
between two media as shown in figure 4-3. The plane containing both
the normal to the interface and the incident direction x' is defined as
the plane of incidence (fig. 4-1). In figure 4-3 the coordinate system has
been drawn so that the y' direction is in the plane of incidence. The
interaction of the wave with the interface depends on the wave orien-
tation relative to the plane of incidence. For example, if the amplitude
vector of the incident wave is in the plane of incidence (amplitude vector
in the y' direction), the amplitude vector is at an angle to the interface.
If the amplitude vector is normal to the plane of incidence (amplitude
vector in z' direction), the incident wave vector is parallel to the interface.
Figure 4-3 shows a plane transverse wave front propagating in the
x' direction. Although the wave will in general bend as it moves across
102 THERMAL RADIATION HEAT TRANSFER

yl

X'

_Plane wave front at

successive times

Medium 1
Interface Medium 2 = y

Ay - _x'/sin

[ "

FIGURE 4-3.- Plane wave incident upon interface between two media.

the interface because of the difference in propagation velocity in the

two media, the wave will be continuous, so that the velocity component
tangent to the interface (y component) is the same in both media at the
interface. This continuity relation will be used in deriving the laws of
reflection.
Consider now an incident wave Ell, _ polarized so that it has amplitude
only in the x' -y' plane (t_g. 4-4) and, hence, is parallel to the plane of
incidence. From equation (4-19c), retaining only the cosine term for
simphcity, the wave is characterized by

Ell, i =EMIl,; cos (tot ntoX'_co

/ (4-27)

From figure 4-4(a), the components of the incident wave in the x, y, z

coordinate system are (components are taken to be positive in the posi-
tive coordinate directions)

Ex, i=-Eii, i sin/3 (4-28a)

Eu, i=EH, i cos/3 (4-28b)

Ez = 0 (4-28c)
 PREDICTIONS BY ELECTROMAGNETIC THEORY 103

Incident wave'_H.i_. /Reflected wave

(a) _ Refractedwave

,--Plane of incidence

ll,i EII,r

.No,ma,X// I .
_ to inier-_r/' I /

Medi_

Medium

(b)

(a) Plane electric field wave polarized in x-y plane striking intersection of two media.
(b) Electric intensity, magnetic intensity, and Poynting vectors for incident wave polarized
in plane of incidence.

FIGURE 4-4.--Interaction of electromagnetic wave with boundary between two media.

Substituting equation (4-27) into equations (4-28) and noting that x',
the distance the wave front travels in a given time, is related to the y
distance the front travels along the interface by
104 THERMAL RADIATION HEAT TRANSFER

x'=y sin/3 (4-29)

as can be seen from figure 4-3, we obtain for the incident components

Ex, i = -- EMII, i sin fl cos [t° (t -- nl y sin /3) ]Co (4-30a)

,cos cos
[ot, nly:in1 ) (4-30b)

Ez, i = 0 (4-30c)

Upon striking the bounding y- z plane between medium I and medium

2, the incident wave separates into a portion ElL, r reflected at angle/3r
and a portion Ell , t refracted at angle X and transmitted into medium 2.
From the geometry shown in figure 4-4, the components in the positive
coordinate directions of the reflected ray evaluated at the interface are
m

(4-31a)

(4-31b)
Elt, r___VMH, r COS /3r COS I6O It nl y sin /3r) ]co

Ez, r = 0 (4--31C)

The direction of Ell, _ was drawn such that Ell, r, Hr, and S_ would be
consistent with the right-hand rule connecting the Poynting vector with
the E and H fields. In a similar fashion from figure 4--4, the components
of the refracted portion of the wave are

Ex't=--EMll't sin x c°s [t° ( t n2y ]Co

sin x)
(4-32a)

Eu't=EMIl't c°s x c°s [t° (t-nzysin X) ]Co

(4-32b)

E_, t= 0 (4-32c)

Certain boundary conditions must be followed by the waves at the

interface of the two media. The sum of the components, parallel to the
interface, of the electric intensities of the reflected and incident waves
must be equal to the intensity of the refracted wave in the same plane.
This is because the intensity in medium 1 is the superposition of the
incident and reflected intensities. For the polarized wave considered
 PREDICTIONS BY ELECTROMAGNETIC THEORY 105

here, this condition gives the following for the equality of the y compo-
nents (parallel to interface) in the two media:

__EMII, r COS _r COS [OJ It nlysin/3r)]co

n Ysi°x)]lco
4_33,
Since equation (4-33) must hold for arbitrary t and y and the angles
/3,/3r, and X are independent of t and y, the cosine terms involving time
must be equal. This can only be true if

nt sin/3=/'/1 sin/3r = n_ sin X (4-34)

which provides that

/3=/3r (4-35)

The angle of reflection of an electromagnetic wave is thus equal to its

angle of incidence (rotated about the normal to the interface through a
circumferential angle of 0= rr). These are the relations that define
mirrorlike or specular reflections as discussed in section 3.5.1.6.2.

Equation (4-34) also yields the following relation between /3 and X:

n, n,-iK,
• = = = (4-36)
sin/3 n2 n2 -- iK2

where the definition of n has been substituted from equation (4-20).

For the general case where K1 and K2 are not zero, equation (4-36)
shows that sin X must be complex since _ and K2 are complex quantities.
This complex ratio of angles can be interpreted to mean that the inter-
action of the incident wave with the interface will result in both phase
and amplitude changes for the refracted wave.
With the cosine terms invblving time equal and by using equation
(4-35), there also follows from equation (4-33)

(EMIl, i cos 3--EMIl, r cos _=--EMII, t COS X)x=0 (4-37)

This can be used to find how the reflected electric intensity is related

295-763 0L--68--8
 THERMAL RADIATION HEAT TRANSFER
106

to the incident value. The refracted component EMIl, t must be eliminated

and to accomplish this the magnetic intensities must be considered.
The magnetic intensity parallel to the boundary must be continuous
at the boundary plane. The magnetic intensity vector is perpendicular
to the electric intensity; since the electric intensity being considered
is in the plane of incidence, the magnetic intensity will then be parallel
to the boundary. Continuity at the boundary provides that

(Hi + Hr = Ht)x=o (4-38)

The relation between electric and magnetic components was shown

in equation (4-25). Although for simplicity this relation was derived for
only the specific components Hz, and Eu,, it is true more generally so
that the magnitudes of the E and/4 vectors are related by

i/_l =____n i/_l (4-39)

IZCo

For both dielectrics and metals the magnetic permeability is very

close to that of a vacuum so that tz -_ go. Then equation (4-38) can be
written as

(n,EMll, _+ nlEM][, r = n2EMI[, ,)x=O (4-40)

Equations (4-37) and (4-40) are combined to eliminate EMll, t and give
the reflected electric intensity in terms of the incident intensity

cos B _1
EMII,_ cos X n,z
(4-41)
EMII,i cos i-Z1 _

COS X n2

If the preceding derivation is repeated for an incident plane electric

wave polarized perpendicular to the incident plane, the relation between
reflected and incident components is

C0$ X nl

EMI, r COS _ _2
(4-42)
EM±, i COS X nl
t--
cos _ n2

The general relations in this section will now be interpreted for the
specific cases of dielectrics and metals.
 PREDICTIONS BY ELECTROMAGNETIC THEORY 107

4.5.1 Incidence and Reflection of a Wave from Dielectric or Transparent

Media (K Negligible Compared to n)

When each of the media either is dielectric or transparent, then

K1= K2--* 0 and equation (4-36) reduces to

sin X= n___ (4-43)

sin/3 nz

This relates the angle of refraction to the angle of incidence by means

of the refractive indices. Equation (4-43) is known as Snell's law. For
the often encountered case when the incident wave is in air (nl _ 1),
then n2 = sin 0/sin X.

For dielectric media, equation (4-41) reduces to

CO$ /3 nl

EMll,r_COS X n2
(4-44)
EMll, i cos/3 _ n_
cos X n2

Then equation (4-43) can be used to eliminate n_/n2 in terms of

sin x/sin/3. With some manipulation using trigonometric identities, the
resulting expression can be cast into the form

= tan (fl -- X)
EMII, i tan (fl+X) (4-45)

Similarly from equation (4-42)

cos _ nl
EM±,r_ COS/3 n2 sin(fl--X)
(4-46)
EM±, i COS X }_n, sin (/3 + X)
cos/3 n2

The energy carried by a wave is proportional to the square of the

amplitude of the wave as shown by equation (4-26). Squaring the ratio
EM, r/EM, i therefore gives the ratio of the energy reflected from a surface
to the energy incident upon the surface from a given direction. This
ratio was defined in section 3.5.1.3 as the directional-hemispherical
reflectivity. Because electromagnetic radiation for the ideal conditions
examined here was shown by equation (4-35) to reflect specularly and
because the electromagnetic theory relations are based on monochromatic
waves, the energy ratio more exactly gives the directional-hemispherical
spectral specular reflectivity as discussed in section 3.5.1.6.2. The
108 THERMAL RADIATION HEAT TRANSFER

spectral dependence arises from the variation of the optical constants

with wavelength.
The values of p_(;_, /3, 0) for incident parallel and perpendicular
polarized components are then obtained as

pi,,,(x,/3, 0)=
\EMIl, i/
(4-47)
, q"
Px±, s( _', /3, 0)_- \E-_l,//

The subscript s denotes a specular reflectivity, and the notation is that

used in section 3.5.1.6.2. Because all reflectivities predicted by electro-
magnetic theory are specular, the subscript s will not be carried from
this point on in order to simplify an already complicated notation. Fur-
ther, because of the assumption of isotropic behavior at the surface
for the ideal surfaces considered, there is no dependence on the angle
0; hence, this variable will no longer be retained.
For unpolarized incident radiation the electric field has no definite
orientation relative to the incident plane and can be resolved into
parallel and perpendicular components that are equal. Then the direc-
tional-kemispherical spectral specular reflectivity is the average of
P_ll(k,/3) and p_z(X,/3). By using equations (4-45) to (4-47), the result is

p_(,k,/3)
= P;'It(X'
!3)+0;,i(X,/3)
2

1 [tau 2 (/3 -- X) 4- sin" (/3 -- X)] ] sin 2 (/3-- X) [! + cos2 (/3 +__X)] (4-48)
-- 2 Ltan 2 (/3 + X) sin" i-/3--_--XX) ] = 2 sin _ (/3 + X) L cos2 (/3 -- X)]

Equation (4-48) is known as Fresnel's equation, and it gives the direc-

tional-hemispherical spectral reflectivity for an unpolarized ray incident
upon a dielectric medium. The relation between X and /3 is given by
equation (4-43).
In the special case when the incident radiation is normal to the inter-
face between the two media, cos /3= cos ×= 1 and equations (4-44)
and (4-46) yield

nl
1----

= = ,,..,_n.,--n, (4-49)
EMIl, i EM±,i 1 + nl n,_J,- nl
_2

The normal directional-hemispherical spectral specular reflectivity is

then
 PREDICTIONS BY ELECTROMAGNETIC THEORY 109

t t (/'t2 -- /_1_ 2

p_., .(k) -- p_.(k, _ : _r: 0): \n2 -4-nl) (4-50)

For a wave entering the dielectric from air (nl _ 1),

, _n2-- 1_2
Px, .(k): \n2 + l/ (4-51)

The foregoing reflectivities are spectral quantities because nl and

nz are functions of k.

4.5.2 Incidence on an Absorbing Medium

When the media have significant K values, the theoretical relations

are of the same form as the dielectric case except that the complex
dielectric constant _ is retained. The angles /3 and X are related by
equation (4-36), and the relations between reflected and incident wave
amplitudes given by equations (4-41) and (4-42) can be used for the
wave interaction at the interface between two metals or between an
absorbing dielectric and a metal.
For a ray incident normal to the interface, equations (4-41) and (4-42)
yield, in an analogous fashion to equation (4-50),

, [(n2--itc2)--(n,--iK1)] 2
(4-52)
Px, _(k) = L(nz - iK=)_ (hi- iKl)J

This is a complex quantity and can be interpreted as giving both the

magnitude and phase change of the reflected wave. The ratio of the
magnitude of the reflected energy to that of the incident energy is ob-
tained by multiplying by the complex conjugate of equation (4-52) to
give

, _ (n2 - nd" + (K_ - ,<_)2

Pk. n(_) (17.2-_-nl)2-[ - (--_2_-K-_ (4--53)

For an incident ray in air (nl = 1, K, _- 0) striking an absorbing mate-

rial (n2, K2), equation (4-53) reduces to

(nz- 1)5 + Kzz

pL .(k) - (nz+ 1)Z+K_ (4--54)

When the material is transparent (K2--_ 0), equation (4-54) reduces to

equation (4-51).
For oblique incidence the directional-hemispherical reflectivity can
be obtained from equations (4-41) and (4-42). For an incident ray po-
] 10 THERMAL RADIATION HEAT TRANSFER

larized parallel to the plane at incidence, equation (4-41) gives the

complex ratio

EMIl. r__ COSX

os, \n.z -- iK.z!
(4-55)
EMII, i cos /3 4- (n,--iu___l_
cos X \n2 -- iK.z/

The reflectivity is obtained as the square of the ratio of the reflected

magnitude to the incident magnitude, which is found by multiplying
equation (4-55) by its complex conjugate. This yields

(n_ cos _-n, cos X)2+ (K2 cos/3-_, cos X) _

P_,II( k, _)=
(n., cos/3+nl cos ×)5+ (K2 cos 13+tq cos X) 2

(4-56)

Similarly for the polarized component perpendicular to the incident

plane

(n2 cos X- n, cos/3) 2 + (K2 cos X--K, cos _)2

(rt2 cos x+n, cos B)2+ (K., cos X+tq cos fl)2

(¢-57)

As before, if the incident beam has no specific polarization, the reflec-

tivity is an average of the parallel and perpendicular components as in
equation (4-48).
To this point in this chapter, the. wave nature of radiation has been
shown from a consideration of Maxwell's equations. Then the interaction
of these waves with nonabsorbing and absorbing media has been dis-
cussed in terms of the refractive index n and the complex refractive index
ft. Now the results will be applied more specifically to a discussion of
some actual radiative properties.

4.6 APPLICATION OF ELECTROMAGNETIC THEORY RELATIONS TO RADIA-

TIVE PROPERTY PREDICTIONS

The electromagnetic theory as applied here to radiative property

prediction has a number of drawbacks that limit its usefulness for prac-
tical calculations. Aside from the many assumptions used in the deriva-
tions, the theory itself becomes invalid when the frequencies being
considered become of the order of molecular vibrational frequencies.
 PREDICTIONS BY ELECTROMAGNETIC THEORY 111

These qualifications restrict the equations used here to wavelengths

longer than in the visible spectrum.
The theory completely neglects the effects of surface conditions on
the radiative properties. This is its most serious limitation, since per-
fectly clean optically smooth interfaces are rarely encountered in prac-
tice. The greatest usefulness of the theory is probably in providing a
means for intelligent extrapolation when only limited experimental data
are available. In the following sections, the equations of electromagnetic
theory that are useful for the prediction of properties will be examined
and the assumptions inherent in their derivation discussed.

4.6.1 Radiative Properties of Dielectric (K --_ O)

The equations to be examined in this section all contain these assump-

tions: (1) The medium is isotropic; that is, its electrical and internal
optical properties are independent of direction. (2) The magnetic per-
meability of the medium is equal to that of a vacuum. (3) There is no
accumulation of static electrical charge. (4) No externally produced
electrical conduction currents are present.
The measured index of refraction of the medium is, in general, a
function of wavelength, and thus any calculated radiative properties
will be wavelength dependent. If, however, the refractive index is
calculated from the permittivity 7 or the dielectric constant K (where
K=T/To), which are not generally given as functions of wavelength,
the spectral dependency is lost. Because of these considerations, no
notation is used in the following equations to signify spectral depend-
ence, but the reader should be aware that such dependence can be
included if the optical or electromagnetic properties are known as a
function of wavelength.
The surfaces are further assumed to be "optically smooth," that is,
smooth in comparison with the wavelength of the incident radiation so
that specular reflections result.
4.6.1.1 Reflectivity.-Under the aforementioned restrictions, the di-
rectional-hemispherical specular reflectivity of a wave incident on a
surface at angle/3 and polarized parallel to the plane of incidence may
be obtained from equations (4-47) and (4-45) as

= ftan x/1 (4-58)

Pil(fl) Ltan _--_j

Similarly, from equations (4-47) and (4-46), for a wave polarized per-
pendicular to the incidence plane
112 THERMAL RADIATION HEAT TRANSFER

, [sin (/3 -- X)]2 (4-59)

p1(/3) = q3¥ j

where X is the angle of refraction in the medium on which the ray

impinges. For a given incident angle/3, the angle X can be determined
from equation (4-43) as

sin X= n__tm
= ____= V_l
(4-60)
sin/3 ne X/_T2

where T is the permittivity, K is the dielectric constant, and n is the

refractive index; the n, T, and K are assumed not to have any angular
dependence.
The reflectivity for unpolarized radiation was shown in equation
(4-48) (Fresnel's equation) to be given by

p,(/3)= lsinz (/3-X)[1

sin 2 (/3+×) + cos
c°s22 (/3-x)J
(/3+X)] (4-61)

EXAMPLE 4-1: An unpolarized beam of radiation is incident at angle

/3=30 ° from the normal on a dielectric surface (medium 2) in air (me-
dium 1). The surface is of a material where /(2 _ 0 and whose index of
refraction is n2=3.0. Find the directional-hemispherical reflectivity
for the polarized components and the unpolarized beam.
Because the incident beam is in air, n_--iK_ _ 1, and from equa-
tion (4-60), n_/n2 = 1/3.0=sin x/sin 30°; therefore, X=9.6 °. The re-
flectivity for the parallel component is, from equation (4-58),

P[I(/3 = 30°) = { [tan (20.4°)]/[tan (39.6°)] }2 = 0.202

and that for the perpendicular component is, from equation (4-59),

p5_(/3=30°) = {[sin (20.4°)]/[sin (39.6°)]}2=0.301

The reflectivity for the unpolarized beam obtained from equation (4-61)
or, more simply here, from the average of the components is

P' (/3 = 30°) = (0.202 + 0.301 )/2 = 0.252

By performing the type of calculation shown in example 4-1 for various

incidence angles and ratios of the indices of refraction, the reflectivity
 PREDICTIONS BY ELECTROMAGNETIC THEORY 113

can be tabulated or presented graphically. This is a directional-hemi-

spherical reflectivity in that it provides all the reflected energy resulting
from an incident beam from one direction. It is a spectral quantity in
the sense that the indices of refraction can correspond to a particular
wavelength if the details of the wavelength dependency are available.
Finally, it is a specular quantity in that it obeys the constraints of equa-
tion (4-35).
4.6.1.2 Emissivity.- After the reflectivity has been evaluated, the
directional spectral emissivity can be found from equation (3-37) as

e' (/3) = 1 --P'(3)

where the body is opaque.

