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National Bureau of Standards

NBS Special Publication 700-1

Industrial Measurement Series

A Primerfor

MassMetrology

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he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The

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Jf Bureau's overall goal is to strengthen and advance the nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a

basis for the nation's physical measurement system, (2) scientific and technological services for industry and

government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety.

The Bureau's technical work is performed by the National Measurement Laboratory, the National

Engineering Laboratory, the Institute for Computer Sciences and Technology, and the Center for Materials

Science.

The National Measurement Laboratory

Provides the national system of physical and chemical measurement;

coordinates the system with measurement systems of other nations and

furnishes essential services leading to accurate and uniform physical and

chemical measurement throughout the Nation's scientific community, in- dustry, and commerce; provides advisory and research services to other Government agencies; conducts physical and chemical research; develops,

produces, and distributes Standard Reference 'Materials; and provides

calibration services. The Laboratory consists of the following centers:

• Basic Standards2

• Radiation Research

• Chemical Physics

• Analytical Chemistry

The National Engineering Laboratory

Provides technology and technical services to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement

capabilities; provides engineering measurement traceability services; develops

test methods and proposes engineering standards and code changes; develops

and proposes new engineering practices; and develops and improves

mechanisms to transfer results of its research to the ultimate user. The

Laboratory consists of the following centers:

Applied Mathematics

Electronics and Electrical Engineering2 Manufacturing Engineering

Building Technology

Fire Research

Chemical Engineering2

The Institute for Computer Sciences and Technology

Conducts research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and economy in Government

operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing

the Federal Information Processing Standards Program, developing Federal ADP standards guidelines, and managing Federal participation in ADP

voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and provides the technical foundation for computer-related policies of the Federal Government. The Institute consists of the following centers:

Programming Science and

Technology

Computer Systems

Engineering

The Center for Materials Science

Conducts research and provides measurements, data, standards, reference

materials, quantitative understanding and other technical information fundamental to the processing, structure, properties and performance of materials; addresses the scientific basis for new advanced materials technologies; plans

research around cross-country scientific themes such as nondestructive

evaluation and phase diagram development; oversees Bureau-wide technical programs in nuclear reactor radiation research and nondestructive evaluation; and broadly disseminates generic technical information resulting from

its programs. The Center consists of the following Divisions:

Inorganic Materials Fracture and Deformation3 Polymers

Metallurgy Reactor Radiation

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Gaithersburg, MD 20899.

2 Some divisions within the center are located at Boulder, CO 80303.

3 Located at Boulder, CO, with some elements at Gaithersburg, MD.
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./VBS Special Publication 700-1 > ^ ^

Industrial Measurement Series

c_

^

A Primerfor

MassMetrology

K.B. Jaeger Measurement Standards Laboratory

Lockheed Missiles and Space Company, Inc. Sunnyvale, CA 94086

and

R.S. Davis Center for Basic Standards National Measurement Laboratory

National Bureau of Standards

Gaithersburg, MD 20899

November 1984

U.S. Department of Commerce

Malcolm Baldrige, Secretary National Bureau of Standards

Ernest Ambler, Director

<
Library of Congress U.S. Government Printing Office Catalog Card Number: 84-601090 Washington: 1984

National Bureau of Standards

Special Publication 700-1

Natl. Bur. Stand. (U.S.), Spec. Publ. 700-1, 85 pages (Nov 1984)

CODEN: XNBSAV

For sale by the Superintendent

of Documents,

U.S. Government Printing Office, Washington, DC 20402
FOREWORD

When the National Bureau of Standards was established more than 80 years ago,

it was given the specific mission of aiding manufacturing and commerce. Today, NBS

remains the only Federal laboratory with this explicit goal of serving U.S. industry

and science. Our mission takes on special significance now as the country is responding to serious challenges to its industry and manufacturing— challenges which call

for government to pool its scientific and technical resources with industry and

universities.

The links between NBS staff members and our industrial colleagues have always

been strong. Publication of this new Industrial Measurement Series, aimed at those

responsible for measurement in industry, represents a strengthening of these ties.

The concept for the series stems from the joint efforts of the National Conference of Standards Laboratories and NBS. Each volume will be prepared jointly by an industrial specialist and a member of the NBS staff. Each volume will be written

within a framework of industrial relevance and need.

This publication, A Primer for Mass Metrology , represents the first of what we anticipate will be a long series of collaborative ventures that will aid both

industry and NBS.

Ernest Ambler, Director

\
I1
ABOUT THE AUTHORS

Klaus Jaeger

Dr. Klaus Jaeger was educated at Syracuse University (B.S., Physics 1965;

Ph.D., High Energy Physics, 1970). He was employed at Argonne National Laboratory

from 1970 to 1980 and Brookhaven National Laboratory from 1980 through 1981. During

these periods Dr. Jaeger was engaged in laboratory research in neutrino and meson interaction physics, bubble chamber experiments, particle beam designs and superconducting magnet technology. In addition to high energy physics his experience includes electronics, computer science and cryogenics.

In January 1982 Dr. Jaeger joined Lockheed Missiles and Space Company, Sunnyvale,

California as a research specialist responsible for upgrading all primary measurement standards required by the organization. Since October 1983 he has been the

supervisor responsible for primary electrical measurement standards. Dr. Jaeger

resides in Saratoga, California.

Richard S. Davis

Dr. Richard Davis was educated at Brown University (B.S., Physics, 1967) and

the University of Maryland (Ph.D., Solid State Physics, 1972). His fields of study

include experimental solid state physics, theoretical and experimental problems in

fluid dynamics, properties of materials at very low temperatures, superfluid properties of liquid helium, electronics, absolute electrical measurements and basic mass metrology.

Dr. Davis joined the National Bureau of Standards in 1972. His laboratory

research programs at NBS have been concerned with the improvement of the measured

value of the Faraday (the fundamental constant of electrochemistry), high voltage

capacitor calibrations, the accurate determination of air density effects in mass measurements and improved methods for mass comparison at the highest levels of

accuracy. Dr. Davis resides in Washington, D.C.

\

v

CONTENTS

Page

Foreword iii

About the Authors v

1. Introduction 1

2. Basic Mass Equations 2

2.1. True Mass and Apparent Mass 8

2.1.1. Example 11

2.1.2. Reference Materials 15

2.1.3. Reporting Apparent Mass 17

2.1.4. Comparing Two Apparent Masses 17

2.1.5. Sources of Error in Buoyancy Corrections 19

2.2. Air Density 21

2.3. Sensitivity Arguments 22

2.3.1. Humidity 23

2.3.2 Temperature 24

2.3.3 Pressure 24

2.3.4. Overall Uncertainty of Air Density 24

2.3.5. Effects of Small Changes in Air Density on the Mass Value .... 25

2.4. Measuring Equipment for Air Density 26

2.4.1. Humidity 26

2.4.2. Temperature 26

2.4.3. Barometric Pressure 26

3. Weighing Methods 28

3.1. Direct Weighing 29

3.2. Direct-Reading Weighing 30

3.3. Substitution Weighing 30

3.3.1. Single Pan Balance 30

3.3.1.1. Single Substitution 30

3.3.1.2. Double-Substitution 35

3.3.2. Two-Pan Equal -Arm Balance 36

3.3.2.1. Single Substitution 36

3.3.2.2. Double-Substitution 36

3.4. Transposition Weighing 37

3.4.1. Single Transposition 37

3.4.2. Double Transposition \ 38

3.5. Weighing on Electronic Balances 39

4. Program Types 41

4.1 . Surveillance 41

4.1.1 Type I 42

4.1.2. Type II 43

4.1.3. Surveillance Limits 49

4.1.4 Identifying Weights Which Have Changed 52

4.2. Calibration 52

4.2.1. Trend Elimination in Direct Reading Balances 53

4.2.2. Designs 54

4.2.3. Statistical Checks 57

5. Conclusion 62

6. References 64

vii
Appendix A ^

Appendix B !!!!!!!!!! Bl

Appendix C ........... ci

Appendix D »!!!!!!!!!! Dl

viii
A Primer for Mass Metrology

K. B. Jaeger

Measurement Standards Laboratory

Lockheed Missiles and Space Company, Inc.

Sunnyvale, CA 94086

and

R. S. Davis

Center for Basic Standards

National Bureau of Standards

Gaithersburg, MD 20899

1 . INTRODUCTION

Much of the mass program at the National Bureau of Standards has been documented

during the last two decades [1-8]. What we attempt here is to explain the relevance

of these publications to mass metrology conducted at laboratories whose standards are calibrated by NBS. We try to begin with basic physics concepts deriving the essential

results with a minimum of rigor. The reader should always consult the appropriate

reference for a more sophisticated treatment.

We have also tried to use a consistent notation. Consequently, our notation may

vary slightly from that found in the references cited above.

Many examples are provided in the belief that understanding of a general case is

much easier if a special case is understood well. Cautionary statements are provided

at places in the text where important practical complications may exist.

