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WED STATES MTMENT OF MMERCE

JLICATION tlN

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JFERENCE

NBS TECHNICAL NOTE 595

Effect of

Ceramic Spectral

Emissivity Variations on the

Computed Luminous Emissivity

of the NBS

Standard of Light

U.S.

RTMENT

OF

MMERCE

53

ational

Bureau

of

ndards
NATIONAL BUREAU OF STANDARDS

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TONAL BUREAU OF STANDARD

JT 6 1971 UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

NBS TECHNICAL NOTE 595

ISSUED SEPTEMBER 1971

Nat. Bur. Stand. (U.S.), Tech. Note 595, 58 pages (Sept. 1971)

CODEN: NBTNA

Effect of Ceramic Spectral Emissivity Variations

on the Computed Luminous Emissivity

of the NBS Standard of Light

William B. Fussell

Heat Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

**»&,„ of **

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, U.S. Government Printing Office, Washington, D.C, 20402.

(Order by SD Catalog No. C 13.46:595). Price 65 cents.

Contents

Page

1 . Introduction 1

1.1. Assumptions of the Sanders-NRC Computer Program 3

1.2. Calculations Required to Compute the Effect of

Ceramic Spectral Emissivity Variations with

Wavelength on the Accuracy of the Apparent Luminous

Emissivities Produced by the Sanders-NRC

Computer Program l±

1.3. Simplified Model of the NBS 1970 Design for the

Standard of Light k1.4. Scheme of the Calculation of the Effect of

Ceramic Spectral Emissivity Variations with

Wavelength on the Accuracy of the Apparent

Luminous Emissivities Produced by the SandersNRC Computer Program for the Simplified Model of

the NBS 1970 Design for the Standard of Light 5

1.5. Outline of the Sanders-NRC Computer Program 9

1.6. Summary of Approximations Used in Computing the

Effect of Ceramic Spectral Emissivity Variations

with Wavelength on the Apparent Luminous

Emissivities Produced by the Sanders-NRC Computer

Program for the Simplified Model of the NBS 1970

Design for the Standard of Light 10

1.7. Fundamental Assumption Relating the Apparent

Luminous Emissivities Produced by the Sanders-NRC

iii
Page

Computer Program and the Apparent Luminous

Emissivities Computed from the Modified NearestNeighbor Approximation U

1.8. Terminology 12

2 . Mathematical Analysis 25

2.1. Zonal Spectral Radiation Balance Equations 25

2.2. Approximate Linearization of the Zonal Spectral

Radiation Balance Equations for Small 6T

1

25

2.3. Nearest-Neighbor Approximation to the Zonal

Spectral Radiation Balance Equations 26

2.4. Zero-Order Approximation to the Zonal Spectral

Radiation Balance Equations 27

2.5. Zero-Order Power Balance Equations 28

2.6. Evaluation of the Fractional Error of the ZeroOrder Approximations for 6TI , 6'., and 6* (Compared

with the Nearest-Neighbor Approximations) 29

2.7. Simultaneous Solution of the Spectral Radiation

Balance Equations for the Base and Wall-Zone 1 31

2.8. Modified Nearest-Neighbor Approximation to the

Power Balance Equations for the Base and

Wall-Zone 1 33

2.9. Luminous Deficiencies for the Base and Wall-Zone

1, Computed: a., from the Luminous Average of the

Spectral Deficiencies; b., from the Spectral

Deficiencies at the Crova Wavelength 3^

iv
Page

3

.

Conclusions 36

4. References 40

List of Figures

Figure 1 . Standard of light ; sighttube geometry 42

Figure 2. Simplified flow chart for the approximate method

of calculating the luminous deficiencies of the

base and wall-zone 1 - of the NBS 1970 design for

the standard of light - that accounts for

variations in the spectral emissivity of the

ceramic with wavelength (see fig. 1; see list of

terminology for definitions of symbols) 43

Figure 3. Simplified flow chart for the approximate method

of calculating the luminous deficiencies of the

base and wall-zone 1 - of the NBS 1970 design for

the standard of light - that neglects variations

in the spectral emissivity of the ceramic with

wavelength (see fig. 1; see list of terminology

for definitions of symbols) 45

Figure 4. Simplified flow chart for the Sanders-NRC computer

program for calculating the luminous deficiencies

of the base and wall zones of the standard of

light; this program neglects variations in the

spectral emissivity of the ceramic with

wavelength (see fig. 1; see list of terminology

for definitions of symbols) 47 v
Page

List of Tables

Table 1. Viewfactor matrix for cylindrical sighttube with

conical base; zone length equals one diameter

(see fig. 1) 49

Table 2. Approximate normalized temperature drops (divided

by the aperture-base viewfactor, F . ) for the base

and wall-zone 1 (see fig. l) of the NBS 1970 design

for the standard of light (made of alumina)

,

computed by two different methods: method 1 accounts

for variations in the spectral emissivity of the

ceramic with wavelength; method 2 neglects

variations in the spectral emissivity of the ceramic

with wavelength (see list of terminology for

definitions of symbols). Fractional errors in

method 2 , assuming method 1 to be correct ^9

Table 3. Approximate luminous deficiencies (divided by the

aperture-base viewfactor, F^g) for the base and

wall-zone 1 (see fig. l) of the NBS 1970 design

for the standard of light (made of alumina)

,

computed by two different methods: method 1 accounts

for variations in the spectral emissivity of the

ceramic with wavelength; method 2 neglects

variations in the spectral emissivity of the ceramic

with wavelength (see list of terminology for

definitions of symbols) . Fractional errors in

vi
Page

method 2 , assuming method 1 to be correct 50

Table 4. Zero-order approximations to the normalized

temperature drops and the thermal and luminous

deficiencies for the base and wall-zones 1 and 2

(see fig. 1) of the NBS 1970 design for the standard

of light (made of alumina) . Fractional errors in

the zero-order approximations, assuming the nearestneighbor approximations to be correct (see list of

terminology for definitions of symbols

)

50

VII

Effect of Ceramic Spectral Emissivity Variations on the

Computed Luminous Emissivity of the NBS Standard of Light

William B. Fussell

A simplified model of the National Bureau of Standards (NBS)

1970 design for the standard of light (platinum point blackbody) is introduced. This model is used to calculate the

apparent luminous emissivity of the base of the sighttube of

the standard of light, and of the wall near the base, by two

different methods. The first method includes the effect of

variations with wavelength in the spectral emissivity of the

ceramic composing the sighttube; the second method does not

include the effect of ceramic spectral emissivity variations.

The results of the two methods are compared and their difference (about 6x10-6) iS found to be negligible for the NBS

1970 design when the sighttube is made of alumina, compared

with other uncertainties and the precision of measurement.

A formula is derived for estimating the apparent luminous

emissivity of the standard of light, given: a., the lengthto-diameter ratio of the sighttube; b., the half-angle of the

conical base; c, , the wall thickness of the sighttube; d.,

the thermal conductivity of the sighttube ceramic ; e .

, the

average thermal and luminous emissivities of the sighttube

ceramic.

Keywords: Ceramic 5 light; photometryj spectral emissivity

j

standard.

1. Introduction

The standard of light is defined to be a blackbody at the freezing

point of platinum (this is a secondary reference point in the International Practical Temperature Scale of 1968 [l] ; its value is 2045 K)

;

the luminance of this blackbody is defined to be 60 candelas per square

centimeter. The physical approximation to this blackbody is also known

Figures in square brackets indicate the literature references at the

end of this paper.
as the standard of light.

In connection with setting up the standard of light at the National

Bureau of Standards ( NBS ) , it is important to calculate , as accurately

as possible, the apparent luminous emissivity of the standard, given

its geometry and the material of which its sighttube is composed (see

fig. 1). C. L. Sanders of the National Research Council (NRC; Canada)

has developed a computer program L2j for this purpose that: a., computes

the temperature distribution inside the sighttube of the standard of

light, given the geometry and the thermal conductivity and average thermal emissivity of the ceramic composing the sighttube; b. f then uses the

temperature distribution of a. to compute the corresponding distribution

of the apparent spectral emissivity inside the sighttube at the appropriate Crova wavelength (about 0.578 micrometer L.3D) » given the average

luminous emissivity of the ceramic. The apparent spectral emissivity at

the Crova wavelength of an area inside the sighttube is assumed to be

equal to the apparent luminous emissivity of this area. This is a valid

assumption if the spectral emissivity of the ceramic composing the

sighttube is constant with wavelength; however, if the spectral emissivity of the ceramic is not constant with wavelength (the spectral emissivity of alumina, for example, varies widely with wavelength), then the

apparent spectral emissivity at the Crova wavelength will differ from

the apparent luminous emissivity.