Angleot emission, !_, (leg

0 Normal
10

}./Emitted

Medium1 [P/_ ray

.\\_\\_1/
Medium2 I//////"

5O

6O

8O

90
.2 .4 .6 .8 LO
Directional emissivity, e'(i5)

FIcultg 4-5.--Directional emissivity as predicted from electromagnetic theory.

114 THERMAL RADIATION HEAT TRANSFER

A graph of the directional emissivity is shown in figure 4-5 for various

ratios of I .,I/In,I, which is a more general ratio than n,,/n, as it includes
the cases where K1 and K2 are not zero. For an incident beam in air
(lh-iI _ 1), the ratio reduces to the absolute magnitude of the complex
refractive index for the material on which the beam is incident. For
an insulating material where s: ,_ n, as is being discussed in this section,
figure 4-5 can be regarded as giving the emissivity of a dielectric into
air when the value for the parameter I_,1/1_,1 is set equal to the simple
refractive index n of the dielectric material. In the following discussion,
figure 4-5 will be interpreted in this sense.
For n= 1, the emissivity becomes unity (blackbody case), and the
curve for this value on figure 4-5 is circular with a radius of unity. As n
increases, the curves remain circular up to about/3 = 70 ° and then begin
to decrease rapidly to a zero value at/3 = 90 °. Thus, dielectric materials
emit poorly at large angles from the normal direction. For angles of
less than 70 °, the emissivities are quite high so that, in a hemispherical
sense, dielectrics are good emitters. It should be emphasized again that
the assumptions used for the present interpretation of Maxwell's equa-
tions restrict these findings to wavelengths longer than the visible
spectrum, as borne out by comparisons with experimental measurements.

From the directional spectral emissivity, the hemispherical spectral

emissivity can be computed from equation (3-5) to be

Ex(h, TA) ---- _x(h, /3, 0, T4) cos/3dto

--7"/"1 _,., ,

Then an integration can be performed over all wavelengths to obtain

the hemispherical total emissivity as given by equation (3-6a). Since the
optical properties are generally not known in sufficient detail so that a
wavelength integration of theoretical Cs can be made, in the theory
spectral Cx values are used for total _ values for lack of anything better.
The integration of ¢'(/3) to evaluate E is complicated by the implicit
relation between X and fl, and, hence, the integration is performed
numerically. The normal emissivity provides a convenient value to
which the hemispherical value may be referenced. The normal emis-
sivity can be computed from equation (4-51) as

E,',= 1 \n_-_-i/ (4-62)

for emission from a dielectric (medium 2) into air. The E_, is shown as
a function of n in figure 4-6(a). Note that normal emissivities less than
about 0.50 correspond to n > 6. Such large n values are not common
 PREDICTIONS BY ELECTROMAGNETIC THEORY 115

1.0

.9

\
.8
\
.?
E
Z

.6

(a)
• 51 3 4
Index of refraction, n

1.04

F.
1.02

I._,
\
m

o • 98
m

.%

0 .94
_ .__._..
(b)
• 92
.5 .6 .? .8 .9 1.0
I
Normal emissivity, (n

(a) Normal emissivity as function of refractive index.

(b) Relation between normal and hemispherical emissivity•

FIGURE 4--6.- Predicted emissivities of dielectric materials .

for dielectrics, so that the curve is not extended to smaller e_,. The ratio
of hemispherical to normal emissivity for dielectrics is provided as a
function of normal emissivity in figure 4-6(b).

EXAMPLE 4--2: A dielectric has a refractive index of 1.41. What is its

hemispherical emissivity into air at the wavelength where the refractive
index was measured?
From equation (4-62) the normal emissivity is

e_= 1 -- (0.41/2.41)2=0.97
116 THERMAL RADIATION HEAT TRANSFER

From figure 4-6(b), e/e_=0.94 and the hemispherical emissivity is

= 0.97 × 0.94 = 0.91.
For a large n the e;, values are relatively low, and with increasing n
the curves shown in figure 4-5 depart more and more from the circular
form of the curve corresponding to n = 1. Figure 4--6(b) reveals that the
flattening of the curves of figure 4-5 in the region near the normal causes
the hemispherical emissivity to exceed the normal value at large n.
For n near unity (el, near 1), the hemispherical value is lower than the
normal value because of the poor emission at large/3 as shown in fig-
ure 4-5.

4.6.2 Radiative Properties of the Metals

The properties of metals were shown to be given by relations of the

same form as for insulating materials. For electrical conductors, how-
ever, the extinction factor K cannot be neglected with respect to the
refractive index n. As will be shown, there are certain simplifying
assumptions that lead to more useful equations than the general results
from the theory. The main difficulty in application of the theoretical
results is that the optical properties for use in these equations are
difficult to obtain: when measured values are available, they are often
inaccurate because of the experimental problems involved in their
measurement.

4.6.2.1 Reflectivity and emissivity relations using optical constants.-

For most metals, the simple index of refraction n and the extinction
coefficient K are quite large at wavelengths longer than those in the
visible region. Because of this fact, the angle of refraction X is quite
small and cos X will be close to unity for incidence from a dielectric
having n near unity. This is shown as follows:
For a metal, the absolute magnitude of the complex refractive index
ratio relating X and /3 can be obtained by multiplying by the complex
conjugate of equation (4-36). This gives, for }_:} _ 1 (incidence through
dielectric or air (medium 1) onto a metal (medium 2)),

sin/3 (4-63a)
Ih2l _--- In2 -- iK_l = X/n._ + K2
2 = sin X

The maximum value of sin/3 is unity: hence, for a given n2 and K2 the
maximum value of sin X is

1
sin X = ,7,, (4-63b)
X/n_ + K22
 PREDICTIONS BY ELECTROMAGNETIC THEORY 117

For the large _ and K2 typical of metals (this will be demonstrated

later by table 4-II), X will have a small value. If V_n_ + K_ has a value
greater than about 3.3, X will be less than 18 °. However, cos 18 ° is about
0.95. Thus, if _n_ + r_ > 3.3, cos X can be set equal to unity with an er-
ror of less than 5 percent.
This fact allows us as a good approximation to set cos X in equations
(4-41) and (4-42) equal to unity; these equations then reduce to, for
incidence through a dielectric,

nl

_ cos/3-G
(4--64a)
EMIhi
cos/3+_
n2

and
1 nl

EM±,r= cos/3 _2
(4-64b)
EM±, i 1 nl

cos /3 n2

where nl is close to unity.

Equations (4-56) and (4-57) give the general reflectivity relations. For
the incident beam from a transparent or dielectric medium, the re-
flectivity components for the metal (cos X = 1) become

PlI(/3) = (n2 if- n, _2 (4---65)

co-;-_/+ _[
and

p_ (fl) = (n2-- n, cos/3)2+ K2z (4-66)

(n2+n, cos /3)2"4-/c22

These expressions are the squares of the real parts of equations (4--64).

For a beam incident through air on a metal with complex refractive

index n2-iK2, these equations reduce to (since the refractive index for
air is n_ = 1 as a very good approximation)

(n2 cos/3--1)2-4 - (K2 COS/3)2

(4--67)
Ph(/3) = (nz cos /3+1)5+ (K., cos /3)5
and
118 THERMAL RADIATION HEAT TRANSFER

(n.,- cos/3)2+ _
p± '(/3) (4--68)
(n2 + COS fl)2+ K2

For an unpolarized beam,

_{_ p
p,(fl)_P':-(fl) Pl](, _)
2 (4-69)

The corresponding emissivity values are found from e' (fl) = 1 - p' (/3),
and these simplify to

4n2 cos
(4-70)
e(l(fl) = (n._+ K._) cos 2 fl+2n2 cos 13+ 1

_(/3) = 4n2 cos/3 (4-71)

cos z/3 + 2n._ cos/3 + n._ + K._

For an unpolarized beam,

_' (/3) = 2 (4-72)

The use of these emissivity relations is demonstrated in figure 4-7

for a pure smooth platinum surface at a wavelength of 2/,tin, and it is
evident by comparison with the experimental data that, although the
general shape of the curve predicted by equation (4-72) is correct, the
magnitude is in error. The data for n and K for platinum, taken from
the Handbook of Chemistry and Physics, 44th Edition (ref. 2), are
n=5.7 and K=9.7.12 A comment as to the difficuhy of the measurement
of the optical properties of metals, perhaps because of the large influence
of metal purity and the ease of contamination, is that the 35th edition
of the Handbook lists values for platinum from an older measurement
for identical conditions as n=0.70 and t¢=3.5. The newer measure-
ments thus differ by a factor of 8 in refractive index and 2.8 in the
extinction factor.

Although the inaccuracy of the optical constants presents a diffÉculty

in the precise evaluation of radiative property values, the theory does
provide an understanding of the directional behavior of the properties.

t2 The reader should be aware that the complex refractive index can be defined in other ways than K = n- iK as used

here. It is also commonly given as n=n--inK and occasionally with a positive sign in front of the extinction factor.
When consulting data references, care should be taken in determining what definition is used so that conversion to the

system used in this report can be carried out if necessary.

 PREDICTIONS BY ELECTROMAGNETIC THEORY 119

Absolute Source
temperature,
T,
oK
.5
V 300 Ref. 16
- O 657 Ref. ll O
_ rt 1400 Ref. 18
._, .4
_r- __ ref. 2(n- 5.7, K- 9.7)
--¢D. .3
_ -- Eq. (4-72) usingciatafrom_ _ V 0 _ -_
I 08 8
2E_ /
II= ,_C

t I J I , I, I , I, I , I, I I
0 10 20 30 40 50 60 70 80 90
Angle of emission, fi, cleg

FIGURE 4-7.-Directional spectral emissivity of platinum at wavelength k = 2 p.m.

For metals, as illustrated by the results for platinum in figure 4-7,

the emissivity is essentially constant for about 40 ° away from the normal
and then it increases to a maximum located within a few degrees of the
tangent to the surface. This angular dependence for emission from
metals is in contrast to the behavior for dielectrics for which the emis-
sion decreases substantially as the angle from the normal becomes larger
than about 60 °.
In table 4-II, the prediction of normal spectral emissivity by using
equation (4-72) with/3= 0 is compared with measured values. All data
are taken from reference 2. A wavelength of _=0.589 gm is used for
some of the comparisons because of the wealth of data available. This
is because of the ease with which a sodium vapor lamp, which emits at
this wavelength, can be employed as an intense monochromatic energy
source in the laboratory. Since this wavelength is in the visible range,
it is in the borderline short wavelength region where the electromagnetic
theory becomes inaccurate.
Comparison of the values in the table 4-II shows the agreement be-
tween predicted and measured ei., to be good, for example, for nickel
and tungsten, but a factor of 4 in error for magnesium. For the cases
of poor agreement, it is difficult to ascribe the error specifically to the
optical constants, the measured emissivity, or the theory itself. Any or
all could contribute to the discrepancy. Most probably the optical con-
stants are somewhat in error, and the experimental samples do not
meet the standards of perfection in surface preparation demanded by
the theory.
120 THERMAL RADIATION HEAT TRANSFER

TABLE 4--1I. -- COMPARISON OF SPECTRAL NORMAL EMISSIVITY PREDICTIONS FROM

ELECTROMAGNETIC THEORY WITH EXPERIMENT

[Data from ref. 2]

Spectral normal
emissivity, ¢_. ,(h)
Metal Wavelength, Refractive Extinction

k, gm index, n coefficient, K
Experi- Calculated
mental from equa-
tion t4-72)

Copper ................ 0.650 0.44 3.26 0.20 0.140

2.25 1.03 11.7 .041 .029
4.00 1.87 21.3 .027 .014

Gold ................... 0.589 0.47 2.83 0.I76 0.184

2.00 .47 12.5 .032 .012

Iron .................... 0.589 1.51 1.63 0.43 0.674

Magnesium .......... 0.589 0.37 4.42 0.27 0.070

Nickel ................. 0.589 1.79 3.33 0.355 0.381

2.25 3.95 9.20 .152 .145

Silver ................. 0.589 0.18 3.64 0.074 0.049

2.25 .77 15.4 .021 .013
4.50 4.49 33.3 .015 .014

Tungsten .............. 0.589 3.46 3.25 0.49 0.455

The hemispherical emissivity for a metal (having complex refractive

index n-iK) in air or vacuum is found by substituting equation (4-72)
into equation (3-5). After carrying out the integration, this yields

[1 + 2n +/12 -4- K2\

4n(n2-K2) tan_l
_=4n-4n 2 log_ _- _-_ .)-_ K ( .K )
rt q- n2 q- K ff

4n 4n 2
n2..f_K2 (/12___ K2) 2 log_ (l+2n+n2+K 2)

4n(K z - n z) K (4-73)
K(nZ-_-K2) 2 tan-1 1+----n

Evaluation of equation (4-73) is difficult because it involves small dif-

 PREDICTIONS BY ELECTROMAGNETIC THEORY 121

1.0

._ ExtinCtion
factor, _

E
oa
D
_o
E
V,n _
L
o
.4 ___ -_

.2 /

0 2 4 6 8 10 12 14 16 18 20
Refractive index, n

1.40

1.30 _,

FFor metals from

_ Jakob, ref. 3)
I. 20

\
I. I0 \
\
1. O0 Eq. 14-731with K - n_ _ '\ ,....
K- 0 (insulators) j

_", (b)
.gQ
0 .2 .4 .6 .8 1.0
Normal emissivity, _

(a) Normal emissivity of attenuating media emitting into air.

(b) Ratio of hemispherical to normal emissivity.

FIGURE 4.--8.--Emissivity of metals as computed from electromagnetic theory.

295-763 OL-68--9
122 THERMAL RADIATION HEAT TRANSFER

ferences of large numbers, and many significant figures must be car-

ried in the calculations.

From equation (4-72), the normal emissivity from a metal into air can
be computed by letting /3=0, and this is shown in figure 4-8(a) as a
function of n and K. Note that, because the velocity of the waves in the
medium must be less than Co, the curve for K= 0 cannot extend below
//_].

It is of interest to compare the hemispherical emissivity with the

normal value. The practical use for this comparison arises from the
fact that it is often the normal emissivity that is measured experimentally
because of the relative simplicity of placing a radiation detector in this
one orientation. With regard to the total amount of heat dissipation,
however, it is the hemispherical emissivity that is desired.

Figure 4-8(b) shows the ratio of hemispherical to normal emissivity

as a function of the normal value. Equation (4-73) divided by E,', has
been plotted for the case where K = n. This is valid at large wavelengths
for many metals as shown in the next section. The curve is seen to be
close to that presented by Jakob (ref. 3) for metals as derived from
approximate equations and to lie somewhat below the curve for insula-
tors (as taken from fig. 4-6(b)) at high normal emissivities.

For polished metals when E_,is less than about 0.5, the hemispherical
emissivity is larger than the normal value because of the increase in
emissivity in the direction near tangency to the surface as was pointed
out in figure 4-7. Hence, in a table listing emissivity values for polished
metals, if the _,] is given, it should be multiplied by a factor larger than
unity such as obtained from figure 4-8(b) to estimate the hemispherical
value. Real surfaces that have roughness or may be slightly oxidized
often tend to have a directional emissivity that is more diffuse than for
polished specimens. For a practical case, therefore, the emissivity ratio
may be closer to unity than indicated by figure 4-8(b).
4.6.2.2 Relation between emissive and electrical properties.-The
wave solutions to Maxwell's equations provide a means for determining
n and K from the electric and magnetic properties of a material. The
relations for n and K are given by equations (4-23). For metals where
re is small, and for relatively long wavelengths, say ho >- 5 txm, the
term ko/(27rcoreT) becomes dominating, and equations (4-23) then reduce
to (the magnetic permeability is taken equal to go)

(4-74)
n-_ K = _] 47rre v re

for all quantities in inks units. If )to is taken in microns and re has the
 PREDICTIONS BY ELECTROMAGNETIC THEORY 123

units of ohm-centimeters (rather than ohm-meters), equation (4-74)

becomes

x/O.OO3ho
n = K = _/ _e (4--75)

This is known as the Hagen-Rubens equation (ref. 4). Predictions of

n and K from this equation can be greatly in error, as shown in table
4-III. Nevertheless, some useful results will eventually be obtained.
With the simplification that n = K, an equation such as equation (4-54)
reduces to the following expression for a material with refractive index
n radiating in the normal direction into air or vacuum:

ca. ,(h) (2.,-e.

= 1 -- \_-_n_ + 2n + (4-76a)

TABLE 4-III. -- COMPARISON OF MEASURED OPTICAL CONSTANTS WITH ELECTROMAGNETIC

THEORY PREDICTIONS

Measured values Spectral normal

emissivity, e_,. ,(h)
n=K
calcu*
Wave- lated

Metal length, Electrical from Calcu-

ho,/.tin resistivity Extinction equa- lated
(at 20 ° C), Refractive coef_- tion Measured from
r,, index, n eient, K (4-75) equa-
(ohm) (cm tion
(a) (4-77)

Aluminum. 12 2.82 × 10 -_ b 33.6 " 76._ 113 a 0.02 0.018

Copper ..... 4.20 1.72 × 10 -_ b 1.92 b 22.8 86 a, e 0.027 ......

4.20 1.72 I, 1.92 b22.8 86 a.015 0.023

5.50 1.72 a 3.16 28.4 98 a.012 .020

Gold ........ 5.00 2.44 × 10 -s a 1.81 32.8 78 a'c 0.031 0.026

Platinum.. 5.00 10× 10 -_ a 11.5 15.7 39 d 0.050 0.051

Silver ....... 4.50 1.63 × 10 ¢ _ 4.49 a 33.3 91 a. e 0.015 0.022

4.37 1.63 b 4.34 L,32.6 9O _, c.015 .022

" Data from ref. 2.

b Data from ref. 14.

Measured at 4 ttm.

d Data from ref. 15.