In general, we have strived to be understandable rather than scholarly where the

combination was elusive.

CAUTION : This document is incomplete. Many extremely important topics—manufacture, specification, cleaning, handling, and storage of weights being among

them—have been omitted. The first two of these subjects are treated in ref. [14].

Our ultimate goal is to outline the way in which mass measurements can be made

to acceptable metro! ogical accuracies. There are three major concepts which must be

explored but which are somewhat interrelated - what one means by "mass;" how one can use a measuring device to determine the mass of an unknown object in terms of an accepted unit, for example, the kilogram; and how to assign a realistic uncertainty

to the results. It will be useful to introduce some basic equations of physics in

order to make our arguments precise. The first major step is to explore the basic

properties of a mass-measuring device, or balance, and then see how the device can be

put to the service of metrology. Mathematically, this means that the first step is

to derive the basic equations of weighing. The remainder of our efforts modify the

basic equations in order to make them practical. We also show their relevance to mass metrology.

1
2. BASIC MASS EQUATIONS

In order to derive the basic equations of weighing, we need a precise notion of

mass. For the purpose of this discussion we will introduce mass through the familiar

equation:

? = mS

(Force = Mass x Acceleration) (1)

That is, each object possesses a property called "mass" which appears in eq. (1) as the constant of proportionality between a force applied to that object and the resulting acceleration of the object. Note that both force and acceleration are vector

quantities: they have a direction associated with them. One should also note that mass is always a positive number (that is, for example, the acceleration is always in

the same direction as the gravitational force, never opposite to it). These comments

about mass are consistent with the qualitative idea of the mass of an object being a measure of the "amount of substance" in the object.

Unfortunately, our intuitive notions of "mass" are often confused with "weight."

Such confusion is unacceptable for science and metrology. Therefore we will now see how the notion of weight differs both logically and practically from that of mass.

We can take it as an experimental fact that over a small plane area of the

earth's surface (|uch as the space occupied by a metrology laboratory) the acceleration of gravity, g, is essentially constant. Since g is a vector quantity, constant

3 implies that, over a small area of the earth's surface, the direction of § is also

practically constant. That is, the acceleration vectors are all parallel and define

the direction "down." In this approximation, we can replace § by a numerical constant, g.

The gravitational force on an object of mass M is then:

F

1 = Mg . (2)

The weight of an object of mass M is defined as F-, . That is, weight is a force, not

a mass.

Using W = F^ , we have

W = Mg . (3)

From what was said about g, it is clear that weight is not a constant property

of matter, but depends on location. Consider, for example, a body of mass M taken

out into deep space such that the gravitational forces are negligible (they are never

zero). In this case, we can approximate (3) as

W = Mg

which holds since

g 0; (M f 0) .

This definition was adopted for international use by the General Conference for

Weights and Measures (C.G.P.M.) in 1901.

2

1
Hence, the body still has mass M but its weight is zero. It is only because most

people live exclusively on the surface of the earth that weight and mass are often

confused.

Let us now consider the action of a scale or balance - the device which we will use to measure mass. There are a great many schemes in use for constructing scales

and balances. All have their strengths and weaknesses but most are designed to

approximate the workings of the following hypothetical device:

This device is a "black box" which has a pan on which to place an object and an indicator which reads zero when the pan is empty and some other number when an object is placed on the pan (fig. 1). Our ideal scale has the following two desirable properties:

1) The indicator always reads the same amount to any desired precision when

the same object is put on the pan at the same conditions of temperature,

barometric pressure, relative humidity, etc.

2) If any two objects are put on the pan, the indicator reading is the sum of

the readings for each object placed individually.

The first property means that the imprecision of the balance readings is zero. The

second property means that the balance is perfectly linear. Later we will show how

to use real balances which only approximate these two important features.

A final property of our idealized balance, which it shares with actual scales

and balances is that the indicator responds to a force on the pan, not a mass. In

other words, our balance would give different readings on the moon than on earth

(though properties 1) and 2) would be unaffected).

When the balance has reached an equilibrium condition, the sum of all forces

acting on the system must be zero. We then have

'

and since the forces acting on the system are known to act in one direction and its

exact opposite (i.e. up and down) we can once again ignore the vector notation and

wri te \

E F.. = (4)

i=l

1

where n = total number of forces acting on the system.

The forces acting on the object placed on the pan of the balance are:

F-j = Mg gravitational force

F

2

, balance force exerted by the balance via deflection of a beam, stretching of a

" spring, or some other method. Obviously this force has to be opposite to F-|

Since acts opposite to F-j , we adopt the convention

F

2 = -k-e-j-g , (5)

3
Figure 1. An idealized scale. The indicating dial is marked in equally spaced

increments. The number pointed to by the needle can be resolved to any desired

precision.

4
where 6, is the balance reading and k is a positive constant (yet to be evaluated)

which converts balance readings to mass units.

In addition to these main forces, we have to consider buoyant, electrostatic,

and electromagnetic forces on the object. Of these three only the first one plays a significant role in usual metrology and we now look at the effect.

Under the influence of gravity, the force on a body submerged in a fluid is

equal in magnitude but opposite in direction to the weight of the displaced medium.

This upward force is known as the "buoyant force." From the definition we see that

F 3

= -mi

where m = mass of displaced medium. The buoyant force was first quantified by

Archimedes in the third century B.C. We also can write

m = Vp

so that

F3

= -Vpm

g (6)

where

V = volume of the displaced medium

= volume of object

Pm

= density of medium e m/V .

With these three forces, we can now write eq. (4) as

F

i + h

+ F

3

= or

F 1+ F 3=-F

2

.

Substituting from eqs. (2), (5) and (6) yields

Mg - Pm

v-g = ke^

(M - PmV)g = ke^ (7a)

(M -

PfnV) = ke

1

. (7b)

It is very important to realize from eq. (7a) that if the weighing is done in a vacuum chamber then the F

3

term drops out and eq. (7a) reduces to

Mg = ke^ . (7c)

Furthermore, the medium or environment can be any fluid--air, water, oil, etc.

For most applications and for practical purposes, the medium is usually air. Even

though air is a mixture of gases and the density is quite small compared to densities

of objects usually measured, the correction term, F

3

» cannot be ignored by metrologists, as we shall see.

Let us now consider the weighing of two objects of mass Ml and M

2

under the same conditions, i.e. in the same medium. We then have from eq. (7a)

g(M

1 -Pm

v

1

) = ke^ (7d)

g(M

2

-pm

V

2

) = ke

2 g . (7e)

5
The density of an object is its mass devided by its volume. That is,

P! = VV

1 where p-j is the density of object 1. We can, therefore, always replace a quantity

such as

by the identical quantity

V1 - Pn/Pl) •

The choice of representation is completely optional, although the proper choice can often simplify a calculation. For the present, we will continue with the volume

representation.

Subtracting eq. (7e) from 7(d) yields

9(M

1 - p^) - g(M2

- pm

V

2

) = g ke

1 - g ke

2 (7f)

and by setting

e = A6

1 we have

g(M

] - p^) - §(M

2

- Pm

V

2

) = gkA9 . (8)

Cancelling g yields

M

1 - M

2

= kA6 +

Pfn (V

1 - V2

) . (9)

Equation (9) is one of the fundamental equations in mass metrology and is known

as the "Weighing Equation." Its importance will become evident in Section 3. For now we simply point out that, using the best balances available, one can determine

AS to much higher accuracy than either 6, or e 2

. Thus eq. (9) turns out to be more

useful than eq. (7b).

In most cases, the laboratory conditions are such that the medium is air, in

which case

PA

=

pm

= density °f a ^ r

and eq. (9) becomes

M

1 - M2

= kA6 + pA

(V

1 - V2

) (10)

Note: When we refer to "weighing" in this text, we will always mean the process

which determines the product k • 6. It should be clear from the above

paragraphs that finding the mass of an object involves more than just

weighing the object. Also, we have no future need to refer to "weight" as defined by (3). From this point on, the term "weight" will be reserved

solely to designate an object manufactured to have particular nominal mass. For example, a "1-g weight" is an object whose mass is close to 1 g.

6
From eqs. (9) and (10) it is obvious the condition

M, - M

2 = kA6

can exist if and only if

pJV, - v

2

) e .

This can only hold if

1) p = i.e., the measurement is done in vacuum

2) V

1 = V

2

.

The first case is usually not encountered in metrology. The second case can be

approximated quite easily. Consider for example two weights of nearly equal mass,

such that

M

1 « M2

,

made from the same billet of alloy. In this case, the densities are equal so that

P-j = P

2 = P •

Since

M, M9

V

1 = — and V2 = —

p

l p2

we have from (10)

M, - M

2

= kAe + pA

M, - M

2

= kA6 + pA

p A

(M

] - M2

) (1 - —) = kA6

P

l-PA

/p

M-j - M

2

- kA6

because pA

« p and M-| - M

2

is already known to be a small number.