The goal of this paper is to calculate approximately this differSpecialized terms and quantities are defined in the list of terminology

at the end of this introduction.
ence between the apparent spectral emissivity at the Crova wavelength

and the apparent luminous emissivity, for the base and the adjacent wall

of the sighttube of the present design for the National Bureau of Standards (NBS) standard of light, composed of alumina. (This will also be

referred to as the NBS 1970 configuration.)

1.1. Assumptions of the Sanders-NRC Computer Program

a. The ceramic composing the sighttube of the standard of light

emits and reflects diffusely .

b. The portion of the outside wall of the sighttube immersed in

freezing platinum is maintained at the freezing point exactly (that is,

temperature gradients in the platinum ingot are ignored).

c. In the computation of the temperature distribution in the interior of the sighttube, the variation with wavelength in the spectral

emissivity of the ceramic may be approximated by substituting, in the

spectral radiation balance equations for the base and the wall zones

(see fig. 1), the average thermal emissivity of the ceramic for the

spectral emissivity.

d. In the computation of the distribution of the apparent luminous

emissivity in the interior of the sighttube, the variation with wavelength in the spectral emissivity of the ceramic in the visible may be

approximated by substituting, in the spectral radiation balance equations for the base and the wall zones (see fig. l) at the Crova waveIn this paper, the terms "diffuse" and "diffusely" refer to reflection

or emission that follows Lambert's cosine law |_4_J«
length, the average luminous emissivity of the ceramic for the spectral

emissivity.

1.2. Calculations Required to Compute the Effect of Ceramic Spectral Emissivity Variations with Wavelength

on the Accuracy of the Apparent Luminous Emissivities Produced by the Sanders-NRC Computer Program

To calculate approximately the effect of the variation with wavelength in the spectral emissivity of the ceramic, on the accuracy of the

apparent luminous emissivity distribution in the interior of the sighttube produced by the Sanders-NRC computer program, it is necessary to

proceed as follows:

a. Calculate the effect of ceramic spectral emissivity variations

with wavelength on the temperature distribution inside the sighttube

computed by the Sanders-NRC program.

b. Calculate the effect of ceramic spectral emissivity variations

with wavelength, in the visible, on the apparent luminous emissivity

distribution inside the sighttube computed by the Sanders-NRC program.

c. Combine effects a. and b. to calculate the total effect, of ceramic spectral emissivity variations with wavelength, on the apparent

luminous emissivity distribution in the interior of the sighttube computed by the Sanders-NRC program.

1.3. Simplified Model of the NBS 1970

Design for the Standard of Light
To enable these approximate calculations to be done efficiently, a

simplified model of the NBS 1970 design for the standard of light is

adopted. The model is simplified as follows:

a. The length-to-internal diameter ratio of the sighttube is taken

to be 20 (the actual value is 19.0).

b. The half-angle of the conical base of the sighttube is taken as

sin"" (0.2) (11.5 ; the actual value is 9.5 ).

c. The interior of the sighttube is divided into 20 wall zones,

each of length equal to the internal diameter, plus the conical base

(see fig. 1).

d. In the spectral radiation balance equations for the base and the

adjacent wall zone (wall zone 1, in fig. l) , the remaining wall zones

are treated as perfect black radiators. This assumption is justified in

the mathematical analysis section of this paper; it is based on the

rapid falloff of the viewfactors between zones with increasing separation

(see the viewfactor matrix given in table 1). For this reason, the

spectral deficiencies (1 minus the apparent spectral eraissivities ; see

the list of terminology at the end of this introduction) of distant

zones - if the deficiencies are not too large - have little effect on

the spectral radiation balance equation of a zone.

1.^. Scheme of the Calculation of the Effect of Ceramic Spectral Emissivity Variations with Wavelength on the Accuracy of the Apparent Luminous Eraissivities Produced by

the Sanders-NRC Computer Program for the Simplified

Model of the NBS 1970 Design for the Standard of Light
a. The spectral radiation balance equations for the base and the

adjacent wall zone (wall zone l) are set up and linearized by the

assumption (justified in the mathematical analysis section of this

paper) that their normalized temperature drops (that is, the temperature drops in the sighttube wall, divided by the freezing point of platinum) are much less than 1.

b. Two simultaneous linear equations for the spectral deficiencies

of the base and wall zone 1 are then derived from a. by the use of the

modified nearest-neighbor model outlined in 1.3.d, above.

c. The pair of simultaneous linear equations, b., are solved for

the spectral deficiencies of the base and wall zone 1.

d. The power balance equations are set up for the base and wall

zone 1, and also linearized. These equations involve the spectral

deficiencies of the base and wall zone 1, computed inc., which appear

in the integrand of an integral with respect to the wavelength from

(nominally) zero to infinity. The power balance equations also involve

the normalized base and wall-zone 1 temperature drops.

e. The power balance equations, d. , are now considered as simultaneous linear equations in the normalized base and wall-zone 1 temperature drops, and are solved for these quantities; the equations are

simplified by neglecting terms involving the square of G (G is l/4 the

ratio of the hemispherical radiative conductance to the thermal conductance through the wall of the sighttube, and is equal to 0.0619 for the

NBS 1970 design for the standard of light if the sighttube is composed

of alumina) and retaining only terms involving the first power of G

(this approximation is justified in the mathematical analysis section).
The solutions for the normalized base and wall-zone 1 temperature drops

then involve integrals, with respect to the wavelength from (nominally)

zero to infinity, of integrands that are rational functions of the cera- mic spectral eraissivity with coefficients that involve the viewfactors.

f

.

Numerical values for the base and wall-zone 1 temperature drops

are obtained by numerical integration of these integrals; to permit

evaluation of the advantage of the conical base configuration over the

flat base configuration, the normalized base and wall-zone 1 temperature

drops are also calculated for the flat base. The results are given in

table 2 (cols. 2 and 5).

g. The values obtained in f . for the base and wall-zone 1 temperature drops include the effect of ceramic spectral emissivity variations

with wavelength; to approximate the values obtained by the Sanders-NRC

computer program, which do not include the effect of ceramic spectral

emissivity variations with wavelength, the ceramic spectral emissivity,

in the solutions, e., for the normalized base and wall-zone 1 temperature drops, is replaced by the average thermal emissivity of the ceramic. Numerical values are then calculated for the normalized base and

wall-zone 1 temperature drops by evaluating the integrals in their solutions. These results are also given in table 2 (cols, k and 7).

h. The difference between the normalized temperature drops calculated in f., and those calculated in g., is approximately the magnitude

of the error in the corresponding quantities, derived from the base and

wall-zone 1 temperatures calculated by the Sanders-NRC computer program,

due to ceramic spectral emissivity variations with wavelength.

i. Return now to the spectral radiation balance equations, b., in
terras of the spectral deficiencies of the base and wall zone 1. The simultaneous solution of the pair of equations, b. , yields solutions for

the spectral deficiencies of the base and wall zone 1 in terms of:

(1) the spectral emissivity of the ceramic at the wavelength

under consideration;

(2) the normalized base and wall-zone 1 temperature drops;

(3) the viewfactors for the aperture as viewed from the base

and wall zone 1;

(4) the viewfactor for the base as viewed from wall zone 1,

and for wall zone 1 as viewed from the base;

(5) the self-viewfactors for the base and wall zone 1;

(6) the partial derivative of the logarithm of the Planck

blackbody spectral radiance function, at the wavelength under consideration and at the freezing point of platinum, with respect to

the logarithm of the temperature.

The quantities (l)-(6) are known, since the normalized base and

wall-zone 1 temperature drops have been calculated in f. and g., above.

To include the effect of variations in the ceramic spectral emissivity with wavelength, in computing the apparent luminous emissivity of

the base and wall zone 1, insert the normalized base and wall-zone 1

temperature drops - derived in f . , above - which include the effect of

ceramic spectral emissivity variations, into the equations derived in

c, above, for the base and wall -zone 1 spectral deficiencies. Multiply

the resulting equations by:

(1) the spectral luminous efficiency function for photopic

vision L5J;

8
(2) the Planck blackbody spectral radiance function for the

wavelength under consideration and the freezing point of platinum.

The resulting equations are then integrated with respect to the

wavelength over the visible spectrum (0.38 - 0.78 micrometer L6]) , and

divided by a normalizing constant, to obtain the luminous deficiencies

of the base and wall zone 1, with the effect of ceramic spectral emissivity variations with wavelength included. These quantities are given

in table 3 (cols. 2 and 5).

j. To compute the luminous deficiencies of the base and wall zone 1

without taking account of ceramic spectral emissivity variations with

wavelength, return to the solutions, c. , for the spectral deficiencies

of the base and wall zone 1. Insert the normalized temperature drops of

the base and wall zone 1 - derived in g., above - which do not include

the effect of ceramic spectral emissivity variations with wavelength,

into the equations derived in c. for the spectral deficiencies of the

base and wall zone 1. Then evaluate these equations at the Crova wavelength by replacing the ceramic spectral emissivity by the average luminous emissivity of the ceramic. The resulting values for the luminous

deficiencies of the base and wall zone 1 are also shown in table 3

(cols, k and 7)» and are assumed to approximate the corresponding values

obtained by the Sanders-MC computer program, since the computational

techniques are similar.