124 THERMAL RADIATION HEAT TRANSFER

Although there is no difficulty in evaluating equation (4-76a), a further

simplification is often made by expanding in a series to give

_, ,,(A.) = 1 -- (2 1 -- --t
/2
2
n"
1 1
n a_2n _
1 "_
2n 6 _-" " ") (4-76b)

Because the index of refraction of metals as predicted from equation

(4-75) is generally large at the long wavelengths being considered
here, _,o > -5 /xm (see table 4-III, column 6), only the first two terms
of the series often are retained, and the normal spectral emissivity is
then given by substituting equation (4-75) to obtain the Hagen-Rubens
emissivity relation

cx,,(_t)=l
_ P'x..(X) _1--
( 2)= 2
1--

¥ re
(4-77)

Data for polished nickel are shown in figure 4-9, and the extrapola-
tion to long wavelengths by equation (4-77) appears reasonable. The
predictions of normal spectral emissivity at long wavelengths as pre-
sented in table 4-III are much better than the prediction of attendant
optical constants•

.3-

•2-- Temperature,
oR
\\_,,, /-2290 (1272)

. - -- ," (IIII)

.o4
Source
•03- _ "',_,_.- 530(294)
J Ref. 1.5 " _.
•02 Ref. 19_ Asgiven "_.
Ref._OJ"In ref. 15 "_
.... Eq. (4-rip

I I I , I ,I,I I i
•01! 2 3 4 5 6 7 8910 20
Wavelength.
_,,pm

FIGURE4-9.-Comparison of measured values with theoretical predictions for spectral

normal emissivity of polished nickel.
 PREDICTIONS BY ELECTROMAGNETIC THEORY 125

The normal spectral emissivity given in equation (4-77) can be in-

tegrated with respect to wavelength to yield a normal total emissivity.
The integration relating spectral and total quantities is given by equation
(3-3b) (modified for a normal emissivity so that/3 = 0):

¢r fo c
ex. n(h,
t
T) tXb (h, T)dk
.t

¢,_(T) -- crT4

Equation (4-77) is only valid for ko > - 5/.tm, so that in performing the
integration starting from h = 0, the condition is being imposed that the
metal temperature is such that the energy radiated from ho = 0 to 5 txm
is small compared with that at wavelengths longer than 5 /xm. Then
substituting equation (4-77) and equation (2-11a) for i_b into the integral
provides

f_ / re ,,_/2 2C_ d_
7r Jo 2_) h_(eC_/_or - 1)
e_,(T) -_
o-T 4

41rC,(Tre),/2 f_ _3._ (4-78)

= (0.003)'/20"C4_ "5 J0 e _- 1 d_

where {= C.,/?toT as was used in conjunction with equation (2-19). The

integration is carried out by use of F functions to yield

e'n(T) _ 41rCl(Tre)l/2
(0.003),/2o.C4.5 (12.27) (4-79)

For pure metals, re is approximately described near room temperature by

re - re. 4_2 (4-80)

where re, 49., is the electrical resistivity, still in ohm-centimeters, eval-

uated at 492 ° R (0 ° C). Substituting equation (4-80) into equation (4-79)
gives the result

47rC, (12.27) /-_. 492

(4-81a)
_(T) -_ (0.003),/2o.C4.5 ,_/--_-_-T

or simply
126 THERMAL RADIATION HEAT TRANSFER

• 24
[3
Platinum
[30 Tungsten} [xNrfment

I
Platinum_Theory (eq. (4-81b)) r-1/0

>_ . 16 --
-c tungslen j

_ .xz--
"_ / t/1//"_

.04 -- _._
DD
I I I I I
.4 .8 1.2 1.6 2.0 2.4 2.8 3,2x103
Temperature, °R

I I I I I I I I I I
0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8x103
Temperature,°K

FIGURE 4-10.-Temperature dependence of total normal emissivity of polished metals.

#(T) _ 0.0193 V_rre,4.q2 T (4-81b)

where T is in °R. This indicates that, for long wavelengths (ho > - 5/zm),
the total emissivity of pure metals should be directly proportional to
temperature. This result was originally derived by Aschkinass (ref. 5)
in 1905. In some cases it holds to unexpectedly high temperatures where
considerable radiation is in the short wavelength region (for platinum,
to near 3200 ° R), but, in general, applies only below about 1000 ° R.
This is illustrated in figure 4-10 for platinum and tungsten (data from
ref. 2).
In figure 4-11, a comparison is made at 100 ° C of the total normal
emissivity from experiment and from equation (4-81b) for a variety of
polished surfaces of pure metals. Agreement is generally satisfactory.
The experimental values are the minimum values of results available
in three standard compilations (refs. 2, 6, and 7).
By using the emissivity from equation (4-81), the total intensity in
the normal direction emitted by a metal is given by

•, metals-
tn, - (n,, metals __ o_ T 5 (4-82)
 PREDICTIONS BY ELECTROMAGNETIC THEORY 127

•10 --

Lineofperfectagreement/
.08

i
Lead
_9 /

®_ .04 -- Tungsten7 ,_0 "rlr°n10

L _Copper _,_0_ _ Platinum
",_Gold/V / _ Zinc
'_ .02 - "_/_ .!Nickel
_'_L _._-Aluminum
/ I'-Sit_ I I I I
.02 .04 .06 .08 .tO
Measured
totalnormalemissivity,
(_

FIGURE4-11. -- Comparison of data with calculated total normal emissivity for polished
metals at 100° C.

This indicates that the normal total intensity is proportional to the fifth
power of absolute temperature rather than the fourth power as with a
blackbody. Again it must be emphasized that many assumptions were
made to obtain this simplified result. If for example more than two
terms had been retained from the series in equation (4-76b), it would
be found that the exact proportionality between normal total intensity
and T 5 no longer holds, although the exponent would still be greater
than 4.
The results of a more detailed computation are given in reference 3,
and these include an integration over all directions to provide hemispheri-
cal quantities. The following approximate equations for the hemispherical
total emissive power fit the results in two ranges:

e(T) = ofT4 (0.751X/_reT - 0.396reT); 0 < reT < 0.2 (4-83a)

and

e(T) =crT4(O.698_reT-O.266reT); 0.2 < rJ< 0.5 (4-83b)

where the numerical factors in the parentheses and those used in speci-
fying the ranges of validity apply for T in °K and re in ohm-centimeters.
The resistivity re depends on T to the first power so that the first term
inside the parentheses of each of these equations provides the T 5
dependency discussed earlier.
128 THERMAL RADIATION HEAT TRANSFER

4.6.3 Summary of Prediction Equations

A summary of equations for property predictions by use of electro-

magnetic theory is given in table 4-IV.

EXAMPLE 4--3: A pohshed platinum surface is maintained at tem-

perature Ta = 400 ° R. Energy is incident upon the surface from a black

enclosure at temperature Ti:800 ° R that encloses the surface. What

is the hemispherical-directional total reflectivity into the direction

normal to the surface?

Equation (3-48) shows that the directional-hemispherical total re-

flectivity can be found from

p_,(TA = 400 ° R) = 1 --a'(TA =400 ° R)

TABLE 4-IV. -- SUMMARY OF EQUATIONS FOR PROPERTY PREDICTION BY ELECTROMAG-

NETIC THEORY

Conditions
Property Equation

Dielectrics (K=0)

Directional reflectivity ............ (4--58), (4-60) Polarized in plane parallel to

plane of incidence.
Directional reflectivity ............ (4--59), (4-60) Polarized in plane perpendicu-
lar to plane of incidence.
Directional reflectivity ........... (4-61), (4-60) Unpolarized.
Normal reflectivity ................ (4-50) Polarized or unpolarized.
Hemispherical emissivity ........ P) Emission into medium having
n_l.

Metals (in contact with transparent medium of unity refractive index)

Directional reflectivity ........... (4-67) Parallel polarized component.

Directional reflectivity ........... (4-68) Perpendicular polarized com-
ponent.
Directional reflectivity ........... (4-69) Unpolarized.
Directional emissivity ............ (4-72) Unpolarized.
Hemispherical emissivity ........ (4--73) Unpolarized.
Normal spectral emis- (4-76a), (4-77) Polarized or unpolarized

sivity. h> _ 5/zm.

Normal total emissivity .......... (4-81b) T< - 1000° R.

See fig. 4-6.

 PREDICTIONS BY ELECTROMAGNETIC .THEORY 129

where OI_,(TA= 400 ° R) is the normal total absorptivity of a surface at

400 ° R for incident black radiation at 800 ° R, that is,

bot ' ,,(h, TA = 400 ° R)i_,b(h, 800 ° R) dk

c_,](Ta = 400o R) ---
foi'Xb(k, 800 ° R) dX

For spectral quantities a_.n(h, Ta=400 ° R)=_. n(h, TA=400 ° R).

From equation (4-77) the variation of re with temperature provides the '
emissivity variation _. n(h, Ta) cc Tla/',-.Then ¢_. n(X, TA = 400 ° R)= _, n(X,
TA = 800 ° R) (400/800) 1/2 and we obtain

• _, .(X, Ta = 800 ° R)tab (X, 800 ° R) dX

o_ (Ta = 400 ° R) =

fo _
t_(h,
.t
800 ° R) dh

_ e'(Ta = 800 ° R)

where the last equality is obtained by examination of the emissivity

definition, equation (3-36). The normal totai emissivity of platinum
at 800 ° R is given by equation (4-81b) as plotted in figure 4-10 as

(TA = 800 ° R) = 0.0193 XFre, 492 X 800 = 0.051

Note that equation (4-81b) is only to be used when temperatures are such
that most of the energy involved is at wavelengths greater than 5/zm.
Examination of the blackbody functions, table V of the appendix, shows
that for a temperature of 800 ° R, about 10 percent of the energy is at
less than 5/.tin so that possibly a small error is introduced.
The reciprocity relation of equation (3-28) for uniform incident
intensity can now be employed to give the final result for the hemi-
spherical-directional total reflectivity,

p,_(Ta = 400 ° R) = 1 -- o_(TA = 400 ° R) _- 1 --_-_9 (TA = 800 ° R)

0.051
--1 0.964
130 THERMAL RADIATION HEAT TRANSFER

4.7 EXTENSIONS OF THE THEORY OF RADIATIVE PROPERTIES

Much work has been expended in improving the theory of the radiative
properties of materials, using both classical wave theory and quantum
theory. A number of authors have successfully removed some restric-
tions which are present in the classical development presented here.
Notable are the contributions of Davisson and Weeks (ref. 8), Foote
(ref. 9), Schmidt and Eckert (ref. 10), and Parker and Abbott (ref. 11),
who all extended the emissivity relations for metals to shorter wave-
lengths and higher temperatures, and of Mott and Zener (ref. 12), who
derived relations for metal emissivity at very short wavelengths on the
basis of quantum relations.
None of these treatments, however, accounts for surface effects.
Because of the difficulty of specifying surface conditions and con-
trolling surface preparation, it is found that comparison of the theory
with experiment is not always adequate for even the refined theories.
In fact, comparison to the less exact but simpler relations given here
is often better. For even the purest materials given the most meticulous
preparation, the elementary relations are often more accurate because
the errors in the simpler theory are in the direction which cause com-
pensation for surface working.
Polarization effects entered into the mathematical description of
electromagnetic waves and wave reflections. A detailed discussion of
these effects is beyond the intent of this publication. A comprehensive
discussion of the analytical methods and technology of polarization
phenomena is given in reference 13.

REFERENCES

1. MAXWELL, JAMES CLERK: A Dynamical Theory of the Electromagnetic Field. Vol. 1

of The Scientific Papers of James Clerk Maxwell. W. D. Niven, ed., Cambridge
University Press, 1890.
2. WEAST, ROBERTC., ED.: Handbook of Chemistry and Physics. 44th ed., Chemical
Rubber Co., 1962.
3. JAKOB,MAx: Heat Transfer. John Wiley and Sons, Inc., 1949.
4. HAGEN,E.; ANDRUBENS,H.: Metallic Reflection. Ann. d. Physik, vol. 1, no. 2, Feb.
1900, pp. 352-375. (See also Hagen, E.: and Rubens, H.: Emissivity and Electrical
Conductivity of Alloys. Deutseh. Phys. Gesell. Verh., vol. 6, no. 4, Feb. 29, 1904,
pp. 128-136.)
5. ASCHKINASS, E.: Heat Radiation of Metals. Ann. d. Physik, vol. 17, no. 5, Sept. 26,
1905, pp. 960-976.
6. HOTTEL, H. C.: Radiant Heat Transmission. Heat Transmission. William H. McAdams,
Third ed., McGraw-Hill Book Co., Inc., 1954, pp. 55-125.
7. ECKERT, E. R. G.; AND DRAKE, R. M., JR.: Heat and Mass Transfer. Second ed.,
McGraw-Hill Book Co., Inc., 1959.
8. DAVlSSON, C.; AND WEEKS, J. R., JR.: The Relation Between the Total Thermal Emis-
sire Power of a Metal and Its Electrical Resistivity. Opt. Soc. Am. J., vol. 8, no. 5,
May 1924, pp. 581-605.
 PREDICTIONS BY ELECTROMAGNETIC THEORY 131

9. FOOTE, PAUL D.: The Emissivity of Metals and Oxides. III. The Total Emissivity of
Platinum and the Relation Between Total Emissivity and Resistivity. NBS Bulletin,
vol. 11, no. 4, 1915, pp. 607-612.
10. SCHMIDT, E.; AND ECKERT, E. R. G.: Ueber die Richtungsverteilung der Waermestrah-
lung yon Oberflaechen. Forschung. Geb. Ing.-Wes., vol. 6, no. 4, July-Aug. 1935,
pp. 175-183.
ll. PARKER, W. J.; AND ABBOTT, G. I,.: Theoretical and Experimental Studies of the Total
Emittance of Metals. Symposium on Thermal Radiation of Solids. NASA SP-55,
1964, pp. 11-28.
12. MOTT, N. F., AND ZENER, C.: The Optical Properties of Metals. Cambridge Phil.
Soc. Proc., vol. 30, pt. II, 1934, pp. 249-270.
13. SHURCLIFF, W. A.: Polarized Light, Production and Use. Harvard University Press,
1962.

14. GARBUNY, M.: Optical Physics. Academic Press, 1965.

15. SEBAN, R. A.: Thermal Radiation Properties of Materials, Part III. (WADD-TR-60-
370, Pt. lII), California University, Aug. 1963.
16. BRANDENBERG, W. M.: The Refleetivity of Solids at Grazing Angles. Measurement
of Thermal Radiation Properties of Solids. Joseph C. Richmond, ed. NASA SP-31,
1963, pp. 75-82.
17. BRANDENBERG, W. M.; AND CLAUSEN, O. W.: The Directional Spectral Emittance
of Surfaces Between 200 ° and 600 ° C. Symposium on Thermal Radiation of Solids.
NASA SP-55, 1964, pp. 313-320.
i8. PRICE, DEREK, J.: The Emissivity of Hot Metals in the Infra-Red. Phys. Soc. (London)
Proc., Sec. A, vol. 59, pt. 1, Jan. 1947, pp. 118-131.
19. HURST, C.: The Emission Constants of Metals in the Near Infra-Red. Roy. Soc. Proc,
Set. A, vol. 142, no. 847, Nov. 1, 1933, pp. 466-490.
20. PEPPERHOFF, W.: Temperaturstrahlung. D. Steinkopf (Darmstadt, Germany), 1956.
 Chapter 5. Radiative Properties of Real Materials
5.1 INTRODUCTION

In this chapter, the general characteristics of the radiative properties

of real materials will be examined. These properties can vary consider-
ably from the idealized cases presented in chapter 4 for "optically
smooth" materials as predicted by electromagnetic theory. The analyti-
cal predictions yield useful trends and provide a unifying basis to
help explain various radiation phenomena. However, the analyses are
inadequate in the sense that the engineer is generally dealing with sur-
faces coated in varying degrees with contaminants, oxide, paint, etc.,
and having a surface roughness that is difficult to specify completely.
Examples of some typical variations of radiative properties as a function
of these and other parameters will be presented in this chapter to
illustrate the types of property variations that can occur. This will
provide the reader with an appreciation of how sensitive the radiative
performance is to the surface condition. In addition to the typical prop-
erties presented, a number of atypical examples will be given in order
to demonstrate that a careful examination of individual properties must
be made to properly select the property values to be used in radiative
exchange calculations.
The discussion in this chapter will be limited to opaque solids, where
opaque is defined to mean that no transmission of radiant energy occurs
through the entire thickness of the body. A composite body such as
a thin coating on a substrate of a different material can have partial
transmission through the coating, but it is assumed for the present dis-
cussion that none of the transmitted radiation will pass entirely through
the substrate. No attempt to compile comprehensive property data will
be made. Extensive but by no means complete tabulations and graphs
of radiative properties have been gathered in references 1 to 6.
As discussed in chapter 4, there are basic differences in the radia-
tive behavior of metals and dielectrics that are evident from electro-
magnetic theory. For this reason the first two sections in this chapter
will deal with these two classes of materials, with metals being dis-
cussed first. Then some special surfaces will be discussed that have
specific desirable variations of properties with wavelength and direction.

5.2 SYMBOLS

,4 area

133
 134 THERMAL RADIATION HEAT TRANSFER

Co speed of electromagnetic radiation in vacuum

e emissive power
Fo- fraction of blackbody energy in spectral range 0- h
P probability function
Q energy rate, energy per unit time
q energy flux, energy per unit time per unit area
re electrical resistivity
T absolute temperature
z height of surface roughness
ol absorptivity
angle measured from normal of surface
Y electrical permittivity
E emissivity
0 circumferential angle
_t wavelength
magnetic permeability
P reflectivity
o Stefan-Bohzmann constant (table IV of the appendix)
O'o root-mean-square height of surface roughness

Subscripts:

A of surface ,4
a absorber
b blackbody condition
c evaluated at cutoff wavelength
e emitted
eq at equilibrium
i incident
max maximum value
n normal direction
R radiator
r reflected
$ specular
spectrally dependent
O--h in wavelength range 0-h

Superscripts:

' directional
" bidirectional

5.3 RADIATIVE PROPERTIES OF METALS

Pure, smooth metals are often characterized by low values of emis-.

sivity and absorptivity, and therefore comparatively high values of re-
 RADIATIVE
PROPERTIES OF REAL MATERIALS 135

flectivity. Figure 4-11 of the preceding chapter demonstrates that the

emissivity in the direction normal to the surface is quite low for a
variety of polished metals. However, low emissivity values are not an
absolute rule for metals; in some of the examples that will be given, the
spectral emissivity rises to 0.5 or larger as the wavelength becomes
short, or the total emissivity becomes large as the temperature is
elevated.

5.3.1 Directional Variations

A behavior typical of polished metals is that the directional emis-

sivity tends to increase with increasing angle of emission/3 (where/3
is the angle measured with respect to the surface normal). This is pre-
dicted by electromagnetic theory and was shown to be true for platinum
in figure 4-7. At wavelengths shorter than the range for which the
simple electromagnetic theory of chapter 4 applies, a deviation from
this behavior might be expected. To illustrate tbis deviation, the direc-
tional spectral emissivity of polished titanium is shown in figure 5-1.

Angleof emission,13,de9
0
15

)o
o .2 ,4 .6 .8
Directionalspectral emissivity,
(_(X,13}

FIGURE 5-1.--Effect of wavelength on directional spectral emissivity of pure titanium.

Surface ground to 16 gin. (0.4/xm) rms. (Data from ref. 6.)
136 THERMAL RADIATION HEAT TRANSFER

At wavelengths greater than about 1 /xm, the directional spectral emis-

sivity of titanium does indeed tend to increase with increasing fl over
most of the fl range. The increase with fl becomes smaller as wavelength
decreases; finally, at wavelengths less than about 1/xm, the directional
spectral emissivity actually decreases with increasing fl over the entire
range of 8. Hence, for polished metals, the typical behavior of increased
emission for directions nearly tangent to the surface can be violated at
short wavelengths.

