NBS TECHNICAL NOTE

Procedures for Precise Determination

of Thermal Radiation Properties

November 1964 to October 1965

A Report to: Air Force Materials Laboratory

J. C. RICHMOND, G. J. KNEISSL,

D. L KELLEY, AND F. J. KELLY

292

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U.S DEPARTMENT OF COMMERCE

National Bureau of Standards
THE NATIONAL BUREAU OF STANDARDS

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UNITED STATES DEPARTMENT OF COMMERCE • Alexander B. Trowbridge, Acting Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

NBS TECHNICAL NOTE 292

ISSUED FEBRUARY 10, 1967

Procedures for Precise Determination

of Thermal Radiation Properties

November 1964 to October 1965

J. C. Richmond, G. J. Kneissl,

D. L Kelley, and F. J. Kelly

A Report to Air Force Materials Laboratory

Research and Technology Division

Air Force Systems Command

Wright-Patterson Air Force Base, Ohio

Project No. 7381, Task No. 738106

(Prepared Under Contract No. DO (33-615)65-1005)

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C., 20402 - Price 50 cents

Abstract

The broad overall objective of this continuing program is to develop equipment and procedures for measuring the important thermal radiation properties of materials, particularly

those used in aircraft, missiles and space vehicles, at temperatures up to the melting point of the most refractory material, and to develop physical standards for checking such equipment

and procedures. During the period covered by this report the specific objectives were

(1) continued development of the laser-source integrating sphere reflectometer, (2) an error

analysis of the shallow cavity technique for measuring normal spectral emittance, and (3) a study of the feasibility of preparing emittance standards for use at temperatures above 1400 °K

(about 2000 °F).

An error analysis of the shallow cavity technique for measuring total normal emittance of ceramic materials at very high temperatures showed that there was an error due to the trans- lucency of the specimens that was as much as +60$ for alumina, and a second error due to thermal gradients in the specimen that was on the order of -10$. Two new techniques were

devised in the hope of greatly reducing the translucency error. Progress was made in developing

codes to compute and correct for the thermal gradients present in the specimen. The laser- source integrating sphere reflectometer for measuring reflectance of specimens at very high

temperatures was extensively redesigned to eliminate errors due to flux reaching the detector on

the first reflection, and to convert the reflectometer from the substitution to the comparison mode.

A literature search was made of techniques for measuring thermal radiation properties of solids

at temperatures above 2500 °K (4000 °F).
Contents

Page

1. Objectives 1

2. Shallow Cavity Procedure for Total Normal Emittance 2

2. 1 Background 2 2.2 Brief Description of Method 2

2. 3 Description of Equipment 5

2. 4 Brief Outline of Work Performed Prior to November,

1964 5

2. 5' Study of Error Sources 8

2. 5. 1 Optical Scattering 812

2. 5. 2 Translucency Errors

2. 5. 2. 1 Theoretical Analysis 12 2.5.2.2 Experimental Procedure "

2.5.3 Temperature Difference Error 18

2.5.3.1 Discussion 18

2. 5. 3. 2 Thermal Gradient Computations 19

2. 5. 3. 3 Results of Thermal Gradient Computations 20

2. 6 New Procedures 27

2. 6. 1 Center Post Technique 27

2. 6. 2 Deep Cavity Technique 29

2. 6. 3 Discussion of New Procedures 30

2. 7 Total Normal Emittance Measurements 33

2. 7. 1 Alumina 33

2. 7. 2 Magnesia 33

2. 7. 3 Zirconia 33

2.7.4 Thoria 36

2. 8 Materials for Emittance Standards 36

2.8.1 Alumina 36

2.8.2 Magnesia 37 2.8.3 Zirconia 37 2.8.4 Thoria 37

2. 8. 5 Metals 38

3. Laser-Source Integrating Sphere Reflectometer 38

3. 1 Background 3$

3.2 Previous Work 39

3. 3 Design Modifications 40

3. 3. 1 Requirements 40
Page

3.3.2 Sphere Geometry 40

3.3.3 Sphere Modifications 41

3.4 Relative Reflectance Measurements 43

3. 5 Absolute Reflectance Measurements 43

3. 6 Modification of Auxiliary Equipment 4(5

3. 6. 1 Laser Source 46

3.6.2 Heater Design 47

3.6.3 Mirror Holder 48

4. Measurements Above 2500°K (4000°F) 48

4. 1 Background 48

4. 2 Literature Survey 49

4.2. 1 Induction Heating 49

4. 2.2 Arc-Image Furnace Heating 50

4. 2. 3 Solar Furnace Heating 50

4. 2. 4 Plasma Torch Heating 51

4. 2. 5 Resistance Heating 51 4.2.6 Special Techniques 51

4. 3 Discussion 51 4.4 The Ellipsoidal Mirror Emissometer 52

4.4.1 Introduction 52 4.4.2 Description of Equipment 53 4.4.3 Other Optical Systems 55 4.4.4 Mathematical Analysis 56 4.4.5 Evaluation of System Parameters 60

5. Summary 62

6. References 63

7. Appendix I 65

7. 1 Interreflection Algebra 65

7. 2 Angle Factors in the Cylindrical Cavity 73

8. Appendix H 77
ILLUSTRATIONS

Figure Page

1. Cross-sectional Drawing of Specimen Furnace 6

2. Schematic Drawing of the Optical System 7

3. Cross-sectional Drawing of Typical Specimens of Non-Conducting and Conducting Materials. 7

4. Cross-section of the Shallow Cavity of a Ceramic Specimen 14

5. Spectral Transmittance Curves for Fused Quartz and Kodak LWP Filters 17

6. Cylindrical Geometry Used in Trial Calculation of Heat Transfer in a Cavity 22

7. Cylindrical Specimen with Shallow Cavity, Having 18 Nodes 22

8. Cylindrical Specimen of Radius d and Length c Containing a Shallow Cylindrical Cavity

of Radius R and Length L 22

9. Cross-sectional Drawing of Specimen Mounted in Susceptor with Graphite Post Comparison Standard 28

10. Cross-sectional Drawing of Specimen with Deep Cavity, Mounted in Susceptor 28

11. Normal Total Emittance of Alumina, Measured by the Shallow Cavity Technique 34

12. Normal Total Emittance of Magnesia, Measured by the Shallow Cavity Technique 34

13. Normal Total Emittance of Zirconia, Measured by the Shallow Cavity Technique 35

14. Normal Total Emittance of Thoria, Measured by the Shallow Cavity Technique 35

15. Schematic Views of Sphere Geometry. 42

16. Schematic Diagram of Ellipsoidal Mirror Emissometer 54

al. Cylindrical Cavity with Subdivided Walls and Base .74

a2. Coaxial Discs of Equal Radii 74

a3. Section Through a Cylindrical Cavity. 74

a4. Section Through a Cylindrical Cavity 76

a5. Section Through a Cylindrical Cavity 76

a6. Total Normal Emittance of Tungsten 78

a7. Total Normal Emittance of Alumina. 80

a8. Total Normal Emittance of Thoria. 80

a9. Total Normal Emittance of Magnesia. 81

alO. Total Normal Emittance of Zirconia. 81

vi
PROCEDURES FOR PRECISE DETERMINATION

OF THERMAL RADIATION PROPERTIES

Joseph C. Richmond, Gerhart J. Kneissl, Douglas L. Kelley

and Francis J. Kelly

An error analysis of the shallow cavity technique for measuring total normal emittance of ceramic materials at very high temperatures showed

that there was an error due to the translucency of the specimens that was

as much as +60% for alumina, and a second error due to thermal gradients

in the specimen that was on the order of -10%. Two new techniques were

devised in the hope of greatly reducing the translucency error. Progress was made in developing codes to compute and correct for the thermal

gradients present in the specimen. The laser-source integrating sphere

reflectometer for measuring reflectance of specimens at very high temperatures was extensively redesigned to eliminate errors due to flux reaching the detector on the first reflection, and to convert the reflectometer from the substitution to the comparison mode. A literature search was made of techniques for measuring thermal radiation properties of solids at temperatures above 2500°K (4000°F).

KeyWords: Emissivity, emittance, high temperature reflectance,

infrared reflectance, radiation properties, reflectance, spectral emittance,

spectral reflectance, thermal radiation, total emittance.

1 . Objectives

The broad overall objective of this continuing program is to develop equipment and procedures

for measuring the important thermal radiation properties of materials, particularly of materials

used in aircraft, missiles and space vehicles, at temperatures up to near the melting points of the

most refractory materials, and to select, prepare and calibrate physical standards for use by other

laboratories in checking their equipment and procedures.

The specific properties of major interest are emittance!/ and reflectance, and the wavelength

range of interest extends from about 0. 25p,m to 35p,m and beyond.

— Emittance is defined as the ratio of the flux per unit area emitted by a specimen to that emitted by

a blackbody radiator at the same temperature and under the same conditions.
Work in previous years has been concentrated in the temperature range of room temperature and

800° to 1400°K (about 1000° to 2000° F), and at wavelengths in the 1 to 15p.m range.

Specific objectives for the current contract year included (1) continued development of the lasersource integrating sphere reflectometer for measuring reflectance of specimens at high temperature,

(2) a study of the errors present in the shallow cavity method of measuring total normal emittance at

high temperatures, methods of quantitatively evaluating such errors and/or methods of eliminating

them, and (3) a study of the feasibility of preparing emittance standards for use at temperatures

above 1400°K (about 2000°F).

2. Shallow Cavity Procedure for Total Normal Emittance

2.1 Background

The equipment for measuring total normal emittance of non-metals by the shallow cavity procedure was designed and constructed on a project sponsored by NASA. Work on this program was

transferred to Air Force sponsorship as of November 1, 1964.

2. 2 Brief Description of Method

The problem of obtaining reliable thermal emittance measurements on non-metals at temperatures above 1800°K, and particularly above about 2000°K, where standard platinum -base thermocouples cannot be used, has never been solved satisfactorily. Non-metals, and particularly the ceramic oxides, generally have low thermal conductivity, relatively high thermal emittance, and are

translucent to appreciable depths below the surface. These properties make it difficult to measure

accurately the temperature of an emitting specimen. The combination of high emittance, which also

implies high heat transfer rates from the material, and low thermal conductivity tends to produce

large thermal gradients in specimens of such materials, and in a specimen that is permitted to

radiate from one surface to surroundings at an appreciably lower temperature, the gradients tend to

be normal to the emitting surface. The translucency of these materials results in the emitted flux

originating in a surface layer of appreciable thickness, rather than from a very thin layer at the surface as in the case of metals and other highly opaque materials. As a further complication, the

thickness of this layer will change with the wavelength of the emitted flux, because it is a function of

the extinction coefficient of the material, which is wavelength dependent for practically all materials.

2
When the flux emitted by a specimen originates from a layer of appreciable thickness within

which a thermal gradient exists, it is difficult to define the temperature of the emitting volume, and

even more difficult to measure it. Even if the effective temperature of the specimen under these

conditions is properly defined and measured, the effective temperature of the emitting volume will be

wavelength dependent, which is an unsatisfactory situation. In addition, it is extremely difficult to

achieve temperature equivalence between a hot specimen and separate blackbody furnace at temperatures above those at which reliable thermocouples are available. An optical or radiation pyrometer

measures brightness or radiance temperature, which is the true temperature for a blackbody, but not

for real materials. For any specimen, the brightness or radiance temperature must be converted to

true temperature, which requires a knowledge of the emittance of the specimen. Since emittance is

the property to be measured, such methods of measuring temperature are not very promising.

One method of overcoming the problems inherent in measurements of emittance of non-conductors at very high temperatures is to use an integral blackbody cavity as a reference. If the reference

and specimen are at the same temperature, it is not necessary to know that temperature very accurately in order to make accurate emittance measurements. Further, it is possible to make an

accurate measurement of the temperature of the blackbody cavity with an optical or radiation pyrometer. The error in emittance measurements arises primarily from the unknown temperature difference between the specimen and reference.

One form of integral blackbody specimen that would appear to offer advantages in measuring the

emittance of nonconducting materials at very high temperature is the so-called shallow cavity. This

cavity consists of a cylindrical hole of small diameter and shallow depth drilled into the specimen

surface. The cavity is on the order of 0.5 mm deep in order to keep the temperature difference between specimen surface and cavity bottom small, so that only a small error will be introduced by

assuming isothermal conditions.

A reference cavity of these dimensions is not a very good blackbody. However, if the cavity

walls are opaque and reflect diffusely, rather than specularly, if the dimensions of the cavity are accurately known, and if the reflectance and emittance of the cavity wall are the same as for the specimen surface, the measured radiance ratio of specimen surface and cavity bottom can be converted to
emittance. The equation used for this conversion was derived by Gouffe r l]—' . The equation for the

emittance of the cavity may be written

e =

c

e [1 + (1 - e ) (a/A - f)] w w

"~i (1 - a/A) + a/A w

(1)

where

We can also write

cavity emittance

e = emittance of cavity walls

1

1 + (L/R)2

L = cavity depth

R = cavity radius

a = area of cavity opening

A = cavity surface area including opening

a/A 1

Efc

2(1 + L/R)

L

s /Lbb =

(2)

in which E^. E, = radiance ratio of surface to cavity

Lg = normal radiance of specimen surface

L = normal radiance of cavity

L^b = radiance of blackbody at specimen temperature

e s = emittance of specimen

If we assume that e = e g

, we can combine Equations (1) and (2) to obtain

E (1 + a/A - f) - a/A

1 - a/A + E

t (a/A - f

)

e = (3)

Equation (3) is used to convert the measured radiance ratios into emittance.

2/— Figures in brackets refer to references in Section 6.
2. 3 Description of Equipment

Equipment for measuring the radiance ratios of shallow cavity specimens was designed and constructed on a contract with NASA. Figure 1 is a drawing of the specimen heater and Figure 2 is a

sketch of the optical system.

The specimen is placed in a susceptor heated by induction, or if the specimen is conducting it

may act as its own susceptor. Typical specimens are shown in Figure 3. The power is supplied by

a 10-kw, 30-MHz radio frequency generator with controlled power output.

The water-cooled field concentrator (Figure 1) is of silver-plated copper. The bell jar is a

Pyrex glass pipe cap with the top and the bottom edge ground and polished to provide vacuum-tight

seals to the 0-rings. The radiant flux emitted by the hot specimen passes out through a 1/2 inch

diameter hole in the top of the bell jar and through a radiation trap and sodium-chloride window to the

focusing optical system. The window is mounted at a slight angle to the top surface of the specimen,

so that any flux reflected from the window will be trapped, and not impinge on the specimen and thus

contribute to the measured flux. The flip mirror above the trap facilitates use of a micro-optical

pyrometer for measuring the temperature of the hot specimen. The bell jar is connected to a vacuum

system, and has a connection through which an inert gas, such as helium or argon, can be introduced.

There is also provision for insertion of a getter filament into the bell jar.

The flux from the hot specimen is emitted vertically, to the focusing optical system (Figure 2),

where a plane mirror at 45° to the vertical directs it horizontally to an off-axis ellipsoid, which

directs it to a vertical plane mirror which directs it back to the detector, located in the second focal

plane. The magnification factor is 2x linear. The detector is mounted on a micrometer stage so that

it can be moved both vertically and horizontally in the focal plane. A platinum diaphragm with a small

circular aperture, mounted on the detector, restricts the view of the detector to a small selected

area of the image of the specimen.

2. 4 Brief Outline of Work Performed Prior to November 1964

The validity of the method depends almost entirely upon the validity of Equation (1). In order to

assign a value to the accuracy of the measurements, it was necessary to check Equation (1) experimentally. This was done, as is described in detail in a recent paper by Kelly and Moore T2],

5
FLIP MIRROR FOR

OPTICAL PYROMETER

MEASUREMENTS

WINDOW HEATER

RADIATION TRAP

(Painted black on inside)

WATER COOLING COIL

O SLOTTED RF FIELD

£X^~ CONCENTRATOR

SUPPORT ROD

(W,Ir,or Al2 3

)

LUCITE CENTERING

RING

1 SCALE ' WATER

IN

WATER

OUT

Figure 1. Cross-sectional Drawing of Specimen Furnace
HEATED J

SPECIMEN jT

(MOUNTED BENEATH ^" ( OPTICAL BENCH)

AMPLIFIER RECORDER

PREAMP THERMOCOUPLE DETECTOR

MOUNTED ON VERNIER FOR SCANNING

MAGNIFIED IMAGE (2X) OF SPECIMEN

90° FOLD CHOPPER 5" SPHERICAL

MIRROR - f/5.7

(6° - 42' OFF

AXIS)

Figure 2. Schematic Drawing of the Optical System.