Relations such as that expressed by eq. (1) or (10) are of use to metrology

only if all variables are expressed in a consistent set of units. In what follows, we will assume that we are working in the International System of Units

(also known as the Systeme International des Unitls, or simply, the S.I.).

7
The unit of mass in the S.I. is the kilogram. By definition, one kilogram is

exactly equal to the mass of an object known as the International Prototype Kilogram, or I.P.K. This object is made of a platinum-iridium alloy and is stored at the

International Bureau of Weights and Measures in Sevres, France. All laboratory mass

standards must ultimately be traceable to the I.P.K.

2.1. True Mass and Apparent Mass

So far, the "mass" we have considered is that found in Newton's basic equation

(i.e., eq. (1)). For that reason, this concept of mass is sometimes called "true

mass". It is a quantity intrinsic to the object alone and not to its situation. We

have also seen that, in a vacuum, the force registered by the balance is exactly

proportional to the Newtonian mass. For this reason the "true" mass is also known as the "vacuum" mass. The terms "mass", "true mass" and "vacuum mass" are interchangeable.

The need to apply buoyancy corrections to mass measurements has led to the

adoption of so-called "apparent" masses. As explained below and in Appendix A, use of apparent mass can simplify the job of both the weight manufacturer and the metrologist. The drawback, however, is that one must learn a new concept (that is,

apparent mass).

Of limited scientific significance, the apparent mass approach nevertheless

often simplified calculations for the metrologist at a time when exact computations

were tedious (either done by hand or through nomographs) and difficult to check.

Today, when every metrologist has access, at the very least, to a digital pocketcalculator, the argument for apparent masses is less compelling. Nevertheless, both

for completeness and because commercial weights are still manufactured on the basis

of apparent mass, we cannot avoid this topic.

The fact is that most mass metrology is carried out: (1) in air; (2) in

laboratories near sea level; and (3) at a temperature of ^20 °C.

3 Air at such conditions has a density nearly equal to 1.2 mg/cm . Fluctuations

about the mean density are known rarely to exceed 3 percent. Thus 97 percent of the

buoyancy correction can be "built into" the calibration of a weight so that small

deviations from the correction are all that need be considered.

Let us carry through an example by defining a reference metal of density p R.

the true mass of an object made of reference material R. (This object need not actually exist.)

the density of the reference material

1.2 mg/cm = the reference density of air. (This is very

nearly the density of air at 20 °C, 50% humidity, and 760 mm Hg pressure.

)

20 °C = reference temperature.

Let Mn = R

p R

=

PQ =

tn

=

If one actually were to weigh an object under vacuum conditions to determine its

mass, one would have to be certain that the object weighed was sufficiently stable--

that is, no gasses normally adsorbed on the surface at atmospheric pressure are vacuumed off, the weight itself does not "out-gas," etc. The equality between vacuum

mass and true mass assumes that the object being weighed is stable.

8
We now define the "apparent" mass of an object as follows: The apparent mass

A T

of an object X is equal to the "true" mass Mp of just enough reference material to

produce a balance reading equal to that produced by X if the measurements are done at

temperature tQ in air of density pQ

(see fig. 2).

Stated another way, we first define specific, unique weighing conditions:

3

i. the air density is 1.2 mg/cm

ii. the temperature is 20 °C (it is necessary to specify temperature because

the volume of a weight depends slightly on temperature.)

Next we imagine a mass comparison between an object X and an assembly of known

objects R. The definition specifies that the R objects shall all have a density

equal to p^. The mass of the R objects is then adjusted until

Ae = e R

- e x

= o .

When these conditions are all satisfied, is by definition equal to the apparent

mass of X. We never really have to carry out the experiment because from eq. (9) we

have

where V

R

is given by V

R

= M^/pn . We can rewrite this relation using density notation:

*

M

J

" P V

X

" M

R + P VR

= kAe

M

X

(1 ' p /pX

) ' M

R

(1 " p /p r)

= kA6

A6 =

T Mx n-P /PX

) _ mA M = -My . (11)

0-P /PR>

X

Note that the denominator is simply a constant whose value is the same for any

3 A

weight, X. Also, for p x greater than Ig/cm (the density of water), My will be

within 0.2 percent of Ml. Furthermore, eq. (11) states that the apparent mass can be

obtained by multiplying the true mass by a buoyancy factor.

The usefulness of apparent mass and hence eq. (11), as well as the subtle

distinctions between true and apparent mass are shown in the next five subsections.

We begin with an example.

so that

and with

9
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I-Xo

LU

t-CC

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CO<

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ID

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IDC Q_

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CO<

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DC

CXII >

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03 z z

LU LU X X

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CO<

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CO<

LU

DC<

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Cl< • X

QJ SZ

+Jc•1

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+Je

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s- o

03CL +-> CL •r— n3 4-O

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+-> 4-J

•(

C c

•r— 4-

QJ

a

"O cu

•(

QJ 4-

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4J

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QJ 4- Cu O CO c CO o

•rTM

03 o

+-> C +J

<L) (T3 CO E

QJ ss_ o

Cl 4-

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S-o -a \— 0)

+->

X

03 o E -Q

ai

QJ

u -C

CO 1

CO

c\j CO

03 a> _Q

SCS ct: CD Cl

•i

u_

10
2.1.1. Example

Using the ideal balance described earlier, we wish to measure the mass ofnan

unknown weight, X, in terms of standard weights, S, of known apparent masses M^'s.

During the actual measurement, done in air of temperature t and air density p^, it was found that a summation of standards produced the same reading on the balance as the unknown mass.

Under what conditions can we set

Solution:

Starting with the balance equation for two masses (eq. (10)).

M

1 - M

2 = kA0 + pA

(V

1 - V2

)

we have

M

X

' M

S

= k(6

X

' 6

S

) + PA

(V

X

' V

= pa<

v

x

- V

since the measurement yielded 6^ = 6<..

Finally, we can write this as

M

X

" M

S

=

PA

AV

* (E1)

Similarly, we have for the apparent masses, using eq. (11):

M

x

(1 - p o/p

x'

M

s

(1 Vps>

(1-Pq/Pr) (1-Pq/Pr)

T T MX MS

< Mx" M

s> "p7 p o + 4

P

(1-Pq/Pr)

By setting ~ =V

X0 • - =vso at20 ° C

pxp

s

we simplify to

MX" M

S (l-p /pR

)

v so> (E2)

11
Substituting for mI - mI from (El), we have

MA MA

PA

( ¥V - p (VXQV

S0 } M„ - = . (E3)

x s n-P /PR

)

Volumes have a slight temperature dependence but if the temperature of the

measurement is close to 20 °C (generally true in the laboratory), then

and

so that (E3) reduces to

v

x s vxo

(1 -Pq/P

r ) (1-PQ/PR

)

The term 1/(1-Pq/p^ can be approximated by 1 for the following reason: Via a Taylor expansion

WP R

Now since pq is the density of air and p R

is the density of a reference metal, we have

p— -3

< 10

J

so that

- = ApAV + ApAV (p /pR

) + 0-P /P R

)

and since ApAV is already a small number, a correction of 10 (or 0.1 percent) is

insignificant. Hence only the first term has to be retained. This argument is

important in that it is used time and again in estimating buoyancy corrections.

With these approximations, we finally arrive at the equation for apparent mass differences

Mj| - = ApAV . (E5)

12
We now would like to find the conditions for which

This implies that ApAV ^ 0.

When one encounters such a relation, one must always decide its precise meaning,

In this particular case the meaning is: ApAV is smaller than the acceptable un- certainty of M^.

/\ For example, suppose needs to be known to one part per million. Then

really means that

ApAV ~

ApAV

< 10

-6

(E6)

A T

But we already know that does not differ from by more than a fraction of a percent (see comments after eq. (11)). We can therefore approximate

"I

Pv = ~

by

Pv *

Now (E6) becomes

Ap AV

< 10

-6

p X

V

X

(E7)

The volume ratio AV/V^ can be written as

AV _ ¥VS

V

X

V

X

'

In the case of different alloys of stainless steel (SS), VY will be close to Vc , so that

A b

AV

< 10

-1

13
We now have

Ap_

pX

x ltr

1 < io"6

(E7')

The density ratio can be written as

Ap .

PA" P

pXP X

where

and

so that

- 8.0 g/cm for SS

pQ

= 1.2 x 10" 3 g/cm

3

Ap _ (pA-1.2x10-)

P X

" 8.0

The inequality (E7') now reduces to

,-3

pA

-1.2x10

8.0

x 10" 1 < 10" 6

which is satisfied as long as

1.28 x 10" 3 g/cm

3 > p. > 1.12 x 10" 3 g/cm3

.

This indicates that the air density during the measurement can be off by as much as

3 0.08 from 1.20 mg/cm and still satisfy condition (E6).

In contrast, the buoyancy correction for true masses is much more critical.

From eq. (El), we want to satisfy the condition that

PA

AV

< 10

-6

or

AV

< 10

-6

Using the same material as before, we have

so that

P—A -1 -6

x 10

1 < 10

8.0

PA

< 0.08 x

10" 3

g/cm

3

14
Note that in order to keep the desired accuracy, the air density during the measurement can only be ignored if it is very small, i.e. vacuum.