1.5. Outline of the Sanders-NRC Computer Program

For a sighttube divided into N+l zones (IM wall zones plus the

base), the Sanders-NRC computer program solves simultaneously - by

9
iteration - the system of N+l power balance equations plus N+l apparent

thermal radiation balance equations for:

a. The N+l interior temperatures of the zones.

b. The N+l apparent thermal emissivities of the zones.

The N+l power balance equations are not linearized in the SandersNRC program, and are solved by Newton-Raphson iteration. After completing steps a. and b., above, the Sanders-NRC program uses the N+l interior temperatures to solve simultaneously - again by iteration - the system of N+l apparent luminous radiation balance equations at the Crova

wavelength for:

c. The N+l apparent luminous emissivities of the zones.

The flow charts shown in figures 2, 3» and k, clarify the distinction between the luminous deficiencies calculated in lA.i, those calculated in l.^.j, and those calculated by the Sanders-NRC computer program.

1.6. Summary of Approximations Used in Computing the Effect

of Ceramic Spectral Emissivity Variations with Wavelength on the Apparent Luminous Emissivities Produced

by the Sanders-NRC Computer Program for the Simplified

Model of the NBS 1970 Design for the Standard of Light

a. All zones are treated as perfect black radiators at the freezing point of platinum except the base and wall zone 1 (modified nearest-neighbor approximation)

.

b. The spectral radiation balance equations and the power balance

equations for the base and wall zone 1 are linearized with respect to

10
the spectral deficiencies and the normalized temperature drops (that is,

these quantities are assumed to be small)

.

c. The normalized temperature drop of the base is neglected in computing the normalized temperature drop of wall zone 1, and vice versa

(that is, only the first power of G is retained in the power balance

equations )

.

1.7. Fundamental Assumption Relating the Apparent Luminous

Emissivities Produced by the Sanders-NRC Computer Program and the Apparent Luminous Emissivities Computed from the Modified Nearest-Neighbor Approximation

i The difference between the luminous deficiencies calculated in

1.4. i, and those calculated in 1.4.j, is assumed to be approximately the

magnitude of the error in the luminous deficiencies calculated by the

Sanders-NRC computer program, due to neglect of the ceramic spectral

emissivity variations with wavelength. This assumption is justified by

an approximate calculation of the errors involved in the modified nearest-neighbor approximation, used to compute the luminous deficiencies of

1.4. i and 1.4. j. An evaluation of the most important terras omitted from

the modified nearest-neighbor approximation shows that these terms do

not significantly affect the results obtained with this approximation.

Table 3 shows that the fractional error due to neglect of ceramic

spectral emissivity variations with wavelength is approximately 2.5$ for

the luminous deficiency of the base, for the NBS 1970 configuration made

of alumina; the corresponding fractional error for wall zone 1 is about

4.4$.

11
Since the quantities in table 3 (except for the fractional errors)

are divided by the appropriate viewfactor for the aperture, as viewed

from the base ( see table 1 for the viewfactor matrix) , the actual luminous deficiencies are quite small; it is found that the luminous deficiencies for the base and wall zone 1, computed by method 1.4. i, which

takes ceramic spectral emissivity variations with wavelength into

-4 -4

account, are 2.44x10 and 0.878x10 , respectively; the corresponding

values for method 1.4. j, which simulates the Sanders-iJRC program and

does not take ceramic spectral emissivity variations into account, are -4 -4

2.38x10 and 0.839x10 , respectively.

Thus the estimated absolute errors produced by the Sanders-NRC computer program, due to neglect of ceramic spectral emissivity variations

with wavelength, in the apparent luminous emissivities of the base and

wall zone 1 of the NBS 1970 design for the standard of light made of

alumina, are on the order of 6xl0~ for the base and 4x10 for wall

zone 1. These uncertainties are insignificant in comparison with the

much larger uncertainties due to other causes.

1.8. Terminology

A quantity with the subscript "j" refers to the interior surface

of the j-th zone of the sighttube of the NBS 1970 design for the standard of light (see fig. 1). "j" runs from "B" (for the base of the

sighttube), through "1" (for the first wall zone), to "N" (for the last

wall zone), and finally to "A" (for the aperture of the sighttube).

The reference temperature for apparent emissivities and for deficiencies is the freezing point of platinum, 2045 K [l].

12
All surfaces axe assumed to emit and reflect diffusely.

A single bar over a radiative quantity denotes the "thermal average" of the spectrum of the quantity; the thermal average is defined as

the average value of the spectrum of the quantity, weighted by the

Planck blackbody spectral radiance function at the freezing point of

platinum, over the wavelength range from (nominally) zero to infinity.

A double bar over a radiative quantity denotes the "luminous average" of the spectrum of the quantity; the luminous average is defined as

the average value of the spectrum of the quantity, weighted by the

product of the Planck blackbody spectral radiance function at the freezing point of platinum and the spectral luminous efficiency function for

photopic vision, over the visible spectrum.

The single apostrophe after a quantity denotes a "normalized" quantity; that is, the value of the quantity lies between zero and 1.

The double apostrophe after a quantity denotes a normalized quantity that has been divided by the viewfactor of the aperture of the

sighttube (for the NBS 1970 design) as seen from the base (this viewfactor is defined below). (The one exception to this rule is Jj"(A.,TF

);

see below.)

The subscript "0" preceding a quantity denotes the "zero-order"

approximation to the quantity (this is explained in the mathematical

analysis section of this paper that follows the introduction)

.

The term "deficiency" refers to 1 minus the corresponding "emissivity".

R, (Outside Radius of the Sighttube): This is about 0.3 cm for the

NBS 1970 configuration.

13
R

2 (Inside Radius of the Sighttube): This is about 0.2 cm for the

NBS 1970 configuration.

L (Length of the Cylindrical Portion of the Sighttube): This is

about 7.5^ cm for the MBS 1970 configuration.

t (Effective Thickness of the Sighttube Wall): This is about

0.081 cm for the NBS 1970 configuration; t is defined mathematically as,

t = R^ln 2-i-n(R-, /R^)

.

8 (Half-Angle of the Conical Base of the Sighttube): This is about

9.5 for the NBS 1970 configuration, but the approximate value of

sin~ (0.2) (11 32 ') is used for the calculations in this paper.

L (Length-to-Inside Radius Ratio of the Sighttube): This is about

38.1 for the NBS 1970 configuration, but the approximate value of 40 is

used for the calculations in this paper; L is defined mathematically as

L = L/R

2

.

F .. (Diffuse Viewfactor of the j-th Zone as Viewed from the k-th

Zone): The diffuse viewfactor of the j-th zone as viewed from the k-th

zone is the fraction of the total radiation from the k-th zone (assuming

it radiates diffusely with uniform radiance) intercepted by the j-th

zone. (The viewfactor matrix for the NBS 1970 configuration is shown in

table 1; this matrix is derived from the basic disc-disc viewfactor

formula given in ref. 7.)

F?R (Aperture-Base Viewfactor for the Flat Base Configuration)

This is defined mathematically by the equation,

F

AB

= FLsin6 ' 14
k (Thermal Conductivity of the Sighttube Ceramic): This is estimated to be 0.0635 W cm" K~ at 2045 K for the high-purity, zero-porosity alumina used in the NBS 1970 design for the standard of light L^»9].

A. (V/avelength)

TF (Freezing Point of Platinum): The value of this secondary reference point in the International Practical Temperature Scale of 1968 is

2045 K [l],

T. (Interior Temperature of the j-th Zone): The average interior

temperature of the j-th zone of the sighttube of the NBS 1970 design

for the standard of light (see fig. 1).

T'. (Normalized Interior Temperature of the j-th Zone): This is

defined mathematically as

I- - T ,/TF

.

6T . (Temperature Drop of the j-th Zone): This is defined mathematically as

6T . - T„ - T .

3 F J

6T» (Normalized Temperature Drop of the j-th Zone): This is defined

mathematically as

6T« - 1 - T«.

3 3

J, (X,T ) (Planck Blackbody Spectral Exitance Function at Wavelength

\ and Temperature T

p

): This is defined mathematically as

J^(X,TF

) = c1

X"5(exp(c

2 ATF )-l)"

1

.

15
c (First Radiation Constant): The most recent value is

3.7i4-18xlO"

16

W m

2 [lO].

c2 (Second Radiation Constant) : The International Practical Temperature Scale of 1968 defines c to be 1.^388xlC~2

m K [l],

J^(^,T

F

) (Normalized Planck Blackbody Spectral Exitance Function at

Wavelength k and Temperature T ) : This is defined mathematically as

J{U,T

F

) = JxU,TF

)/oT

F

'.

a (Stefan-Boltzmann Constant) : The most recent value is

5.6696xl0~8 W m"

2^

[lo],

(Note that a, c-,, and c2

, are related by the equation,

a = (n2f /l5)c1

c^;

note also that

/^(X,T

F

)dX = 1.)