SUSCEPTOR

(Pt.Ir.or Graphite)

SPECIMEN

-SUPPORT ROD ^ (W,Ir,or Al 2 3

) K

- 5mm

SPECIMEN

NONMETAL METAL

Figure 3. Cross-sectional Drawing of Typical Specimens of Non-Conducting

and Conducting Materials
The more important findings were:

(1) When the wall material was a good diffuse reflector, the Gouffe equation predicted the reflectance of shallow-cylindrical cavities to 0. 01 in reflectance units.

(2) When the wall material was a somewhat specular reflector (poor diffuser), the equation still

predicted with good accuracy (± 2%) the reflectance of a cavity with a L/R ratio (depth to radius) of

0.51.

(3) The Gouffe equation can be used to convert measured radiance ratios to emittance without

significant error for all materials that are good diffuse reflectors. Fortunately, most non-metals

are good diffuse reflectors.

The capabilities of the equipment were demonstrated, and data were obtained on a number of

specimens [3],

2. 5 Study of Error Sources

Realistic estimates of the errors present in the measurements made with the equipment

described above could not be made without separate evaluations. There are several sources of bias

in addition to the normal statistical error of measurement. They result primarily from imperfections in the optical system and failure of the experimental conditions to conform exactly to the assumptions on which the Gouffe analysis is based. The three most important error sources can be

designated (1) errors due to scattering and aberrations in the optical system, (2) errors due to trans

lucency of the cavity walls, and (3) errors due to thermal gradients.

2. 5.1 Optical Scattering Errors

The optical scattering errors result from imperfections in the optical system. The fluxes from

the specimen surface and reference cavity pass through the optical system simultaneously. The flux

reaching any given small area A' on the image corresponding to the detector aperture will consist of

the flux emitted by the corresponding area, A, on the specimen and focused on area A', plus the flux

originating from other portions of the specimen or susceptor and deviated to area A' by scattering

and imperfections in the optics. In order to evaluate this error, certain basic assumptions must be

made:

1). Only small -angle deviations need be considered. Any deviations that cause the emitted flux

8
to miss the image entirely will be indistinguishable from absorption.

2). The fractions of the flux scattered or absorbed are a function of wavelength, and hence of

the wavelength distribution of flux in a measurement of total normal emittance, and the relative sizes

of the specimen, cavity and detector aperture. Even though these fractions are wavelength-dependent,

they will not change appreciably from one material to another if the wavelength distribution of

emitted flux does not change appreciably. These fractions are independent of the flux density.

3). The net effect of the small angle deviations is an increase in the measured radiance due to

scattered radiation reaching the detector from areas other than that being directly observed. Flux

from the susceptor, specimen and cavity is scattered over the entire image. The contribution of the

cavity to the total scattered flux will be small, because the ratio of cavity area to specimen area is

about 0. 01. Thus the cavity contribution will be only 1 to 2 percent of a total error that is on the

order of 5 percent of the measured value. No significant error will be introduced by ignoring the

cavity contribution to the total error.

On the basis of the above assumption, the error term can be related to the "true" radiance of

the specimen surface for specimens that can be heated directly by induction. Some of the emitted

flux is lost in the system, due to absorption, scattering and aberrations. From assumption (2) above,

the fraction a, of the flux emitted by area A of the specimen will be focused on the detector, when A

is the area of the detector aperture projected through the optical system onto the specimen. The

fraction (1 - a) of the emitted flux is lost. Flux will also reach the detector from all other points on

the specimen. If s is the fraction of the surface radiance that reaches the detector by scattering, the

correction factor for both surface and cavity is sA'GK where A' is the area of the detector aperture

(which determines the solid angle subtended to the aperture from the specimen) Lg is the radiance of

the specimen surface, and G is the total area of the specimen. The measured ratio, E, is then

aA'L + sA'GL

s E =

aA'L, + sA'GL, <4)

setting K= -•?

,
L

s (1 + K)

(5) L + KL

c s

L

s .. - E

or EL = L (1 + K - EK). Since E

t = —, then E

t - ~+ K (l - Ej (6)

c s q

. \ /

All that remained was to evaluate the constant K, the scattering factor, in order to evaluate the

true radiance ratio, E

t

, from the measured E.

It can be seen in Equation (5) that if the cavity can be made nonradiating, L =0, then K can be

evaluated. The cavity was, in effect, made nonradiating by replacing it with a hole of the same

diameter completely through the specimen, and placing a cooled, nonreflecting surface below the hole.

Under these conditions any flux reaching the detector when it is focused on the hole must be scattered

from the specimen surface. When L = 0, Equation (5) reduces to E = (K + 1)/K. If we designate

the radiance ratio measured under these conditions as E , then

o

-1 K = (E - 1) (7) o

The factor K was found to be about 0. 05 for a specimen of oxidized Inconel of the diameter of a

normal susceptor as shown in Figure 3, with a 0. 5 mm hole, and using a diaphragm with a 0. 6 mm

hole over the detector.

The factor K is primarily a geometric function of the specimen size, cavity size, and detectoraperture size. It may also be influenced to some extent by the specimen temperature, and probably

to a much lesser extent by the spectral emittance of the specimen. It should be independent of the

total emittance. The value of 0. 05 should be typical for the conditions quoted.

The above analysis is based on the assumption that the amount of scattered flux reaching the

detector is the same whether the detector is focused on the center cavity or on the specimen. This

assumption was checked by preparing a nickel specimen with two 0. 5 mm holes completely through it.

One hole was at the center of the specimen, and the other was midway between the center and edge.

No significant difference in the amount of scattered flux was detected when the detector was focused

10
alternately on the two holes, regardless of the orientation of the specimen.

In the more general case, illustrated for the nonmetallic specimen in Figure 3, there will be a

hot susceptor surrounding the specimen and some of the scattered flux reaching the detector may

originate from the susceptor. An attempt was made to block out such scattered flux by baffling in the

optical system, but without success.

This case can be analyzed by the same procedure used for deriving Equations (4), (5) and (6). In

this case

aA'L + sA'GL + s'A'G'L

p _ S S U / » ^ aA'L + sA'GL + s'A'G'L (S)

C S U

where s' is the fraction of the susceptor radiance that reaches the detector by scattering, E is the

measured-radiance ratio, G' is the area of the susceptor, and L is the radiance of the susceptor.

All other terms are as in Equations (4), (5) and (6). If we set K' = s'G'/a,

L

= (1 + K) + K'L

Ei =

L + KL + K'L (9)

c s u

The two constants can be evaluated as before. In this case K is found by heating a specimensized sample with a cavity-sized hole. The radiance ratio, E , of cavity flux to surface flux is

measured, and K is computed from Equation (7). K' is found by heating a susceptor-sized sample

with a specimen-sized hole, and the radiance ratio, E , is measured, and K' is computed from

Equation (7).

The true radiance ratio, E = L /L , is required for substitution in Equation (3) to obtain the

I s c

emittance of the specimen. Solving Equation (9) for E gives

Ei - K' L /L (1 - Ei)

Ex =

u' c ' — (10)

't 1 + K - RE,.

The ratio L /L is required to solve Equation (10). This is the true ratio of susceptor radiance

I. to cavity radiance Lc

. The measured value of this ratio, E2, is given by

aA'L + sA'GL + s'A'G'L

v _ u § £ (11)

** ~ aA'L + sA'GL + s'A'G'L

c s u

or

11
L (1 + K') + KL

E = -H . & (12)

2 L + KL + K'L

c s u

solving for L /L

u c L^ Eg - K (L

s/Lc

) (1 - Eg ) L ~~ 1 + K' - K% (13)

but L /L = E,, thus

s c t L Ea -KE. (1 -EJ

c

substituting the above in Equation (10) gives

Ej - K' (£., - EJ

E

t ~1 + K (1 - Ei) + K> (1 - E2 ) (15)

The correction factor K was evaluated for an oxidized-nickel specimen of the same diameter as

a normal nonmetallic specimen, having a cavity -sized hole through its center. A value of 0. 113 was

obtained. A similar evaluation of K' was made on an oxidized-nickel specimen of the same diameter

as a normal susceptor, having a specimen-sized hole through its center. A value of 0. 016 was

obtained. These values are considered to be typical for any specimen and susceptor combination, and

will be used to correct measured total emittances for the scattering error by use of Equation (15).

2. 5. 2 Translucency Errors

2. 5. 2. 1 Theoretical Analysis

The basic equation for attenuation of flux in traversing a material is I = I e

-^ (16)

where I is the radiant intensity after traversing a thickness x of the material, I is the radiant

intensity at x = 0, e is the base of natural logarithms and k is the extinction coefficient. The extinction coefficient is made up of the absorption coefficient a and the scattering coefficient S,

k = /a (a + 2S)

( 17 )

The absorption coefficient a is large for opaque materials, such as metals, but is generally much

lower for nonmetallic materials such as ceramic oxides. The absorption coefficient for such

12
materials varies markedly with wavelength, from a low value at short wavelengths to a high value at

long wavelengths. The transition usually occurs rather abruptly, and Is called the absorption edge.

For most ceramic materials, the absorption edge occurs In the 3-8 micron range.

The absorption coefficient a is a property of a material, and Is strongly Influenced by minor

constituents, lattice defects, etc. The scattering coefficient S Is primarily a function of the physical

state of the material —grain size, porosity, etc. A highly opaque material has a high absorption coefficient, and hence all of the Incident flux

penetrating the surface will be absorbed within a very thin surface layer, and all of the internally

generated flux that Is emitted will originate within this same very thin surface layer. The thickness,

r, of this layer is somewhat indefinite, because as Equation (16) shows, the absorption is an exponential process. About 5% of the initial flux will remain after traversing a thickness of material

of 3/k, and about 1. 8% after traversing a thickness of 4/k. On this basis, an arbitrary value of

r = 3/k is not unreasonable. When k is large, r is so small that no appreciable error is introduced

by considering the processes of emission, reflection and absorption to occur at the surface. Essentially all of the flux emitted by an infinitesimal area of surface will originate in the material below

that area, and essentially all of the reflectance will occur at the material -air interface.

For translucent materials, r is large enough that emission, reflection and absorption can no

longer be considered to occur only at the surface. The flux emitted at an Infinitesimal area on the

surface of a specimen will originate within a spherical volume of the material with radius r centered

at the infinitesimal area, and the flux incident on the plane surface of a specimen in an infinitesimal

area will be internally scattered within the material and emerge from a circular area of radius r

centered on the Incident area.

Since the Gouffe analysis is based on the assumption that the walls of the cavity are opaque,

the translucency of the specimen will introduce an error. The shallow-cavity method is based on

measurement of the radiance ratio of two areas on the specimen. One area, A in Figure 4, is at the

center of the bottom of the shallow cavity, and the other, B, of equal size, is on the flat top of the

specimen. The radiance of area A is due to the flux emitted from area A, that emitted by the walls of

the cavity and reflected from A, and that emitted by the bottom of the cavity, reflected from the walls

13
Figure 4. Cross-section of the Shallow Cavity of a Ceramic Specimen.

to A and reflected by A. The radiance of area B is due solely to flux emitted from area B.

Figure 4 shows two infinitesimal areas on the side of the cavity and the spherical volumes of

radius r surrounding these areas from which the emitted flux will originate. The flux incident on the

indicated areas will be reflected from the area bounded by the intersection of the cylindrical -cavity

wall and the sphere of radius r.

It can be seen intuitively from Figure 4 that if r is less than the distance from area B to the

cavity or edge of the specimen, the radiance of area B will not be affected by the translucency, and

the directly emitted portion of the radiance of area A will not be affected. The emittance and reflectance of an infinitesimal area on the cavity wall will be reduced in a ring of height r below the top

of the cavity, by a fraction that approaches 1/2 as the area approaches the top of the cavity, and the

radiance of the top of the specimen will be affected within a circular band of width r surrounding the

cavity.

The effects of translucency can be summarized as follows: (1) the reflectance of a band of width

r at the top of the cavity will be reduced, (2) the emittance of a band of width r at the top of the

cavity will be reduced, (3) the emittance of a circular band of width r surrounding the cavity will be

reduced, and (4) the apparent emittance of the circular band of width r surrounding the cavity will be

14
increased due to flux incident on the cavity wall that is transmitted by the specimen and emitted in the

circular band. Effects (3) and (4) are compensating, and are small when the volume of the cavity is

small, and hence will be ignored, since area B is usually separated from the cavity by a distance

greater than r.

The measured radiance ratio, E, can be related to the true radiance of the cavity, L , and the

c specimen Lg as L

E =

L - V- Tl c

s (18)

where vis the correction for the flux not emitted by the band at the top of the cavity, and in is the

correction for the decreased reflectance of the band at the top of the cavity. The net effect is that the

measured Lc

is low. The two corrections are related, and can be combined into a single correction

factor, g.

E = LS /(LC - g) (19) Asr-0, g - 0, or more realistically, g - as r « L, where L is the depth of the cavity.

2. 5. 2. 2 Experimental Procedure

The experimental procedure used to evaluate g in equation (19) is based on the assumption that

the normal spectral emittance measured with the rotating cylinder equipment [4] is essentially free

from errors that are present in the values obtained by the shallow cavity technique. This assumption

appears to be valid, because in the rotating cylinder method the radiance of a specimen, heated under

conditions such that there is essentially no thermal gradient normal to the surface, is measured and

compared to that of a laboratory blackbody furnace at the same temperature and under the same conditions. If the rotating cylinder method measures the true normal spectral emittance of a material,

the data can be integrated to obtain the true normal total emittance, e

tt

- Then if the specimens for

the two methods are identical in composition and surface finish, the difference in the measured values

can be taken as the error of measurement in the shallow cavity technique.

The error of measurement evaluated by comparison of the rotating cylinder and shallow cavity

values would be made up of the scattering error, the translucency error and the thermal gradient

error. The scattering error can be separately evaluated and corrected for, as discussed previously.

15
However, the translucency error and the thermal gradient error are expected to be opposite in sign,

and cannot be separately evaluated by these measurements alone.

It is possible, by means of selective filters, to measure the normal emittance of the shallow

cavity specimen over a wavelength band where the specimen is translucent, and hence where the

translucency error would be expected to be large , and also at another wavelength band where the

specimen is essentially opaque, and hence where the translucency error would be very small or completely absent. In this way, by assuming that the thermal gradient error is independent of wavelength (this is only a crude approximation at best), the two error terms can be separately evaluated.

Two filters were used to separate the flux emitted by an alumina specimen into two portions, as

mentioned above. The spectral trans mittances of the filters are shown in Figure 5. Filter fx is of

fused quartz, and has high transmittance at wavelengths from the ultraviolet out to about 4. 2 M.m

essentially opaque at wavelengths beyond about 4. 6 u.m. It thus transmits over the wavelength range

where many ceramic materials are translucent. The second filter, fg is opaque at wavelengths

shorter than about 7. 3u.m, and has high transmittance from about 7. 9 to 14. 3u.m, where most ceramic

materials are essentially opaque.

The normal total emittance of an alumina specimen was measured at 1600°K with no filter in the

optical path, and then with filters fx and fg respectively, in the path. The measured emittances

were corrected for the scattering error, by the procedure that was previously described. The

normal spectral emittance obtained on a specimen of the same material by the rotating cylinder

method at 1600°K was used to compute the true total normal emittance of the alumina at 1600°K by use

of the following equation

e _ 2LL_E------- (20) tt J*. Z,

? L,T,AX JX; w

where &. is the "true total" normal emittance of the alumina, L, , is the spectral radiance of a blackbody at 1600°K, e, is the normal spectral emittance of alumina at 1600°K, t, is the transmittance of

the filter (fx or i^ ) used (when no filter is used t, = 1. 00 at all wavelengths) A^ = 0. 5u.:m, since the

products were computed at 0. 5n.ni wavelength intervals, and \x and \B include all wavelengths at 16
1 1 1 1 1 1 1 1 1

—

DC

LU

ti / —

— 3 /

—

— < 1 —

— —

—

orLU

!j

Ll

—

—

QUARTZ 1 —

— III 1 1 1 1

_ lO

ro

LU

>

<

- m

- ro

o o o O o

o 00 CD ^r CO

o

f*

"8

C

ua o 3&

o 0) CD e

in I

M X O h- to 


-z.

0)3

LU U

£

COc

3

H

73

+>o

cua

0)

1-4

001 X 30NVllllAISNVdl

17
which LkiT\ is significant.

The "true total" emittance, e.. , computed from the normal spectral data, was 0. 25 for the case

where no filter was used, 0. 08 when the quartz filter (fa ) was used, and 0. 94 when the Kodak LWP

filter (fg) was used. The values measured with the shallow cavity equipment were 0. 31 with no filter,

0.17 with the quartz filter (fx ) and 0. 86 with the LWP filter (fg). All values were corrected for the

scattering error, as previously discussed.