This has been a long example in which several generally useful techniques have

been introduced. It is, therefore, worthwhile to look back and summarize what this

example also teaches about apparent mass. The most important inference we can draw

is that, for true mass comparisons, we almost always need to account for air buoyancy

even though we might be able to use an assumed value for the air density (recall that

air density is usually constant in a given laboratory to about ±3 percent). For

apparent mass comparisons, the air buoyancy correction is much less important and

can, therefore, sometimes be ignored. Apparent mass works the way it does,because

the formalism already has an assumed value for air density (i.e. 1.2 x 10 g/cm^)

built into it. In measurement situations where this assumed value is not accurate

enough, the buoyancy term in eq. (E5) becomes important. This eventuality occurs at

precisely the same point when p« in eq. (El) can no longer be sufficiently well

-3 3 approximated by 1.2 x 10 g/cm .

2.1.2. Reference Materials

So far we have not yet defined a reference material. At present, two different

apparent mass bases are utilized by the NBS. One of them, the older one, is called

''normal brass" and was the logical choice when most laboratory weights were made of

brass.

Normal brass is defined by

P BQ

= 8.4 g/cm

3

at °C

-5

otg = 5.4 x 10 /°C = coefficient of cubical expansion

3

P = 1.2 mg/cm = air density at 20 °C.

These parameters together with the expression for volumetric expansion

V

B

(t) = V

BO[l+aB (t-0 °C)] (12)

and the corresponding density correction

P B (t) = pBO/[l+a

B

(t-0 °C)]x (13)

completely determine the parameters at 20 °C. Note from above that p BQ

is given at

°C and therefore has to be determined for 20 °C via eq. (13) to yield

p R (20) = = 8.390938 g/cm

3

. (13')

[l+5.4xl0"

5x20]

The second apparent mass basis is referred to an arbitrary material with the

following density:

pA = 8.0 g/cm

3

at 20 °C . Once more,

p

= 1 .2 mg/cm3

at 20 °C.

Note that this basis does not require any expansion coefficient since all

parameters are^ defined at 20 °C. It is therefore comparable to the basis defined for

normal brass via eq. (13* ).

15
It has now become common practice to report masses and their uncertainty in the

true mass basis and then quote corrections to nominal values for "normal" brass and

3 for p« = 8.0 g/cm bases. Clearly, the latter apparent mass basis is beginning to

be preferred by many laboratories since most weights used in a set are made of

stainless steel which has a density very close to 8.0 at 20 °C. Thus the correction

term in eq. (11) is small, making the apparent mass close to the true mass (see

Appendix A.

)

We are now in a position to write eq. (11) in the "normal brass" (p b

) and

in the "8.0" (pg) basis. Starting with eq. (11), we have

MA M

T

(1 - p /p

X

} My = My

and substituting for p R

with pg from (13), we obtain

A T (1-P /Pw)

M

X

= M

X

pQ

[l+a

B

(t-0 °C)]

0-

P B0

or

M

'x = Mx ~

Pn

1 - [l+20a

B

] P B0

where the denominator is just a constant equal to 0.9998569886.

Similarly, we use

3

for the 8.0 g/cm basis to yield

PR

=

p6

A

= T (1-Pp/Px)

j X X (1-Pq/po)

where the denominator is once again a constant exactly equal to 0.99985000. It is

interesting to note that apparent masses in the two bases we have examined, eqs.

(14) and (15), have a constant ratio

„A

( M

j BRASS = j t 0000Q699 _

< H\.o

That is, the apparent masses of the same weight in the two bases differ by only

^.0007 percent.

16
Equation (15) indicates a very important point. If the object being weighed

has a density p^ equal to 8.0 g/cm (i.e. p x

= Pq) at 20 °C then

For the brass basis, eq. (14), a similar argument has to be treated with more

caution. In this case, if p^ is equal to the density defined by eq. (13'), then

n

x

1 'x

*

2.1.3. Reporting Apparent Mass

Sometimes the apparent mass correction in a given basis is required. This is

defined as

Correction (mJ) = - N

x (16)

where is the nominal value of the weight.

2.1.4. Comparing Two Apparent Masses

We now derive an equation similar to (10), appropriate for apparent mass differ- ences. Using eq. (11) we have for mass 1

M

A

M

l T - p V01

"l

" 1-Pq/Pr

where we have used mT/p-j = Vq-j .

Similarly we have for mass 2

MA

M

2

T- p V

02

M

2

= 1- P /P R Subtracting the second from the first yields

M

A ,A _ ">2WV02>

X

1 2

] -Po/pR

From eq. (10), we have the true mass difference of the two weights

m{ - mJ = kAe + pA

(v rv

2

)

so that, upon substitution, we obtain

„A . kA , ^rt1*!^ . (17)

Now, employing eq. (12), we can use

17
V

l = V

01 H+a^t-tQ)]

and

V

2 = V

Q2 [l+a

2 (t-t )j

where Vq-j and Vq

2 are the respective volumes at tg = 20 °C.

Substituting for V-| and V

2

into (17) yields

- = kA6 - p (VorV02 ) + pA [V^+VQ^^t-20) - VQ2 -V02a

2 (t-20)]

where we have set

t

Q

= 20 °C

and have approximated (I-Pq/Pr)"^ by 1 ( see derivation of (E5)).

Simplifying further gives

^ - Nl£ = kA6 + (pA

-p )(Vor

V

02 )+pA[(t-20)(V01 ar

V

02

a

2 )] . (18)

Equation (18) is a fully corrected formula for calculating mass differences in the

apparent mass basis, except for the one approximation mentioned above. At this

point it is worthwhile to consider a specific example in order to get a feeling for

the magnitude of the terms in eq. (18). Assume that two weights of nominal value 2g

have the following properties:

V

Q1

= 0.2564 cm

3

V

Q2

= 0.1884 cm

3

also, a, « a

2

= 4.5 x 10" 5

/°C;

3

PA

= 1.17 mg/cm

3

Pq = 1.2 mg/cm .

We then find that

(pA

- P )(V

01

- V

Q2 ) = -0.03 x 0.068 = -2.04 x 10' 3

mg

and

PA

(t - 20)(V

Q1 a1 - VQ2a2

) = 1,17 (t - 20) x 3.06 x 10" 6

= (3.58 x 10" 6

)(t - 20) .

Considering an extreme condition such that t is 10 °C, we have for the latter term

= -3.58 x 10~ 5

mg .

Combining this term with the first we have

-2.04 x 10" 3

- 0.0358 x 10" 3

= -2:08 x 10"3

mg

18
so that the second term (under extreme conditions) contributes only 0.04 ug to the

overall buoyancy correction. Usually the (t - 20) term is of the order of °C to

4 °C so that the effect of the second term on the overall buoyancy correction of the

above weights is even less. Hence in most cases the second term can be safely

ignored and eq. (18) can be approximated by

M* - = kA6 + (pA

- pQ

)(V

01

- V

02 ) . (19a)

For true masses we start with eq. (10) and write

mJ - Mg = kA6 + pA (V^Vg) .

Employing the same volume expansion formulae used for the apparent masses (after eq.

(17)), we derive

M]

- m£ = kA6 + pA

(V

Q1

- V

02 ) + PA

(t-t

Q

)

x (V

01

a

1 - V02a

2

) . (10')

Once again the second term is very small compared to the first (see arguments above)

and the relation can usually be approximated by

m{ - M

2 = kA6 + pA

(V

Q1

- V

02 ) . (19b)

A quick comparison between eqs. (19a) and (19b) shows that the apparent mass differ- ence has a correction term which contains the factor (pa

- Pq) whereas for true masses the corresponding factor is p/\. This then shows that the former correction

is much smaller. In both cases the unknown parameter is pA

-

2.1.5. Sources of Error in Buoyancy Corrections

It is important to determine the sources of error in buoyancy corrections in

order to make the best measurements possible with the least amount of effort and

expense. Let us look back to eqs. (19a) and (19b) and write the most important

buoyancy terms:

True Mass Apparent Mass

pA

(V

01

V

02 } (pA

P )(

XV

01 " V02 )

Since pQ is defined exactly, uncertainties in the mass difference measurements

arising from buoyancy corrections are due to errors in pA , Vq-j , and Vq

2

- Let us denote the uncertainty in a quantity by a preceding 6 (i.e., the uncertainty in pA

is 6pA

). Then, from eqs. (19a) and (19b) the sources of uncertainty in the buoyancy

corrections are

True Mass Apparent Mass

6PA

(V

Q1

- V

02 ) 6PA

(V

Q1

- V

Q2 )

6V

01 PA

6V

Q1 (PA

- Pg)

6V

02 pA

6V

Q2 (pA

- pQ

)

where we will assume that 6Vnq , <5V n9 , and 6p« are uncorrected.