5.3.2 Effect of Wavelength

In the infrared region, it was shown in chapter 4 that the spectral

emissivity of metals tends to increase with decreasing wavelength. This
trend remains true over a large span of wavelength as illustrated for
several metals in figure 5-2 which gives the spectral emissivity in the
normal direction. For other directions, the same effect is illustrated
in figure 5-1 except at large angles from the normal where curves for
various wavelengths may cross. The curve for copper in figure 5-2 pro-
vides an exception as the emissivity remains relatively constant with
wavelength.
At very short wavelengths, the assumptions upon which the sim-
plified electromagnetic theory of chapter 4 are based become invalid.

% Metal Temperature,

-_ °R (°K)
-- _ /- Molybdenum
2000(1111)
m
Iron 2370(1317)

t-.
.,<"
w .2

! ,'
I I

S _ /
.1 -- I ]

t- Nickel 21(:0(1200)
|
b
z

,,-Copper

.o_ I j l ,lilll l
2 4 6 8 I0 20
Wavelength, ),, pm

FIGURE 5--2. -- Variation with wavelength of normal spectral emissivity for polished metals.
(Data from Seban (ref. 15).)
 RADIATIVE
PROPERTIES OF REAL MATERIALS 137

Indeed, most metals exhibit a peak emissivity somewhere near the visible
region, and the emissivity then decreases rapidly with further decrease
in wavelength. This is illustrated by the behavior of tungsten in figure
5-3.

5.3.3 Effect of Surface Temperature

The Hagen-Rubens relation (eq. (4--77)) showed that, for wavelengths

that are not too short (h > -5 pm), the spectral emissivity of a metal
is proportional to the resistivity of the metal to the one-half power.
Hence, we can expect the spectral emissivity of pure metals to increase
with temperature as does the resistivity, and this is found to be the
case in most instances. Figure 5-3 is an example for the hemispherical
spectral emissivity of tungsten. The expected trend is observed for
> 1.27 ftm. Figure 5-3 also illustrates a phenomenon characteristic
of many metals as discussed in reference 7. At short wavelengths (in
the case of tungsten h < 1.27 /zm), the temperature effect is reversed
and the spectral emissivity decreases as temperature is increased.
The observed increase of spectral emissivity with decreasing wave-
length for metals in the infrared radiation region (wavelengths longer
than visible region) as discussed in section 5.3.2 accounts for the increase
in total emissivity with temperature. With increased temperature,
the peak of the blackbody radiation curve (fig. 2-6) moves toward
shorter wavelengths. Consequently, as the surface temperature is
increased, proportionately more radiation is emitted in the region of
higher spectral emissivity, which results in an increased total emis-

.5

.4

)
._u

, I i I I Ill I ,
'2 .3 .4 .6 .8 1 2 3
Wavelength, k, IJm

i FIGURE

295-763
5-3.--Effect

0L-68-- 10
of wavelength
emissivity
and surface
of tungsten
temperature
(ref. 16).
on hemispherical spectral
138 THERMAL RADIATION HEAT TRANSFER

"8r
_" .6

.4
0

_ 7M_nesiurn
.2-- P_ncone_=

- , [ I I ,
400 800 1200 1600 2000
Temperature, TA, °F
[ I I I I [ I
200 400 600 800 1000 1200 1400
Temperature, TA, °K

FIGURE 5-4.--Effect of temperature on hemispherical total emissivity of several metals

and one dielectric. (Data from Gubareff et al. (ref. 1).)

sivity. Some examples are shown in figure 5-4. Here the behavior of
metals is contrasted with that of a dielectric, magnesium oxide, for
which the emissivity decreases with increasing temperatures.
The next two factors to be discussed are surface roughness and sur-
face impurities or coatings. These can cause major deviations from the
electromagnetic theory predictions of chapter 4.

5.3.4 Effect of Surface Roughness

If the surface imperfections present on a material are much smaller

than the wavelength of the radiation being considered, the material
is said to.be optically smooth. A material that is optically smooth for
long wavelengths may be comparatively quite rough at short wavelengths.
The radiative properties of optically smooth materials can be predicted
within the limitations of electromagnetic theory as discussed in
chapter 4.
For wavelengths that are very short in comparison with the degree
of roughness, the directional distribution of emitted or reflected energy
is governed chiefly by the roughness. If the orientation and constitution
of the roughness is specified, it is possible in certain cases to predict
analytically these directional distributions. Such a case for parallel
grooves will be noted in section 5.5.2.
Various attempts at predicting the effect of surface roughness on the
 RADIATIVE PROPERTIES OF REAL MATERIALS 139

radiative properties of metals have been made. All must be viewed as

preliminary prohings of an extremely complex subject, and none are
satisfactory over the entire range of variables encountered for engineer-
ing surfaces.
A chief stumbling block is in the precise definition of surface charac-
teristics for use in an analysis. Perhaps the most common way of charac-
terizing surface roughness is by the method of preparation (lapping,
grinding, etching, etc.) plus a specification of root-mean-square (rms)
roughness. The latter is usually obtained by means of a profilometer,
which is an instrument that traverses a sharp stylus over the surface
and reads out the vertical perturbations of the stylus in terms of a
rms value. It does not account for the horizontal spacing of the roughness
and gives no indication of the distribution of the size of roughness
around the rms value. At present, there is no generally accepted method
of accurately specifying surface characteristics, and none of those
mentioned in this paragraph are adequate for prediction of radiative
properties.
A few of the analytical approaches taken in the face of the afore-
mentioned difficulty will now be mentioned. Davies (ref. 8) has ex-
amined the reflecting properties of a surface with roughness that is
assumed to be distributed according tO a Gaussian (normal) probability
distribution, specified as a probability p(z) of having a roughness of
height z given by

p(z) exp --

where O-o is the rms roughness. Using this distribution and the assump-
tions that the individual surface irregularities are of sufficiently small
slope that shadowing can be neglected, that the material is a perfect
electrical conductor, and that o'o is very much smaller than the wave-
length of incident radiation h, Davies was able to derive relations that
predicted the distribution of reflected intensity. The reflected distribu-
tion was found to consist of a specular component and a component
distributed about the specular peak.
A similar derivation, with tro assumed much larger than _, again
yielded a distribution of reflected intensity about the specular peak, this
time of larger angular spread than for the case of O'o _ _,. This would
be expected since the surface should behave increasingly like an ideal
specular reflector as the roughness becomes very small compared with
the wavelength of the incident radiation. Davies' treatment is found
to be very inaccurate at near grazing angles because of the neglect of
shadowing.
Porteus (ref. 9) extended Davies' approach by removing the restrictions
140 THERMAL RADIATION HEAT TRANSFER

on the relation between ¢ro and h and including more parameters for
specification of the surface roughness characteristics. Some success in
predicting the roughness characteristics of prepared samples from
measured reflectivity data was obtained, but certain types of surface
roughness led to poor agreement. Measurements were mainly at normal
incidence, and the neglect of shadowing makes the results of doubtful
value at near grazing angles.
A more satisfactory treatment has been given by Beckmann and
Spizzichino (ref. 10). Their method includes the autocorrelation distance
of the roughness in the prescription of the surface. This is a measure
of the spacing of the characteristic roughness peaks on the surface. The
method gives somewhat better data correlation than the earlier analyses.
Some observed effects of surface roughness are shown in figures 5-5
and 5-6. The former shows the directional emissivity of titanium at a
wavelength of 2 gm for three surface roughnesses, the maximum rough-
ness being 16 microinches (/xin.). Since 2/xm is equal to 78.7/xin., the

Angle of emission, I_, deg

0
15

3O

75

0 .2 .4 .6 .8 90

Directional spectral emissivity, E_(;_= 2 _tm,13)

FIGURE S-5. -- Effect of surface finish on directional spectral emissivity of pure titanium.
Wavelength, 2 tsm (78.7/_in.). (Data from ref. 6.)
 RADIATIVE PROPERTIES OF REAL MATERIALS 141

. .- I.0_ roughness,
Mechanical

•
_ ÷ .8
o
,_= o .17 / ,.,.,,,,,f-
•- o .315 p_ _lff
%
• -hJ =
= _ k .4i
_ +
,

.2= _" .2--

2 4 6 8 10 12 N
Wavelength, ),, pm

FIGURE 5-6.-Effect of roughness on reflectivity in specular direction for ground nickel

specimens. Mechanical roughness for polished specimen, 0.015 /zm. (Data from
ref. 17.)

wavelength of the radiation is significantly larger than the surface

roughnesses. Hence, relative to this wavelength the specimens are
smooth. As a result, the emissivity changes only a small amount as the
roughness varies from 2 to 16/_in.
Figure 5-6 provides the reflectivity of nickel for energy reflected into
the specular direction from a beam incident at an angle 10 ° from the
normal. In this figure, the reflectivities of the rough specimens are
expressed as a ratio to the reflectivity of a polished surface in order to
exhibit the effect of roughness. The polished surface used for com-
parison had a roughness about 10 times less than that of the rough
specimens. A high value of the ordinate thus means that the specimen
is behaving more like a polished surface. Data are shown for ground
nickel specimens with four different roughnesses. The reflectivity rises
as wavelength is increased because for a given roughness the surface
is more smooth relative to the incident radiation. As expected, for
a fixed wavelength the reflectivity for the specular direction decreased
as the roughness was increased.

5.3.5 Effect of Surface Impurities

Impurities in this context include contaminants of any type which

cause deviations of the surface properties from those of an optically
142 THERMAL RADIATION HEAT TRANSFER

1--

2--

A
o

,<-
---,< o \
>_

E .1 --

O Data from ref. 6 _ Unoxidizecl

D Data from ref. 18 J
Hagen-Rubens relation (eq. (4-77))
g
------ Surface oxidized to layer thickness
of 0.06 pm (ref. 18)

.01
I , I,l,l,I i , l,l,l,I i , L,L,I,I
.1 1 l0 100
Wavelength, X, pm

FIGURE 5-7.- Effect of oxide layer on directional spectral emissivity of titanium. Emission
angle, 25°; surface lapped to 2 gin. (0.05 /_m) rms; temperature, 530 ° R (294 ° K).

1.0_

.9
.8
L

.7

•_ .6
E

-_ _ _._ _ Asr_ceived
.4 d

E
0
.3 -
.6 1.0 1.4 1.8 2.2 2.6 3.0
Wavelength, X, pm

FIGURE 5-8.--Effect of oxidation on normal spectral emissivity of Inconel X. (Data from

ref. 5.)

smooth pure metal. The most common contaminants are thin layers of
foreign materials deposited either by adsorption, such as in the case of
water vapor, or by chemical reaction. The common example of the
latter is the presence of a thin layer of an oxide on the metal. Because
dielectrics, as will be discussed in section 5.4, have generally high
 RADIATIVE PROPERTIES OF REAL MATERIALS 143

values of emissivity, an oxide or other nonmetallic contaminant layer

will usually increase the emissivity of an otherwise ideal metallic body.
Figure 5-7 shows the directional spectral emissivity of titanium at
an angle of 25 ° to the surface normal. The data points are for the unoxi-
dized metal, and the solid line is the ideal emissivity predicted from
electromagnetic theory. The dashed curve shown above the data points
is the observed emissivity when an oxide layer only 0.06 ftm in thickness
is present. The emissivity is seen to be increased by a factor of almost 2
from that of the pure material over much of the wavelength range.
Figure 5-8 shows a similar large increase in the normal spectral emis-
sivity of Inconel X for an oxidized surface as compared with that for the
polished metal.
Figures 5-9 and 5-10 illustrate the effect of an oxide coating on the
hemispherical total emissivity of copper and the normal total emissivity
of stainless steel. The details of the oxide coatings are not specified,
but the large effect of surface oxidation is apparent. A more precise
indication of oxide coating effect is shown in figure 5-11 where the
hemispherical total emissivity of aluminum is given. An oxide thickness
of a few ten-thousandths of an inch provides a very substantial emissivity
increase.

1.0--

Black oxide
A

"_ .8

Heavily oxidized

•_ .6 Lightly oxidized
E
o_

¢= .4

E
_: .2

Polished (pure)
--I I I I I
3]0 400 60O 800 1000
Temperature, TA, °F

I I I I I I
300 400 500 600 700 800
Temperature, TA, °K

FIGURE 5--9. - Effect of oxide coating on hemispherical total emissivity of copper. (Data from
Gubareff et al. (ref. 1).)
144 THERMAL RADIATION HEAT TRANSFER

1.0 -- Sandblasted, weathered,

__ and oxidized at 1500° F-7
.9 i
A
___-o--_-_
.7

"_ .6
Acid treated ',
E .5 and weathered _'
tD

"_ .4
-Polished
_s
,;-Unpolished
E

z
.1-- _ I
100 300 500 700 9OO
Temperature, TA, °F
1 I I I I
3OO 400 500 600 70O
• Temperature. TA, °K

FIGURE 5-10. - Effect of surface condition and oxidation on normal total emissivity of
stainless steel type 18-8. (Data from ref. 5.)

.8 B

._> .6

•_ .Z

I I I I
0 10 20 30 40x10 -5
ThicKness o_;coating, in.

I I I I I I
0 2 4 6 8 10
ThicKness of coating, pm

FIGURE 5-11.- Typical curve illustrating effect of electrolytically produced oxide thickness
on hemispherical total emissivity of aluminum. Temperature, 100 ° F. (Data from Gubareff
et al. (ref. 1).)

Figure 5-12 shows approximately the directional total absorptivity

of an anodized aluminum surface for radiation incident from various fl
directions and originating from sources at various temperatures. The
quantity p'_(fl) is the fraction of the incident energy that is reflected into
 RADIATIVE PROPERTIES OF REAL MATERIALS 145

Angle of incidence, fi, deg

0 .5 1.0 1.5 2.0

El-pi_ll - pl,n_

FIGURE 5-12.--Approximate directional total absorptivity of anodized aluminum at room

temperature relative to value for normal incidence. (Redrawn from data of Munch
(ref. 19).)

the specular direction; hence, 1--p's(13) is the fraction of the incident

energy that is absorbed plus the fraction of the incident energy reflected
into directions other than the specular direction. For the specimens
tested, only a few percent of the energy was reflected into directions other
than the specular direction. Thus, in figure 5-12, the quantity 1-p',(/3)
can be regarded as a good approximation to the directional total ab-
sorptivity. The curves have all been normalized to pass through unity at
/3 = 0; hence, it is the shapes of the curves that are significant. At low
source temperatures, the incident radiation is predominantly in the long
wavelength region. This incident radiation is barely influenced by the
thin oxide film on the anodized surface; consequently, the specimen acts
like a bare metal and has large absorptivities at large angles from the
normal. At high source temperatures where the incident radiation is
predominantly at shorter wavelengths, the thin oxide film has a significant
effect and the surface behaves as a nonmetal where the absorptivity
decreases with increasing/3.
The structure of the surface coating can also have a substantial
effect on the radiative behavior. Figure 5-13 shows the hemispherical
spectral reflectivity of aluminum coated with lead sulfide. The mass of
146 THERMAL RADIATION HEAT TRANSFER

1.00 m

f
A /

.so- "-unc_t_ _ _ _ - -
aluminum
T . . __ .

.40-- I _ _ _ _

I fo.o-,_mcobic
cryst_i_
E
H

I I I I I I I I I J
u.4 .6 .8 l 2 4 6 8 10 12 14
Waveteng|h, X, pm

FIGURE 5-13.--Hemispherical spectral reflectivity for normal incident beam on aluminum

coated with lead sulfide, Coating mass per unit _urface area, 0.68 mg/cm 2. (Data from
ref. 20.)

the coating per unit area of surface is the same for both sets of data
shown. The difference in crystal structure and size causes the reflectivity
of the coated specimens to differ by a factor of 2 at wavelengths longer
than about 3 gm.

5.4 RADIATIVE PROPERTIES OF OPAQUE NONMETALS

Roughly speaking, nonmetals are characterized by large values of

total hemispherical emissivity and absorptivity at moderate temperatures
and, therefore, generally small values of reflectivity in comparison
with metals. For a clean optically smooth surface, several results were
arrived at in chapter 4 by use of the simplified electromagnetic theory
presented there. These provide the following generalizations (bearing
in mind the rather stringent assumptions of the theory): the directional
emissivity will decrease with increasing angle from the surface normal;
wavelength dependence is often weak, as it enters the predicted prop-
erties through the refractive index which varies slowly with wavelength
for many nonmetals; finally, the temperature dependence of the prop-
erties of nonmetals will also be small, since temperature also enters
the prediction only through the refractive index which is usually a weak
function of temperature.
The difficulty with these generalizations is that most nonmetals
cannot be polished to the degree necessary to allow their surfaces to be
considered ideal, although some common exceptions exist such as glass,
 RADIATIVE PROPERTIES OF REAL MATERIALS 147

large crystals of various types, gem stones, and some plastics (some of
these are not opaque materials like those being discussed here). As a
result of having such nonideal surface finishes, many nonmetals, in
practice, deviate radically from the behavior predicted by electro-
magnetic theory.
Available property measurements for nonmetals are much less
detailed than for metals. Specifications of the surface composition,
texture, and so forth, are often lacking. Table 5-i (taken from ref. 1)
illustrates this, as the type of wood, texture of the brick, and compo-
sition of the oil paint are unspecified. This table does reveal the large
emissivity values that many of the nonmetal materials have at room
temperature.

TABLE 5--|.--NORMAL TOTAL EMISSIVITY OF

NONMETALS AT ROOM TEMPERATURE (68 ° F)

[Data from Gubareff et al. (ref. 1)l

Material

Brick ................................... 0.94

Lampsoot .............................. .95
Oil paint ................................ .89 to .97
Roofing paper ......................... .91
i Hard rubber ........................... .92
Wood .................................... .8 to .9

An effect which complicates the interpretation of the measured

properties of nonmetals is that radiation passing into such a material
may penetrate quite far (this is evident for visible wavelengths in glass
as an example) before being absorbed. A specimen must be of sumcient
thickness to absorb essentially all the radiation that enters it, if it does
not, it cannot be considered opaque and transmitted radiation must be
accounted for. Often, samples of nonmetals such as paints are sprayed
onto a metallic or other opaque base (substrate), and then the properties
of the composite are measured. If in such a case it is desired to have the
surface behave completely as the coating material, the thickness of the
nonmetal coating must be sufficient to assure that no significant radiation
is transmitted through the coating. Otherwise, when making a reflectivity
measurement, some of the incident radiation will be reflected from the
substrate and then transmitted again through the coating to reappear
as energy measured by the instruments. The measured data will then
be a function of both the coating material and the substrate.
14,8 THERMAL RADIATION HEAT TRANSFER

i,I .--

,I II/
, ! j, _Substratewith

Coating
roughness

/ / I ,¥ thickness,

' filled by zinc oxide

_ 2 _ .10[. Filled substrate

_ .20|pluscoating
I I I I 4iJ' 1 I
0 2 4 6 8 lO 12 14
Wavelength, ),, I_m

FIGURE 5-14.--Emissivity of zinc oxide coatings on oxidized stainless steel substrate.