The error at the long wavelengths (fg filter) was about 8. 5% of the "true" value, and is considered to be due almost entirely to the thermal gradients that were present. A temperature difference of about 33°K could account for this error, if total emittance were being measured, in which

case the total radiance of a blackbody, L = - T

4

, where a is the Stefan-Boltzmann constant,

rr = 3.14159, and T is absolute temperature. Since filter fg transmits only in the 7. 3 - 14. 3n,m range,

the radiance in this wavelength range from a blackbody at 1600°K is more nearly approximated by

L, (fg) = (o/tt)T *

. On this basis, the temperature difference is on the order of 150°K. Such a temperature difference would cause the readings in the short wavelength region to be low by about 38%.

On this basis, the error observed in the readings with the fx filter of +53% would be made up of a +91%

error from translucency and a -38% error from the temperature difference. The readings with no

filter had a net error of +24%, which would be made up of an error of about +61. 5% from the translucency, and -37. 5% from the temperature difference. These errors are intolerably large unless

accurate corrections can be made.

2.5.3 Temperature Difference Error

2.5.3.1 Discussion

A basic assumption in the derivation of the Gouffe equation is that the specimen is at uniform

temperature. This is at best only an approximation, even for materials of high thermal conductivity

and low emittance. When a cylindrical specimen is heated from the bottom and sides, and cooled

from the top, there will be a temperature drop from the outside to the center of the cylinder (a radial

thermal gradient) and a temperature drop from the hot end to the cold end (an axial thermal gradient).

The presence of the cavity at the center of the cooled end causes some slight perturbation of the

gradients.

18
If only the radial thermal gradient were present, the cavity would be cooler than the area of the

specimen surface viewed in an emittance measurement, and the measured emlttance values would be

too high. If only the axial thermal gradient were present, the bottom of the cavity would be hotter

th an the top and hotter than the area of the specimen surface viewed in an emlttance measurement,

and the measured emlttance would be too low. Thus the effects of the two gradients tend to compensate each other. Preliminary measurements indicate that the axial gradient is much larger than the

radial gradient, and that the net effect is to give a measured emittance that is too low. The effect is

more complex than this simple analysis would Indicate, however. The gradients produce two different effects.

First, since the cavity walls are not isothermal, the cavity would not emit as a blackbody even

if It were completely enclosed. There would be net radiant heat transfer within the cavity, and the

spectral distribution of flux in the cavity would be different from that in an otherwise identical Isothermal cavity. This is not too serious for total emlttance measurements, since the detector

measures only the total amount of flux emitted, and not its spectral distribution. However, it implies

that an effective temperature of the non-isothermal cavity must be defined as the temperature of an

otherwise Identical isothermal cavity having the same radiance. This effective temperature will be

somewhat lower than the actual temperature of the bottom of the cavity, but not as low as that of the

top of the cavity wall.

Second, the effective temperature of the cavity will be higher than that of the area of the specimen surface viewed in an emittance measurement. The percentage error from this source will be

approximately equal to four times the percentage difference in absolute temperatures of specimen and

cavity.

The problem of correcting for the temperature difference error involves first an evaluation of

the thermal gradients, and then computation of the effective temperature of the cavity.

2. 5. 3. 2 Thermal Gradient Computations

Several possible methods of computing the thermal gradients in the shallow cavity specimens

were considered, and a finite -difference approach was selected as the most promising. In this

method, the specimen is divided into a fixed number of small volumes called nodes, the exact number

19
selected being determined by the accuracy desired and the time available for computation. A temperature is assigned to each node and the heat transfer between each node and each of the surrounding

nodes is computed. If the net heat transfer to or from a node is not zero, Its temperature is adjusted

to reduce the net heat transfer, and the computations are repeated until a temperature distribution is

found for which all of the net heat transfers are zero. The thermal gradients are then computed from

the temperature distribution.

It should be noted that radiant heat transfer is involved for nodes with exposed surfaces. When

nodes view other nodes, as on the interior surface of the cavity, multiple reflections are involved,

which will be discussed In more detail.

Various schemes have been used in the past for calculating the heat transfer within a diffusely

reflecting and emitting cavity. All of these methods somehow reduce to the solution of N linear

equations of N unknowns —or its equivalent—the inversion of an N x N matrix. The procedures involved in radiant transfer calculations are well known; however, the clarity of the concepts involved

may perhaps be enhanced when they are expressed in terms of transformations on an N-dimensional

vector space. See Appendix I.

2. 5. 3. 3 Results of Thermal Gradient Computations

The temperature distribution in a specimen of simple geometry, shown in Figure 6, was computed by hand. Since the specimen has complete symmetry about the axis corresponding to the center

of the cylinder, the thermal gradients, in the three-dimensional specimen can be represented in a two

dimensional network. The temperature of volumes 5 through 10 were set at 2000°K, and the specimen

was assumed to emit to a heat sink at 0°K. Under these conditions, the temperature of a volume is

fixed when the total heat flux, by conduction and radiation into the volume, exactly equals that out of

the volume. Four simultaneous equations in four unknowns can be set up for the four volumes, as

follows

:

V

2 °

i Sf _ ^ $ *\ i /T _ T , (2D

P T

4 +\ c—MT. - Ti) P ,-1 1-1

(i = 1, • • • 4)

20
where the subscripts i and j refer to the i-th and j-th volumes respectively. (In this case i and j are

the numbers of any two of the volumes 1 through 4). e is the emittance, p is the reflectance and k is

the thermal conductivity of the material, o is the Stefan-Boltzmann constant. A^ is the exposed surface area of volume i, A. . is the contact area between volumes i and j, P, * is the i, J element of a

matrix, P, which is similar to the F matrix discussed previously, T is absolute temperature, and

h, . is the distance between the centers of the i and j volumes.

The four unknowns in the equations are Tx

, T?, T3 , and T4

. The equations are rather complex

for direct solution, and are most easily solved by a relaxation or iteration procedure. The first

step is to arbitrarily assign values to the Tj's, and compute the s^'s. The sj's represent the net

heat transfer to the respective volumes, and are called residuals. A positive s. indicates that there

is net heat gain by the i volume, and that its temperature is too low for its environment; a negative

s. indicates that there is net heat loss by the i volume, and that its temperature is too high for its

environment. The volume having the largest j s | is farthest from equilibrium. The temperature of

this volume is adjusted in predetermined small increments until | s. | reaches a minimum. Then all

of the s 's are recomputed, using the new T. for volume i and the old T.'s for all other volumes. This

process is repeated until all s/s are at a minimum. At this point, a change in any T, by the preselected increment will increase the thermal imbalance in the system. It is then highly probable that

the computed T.'s are within a few increments of the values that would satisfy equation (21) exactly.

The temperatures of elements 1 through 4 were computed by hand for the specimen shown in

Figure 6. The temperatures of elements 5 through 10 were fixed at 2000°K, and the specimen was

assumed to have an emittance of 0. 3, a reflectance of 0. 7 and a thermal conductivity of 0.42 Joules

cm~2sec. (°K cm"')", typical values for alumina. The results are shown in Table 1. A computer code was written to perform the same computation, and after some de-bugging, the

same results were obtained, as indicated in Table 1.

21
—»-| Imm [ml

I ! 10

i 2 9

I

i 3 ! 8

(—+ 1—

5 6 7 I

i liolij

2j_9ij2

" 6l7_|l4

-5

-k- •18 IT! 16115 i l"i ,w

l

Figure 6. Cylindrical Geometry Used in Trial Calculation of Heat Transfer in a Cavity

Figure 7. Cylindrical Specimen with Shallow

Cavity, Having 18 Nodes

SHALLOW CAVITY SPECIMEN

K-2R-1

T

1

* X

:l X x K x

xX

X*x

XX

x

X

X

XX

X

y x x « rfx >« M »< w w>< w>t

X

X

X

X

rxxxxxxxxx x

1 !:

X. X X X X X X X. X X X , x x x * x X X X X X X » XXXX^XXXXXM

ttXXXXXXXXXX

X X

X

HX

XX

X

XX

X

X

XX

KX

XX X *

* x ?

X X

X A X. » x X * » i«ij7

H»

X J! X jl X X. X X

X X

x x k X x 4

X X

X X

X x i

X X ^ X X * x x

X X

X x A X xx-:

X A it X X x r- > X X X X

x xxx x xxi

J-1,27

1

a, h

^pa 2 h

Figure 8. Cylindrical Specimen of Radius d and Length c Containing a Shallow

Cylindrical Cavity of Radius R and Length L

22
Table 1

Temperature Distribution for Specimen Shown in Figure 6.

Hand & Computer Code

Tx 1958°K

T? 1981°K

T3 1993°K

T4 198 7°K

The number of incremental volumes was then increased to 18, with the geometry shown in

Figure 7. The same values of e, p and k were used as before, and the temperatures of elements 11

through 18 were fixed at 2000°K. The temperatures obtained with the computer are given in Table 2.

Table 2

Temperature Distribution for Specimen Shown in Figure 7.

Ti 190 30K Te 1987°K

T2 1944°K T7 1991°K

T3 1972°K T8 1983°K

T4 1967°K T9 1966°K

T5 1985°K T10 1925°K

It is obvious from a comparison of the data in tables 1 and 2 that the number of elemental

volumes must be greatly increased before the computed temperature distributions will approach

those encountered in real specimens. It will be noted in Figure 7 that elements 3, 5, 6, 7, 8 and 9

have no exposed surface area, hence A. in equation 21 is zero, and the first term drops out. Under

these conditions equation 22 can be used to compute thermal gradients within the specimen. This

equation expresses the temperature T at a point r, z within a cylinder in terms of the radial distance

r from the center axis of the cylinder and the vertical distance, z, from the top end of the cylinder.

The equation is

_9f T +

_1 _3T + 3f T = (22)

^r2

23
and applies when steady state heat transfer exists In the specimen. The boundary conditions at the

surfaces of the specimen must be supplied in order to solve equation 22. A computer code is being

developed to solve the equation, based on the geometry of Figure 8. The code will be general in form,

so that it can be applied to other specimen configurations to be discussed later.

In order to solve equation 22 by a finite difference approach, a series of node points must be

located throughout the specimen, and the temperature at each node must be written in terms of the

temperatures of the surrounding nodes. Each node represents a volume that surrounds it, having

dimensions, in the plane shown, extending half way to each surrounding node.

For any specimen having complete axial symmetry, the thermal gradients can be represented in

a two-dimensional lattice, as in Figures 6 and 7. Such a generalized lattice is shown in Figure 8, in

which the nodes are spaced a distance h apart on a aquare lattice, in the c and d directions, parallel

to and normal to the specimen axis, respectively. Each node is identified by an i, j index, in which i

represents the column, numbered outward from the specimen axis, and j represents the row,

numbered downward from the top of the specimen. For the lattice shown it is assumed that the depth

L and radius R of the cavity are evenly divisible by h, but that the height c and radius d of the specimen are not necessarily divisible by h. When the dimension d is not evenly divisible by h, the distance between i and i -1 will be some fraction Qfi of h, and when the dimension c is not evenly

max max

divisible by h the distance between j and j -1 will be some fraction a~ of h. max max

There will be a separate equation relating the temperature of each node to the temperatures of

the surrounding nodes. These equations will be similar for nodes with similar surroundings. Hence

it is convenient to divide the lattice into zones such that the same equation applies to all of the nodes

in that zone. In order to describe these zones mathematically, the following symbols will be used.

i cav

= The i-index of the column of nodes on the cavity wall

j cav

= The j -index of the row of nodes on the cavity bottom

i = The i-index of the column of nodes on the outer specimen wall max ^

Jj = The i-index of the row of nodes on the specimen bottom max J ^

The eleven regions can be described as follows:

Rl - i < imax> J = Jmax ~ These are the nodes on the specimen bottom.

24
R2 - 1 < i < imax -1, j = j max

- 1 - These are the nodes in the row immediately above the speci- men bottom, except for the node at the center and the two nodes at the outer surface.

R3 - i = 1, J„„„< j < j -1 - These are the nodes in the center column, at least two rows

above the specimen bottom and below the cavity bottom.

R4 - i £ i cay

and J = j cav, or i = i cav and j < j cav . These are the nodes on the cavity wall.

R5 - i < 1 and j < J ca

„. These are the nodes inside the cavity. Since these nodes are in air

or free space, it can be considered that they do not exist.

R6 - i < i < lTr|Hy ) j = 1. These are the nodes on the top of the specimen.

R7 - i = i and 1 < j < J_ nv.. These are the nodes on the outer surface of the specimen.

R8-i = i - 1, 1 < 1 < j -1. These are the nodes in the first row in from the outer sur- max ' i J max

face of the specimen, below the top row and at least two rows above the bottom.

R9-i <i<i -1, 1< j< JTM„ -1 or 1< i< i . 1 < j< ]„„ -1- These are the nodes in cav max •'max cav' 'cav J J max

the interior of the specimen at least a distance h from the top and axis of the specimen and cavity

walls, and at least a distance 2 h from the specimen bottom or side.

RIO - i = i -1 and 1 = 1 -1. This is the node Just inside the bottom outside corner of the max J J max J

specimen.

Rll -1 = 1, and j = J max

-1« This is the node in the next to the bottom row along the axis of the

specimen.

The equations that apply in these regions are as follows:

Rl - No equation - all nodes in this region are assumed to be at the same predetermined temperature.

R2: T

E* \i " I

TSJ

+ sKi - i + \l * i]

(23)

(24)

25
R4: kg = F eaT* (25)

R5: There Is no specimen material at the location of the nodes, hence no equation is required.

U: TM

- [. g

_ Wl. A

,a ]-l[3M

(Ii;i)

. . *

Iij2 j

(26)

R7: Nodes in this region are at the same predetermined temperature as those in region Rl.

R8: Ti,J = 2^ + &±

- h (or8 - l) t

Ti,j + 1 + Ti,j - 1 +

(27 )

iTToTriT

^Ti - l,j + T

i + M

} + 2^57 <~ "*T

i-l,J + T

i ^ lj\

R9: Ti,J = UT

± + 1,J + Ti-i,j + T

i,o-+ 1 + Ti,j - i J +

(28)

h - 8R? IT. , _ . - T. _ . l L i + l,j i - l,jj

2Q?1

Qfs R

i

2 R10: Ti,j = mta+cfc) -hai(as - I) t QTiCl + ttl ; as (l + ^7T < 2S >

«iorB (l + aj ^

j + QflQf s (l + Qf2 ) T± j _ x + ax («! + l) T

± + L j +

a9 T. . . - T. , , . «. (1 + «» ) T, , M J i-^iJ L+ iiJ

20-13 hR,

' *i,J + U

i

r orT. . + T. _ Rll: T. . + r^rr 2T . + ^uLZ-1 i»J +

3: | (30) i,j 2ax + 1 I 2, J

Qf1 (a1 + lj J

In the above equations* Tf

. . is the temperature of the i, j node on the previous iteration.

R. = h(l-l), and 3T/£n is the vector thermal gradient normal to the surface. All other terms have

been defined previously.

The computation of the temperature distribution In a specimen by this procedure requires the

simultaneous solution of i j equations in i j unknowns. This is obviously impossible except by computer techniques. Again an iterative technique is used, as was previously explained. This method of

solving the conductive heat transfer problem has been used many times in the past with good

results [ 5l

Some progress has been made in preparing a computer code to perform the above computation,

but much work remains to be done.

26
2. 6 New Procedures

Because of the large errors found in the shallow cavity technique, two new techniques were developed that use the same basic equipment, but for which the errors were expected to be much smaller.

The new techniques involve comparison of the flux emitted by the hot specimen to that emitted

by (1) a comparison standard of known emittance, at the temperature of the specimen, or (2) by a

deep cavity In the specimen that approaches a blackbody radiator. These will be referred to as the

center post and deep cavity techniques. Both techniques require a specimen in the form of a thick

walled cylinder, of the same outside diameter as the shallow cavity specimen, and an inside diameter

slightly larger than that of the cavity in the shallow cavity specimen.

2. 6. 1 Center Post Technique

In the center post technique the susceptor has a center post of the same diameter as the center

hole In the specimen, as shown in Figure 9. The top of the post is flat, and serves as the comparison standard of known emittance. The center post also performs a second function in heating the

specimen, and thus reducing the thermal gradients. The center post is both a good thermal conductor

and a good electrical conductor. Heat is conducted by the post from the outer susceptor to the specimen, and heat Is generated by induction in the post itself.

The material first chosen for use as the susceptor with the center post technique was highpurity graphite. * This material was chosen because of its high melting point, good electrical properties, high opacity and high emittance. Tests showed that it worked very well at temperatures of

1400 and 1600°K, but that it reacted with or diffused into the ceramic specimens at higher temperatures.