19
In the calculation of true mass differences, uncertainty in the volume of the masses is much more important than it is for differences in apparent mass as can be seen by comparing similar terms:

(«V

01 + 6V02 )pA

> («V

01 + <SV02 )(pA

-p ) .

Also we see that any uncertainty in Vq-j or V

Q2

will lead to systematic uncertainty

in mass measurements (even if Vq-j = Vq

2

).

Example :

Two weights produce the same reading on an ideal balance In air of p^ = 1.17 mg/cm

and t = 20 °C. Both weights are said to be of density 7.89 g/cm

3

. Find

1) their true mass difference

3

2) their apparent mass difference against Pq = 8.0 g/cm

3) the errors in these numbers due to uncertainty in the volumes.

The weights are marked "1 kg".

Answers:

1 ) From eq. (19b)

t P/\ t Pa

M (1- ) - Ml (1- -^—) = k-A0 =

1 7.89 c 7.89

because A9 is given as 0.

Hence m{ - = 0.

2) From eq. (19a)

A' M

2 " <PA " po^V

01

-V

02 ) + kA6 =

since A0 = and Vq^ = Vq

2

(because m| = M

2

and p^ = Pq 2

)-

3) the weights could have come from different lots of metal. Since no uncer- tainty is given for density, we must assume that

Pq = 7.89 ± 0.005 g/cm

3

,

i.e. the density is uncertain in the first unreported decimal place.

.As a worst case, let

P^ = 7.895 and p^ = 7.885 .

Then

20
6(M{ - Mj) - PA

* 0.00117 x 1000

* 0.00003 x 1000

Note that the nominal mass value of each weight is accurate enough to use in estimating the uncertainties in question.

CAUTION : In calibration certificates, it is normal practice not to include in

the assignment of systematic errors any contribution from volume uncertainties: The

volumes, usually calculated from densities provided by the weight manufacturers, are assumed to be without error. For the most critical work, volumes of individual

weights must actually be measured. (Since 1982, NBS has been routinely measuring the

volume of single-piece 1 -kilogram weights which require the best possible mass calibration.)

We have yet to examine the effects on mass measurements of uncertainty in p..

For mass differences, we have shown above that the errors are approximately

<5pa (Vqi - Vpg). Therefore, unless = we mst be concerned with uncertainty

in p^. The calculation of p^ along with an analysis of the attendant uncertainties

are presented in the next section.

2.2. Air Density

We have seen from the derivations in Section 2.1 that the correction to the mass determination requires the precise knowledge of the air density (p^) if the weighing

is done in an air environment. (We will assume from now on that we are always working in air.) The buoyancy correction in eq. (19b) requires p^ for true masses,

whereas for apparent masses we require (p^-Pq) in ecl- (19a). It should be remembered

that Pq is defined as pQ = 1.2 mg/cm^ which is the approximate value for 20 °C with

1 atmosphere of pressure with 50% relative humidity.

The usual approach to determining p. is to calculate the density of air based on

an "equation-of-state." To use this equation, one must supply ambient values for

temperature, barometric pressure, and relative humidity. For extremely exacting

work, the concentration of carbon dioxide in the laboratory ambient is also required.

Appendix B contains a derivation of the equation-of-state based on the work of Jones

[2J. This formulation is close to those which are found in handbooks or older

references but has three unique virtues:

21
1. It is based on the most recent reference data available.

2. It has been derived in such a way that uncertainties associated with the

equation itself are known and stated.

3. Virtually the same equation has been endorsed for international use by the

Consultative Committee for Mass of the International Committee of Weights

and Measures (C.I. P.M.). 3

The interested reader is urged to consult Appendix B and especially Ref. [2] for

the complete derivation. We state here that4

(iQLj =

°- 0034848 . (p _ . 0037960 . u

• e_) ( 20a) M

3 (t+273.15)

s

where t = temperature in °C

P = barometric pressure in pascals

(133.322 Pa = 1 mm Hg)

U = relative humidity in %

e<. = saturation vapor pressure.

e

s

= 1.7526 x 10

11

exp (-5315.56/(273.15 + t))Pa

is an excellent approximation.

If barometric pressure is measured in mm Hg, then

pA

(incLj = 0-46460

(P _ . 0037960 u

. e j (20b)

3 (t+273.15)

cm 1

o where e

s

= 1.3146 x 10 exp (-5315.56/(273.15 + t)) mm Hg is an excellent approximation.

2.3. Sensitivity Arguments

The major correction normally required for mass determination is due to air

buoyancy. As is evident from eqs. (19a) and (19b) the only parameter that has to be

determined very carefully, aside from the volumes of the weights being compared, is

the air density. Equation (20) clearly indicates that three parameters affect the

air density, namely humidity, barometric pressure, and temperature. We will see in

Section 2.3.5 that an overall uncertainty in of ^0.0030 mg/cm , i.e. ^0.25 percent

of p^ near sea level, is usually acceptable. In Appendix C we see that the total

uncertainty in p^ is the square root of the sum of the squares of the individual

uncertainties arising from measurements of humidity, temperature and pressure, (U, t

and P). Thus one way of assuring that the total uncertainty of p^ is within acceptable limits is to make sure that the individual uncertainties from U, t and P are each less than 1//3 of the total acceptable uncertainty. That is, we set a goal that

3

the contribution to uncertainty in p« from U, P, or t should be less than 0.0017 mg/cm . We will see in Section 2.4 that this goal can be met or exceeded with a modest investment in equipment. We now show the relative importance of measurements of humidity,

3The CCM recommends that this equation referred to as "Euqation for the determination

of the density of moist air (1981): in anticipation of improved reference data. There

are some stylistic differences in the CCM version of [2] but the calculated values are completely consistent at meaningful levels of precision.

Equations (20a) and (20b) are somewhat simplified (see Appendix B). Weighing errors

due to the simplifications are usually very small and are estimated in Table 5, p. 427

of ref. [2].
temperature, and pressure to the uncertainty in air density; we also indicate now accurately these must be measured to achieve our goal of 0.0017 mg/cm

3

for each one.

2.3.1. Humidity

It is easiest to discuss the effects of humidity (U) via examples. For practical purposes we can consider as standard a condition when t = 20 °C, P = 760.00 mm Hg

and U = 50 percent.

Example 1

Consider t = 20 °C, P = 760.00 mm Hg, U = 50 percent (i.e. standard condition). Using eq. (20b), we find

PA

= 1.2045 - 0.0053 = 1.1992 mg/cm

3

. 3

We note that the humidity correction, 0.0053 mg/cm , corresponds to a correction of only 0.44 percent in p^.

Example 2

Same conditions as in Example 1 except t = 30 °C. We calculate

PA

= 1.1648 - 0.0093 = 1.1555 mg/cm3

and the humidity corrections is ^0.8 percent of p^.

Example 3

Same as Example 1 except U = 60 percent.

PA

= 1.2045 - 0.006330 = 1.1982 mg/cm

3

so that the humidity correction is 0.5 percent.

Summarizing the results on humidity we can make the following observations:

i. The magnitude of the humidity correction to the air density at standard

conditions is ^0.4 percent,

ii. By increasing the temperature from 20 °C to 30 °C, the humidity correction

amounts to 0.8 percent of the air density. Thus even under these high

temperatures, very unlikely in a temperature-controlled laboratory, the

humidity correction term is still quite small,

iii. A change of 10 percent in the relative humidity (i.e. U = Un+10) changes

1 1992-1 1982

the air density by only 0.08 percent (i.e. '-| -[992* )•

We can conclude, therefore, that if we want to know the air density to within

3 0.0017 mg/cm then the relative humidity should be known to ±16 percent. (If a safety factor of 4 is required then the relative humidity should be known at the ±4

percent level )

23
2.3.2. Temperature

The effects of temperature on the humidity term were already demonstrated in

Example 2. In particular we note that a 10 °C change, yielded a humidity correction

of 0.8 percent.

The most sensitive effect of temperature on calculated air density, however, occurs in the first term

0.46460 n

PA

= P

rt (t+273.15)

From Example 1, we note that at standard conditions the first term yields

p = 1.2045 mg/cm3

Example 4

P = 760.0 mm Hg, t = 21 °C

In this case the first term of eq. (20b) gives

PA

= 1 .2004 mg/cm3

3 and pA has changed from p by 0.0041 mg/cm or 0.34 percent.

3

To achieve an uncertainty of 0.0017 mg/cm , we must measure the temperature

accurately to ^±0.4 °C (which reduces to ±0.1 °C for a safety factor of 4.)

2.3.3. Pressure

The arguments for the pressure uncertainty follow those of the temperature.

Once again we only have to consider the first term of the density equation

0.46460 D PA

= P

rt (t+273.15)

For standard conditions of t = 20 °C, and P = 760.0 mm Hg

p = 1 .2045 mg/cm3

Example 5

P = 764.00 mm Hg

t = 20 °C

In this case pA

= 1.2108 and has changed from p by 0.53 percent.