N(X,T ) (Exponent of Temperature Dependence of the Planck Blackr body Spectral Exitance Function at Wavelength A and Temperature T?

)

This is defined mathematically as

NU,T

F

) = d[ln Jx (X,TF

)]/d[ln Tf

].

e (Emissivity) : The emissivity of an isothermal radiating surface

is the ratio of the power radiated (but not reflected) by the surface to

a specified receiving surface, to the corresponding power radiated by a

perfectly black surface at the same temperature and having the same configuration.

16
e(X.) (Spectral Emissivity) : The spectral emissivity of an isothermal radiating surface at wavelength k is the ratio of the power radiated

(but not reflected) by the surface in a small wavelength interval to a specified receiving surface, to the corresponding power radiated by a

perfectly black surface at the same temperature and having the same configuration.

r(X) (Spectral Reflectivity) : This is defined mathematically as

r(X) « 1 - e(X).

e (Average Thermal Emissivity) : This is defined mathematically as

e - J^eU)Jj»tt,TF

)dX;

e is found to be 0.223 for the ceramic (alumina [ll]) used in the NBS

1970 design for the standard of light.

r (Average Thermal Reflectivity) : This is defined mathematically as

r = 1 - e;

thus r is 0.777 for the ceramic (alumina) used in the NBS 1970 design

for the standard of light.

Wt (Apparent Emissivity of the j-th Zone) : The apparent emissivity

of a radiating surface (not necessarily isothermal) is the ratio of the

total power radiated and reflected (from the surroundings) by the surface to a specified receiving surface, to the corresponding power radiated by a perfectly black surface having the same configuration and isothermal at the given reference temperature (T_, for the standard of

light).

17
W'.(X) (Apparent Spectral Emissivity of the j-th Zone): The apparent

spectral emissivity of a radiating surface (not necessarily isothermal)

at wavelength X is the ratio of the total power radiated and reflected

(from the surroundings) by the surface in a small wavelength interval to

a specified receiving surface, to the corresponding power radiated by a

perfectly black surface having the same configuration and isothermal at

the given reference temperature (T^, for the standard of light).

6!(X) (Spectral Deficiency of the j-th Zone): The spectral deficiency of a radiating surface (not necessarily isothermal) at wavelength

A. is 1 minus the apparent spectral emissivity of the surface at this

wavelength; that is,

6jU) =1 -WjU).

W! (Apparent Thermal Emissivity of the j-th Zone): This is defined

J mathematically as

W« - /qW«(X)J£(X,Tf)cIX.

o"! (Thermal Deficiency of the j-th Zone): This is defined mathematj

ically as

61 - 1 - W«

J J

N (Average Thermal Temperature Exponent) : This is defined mathematically as

N - /QN(X,T

F)J£U,TF

)dX = k.

e

1 (Temperature-Exponent Weighted Average Thermal Emissivity): This

18
is defined mathematical 1 y as

e« - /oe(X)N(X,TF)J

;i (X,TF)dX/(^);

e* is found to be 0.204 for the ceramic (alumina jJLl]]) used in the NBS

1970 design for the standard of light.

2

e' (Temperature-Exponent Weighted, Average Thermal Square of the

Emissivity) : This is defined mathematically as

ef - /Qe

2

(X)N(X,TF)Jx(X,TF)dX/(^);

e' is found to be 0.230 for the ceramic (alumina [ll]) used in the NBS

1970 design for the standard of light.

G (l/4 the Ratio of the Hemispherical Radiative Conductance to the

Thermal Conductance through the Wall of the Sighttube) : This is defined

mathematical ly as the dimensionless quantity,

G - QT^t/k;

G is found to be 0.0619 for the ceramic (alumina) and the wall thickness

used in the NBS 1970 design for the standard of light.

6!(\) (Zero-Order Spectral Deficiency of the j-th Zone): This is

defined mathematically (see the mathematical analysis section of this

paper) as

Q

6»(X) = F

A [rU) + e(X)N(X,TF

)Ge(l-rF

Jj

)-1

][l - r(X)F

Jj

]

_1

.

E.(X) (Approximate Fractional Error of the Zero-Order Spectral

Deficiency of the j-th Zone, Compared with the Corresponding NearestNeighbor Spectral Deficiency): This is defined mathematically (see the

19
mathematical analysis section of this paper) as VX) -^

tt)(,A.^F^.J +,

A.J*lVl.J)]

x[FAj (r(X) + e(X)N(Ak ,TF)Ge(l-?F

jJ

)"1]-±

.

6! (Zero-Order Thermal Deficiency of the j-th Zone): This is

defined mathematically (see the mathematical analysis section of this

paper) as

n

6« = FA .[? + ^(l-rF. -)"1!! - rF. J"

1

.

E. (Approximate Fractional Error of the Zero-Order Thermal Deficiency of the j-th Zone, Compared with the Corresponding Nearest-Neighbor Thermal Deficiency): This is defined mathematically (see the mathematical analysis section of this paper) as

E"j " [?2(FA,j-lF

j-l.j + F

A, j+l

F

j+l,j>IV? + **Wjj

)'1]"1

-

6T! (Zero-Order Normalized Temperature Drop for the j-th Zone):

This is defined mathematically (see the mathematical analysis section of

this paper) as

n

6T« = FA .Ge(l - rF..)"1

.

E.(_6T') (Approximate Fractional Error of the Zero-Order Normalized

Temperature Drop of the j-th Zone, Compared with the Corresponding Nearest-Neighbor Normalized Temperature Drop) : This is defined mathematically (see the mathematical analysis section of this paper) as

20
F

A, J+i

F

J+i.j(1-fF

j +i, J+i

) "1l1 - *

M

3-

6'! (Thermal Deficiency of the j-th Zone, Divided by the ApertureBase Viewfactor) : This is defined mathematically as

6"3 " S

]

/FAB6T'! (Normalized Temperature Drop for the j-th Zone .Divided by the

iJ Aperture-Base Viewfactor) : This is defined mathematically as

6T'!(e) (Normalized Temperature Drop for the j-th Zone, Divided by

the Aperture-Base Viewfactor, Computed from e): 6T"(e) and 6T"(e) are

computed from the modified nearest-neighbor power balance equations (see

the mathematical analysis section of this paper) with the spectral emissivity of the ceramic, e(A.), replaced by the average thermal emissivity,

e. 6T£(e) and 6T"(e) are assumed to approximate the corresponding temperature drops computed by the Sanders-NRC computer program, since the

computational procedures are similar; these quantities do not take into

account variations in the spectral emissivity of the ceramic with wavelength.

6T'!(e(A.)) (Normalized Temperature Drop for the j-th Zone, Divided

by the Aperture-Base Viewfactor, Computed from e(X)): 6T"(e(X)) and

6T"(e(X)) are computed from the modified nearest-neighbor power balance

equations, obtained by numerical integration with respect to the wavelength from (nominally) zero to infinity of the appropriate rational

21
functions of the ceramic spectral emissivity (see the mathematical analysis section of this paper) ; these quantities do take into account variations in the spectral emissivity of the ceramic with wavelength.

V(X.) (Spectral Luminous Efficiency Function for Photopic Vision)

:

The numerical values of V(X) are given in reference 6.

jr(X,T

F

) (Planck Blackbody Spectral Exitance Function at Wavelength A. and Temperature T-,, Normalized for "Light Watts" per Square

Centimeter) : This is defined mathematically as

J£U,T

F

) = 3.570ifc

i

r5(exp(c

2 ATF

)-l)"

1

.

(Note that Jj*(X,T_) is an exception to the rules of terminology

given at the beginning of this section on terminology; also note that

/"vU)J£U,T

F

)dX = 1,

since

3.5704- = K

m

/60n.)

K (Maximum Luminous Efficacy for Photopic Vision): The currently

accepted value is 673 1m W~ [l2~],

N (Average Luminous Temperature Exponent) : This is defined mathematically as

N = /qNU,Tf)VU)J£U,Tf

)cIA = 12.15.

A.c (Crova Wavelength at Temperature T

F

): This is defined mathematically by the equation,

NUC

,TF

)

«= N;

22
Xp is found to be 0.578 micrometer [J3J.

e (Average Luminous Emissivity) : This is defined mathematically as

e - /~eU)V(A)J»(X,TF

)dX;

e is found to be 0.3^ for the ceramic (alumina [13]) used in the NBS

1970 design for the standard of light.

r (Average Luminous Reflectivity) : This is defined mathematically

as

r = 1 - e.