Tungsten was next used as the susceptor for the center post technique. It has a very high

melting point, good electrical properties, and many measurements of its emittance are reported in

the literature. The maximum temperature at which a tungsten susceptor could be used was found to

vary with the composition of the specimen being measured.

With the center post technique there is no cavity, and thus the errors due to translucency of the

cavity walls and the thermal gradient along the cavity walls will be eliminated. There will still be a

c Mfr. designation ATJ, National Carbon Co.

27
SPECIMEN

SUSCEPTOR

Figure 9. Cross-sectional Drawing of specimen Mounted in Susceptor with Graphite

Post Comparison Standard

SPECIMEN SUSCEPTOR

Figure 10. Cross-sectional Drawing of Specimen with Deep Cavity, Mounted in Susceptor

28
temperature difference between the center post and the area of the specimen viewed for measurement.

If E Is the ratio of the specimen radiance to that of the center post, we can write

e = h. =

e s n o Ts 4

Ic ec

TI o Tc 4

" (31)

In which Lj is the normal radiance of the specimen, Lj is that of the post, e, is the normal total

emittance of the specimen and ec is that of the center post, tt and o are constants which cancel out,

Ts Is the absolute temperature of the specimen and Tc is that of the center post. Solving for es we

get T

*

G s = E e

c T7 (32)

If Tc = Ts

, the only corrections that need be made are those for ec and the scattering error

which has been discussed previously. Tc can be measured accurately with ah optical pyrometer,

since the normal spectral emittance at the optical pyrometer wavelength of the center post is known.

If the corresponding emittance of the specimen is known, Ts can also be measured directly. If it is

not known, Ts can be computed by the techniques previously discussed, or estimated by techniques

that will be discussed later.

2. 6. 2 Deep Cavity Technique

In the deep cavity technique a susceptor of the type used with the shallow cavity technique is used,

as shown in Figure 10. The cylindrical hole through the specimen forms a deep cavity that is a

reasonable approximation of a true blackbody, especially if the bottom of the cavity is of a high

emittance material. In this case the normal spectral emittance of the cavity is taken as 1.00, and Tc

can be measured accurately with an optical pyrometer. The major uncertainty in this procedure is

the evaluation of Ts . The temperature difference between Ts and Tc will be much larger than in

either the shallow cavity or center post techniques. It will thus be necessary to correct for the

temperature difference error. The translucency error is not completely eliminated by the deep

cavity technique, but its magnitude, compared to that for the shallow cavity technique, is greatly

reduced. These errors will be discussed in more detail in the following section.

Equation (32) is modified for the deep cavity technique to

29
T 4

G = K -C~

8 T

s4 (33)

since in this case ec is assumed to be 1. 00. A graphite susceptor was used for the first measurements by the deep cavity technique. This

had a special advantage for this method, because the high emittance of the graphite bottom of the cavity

made it a near perfect blackbody. However, graphite had the same disadvantages here as when used

with the center post technique - diffusion of graphite into the specimen, and reaction with the specimen, particularly at high temperatures.

Tungsten susceptors were used in the later work. Since tungsten is easily oxidized, it was

necessary to work in an oxygen-free atmosphere or in a vacuum of better than 10-b torr. Because of

the relatively low emittance and specular reflectance of tungsten, which formed the bottom of the

deep cavity, the inside of the cavity was lined with nickel oxide, to increase the emittance and diffuseness of the cavity to a point where it approached a true blackbody. Unfortunately the nickel oxide

decomposed at temperatures above about 1800°K, and nickel metal was deposited on the bell jar and

sodium chloride window. As a result, measurements by this technique were restricted to temperatures below about 1700°K. Another coating having high emittance plus greater thermal stability in

vacuum at high temperatures is being sought.

2. 6. 3 Discussion of New Procedures

The two new techniques have advantages over the shallow cavity technique in that the only

serious errors are the scattering error and the temperature difference error. The scattering error

is a function of the optical system, and can be evaluated and corrected for.

The temperature difference error is the largest source of uncertainty in either method. An

accurate correction for this error can be made if the temperatures of the specimen at the area viewed,

and the temperature of the center post or deep cavity, can be accurately evaluated. The temperature

of the center post can be measured with an optical pyrometer, because its normal spectral emittance

at the optical pyrometer wavelength is known. That of the deep cavity can also be measured, because

its emittance is taken as 1.00.

Evaluation of the temperature of a partially translucent specimen is more complex. Such a

30
specimen emits from a spherical volume of radius r, within which a thermal gradient exists, as

previously explained and Illustrated in Figure 4.

The total flux emitted from an elemental area of surface will be made up of the sum of the

fluxes originating in elemental volumes within the specimen at distances less than r from the surface

element, each emitting at its own temperature. Each of these fluxes is attenuated logarithmically,

as Indicated in equation 16, In traversing the specimen to reach the surface element. Hence the

specimen has no unique temperature at which it Is emitting, but rather a range of temperatures,

which is determined by the magnitudes of the thermal gradient and extinction coefficient. The extinction coefficient, k, varies with wavelength for most materials.

For any non-Isothermal translucent specimen there will exist an effective temperature, which is

defined as the temperature of an otherwise identical isothermal specimen that is emitting at the

same rate. Such an effective temperature can be computed for the combination of thermal gradient

and extinction coefficient that will exist at any one wavelength. The computation becomes more

complex when a large wavelength range Is involved, over which the extinction coefficient varies

widely, as in the case of measurements of total emittance. To further complicate the problem, the

extinction coefficient, k, will usually also vary with temperature, although such variations are usually

much smaller than the variations with wavelength.

The brightness temperature, measured with an optical pyrometer, of a non-isothermal specimen,

is the same as that of an otherwise identical Isothermal specimen at the same effective temperature

for the optical pyrometer wavelength, as defined above. Also the radiance temperature, measured

with a total radiation pyrometer, of such a specimen is the same as that of an otherwise identical

Isothermal specimen at the same effective temperature for total emitted flux. The brightness

temperature or the radiance temperature can be converted to true temperature if the appropriate

emittance Is known. Unfortunately, however, this is usually not the case, since the total normal

emittance is the property being measured.

For most ceramic materials the optical pyrometer temperature will be a better approximation

of the effective temperature for total emitted flux than will the surface temperature.

There are then, three basic techniques that can be used to evaluate the effective temperature of

31
the specimen (1) to compute it by the techniques previously discussed, which requires a knowledge of

the thermal conductivity and extinction coefficient of the specimen, (2) to measure it with an optical

pyrometer or total radiation pyrometer, which requires a knowledge of the appropriate emittance, or

(3) to measure the temperature distribution across the top of the specimen in terms of brightness

temperature (optical pyrometer temperature uncorrected for emittance) and assume that the observed

differences in brightness temperatures are equal to the corresponding differences in true temperatures. Technique No. 3 is the most easily applied, because it requires no prior knowledge of the

properties of the specimen, but it is also probably the least accurate of the three techniques.

The translucency error is largely eliminated in both new techniques. It arises from a net transfer of radiant flux across the interface between the center post or deep cavity and the specimen. If

there is net transfer of radiant flux into the specimen, the measured apparent emittance of the specimen will be too high in a ring of width r around the top of the post or cavity, the apparent emittance

of the post or cavity will be too low, and the measured emittance will be too high. If the net transfer

of radiant flux is in the opposite direction, the measured emittance will be too low.

If the center post was a perfect mirror, there would be no net transfer across the interface, and

no error. Also, if the specimen and center post are at the same temperature, there will be no net

transfer of flux across the interface. Flux will be crossing the interface in both directions, but the

amounts will be exactly the same, because the transmission of the interface along any optical path

will be identical in the two directions, and the incident radiance in the two directions will be equal on

the two sides. Hence we can say that there is no translucency error in the center post technique.

There is no net transfer of radiant flux across the wall of a blackbody cavity, since by the

definition of a blackbody the absorbed flux is exactly equal to the emitted flux. Thus if the deep cavity

were a true blackbody, there would be no net transfer of radiant flux across the wall of the cavity, and

hence no translucency error. Unfortunately the cavity departs appreciably from blackbody conditions

especially near the top of the specimen. The net effect is that there will be net transfer of radiant

flux into the cavity from the specimen, and the apparent emittance of the specimen is low in a ring of

width r around the top of the cavity. Because of the depth of the cavity the effect of the translucency

error on the apparent emittance of the cavity bottom will be negligibly small, and can be ignored.

32
Thus the net translucency error will be negligible if the portion of the specimen viewed for measurement is more than the distance r from the edge of the deep cavity.

2.7 Total Normal Emittance Measurements

Data were obtained on specimens of four materials by all three techniques, shallow cavity,

center post, and deep cavity. So far only the data taken by the shallow cavity method have been

reduced and no correction has been made for the translucency error or the temperature difference

error. The four materials were alumina, thoria, magnesia and zirconia. The tungsten used for the

susceptor was also measured. See Appendix n.

2.7.1 Alumina

The normal spectral emittance of two specimens of alumina was measured at temperatures from

1400 to 1900°K. One specimen had consistently higher emittance than the other, as shown in Figure

11. The average value for the two specimens checks data obtained previously. The alumina specimens

were difficult to polish satisfactorily. The individual grains were relatively large and very hard and

the bond between grains was relatively weak. As a result there was a tendency for the grains to

break loose during polishing, leaving a rough, non-uniform surface. The observed difference in the

emittance of the two specimens that were measured is ascribed primarily to the known difference in

surface finish of the two specimens, rather than to a difference in composition.

2.7.2 Magnesia

The normal spectral emittance of two specimens of magnesia was measured at temperatures

from 1400 to 1800°K. This material Is easy to machine, but very fragile. Again a consistent difference was observed in the emittance of the two specimens.

The data shown in Figure 12 agree very well with published values by Olson and Morris [6] and

Pears [7].

2. 7. 3 Zirconia

The normal spectral emittance of three specimens of zirconia was measured at temperatures

from 1400 to 1800°K. The tungsten susceptor reacted with the zirconia at temperatures above about

1850°K, causing a reduction of Zr0? to Zr metal.

The zirconia has low thermal conductivity and had the largest thermal gradients observed on

33
5

.4

Uio

z

<

2 3

UI<5

oro

2 2

_i<o

ALUMINA

1400 1500 1600 1700 1800 TEMPERATURE (DEGREES K) 1900 2000

Figure 11. Normal Total Emittance of Alumina, Measured by the Shallow Cavity Technique

o

£.3

2 2

<o

11

' 5

: <

! ! !

MAGN ESIA

1300 1400 1500 1600 1700

TEMPERATURE (°K)

1800 1900 2000

Figure 12. Normal Total Emittance of Magnesia, Measured by

the Shallow Cavity Technique

34
z

<K

P. .4

1

\

\

I \\ -Physical change

\ \

\

\\

\

\

ZIRCONIA

1400 1500 1600 1700 1800 TEMPERATURE (DEGREES K) 1900 2000

Figure 13. Normal Total Emittance of Zirconia, Measured by

the Shallow Cavity Technique

.5

.4

1 ]

o

z

<

K

t-2.3

_l<

z

<ro

z

_i.2 <

i-o

iTH0R IA

1400 1500 1600 1700 1800 1900 TEMPERATURE (DEGREES K) 2000 2100

Figure 14. Normal Total Emittance of Thoria, Measured by

the Shallow Cavity Technique

35
any of the four materials tested. Specimens broke If heated too rapidly. The readings from one spot

to another on one specimen varied more than for any other material tested but the average values for

the three specimens agreed well, as indicated in Figure 13. The data in Figure 13 agree well with

results published by Cox T8], Olson and Morris [6] and Pears [7].

2.7.4 Thoria

Measurements were made on four specimens of thoria, at temperatures from 1500 to 2000°K.

There was no indication that the specimens would not withstand heating to higher temperatures, but

there was not sufficient power available to reach higher temperatures with the available power source.

Some fogging of the sodium chloride window of the bell jar enclosing the hot specimen occurred at the

temperature of 2000°K, which is believed to be from tungsten oxide produced by reaction of the susceptor with traces of oxygen in the vacuum system, which was at a pressure of about 10_£

torr.

The emittance was relatively constant over the surface of any one specimen but there was

appreciable variation from specimen to specimen, as shown in Figure 14. The data plotted in

Figure 14 agree well with similar data published by Pears [7 J at 1500°K, but he reports essentially

constant emittance at temperatures from 1500° to 1900°K, with a sharp increase in emittance at

higher temperatures.

2. 8 Materials for Emittance Standards

The four materials referred to above had been selected as possible candidate materials for use

as emittance standards in the 1400-2500°K temperature range (approximately 2000—4000°F). On the

basis of experience to date, none of the materials is an outstanding choice for this use.

2.8.1 Alumina

The melting point of alumina is only slightly above 2300°K, but the fact that it is readily available and relatively easy to fabricate make it appear suitable for use as a standard at temperatures up

to perhaps 2200°K. The specimens that were tested did not come up to expectations. This may be

due to the physical structure of the specimens that were tested, which made it difficult to produce a

smooth surface by conventional grinding and polishing techniques, and thus contributed to the observed differences in emittance of specimens that were supposedly identical.

36
2. 8. 2 Magnesia

The high melting point, 3073°K, of magnesia would appear to make it suitable for use as a hightemperature emittance standard. It is relatively easy to fabricate specimens by machining since the

material is relatively soft. However, it is perhaps too fragile, since it will break with rough handling.

The emittance appears to be quite uniform, not only from one point to another on a specimen, but

from specimen to specimen.

The major disadvantage of magnesia is its chemical instability. It will react slowly with moisture

and carbon dioxide in the atmosphere at ambient temperatures, to form Mg(OH)., and MgCOa , with an

appreciable increase in volume. At high temperatures, above about 1700°K, it reacts with many

materials with which it comes in contact, and particularly with the tungsten susceptor. It also has

a relatively high vapor pressure at high temperatures. The normal spectral emittance of MgO was

found to vary only slightly with temperature in the 1400-1800°K temperature range.

2. 8. 3 Zirconia

Zirconia is by far the hardest of the materials tested. It is very brittle, and very difficult to

machine. As mentioned previously, it has low thermal conductivity, and must be heated and cooled

slowly to prevent fracture from thermal shock. It reacts with tungsten at temperatures above about

1800°K. The emittance varies appreciably over the surface of a specimen by as much as 0. 25 in

emittance. There is some indication that the emittance of the material changes as a result of heating.

This is not surprising in view of the known phase change which occurs in the pure material on heating,

which is inhibited by the addition of a stabilizer such as CaO.

2.8.4 Thoria

Thorla is relatively easy to machine, and the specimens do not vary appreciably in emittance

over the surface. Although there was appreciable variation in emittance from specimen to specimen

of those tested, the values for any one specimen were reproducible on repeated measurements. No

reaction with the tungsten susceptor was noted at temperatures below 2100°K.

Thorla has another possible disadvantage as a standard because of its slight radioactivity. This

requires special precautions on the part of those machining specimens to avoid breathing or ingestion of the dust.

37
2.8.5 Metals

Of the metals available for use as standards In the 1400-2500°K temperature range, only iridium,

molybdenum, niobium, osmium, rhenium, ruthenium, tantalum, and tungsten have melting points

high enough to make them suitable for use over the entire temperature range. Of these, iridium

and rhenium have better oxidation resistance than the others, and of the readily oxidized metals

tungsten has the highest melting point and lowest vapor pressure at high temperatures. On this basis,

iridium and tungsten have been selected for further tests.

3. Laser-Source Integrating Sphere Reflectometer

3. 1 Background

Reflectance, which is defined as the ratio of reflected flux to incident flux, is nearly independent of temperature, while the emitted flux, which is measured in an emittance determination, is a

very strong function of temperature, particularly at wavelengths below the peak of the Planck curve.

Hence errors due to errors in temperature measurement should be much smaller in reflectance

measurements than in direct measurements of emittance.

The basic problem in measuring reflectance of specimens at high temperatures is to discriminate between the flux reflected by the specimen and that emitted by the hot specimen. The classical

method of discriminating is to chop the incident flux, and to use synchronous a. c. amplification of the

signal from the detector, which essentially ignores the unchopped flux emitted by the hot specimen.

This system is most satisfactory when the a. c. signal from the detector produced by the chopped reflected flux is large compared to the d. c. signal produced by the unchopped emitted flux, and fails

completely when the d. c. signal is appreciably larger than the a. c. signal. For this reason conventional thermal sources are not very satisfactory for use in reflectance measurements of specimens at very high temperatures, particularly for measuring spectral reflectance.

A laser emits a high-intensity, highly collimated beam of flux over a very narrow wavelength

band and hence should be an ideal source for use in high-temperature reflectance measurements.

Preliminary computations show that the 1 mW output of a helium-neon continuous -wave gas laser is

equivalent to the flux that would be produced in the same wavelength band by a blackbody radiator at

a temperature of about 1, 200, 000°K.