Surface temperature, 880+_8 ° K. (Data from ref. 11.)

In emissivity measurements of a coating material, the coating must be

thick enough that no emitted energy from the substrate penetrates the
coating. A good illustration is given by Liebert (ref. 11). He examined the
spectral emissivity of zinc oxide on a variety of substrates, using various
oxide thicknesses. The effect of coating thickness on the emissivity of the
composite formed by the zinc oxide coating and a substrate of approxi-
mately constant normal spectral emissivity is shown in figure 5-14.
The effect of increasing the coating thickness becomes small in the
range of 0.2- to 0.4-millimeter (ram) thickness, indicating that the emis-
sivity of the zinc oxide alone is being approached.
We will now examine the effects of wavelength, temperature, and
surface roughness on the radiative properties of dielectrics and then
briefly examine the radiative properties of semiconductors.

5.4.1 Spectral Measurements

Compared with metals, there are relatively few detailed spectral

measurements for dielectrics. Figure 5-15 shows the hemispherical-
normal spectral reflectivity for three paint coatings on steel. From
KirchhoWs law and the refleetivity reciprocity relations, we can regard
the difference between unity and these reflectivity values as the normal
spectral emissivity. The three paints shown all exhibit somewhat
ditterent characteristics. White paint has a high reflectivity (low emis-
sivity) at short wavelengths, and the reflectivity decreases at longer
wavelengths. Black paint, on the other hand, has a relatively low reflec-
tivity over the entire wavelength region shown. Using aluminum powder
 RADIATIVE PROPERTIES OF REAL MATERIALS 149

,-Aluminized silicone paint {0.001 in.

i

"_ / (25pm) thick)onInconelX

' I / [-Black enamel (0. 0006 in.
/ • (15pro)thick)onstee_

I 1 I I I
5 10 15 20 25
Wavelength, X, prn

FIGURE 5-15.--Spectra] reflectivity of paint coatings. Specimens at room temperature.

(Data from ref. 21.)

in a silicone base as a paint increases the reflectivity as would be ex-

pected for the more metallic coating. This particular specimen of
aluminized paint acts approximately as a "gray" surface since the
properties are reasonably independent of wavelength. Because of the
large variation in spectral emissivity at short wavelengths, the gray
approximation would be poor for the white paint unless very little of the
participating radiation were at the shorter wavelengths.
Figur e 5-16 illustrates that at the short wavelengths in the visible
range the reflectivity for some nonmetals may decrease substantially.

l. O0

.80

• 70
=6

t-

.40 _-'-

.20

.1.o t
t_ _--]_Visiblera?e I I I I
.6 1.0 1.4 1.8 2.2 2.6 3.0
Wavelength, X, _tm

FIGURE 5-16•--Spectral directional-hemispherical reflectivity of aluminum oxide. Incident

angle, 9°; specimens at room temperature. (Data from ref. 5.)
150 THERMAL
RADIATION
HEAT TRANSFER

This behavior is very important when considering the suitability of a

specific nonmetallic coating for reflecting radiation from a high temper-
ature source where much of the energy will be at short wavelengths.

5.4.2 Variation of Total Properties With Temperature

The effect of surface temperature on the total emissivity of several

nonmetallic materials is shown in figures 5-17 to 5-19. Both increasing

O Ceramic coating on stain-

less steel Hemispherical total
Zirconia (opaque coating) emissivity
on Inconel
A Silicon carbide coating on
graph ire
Magnesium oxide refractory
O Aluminized silicone paint

(0. 001 in. (25 IJm) thick) Normal total emis-

on titanium
I> Black heat-resistant paint sivity
(0. 0006 in. (15 IJm) thick)
on steel
<3 White paint on stainless
steel
1.0

.9

.8

.7

.6

.5
E

m
t_

.4

.3_

2O0 400 600 800 1000 2000

Temperature, TA, °F
I I I [ I I 11111
400 500 600 800 1000 1200 1400
Temperature, TA, °K

FIGURE 5-17.-Effect of surface temperature on emissivity of dielectrics. (Data from

Gubareff et al. (ref. 1).)
 RADIATIVE
PROPERTIES
OFREAL
MATERIALS 151

. .8 _ OA
_" .7 _ ,,--Norton

'_ .._ .6 603

• I °%-oo0
:" I I [ i I I i
• -460 0 500 1000 1500 2000 2500 3000
Temperature, TA, °F

l I I I I
0 500 1000 1500 2000
Temperature, TA, °K

FIGURE 5-18.-Effect of surface temperature on normal-total emissivity of Muminum

oxide. (Data from ref. 5.)

. .9q r'-.

• ,s _ _ A Magnesium stabilized
8 -_'xO O Calcium stabilized
_.6- zx
,5 --

z 4 -- O

"-400 0 500 I000 1500 2000 2_)0 3000

Temperature, TA, °F

I I I I I
0 500 1000 1500 20O0
Temperature, TA, °K

FIGURE 5--19.- Effect of surface temperature on normal total emissivity of zirconium oxide.
(Data from ref. 5.)

and decreasing emissivity trends with temperature are observed. Some

of these effects may be caused by the fact that the dielectric coating is
rather thin; hence, the properties are influenced by the temperature and
spectral characteristics of the underlying material (substrate). For
example, as shown in figure 5-17, magnesium oxide refractory exhibits
a significant emissivity decrease with increasing temperature. For a
silicon carbide coating on graphite, however, the emissivity increases
with temperature; this may be partly caused by the emissive behavior
of the graphite substrate, which was shown in figure 5-4 to increase with
temperature.
White and black paint both have high emissivities for the temperature
range shown as is typical for ordinary oil-base paint. Aluminized paint is
considerably lower in emissive ability since it behaves partly like a metal.
152 THERMAL RADIATION HEAT TRANSFER

1.0 --

.8--

5
.6

E
.4 _ O Weathered asphalt

• I , i ,i,,,l i , I
• 2100 200 400 600 800 lO00 2000 4000 6000 10 000
Temperature of incident radiation source, °F

I I , I ,1,1 I , I ,I
3G0 4O0 6O0 8O0 1000 200O 4OOO 6000
Temperature
ofincident radiationsource,°K

FIGURE 5-20.-Norma| total absorptivity of nonmetals at room temperature for incident

black radiation from source at indicated temperatures. (Data from Gubareff et al.
(ref. 1).)

Note that the emissivity for aluminized paint in figure 5-17 is about
one-half that in fgure 5-15. This further emphasizes the wide variation
in properties that can be found for samples having the same general
description. For applications where the property values are critical, it
may often be necessary to make radiation measurements for the specific
materials being used.
Figure 5-20 gives the normal total absorptivity of a few materials for
blackbody radiation incident from sources at various temperatures.
White paper is shown to be a good absorber for radiation emitted at low
temperatures but is a poor absorber for the spectrum emitted at tempera-
tures of several thousand degrees Fahrenheit. It is thus a reasonably good
reflector of energy incident from the Sun. An asphalt pavement or a
gray slate roof, on the other hand, absorbs energy from the Sun very well.

5.4.3 Effect of Surface Roughness

In figure 5-21 the bidirectional total reflectivity of typewriter paper

is shown for three different angles of incidence. For an ideal (polished,
smooth) surface, a specular peak would be expected with the angle of
reflection and the angle of incidence symmetric about the normal;
obviously, the surface finish of typewriter paper is not ideal, since the
reflected intensity occupies a rather large angular envelope around the
direction of specular reflection.
 RADIATIVE PROPERTIES OF REAL MATERIALS 153

Angleofreflection,I_r, deg
0

-30 30

-45 45

-60 60

-75 ?5

Bidirectional
totalrellectivlty,P"(I_,
6,gr,Or"0 + x)

FIGURE 5-21. - Bidirectional total reflectivity of typewriter paper in plane of incidence.

Source temperature, 2120° R (1178° K). (Replotted from data of Munch (ref. 19).)

The type of curves shown in figure 5-21 has suggested character-

izing reflected energy as a combination of a purely diffuse plus a purely
specular component. This type of approximation has merit in some cases
and results in a simplification of radiant interchange calculations in
comparison with the use of exact directional properties (refs. 12 and 13);
in other cases, however, the approximation would fail completely. An
example is shown in figure 5-22. This figure shows the observed bi-
directional total reflectivity for visible light reflected from the surface
of the Moon. These particular curves are for the mountainous regions,
but very similar curves are obtained for other areas. The interesting
feature of these curves is that the peak of the reflected radiation is back
into the direction of the incident radiation. This peak is located at a
circumferential angle 0 of 180 ° away from where a specular peak would
OCCUr.

A moment's thought will confirm that curves of this type must charac-
terize the lunar reflectivity. At full moon, which occurs when the Sun,
Earth, and Moon are almost (but not quite) in a straight line (fig. 5-23),
the Moon appears equally bright across its face. For this to be true, it
follows that an observer on Earth sees equal intensities from all points
on the Moon. However, the solar energy incident upon a unit area of the
lunar surface varies as the cosine of the angle 13 between the Sun and the
normal to the lunar surface. The angle/3 varies from 0 ° to 90 °, as the posi-
tion of the incident energy varies from the center to the edge of the lunar
disk. To reflect a constant intensity to an observer on Earth from all
observable points on the lunar surface therefore requires that the

295-7630L-68--11
154 THERMAL RADIATION HEAT TRANSFER

¢o.

o_

¢3t
¢o

i
 RADIATIVE PROPERTIES OF REAL MATERIALS 155

Intercepted
incidentenergy
i'Sun dAcos(3d_sun

i'Sun_Sun
!ntensity reflectedtoEarth
=Suncosfi Cl_Su n p"(l_,13
r)

FIGURE5-23.--Reflected energy at full moon.

product p"(_, /3r) cos /3 be constant. Consequently, the value of the

bidirectional reflectivity in the direction of incidence must increase
approximately in proportion to 1/cos /3 (shown by the dashed line in
fig. 5-22) as the angle of incidence increases. This change in reflectivity
with angle of incidence will compensate for the reduced energy incident
per unit area on the Moon at the large angles. The reflectivity behavior
is confirmed by the curves in figure 5-22. Hence, the fact that the Moon
appears uniformly bright does not imply that it is a diffuse reflector. If
the Moon were diffuse, it would appear bright at the center and dark at
the edges.

5.4.4 Semiconductors

Semiconductors are arbitrarily considered here along with the non-

metals, but they behave partly as metals. Liebert (ref. 14) has shown that
their radiative properties can be determined through electromagnetic
theory by treating semiconductors as metals with high resistivity. In
figure 5-24, the normal spectral emissivity of a silicon semiconductor is
shown. The Hagen-Rubens relation shown for comparison is based on
the dc resistivity measured for the same sample, one of the few cases
where such comparable emissive and electrical data are available. Agree-
ment does not become good until wavelengths are reached that are much
156 THERMAL RADIATION HEAT TRANSFER

m
1.0
Experimental
m m -- Hagen-Rubens relation
.8
E
_o A .6

.4

I i I I I I I I I
4 8 12 16 20 24 28 32 36
Wavelength, },,IJm

FIGURE 5-24.--Normal spectral emissivity of a highly doped silicon semiconductor at

room temperature. (Data from ref. 14.)

greater than those giving agreement for metals. This difference in range
of agreement can be traced to the following assumption used in deriving
the Hagen-Rubens equation (see section 4.6.2.2):

( Xo
2¢rcorey/ >> l

For semiconductors, where the resistivity is larger than it is for metals,

this inequality cannot hold until a range of wavelengths larger than those
for metals is reached.
The shape of the curve measured for silicon (fig. 5-24) resembles
what would be expected for a polished metal (see, for example, the
tungsten data in fig. 5-3). The emissivity increases with decreasing
wavelength over much of the measured spectrum, with a peak occurring
at shorter wavelengths. However, most of the features of the semi-
conductor curve occur at longer wavelengths than for a metal: the peak
emissivity, for example, is well outside the visible region.
Liebert (ref. 14) was also able to show excellent agreement between
the measured emissivity and predictions from electromagnetic theory
which included the effects of free electrons and was more sophisticated
than that discussed in chapter 4. The theoretical equations were eval-
uated by using required physical properties that were measured from
the specific samples on which the emissivity measurements were made.

5.5 SPECIAL SURFACES

For engineering purposes, it is often desirable to tailor the radiative

properties of surfaces to increase or decrease their natural ability to
absorb, emit, or reflect radiant energy. This can be done to provide two
general types of behavior, a desired spectral performance or desired
directional characteristics.
 RADIATIVE
PROPERTIES OF REAL MATERIALS 157

5.5.1 Modification of Spectral Characteristics

In applications of surfaces for use in the collection of radiant energy,

such as in solar distillation units, solar furnaces, or solar collectors for
energy conversion, it is desirable to maximize the energy absorbed by a
surface while minimizing the amount lost by emission. In solar thermi-
onic or thermoelectric devices, it is desirable to maintain the highest
possible equilibrium temperature on the surface exposed to the Sun.
Here again a maximum-collection, minimum-loss performance is needed.
Later in this section the condition will be discussed where it is desirable
to keep a surface cool when it is exposed to the Sun. In the latter case,
it is desirable to have a maximum solar reflection accompanied by a
maximum radiative emission from the surface.
For purposes of solar energy collection, a black surface will, of
course, maximize the absorption of incident solar energy, unhappily,
it also maximizes the emissive losses. However, if a surface could be
manufactured that had an absorptivity large in the spectral region of
short wavelengths about the peak solar energy, yet small in the spectral
region of longer wavelengths where the peak emission would occur, it
might be possible to absorb nearly as well as a blackbody while emitting
very little energy. Such surfaces are called "spectrally selective." One
method of manufacture is to coat a thin, nonmetallic layer onto a metallic
substrate. For radiation with large wavelengths, the thin coating is essen-
tially transparent, and the surface behaves as a metal yielding low values

1.0
Siliconoxideonaluminum
(ref. 23)
------ Siliconoxide-germanium-
A
copper(ref. 24)
.8 ----- Idealselectivesurfacefor
incidentsolarenergyto
driveCarnotcycle(ref. 2.5)
:> .6 l ---- Assumed selectivesurface
l for exampleproblem
l
I
.4

t_
l
E
z .2

I , I ,I,1,1 I
.2 .4 I 2 4 10 20
Wavelength,
k, pm

FIGURE 5-25.- Characteristics of some spectrally selective surfaces.

158 THERMAL RADIATION HEAT TRANSFER

for the spectral absorptivity and emissivity. At short wavelengths, how-

ever, the radiation characteristics approach those of the nonmetallic
coating so that the spectral emissivity and absorptivity are relatively
large. Some examples of material behavior of this type are shown in
figure 5-25.
An ideal solar selective surface would absorb a maximum solar
energy while emitting a minimum amount of energy. The surface would
thus have an absorptivity of unity over the range of short wavelengths
where the incident solar energy has a large intensity. At longer wave-
lengths, the absorptivity should drop sharply to zero. The wavelength
_,c at which this sharp drop occurs is termed the cutoff wavelength.

EXAMPLE 5--1: An ideal selective surface is exposed to a normally

incident flux of radiation corresponding to the average solar constant
qi, where qi =442 Btu/(hr)(sq ft). The only means of heat transfer to
or from the surface is by radiation. Determine the maximum equilib-
rium temperature Teq corresponding to a cutoff wavelength of _c = 1/zm.
(Note that the energy arriving from the Sun can be assumed to have a
spectral distribution proportional to that of a blackbody at 10 000 ° R.)

Since the only means of heat transfer is by radiation, the radiant

energy absorbed must be equal to that emitted. Since we have specified
an ideal selective absorber, the hemispherical emissivity and absorptivity
are given by

_x(X) = a_(X) = 1, 0 _< _ <

and

The energy absorbed by the surface per unit time is

Qa = (1)Fo-_c(Tn)qiA

where Fo-_(TR) is the fraction of blackbody energy in the range of wave-

lengths between zero and the cutoff value, for a radiating source at
temperature TH. In this case, TH is the effective solar radiating tempera-
ture of about 10 000 ° R. Similarly, the energy emitted by the selective
surface is

Q_= (1)Fo-_c( T_q)crT_qA

Equating Q_ and Qa yields

 RADIATIVE
PROPERTIES OF REAL MATERIALS 159

qiFo-xc( Tn )
T_qF°-xc ( req) = o"

For the chosen value for he, all terms on the right are known, and we
can solve for Teq by trial and error. The equilibrium temperature for
he= 1 gm, as specified in the problem, is 2400 ° R. Values of Teq corre-
sponding to other values of h_ are given in the following table:

Cutoff Equilibrium
wavelength, temperature,
Teq,
gm oR

0.6 3250
.8 2750
1.0 24OO
1.2 2150
1.5 1890
712

For a blackbody (h¢---_ _), the equilibrium temperature is 712 ° R; this

is the equilibrium temperature of the surface of a black object in space
near the Earth's orbit when exposed to solar radiation and with all other
surfaces of the object perfectly insulated. The same equilibrium tem-
perature is reached by a gray body, since a gray emissivity would
cancel ou_ of the energy balance equation.
As smaller values of hc are taken, Teq continues to increase even
though less energy is absorbed, because it becomes relatively more
difficult to emit energy as h_ is decreased.
A common measure of the performance of a given selective surface
is the ratio of the directional total absorptivity c_' (/3, 0, TA) of the surface
for incident solar energy to the hemispherical total emissivity of the
surface e(TA). The ratio a'/_ for the condition of incident solar energy
is a measure of the theoretical maximum temperature that an otherwise
insulated surface can attain when exposed to solar radiation. The
significance of _'/_ is shown as follows.
The energy absorbed per unit time by any surface when exposed
to a directional incident intensity is given by

dQ'_(fl, O, Ta) =a'(/_, 0, TA)dQ_(_, o) (5-1)

For the case of solar energy with a flux of qi = 442 Btu/(hr)(sq ft), incident
160 THERMAL RADIATION HEAT TRANSFER

from direction (/3, 0) on a surface element dA, this can be written as

dQ'_(fl, O, TA) =t_'(fl, O, TA)qidA cos fl (5-2)

The total energy emitted per unit time by the surface element is
given by

dQ_ =e(Ta)dA =¢(TA)o'T] dA (5-3)

If the only energy absorbed by the surface in question is that given by

equation (5-2) and the surface only loses energy by radiation, the
emitted and absorbed energies as given by equations (5-3) and (5-2),
respectively, may be equated to give

or' (fl, O, Teq) = ¢rT_4q (5-4)

(Tea) qi cos 13

where Teq is the equilibrium temperature that is achieved. Thus, the

ratio a'(/3, 0, TA)/_(TA) is a measure of the equilibrium temperature of
the element. Note also that the temperature at which the properties a'
and _ are selected must be the equilibrium temperature that the body
attains. In practice, temperature dependence of the properties is often
assumed small so that this restriction can be somewhat relaxed.
The most common case considered is when the solar radiation is
incident in the direction normal to the surface. Equation (5-4) becomes

aL(T_q)= (rT4_ (5-5)

_ ( Teq ) q_

Equation (5-5) shows that the smaller the value of a',J_ that can be
reached, the smaller will be the equilibrium temperature. For a cryo-
genic storage tank inspace, a'_/_ should be as small as possible. In
practice, values of a'_/_ in the range 0.20 to 0.25 can be obtained.
To attain high equilibrium temperatures, c_;_/_ should be as large
as possible. Polished metals attain a'_/_ values of 5 to 7, while specially
manufactured surfaces have values of a;,/E approaching 20.
The upper limit of a'_/s is established by the thermodynamic argument
that the equilibrium temperature of the selective surface cannot exceed
the effective solar temperature of about 10 000 ° R. Substituting this solar
temperature value into equation (5-5) gives

max -- o'(10 442000) 4_ 3.87 × 104 (5-6)

(Teq)
 RADIATIVE PROPERTIES OF REAL MATERIALS 161

Attaining anything even close to this value of a_/t is far beyond the
present state of the art.