Thus in order to know pA

to 0.0017 mg/cm3 we will need to have accurate baro- metric measurements to ±1.1 mm Hg (which reduces to ±0.3 mm Hg for a safety factor of

2.3.4. Overall Uncertainty of Air Density

We have seen in eq. (20) that the air density is a function of the temperature,

pressure, and humidity

24
PA

= pA

(t,P,U) .

To evaluate the overall uncertainty of pa

, we assume uncorrelated errors and develop

a final value in Appendix C:

^ 0.25 percent . PA

Note in particular that this value was obtained using the errors established in

Sections 2.3.1, 2.3.2, and 2.3.3 of this section (with no safety factor).

2.3.5. Effects of Small Changes in Air Density on the Mass Value

Having established the sensitivities of U, t, P on pa

, the air density, we can now proceed and see what effect a small change of pA

has on the mass determination.

The arguments can best be carried through with the following example:

Assume that we are comparing the masses of two 1 kg weights.

3 Weight #1 has density of 8.4 g/cm

3 Weight #2 has density of 7.5 g/cm

3 Air density pa

= 1.20 mg/cm

If the air density is known to an accuracy of 0.25 percent, what uncertainty does

this cause in the mass difference calculations?

Answer : We found in Section 2.1.5 that the uncertainty for true as well as apparent

mass was given as

uncertainty = ±6p

A

(V

01

-V

Q2 )

= ±6pA

N (i- - L_)

p

l p2

where we have approximated the volumes by

V

] * N/p-, ; V

2 * N/p2

with N being the nominal mass of each weight.

Substituting numerical values, yields

Uncertainty = ±0.0025 x p fl x 1000 (— —

)

A

8.4 7.5

= ±0.0025 x 1.2 x 1000 (-

1 8.4 7.5

= ±0.043 mg.

25
This is about one-half the calibration uncertainty limit provided by the NBS for

3 high-quality 1 -kg weights. The span 7.5 to 8.4 g/cm includes the densities of 1 -kg

weights used in routine fine work.

It is usually true that the most stringent requirements placed on buoyancy

measurements are in the precise comparison of kilograms. An uncertainty in pA of

0.25 percent is usually sufficient even in this case.

CAUTION : Some weights are manufactured for purposes other than general mass

metrology. For instance, a scientist may have a 1 -kg aluminum weight which needs the

best calibration possible. In such special situations, the requirements and limitations placed on buoyancy work must be examined on a case-by-case basis.

2.4. Measuring Equipment for Air Density

The previous section yielded information on the accuracies required for the

three parameters needed for the air density. In this section we discuss briefly the

types of equipment required to meet our accuracy goals.

2.4.1. Humidity

This parameter is usually measured with a psychrometer or hygrometer. Several

commercial instruments can provide accuracies of ±2% relative humidity over a wide

range such as 10-80%. The calibration of such instruments can be verified at fixed

points by the user by means of an in-laboratory calibration station. Since an accuracy of ±16% (or ±4% using a "safety factor of 4") is required for mass metrology,

these instruments are more than adequate.

2.4.2. Temperature

An accuracy of ±0.4 °C (or ±0.1 °C for a "safety factor of 4") is required.

Simple, mercury-filled, glass thermometers can provide reading accuracies up to ±0.1

°C. More elaborate units can yield ±0.01 °C.

The user should be very careful in taking temperature readings. Generally, the

physical presence of the metrologist tends to warm up the balance with respect to the

rest of the room. It is therefore essential that, no matter what form of thermometer

is used, its sensor be placed as close as possible to the balance pan. Also, every

attempt should be made to ensure that objects being weighed are in thermal equilibrium with the balance. The latter requirement is especially important when comparing

weights of large surface area or of different geometries.

2.4.3. Barometric Pressure

The required accuracy is ±1.1 mm Hg (or ±0.3 mm Hg for a "safety factor of 4").

One can obtain mercury manometers or aneroid barometers with reading accuracies of

±0.05 mm Hg. Since this is sufficient for usual mass calibrations, one does not have

to push for higher accuracy instruments.

Present technology can provide defining instruments with overall uncertainty of

±0.01 mm Hg. Since such devices are the most accurate on the market, they are generally used in calibration of other manometers or aneroid barometers which con- sequently would have greater uncertainty.

26
Many laboratories possess either Fortin-type or aneroid barometers. These

require calibration against some defining instrument—such as a mercury manometer or

a piston gage; but such defining instruments are costly and may not be available in

the laboratory. On the other hand, aneroid and Fortin-type barometers are best

calibrated in place.

A solution to this problem has recently been demonstrated [9]. Two weights of

well -known and nearly-equal mass but having very different volumes are compared in

the laboratory on a sensitive balance. The measured difference in balance readings

between these two weights, kA6, determines the air density in the balance enclosure

via eq. (9). The air density is also calculated from eq. (20) using a calibrated

thermometer, a calibrated hygrometer, and the uncalibrated barometer. The difference

in the two measurements of air density serves to calibrate the barometer. Calibration

uncertainties of less than 1 mm Hg (three standard deviations plus known systematic

errors) have been demonstrated.

In summary, we note that all the requirements for air density measurements can

be met with presently available instrumentation. If, however, the accuracy of ±0.25

percent for has to be improved, then the limiting instrument will be the pressure

gage. (It does not make sense to improve temperature and humidity resolution by an order of magnitude, if the pressure cannot be read with higher precision.)

27
3. WEIGHING METHODS

Until this point, we have assumed that all mass comparisons have been carried

out on the idealized balance defined in Section 2. In this section we develop the means of evaluating the constant, k of eq. (8). We also extend our analysis to

actual balances currently in use. Just as one need not be a mechanical engineer to

drive a car, so one need not have a detailed knowledge of balance design in order to

carry out successful mass measurements. Nevertheless, a general knowledge of balance

design is often useful. The review paper by Schoonover [10] provides an excellent

introduction to this subject.

The idealized balance of Section 2 had no imprecision. Unfortunately, this is

not the case for real balances. By saying a balance has imprecision, we mean two

distinct contributions to uncertainty:

1) Resolution. A balance observation can be no more certain than one's

ability to resolve the least-significant digit of the read-out.

2) Reproducibility. A balance observation can be no more certain than the

tendency of the balance to produce identical read-outs under identical

conditions. The lack of this property is called "scatter."

For balances used in the best metrology, scatter is by far the more important of

the two. What we loosely term the scatter of the balance is not entirely intrinsic

to the balance. That is, it may depend on the type of table on which the balance is

placed, the type of air-conditioning in the laboratory, the skill of the balance

operator, etc. We need a measure of the scatter which includes all these effects.

Such a measure is provided by the "process standard deviation" [7,8].

The standard deviation for a particular weighing process is easily estimated by

repeating the process a number of times. (Ten repetitions are usually sufficient to

give a respectable estimate.) Two weights, representative of the weights normally

measured on a particular balance, are intercompared n times by one of the methods

described below. The n values obtained are a-,, a

9 , aq

, a -, , a . The average

of the n values is

1 c J n n

n.£ a,

a = inl-L

n and the standard deviation (a) of the balance is estimated to be

n \ 1/2

\^—2

I. n-1

with s -* o as n -+• «>. (The estimate s approaches the actual standard deviation a as

n is made larger.)

The standard deviation should be estimated in this way at regular intervals in

order to spot trends or shifts in the evolution of s with time. A slow rise in s could signal, for example, a slow deterioration of the balance. An abrupt shift

could signal a problem with the balance, a change in the air-conditioning system of

the laboratory, or some other problem which requires attention.

28
Before actually discussing various types of weighing procedures, we have to

define some terms required in the descriptions. The following symbols will be

util ized.

M = any unknown weight with true mass m"*" whose value has to be determined

C = a counterpoise (or counterweight) with true mass C^.

This weight is usually required on beam balances. It can be of any stable

shape, form and composition. However, it has to be adjusted so that the

scale (balance) does not deflect past its maximum allowable value when

loaded with a weight. This mass appears in equations which are combined in

such a way that C

T drops out of the final result.

S = a standard weight with true mass S^.

This is a known mass against which the unknown is measured. It is always

assumed that in any process is known and is traceable to the NBS. In

the following description S can be of a single unit or can be made up of

S = Z S. such that the sum S roughly equals M or C . i=l

1

A = a sensitivity weight with true mass a"^"

This mass is usually of very small magnitude. In principle it is only used

to measure the deflection of the scale per unit mass. This mass is very

critical. It must be well calibrated and traceable to the NBS, although

such calibration is a relatively simple matter.

Of the various weighing procedures, we will mention Direct Weighing, DirectReading Weighing, and discuss Substitution and Transposition Methods.

3.1 . Direct Weighing

This type of weighing is seldom used in the mass calibration program. It

requires the use of an equal -arm balance. Essentially, the reading steps are as follows:

1. Release brake and take null reading with both pans empty.

2. Load one pan with M and balance with S masses on the opposing pan until the

null position is once again reached.