38
One disadvantage of the laser source is that It operates at only a few specific wavelengths. However, for materials that do not undergo changes in structure or phase as a result of heating, the

general shape of the spectral reflectance curve does not change appreciably with temperature, so

that it is possible to interpolate and extrapolate the high-temperature reflectance curve of a

material on the basis of accurate measurements made at a few wavelengths and the complete roomtemperature reflectance curve.

3. 2 Previous Work

The laser-source integrating sphere reflectometer was designed and built to measure spectral

directional hemispherical reflectance of specimens at temperatures up to 2500°K. The original 35. 5

cm (14 in.) integrating sphere was designed for use in the substitution mode. There was only one

sample port, and the specimen and comparison standard were alternately placed over the specimen

port for measurement.

Performance tests showed that the error in reflectance measured with the original equipment

was on the order of ±0.02 in reflectance units, which for dark specimens was as much as 50% of the

measured value, and it was not possible to accurately predict or correct for the error.

The reflectometer reading obtained when a specimen is at the specimen port is related to the

reflectance of the specimen, and when the specimen is replaced with a comparison standard of

known reflectance, the reading obtained is related to the reflectance of the standard. The assumption is made that the constant relating instrument reading to reflectance is the same for the two cases,

so that the reflectance of the specimen is computed by correcting the ratio of observed reflectometer

readings for the known reflectance of the standard. The error arises from the fact that the constant

relating reflectometer reading to reflectance is a function of the geometric distribution of reflected

flux in the integrating sphere.

The detector views a small portion of the sphere wall. If the sphere wall is a perfect diffuser,

which it approximates, the same fraction of the flux incident on the viewed area of the sphere wall

will always be received by the detector. However, the fraction of the reflected flux that reaches the

detector is larger for the flux that is directly Incident on the viewed area, and thus reaches the

detector after a single reflection from the sphere wall, than for the reflected flux that is directly

39
incident outside the viewed area, and thus must undergo two reflections from the sphere wail before

reaching the detector. The maximum error from this source can be evaluated by comparing readings

obtained with a good mirror oriented so that the specularly reflected beam falls, first, entirely inside the viewed area, and second, falls entirely outside the viewed area. The difference between the

two readings, divided by the smaller reading, is the maximum possible percentage error from the

source. For any specimen the fraction of the reflected flux that is directly incident on the viewed

area is a function of the geometric distribution of reflected flux. If this fraction is the same for the

specimen and reference standard, there is no error from this source.

3. 3 Design Modifications

3.3.1 Requirements

Because of the somewhat disappointing performance, the basic design of the reflectometer was

reviewed. The requirements to be met are as follows:

a. The reflectometer should measure reflectance accurately, regardless of the geometric distribution of reflected flux from the sample or comparison standard.

b. The reflectometer should be capable of measuring reflectance in absolute units or relative to

a comparison standard.

c. The specimen heater should not disturb the sphere configuration, and should be capable of

heating the specimen to at least 2500°K.

d. The reflectometer should be adapted for operation with air, vacuum or inert atmosphere in the

sphere.

3. 3. 2 Sphere Geometry

In order to discuss the modifications of the sphere, it is first necessary to describe its geometry.

The sphere is made up of two hemispheres, which are joined with bolted flanges having an 0-ring seal.

The principal axis of the sphere is defined as the diameter normal to the plane through the joint

connecting the hemispheres. The primary plane of the sphere is defined as the plane containing the

principal axis and the center of the entrance port, and the secondary plane of the sphere is defined as

the plane through the principal axis and normal to the primary plane. The apex of all angles mentioned

is the center of the sphere, and angles are measured from the principal axis.

40
The entrance port was located in the upper hemisphere, and centered 11° from the principal

axis. In the original design the specimen port was located in the lower hemisphere and centered on

the principal axis. There were two detector ports, one in each hemisphere. The one in the lower

hemisphere was located 60° from the principal axis. Since the two hemispheres could be rotated

relative to each other, this port could be positioned in either the principal plane, where it was

identified as No. 1, or in the secondary plane, where it was identified as No. 2. The detector port

in the upper hemisphere, identified as No. 3, was centered 45° from the principal axis in the primary

plane, nearest the entrance port.

3. 3. 3 Sphere Modifications

The sphere design was modified to permit operation in the comparison mode, as shown in

Figure 15. Only the detector port identified as No. 3 was used, and the field of view of the detector

was restricted to a small area of the sphere wall diametrically opposite the detector port. The

sample specimen port was replaced by two ports, one for the specimen and one for the comparison

standard. These ports were located in the lower hemisphere, in the secondary plane, centered 20°

on either side of the principal axis. They are thus located symmetrically with respect to the entrance

port and detector port.

Shields were introduced into the sphere to prevent reflected flux from impinging directly on the

area viewed by the detector. Since the area viewed by the detector is located near the specimen

ports, small shields could be used, which do not appreciably disturb the sphere configuration. Because it is sometimes desirable to operate the reflectometer without the shield for the comparison

standard, it was attached to the sample holder rather than the sphere itself. A sample holder without the shield can be used if desired.

Restriction of the field of view of the detector to a small area of the sphere which is shielded

from the sample eliminates the effects of variations in the geometric distribution of reflected flux on

the measured reflectance. All of the flux reflected from the sample must be reflected at least twice

by the sphere before reaching the detector. If the sphere surface is a perfect diffuser the reflected

flux will be uniformly distributed over the surface of the sphere after the first reflection from the

sphere. If the sphere surface is also uniform in reflectance, a constant fraction of the flux that is

41
I C_5

CO

QC /

Ov ./ I__ cco

Q_

rr Q_ o «=t l

QC C_3 |— l_l_l | 1 LXJ

~T~ C=) o>1 uVSo

<a O

0)M&SO

0)

5,

to

42
reflected by the specimen will reach the detector, regardless of its geometric distribution. The

barium sulfate coating used on the sphere is uniform in reflectance and is also a very good diffuser

at wavelengths shorter than about 2 microns.

3. 4 Relative Reflectance Measurements

The modified sphere is well suited to measurement of relative reflectance. This type of

measurement will be particularly well suited to evaluation of the thermal coefficient of reflectance.

The reflectance of the hot specimen will be measured relative to a nearly identical specimen of the

same material at room temperature, and the change in reflectance produced by the change in temperature can be measured directly.

Elimination of the error due to variations in the geometric distribution of reflected flux permits

use of a high reflectance mirror as the comparison standard. The absolute reflectance of such a

standard can be calibrated very accurately (to about 0.1%) by a multiple reflection technique, hence

this procedure can be used to obtain accurate absolute reflectance.

The ratio of the detector outputs when the two specimens are irradiated will be the ratio of

their reflectances, to an accuracy of better than 1%. The only errors that have not been eliminated

are those due to flux losses out the entrance port and detector port. Such losses are small, being

approximately equal to the ratio of the total area of the openings to the total area of the sphere,

which is 0. 0027. This fraction of the flux reflected by the specimen and standard would be lost, and

hence the total error from this source would be appreciably less than the figure quoted. For a specimen and standard with the same geometric distribution of reflected flux the losses would be the same,

and the error from this source would be zero.

3. 5 Absolute Reflectance Measurements

A literature survey revealed that of the many possible ways to measure absolute reflectance

with an integrating sphere reflectometer, that described by McNicholas [9] appeared particularly

well suited for use with the present sphere. In this method, the comparison standard is replaced by

a spherical cap of the same radius as the sphere, and coated with the sphere coating, but without the

shield, so that the area of view of the detector is no longer shielded from the comparison standard.

Under these conditions, the ratio of the detector outputs when the specimen and the sphere coating

43
are alternately irradiated Is the absolute reflectance of the specimen.

Let F be the incident flux, Ag the area viewed by the detector, A the total area of the sphere,

p s the directional hemispherical reflectance of the specimen, p w the reflectance of the sphere wall,

and L'^'.cp') the reflected radiance in the direction e'^'.

Flux F Is incident on the specimen from the direction 0,cp. The reflected flux incident on a

small area dA of the sphere in the direction 8

, , Cp' from the specimen is

L'(e , ,co

, ) Air

cos e

,(iu

where da* is the elemental solid angle subtended by dA at the sample, and A, is the irradiated area

ir

of the specimen, and that leaving dA is

P w

L'(6«,

(r

') A^cose'du,

Since we assume the sphere coating to be a perfect diffuser, the fraction of this flux reacing Aq , the

area of the sphere viewed by the detector, Is equal to the geometric view factor between areas dA and

Ajj , F^^ a . Thus the irradiance of \ due to the flux incident on dA is

1/A p l/(e',cp')A. cose'duu (34)

As shown by Taylor T10] and many others, the irradiance is increased by the factor (1 -

p )"-

due to multiple reflections within the sphere. Thus the irradiance of Aq from the flux reflected from

^

^-A,' 18 pj>(l - P )] _1 L'(9',cD')dAir

cose'duj (35) "AD - dA w

L

and the total irradiance L» _ a due to all the flux reflected from A^r

is

Pw ? L

A. - A = ATl~"p"7 Wo" L'(9", cp') cose'du (36)

ir w

which produces a reflected radiance over the area Aj, of L's

t, = i -T-A A. \ 2Q

n L'(9»,cp') cosG'du^ (37) L

s n All - pw)

irJ°

But, by definition, the total flux reflected from the sample, F', is

F' = A. Jq

71 L'(3',cd') cose'duc (38)

44
hence L' s is also given by

P 2 L

s -« Arr~p-y F

' (39) w

In order to evaluate the flux, F , incident on the specimen, we illuminate an area of sphere

wall (the coated spherical cap in the comparison standard port) that is not shielded from Aq . The

flux leaving the sphere wall is F pw. The fraction of this flux reaching Aj, is determined by the geometric view factor from A. to Aj = \/A. The irradiance of A,, on the first reflection is thus

Pw

F

o

A /A • l/Ao = -p. (40)

Again the irradiance is increased by multiple reflections, by the factor . . Thus the total

1 - Pw

irradiance L. . is

T Pw

LAir

- A, - A(l-~p~)- F

o

< 41 >

and the total radiance of \ , L' , will be

T

, -_L ...

Pl- Fn

(42) Lo Tl A(l - P w

) °

The ratio of the radiances of \ for the two conditions (eqs. 39 and 42) is thus

1 Pw F' ttA ( 1 - Pw) F' " ---~ • p^---—W- =-p- = pe,cp (43) n A(! " P w) w

F

o Fo

since by definition F'/F is the directional hemispherical reflectance, p(6,cp) of the specimen.

Certain assumptions were made in the above derivation. First, the sphere surface was assumed

to be a perfect diffuser. Second, the ratio of the total area of sphere ports to total area of the sphere

was assumed to be zero. As noted previously, this ratio was about 0.0027. Third, equation (38)

ignores that fraction of the reflected flux directly incident on the shield that is absorbed on the first

reflection. As a rough approximation, on the order of 2% of the flux reflected by a diffusely reflecting specimen will be incident on the shield, and of this perhaps 5% will be absorbed. The total error

45
from this assumption is thus 0. 001 of the reflected flux.

From the above, it is evident that the shield should be as small as possible, and of the highest

possible reflectance. If the shield were a perfect mirror, there would be no error, since the flux

would then merely be redistributed over the sphere surface, without loss.

The errors in the measurements will be analyzed in more detail in a future report.

Since the fraction of the reflected flux intercepted by the shield is greatest for a diffusely reflecting specimen, and is zero for a perfectly specular reflector, limits can be placed on the size of

the error. This error will be present only in the absolute reflectance measurements. When the

relative reflectance is measured for the case of a specimen and standard having identical geometric

distributions of reflected flux, the losses compensate and there is no error. This will be true in

measuring the reflectance of a hot specimen relative to an identical specimen at room temperature.

3. 6 Modification of Auxiliary Equipment

3. 6. 1 Laser Source

One continuing problem in operation of the reflectometer has been fluctuation of the intensity of

the laser beam. Use of a highly-regulated constant voltage power supply, and controlled variation of

the input voltage, showed that the observed fluctuations were not related to variations in input voltage

to the laser. The fluctuations in output when operating at the 0. 6328 micron wavelength vary in

amplitude from about 2 to 5 percent, and in period from about 10 minutes to an hour or more. The

amplitude of the fluctuations is considerably larger at the longer wavelengths, up to as much as 50

percent in some cases. A method was devised to correct automatically for the fluctuations in the output of the laser. The

laser tube produces beams at both ends of the tube. The beam from the back of the tube is used to

monitor the output, since the two beams are always proportional. The beam is chopped, attenuated

and then monitored with a lead sulfide detector, and the signal amplified by a synchronous amplifier,

and then mechanically rectified to produce a d. c. potential that is used as the reference potential

across the slidewire of a potentiometer recorder. The beam out the front of the tube is used as the

source for the reflectometer, and the signal from the detector of the reflectometer is amplified,

rectified and balanced against the reference signal. In this way the recorded signal is automatically

46
corrected for fluctuations In power output of the laser.

The preamplifier used with the lead sulfide detector for the reference beam was designed for

use with a detector with a dark resistance of 6 megohms, but the dark resistance of the cell used was

only L 6 megohms. It was necessary to modify the preamplifier by shunting the load resistor In the

preamplifier to approximately equal the dark resistance of the lead sulfide detector. In addition, the

d. c. power supply of the main amplifier was used to supply the bias voltage to the cell. The +240 volt

power supply was connected to ground through a 100,000 ohm precision potentiometer, with a capacitor in parallel with the potentiometer to eliminate the ripple in the power supply. Thus any voltage

from to 240 volts could be put across the cell and load resistor. With these modifications, the preamplifier operated satisfactorily.

The entire system is now being checked for noise and linearity.

3. 6. 2 Heater Design

In the original design the specimen was heated by means of resistance heating elements mounted

In the specimen holder. With this arrangement specimens could be heated in air to a temperature of

about 1800°K„ Resistance heating elements for use to higher temperatures generally require use of a

non-oxidizing atmosphere surrounding the heating element, and special refractories to support the

heating element and specimen. Such refractories must be not only stable at the operating temperature in the non-oxidizing atmosphere, but also non-reactive with the heating element and specimen.

Materials for such an application are not readily available, particularly for use at temperatures of

2500°K or above. Induction heating eliminates many of these problems, and has been selected for

heating specimens to the higher temperatures.

The induction heater should not disturb the sphere configuration, should be capable of heating the

specimen to at least 2500°K (4000°F), and should not heat the sphere wall appreciably. Tests were

made with an induction heater comprising a 3 in. diameter slotted copper disc concentrator, with a

center hole just large enough to clear the susceptor, with the first turn of the induction coil soldered

to the concentrator to provide some water cooling. A stainless steel plate with a 3 in. diameter hole

was mounted above the heater, and a nickel susceptor mounted In the heater was held at 1400°K for

several minutes. Only slight heating of the stainless steel was observed, and this amount of heat

47
could be removed easily by water cooling.

In the final design the induction coil is in the form of a flat spiral of 3/16 in. copper tubing.

The first turn is silver soldered to a copper plate, l/l6 in. thick, that is shaped to the radius of the

sphere. The copper plate and a bakelite ring 1/2 in. wide separating the copper plate from the stainless steel sphere form a portion of the sphere wall and are coated with barium sulfate. The susceptor

is of high-purity tungsten, and is centered in the hole in the slotted copper concentrator. It is supported by a tube of stabilized zirconia. The entire heater assembly is housed in a vacuum-tight

enclosure, so that measurements can be made in vacuum or inert atmosphere if desired. The portion

of the stainless steel sphere wall surrounding the heater is water cooled by means of copper tubing

soldered to the outside of the sphere.

3. 6. 3 Mirror Holder

In operation the incident laser beam must be directed alternately onto the specimen and the

comparison standard. In order to avoid the necessity of realigning the mirror each time it is moved,

a new mirror holder was designed and built. The mount rotates freely about an axis which is the

intersection of the front surface of the mirror and the principal plane of the sphere. Adjustable stops

are provided to position the mirror so that the incident beam is centered alternately on the specimen

and comparison standard as the mirror contacts the two stops. The mirror is held in contact with

the stop by means of magnets attached to the holder, which are attracted by the steel stops. The

position of the holder about a second axis of rotation, perpendicular to the first and also in the front

surface of the mirror, can be adjusted and firmly fixed by means of adjustable screws. These two

adjustments permit the incident beam to be centered accurately on the specimen and comparison

standard, and changed quickly from one to the other, while assuring that the beam will always return

to the same preset position when the holder is in contact with one of the stops.