W! (Apparent Luminous Emissivity of the j-th Zone): This is defined

mathematically as

W! - /~W«(X)V(X)J«(X,TF

)dX.

6*. (Luminous Deficiency of the j-th Zone): This is defined mathematically as

6'. - 1 - W!.

J 3

Q

b\ (Zero-Order Luminous Deficiency of the j-th Zone): This is

defined mathematically (see the mathematical analysis section of this

paper) as

E . (Approximate Fractional Error of the Zero-Order Luminous Deficiency of the j-th Zone, Compared with the Corresponding Nearest-Neighbor Luminous Deficiency): This is defined mathematically (see the mathe23
matical analysis section of this paper) as

6'! (Luminous Deficiency of the j-th Zone, Divided by the Aperture-

%J Base Viewfactor) : This is defined mathematically as

6" «= 6'/F

j j

/fAB'

6 '(6) (Zero-Order Luminous Deficiency of the Base, as a Function

of the Half-Angle, 6, of the Conical Base of the Sighttube): This is

defined mathematically (see the concluding section of this paper) as

6^(9) = F^sineC? + NeGeXe+rsine^Xe + rsinG]"1.

6'!(e) (Luminous Deficiency of the j-th Zone, Divided by the Aperture-Base Viewfactor, Computed from e): 6"(e) and 6"(e) are computed

from the modified nearest-neighbor spectral radiation balance equations

(see the mathematical analysis section of this paper) at the Crova wavelength, with the spectral emissivity of the ceramic, e(X), replaced by

the average luminous emissivity, e. 6"(e) and 6" (e) are assumed to

approximate the corresponding quantities derived from the Sanders-NRC

computer program, since the computational procedures are similar; these

quantities do not take into account variations in the spectral emissivity of the ceramic with wavelength.

6'!(e(X.)) (Luminous Deficiency of the j-th Zone, Divided by the

s= = Aperture-Base Viewfactor, Computed from e(X)): 6£(e(A.)) and 6"(e(X))

are computed from the modified nearest-neighbor spectral radiation balance equations, weighted by the product V(A.)Jj!(X,TF

) , and integrated

24
with respect to the wavelength over the visible spectrum (see the mathematical analysis section of this paper) ; these quantities do take into

account variations in the spectral emissivity of the ceramic with wavelength.

2. Mathematical Analysis

2.1. Zonal Spectral Radiation Balance Equations

The spectral radiation balance equation for the j-th zone is

N

W JU) - eU)[J

x

(X,T )/JxU,TF

)] +

kfeFkJ

W

k

(X)r(X). (D

That is, the apparent spectral emissivity of the j-th zone at wavelength X, referred to temperature T

p

, is the corresponding spectral

emissivity of the j-th zone plus the sum of the corresponding reflected

apparent spectral emissivities of all zones (including the j-th zone

itself)

.

2.2. Approximate Linearization of the Zonal Spectral Radiation Balance Equations for Small 6T 1

.

If the normalized base and wall-zone temperature drops (that is,

the 6T 1.) are very small (table 2 shows the base and wall-zone 1 normalized temperature drops to be of order of magnitude 10" for the NBS 1970

configuration), then J,(X,T.) may be accurately represented by the first

K 3

two terms of its Taylor series expansion about T„. That is,

[JxU.T.)/J

x

(X t T

F

)] - 1 - 6TjNU,T

F

). (2)

25
Now substitute for the left side of eq (2), as it appears in eq

(1), the right side of eq (2). Further substitute

W'(X) =1 - 6'U), W£(X) « 1 - 6£U),

into eq (l). The result is

1 - 6«(X) = e(X)[l - 6T«NU,T

F

)] + J^t^jj. ' 6

k^)>U). (3)

Now introduce the relation,

JUj - x - V w

That is, the surroundings viewed from zone j are completely covered by

the base, the N wall zones, and the aperture.

Equation (3) may be transformed by the use of eq (4) into

6«U) = eU)6T«NU,TF

) + rU)|>

AJ + kfeF

k .j 6£(^J. (5)

In other words, the spectral deficiency of the j-th zone is the sum of:

a. The spectral deficiency due to the temperature drop in the wall

of the j-th zone.

b. The spectral deficiency due to the (diffuse) reflection of the

aperture in the j-th zone.

c. The spectral deficiency due to the reflection of the spectral

deficiencies of all the zones (including the j-th zone) in the j-th

zone.

2.3. Nearest-Neighbor Approximation to the

Zonal Spectral Radiation Balance Equations

26
It is clear from the viewfactor matrix (table 1) that the viewfactors between the zones fall off rapidly as the separation between the

zones increases. Hence, if the zonal spectral deficiencies of the base

and the adjacent wall zones are of the same order of magnitude (this

assumption is justified below in sec. 2.4), it is justifiable to consider

only the nearest-neighbors in approximating eq (5) for the zonal spectral radiation balance equations. Of course, the spectral deficiency of

the aperture cannot be neglected, since it is nearly equal to 1 (that

is, the effective temperature of the environment seen through the aperture is much less than the freezing point of platinum) . Thus a good

approximation to the zonal spectral radiation balance equation, eq (5)»

is

6UX)[l - r(X)F ] = eU)6T«N(X,T

F

) +

rU)[FA . V,06

J-1U) + Fj+l,J

6kU)]- (6)

The coefficient, [1 - r(X)F..]] t is due to interreflections within the

zone.

2,k, Zero-Order Approximation to the Zonal

Spectral Radiation Balance Equations

To estimate the relative order of magnitude of the 6UX) and the

6T1, the zero-order approximations, 6U*-) and

o6Ti' are use;Cul « The

zero-order approximations are defined by the zero-order zonal spectral

radiation balance equation,

6jU)[l - rU)Fjj] - e(X)

Q6TjN(X,TF

) + r(X)F

AJ

. (?)

27
The fractional error in _6 f.(X), compared with 6l(X), is roughly

J J jU) =Cr2U)(FA$ .^F.^ . + F

Af j+1F.

+lf .)]

x[rU)F

AJ

+ eCXj^TjNCX.Tp)]"1

. (8)

2.5. Zero-Order Power Balance Equations

The exact power balance equation for the j-th zone is

6T..k/t = /"eU)J

xU,T..)dX - /Qe(X) kgB

F

kj

W^(A.)Jx

(X,TF)dX. (9)

Equation (9) simply states that the heat flow through the sighttube wall

for a given zone (per unit interior area) is equal to the radiant power

emitted by the zone minus the radiant power received by the zone. In

other words, at equilibrium the net power input to the zone must be

zero.

Next divide eq (9) through by oTF

, linearize, and substitute

W«U) =1-*£U), JUej-l-V

•(G-1 + l^e') = eFA

. + A(X)J«(A.,Tt,) 1 ,ZuF1 , 4 6«(X)dX. (10)

into it to get

The nearest-neighbor approximation to eq (10) is

ai^G-1 + «) = sf

a

. + S

m

eMJia,Tf)lF._1 j'._1M +

The zero-order approximation to eq (10) is

28
^T'CG"1 + kS') = eFAj + /oe(A.)J

J »(X,TF

)F

jj

( 6i(X))dX. (12)

Now substitute into eq (12) the value of &U^) from eq (7) to get

^Tp}-1 + 4e« - F

JJ

/"e

2

(X)N(X,TF

)(l - r(X)F

Jj

)-1

J

J»(X,TF )dX]

- F

A/oeU)(1 " r(X)F

jj

)"1J^(^,T

F

)dA. (13)

For the NBS 1970 configuration (composed of alumina), G"1

is 16.16

and 4e* is 0.816; furthermore, the magnitude of the remaining term in

the coefficient of 6T«. in eq (13) is roughly \J^^T . ./(1-rF . .) J. Thus

for the base (F. . = 0.8) the magnitude of this term is 0.W?, while for

the wall zones (F. . 0.5858) it is 0.227. It is clear, therefore, that

the term G in the coefficient of

Q

6T^ in eq (13) is much larger than

the remaining terms. Hence a good approximation is

6T« - GFAj/"e(\)(l - r(X)F

Jj

)-1

J

;«(X,TF

)dX. (14)

Finally, numerical integration of the integral on the right side of

eq OM) shows that there is less than 8% fractional error (for alumina

and the viewfactors shown in table l) in approximating e(X) and r(X) in

the integrand by e and r, respectively, to obtain

G

aT- -GF^/d-5^). (15)

2.6. Evaluation of the Fractional Error of the ZeroOrder Approximations for 6T 1

. , 6' and 6! (ComJ J J pared with the Nearest-Neighbor Approximations)

Return to eq (7) , the zero-order zonal spectral radiation balance

29
equation, and substitute the value of Q

6T'. given by eq (15) into eq (7);

the result is

Q6«U)[l - r(X)F^] = FAj[rU) + e(X)N(A.,TF

)Ge(l-rF

jj

)-1

]; (16)

eqs (15) and (16) point out the importance of the viewfactor of the

aperture as viewed from zone j , F. . , in determining the magnitudes of

6T! and 6t(X).