EXAMPLE 5--2: The properties of a real SiO-A1 selective surface can be

approximated by the long-dashed curve of figure 5-25. (It is assumed
that the long-dashed curve can be extrapolated toward _= 0 and h = _.)
What is the equilibrium temperature of the surface for normally incident
solar radiation when the only heat transfer is by radiation? What is
a',,/_ for the surface? Describe the spectra of the absorbed and emitted
energy at the surface. (Assume normal and hemispherical emissivities
are equal.)

As in the derivation of equation (5-5), we equate the absorbed and

emitted energies. The emissivity has nonzero values on both sides of
the cutoff wavelength, so that

Oa'= eo-_cFo-xc( TR)qiA + eXc-=Fxe-®( TR ) qiA = a'nqiA

and

Qe= E0-x_0-_c(Teq)_Te_A + _c___= (Te0)_T_$A= _rT_ct

Equating Qe and Q_ gives

{0.95Fo-x_(TR) + 0.0511 -Fo-x_(Tn)]}q_

= {0.95Fo-x_(T_q) + 0.0511 -- Fo-x_(T_q) ] }trT_q

Solving by trial and error as in example 5-1, we obtain for _ = 1.5/xm,

Teq=1430 ° R. For qi=442 Btu/(hr)(sq ft), equation (5-5) gives
a'Je=or(1430)4/442 = 16.2. The small difference in properties in this
example from the properties of an ideal selective surface produced a
significant change in Teq, which from the previous example was 1890 ° R
for an ideal selective surface having the same cutoff wavelength. The
spectral curves of absorbed and emitted energy are shown in figure 5-26.
The spectral curve of incident solar energy is given by

ex.i(h, Tn) _ exb(h, TR)

It has the shape of the blackbody curve at the solar temperature, but it
is reduced in magnitude so that the integral of ex, i over all h is equal
to q_, the total incident solar energy per unit area. Multiplying this
curve by the spectral absorptivity of the selective surface gives the
spectrum of the absorbed energy. The spectrum of emitted energy
162 THERMAL RADIATION HEAT TRANSFER

l@-
-- ------ Normal incident solar radiation
- ---- Energy emitted by blackbodyat 1420° R (789° K)
I:'/"/'/721 Energy absorbedby selective surface
103 _-- - _ Energy emitted by selective surface

- A

/ ,,

ff a:J

= 102 _=

-- m

101_
lo-i lo0 1oI 1o
2
Wavelength, ;_, pm

FIGURE 5--26.--Spectral distribution of energy absorbed and emitted by example selective

absorber.

Thickness
of sheet,
cm
1.C-- co

.6

._ ,<- .8-- _

.2
"4 f "
-w _2:_11 I I I I I
0 I 2 3 4 5 6 7 8
Wavelength, ;_, pm

FIGURE 5-27.- Emittance of sheet s of window glass at 1000 ° C. (Data from Gardon (ref. 26).)
 RADIATIVE
PROPERTIES OF REAL MATERIALS 163

is that of a blackbody at 1420 ° R multiplied by the spectral emissivity

of the selective surface. The integrated energy under the spectral
curves of absorbed and emitted energy are equal, ahhough this is not
obvious from the log-log plot.

EXAMPLE 5-3: A selective surface having spectral characteristics as

given in the previous example is to be used as a solar energy absorber.
The surface is to be maintained at a temperature of T_ = 712 ° R by ex-
tracting energy to be used in a power generating cycle. If the absorber
is placed in orbit around the Sun at the same radius as the Earth, how
much net energy will a square foot of the surface provide? How does this
energy compare with that supplied by a black surface at the same
temperature?

The net energy extracted from the surface is the difference between
that absorbed and that emitted. The absorbed energy flux is as in
example 5-2

qa= {0-95F0-s c (Ts) + 0.0511 -- F0_s c (TR)] }qi

= [0.95 (0.869) + 0.05 ( 1 - 0.869) ] 442

Btu
= 368
(hr)(sq ft)

The emitted flux is

qe = {0.95Fo-xc(Ta) + 0.05 [1 -- Fo-xc (Ta) ] }crTg

= {0.95 x (- 0) +0.0511-- (-- 0)]}0.1712x 10 -s× (712) 4

Btu
= 22.0
(hr)(sq ft)

and the net energy that can be used for power generation is (368-22)
= 346 Btu/(hr)(sq ft). For a black or gray body, the equilibrium tempera-
ture was found in example 5-1 as 712 ° R, so that the net useful energy
that could be removed from such a surface would be zero.

Although it is partly a transmission effect, it is worth mentioning the

characteristic ability of a glass enclosure, such as a greenhouse, to trap
solar energy. A glass plate can also be used to cover a surface in order
to increase the efficiency of the surface as a solar absorber. The reason
for this is that many types of glass are spectrally selective with regard
to their transmission of radiation. Figure 5-27 shows the emittauce
164 THERMAL RADIATION HEAT TRANSFER

(being an extensive property) of window glass sheets of various thick-

nesses, and the form of the curves with regard to variation with wave-
length is opposite to that in figure 5-25. The cutoff wavelength is about
2.7/_m; this means that the glass has a low absorptance for solar radia-
tion which is primarily at the short wavelengths, and consequently inci-
dent solar radiation passes readily into a glass enclosure. The emission
from objects at ambient temperature inside the enclosure is at long
wavelengths and is trapped because of the high absorptance (poor
transmission) of the glass in this spectral region.

Another application in which spectrally selective surfaces can be

employed to advantage is where it is desirable to cool an object that is
exposed to incident radiation from a high-temperature source. The most
common situation would be objects exposed to the Sun, such as a gaso-
line storage tank, a cryogenic fuel tank in space, or the roof of a building.
A highly reflecting coating could be utilized, such as a iJolished metal.
This would reflect much of the incident energy, but would be poor for
radiating away any energy that was absorbed or generated within the
enclosure (e.g., an enclosure filled with electronic equipment). Also,
some metals have a tendency toward lower reflectivity at the shorter
wavelengths; this is shown, for example, for uncoated aluminum in
figure 5-13. For some applications, it may be advantageous to use a
material that is spectrally selective; white paint as shown in figure 5-28

1.0 --

.8

re, .6 --

,4

.2

0 , I, 1,1 , I
.4 .6 .8 1 2 4 6 8 10 20 4o
Wavelength, _,, _m

FIGURE 5-28.-Reflectivity of white paint coating on aluminum. (Replotted from Dunkle

(ref. 27).)
 RADIATIVE PROPERTIES OF REAL MATERIALS 165

is an example. This will not only reflect the incident radiation which is
predominantly at short wavelengths but will also radiate well at the
longer wavelengths characteristic of the relatively low temperature of
the body.

5.5.2 Modification of Directional Characteristics

As discussed in previous sections of this chapter, the roughness of a

surface can have profound effects on the radiative properties and will
indeed become a controlling factor when the roughness is large in com-
parison with the wavelength of the energy being considered. This leads
to the concept of controlling the roughness in order to tailor the direc-
tional characteristics of a surface.
If the surface is used as an emitter, the surface might be roughened
or designed in such a way as to emit strongly in preferred directions,
while reducing emission into unwanted directions. Commercial radiant
area heating, equipment would operate more efficiently using such sur-
faces by directing energy to the areas where it is most needed. The most
common device for controlling the directional distribution of electro-
magnetic radiation in the visible region is called a "lamp shade."
If the directional surface were primarily used as an absorber, then,
using a solar absorber as an example, we might make it strongly absorb-
ing in the direction of incident solar radiation but as close as possible to
nonabsorbing in other directions. The surface would, because of Kirch-
hoff's law for directional properties, emit strongly toward the Sun, but
weakly in other directions. The surface would absorb the same energy
as a nondirectional absorber since the incident energy is only from the
direction of the Sun but would emit less energy than a surface that
emits well into all directions.
The characteristics of one such surface are shown in figure 5-29. The
surface has very long grooves of angle 18.2 ° running parallel to each
other. A highly reflecting specular coating is placed on the side walls
of each groove, and a black surface is placed at the base of each groove.
The solid line gives the behavior predicted by analysis of such an ideal
surface, while the data points show experimental results for an actual
surface. It is seen that the directional total emissivity is very high for
angles of emission less than about 30 ° from the surface normal. It then
drops rapidly as the angle becomes larger. Many other such surface
configurations exhibit similar characteristics.

EXAMPLE 5--4: Suppose that a directional surface has a directional

total emissivity given by

¢(fl) = 1 0 _</3 _< 30 °

_(/3) =0 /3>30 °
166 THERMAL RADIATION HEAT TRANSFER

Angle of emission, 13, deg

0
15

90
0 .2 .4 .6 .8 1.0
Directional emissivity, E'(_)

FIGURE 5-29.--Directional emissivity of grooved surface with highly reflecting specular

side walls and highly absorbing base; diD = 0.649.

For solar radiation incident normally on such a surface on Earth with

no other heat exchange except by radiation from the directional surface,
what is the equilibrium temperature of the surface? How does this
temperature compare with that achieved by a black surface?

The absorptivity of this surface for normal incident radiation is unity.

Therefore, the absorbed energy per unit time is

Q, = (1) qM

where qi is again the solar constant for an object at a distance equal to

the mean radius of the Earth's orbit from the Sun (qi = 442 Btu/(hr)(sq ft)).

The energy emitted by the body when it is at thermal equilibrium is

 RADIgTIVE PROPERTIES OF REAL MATERIALS 167

Qe == EorT4q/1

where e is the hemispherical total emissivity given by equation (3--6b) as

1 C
dco
E(Te) =_ |J_ _' (_, O, Teq) cos/3

For this problem, e becomes

sin fl cos fl d E = 0.25

Equating Qa and Qe for radiative equilibrium gives

Te°= (qi)l/4
_ = ( 442
0.25 × 0.1712 × 10-_] ,_,/. = 1005° R

This is larger than the equilibrium temperature of a black or gray dif-

fuse body of 712 ° R as shown in example 5-1.
Note that equation (5-5) can be used for the a_/¢ of directional as
well as speetrally selective surfaces. For the surface used in this ex-
ample, a_/¢=4.0. Combining selective and directional effects would
be a way for obtaining considerably increased values of a_/_ for a given
surface.
It should not be inferred that the directional distribution of emis-
sivity assumed in this example corresponds to that of the parallel
grooved surface in figure 5-29. In the case of figure 5-29, there is a
strong dependence on the angle 0, which has been ignored in this
example.

5.6 CONCLUDING REMARKS

The radiative property examples discussed in the present chapter

have illustrated a number of the features that may be encountered when
dealing with real surfaces. Certain broad generalizations could be at-
tempted. For example, the total emissivities of dielectrics at moderate
temperatures are larger than those for metals, and the spectral emis-
sivity of metals increases with temperature over a broad range of wave-
lengths. However, these types of rules can be misleading because of the
large property variations that can occur as a result of surface roughness,
contamination, oxide coating, grain structure, and so forth. The presently
available analytical procedures cannot account for all these factors so
that it is not possible to predict radiation property values directly except
168 THERMAL RADIATION HEAT TRANSFER

for surfaces that approach ideal conditions of composition and finish.

By coupling analytical trends with observations of experimental trends,
it is possible to gain some insight into what classes of surfaces would be
expected to be suitable for specific applications and how surfaces may
be fabricated to obtain certain types of radiative behavior. The latter
includes spectrally selective surfaces which are of great value in a num-
ber of practical applications such as the collection of solar energy.
Some other factors affecting radiative properties that evade prediction
are outside the range of interest in this work, but they should be men-
tioned. For example, it is well known that exposure to ultraviolet radia-
tion; cosmic rays: neutron, gamma, and proton bombardment: and the
solar wind can all cause significant changes in radiative properties. For
design of spacecraft, these effects are of major concern.
Finally, some comment on the measurement of radiative properties
should be given. It has been noted that few precise measurements of
directional spectral properties have been made. The reason for this
lies in one of the many practical difficulties involved. This is that for a
directional measurement the energy available for detection at a small
solid angle centered about a given direction is itself small. If only the
portion of this small energy that lies within a wavelength band is then
measured to obtain directional spectral values, an even smaller energy
is available for detection. Minor absolute errors in the measurement of
the energy can then lead to large percentage errors in the directional
properties being determined. Further, the sheer magnitude of the amount
of data generated for such combined directional spectral properties
precludes its gathering unless a very specific problem requires it. These
and similar practical problems make the field of thermal radiation prop-
erty measurement a most exacting and difficult one.

REFERENCES

1. GUBAREFF, G. G.; JANSSEN, J. E.; AND TORBERG, R. H.: Thermal Radiation Properties
Survey. Second ed., Honeywell Research Center, 1960.
2. SVET, DARII IAKOVLEVICH: Thermal Radiation; Metals, Semiconductors, Ceramics,
Partly Transparent Bodies, and Films. Consultants Bureau, 1965.
3. GOLDSMITH, ALEXANDER; AND WATERMAN, THOMAS E.: Thermophysical Properties
of Solid Materials. (WADC TR 58-476), Armour Research Foundation, Jan. 1959.
4. HOTTEL, n. C.: Radiant Heat Transmission. Heat Transmission. William H. McAdams,
Third ed., McGraw-Hill Book Co., Inc., 1954, pp. 472-479.
5. WOOD, W. D.; DEEM, H. W.; AND LUCKS, C. F.: Thermal Radiative Properties. Plenum
Press, 1964.
6. EDWARDS, D. K.; AND CARTON, IVAN: Radiation Characteristics of Rough and Oxidized
Metals. Advances in Thermophysical Properties at Extreme Temperatures and
Pressures. Serge Gratch, ed., ASME, 1965, pp. 189-199.
7. SADYKOV, B. S.: Temperature Dependence of the Radiating Power of Metals. High
Temp., vol. 3, no. 3, May-June 1965, pp. 352-356.
 RADIATIVE PROPERTIES OF REAL MATERIALS 169

8. DAVIES, H.: The Reflection of Electromagnetic Waves from a Rough Surface. Inst.
Elect. Engrs. Proc., IV, vol. 101, Aug. 1954, pp. 209-214.
9. PORTEUS, J. O.: Relation Between the Height Distribution of a Rough Surface and the
Reflectance at Normal Incidence. Opt. Soc. Am. J., vol. 53, no. 12, Dec. 1963, pp.
1394-1402.
10. BECKMANN, P.; AND SPIZZICHINO, A.: The Scattering of Electromagnetic Waves From
Rough Surfaces. The Macmillan Company, 1963.
11. LIEBERT, CURT H.: Spectral Emittance of Aluminum Oxide and Zinc Oxide on Opaque
Substrates. NASA TN D-3115, i965.
12. SAROFIM, A. F.; AND HOTTEL, H. C.: Radiative Exchange Among Non-Lambert Sur-
faces. J. Heat Transfer, vol. 88, no. 1, Feb. 1966, pp. 37--44.
13. SPARROW, E. M.; AND LIN, S. L.: Radiation Heat Transfer at a Surface Having Both
Specular and Diffuse Reflectance Components. Intl. J. Heat Mass Transfer, vol. 8,
May 1965, pp. 769-779.
14. LIEBERT CURT H.: Spectral Emissivity of Highly Doped Silicon. Paper No. 67-302,
AIAA, Apr. 1967.
15. SEBAN, R. A.: Thermal Radiation Properties of Materials. Part IlL (WADD TR-60-370,
Pt. III), California Univ. (Berkeley), Aug. 1963.
16. DEVos, J. C.: A New Determination of the Emissivity of Tungsten Ribbon. Physica,
vol. 20, 1954, pp. 690-714.
17. BIaKEBAK, R. C.; AND ECKERT, E. R. G.: Effects of Roughness of Metal Surfaces on
Angular Distribution of Monochromatic Reflected Radiation° J. Heat Transfer, vol.
87, no. 1, Feb. 1965, pp. 85-94.
18. EDWARDS, D. K.; AND DEVOLo, N. BAYARD: Useful Approximations for the Spectral
and Total Emissivity of Smooth Bare Metals. Advances in Thermophysical Prop-
erties at Extreme Temperatures and Pressures. Serge Gratch, ed., ASME, 1965,
pp. 174-188.
19. MUNCH, B.: Directional Distribution in the Reflection of Heat Radiation and its Effect
in Heat Transfer. Ph.D. thesis, Swiss Technical College of Zurich, 1955,
20. WILLIAMS, D. A.; LAPPIN, T. A.; AND DUFFle, J. A.: Selective Radiation Properties
of Particulate Coatings. J. Eng. Power, vo]. 85, no. 3, July 1963, pp. 213-220.
21. OHLSEN, P. E.; AND ETAMAD, G. A.: Spectral and Total Radiation Data of Various
Aircraft Materials. Rep. No. NA57-330, North American Aviation, July 23, 1957.
22. ORLOVA, N. S.: Photometric Relief of the Lunar Surface. Astron. Zhur., vol. 33, no. 1,
Jan./Feb. 1956, pp. 93-100.
23. LONG, R. L.: A Review of Recent Air Force Research on Selective Solar Absorbers.
J. Eng. Power, vol. 87, no. 3, July 1965, pp. 277-280.
24. HIBBARD, R. R.: Equilibrium Temperatures of Ideal Speetrally Selective Surfaces.
Solar Energy, vol. 5, no. 6, Oct.-Dec. 1961, pp. 129-132.
25. SHAFFER, L. H.: Wavelength-Dependent (Selective) Processes for the Utilization of
Solar Energy. J. Solar Energy Science Eng., vol. 2, no. 3-4, July-Oct. 1958, pp. 21-26.
26. GARDON, ROBERT: The Emissivity of Transparent Materials. Am. Ceramic Soc. J.,
vol. 39, no. 8, Aug. 1956, pp. 278-287.
27. DUNKLE, R. V.: Thermal Radiation Characteristics of Surfaces. Theory and Funda-
mental Research in Heat Transfer. J. A. Clark, ed., Pergamon Press, 1963, pp. 1-31.
28. PERLMUTTER, MORRIS; AND HOWELL, JOHN R.: A Strongly Directional Emitting and
Absorbing Surface. J. Heat Transfer, vol. 85, no. 3, Aug. 1963, pp. 282-283.
29. BRANDENBERG, W. M.; AND CLAUSEN, O. W.: The Directional Spectral Emittance of
Surfaces Between 200 ° and 600 ° C. Symposium on Thermal Radiation of Solids,
S. Katzoff, ed., NASA SP-55 (AFML-TDR--64-159), 1965.