4. Measurements Above 2500°K (4000°F)

4. 1 Background

The overall objectives of this program were to make a critical analysis of all methods of

measuring emittance at temperatures in the range of 2 500 to 3600°K or methods which can be extended to this temperature range. The precision, accuracy, limitations and improvements to obtain

higher accuracy were to be evaluated, along with definitions of future programs, resources, and

48
time requirements for techniques which warrant further development in this temperature range.

4. 2 Literature Survey

As a part of the study of the feasability of making direct measurements of emittance of specimens in the 2500 - 3600°K(4000 - 6000°F) temperature range, a literature survey was made in order

to critically evaluate work that has been done in this temperature range. For convenience, the procedures will be classified by the method of heating the specimen.

4. 2. 1 Induction Heating

Clayton Til] used a rotating specimen inside an induction-heated furnace to measure emittance

and transmittance of materials at high temperatures. Although the method is reported to be suitable

for use at temperatures up to 2790°K (4500°F), no data are reported at temperatures above 2388°K

(2800°F). The reported error in total normal emittance was 13 percent.

Pears [12] heated his specimens by induction, directly if the specimen was an electrical conductor, or by means of a tungsten or tantalum disc, 0.010 in. thick, used as a susceptor. The specimen, in the form of a disc 1/2 to 1 1/2 in. in diameter and 1/16 to 1/4 in. thick, rested on the susceptor disc. The total radiance of the hot specimen was compared to that of a blackbody cavity at the

same temperature. Temperature was measured with a thermocouple at the lower temperatures, and

with an optical pyrometer at temperatures above 1480°K(2500°F). The true temperature was computed from the brightness temperature measured with an optical pyrometer and the total radiance

measured with a radiometer, on the basis of the graybody assumption. Errors are reported to be less

than 12 percent.

Riethof and DeSantis [13] used the deep-cavity technique to measure the emittance of conducting

specimens that were heated by induction. A specimen in the form of a cylinder 1/2 in. long and 1/4 in.

in diameter, with a cavity . 020 in. in diameter and 0. 080 in. deep in the center of one face, was

heated in vacuum or controlled atmosphere by induction, and the radiance from the surface of the

specimen was compared to that of the cavity to obtain the emittance. Measurements were made of

both total emittance and spectral emittance from 0. 4 to 4. microns, at temperatures up to 3400°K

(5660 F). No estimate of the accuracy was given.

49
4. 2. 2 Arc-Image Furnace Heating

An arc-image furnace consists of two ellipsoidal mirrors mounted on a common axis facing each

other so that their second focal points coincide. An image of a carbon arc, located at the first focal

point of one mirror, is formed on the specimen at the first focal point of the second mirror. The

specimen is heated entirely by radiation.

Comstock [14] used an arc-image furnace to heat and irradiate a specimen for measurement of

reflectance at high temperature. The incident flux, and the reflected plus the emitted flux were

sampled by an ingenious light pipe that traversed the image of the source or specimen, formed at the

common second focal point of both mirrors. A chopper, between the source and the scanning light

pipe, interrupted the incident flux periodically so that the emitted flux could be separately evaluated.

Measurements were made at temperatures up to about 4000°K (6700°F). Reproducibility was reported as ± 4 percent, but no estimate of accuracy was made.

Null and Lozier [15] used an arc-image furnace to measure the reflectance of the hot crater

face of the positive electrode of an electric carbon arc, at 3800°K (6440°F). In this case a hole in

the ellipsoidal mirror permitted the specimen to be viewed at an angle of 45° to the normal. Rotating

choppers facilitated measurement of reflected and emitted flux separately. The precision of measurement is reported as ± 0. 004 in reflectance, but no estimate of accuracy is reported.

Wilson [16] used equipment similar to that described above and used by Comstock [14], to

measure the emittance and reflectance of ablation chars, carbon, graphite and zirconia to 3700°K.

Errors are discussed, but are not quantitatively evaluated.

4. 2. 3 Solar Furnace Heating

The solar furnace is conceptually very similar to the arc-image furnace discussed above, except that the arc source is replaced by the sun, and a simple parabolic mirror is used to concentrate

the solar energy on the specimen,

Blau, et. al. [17] and Laszlo, Gannon and Sheehan [18] used solar furnaces to measure reflectance of samples at very high temperatures. The basic technique was similar to that used with

the arc-image furnace, except that rotating shutters, rather than a traversing light pipe, were used

to faciUtate measurement of incident, reflected and emitted flux. Blau et. al. reported poor results

50
because of instability of incoming solar flux. Laszlo et. al. apparently solved some of the problems,

but reported no data.

4.2.4 Plasma Torch Heating

Cox [19] heated a disc-shaped specimen about 3/4-in. thick with a plasma torch on the front

surface and a propane torch on the rear surface. The temperature of the specimen was measured at

three depths below the surface, by means of an optical pyrometer sighted on the bottom of cylindrical

cavities extending to the center of the specimen and thermal gradients in the specimen were computed

by standard heat-transfer techniques from the measured temperatures and the observed external temperature profile of the specimen. Data are reported to 4600°F (about 2800°K). Errors are estimated

to be ± 15 to 20 percent.

4. 2. 5 Resistance Heating

Moore, Stetson and Metcalfe [20] measured total hemispherical emittance of ceramic coatings

applied to a tungsten filament that is heated by passing a current through it. The emittance is computed from the observed power input to and temperature of a short segment near the center of the

specimen. Temperature is computed from the observed resistance of the tungsten wire. Data to

2800°K (4800°F) are reported for yttria-stabilized hafnia.

4. 2. 6 Special Techniques

There are several additional heating techniques that can be used to attain very high temperatures

that apparently have not yet been applied to measurements of thermal radiation properties. These include electron beam heating, and dielectric heating. There are serious difficulties to be overcome

before either of these techniques can be used for such measurements.

4. 3 Discussion

None of the methods described in the literature appears to be ideally suited to measurement of

the thermal radiation properties of non-metallic materials at temperatures above 2500°K (4000°F).

For direct emittance measurements, the major problem concerns temperature measurement.

Non-metallic materials in general are poor thermal conductors, and somewhat translucent. Large

thermal gradients will be present in the specimen unless extreme precautions are taken to assure

uniform heating. Even if the specimen is heated uniformly, temperature measurement is a serious

51
problem, because optical and radiation pyrometers are essentially the only methods available for

temperature measurement in this range, and they depend upon a knowledge of the emittance of the

specimen.

For reflectance measurements the required precision of temperature measurement is much less

stringent. The major problem here is to measure accurately the relatively small amount of reflected flux in the presence of the relatively large amount of emitted flux. This can be done in the

case of specimens heated by radiation, as in the arc-image or solar furnace, where the incident

flux is greater than the emitted flux, but the changing temperature of the specimen as the incident

flux is chopped reduces the accuracy of the method. Use of a high-intensity, narrow wavelength

source, such as a laser, is advantageous at the wavelengths where such sources are available.

Studies are continuing In this area to complete the critical analysis of all the ultra-high-temperature emittance measurement techniques.

An ellipsoidal mirror emissometer described below appears capable of yielding results at

least as accurate as the best that can be obtained by other methods reported in the literature.

4.4 The Ellipsoidal Mirror Emissometer

4.4.1 Introduction

Essentially all radiometric procedures for measuring emittance consist of measuring the ratio

of the radiance of a hot specimen to that of a blackbody radiator or other comparison standard of

known emittance that is at the same temperature. Practically all of the serious problems associated

with such measurements at very high temperatures are related to the generation, control and

measurement of the very high temperatures.

As was mentioned in an earlier section of this report, it is even difficult to define the temperature of a specimen that is partially translucent if it is heated under conditions such that there is a

thermal gradient normal to the emitting surface.

The critical requirement for temperature measurement and control is to ensure that the hot

specimen and the comparison standard are at the same temperature within very close limits. The

temperature of a laboratory blackbody furnace can be measured accurately with an optical pyrometer,

but the brightness temperature of the specimen must be corrected for the normal spectral emittance

52
of the specimen in order to convert it to true temperature. The error in the measured total emittance (Ae) due to a difference in the temperatures of the specimen (Tg

; and biackbody (T^) can be expressed as follows:

* 4(Ts " Tb)

Ae = (44) b

The method described in this section is based on the fact that the emittance of a test specimen

depends on the ratio of two quantities, the radiance of the specimen enclosed in a biackbody cavity at

the same temperature as the specimen and the radiance of the specimen enclosed in a biackbody

cavity at a much lower temperature. This ratio of the radiance of the specimen under the two conditions is the emittance of the specimen, if the temperature of the specimen is the same under the

two conditions. The essential condition for the specimen to appear to be in a biackbody enclosure is

that the total reflected and emitted radiant flux leaving the surface be the same as that incident on

the surface. This could be achieved by surrounding the specimen either with walls at its temperature, forming a true biackbody cavity, or with perfectly reflecting walls, which will form a pseudoblackbody cavity. The latter approach is used in the instrument to be described.

4. 4. 2 Description of Equipment

The ellipsoidal mirror emissometer consists of five components: (1) a specimen and suitable

specimen heater; (2) a self-irradiator, or an optical system for returning to the specimen the flux

emitted by it, (3) a chopper or other means of intermittently blocking the emitted flux from returning

to the specimen; (4) an optical system for viewing the specimen both when it is self-irradiated and

when not irradiated; and (5) a suitable detector-araplifier-recorder combination to measure the flux

emitted by the specimen during the periods when it is self-irradiated and not irradiated, and record

the ratio of the two signals.

The detector-amplifier-recorder combination may incorporate a suitable spectrometer, if it is

desired to measure spectral emittance.

One possible system incorporating the above components is shown in Figure 16. The specimen

is mounted at the first focal point of an ellipsoidal mirror, and a mirror chopper is placed at the

second focal point, normal to the axis of the ellipsoid. A cooled biackbody cavity behind the chopper

53
U

0)

+->

<ti £o

<n

/.

•r-l £wuo

13O

0!) W£

bfi 0]£ou

54
absorbs the flux from the specimen when the chopper blade Is out of the path. When the mirror

chopper is in the path, nearly all of the flux emitted by the specimen is returned to it. The surface

of the specimen is viewed through a small hole in the ellipsoidal mirror. The detector amplifier

system measures the ratio of the radiances of the specimen when the chopper is in and out of the

optical path. If there were no losses in the system, this ratio would be the normal emittance of the

specimen. It is possible to correct for the losses, and thus to compute the emittance from the

measured ratio.

During the period when the mirror chopper is in the optical path, most of the radiant flux leaving

the specimen is returned to it, and there is very little net heat loss from the emitting surface, hence

the thermal gradients normal to the emitting surface are much smaller than they would be for a

freely emitting specimen at the same temperature.

4. 4. 3 Other Optical Systems

Several alternative optical systems could be used to return the emitted flux to the specimen.

These include various combinations of paraboloidal, ellipsoidal, spherical and plane mirrors. The

essential features are (1) that essentially all of the radiant flux emitted by the specimen be returned to

it during a portion of a cycle, (2) that essentially none of the flux emitted by the specimen be returned

to it during a portion of a cycle, and (3) that means be provided, such as a mirror or other type of

chopper, to alternate between conditions (1) and (2) at a rate such that the specimen remains essentially in thermal equilibrium during an entire cycle. A chopping rate of 11 or 13 cycles per second

is considered adequate to accomplish this in most cases.

It is desirable to have the fraction of the cycle during which the emitted flux is absorbed small

compared to the fraction when it Is reflected back to the specimen, in order to minimize the heat

transfer from the specimen, and thus the thermal gradient normal to the surface.

For operation in vacuum or an inert atmosphere, it will be necessary to enclose the specimen

and susceptor in a gas-tight enclosure. From several standpoints, the errors will be less if the

ellipsoidal mirror and chopper are also in the enclosure, in order to avoid reflection and absorption

by the enclosure.

55
4. 4. 4 Mathematical Analysis

Not all of the flux leaving the surface of the specimen will be returned to it by the optical system.

The major losses are:

1) Absorption by the mirror.

2) Loss due to shading by the specimen heater and support.

3) Loss out the viewing aperture in the ellipsoidal mirror.

4) Loss due to flux scattered by the mirrors.

The losses are all related to the specific optical system used, and will be essentially independent of the specimen that is being measured, assuming that specimens are of the same dimensions.

Thus it is possible to include all of these losses in a simple loss factor, which for convenience is

expressed as (l-l), where I is the fraction of the flux leaving the specimen that returns to it after one

passage through the system. The emitted spectral radiance of the hot specimen, L. , will be

where e(X) is the spectral emittance of the specimen at wavelength X, and L, , is the spectral radiance of a blackbody radiator at the same temperature and under the same conditions. Of this emitted

radiance, -L, e(\) L, , will return as irradiance after one passage through the system. Of this returning irradiance, a fraction e(X) will be abosrbed, and the fraction [1 - e(\)~\ will be reflected, so

that the reflected flux at this point will be l

x

e(\) ri-e(^)] L. b

„ Of this the fraction l

x will again

return to the surface, so that the irradiance after two passages through the system will be

l-^e(\) [1 -e(X) ] L^ b, and of this a fraction [1-e (X.)] will again be reflected, and so forth. Thus

we get as the total radiance of the specimen, L,\,s,t

H, s,t = e < X)LA,b + ^ ri " e (^ eXLx.b+V^ 1 -e(A)]2 e(X)LX)b +^3 [1 - e(X)]3

e(X) LAb +t4 n-e(X)]4 e(X)L^ b

+ - • (46)

This is an infinite series that reduces to

L. . = -^-b- (47)

A > s ' 1 1 -IJl - e(X)]

56
It can be seen from equation (47) that if l-^ = 1, or in other words, if there is no loss in the

optical system,

e(MLx b

L = — = Lx

. (48)

X, s,t e(X) X > b

which is the basis for the method. If the specimen is a diffuse reflector, the radiances L. . and

L^ s will be sampled by the detector through the small hole in the ellipsoidal mirror.

When the plane mirror at the second focal point of the ellipsoidal mirror is replaced by a complete absorber, none of the emitted flux is returned to the specimen, and the radiance L. of the

specimen then is expressed by equation (45), above.

The measured ratio of the radiances under the two conditions will be

iiB. = h* = i _ i ri _ e(X) ] (49)

L

x,s,t

e{X) H,b x

If we set Lg X /Lg x ^ = Rx

, we can solve for e(X) to get

RX-l+lx

e{X) = (50)

h

The uncertainty in e(X) can be expressed in terms of the uncertainties in R and I, as ^) ._ + _

f_ir

Assuming e(X) = 0. 50, R, = 0. 60, and I. = 0. 80, the error in e(X) due to an uncertainty of 0. 01 in

Rx and 0. 01 in lx will give an uncertainty of e(X) of less than 0. 02, or 4%. Precision of this order

should be attainable with careful work.

The above analysis was based on the use of radiances, L. Radiance is defined as flux, $, per

unit area of surface, A, projected normal to the direction of propagation, and per unit solid angle, id.

Hence radiance, L, is defined as

L = d

2

§ / (cos 6 dAduu) (52)

where e Is the angle between the direction of propagation and the normal to the emitting surface.

57
To make equations (45-49) rigorously correct, each radiance, L, should contain an area factor

A, and a solid angle factor, oi

.

$ = Ja L l cos e dx ^

(53)

All of the equations (45-49) have been simplified by omitting the integral factors.

A more rigorous analysis of the system shows that there will be a contribution to the total

radiance of the specimen from the flux emitted by the specimen and susceptor and reflected directly

back to the specimen by the ellipsoidal mirror. This requires modification of equations (45) and (47).

The diffuse configuration factor from the specimen to its image reflected in the ellipsoidal

mirror is defined as fx and that from the susceptor to the image of the specimen is defined as fj.

Aj is the area of the specimen, and A^ is the area of the susceptor. uu v is the solid angle of flux

received by the detector. The specimen is assumed to be a diffuse emitter and reflector. The susceptor is assumed to be a diffuse emitter, but a specular reflector. The radiance of the susceptor is

defined as 1^ .