The approximate fractional error in Q

6«(A.), compared with the nearest-neighbor approximation to 6!(X), eq (6), can now be evaluated by

substituting into eq (8) for this fractional error the value of

Q

6T*

given by eq (15). Thus one derives

E

j( X) =[r2 (X)/FA .lFAf ._lF .,lf

. + F

Af

. +1F.

+lfj ]

x[rU) + e(X)N(X,TF

)Ge(l-rF

jj

)-1]"1

. (1?)

The thermal average of E.(A.), E., may be roughly computed from eq

(17) as

i. - e

?^

1p&A.^rw.j + V^iW1

X[r + ^G(l-rF J4]-1

; (18)

and the luminous average of E.(X), E., may be similarly computed as

J J

x[r + fed-fF..)"1

]

-1

. (19)

One may also estimate the fractional error in the zero-order normalized temperature drop of the j-th zone,

Q

6T' , compared with the near30
est-neighbor approximation to the normalized temperature drop, by substituting q^-jU) for 6^(1), Q

6«(X) for 6J(X), and &^+1U) for

6t +1(^.), in eq (11) for 6T', and proceeding as in the calculation of

6T'. but retaining the terms involving

Q n_iM and &*

+1(A.). The

result is

F

A. J+lVl. J

( "V^V.j+i

rl]

- (20)

Now insert into eqs (18), (19), and (20), for E., E., and E .( Q

6T'.)

,

respectively, the values given in the list of terminology at the end of

the introduction to this paper, for the NBS 1970 configuration for the

standard of light; also insert the appropriate viewfactors from table 1.

The resulting values for E., E., and E.( 6Tt), are shown in table k.

(Note that the effect of ceramic spectral emissivity variations with

wavelength is not included in the fractional errors given in table k,

which account only for the omission of nearest-neighbor effects.) It is

clear from table 4 that the zero-order approximation is good for the

base, roughly satisfactory for wall-zone 2, and definitely not satisfactory for wall-zone 1.

2.7. Simultaneous Solution of the Spectral Radiation

Balance Equations for the Base and Wall-Zone 1

It is clear from the results of table k that a valid approximation

to the nearest-neighbor spectral radiation balance equation for wallzone 1 is obtained by setting the spectral deficiency of wall-zone 2

31
equal to zero , since the spectral deficiency of wall-zone 2 as viewed

from wall-zone 1 is less than 20$ of the spectral deficiency of the base

as viewed from wall-zone 1. In fact, an approximate calculation of the

fractional error in the spectral radiation balance equation for wallzone 1, due to omitting the spectral deficiency of wall-zone 2, indicates this error is less than 8$ for both the thermal and the luminous

deficiencies. This approximation is denoted the "modified nearestneighbor" approximation.

In the modified nearest-neighbor approximation, the spectral radiation balance equations for the base and wall-zone 1 become,

6£(X)[l - rU)FBB] = e(X)6T^N(X,T

F

) + rU)^ + F^U)], (21)

and

6jU)[l - rU)F

i;L] «= e(\)6Tp(X,T

F

) + r(X)[FA1 + F^U)], (22)

respectively.

The pair of spectral radiation balance equations above may be

solved simultaneously for 6'(X) and 6»(X). The result is

6

B U) = [r(X)(F

11

F

AB

e(X) +FA1F

:iB r(X) +2FAB

FB1) +

N(X,T

F)e(X)((F11eU)+2FB1)6T^ + F^rU^T^j/OU)] (23)

(where the identity, 1 - F^ <= 2FB1 , has been used), and

6' (A) = [r(X)(F

BB

FA1e(X) +F

AB

FB1r(X) +FA1(l-FBB )) +

N(X,TF)e(X)((FBB

e(X)+l-FBB

)6Tj + FB1rU)6T£)]/[D(X)], {2k)

32
where

D(X) = [e (J.)(FBBFn

-FB1F

]B) + e(A.)(FBB+

F

11-2F

BBF11+2FB1F

:1B

) +

((i-F

BB

)(i-Fn) - rmrB)]. (25)

2.8. Modified Nearest-Neighbor Approximation to the Power

Balance Equations for the Base and Wall-Zone 1

By reasoning similar to that used in section 2.7 to justify omitting the contribution of the spectral deficiency of wall-zone 2 to the

spectral radiation balance equation for wall-zone 1, the spectral deficiency of wall-zone 2 can also be omitted from the power balance equation for wall-zone 1. Thus the modified nearest-neighbor approximation

for wall-zone 1 leads to the power balance equations,

bT^G'1 + «•) - eF^ + QeMJ^X^iF^^k) + F-^S-jU) ]dX , (26)

and

bl^G"1 + 4e») - eFA1 + /oe(X)J£(X,TF)i>B1 £(X) + F^ jU)]dX, (27)

for the base and wall-zone 1, respectively.

Furthermore, by reasoning similar to that used in section 2.5 to

2 justify omitting terms of order G from the zero-order power balance

2 equations for the base and wall-zone 1, terms of order G may also be

omitted from the corresponding modified nearest-neighbor power balance

equations. Thus, after substituting the values of &£(*•) and 6jU) given

by eqs (23) and (2*0 into eqs (26) and (27) , one derives the approximate

modified nearest-neighbor power balance equations for the base and wall33
zone 1,

[0.8279-0.W3rU)]

[6T"/G] = e + / neU)rU)J«U,Tj - dX, (28) b K

* [l-1.3858r(X)+0.W3r2(X)]

and

[0.2861-0. 0586rU)]

[6T»/G] - 0.1349e + /ne(X)r(X)J«(X,Tj- - dX, (29) 1 U A * [l-1.3858r(X)-K).W3r2

(X)]

respectively.

Numerical integration of eqs (28) and (29) leads to the values

given in table 2. The values for sinG = 1 refer to the NBS 1970 configuration with a flat base (the actual base is conical, but the flat base

values are included for comparison) . Note that the normalized base and

wall-zone 1 temperature drops have been divided by the viewfactor of the

aperture as viewed from the base, F.g (approximately equal to sinG/L ).

For the conical base (sin6 «= 0.2), F^ is approximately 1/8000; for the

flat base, F^ is approximately l/l600. That is, for the conical base,

6T^(e(X)) - 0.759xl0"

/*G, 6T»(e(X)) - 0.278xlO'J4

'G;

while for the flat base,

6Tj»(eU)) - 1.7*Jxl0"\r, 6Tj(e(X)) = 0.529x10""^}.

2.9. Luminous Deficiencies for the Base and WallZone 1, Computed: a., from the Luminous Average

of the Spectral Deficiencies; b., from the

Spectral Deficiencies at the Crova Wavelength
a. Luminous Deficiencies Computed from the Luminous Average of the Spectral Deficiencies

By definition, the luminous deficiencies of the base and wall-zone

1 are

6£ = /~6'(X)V(X)J»(X,T

F

)dX, (30)

and

6j «= /~6«(X)V(X)J»(X,TF

)dX, (31)

respectively.

Since 6Tg(e(X)), 6T»(e), 6T£(e(X)), and 6T"(e), have now been calculated, eqs (23) and (2^) for &'(X) and 6'(X) are therefore completely

determined as functions of wavelength. Hence 6' and 6' may be obtained

from eqs (30) and (31) by numerical integration. Because 6T'(e(X)) and

6T'(e(X)) take account of the spectral variations in e(X), these quantities - rather than 6T'(e) and 6T'(e) - are substituted into eqs (23)

and (2*0 for 6«(X) and 6«(X). 6«(X) and 6'(X) are then substituted into

eqs (30) and (31) to obtain 6' and 6 ' . Since 6' and 6* computed in this

D X a X

manner do take account of the spectral variations in e(X) , they are

ts ex denoted 6»(e(X)) and 6'(e(X)). The results of these calculations are

i> x

shown in table 3.

b. Luminous Deficiencies Computed from the Spectral Deficiencies at the Crova Wavelength

As mentioned in the introduction to this paper (sec. 1.1. c), the

Sanders-NRC computer program uses the average thermal emissivity of the

35
sighttube ceramic, e, to compute the temperatures of the base and the

wall zones. An approximation to this procedure is simply to replace

e(X) by e in the power balance equations for the base and wall-zone 1

(eqs (26) and (27), respectively), and proceed as before to compute

6T^(e) and 6TJ(e). (These quantities are denoted 6T£(e) and 6T"(e) to

indicate that e instead of e(X) has been used in their calculation;

6T"(e) and 6T"(e) do not take account of the spectral variations in

e(X).) Table 2 presents 6Tg(eU)) t 6T»(e) , 6T£(eU)), and 6T»(e).