°.

295-763 0L-68-- 12
 Appendix
Tables of conversion factors between the mks and other common
systems of units are given in tables I to III of this appendix. In table IV,
accurate values of the various radiation constants are given in both mks
units and the common English engineering units. Finally, table V lists
various blackbody properties as functions of the variable hT, again in
both mks and English engineering units.
With regard to table V, Pivovonsky and Nagel (ref. 3) and Wiebelt
(ref. 4) have presented polynomial curves fitted to the function F0-xr.
These curve fits can be quite useful for computer solutions of various
types of radiation problems. Wiebelt recommends use of the following
polynomials:

15 e-////_

F°-_r= -- 2 {[(mv+ 3)mv+6]mv+6}, v >_ 2

¢r4 m 4
//1=1, 2, . . .

F0_xr = 1 --_v15 3{1

_-_-_ v 60v_ 5040v44-272160
v6 133_600vS ) , v < 2

where

and C2 is given in table IV. The series is carried out to a sufficient number
of terms to gain the desired accuracy.

REFERENCES

l. MECHTLY, E. A.: The International System of Units. Physical Constants and Conversion
Factors. NASA SP-7012, 1964.
2. GUnAREFF, G. G.; JANSSEN, J. E.; AND TOanERG, R. H.: Thermal Radiation Properties
Survey. Second ed., Honeywell Research Center, 1960.
3. PIVOVONSKY, MARK; AND NAGEL, MAX R.: Tables of Blackbody Radiation Functions.
Macmillan Book Co., 1961.
4. WIEBELT, JOHN A.: Engineering Radiation Heat Transfer. Holt, Rinehart & Winston,
1966.

171
172 THERMAL RADIATION HEAT TRANSFER

=o
.._ X S&xx_B$ _

..=-

ii
X &Bxx$$_ 22
§

X $&x_2. ' '

[-

ae
_._-_ _:
o.
O
[-

r_-

mm

C:

I
_x_-×××x ××

[..
i

_ o

_ _ X X X X X X X _<

¢-1 _0 u'_ ¢-I ¢",1 ¢',1

: : : : : : : : I(
 APPENDIX 173

XXXXX_

X X X _:_,'-_ X

0_

N
z

< E_ ?TT?
z
_XXXX
£
i.

_XXXXX

<

iiiiii
ii::iii
i!iii::
• ° .... [..

::iiiil
i II : : : .

: ._ . : : .
174 THERMAL RADIATION HEAT TRANSFER

TABLE |II.--CONVERSION FACTORS FOR ENERGY FLUX

cal/(sec)(cm 2) Btu/(hr)(ft _) W/m z erg/(sec)(cm 2)

1 cal/(sec)(cm2) a=. .......... 1 1.329 x 104 4.187 x 104 4.187 x 107

I Btu/(hr)(ft 2) =. .............. 7.525 x 10 -_ 1 3.152 3.152 x 10 a
1 W/m _=. ..................... 2.388 x 10 -_ 0.3174 1 103

1 erg](sec)(cm 2) =. ........... 2.388 x 10 -s 3.174 x 10 -4 1 0 =a 1

abased on International Steam Table.

TABLE IV.--RADIAT1ON CONSTANTS a

Symbol Definition Value

C1 ................. Constant in Planck's spectral energy. 0.18892 x 10s; 0.59544 x 10 -12

distribution, (BtuXttm4)/(hrXft2);
W/em z
C 2 ................. Constant in Planck's spectral energy 25898; 1.4388
distribution, (gm)(°R); (cm)(°K)
C 3 ................ Constant in Wien's displacement law 5216.0; 0.28978
(gm)(°R); (cm)(°K)
O'calcula_ d ........ Calculated Stefan-Bohzmann con- 0.1712 x 10-s; 5.6693 x 10 -12
stant, Btu/(hr)(ftz)(°R_ -,
W/(cm2)(°K4)
O'experimental ..... Experimental Stefan-Bohzmann 0.173 × lO-S; 5.729 × 10 -lz
constant, Btu/(hr)(ft2)(°R4);
W/(cm2)(°K 4)

a Recommended" values from ref. 2.

 APPENDIX .175

TABLE V. -- BLACKBODY FUNCTIONS

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference
product, _.T power of temperature, e_/T s Blackbody between

fraction, successive
I Fo-xr FO-kT

Btu W values, AF

(_m)(°R) [ (#t m)(°IO (hrXsq ft)(gm)(°R 5) (cm_)(gm)(°K s)

1000 555.6 0.000671 × 10 -'5 0.44)0 × 10 -20 0.170 × 10 -7 0

1100 611.1 .00439 .261 × 10 -'9 .136 × 10 -_ .119 × 10 -s
1200 666.7 .0202 .120 X 10 -ts .756 × 10 -s .620 × 10 -s
1300 722.2 .0713 .424 × 10-tS .3i7 × i0 -5. .241 × 10 -s
1400 777.8 .204 .00122 × 10 -t5 .106 × 10 -4 .748 × 10 -s

1500 833.3 .496 × 10 -Is .00296 x 10 -t5 .301 X 10 -4 .194 × 10 -4

1600 888.9 1.057 .00630 .738 × 10 -4 .437 x 10-4!
1700 944.4 2.023 .01205 .161 × 10 -3 .876 x 10 -4
1800 1000.0 3.544 .02111 .321 x 10 -s .00016
1900 1055.6 5.767 .03434 .589 × 10 -_ .00027

2000 1111.1 8.822 × 10 -Is .05254× 10 -_s .00101 .OOO42

2100 1166.7 12.805 .07626 .00164 .00063
22OO 1222.2 17.776 .10587 .00252 .00089
23OO 1277.8 23.746 .14142 .00373 .00121
2400 1333.3 30.686 .18275 .00531 .00158

25O0 1388.9 38.526 × 10 -_ .22945 × 10 -'s .00733 .00202

2600 1444.4 47.167 .28091 .00983 .00250
2700 1500.0 56.483 .33639 .01285 .00302
2800 1555.6 66.334 .39505 .01643 .0O358
29OO 1611.1 76.571 .45602 .02060 .00417

3000 1666.7 87.047 × 10 -_5 .51841 × 10 -is .02537 .0O477

3100 1722.2 97.615 .58135 .03076 ,00539
3200 1777.8 108.14 .64404 .03677 .00600
3300 1833.3 118.50 .70573 .04338 .00661
3400 i888.9 128.58 .76578 .05059 .00721

3500 1944.4 138.29 × 10 -l_ .82362 × 10 -is .05838 ._779

3600 2000.0 147.56 .87878 .06672 .00834
3700 2055.6 156.30 .93088 .07559 .00887
3800 2111.1 164.49 .97963 .08496 .00936
39OO 2166.7 172.08: 1.0248 .09478 .00982

4O0O 2222.2 179.04 × 10 -_5 1.0663× 10 -Is .10503 .01025

4100 2277.8 185.36 1.1039 .11567 .01064
420O 2333.3 191.05 1.1378 .12665 .01099
43OO 2388.9 196.09 1.1678 .13795 .01130
44OO 2444.4 200.51 1.1942 .14953 .01158
176 THERMAL RADIATION HEAT TRANSFER

TABLE V. -- BLACKBODY FUNCTIONS -- Continued

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference
product, AT power of temperature, e_b/T 5 Blackbody between
_action, successive
Fo-_T FO-_T
Btu W vMues, AF
(gm)(°R) ! (p,m)(°K)
(hr)(sq ft)(gm)(°R 5) (cm2)(/_m)(°K 5)

4500 2500.0 204.32 × 10 -15 1.2169 × 10 -15 0.I6135 0.01182

4600 2555.6 207.55 1.2361 .17337 .01202
4700 2611.1 210.20 1.2519 .18556 .01219
4800 2666.7 212.32 1.2645 .19789 .01233
4900 2722.2 213.93 1.2741 .21033 .01244

5000 2777.8 215.06× 10 15 1.2808 × 10 -15 .22285 .01252

5100 2833.3 215.74 1.2848 .23543 .01257

5200 2888.9 216.00 1.2864 .24803 .01260

5300 2944.4 215.87 1.2856 .26063 .01260
5400 3000.0 215.39 1.2827 .27322 .01259

5500 3055.6 214.57 × 10 is 1.2779× 10 15 .28576 .01255

5600 3111.1 213.46 1.2713 .29825 .01249
5700 316_7 212.07 1.2630 .31067 .01242
5800 3222.2 210.43 1.2532 .32300 .01233
5900 3277.8 208.57 1.2422 .33523 .01223

6000 3333.3 206.51 × 10 -15 1.2299 × 10 -15 .34734 .01211

6100 3388.9 204.28 1.2166 .35933 .01199
6200 3444.4 201.88 1.2023 .37118 .01185
6300 3500.0 199.35 1.1872 .38289 .01171
6400 3555.6 196.69 1.1714 .39445 .01156

6500 3611.1 193.94 × 10 -15 1.1550 x 10 -15 .40585 .01140

6600 3666.7 191.09 1.1380 .41708 .01124
6700 3722.2 188.17 1.1206 .42815 .01107
6800 3777.8 185.18 1.1029 .43905 .01089
6900 3833.3 182.15 1.0848 .44977 .01072

7000 3888.9 179.08 X 10 -15 1.0665 × 10 -15 .46031 .01054

7100 3944.4 175.98 1.0481 .47067 .01036
7200 4000.0 172.86 1.0295 .48085 .01018

7300 4055.6 169.74 1.0109 .49084 .01000

7400 4111.1 166.60 0.99221 .50066 .00981

7500 4166.7 163.47 × 10 -Is .97357 × 10 -15 .51029 .00963

7600 4222.2 160.35 .95499 .51974 .00945

7700 4277.8 157.25 .93650 .52901 .00927

7800 4333.3 154.16 .91813 .53809 .00909

7900 4388.9 151.10 .89990 .54700 .00891

 APPENDIX 177

TABLE V. -- BLACKBODY FUNCTIONS-- Continued

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference
Blackbody
product, _T power of temperature, e_/T 5 between
_action,
successive
Fo-kr
Fo-xT

Btu W values, AF
(pmX°R) ! (p, mX°K)
(hrXsq ft)(gm)(°R s) (cm2X/zm)(°K 5)

8000 4444.4 148.07 × 10 -'5 0.88184 × 10 -_5' 0.55573 0.00873

8100 4500.0 145.07 .86396 .56429 .00855
8200 4555.6 142.10 .84629 .57267 .00838
8300 4611.1 139.17 .82884 .58087 .00821
8400 4666.7 136.28 .81163 .58891 .00804

8500 4722.2 133.43 x 10 -15 .79467 × 10 -_ .59678 .00787

8600 4777.8 130.63 .77796 .60449 .00771
8700 4833.3 127.87 .76151 •61203 .00754
8800 4888.9 125.15 .74534 .61941 .00738
8900 4944.4 122.48 .72944 .62664 .00723

9000 5000.0 119.86 x 10 -'s .71383 × 10 -'5 .63371 .00707

9100 5055.6 117.29 .69850 .64063 .00692
9200 5111.1 114.76 .68346 .64740 .00677
9300 5166.7 112.28 .66870 .65402 .00662
9400 5222.2 109.85 .65423 •66051 .00648

9500 5277.8 107.47 × 10 -'5 .64006 × I0 -'5 .66685 .00634

9600 5333.3 105.14 .62617 .67305 .00620
9700 5388.9 102.86 .61257 .67912 .00607
9800 5444.4 100.62 .59925 .685O6 .00594
9900 5500.0 98.431 .58621 .69087 .00581

10000 5555.6 96.289 x 10 -'5 .57346 X I0 -'5 .69655 .OO568

10100 5611.1 94.194 .56098 .70211 .00556
10200 5666.7 92.145 .54877 .70754 .00544
10300 5722.2 90.141 .53684 .71286 .00532
10400 5777.8 88.181 .52517 .71806 .00520

10500 5833.3 86.266 × 10 -15 .51376 × 10 -'s .72315 .0O5O9

10600 5888.9 84.394 .50261 .72813 .00498
10700 5944.4 82.565 .49172 .73301 .00487
10800 6000.0 80.777 .48107 .73777 .00477
10900 6055.6 79.031 .47067 .74244 .00466

11000 6111.1 77.325 × 10 -_ .46051 × 10 -_s .74700 .OO456

11100 6166.7 75.658 .45059 .75146 .00446
11200 6222.2 74.031 .44089 .75583 .00437
11300 6277.8 72.441 .43143 .76010 .00427
11400 6333.3 70.889 .42218 .76429 .00418
 THERMAL RADIATION HEAT TRANSFER
178

TABLE V. -- BLACKBODY FUNCTIONS-- Continued

Wavelength- Blackbody hemispherical spectral

Difference
temperature emissive power divided by fifth
Blackbody between
product, _,T power of temperature, exb/T 5
fraction, successive

FO-_T
Fo-xr

Btu W values, AF
_mX°R) mnX°K)
(hrXsq ft)(pmX°R 5) (cm2Xp, m)(°K _)

0.00409
11500 6388.9 69.373 × 10 -_5
.40434 .77238 .00401
11600 6444.4 67.892
.39573 .77630 .00392
11700 6500.0 66.447
.38732 .78014 .00384
11800 6555.6 65.036
.37912 .78390 .00376
11900 6611.1 63.658

.37111 × 10 -15 .78757 .00368

12000 6666.7 62.313 × 10 -15
.36328 .79117 .00360
12100 6722.2 60.999
.35565 .79469 .00352
59.717
12200 6777.8 .00345
58.465 .34819 .79814
12300 6833.3 .00338
57.242 .34091 .80152
12400 6888.9

.33380 × 10 -15 .80482 .00331

12500 6944.4 56.049 × 10 -15
.32687 .80806 .00324
12600 7000.0 54.884
.32009 .81123 .00317
12700 7055.6 53.747
.00310
52.636 .31348 .81433
12800 7111.1
.30702 .81737 .00304
12900 7166.7 51.552

.30071 × 10 -15 .82035 .00298

13000 7222.2 50.493 × 10 -1_
.00292
49.459 .29456 .82327
13100 7277.8
.28855 .82612 .00286
13200 7333.3 48.450
.28268 .82892 .00280
13300 7388.9 47.465
.27695 .83166 .00274
13400 7444.4 46.502

.00269
45.563 × 10 -15 .27135 × 10 -15 .83435
13500 7500.0
.26589 .83698 .00263
13600 7555.6 44.645
.26055 .83956 .00258
13700 7611.1 43.749
.25534 .84209 .00253
13800 7666.7 42.874
.25024 .84457 .00248
13900 7722.2 42.019

.00243
41.184× 10 -_5 .24527 × 10 -_5 .84699
14000 7777.8
.24042 .84937 .00238
14100 7833.3 40.368
.23567 .85171 .00233
14200 7888.9 39.572
.00229
38.794 .23104 .85399
14300 7944.4
.00224
38.033 .22651 .85624
14400 8000.0

.00220
37.291 × 10 -Is .22209× 10 -15 .85843
14500 8055.6
.21777 .86059 .00216
14600 8111.1 36.565
.21354 .86270 .00211
14700 8166.7 35.856
.00207
35.163 .20942 .86477
14800 8222.2 .00203
.20539 .86681
14900 8277.8 l 34.487
 APPENDIX 179

TABLE
V.--BLACKBODY
FUNCTIONS--
Continued

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference
product, kT power of temperature, e_/T 5 Blackbody between
fraction, successive
Fo-_r Fo-_T
Btu values, AF
(gmX°R) (g reX°K)
(hrXsq ft)(pmX°R 5)

15000 8333.3 33.825 × 10 -'5 0.20145 × 10 -'5 0.86880 0.00199

15100 8388.9 33.179 .19760 .87075 .00196
15200 8444.4 32.547 .19383 .87267 .00192
15300 8500.0 31.929 .19016 .87455 .00188
15400 8555.6 31.326 .18656 .87640 .00185

15500 8611.1 30.736× 10 -'5 .18305 x 10 -'5 .87821 .00181

15600 8666.7 30.159 .17961 .87999 .00178
15700 8722.2 29.595 .17625 ._173 .00174
15800 8777.8 29.043 .17297 .883_ .00171
15900 8833.3 28.504 .16976 .88512 .00168

16OOO 8888.9 27.977 x l0 -15 .16662 x 10 -z5 .88677 .00165

16100 8944.4 27.462 .16355 .88839 .00162
16200 90OO.O 26.957 .16055 .88997 .00159
16300 9055.6 26.464 .15761 .89153 .00156
16400 9111.1 25.982 .15474 .89306 .00153

1650O 9166.7 25.510 × 10 -'s .15193 x 10 -'5 .89457 .00150

16600 9222.2 25.049 .14918 .89604 .00148

16700 9277.8 24.597 .14649 .89749 .00145
16800 9333.3 24.156 .14386 .89891 .00142
16900 9388.9 23.723 .14129 .9O031 .00140

17000 9444.4 23.301 x 10 -'5 .13877 x 10 -_5 .9O168 .00137

17100 9500.0 22.887 .13630 .90303 .00135
17200 9555.6 22.482 .13389 .90435 .00132
173OO 9611.1 22.085 .13153 .90565 .00130
17400 9666.7 21.697 .12922 .90693 .00128

17500 9722.2 21.318 × 10 -'5 .12696x 10 -'5 .9O819 .00126

17600 9777.8 20.946 .12475 .90942 .00123
17700 9833.3 20.582 .12258 .91063 .00121
17800 9888.9 20.226 .12046 .91182 .00119
17900 9944.4 19.877 .11838 .91299 .00117

18000 10000.0 19.536 x 10 -'5 .11635 x 10 -'5 .91414 .00115

18100 10055.6 19.201 .11435 .91527 .00113
18200 10111.1 18.874 .11240 .91638 .00111
18300 10166.7 18.553 .11049 .91748 .00109
18400 10222.2 18.239 .10862 .91855 .00107
180 THERMAL
RADIATION
HEATTRANSFER
TABLE
V.--BLACKBODY FUNCTIONS--Continued

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference
Blackbody
product, _T power of temperature, e_/T 5 between
_action, successive
Fo--_T
Fo-AT
Btu W values, /iF
(p,m)(°R) (p, m)(°K)
(hr_(sq fl)(_mX°R 5) (cm2X/_m)(°K 5)

,0.10679 x 10 -_ 0.00106
18500 10277.8 17.931 x 10 -15 0.91961
18600 10333.3 17.630 .10500 .92064 .00104
18700 10388.9 17.335 .10324 ,92166 .00102
18800 10444.4 17.045 .10151 .92267 .00100
18900 10500.0 16.762 .09983 .92365 .00099