When the mirror chopper is out of the path and is not reflecting flux back to the specimen, the

radiance of the specimen will consist of the emitted flux, e(A)L, . plus the flux emitted by the susA, D,

ceptor and reflected directly back to the specimen. Since the susceptor is a specular reflector, flux

from the susceptor and specimen reflected back to the susceptor will be reflected out of the optical

path and lost. Since the specimen is a diffuse reflector, there will be multiple reflection between the

specimen and ellipsoidal mirror. To simplify the derivation, we will consider separately the flux

originating from the specimen and susceptor. The flux reaching the detector that was emitted by the

specimen is L , AjOiy (specimen). After one reflection from the ellipsoidal mirror it is

L

s a^^ < sPec-) = G M hi \ \ ^v + e(\) 1^ ^A. aV fiPiD - e(\)]

and after multiple reflections reduces to e(A)L

b ^A3 uv Lo i As u'v ( sPec. ) = ---• -- (54)

S > X ^ 1 -fa Pi Ll -e(X)]

The flux reaching the detector that was emitted by the susceptor is L Ajov (susceptor). After the

first reflection from the ellipsoidal mirror it is

58
Ls ^u)v (susceptor) = Aj L^p,^ ri-e(X)] + A^ Lu^p" uu v [1 -e(X)]2

and after multiple reflections reduces to

Aui^^Pi "v r i - e(M]

L . A. it, (susceptor) = i^r\

S ' X l-flPl

r l-e(\)]

(,J)

Combining equations (54) and (55)

e(^)Lb x Asa- v + Aj IAJ fsPl Iv Tl - e(X)]

L A, uu v = ' (56)

s ' i-flPl ri- e(x)]

or

e(X)L

b x \ + AuL^f^ri - e(X)] ^ L

S) x = l

: ( 57 > 1 -flPl [l-G(X)]

When the mirror chopper is in the optical path, a fraction, l^, of the flux leaving the specimen,

which will be L . Ayir, will be returned to the specimen, and of this a fraction [1 -e(X)] will be

S A

diffusely reflected into the hemisphere, and a fraction uu v /rr of the reflected flux will reach the detector. Hence after one reflection, we can write for the flux emitted by the specimen

^^^^(spec.) = e(X)Lb>x As

u) v + e(X)Lb ^As

TT u> v /n [f, Pl [1 -e(X)] + lxTl - e(X)]}

and after multiple reflections this reduces to

eMl^x-Asd),

L x *A. uuv(spec) = '- (58)

' ' 1 - (fiPi +*J[l-e(\)]

Similarly, the contribution from the flux emitted by the susceptor will be

L . An, (susceptor) = J^SlAUbSk-J^H^L (59) s,M^ y

1 - (flPl +l

x

) [1 -e(X)]

Combining equations (58) and (59) we get,

e(X)L A= u> v + fslAjAuPifl - e(X)] ,, v

S - X 't^ 1- (*, Pl + ^) n-e(X)] * '

and the measured ratio, E, , becomes

L

s,X

As a>v 1 ~ ( fiPi + V [1 _e ( X ) ] R

^ = "

f:s"\"t^ U)v = "i~-T7i"i "e(MT (61)

59
Solving for e(k) \ - (1 - fi Pi ) (l - Rx ) e(M = —7—"J---/K—;; (62) \ - h Pi (% - 1)

It should be noted in equation (62) that only p 1

, fx and ^ need be evaluated.

4.4.5 Evaluation of System Parameters

Reasonably good values of the parameters px , fx and l, may be computed if the geometry of the

system and the reflectances of the ellipsoidal and plane mirrors are known. For the system under

consideration, the following geometry for the system will be assumed.

Focal length of the ellipsoid = 2c = 17 in.

Latus rectum = S = 6.166 in.

Eccentricity = e = 0. 70103

Semi-major axis = a = 12. 125 in.

Heater coil radius = b^ = 1. 00 in.

Susceptor radius = h, = 0. 50 in.

Viewing port radius = 0. 18 in. , centered on the major axis of the ellipse.

Spectral reflectance of ellipsoidal mirror = px (k) = 0. 97.

Spectral reflectance of plane mirror = p s (X) = 0. 97

In addition, two angles will be defined as follows. 8 is the angle of a beam of radiation from the

normal to the specimen at the first focal point, and a is the angle of a beam of radiation from the

normal to the plane mirror at the second focal point.

The above geometrical characteristics of the system were selected for the analysis because they

can be readily achieved without excessive cost. Improved operation can be achieved by selection of

more favorable geometry, but the above will suffice for this preliminary analysis.

The flux losses, represented by (1 - -C^ ), must include the flux lost out the viewing port, plus

the flux absorbed by the mirrors and that blocked by the specimen heater and supports.

The fraction 6 X of the flux leaving the specimen at the first focal point that is blocked by the

heater, susceptor and specimen, from returning to the specimen from the plane mirror at

the second focal point, is given by

60
= _i p Pi cos (3 cl . = sin2 Pi (63)

P x is defined as the angle to the major axis of the ellipsoid at which a ray leaves the first focal point

of the ellipsoid to just graze the edge of the heater when reflected by the ellipsoidal mirror. The

angle ax is the angle to the major axis of the ellipsoid at the second focal point, and can be computed

as

tanc*! = 1^/20 (64)

3 and a for any ray are related by

sln g = (L-gD siP QT (65)

(1 - e" - 2e cos a)

The hole loss and the flux blocked by the support can be computed in a similar manner. In the

ellipsoid described here, the hole loss is included in the shading loss, but is included to indicate its

magnitude. The various parameters have been computed to be

b 1 = 0.106 = fraction blocked by specimen, susceptor and heater.

6 9 = 0.015 = fraction blocked by supports.

6 3 = 0. 0025 = fraction passing out the viewing port in the ellipsoidal mirror.

6 4 = 0. 007 = fraction returned to specimen directly by ellipsoidal mirror = fx in

equations (54-62).

6 5 = 0.025 = fraction going to the susceptor.

The value of 6 4is low because the viewing port is located in the area of the mirror that would

normally reflect flux back to the specimen. For a mirror in which the viewing port is outside of this

area, the value would be increased by the amount of 5 3

. The view factor fg in equations (54-62) can

be computed from 6 5 and the areas of the susceptor, Ay, and specimen, Aj , as

U = 6^ /A, (66) We can compute JL from the above parameters as

A

l

x = (1-6! -6 2 )p2 Ps (67)

Substituting the given values in equation (67), ^ = (1-0. 106 - 0. 015) 0. 973 = 0. 802. Thus all

of the system parameters required for solution of equation (62) have been obtained. Construction of

61
an ellipsoidal mirror emissometer will be financed on NBS funds.

5. Summary

The shallow-cavity technique for measuring emittance was found to be subject to unacceptably

large errors in measurements made on partially translucent specimens of nonmetallic materials, and

especially on ceramic oxide materials. The translucency error was found to be as large as +60%,

and the error due to thermal gradients in the specimen was found to be on the order of -10%. Two

new techniques that show promise of greatly reducing these errors were developed, and preliminary

tests indicate that the translucency error is largely eliminated. Corrections can be made for the

temperature difference error, if the magnitude of the thermal gradients is known. Considerable

progress was made in developing computer codes to evaluate the thermal gradients from the thermal

conductivity and emittance of the specimen.

The laser-source integrating sphere reflecto meter, as originally designed, was found to give

reflectance data at 0. 6328 micron that was in error by about ± 0. 02 in reflectance. This was as

much as a 20 percent error for specimens of low reflectance. The integrating sphere and associated

equipment has been redesigned to improve accuracy and precision. Final evaluation awaits completion of the modifications.

A survey of the literature on techniques for measuring thermal radiation properties of solids

in the 4000-6000°F (2500° to 3600°K) temperature range indicates that measurements to an accuracy

of better than 10 to 20 percent are very difficult to make. A newly-designed ellipsoidal mirror

emissometer shows promise of giving accuracies at least as good as those available by the best

alternative technique.

62
6. References

(1] Gouffe, A., "Temperature Corrections of Artificial Blackbodies, Taking Multiple Internal Diffusion into Consideration. " NRL Trans. 429, translated from Revue d'Opt. 24_1 - 10 (1945).

[2] Kelly, F. J. and Moore, D. G. , UA Test of Analytical Expressions for the Thermal Emlssivity

of Shallow Cylindrical Cavities. " Applied Optics 4 31 (1965).

[3] Moore, D. G. , "Investigation of Shallow Reference Cavities for High-Temperature Emlttance

Measurements. " Measurement of Thermal Radiation Properties of Solids, J. C. Richmond,

Ed. , NASA SP 31 pp 515-525. (1963).

[4] Clark, H. E. , and Moore, D. G. , "Method and Equipment for Measuring Thermal Emlttance

of Ceramic Oxides from 1200° to 1800°K. " Symposium on Thermal Radiation of Solids,

S. Katzoff, Ed. NASA SP 55, ML-TDR 64-159, pp 241, 257 (1965).

[5] McCracken, D. D. , and Dorn, W. S. , Numerical Analysis and Fortran Programming.

[6] Olson, O. H. , and Morris, J. C, "Determination of Emlssivity and Reflectivity Data on Aircraft Structural Materials," WADC TR 56-222, Part 3, 1959.

[7] Pears, C. D. , "The Thermal Properties of Twenty Six Solid Materials to 5000°F or Their

Destruction Temperatures. " ASD-TR 62-765 Jan. 1963.

[8] Cox, R. C. , "Emlttance of Zirconia to 4600°F. " Chance-Vought Corp. Report 3/14000/2R19

April 1962.

[9] McNicholas, H. J., "Absolute Methods in Reflectometry. " NBS Jour. Res. J. 29-73 (1928).

[10] Taylor, A. H. , "Measurement of Diffuse Reflection Factors and a New Absolute Reflecto meter.

Scientific Papers of NBS 16^421-436 (1920).

[11J Clayton, W. A. , "A 500°F to 4500°F Thermal Radiation Test Facility for Transparent

Materials. " Measurement of Thermal Radiation Properties of Solids, NASA Special Publication SP-31, pp 445-460.

[12] Pears, C. D. , "Some Problems in Emlttance Measurements at the Higher Temperatures and

Surface Characterization. " Measurement of Thermal Radiation Properties of Solids, NASA

Special Publication SP-31, pp 541-551.

63
[13] Rlethof, T. R. , and DeSantis, V. J., "Techniques of Measuring Normal Spectral Emissivity

of Conductive Refractory Compounds at High Temperatures. " Measurement of Thermal Radiation Properties of Solids, NASA Special Publication SP-31, pp 565-583. (1963).

[14] Comstock, D. F. , Jr., "A Radiation Technique for Determining the Emittance of Refractory

Oxides. " Measurement of Thermal Radiation Properties of Solids, NASA SP-31, pp 461-468.

(1963).

[15] Null, M. R. , and Lozier, W. W. , "Measurement of Reflectance and Emittance at High Temperatures with a Carbon Arc - Image Furnace. " Measurement of Thermal Radiation Properties of Solids, NASA SP-31, pp 535-539 (1963).

[16] Wilson, R. G. , "Hemispherical Spectral Emittance of Ablation Chars, Carbon and Zirconla

(to 3700°K). " Symposium on Thermal Radiations of Solids, NASA SP 55, pp 259-275. (1965).

[17] Blau, H. H. , Jr. et. al. , "High Temperature Thermal Radiation Properties of Solid

Materials. " Air Force Report AF CRC TN-60-146, October, 1960.

[18] Lazlo, T. S. , Gannon, R. E. , and Sheehan, P. J., "Emittance Measurements of Solids Above

2000°C. " Symposium on Thermal Radiation of Solids, NASA SP-55, pp 277-286. (1965).

[19] Cox, R. L. , "A Technique for Measuring Thermal Radiation Properties of Translucent

Materials at High Temperature. " Measurement of Thermal Radiation Properties of Solids,

NASA SP 31, pp 469-481. (1963).

[20] Moore, V. S. , Stetson, A. R. and Metcalfe. A. G. , "Emittance Measurements of Refractory

Oxide Coatings up to 2900°K. " Measurements of Thermal Radiation Properties of Solids,

NASA SP 31, pp 527-533. (1963).

[21] Birkhoff, G. and Mac Lane, S. , A Survey of Modern Algebra, p 216.

[22] Walsh, J. W. T. , Proc. Phys. Soc. (London) 32, 59 (1920).

64
7. Appendix I. Computation of Thermal Gradients

7. 1 Interreflection Algebra

There are three basic types of relations that must be expressed. They are in the following

equations.

B(x) = p(x) H(x) (al) = r H(x) -(x) dA(x)

*a •> c (a2)

H(x) = f B(x) K(x x) dA(x ) (a3) ~ • c ~ ~o,~ ~o where B(x) is the flux per unit area leaving an element of area dA(x) located at x, p(x} is the reflectance of the surface at x, H(x) is the flux per unit area incident on the surface at x, Q is the

amount of flux absorbed in a cavity when the incident flux per unit area H(x) falls on each point x, and

[ indicates the integral over the cavity. K(x , x) is the diffuse angle factor or "geometric factor"

between an element of area dA(x Q

) from which flux is diffusely radiating and the element of area

dA(x) which receives the flux.

Each one of the functions H(x), p(x), B(x), and a(x) can be thought of as vectors in an infinite

dimensional vector space, and each can be expressed in terms of some complete set of basis functions

(basis vectors) in that space. Since this functional vector space has an infinite number of dimensions,

an infinite number of basis vectors are required to span the space. Thus, a given function may require an infinite number of component basis functions for its expression. Obviously no one can do a

practical calculation with an infinite number of components for each vector, so it has been customary

to reduce the number of dimensions of the space to some finite number N when an}' practical calculation is done. This is a familiar procedure in phj^sics and engineering. For example, if one wants

to express a complex periodic function in terms of its Fourier components, one, in general, must

use an infinite number of components. If, however, one has some practical calculation to do, it is

necessary to stop with some finite number of components.

An interreflection calculation is very similar. Strictly speaking, equations (a 1), (a2), and (a3)

are statements about each one of the infinite number or mathematical points on a cavity surface : but,

to get practical results, it is customary to express the infinitely fine-grained information in the

function B(x) in terms of a coarse-grained N dimensional vector B, and instead of considering the

65
infinitesimal surface element dA(x), we consider a number of finite surface areas AA arranged in

vector form AA. It is the same with H(x), p(x), a(x), or any function of x.

Hs Bs AAS

(a4)

H(x)- ' = H B(x)-. * = B dA(x)^ = M

HN \ ^N

In tne following we will denote the N-dimensional column vector corresponding to any function

w(x) by the vector symbol w. The component w. is an average value of the function w(x) on the area

AA. ; mathematically it is just

P w(x) dA(x)

1 1

With these kinds of vectors in the vector space, it is desired that the theory be a complete analogue

of the functional theory. Thus equations (al), (a2) and (a3) must be expressible in a vector form. In

order to do this, it will be very helpful if we first consider the set of basis vectors that will be used.

The most convenient set of N basis vectors -j T| . I

1

. ~ is the one which gives us B = I.B.Tj..

Then the basis vectors10have the ordinary representation

7 10 ^

=

'. Up = ; h =

; < a6 >

o o i and the vector B has (for example) the following form.

BiB„ B = • (a7)

The vector space analogy of Equation (al) is

B = p H (a8)

*i \h = h^L^

^^ = i?J

P i H

j 3l 3d (a9)

66
In the above equations a vector valued multiplication operates in the vector space. This type of

vector multiplication must be commutative, distributive and associative, because it must be the analog

of the ordinary multiplication of functions. In addition the multiplication operation must be linear in

the sense that if a and b are any real numbers and if u, v, and w are any vectors,

then u (av + bw) = a (u v) + b (uw) (alO)

and (av + bw) u = a (vu) + b (wu) (all)

A vector space having a vector valued multiplication operation which is linear, associative, and distributive is known as a linear algebra.

A linear multiplication operation in a vector space can be specified by giving the rule for the

multiplication of the set of basis vectors amongst each other! that is

xy = X . x. f). y. 1). = X . x. y. 1). 7] (al2) ~~ i, ] i -l J ~J i,

J i i ~i ~0

and if all the vectors HJL are known then the multiplication operation is completely specified. In the

case which we have under discussion the multiplication table for the basis vectors is quite simple

(Table 1). The rule is h Hj = 6 i A (al3)

where 6 . . is the Kronecker delta

= 1 if and only if i = j *i = if and only if i 4 j. Table 1.

J> J> J33

J> $

V

-IS I?

Tfe T],

Also there exists a vector 1, analgous to the scalar 1 such that for any vector w

1 w = w 1 = w ( al5)

67
Written out longhand the vector Ms

11

1

(al6)

For every vector w such that none of the components of w is equal to zero, there exists a vector w

_1 such that w w

_1 =1. The vector w

_1 can be easily constructed when w is known. If w = II- w. 71..

then w

-1

= E. (l/w.) T).. It is easy to see that if any w. is equal to zero, no finite inverse vector for w exists, since the magnitude of one of the components of w

-1 would be l/O.

One of the more important theorems that one can prove about a linear algebra is that for each

N-dimensional linear algebra there exists an isomorphic set of N x N linear transformations (matrices)

such that the operations of vector addition and multiplication and also multiplication by a scalar

(real number) are preserved under the isomorphism T21]. (An isomorphism is a one-to-one correspondence between two sets such that the structure of certain operations in those sets is preserved).

It is not difficult at all to find the set of matrices which is isomorphic to the set of vectors

under consideration.