We have as final result, if buoyancy corrections are insignificant,

T

n T

M = E S,

1

i=l

1

29
3.2. Direct-Reading Weighing

This weighing procedure requires a one-pan balance that reads directly in S.I. mass units or some other commercially recognized unit (e.g. pounds or carats). The

procedural steps are as follows:

1. Null the balance without any load on the pan.

2. Place M on pan and record the mass directly.

This method is obviously quite simple but usually is not accurate enough for metrology. It is appropriate for scales most commonly used in stores. Nevertheless, the

method is quite similar in principle to that described in Section 3.4.5., below.

3.3. Substitution Weighing

This weighing procedure is the usual one employed for both single pan or equal - arm balances. The basic idea is the comparison of an unknown mass with a standard

mass and a sensitivity mass.

3.3.1. Single Pan Balance

Known as "direct-reading analytical balances," they contain built-in weights

which are usually manipulated by external knobs or dials. These we will refer to as

"dial weights." The remaining balance reading is indicated by the rest point of a moving optical screen or by an automatic electronic display. This part of the

balance output is called the "screen reading." Thus each balance indication is the

summation of the dial weights and the screen reading. Calibration of the dial

weights is unnecessary for the measurements described below. The majority of balance

operations outside metrology (e.g., chemistry, metallurgy, etc.) do rely on the

accuracy of the built-in dial weights, however. For calibration of these weights the

reader is referred to Ref. [11] and Appendix A.

3.3.1.1. Single Substitution

To find the mass difference between two weights of nominally equal value, we

will first describe the method of "single substitution". For the moment, let us assume that the weights are matched closely enough so that, when placed on the

balance, they both require the same setting of dial weights. Therefore they differ

only in their screen readings. If we look into the balance (fig. 3), we see that the

pan is suspended from a beam which pivots about a very good bearing (the fulcrum,

usually a knife on a flat). On the end of the beam opposite to the pan, is a fixed

counterpoise C. (There will also be either a servo-motor on this end or a dashpot to

damp the beam oscillations. These details are unimportant to our derivation.) Also

suspended from the pan-end of the beam are the built-in dial weights. Thus when an object is placed on the pan, dial weights are removed so that the force of the total

assembly on the pan side (object being weighed plus remaining dial weights) balances

the counterweight as closely as possible. The remaining imbalance causes the beam to

rock slightly out of its horizontal position. The angle of tilt is proportional to

the remaining force imbalance. In a mechanical balance, this angle is directly

proportional to the screen readings. In a servo-controlled balance, a motor is used

to drive the beam back to horizontal. The screen readings are then proportional to

the average force generated by the motor.

30
31
In the balances just described, we can define the following quantities:

ZdT = the true mass of the dial weights removed

EV.. = the volume of the dial weights removed

6-j = the screen reading

6q = the screen reading when the pan is empty and the dial weights are replaced

(this was always zero for the ideal balance of Section 2)

k = the proportionality constant between screen reading and mass.

(Recall the previous definitions for M, S, A.) We also recall from eq. (8) that

for two weights, M-| and with volumes V-| and the weighing equation can be

written as (using true masses)

With

g (M^-PrnV = 9 ( M

2

T

-PmV

2

} +

5 k(A6)

Pm

= PA= air density;

(A6) = e

] - e 2

; and setting

we arrive at^

mJ = edT

v

2 = zv.

g(Nlj - p^) = (zdT - pA EV.)g + gk(e

1 - e )

where 8q = 6^ in this notation.

The following observations can be made:

t The counterpoise weight does not_enter into the final result.

t The gravitational acceleration, g, can be cancelled from both sides of the

above balance equation,

t The above equation is identical to that for the true mass difference

between weight #1 and the removed dial weights (see Appendix A).

In single substitution, we compare a weight M with a standard weight S. Either M

or S may actually be a summation of weight pieces. A small calibration weight,

sensitivity weight A, is also required.

A full derivation would involve a balance equation for each weighing operation (i.e.

the pan unloaded and the pan loaded). These equations would each involve the counterpoise C. Combining these relations would then give the desired result.

32
The steps in single substitution are:

1) Load M and record screen reading 61

2) Remove M

3

J Load S and record screen reading 62

4) Add A to S and record screen reading 83

In general we adjust S and choose A so that

The same dial weights are used in steps 1, 3, and 4.

t Je^ - ©2

J ^ s at l east f°ur times as large as leg - 9-J

• (63 - 6

2

) °° V5 the total range of the screen.

From the above steps 1, 3 and 4, we have the following relations:

ke

1 = (m

T

-pa

vm

) - (zdT-p^.) + ke (22)

ke

2

= (s

t

-pa

v s

) - (zdT-pa

zv.) + ke , (23)

ke

3 = (s

t

-pa

v

s

) + (a

t

-pa

va

) - (zdT-p

a

zv.) + ke . (24)

The symbol = indicates that we are now assuming that some or all of the measured

values have a non-negligible uncertainty.

Subtracting (23) from (22) yields

M

T

- S

T

= PA

(VM-V

S

) + k(6 r62) . (25)

Subtracting (24) from (23) yields

, _ aT

-Pava (9 ,,

k = 7 r • (26)

(

e 3-6

2

)

Finally, substituting for k from (26) into (25) yields

A

T

- V

m

t

-st

= pA

(vrv

s

) + /

Pa

\

(e r

e 2

) . (27)

(

6 3-e 2

)

Note that this equation is identical to (10), the crucial difference being that

we have found the value for the proportionality constant k.

One may wonder why k must be evaluated for every measurement if it is a constant. The answer is that for most sensitive balances, k is truly constant only for

short periods of time and only over reduced regions of the range of the screen.

In the derivation above we choose A so that 63-62 is about 20 percent of the screen range (fig. 4). In this reduced region, k will be sufficiently constant for

metrology if one is using well-made balances.

33
F M FS

F S+A

FORCE ON BALANCE

Figure 4. Balance indications as a function of the force on the pan. The response

shown is that of the screen when the dial weights are not changed.

34
3.3.1.2 Double-Substitution

In the derivation of single-pan substitution given above, we have treated 6q as

a constant. This is the same as assuming that the balance indication does not drift

with time. But this is rarely the case. Usually, however, the zero-drift in the

balance will be approximately linear over a short period of time. The method known

as single-pan double-substituion has the following two advantages over singlesubstitution: • linear drifts in the balance zero are eliminated if there is an equal time

between successive observations; • a second estimate of the mass difference between M and S is obtained.

Both benefits are obtained by a single additional measurement:

5) Remove S from pan, leave A where it is

6) Place M on the pan and record secreen reading 6*. The last step gives the

additional balance relation

ke

4

= (m

t

-pa

vm

) + (a

t

-pa

va

) - (ed]"-p

a

zv.) + ke . (28)

Substracting (24) from (28) yields

m

t

- sT

= pa

(vm

-v

s

) + k(e

4 -e3

) .

Substituting for k from eq (26) then gives

T T

(aT - pA

V

A

)

* 3~ 2'

Finally, by adding (29) and (27) we obtain

T T

( aT-PaV

pa

(vm-v

s

) + -j^^ (e r

e 2 -e 3+e 4

) . (30)

\

One might wonder how eq. (30) would have to be changed if 6q were drifting

linearly with time instead of being constant. The answer is: the term (

t ~ e 2~e 3+e 4^

would still be strictly correct; the term (63-62) would not be strictly correct.

However, provided that the drift among readings is small compared to (63-62), the

error from this source will be negligible. This will in actuality always be the

case. If it were not, the balance would be drifting so badly that no stable readings

could be obtained. If instead of (8 3

-6

2

) one substitutes the term (363-362+0^ -e 4

)/2

,

linear drift in the sensitivity calculation will be removed [12]. This more com- plicated formulation is generally used when data are being analyzed by digital

computer software.

35
CAUTION : If 6-j or

2

is very close to zero, balance drift may cause readings to

become negative or to go off scale. To avoid this problem, a small, uncalibrated

"tare weight" can be added to the pan during all measurements. Mathematically, this

is equivalent to changing 0q by a constant amount and so has no effect on the final

results.

3.3.2. Two-Pan Equal-Arm Balance

This is the oldest type of precision balance. Its basic design is represented

by the "scales of justice" or by the zodiac sign for Libra.

3.3.2.1. Single Substitution

This method is very similar to single-substitution on a single-pan balance. The

read-out provided by an equal -arm balance corresponds to just the screen-reading of

the single-pan balance. An uncalibrated counterpoise which has nearly the same mass

as M and S is placed on one of the pans throughout the measurement sequence. The

size of the counterpoise is selected to ensure that the balance indication will be on scale with M on the second pan. The weights M, S and a sensitivity weight A are placed on the second pan in nearly the same sequence used in the one-pan case.

The required steps are as follows:

1) Place C on first pan, M on second pan and record indicator reading 6-i

.

2) Remove M

3) Place S on second pan and record indicator reading (Recall that S is

a summation of calibrated weights which is chosen to be near the scale

indication of M.)