19000 10555.6 16.484 × 10 -15 .09817 × 10 -is .92462 .00097

19100 10611.1 16.212 .09655 .92558 .00095
19200 10666.7 15.945 .09496 .92652 .00094
19300 10722.2 15.684 .09341 .92744 .00092
19400 10777.8 15.428 .09188 .92835 .00091

19500 10833.3 15.177 × 10 -15 .09039 × 10 -15 .92924 .00089

19600 10888.9 14.931 .08892 .93012 .00088
19700 10944.4 14.690 .08749 .93098 .00086
19800 11000.0 14.453 .08608 .93183 .00085
19900 11055.6 14.221 .08470 .93267 .00084

20000 11111.1 13.994 × 10-15 .08334 × 10 -x5 .93349 .00082

20200 11222.2 i3.553 .08071 .93510 .00161
20400 11333.3 13.128 .07819 .93666 .00156
20600 11444.4 12.720 .07575 .93816 .00151
20800 11555.6 12.327 .07341 .93963 .00146

21000 11666.7 11.949 x 10 -15 .07116 × 10 -15 .94104 .00142

21200 11777.8 11.585 .06899 .94242 .00137
21400 11888.9 11.234 ,06691 .94375 .00133
21600 12000.0 10.897 .06490 .94504 .00129
21800 12111.1 10.572 .06296 .94629 .00125

22000 12222.2 10.258 × 10-15 .06109 × 10 -is .94751 .00122

22200 12333.3 9.956 .05930 .94869 .00118
22400 12444.4 9.665 .05756 .94983 .0fill5
22600 12555.6 9.384 .05589 .95094 .00111
22800 12666.7 9r114 .05428 .95202 .00108

23000 12777.8 8.852 × 10 -15 .05272 × 10 -Is .95307 .00105

23200 12888.9 8.600 .05122 .9.5409 .00102
23400 13000.0 8.357 .04977 .95508 .00099
23600 13111.1 8.122 .04837 .95604 .00096
23800 13222.2 7.895 .04702 .95698 .00093
 APPENDIX 181

TABLE V. -- BLACKBODY FUNCTIONS-- Continued

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference
product, J_T power of temperature, exv/T s Blackbody between
fraction, successive
Fo-xT F0-kT
Btu W values, AF
(/zmX°R) (ttm)(°K)
(hrXsq ft)(panX°R s) (cm2)(gm)("K9

24000 13333.3 7.676 × 10 -_5 0.04572 × 10 -15 0.95788 0.00091

24200 13444.4 7.465 .O4446 .95877 .00088
24400 13555.6 7.260 .04324 .95963 .00086
246O0 13666.7 7.063 .042O6 .96046 .00084
24800 13777.8 6.872 .04092 .96128 .00081

250O0 13888.9 6.687 x 10 -'s .03982 × 10 -'s .96207 .00079

25200 14OOO.0 6.508 .03876 .96284 .00077
25400 14111.1 6.336 .03773 .96359 .00075
25600 14222.2 6.169 .03674 .96432 .00073
25800 14333.3 6.007 .03577 .96503 .00071

26OO0 14444.4 5.850X 10 -15 .03484 × 10 -'5 .96572 .00069

26200 14555.6 5.699 .03394 .96639 .00067
26400 14666.7 5.552 .033O7 .96705 .00066
26600 14777.8 5.410 .03222 .96769 .00064
268O0 14888.9 5.273 .03140 .96831 .OO062

27000 15000.0 5.139 × 10 -'5 .03061 × 10 -'5 .96892 .00061

27200 15111.1 5.010 .02984 .96951 .O0059
27400 15222.2 4.885 .02909 .97OO9 .0O058
27600 15333.3 4.764 .02837 .97O65 .0O056
27800 15444.4 4.646 .02767 .97120 .00055

28000 15555.6 4.532 x 10 -x5 .02699 > 10 -'5 .97174 .00054

282O0 15666.7 4.422 .02633 .97226 .00052
28400 15777.8 4.315 .02570 .97277 .00051
286O0 15888.9 4.211 .02508 .97327 .OO050
28800 16OOO.0 4.110 .02448 .97375 .00049

29000 16111.1 4.012 × 10 -'5 .02389 × I0 -'5 .97423 .00047

29200 16222.2 3.917 .02333 .97469 .00046
29400 16333.3 3.824 .O2278 .97514 .0(045
29600 16444.4 3.735 .02224 .97558 .0000
29800 16555.6 3.648 .O2172 .97601 .00043

3OOO0 16666.7 3.563 × 10 -'5 .02122 × I0 -'5 .97644 .00042

30200 16777.8 3.481 .02073 .97685 .0OO41
3O4OO 16888.9 3.401 .02026 .97725 .00040
306O0 17OOO.0 3.324 .01979 .97764 .00O39
30800 17111.1 3.248 .01935 .978O2 .OO038
182 THERMAL RADIATION HEAT TRANSFER

TABLE V. -- BLACKBODY FUNCTIONS -- Continued

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference
product, AT power of temperature, exb/T 5 Blackbody between
_action, successive
Fo-_T Fo-AT
Btu W vMues, AF
(gm)(°R) (/xm)(°K)
(hr)(sq ft)(gmX°R 5) (cm_)(ttmX°K _)

31000 17222.2 3.175 x 10 -15 0.01891 x 10 -15 0.97840 0,000__7

31200 17333.3 3.104 .01849 .97877 .00037
31400 17444.4 3.035 .01807 .97912 .00036
31600 17555.6 2.967 .01767 .97947 .OO035
31800 17666.7 2.902 .01728 .97982 .00034

32000 17777.8 2.838x 10 -15 .01690 x 10 -15 .98015 .OO033

32200 17888.9 2.776 .01653 .98048 .00033
32400 18000.0 2.716 .01618 .98080 .00032
32600 18111_1 2.657 .01583 .98111 .O0031
32800 18222.2 2.600 .61549 .98142 .00031

33000 18333.3 2.545 x 10 -is .01515 x 10 -15 .98172 .00030

33200 18444.4 2.490 .01483 .98201 .00029
33400 18555.6 2.438 .01452 .98230 .00029
33600 18666.7 2.386 .01421 .98258 .0OO28
33800 18777.8 2.336 .01392 .98286 .00O28

34000 18888.9 2.288x 10 -15 .01363 × 10 -is .98313 .0OO27

34200 19000.0 2.240 .01334 •98339 .00026
34400 19111.1 2.194 .01307 .98365 .00026
34600 19222.2 2.149 .01280 .98390 ,00025
34800 19333.3 2.105 .01254 .98415 .00025

35000 19444.4 2.062 x 10 -15 .01228 x 10 -Is .98440 .OOO24

35200 19555.6 2.021 .01203 .98463 .00024
35400 19666.7 1.980 .01179 • 98487 .00023
35600 19777.8 1.940 .01156 .98510 .00023
35800 19888.9 1.902 .01133 .98532 .00022

36000 20000.0 1.864x 10 -15 .01110 × 10 -15 .98554 .OOO22

36200 20111.1 1.827 .01088 .98576 .00022
36400 20222.2 1.791 .01067 .98597 .00021
36600 20333.3 1.756 .01046 .98617 .00021
36800 20444.4 1.722 .01026 .98638 .00020

37000 20555.6 1.689 x 10 -1_ .01006 x 10 -15 .98658 .0O020

37200 20666.7 1.656 .00986 .98677 .00020
37400 20777.8 1.624 .00967 .98696 .00019
37600 20888.9 1.593 .00949 .98715 .00019
37800 21000.0 1.563 .00931 .98734 .00018
 APPENDIX 183

TABLE V.- BLACKBODYFUNCTIONS--Continued

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference
power of temperature, ex_/TS Blackbody between
product, XT
fraction, successive
Fo-_,T
Fo-xT
Btu W values, AF
(p.mX°R) (/zmX°K)
(hrXsq ft)(/zmX°R s) (cm_)(_mX°K _)

380OO 21111.1 1.533 × 10 -Is 0.00913 × 10 -_s 0.98752 0.00018

38200 21222.2 ! ,505 .._t_396_ .98769 .00018
384OO 21333.3 1.476 .00879 .98787 .00017
38600 21444.4 1.449 .00863 .98804 .00017
38800 21555.6 1.422 .00847 .98821 .00017

39000 21666.7 1.396 × 10 -15 .00831 × 10 -Is .98837 .00016

39200 21777.8 1.370 .00816 .98853 .00016
39400 21888.9 1.345 .00801 .98869 .00016
39600 22000.0 1.320 .00786 .98885 .00016
1•296 .00772 .98900 .00015
39800 22111.1 ]
40O00 22222.2 1.273 × 10-15 .00758 × 10 -15 .98915 .00015
42000 23333.3 1•065 .00634 .99051 .00136
44000 24444.4 .898 ,00535 .99165 .00114
46000 25555.6 .762 .00454 .99262 .00097
48000 26666.7 .651 .00388 .99344 .00082

50000 27777.8 .560 × 10-Is _.00333 × 10 _ls .99414 .00071

52OO0 28888.9 .484 .00288 .99475 ,00061
54000 30000.0 .420 .00250 .99528 .00O53
56OOO 31111.1 .367 .00218 .99574 .00046
58OOO 32222.2 .321 .00191 .99614 '.00040

60000 33333.3 .283 × 10 -Is .00168 X 10 -Is .99649 .00035

62000 34444.4 .250 .00149 .99680 .00031
64000 35555.6 .222 .00132 .99707 .00027
66000 36666.7 •197 .00117 .99732 .00024
68OO0 37777.8 •176 .00105 .99754 .00022

70000 38888.9 .158 × 10 -15 .940 × 10-Is .99773 .00019

72OOO 4OOOO.O •142 .844 .99791 .00017
74000 41111.1 •128 .760 .99806 .00016
76OOO 42222.2 .115 ,687 .99820 .00014
78000 43333.3 .104 .622 .99833 .00O13

80000 44444.4 .0948 × 10 -_s 564 X 10-Is .99845 .O0O12

820O0 45555.6 .0862 •513 .99855 .00010
84000 46666.7 .0786 .468 .99865 .00010
86OO0 47777.8 .0718 •428 .99874 .00009
880OO 48888.9 ".0657 .391 .99882 .00008
184 THERMAL RADIATION HEAT TRANSFER

TABLE V. -- BLACKBODY FuNCTiONS--Concluded

Wavelength- Blackbody hemispherical spectral

temperature emissive power divided by fifth Difference

product, XT power of temperature, e_b/T 5 Blackbody between

fraction, successive
FO-_,T Fo-AT
Btu W values, AF
(/xm)(°R) (gm)(°K)
(hrXsq ftX/zm)(°R 5) (cm2)(/zmX°K '_)

90OOO 50000.0 0.0603 × 10 -15 0.359 × 10- is 0.99889 0.00007

92000 51111.1 .0554 .330 .99896 .00007

94000 52222.2 .0510 .304 • 99902 .00006

.0470 .280 .99908 .00006

96000 53333.3
.0434 .259 .99913 .00005
98000 54444.4

100000 55555.6 .0402 × 10 -15 .239 × 10 _s .99918 .OOOO5

 Index
absolute temperature, 21
absorbing media in electromagnetic theory, 9, 98, 109, 116

absorptiv!ty,
definition, 64
directional spectral, 49, 64
directional total, 49, 66
h_ ".--L -: -I _1 A_ r_
• ,_m1_t,,,e,,¢_l spectsea, ,_v, o
hemispherical total, 49, 68
relation to emissivity, 66, 67, 68, 69, 70
relation to reflectivity, 84
Angstrom, 8
angular frequency, 97
approximate spectral distributions, 25
bidirectional reflectivity,
reciprocity relation, 73
spectral, 50, 72
total, ,50, 80
black, 9, 11
blackbody,
cavity, 37
definition, 11
emission into solid angle, 20
emissive power,
hemispherical, 19
definition, 18
angular dependence, ]3, 18
emission within wavelength interval, 29
fourth power law, 28, 36
historical development, 43
manufacture of, 36
Planck's radiation law, 21, 35, 43
properties, 1], 37
spectral distribution of intensity, 21, 23, 35
summary of properties, 38
tables of emission, 1"15
total intensity, 16, 27
Boltzmann, Ludwig, 44
cavity, blackbody, 37
complex refractive index, 36, 99, 118
conduction,
equation, 4
Fourier's Law of, 3
conductors, electrical,
radiative properties of, 98, 109, 116, 120, 122, 134

295-763 OL-68--13 ]85

186 THERMAL RADIATION HEAT TRANSFER

constants, radiation, 35, 174

convection, 5
conversion factors, 172-174
cosine law, 18, 42
cutoff wavelength for selective surface, 158
dielectrics,
radiative properties of, 108, 111, 115, 145
dielectric constant, 111
diffuse surfaces, 18, 77
directional,
surfaces, 165
absorptivity, 49, 64, 67, 145
emissivity, 48, 55, 57, 113, 135, 140
reflectivity, 50, 51, 75, 81,111,153 4
directional-hemispherical reflectivity,
spectral, 50, 75, 111
total, 50, 81
displacement law, 26
Draper point, 24
electromagnetic radiation, 16, 89
electromagnetic spectrum, 6
electromagnetic theory,
metals, 9, 98, 109, 116, 120
nonmetals, 107
simplifying restrictions, 89, 110
summary table of property predictions, 128
table of units, 92
electromagnetic waves,
characteristics, 92, 96
energy, 100
speed, 7, 96
electric intensity, 92
electrical conductors, 98, 109, 116
electrical resistivity, 92, 125
relation to emissivity, 122
emissive power,
blackbody, 18
directional, 20
spectral, 20
total, 28, 36
emission,
blackbody, 15, 175
metals, 120, 127
emissivity,
definition, 55
directional spectral, 48, 55, 113, 135
directional total, 48, 57
electromagnetic theory predictions, 113, 116
hemispherical spectral, 48, 59
hemispherical total, 48, 59
metals, 116, 120, 126
nonmetals, 113, 115, 146
semiconductors, 155
 INDEX 187

energy,
in electromagnetic wave, 100
extinction eoef_cient, 98

field intensity,
electrical, 92
magnetic, 92
Fourier conduction law, 3
Fresnel equation, 108
frequency, 22
frequency form of Planck's distribution, 22
frequency-wavelength relation, 22
gray surface, 67, 70
greenhouse effect, 163
grooved surface, !65
Hagen-Rubens equation, 124
hemispherical,
absorptivity, 49, 67, 68
emissivity, 48, 59
reflectivity, 50, 76, 82
hohlraum, 37
incidence of wave on interface,
dielectrics, 107
conductors, 109
integral equations in radiative transfer,.5
impurities on surface, 141
infrared radiation, 8
index of refraction,
complex, 36, 99, 118
simple, 7, 36, 97
insulators, 107
intensity,
blackbody, 15
fifth power temperature dependence for metals, 126
maximum blackbody, 29
spectral, 15, 21
total, 16, 27
isotropy within an enclosure. 12
Jeans, Sir James, 44
Kirchhoff's Law, 65, 67, 71
table of restrictions, 70
Lamber's Cosine Law, 18, 42

Laplace equation, 3
light,
speed in a vacuum, 7, 96
speed in media. 7
lunar reflected radiation, 153

magnetic permeability, 92
in vacuum, 92

magnetic intensity, 92
Maxwell, James Clark, 89
188 THERMAL RADIATION HEAT TRANSFER

Maxwell's equations, 91
metals,
electromagnetic theory, 98, 116, 127
emissivity, 116, 134
reflectivity, 117
micron, 8
Moon,
reflectivity of, 153
nonmetals, 107, 111,146
notation, 53
opaque substances, 9, 47, 146
optical constants,
relation to electrical and magnetic properties, 99, 123
optically smooth surface, 89, 138
permeability, magnetic, 92
permitivity, electric, 92
phase change in reflection, 105, 109
photon, 5
Planck, Max, 44
spectral distribution, 21, 35, 43
plane of incidence, 101
plane wave, 102
polarization of electric and magnetic waves, 66, 102, 130
parallel and perpendicular, 101
Poynting vector, 100
properties,
prediction by electromagnetic theory, table, 128
metals, 116, 134
nonmetals, 111,146
semiconductors, 155
quantum theory, 5
radiation,
constants, 174
infrared, 8

spectrum, 6
thermal, 1, 6
ultraviolet, 6
visible, 6, 8
radiation laws,
blackbody formulas, 38
Lambert's cosine law, 18
Planck's law, 21
Stefan-Boltzmann law, 28, 36
Wien's displacement law, 26
Wien's spectral distribution, 26
Rayleigh, Lord, 44
Rayleigh-Jeans distribution, 26
recriprocity of reflectivities, 73, 75, 81, 82
table of restrictions, 83
reflectivity,
angular dependence, 153
bidirectional, 50, 72, 80
 INDEX 189

diffuse, 77
directional-hemispherical, 50, 75, 81, 111
hemispherical, 51, 76, 82
hemispherical.directional, 51, 75, 82
polarization, 108, 110
reciprocity, 73, 75, 81, 82
relation to absorptivity and emissivity, 84
spectral, 72
specular, 78
table of reciprocity relations, 83
total, 80
refraction, 101
refractive index,
complex, 8, 36, 99, 118
relation to electrical and magnetic properties, 99
simple, 7, 36, 97
resistivity,
dependence on temperature, 125
relation to emissivity, 124, 126
roughness of surface,
analysis of effect, 139
effect on properties, 138, 152
selective surfaces,
cutoff wavelength, 158
for collection of radiation, 157
for emission, 164
glass enclosm'e, 163
semiconductors. 155
Snell's Law, 107
solar radiation, 24, 157, 164
spectrally selective surfaces, 157
spectrum, electromagnetic. 6
specular surfaces, 78
speed of electromagnetic wave, 7
Stefan, Joseph, 44
Stefan-Bohzmann Law, 28, 36, 42
Stefan-Bohzmann constant, 28, 174
surfaces, effect on properties of,
purity, 141
roughness, 138, 152
surfaces, selective, 157
tables,
blackbody, 34, 175
conversion factors, 172-174
electromagnetic theory inks units, 92
Kirehhoff's Law relations, 70
radiation constants, 174
reciprocity relations, 83
temperature,
effect on properties, 127, 137, 150
thermal radiation, 1
total radiation, 14, 16
]90 THERMAL RADIATION HEAT TRANSFER

visible radiation, 8
wavelength, 1, 8
at maximum blackbody emission, 27, 36, 43
wave, electromagnetic, 96
wave equation, 95
wave number, 22
wave number form of Planck's distribution, 22
wave propagation, 92
dielectric, 93
conductor, 98
wave versus quantum model, 5
Wien, Willy, 44
Wien's displacement law, 26, 36. 43
Wien's spectral distribution, 26
relation to Planck's law, 26

U,S. GOVERNMENT PR_NT|NG OFFICE : |lt_ OL_Zf_S-Tg3