The vector equation (in long form) is:

Pi Hx

PS HP

Hi

(al7)

PN

HN

Pi Hx

p 2 HP

PN

hn

product of diagonal matrices.

Pi o Hi .

(^ • H9 •

, ; (al8)

. PN

. *s

pN

HN

It is easy to see that the matrix multiplication in equation (a8) does essentially the same thing with the

i diagonal elements of the matrices as the vector multiplication in equation (a7) does with the i

components of the vectors. One can see intuitively that an isomorphism exists and the proof of this is

68
quite easy also.

It is easy to show that vector addition and scalar multiplication are preserved under this isomorphism. We can denote the isomorphism by t and its inverse t-1 such that

Wl win v, (al9)

W

N

•l

W

N

Wi . . wP

.

and wx

. . U Wjw? T =

_ (320)

. . . wN w

N

If we denote vector by a vector symbol w, then we can denote the diagonal matrix which corresponds to

w by t (w); however, the notation would be somewhat simplified if one simply denoted the operator by

removing the vector symbol, for example

t(w) =w. (a21)

It is a useful property of these diagonal matrices that the operation of one, say w, by ordinary matrix

multiplication on a vector v gives the same result as the multiplication of the original vector w on the

vector y, i. e. for any vectors v, w

(a22) wv = wv = t(wJv

The next operation for which we wish to find an analogue is shown in equation (a2). In this equation two functions H(x) and »(x), (or three functions if we consider dA(x) as a function) are operated on

so as to give a scalar (single number) result Q . a

In vector space language this is denoted by [A, B, C].

Q = [a-, H, AA] (a23 )

This is called a trilinear form because it maps three vectors, a, H, and _AA, into a scalar Q in a

linear fashion, i. e. , linear in the sense that for any scalars Y, X, and vectors w, x, y, and z, [yw + be, y, z] = y[w, y, z] + \[x, y, z]

(324)

[w, Yx + \y, z] = y[w, x, z] + \[w, y, z_]

[w, x, Y7 + ^z] = Y[w, x, z] + X[w, x, z]

69
In addition the trilinear form that is considered here has the special property that it is equivalent to

a bilinear form on two vectors, one of which is the vector product of the other two.

Q = [a, HfiA] (a25) a ~ ~ ~

Also it can be considered as a linear functional of the one triple product vector

Q = [aHM] (a26)

This final operation—the linear functional shown in equation (a26) — is perhaps the most useful and

most easily understood operation. Q is just the sum of all of the components of the vector (orHAA).

The set of all possible linear functionals on a vector space constitutes a vector space, which is

closely related to the original vector space. The vector space of linear functionals is known as the

dual space of the original vector space, and each vector in the original vector space has a conjugate

vector (or linear functional) in the dual space. In quantum mechanics this is symbolized by the

famous bra or ket notation of Dirac. A bra is the symbol < | , and a ket is the symbol | >. A ket

stands for a vector; and if the vector has a name, say a, then the ket is written as |a>. Similarly a bra

stands for a linear functional or a vector in the dual space to the original vector space. The vector

isomorphic to the ket |a> is the bra <a | . Since the scalar field onto which these linear functionals

map the vectors is the field of real numbers, we do not have to worry about the complications that

ordinarily arise in quantum mechanics because a complex scalar field is used. In order to construct

i=N

the basis of the dual space, one constructs the set of linear functionals i < y. | k which are such

that any one of these operating on its conjugate basis vector |T].> gives the value 1 and operating on

any other basis vector gives the value 0. Thus the set of basis vectors j <y. | I in the dual space is

constructed such that

where 6^ is the Kronecker delta function. In terms of these operators then we have

Qa = <1 | orHAA^ = (E.<Y. | ) (E. a H. M. | T) .>)

J J J J J

=

i£i

<*iH

i M.<' y. | T|.-> ,E or. H LA. 6. . = T. a. H. AA

.

In terms of the operators corresponding to the vectors, this equation is

Q = <1 | aH^A | 1>

since a, H and <±A are symmetric (Hermetian) operators

70

(327)

(a28)

(a29)
= <a I H | aA> (a30)

A second way of expressing equation (a2) is completely in terms of the operators a, H and dA

and the trace operation.

Q^ = trace (aHzJl) (a31)

where the trace of a matrix is the sum of its diagonal terms, for example

trace M = trace

M

ll

M

12 "13

M M

21 22 23

Mw M M

31 32 33

M + M + M

ll

W

22 33

(a32)

thus

ffj Hx AA,. o-gHgAAg

= £ a.H.M. ..ii l

(a33)

QA^K

Finally we want to express equation (a3) in vector form. This is not difficult since the operator

Jc K(x x)dA(x J acting on the function B(x J is a linear transformation; that is, it obeys the following

relation:

$c [^ U ) + PB2 (xo )] K(xo ,x) dA(xp ) = a[

c B, U ) K U ,x) dA(xp )

( a34 > + Pj

c B? U) K(^,x) dAU)

For any numbers o?, 3, and functions Bx , Bg.

The function B(x) is of course represented by the vector B and the operation fc B(x) K(x

rf x)

dA(x ) is represented by performing a linear transformation on B. A theorem states that any linear

transformation on a vector in an N-dimensional vector space may be represented by the multiplication of the vector by an N x N matrix. One method of calculating the elements of this matrix is to

consider the case in which each area AA. is uniformly radiating a constant diffuse flux per unit area j=N

B.. Then the cavity surface integral is broken into N integrals over the sub areas -j AA,' . of the

(a3) becomes:

N N H(x) =£ f B%,x) dA(xJ =EB J K^ ,x) dA(xo ) ( a35)

j=l AA, J i=l J M.

cavity. Equation (a3) becomes

N

71
If we evaluate the average flux incident on each AA. we have

J^. H Wd%) (a36)

This gives us

H . __

N WaA K(^o^)dA(^o )dA W

H. = T B. i 1- (a37)

i . . J AA.

If we define the matrix K by

J AA J AA .K(xq ,x)dA(xo )dA(x)

K = (K ) = i i-~- - .__ (a38)

J-J tin., tin .

then we have, from equation (a37)

H = K(M B) (a39)

and since K^ , x) = K(x, Xp ), we have K., = K-j.

There is another useful way to look at this problem. Under the above conditions the total flux

leaving the j-th sub area is AA.B. and the amount of flux reaching the i-th area from the j-th area is

just AA.K. .AA.B.. so the fraction F- j of the flux leaving AA. and arriving at AA. is given by

F. . = AA.K. . AA

.

(a40)

F. . = M.K. . = -rr* F. . i-J J ij AA

± j-x

In matrix form this is (a41) F = KAA

and equation (a3) has the analogue

H = FB (a42)

The net local heat loss q^ at each AA^ on the cavity surface must be known for the thermal

gradient calculation. One has the following equations:

B = eaT4 + PH

< a43)

a = eoT* - eH < a44 >

H = KAA B

< a45)

72
Combining (a43) with (a45), we obtain

B = ecrT4 + pKMB (a46)

This gives

(1 - pKAA)B = G^T4 (a4?)

Physical reasoning tells us that the inverse matrix (1 - pKAA)-1 must exist, so we can write

B = (1 - pK^O-^T4 (a48)

The above equation is the solution for B in terms of T* and geometry. To complete the derivation,

we need an expression for q in terms of B and T

4

. Equation (a43) gives us

H = p-i(B - eoT4

) (a49)

Substituting this result into equation (a44) one obtains

£ = en-T

4 - ep

_1 (B - e^T4

) (a50)

Taking B from equation (a48), one obtains

^ = eoT4 - ep

_1 r(l - pKMr'eoT4 - eoT4

] = {l - ep

_1 [(l - pKM)" 1 - l]}eaT* < a51 > = 3 eoT4

where

3 = {l + ep

_1

[1 - (1 - pKM)" 1 ]}

(a52)

Thus we can obtain the complete transfer solution matrix 3 from a knowledge of K and e. Equivalent

prescriptions for obtaining this matrix have been given numerous times in the literature. In the

present development the roles of matrix multiplication and inversion have been emphasized and

the isomorphism between the vector £ and the matrix operation e(or t (£,)) were described. This may

help to clarify the mathematical situation because any equation involving expressions like those in

equations (al), (a2), or (a3) can be written down quickly in terms of vectors, linear transformations,

linear functionals, or traces.

7. 2 Angle Factors in a Cylindrical Cavity

The angle factors between various areas (AA.) on the surface of a cylindrical cavity are

relatively easy to evaluate. An expression for the fraction F. . of the diffuse flux leaving a given

area, AA., which falls directly on any other area, AA., is needed. The cavity base is divided into a

series of annular rings and the wall is divided into a series of cylindrical rings (Figure al).

The angle factor between any two cylindrical rings of equal width, h, can be expressed as

73
h

-Hhk

^R^

Figure al. Cylindrical Cavity with

Subdivided Walls and Base

DISC

DISC 2

Figure a2. Coaxial Discs of Equal Radii

T

h

v,

s^

K- R -H

Figure a3. Section Through a Cylindrical Cavity

74
F3 (d,R,h) = Fg (d - 0.5h,R,h) - Fs (d + 0.5h, R, h) (a53)

where d is the distance between the centers of the two rings, and R is the radius of the cylinder.

In the above equation, the function F?

(d, R, h) is given by

FF (d,R,h) = ~ FiU.R.h) (a54)

where

F;i(d,R,h) = F

Q

(d - 0.5h,R) - F

Q (d + 0.5h,R) (a55)

and

F (d,R) = [d2 + 2R2 - (d2 + 2R2

) 2 - AR4 ]^ (2R2 )" 1 (a56)

The derivation of the above series of functional relationships proceeds as follows: F (d, R)

gives the fraction of the diffuse flux, leaving a circular disc of radius R, which is incident on a

parallel circular disc of equal radius when the two discs are a distance d apart on the same axis

(Figure a2). This equation is given by Walsh F22] for discs of unequal radii.

Fj (d, R h) is the fraction of the flux leaving disc 1 of radius R and reaching cylindrical ring 2

of radius R and width h whose center is located a distance d above the center of disc 1 (Figure a3).

This fraction can be considered to be the difference between the fraction reaching disc 3 and that

reaching disc 4 (Figure a3), and this is expressed in equation (a55). The fraction F2

(d, R, h) of the

flux leaving ring 2 and falling directly on disc 1 (in Figure a3) is easily calculated from equation (a40).

Finally, the fraction of the flux leaving cylindrical ring 1 and reaching cylindrical ring 2 is given by

F3(d, R, h) in equation (a53) above; F3

(d, R, H) is expressed as the difference between the fraction

leaving ring 1 and arriving at disc 3 and the fraction leaving ring 1 and arriving at disc 4 (Figure a4).

Similar reasoning gives the following series of functional relationships for the fraction

F3 (R, h, d, r) of the flux leaving an annular ring 1 of width h and mean radius r on the base of a cylindrical cavity and falling incident on cylindrical ring 2 of width h located a mean distance d above the

cavity base (Figure a5).

F3 '(R,h,d,r,h) = F-'(R,d - 0.5h,r,h) - Fs '(R,d + 0.5h,r,h) ( a57 >

Fs '(R,h,r,h) = ffil'(M,rt h) (a58) [(r + 0.5h)2 - (r - 0.5h)2

]

75
?1

' (R,d,r,h) = F ' (R,r + 0. 5h,d) - F ' (R, - 0. 5h,d) (a59)

F (d,R,r) = {d2 + R

2 + r

2 - [(d2 + r2 + R

2

)a - ^r2 F} (2R2 T

1

The fraction F4 (R, h, d, r, h) of the flux leaving cylindrical ring 2 of the width h located a mean

distance d above the base of the cylindrical cavity and incident on an annular ring of width h and mean

radius r on the cavity base is given by

[(r + 0.5h)2 - (r - 0.5h)s

] F4 '(R,h,d,r,h) = — i F3 '(R,h,d,r,h) (a61)

2Rh

These functional relationships have been incorporated into a subroutine of the computer program

that will calculate the temperature distribution in the cylindrical specimen. This subroutine is

currently under development.

A.

h/2

T

h/2

T

£ T

h

±

T

h

I \

/Jl. hT

Figure a4. Section Through a Cylindrical Cavity Figure a5. Section Through a Cylindrical Cavity

76
8. Appendix II

Measurements were made of the total normal emittance of several materials by the deep cavity

and center post modifications of the shallow cavity technique after the end of the period covered by

this report. In order to make the best values for these properties available without delay, the data

are included in this appendix.

The emittance of the tungsten used for the susceptor was measured by the deep hole method.

These data are presented in Figure a6. The hole was coated with silicon carbide to make its

emittance nearly one. Since tungsten is a metal, it is not translucent and the temperature is nearly

uniform throughout the sample. The total error in this measurement is estimated to be less than

2%.

The data for alumina, thoria, magnesia and zirccnia obtained by the center post and deep

cavity procedures are compared in figures a7, a8, a9 and alO with data obtained by the shallow

cavity procedure, and by integration of spectral data on similar specimens measured by the rotating

cylinder procedure. Several trends are evident. In general, the shallow cavity procedure gave the

highest values of emittance, and the comparison post method gave the lowest values.

Except for the case of thoria, where the specimens measured by the rotating cylinder method

were appreciably different from those measured by the other procedures, the data obtained by integration of spectral values are only slightly higher than those obtained by the comparison post method.

All of the data have been corrected for the optical scattering error. The shallow cavity data

have not been corrected for either the translucency error or the temperature difference error.

The deep cavity data should be essentially free from the translucency error, but have not been

corrected for the temperature difference error. The comparison post data should be completely free

from the translucency error, and have been corrected for the temperature difference error on the

basis of optical pyrometer temperature measurements.

It can be concluded from the figures that the data obtained by the comparison post technique

are more nearly accurate than those obtained by the shallow cavity or deep cavity procedures.

77
oow

CVJ oo

CMoo

rO

00O

00

00o c

oo -t-> CO ' ' C

*: o H o . o

00

UJ CDo 3 & <

-4-> o tr £

en

UJ

0_5 w£

UJ

u

1- o O 2 o ,—.

00

O

E-"

CO o a o CD r-oo

IDooinOO

ro oo 30NV11IKI3 1VWH0N "IV101

78
The ceramic oxide specimens were obtained from the Laboratory Equipment Corporation,

St. Joseph, Michigan. A chemical analysis was supplied by the vendor and is presented in Table IB.

The porosity of the specimens was alumina 30$,Thoria 13$, Magnesia 30$ and Zirconia 20$ by

volume. The tungsten was supplied by Wah Chang Company, New York, N. Y.

Table IB. Vendor's Chemical Analysis of Specimen

Materials in Percentage by Weight

A1

2 3 Zr0

2 +Hf0

2 Th0r

SiO,

TlOr

Alumina-2

99.+

a/ ThoriaV Magnesia

0. 1

0. 10

99.+

0.05 0.4

CaO 0. 07 trace 0. 1

MgO 0.02 99.2

Fe2°3 0.40

a/ Also, 0. 02$ Na

2 0$.

0. 1

Zirconia

0. 15

94.2

0.60

0.30

4. 50

0.07

0. 10

b/ Also, trace amounts of the oxides of phosphorus, iron, and

the rare earths as well as a minor amount of alkali oxides which was expressed in the analysis as 9. 10 $ of combined Na, K, and

Li.

79
ALUMINA

1 1 r

1 1 1 SHALLOW CAVITY o DEEP HOLE D • COMPARISON POST A INTEGRATED SPECTRAL DATA D ^\ D

FROM ROTATING CYLINOER

—

D

D

-

- -

-

o

o A

A D ~^>*C -

• •

1 1 1 1 ! 1 1300 1400 1500 1600 1700 1800 TEMPERATURECK)

1900 2000

S 3

ol1400

THORIA —

I

Q SHALLOW CAVITY O DEEP HOLE

• COMPARISON POST & INTEGRATED SPECTRAL DATA FROM ROTATING CYLINDER

l

!

! 1600 1700 1800 TEMPERATURE (°K> 1900 2000 2100

Figure a8. Total Normal Emittance of Thoria

Figure a7. Total Normal Emittance of Alumina

MAGNESIA

D SHALLOW CAVITY o DEEP HOLE

• COMPARISON POST A INTEGRATED SPECTRAL OATA FROM ROTATING CYLINOER

1300 1400 1500 1600 1700 1800 1900 2000 TEMPERATURECK)

Figure a9. Total Normal Emittance of Magnesia

< .3 5

ZIRCONIA

D SHALLOW CAVITY O DEEP HOLE

• COMPARISON POST A INTEGRATED SPECTRAL DATA FROM ROTATING CYLINDER

1300 1400 1500 1600 1700 1800 TEMPERATURE (*•<)

1900 2000

Figure alO. Total Normal Emittance of 7irconia

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