After computing the temperatures of the base and the wall zones,

the Sanders-NRC program uses these values in conjunction with the average luminous emissivity of the sighttube ceramic, e, to compute the

spectral emissivities of the base and the wall zones at the Crova wavelength, X„. As explained in the introduction, the Sanders-NRC computer

program assumes that the apparent spectral emissivities at the Crova

wavelength are equal to the corresponding apparent luminous emissivities .An approximation to this procedure is to replace e(X) by e, r(A.) by

r, N(X,T

p

) by 3, and 6T£ and 6T^ by 6T^(e) and 6Tj(e), respectively, in

eqs (23) and (2^) for &£M and 6'(X). The resulting expressions for

6'(X) and 6«(X) - when substituted into eqs (30) and (31) - yield values

for 6 • and 6,1 . Since 6 ' and 6* computed in this manner do not take

account of the spectral variations in e(X), they are denoted 6 '(e) and

6 *(e). The results of these calculations are also shown in table 3.

3. Conclusions

36
It is concluded from the results presented in table 3, that the

effect of the variations in the spectral eraissivity of the sighttube

ceramic with wavelength, on the accuracy of the apparent luminous emissivities of the base and the adjacent wall of the sighttube, as computed

by the Sanders-NRC computer program, is negligible for the NBS 1970

design for the standard of light, composed of alumina, compared with the

other uncertainties and the precision of measurement. (An approximate

calculation of the errors involved in the use of the modified nearestneighbor approximation, which was used to derive the results given in

tables 2 and 3t wa-s obtained by evaluating the most important terms

omitted from this approximation. The calculation shows that the errors

produced by the use of the modified nearest-neighbor approximation do

not significantly affect the results of table 2 or table 3.)

Inspection of the spectral emissivity variations with wavelength of

alumina, magnesia, and thoria (given in refs. 11 and 13), shows that

alumina has the greatest variation with wavelength of these 3 ceramics.

This is true for both the visible range and the range containing most of

the radiated energy of a 2045 K blackbody (roughly 0.5 to 12 micrometers

for 99$ of the radiated energy). Thus the differences shown in table 3

between 6^(e(A.)) and &g(e), and between 6»(e(A.)) and 6»(e), for the NBS

1970 configuration made of alumina, would probably be less if the calculations in this paper were carried through for magnesia or thoria.

The results of table 3 also demonstrate that the sighttube with the

conical base produces smaller luminous deficiencies than the corresponding sighttube with a flat base (the cylindrical portions having equal

length-to-diameter ratios) . This is shown by multiplying the values

37
given in table 3 for 6£(e(X)) and 6£(eU)) by the appropriate aperturebase viewfactors (1/8000 for the conical base, l/l600 for the flat base)

to derive: for the conical base,

6£(eU)) » 2.^xl0~\ o£(eU)) = 0.88x10"^;

while for the flat base,

6£(eU)) = 5.50xl0~\ 6J(eU)) = 1.68x10"^.

Thus the luminous deficiencies for the conical base geometry are

roughly l/2 the corresponding values for the flat base. Therefore, the

above luminous deficiencies are not proportional to sin6, since the

value of sin6 is l/5 for the conical base and 1 for the flat base.

The effect of the conical base, in reducing the luminous deficiency

of the base of the sighttube of the MBS 1970 design for the standard of

light, can be estimated from the zero-order approximation. Referring to

eq (16) in section 2.6, and approximating the zero-order value of 6* by

S

B

* FJ-9 + NSSSd-rFgg)-1]/^ - SF

BB ], (32)

one derives

[o^(0)/o6^(n/2)] - [sin6(r + NetreCe+rsinG)"1

)]

X[(e + rsinG)(r + fee)]"1

. (33)

When the values appropriate to the NBS 1970 design made of alumina

(given in the list of terminology at the end of the introduction) are

substituted into eq (33) » it is found that

38
^(e)/ 6^(TT/2) « 0A8;

this is close to the more accurate value (from col. 2 of table 3),

6^(e)/6^(TT/2) - QA5.

It can also be seen from eq (33) that if rsin6 » e and r » NetJe,

then approximately,

Sj(e)/ %(it/2 ) - ijr.

In other words, the effect of a high average luminous reflectivity of

the ceramic composing the sighttube is to eliminate (for sin9 not too

small) the advantage of the conical base which holds for low values of

ceramic luminous reflectivity. For if rsinG « e, but still r » Nitre,

then approximately,

^(e)/ 6^(n/2) = sin9/e\

The basic conclusion of this report is that the error introduced

into the values of the apparent luminous emissivities of the base and

the adjacent wall of the NBS 1970 design for the standard of light, made

of alumina, as computed by the Sanders-NRC computer program, due to

neglect of ceramic spectral emissivity variations with wavelength, is of

the order of 5x10 (see sec. 1.7). This is negligible, compared with

the uncertainties in the computed values produced by uncertainties in

the thermal and radiative properties of alumina, and compared with the

uncertainties which present techniques achieve in photometric intercoraparisons.

39
4. References

£l] The International Practical Temperature Scale of 1968, Metrologia

£, 35 (April 1969).

j_2J Sanders, C. L., Thermodynamic Considerations in Realizing the Primary Standard of Light, Metrologia 2, 119 (Oct. 196?).

L3] Smithsonian Physical Tables, 9th ed., p. 96 (Smithsonian Institution, Washington, D. C, 1964)

.

|V] Walsh, J. W. T., Photometry, 3rd ed., p. 137 (Dover Publications,

Inc., New York, N. Y., 1958).

[5] American National Standard Z7.1 - 1967, p. 5 (Illuminating Engineering Society, New York, N. Y., 1968)

.

[6] Reference 5, p. 34.

[7] Hottel, H. C„ Sarofira, A. F., Radiative Transfer, p. 53 (McGrawHill Book Co., Inc., New York, N. Y., 1967)

.

[8] Powell, R. W. , Ho, C. Y., Liley, P. E., Thermal Conductivity of

Selected Materials, p. 72, National Standard Reference Data

Series, Natl. Bur. Std. (U.S.), 8 (Nov. 1966)

.

[9] Coors Porcelain Co., Catalog 201, p. 16 (Golden, Colorado, 1968)

.

[10] Revised List of Physical Constants, Natl. Bur. Std. (U.S.), Tech.

News Bull. i£, 71 (March 1971).

[ll] Clark, H. E., Moore, D. G., A Rotating Cylinder Method for Measuring Normal Spectral Emittance of Ceramic Oxide Specimens from

1200 to 1600 K, J. Res. Natl. Bur. Std. (U.S.) 2PA, 393 (Sept.-

Oct. 1966).

[l2] Color Temperature, Luminous Efficacy and the International Practical Temperature Scale of 1968, Natl. Bur. Std. (U.S.), Tech.

40
News Bull. *&, 206 (Sept. 1970).

[l3] Liebmann, G., Die Temperaturstrahlung der Ungefarbten Qxyde im

Sichtbaren, Zeitschrift fur Physik 63, if04 (1930).

kl
TOP

w

YA

YA

YA

T

5

APERTURE

m

'A

YA

YA

YA

wall SECTION

ZONES A_A

BASE

Figure 1. Standard of light; sighttube geometry.

42
Figure 2. Simplified flaw chart for the approximate method of calculating the luminous deficiencies of the base and wall-zone 1 - of the NBS 1970 design for the standard of light - that

accounts for variations in the spectral emissivity of the cer- amic with wavelength (see fig. 1; see list of terminology for

definitions of symbols).

1. Data Input

L, 1^, R

2

, 6, k, T

p

, e(X), N(X,TF), J£U,T

p

), JjU.Tj.), V(X)

2. Zone Partition

Subdivide interior of sighttube into N wall zones

(plus the base); compute viewfactor matrix (F ).

3. Spectral Radiation Balance Equations for Base and Wall-Zone 1

a. Set up the spectral radiation balance equations

for the base and wall-zone 1.

b. Make modified nearest-neighbor approximation.

c. Solve spectral radiation balance equations for

base and wall-zone 1 for 6"(X) and 6"(A).

(cont. next page)

^3
(cont. from prec. page)

k. Power Balance Equations for Base and Wall-Zone 1

a. Set up power balance equations for the base and

wall-zone 1.

b. Make modified nearest-neighbor approximation.

2

c. Discard terms of order G .

d. Integrate the power balance equations with respect

to wavelength from zero to infinity (nominally) to

obtain 6T"(e(X)) and 6T"(e(X)).

a J.

5. Luminous Deficiencies of Base and Wall -Zone 1

a. Substitute 6Tg(e(X)) and 6T"(eU)) into the equations for 6"(X) and 6"(X).

b. Integrate 6»(X)V(X)J^(X,T

F

) and 6£U)VU)J£(A,T

F

)

with respect to wavelength over the visible spectrum to obtain o^(eU)) and 6J(e(X)).

44
Figure 3. Simplified flow chart for the approximate method of calculating the luminous deficiencies of the base and wall-zone 1 - of the NBS 1970 design for the standard of light - that

neglects variations in the spectral emissivity of the ceramic

with wavelength (see fig. 1; see list of terminology for definitions of symbols).

1. Data Input

L, R^ R

2

, G, k, T

F

, e, e, N.

2. Zone Partition

Subdivide interior of sighttube into N wall zones

(plus the base); compute viewfactor matrix (F ., ).

3. Spectral Radiation Balance Equations for Base and Wall-Zone 1

a. Set up the spectral radiation balance equations

for the base and wall-zone 1; e(X) is replaced

by §.

b. Make modified nearest-neighbor approximation.

c. Solve the spectral radiation balance equations for

the base and wall-zone 1 for 6"(A.) and 6"(X).

(cont. next page)

^5
(cont. from prec. page)

k. Power Balance Equations for the Base and Wall-Zone 1

a. Set up power balance equations for the base and

wall-zone 1.

b. Make modified nearest-neighbor approximation.

c. Discard terms of order G .

d. Solve the power balance equations for the base

and wall-zone 1 for 6T£(e) and 6T"(e).

5. Luminous Deficiencies of Base and Wall-Zone 1

a. Substitute 6T"(e) and 6T"(e) into the equations

D 1

for 6"(X) and 6"(X); e(X) is replaced by e.

b. Evaluate 6"(X) and 6"(X) at the Crova waveis i

length (X ) to obtain 6»(e) and 6«(e).

46
Figure 4. Simplified flow chart for the Sanders-NRC computer program

for calculating the luminous deficiencies of the base and

wall zones of the standard of light; this program neglects

variations in the spectral emissivity of the ceramic with

wavelength (see fig. 1; see list of terminology for definitions of symbols).

1. Data Input

L, R^, R^, 9, k, Tp,, e, e, A.„,

2. Zone Partition

Subdivide interior of sighttube into N wall zones

(plus the base); compute viewfactor matrix (F., ).

3. Thermal Radiation Balance Equations and Power Balance Equations

a. Set up N+l spectral radiation balance equations for

the base and N wall zones; e(X) is replaced by e.

b. Set up N+l apparent thermal radiation balance equations for the base and the N wall zones by integrating the corresponding spectral radiation balance

equations with respect to wavelength from zero to

infinity (nominally).

c. Set up N+l power balance equations for the base and

the N wall zones.

d. Solve the system of 2N+2 equations (b. plus c.)

simultaneously, by iteration, for the N+l interior

temperatures and the N+l apparent thermal emissivities of the base and the N wall zones.

(cont. next page)

47
(cont. from prec. page)

k. Apparent Luminous Emissivities of Base and Wall Zones

a. Substitute the N+l interior temperatures of the

base and the N wall zones into the corresponding

spectral radiation balance equations; e(X) is replaced by e.

b. Solve the system of N+l equations (a.) simultaneously, by iteration, at the Crova wavelength (\„)

for the N+l apparent luminous emissivities of the

base and the N wall, zones.

46
Table 1. Viewfactor matrix for cylindrical sighttube with conical base; zone length equals one diameter (see fig. 1)

.

Viewed

Zone Base

Viewing Zone

Wall 1 Wall 2 Wall 3 Wall N Aperture

Base

Wall 1

Wall 2

Wall 3

Wall N

Aperture

(L)"2 1 - sine 0.2071 0.0290

* 0.0073 a

(L-l)~3

sin6(0. 828*0 0.5858 0.1781 0.0216 6(L-2)"

Jl' 4(L-1)~3

sine(0.1158) 0.1781 0.5858 0.1781 6(L-4)~4

4(L-3)~3

sin6(0.0294) 0.0216 0.1781 0.5858 6(L-6)~if 4(L-5)~3

4sin9(L-l)-3 6(L-2)"4 6(L-4)~4 6(1-6)"^ 0.5858 0.8284

sine(L)"2 (L-l)~3 (L-3)"3 (L-5)"3 0.2071 0.0000

Hfiewfactors with E (L=L/R£) are approximate; L is equal to 40 for the

simplified model of the NBS 1970 design for the standard of light.

Table 2. Approximate normalized temperature drops (divided by the aperture-base viewfactor, F^b) for the base and wall-zone 1 (see

fig. 1) of the NBS 1970 design for the standard of light (made

of alumina) , computed by two different methods : method 1

accounts for variations in the spectral emissivity of the

ceramic with wavelength; method 2 neglects variations in the

spectral emissivity of the ceramic with wavelength (see list

of terminology for definitions of symbols). Fractional errors in method 2, assuming method 1 to be correct.

Normalized Base Tempera- ture Drop . Divided by F&q

Normalized Wall-Zone 1 Temper- ature Drop . Divided by F^g

( Fractional ( Fractional

sine (Method l) Error in (Method 2) (Method l) Error in (Method 2)

Method 2) Method 2)

6T£(e(X)) 6T»(e) 6T»(e(X)) 6T»(e)

0.2 0.6073G 0.122 0.6815G

1.0 0.2783G 0.022 0.2844G

0.2225G 0.146 0.2549G

0.0846G 0.123 0.0950G

4-9
Table 3. Approximate luminous deficiencies (divided by the aperturebase viewfactor, F^g) for the base and wall-zone 1 (see fig.

1) of the NBS 1970 design for the standard of light (made of

alumina) , computed by two different methods : method 1 accounts

for variations in the spectral emissivity of the ceramic with

wavelength; method 2 neglects variations in the spectral emissivity of the ceramic with wavelength (see list of terminology

for definitions of symbols). Fractional errors in method 2,

assuming method 1 to be correct.

Base Luminous Deficiency . Divided bv F^g

Wall-Zone 1 Luminous De- ficiency . Divided by F^g

(Fractional (Fractional

sin9 (Method 1) Error in (Method 2) (Method l) Error in (Method 2)

Method 2) Method 2)

«g(e00) 6g(S) 6«(e(X))

6J(5)

0.2 1.953 0.025 1.905 0.702 0.044 0.671

1.0 0.880 0.010 0.872 0.269 0.028 0.261

Table 4. Zero-order approximations to the normalized temperature drops

and the thermal and luminous deficiencies for the base and

wall-zones 1 and 2 (see fig. l) of the NBS 1970 design for

the standard of light (made of alumina) . Fractional errors in

the zero-order approximations, assuming the nearest-neighbor

approximations to be correct (see list of terminology for definitions of symbols )

.

Zero-Order

Normalized Zero-Order

Tempera- Thermal

Zone ture Drop Deficiency

Fractional

Error in Fractional Fractional

Zero-Order Error in Error in

Zero-Order Normalized Zero-Order Zero-Order

Luminous Tempera- Thermal Luminous

Deficiency ture Drop Deficiency Deficiency

W

i

6«

j

6'

j

E.( 6T.) E.

J E

Base a0.046(-4) 2.68(-4) 2.13(-4) 0.014 0.017 0.012

Wall 1 0.004(-4) 0.25(-4) 0.21(-4) 1.60 1.32 0.98

Wall 2 0.005(-4) 0.29(-4) 0.24(-4) 0.28 0.27 0.20

^he symbol "(-4)" indicates

by 10-^.

the preceding number is to be multiplied

i- U. S. GOVERNMENT PRINTING OFFICE : I 971 —Wl -325/61

50
form NBS-114A d-71)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS-TN-595

2. Gov't Accession No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE i Effect of Ceramic Spectral Emissivity

Variations on the Computed Luminous Emissivity of the NBS

Standard of Light

5. Publication Date September 1971

6. Performing Organization Code

7. author(s) William B. Fussell, Physicist }. Performing Organization

10. Project/Task/ Work Unit No. 2211125

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as No. 9.

13. Type of Report & Period Covered not applicable

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) A simplified model of the National Bureau of Standards (NBS) 1970 design for the standard of light (platinum point blackbody) is introduced. This model is used to calculate the apparent luminous emissivity of the base

of the sighttube of the standard of light, and of the wall near the base, by two different methods. The first method includes the effect of variations with wavelength

in the spectral emissivity of the ceramic composing the sighttube; the second method

does not include the effect of ceramic spectral emissivity variations. The results

of the two methods are compared and their difference (about 6x10"°) is found to be

negligible for the NBS 1970 design when the sighttube is made of alumina, compared

with other uncertainties and the precision of measurement.

A formula is derived for estimating the apparent luminous emissivity of a standard of light, given: a., the length-to-diameter ratio of the sighttube; b., the halfangle of the conical base; c, the wall thickness of the sighttube; d. , the thermal

conductivity of the sighttube ceramic ; e . , the average thermal and luminous emissivities of the sighttube ceramic.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Ceramic; light; photometry; radiometry; spectral emissivity; standard.

18. AVAILABILITY STATEMENT

[XI UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

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