4) Place the sensitivity weight A on that pan which will cause the indicator

to move toward the center of the scale. Record indicator reading 63.

It is convenient to choose as the second pan (i.e. the one on which M and S are placed) the one for which increasing increments of load give increasingly positive

indicator readings.

Going through detailed arguments similar to those for single-pan, single substitution, we arrive at

T T

( aT

-PaVA> m - s

= pa%v s) + I7TT (e r

e 2> (31)

| 3"e

2|

The absolute value j^-^j comes about because the A weight might be placed on either of the two pans. Note that if the arms of the balance are not equal, the

sensitivity weight must be placed on the second pan.

3.3.2.2. Double-Substitution

As in the case of the single-pan balance, one additional reading is taken for

double substitution.

5) Remove S from the balance; leave A where it is,

6) Place M on the second pan and record the indicator reading e-.

36
By obvious extension of the preceding arguments, we arrive at the equation for

double-substitution on an equal -arm balance:

PA

CVM

-V

S

) + T—~

J- — • (32)

Again, to be strictly correct in the presence of linear drfit, the term 63-62 can

be replaced by (363-362+61 -6^)/2. Double substitution in this case has the same advantages as weighing on a single-pan balance.

In summary, we have derived the following equations for the four types of

substitution weighing we have discussed:

Balance Type Substitution Type Equation

Single-Pan Single (27)

Single-Pan Double (30)

Two-Pan Single (31)

Two-Pan Double (32)

3.4. Transposition Weighing

Two-pan, equal -arm balances can also be used to do transposition weighing.

This method does not require a separate counterpoise and also doubles the balance

sensitivity.

3.4.1. Single Transposition

The following steps are required:

1. Place M on one of the pans, e.g., the left pan and S on the right pan. S

should be adjusted so that the indicator reads on scale. Record the indication

2. Remove M and S.

3. Replace M and S on the balance but in transposed position (that is, M on the

right pan and S on the left pan). Record the reading 9o4. Add A to that pan which will cause the indicator to deflect towards the center

of the reading scale. Record the indication 63.

The three indications give the relations:

M

T

- sT= pa

(vm-v

s

) + k(e r

e ) (33)

S

T

- M

T

= PA

(V S

-V

M

) + k(6

2

-6 ) (34)

and

sT

- m

t = pa

(v s-vm

) ± (a

t

+pa

va

) + k(e

3

-e ) . (35)

Here again, 6q is the balance indication when both pans are empty. The ± sign

is used to indicate that, in the last equation, the sign of (A

T

-pA

V.) depends on which pan held the sensitivity weight.

37
Subtracting (34) from (35) yields

k = -i—^4 . (36)

9 3"6

2

where, once again, the absolute value was used to take account of whichever pan the

sensitivity weight is placed on.

By subtracting (34) from (33), we have

m

t

- sT

= pa

(vm

-v

s

) + |

(e re

2

) . (37)

Finally, we can substitute from (36) into (37) to obtain

M - S - PA

(VM-V

S

) + j—-j . (38)

|v 3-w

2 j 2

CAUTION : In transposition weighing, attention has to be paid to the sense of

the balance scale. We have assumed above that the balance indication becomes more positive as the mass on the left pan is increased. For scales of opposite sense,

the plus sign of eq.(38) should be changed to minus. Also, if the arms of the

balance are insufficiently "equal," transposition weighing will result in error.

3.4.2. Double Transposition

The procedure for this method follows that outlined above but with the following additions:

5) Remove M and S, leaving A where it is.

6) Place M on the left pan and S on the right pan. Record indication 6^.

These steps provide us with the additional equation

m

t

- sT

= pa

(vm-v

$

) ± (a

t

-pa

va

) + k(e

4

-e ) (39)

where the ± sign is needed once again since the location of the A weight is ambiguous .

By subtracting (35) from (39) we have

M

T

- sT= pA

(vM-v

s

) + \

(e4-e

3

)

and substituting for k from (36) yields

T T 1 (aT

"PaV

a)

M

T

- S

T

= PA (vM

-v

s

) + 1

r y (e

4

-e

3

) . (40)

38
Adding (40) to (38) finally results in

m

t

- sT

= pA

(vM-v

s ) (e r

e 2 -e 3+e 4

) . (41)

4 e 3 -e

2

Again the reader is cautioned that: the plus sign in front of the second term applies

only to balances whose indications become more positive as the mass in the left pan

is increased; in the presence of linear drift, the term 6,-e« can be replaced by

(3e

3

-3e

2+e

1 e 4

)/2.

J c

3.5. Weighing on Electronic Balances

The most recent development in mass technology is the appearance of the com- pletely electronic balance. This can be thought of as a single-pan balance whose

screen span has become so large that dial weights are no longer necessary. These

balances are rugged, easy to use, and most are directly interfaceable to computers or

data loggers. Unfortunately at present they rarely attain the precision achieved by

the best mechanical balances. For metrology purposes, these balances are used just

as the single-pan balances discussed above and the same equations apply. If and only

if experience verifies that the constant k is indeed constant , can weighing procedures be simplified so that the sensitivity weight need not be measured for every

substitution or double substitution.

The results we have just derived can be generalized. First, although only true mass has been dealt with, the reader should by now be able to generate the corre- sponding equations for apparent mass. Next, since S is assumed to be an assembly of

calibrated weights, all the measurements described in this section provide an estimate for m"^, although they do not constitute an actual calibration.

Recall that we had (eq. (10))

M

-T =

. SJ + pA

(VM

-V

S

) + kA6 (10') or equivalently

T S -pi.Y c+kA0

M = SL2 . (10")

1-PA

/P

Usually, the manufacturer of a weight supplies the buyer with the weight's density so that formulation (10") is more appropriate. Equation (10

l ) is often used in computer

analysis of data, however. In this case, one estimates as V

M

N/p, where N is

the nominal mass value of M. Equation (10

1 ) then gives a first order approximation

for M

T

. This first estimate can be used to find a better value for VM

and the

process repeated until convergence is reached.

If S and M are both unknown and not sufficiently close in mass, known weights

can be added to either S or M. These added weights are then included in the weighing

equations.

39
Example

Sometimes M itself will be a summation of weights. Let us say, to take a specific example, that M is the sum of three weights whose nominal values are 50 mg,

30 mgj and 20 mg. Furthermore, the weight manufacturer specifies that the 50-mg and

3 6

30-mg weights are made of tantalum (p = 16.6 g/cm ; a = 20 x 10 /°C) but the 20-mg

3 6

weight is made of aluminum (p = 2.7 g/cm ; a = 69 x 10" /°C). What is the effective

density of the summation and what is its effective coefficient of expansion?

Answer:

As a first approximation, assume the mass of each weight is equal to its nominal

value. Therefore, the total volume is approximated by

050 , 0.030 , 0.020 _ n moooi TM3

7^6 + -T676

+ ^7-~ °- 012227 cm

The effective density is then approximated by

n - m/v - .050 + .030 + .020 q 10 .3

p = M/V 0.012227 8.18 g/cm

The effective coefficient of expansion is a weighted average—the coefficients of

each weight are added in proportion to the volume of the weight to the total volume:

20 x lo"

6 x M50 + 20 x 10"6 x + 69 x 10" 6 x ^0

16J 16^6 2.7

a = .050

+ 0.030 0.020

16.6 16.6 2.7

= 50 x 10" 6

/°C

After mass values for the weights comprising M have been found, the metrologist

should verify that the approximations used to estimate p and a were adequate. Only

very rarely will this not be the case, requiring an iteration: the mass values

obtained for the weights comprising M are used to find new effective values of p and

a; these latter two values are then used to calculate improved mass values.

40
4. PROGRAM TYPES

We will discuss two distinct programs in mass metrology: surveillance and

calibration. As the name implies, surveillance attempts to monitor a calibrated

assembly or set of weights. Surveillance looks for signs that one or more members

of the set may have changed since the last calibration. Calibration, of course,

attempts to assign the best possible value of mass to a weight by relying on a chain of traceability to universally recognized standards. Assignment of a realistic uncertainty to a calibration value is equal in importance to the calibration

value itself. Several additional definitions will be useful.

A standard mass S^ was already defined in Section 3. We also have:

SC: Check standard weight with true mass (SC)"^.

This mass is basically required to monitor in-house procedures and accuracies.

Usually (SC) is a calibrated standard S that has been added to a set during calibration. By measuring (SC) (treated as an unknown M) versus S and comparing the

results with the known value one can then determine if the measuring process is

either in or out of control. This type of self check should be done periodically

and the values for (SC) should be monitored continuously via trend charts (see

below)

Set: A group of weights in a specific order. To be of maximum utility, sets usually

cover several decades of nominal mass; e.g., 1 mg to 100 g. Further smaller groups

(called subsets) of the complete set can be selected to produce any nominal value

within the span of the set in increments of the smallest set members. What this means is that, taking the example of a 100 g to 1 mg set, any of the nominal values

0.001 g, 0.002 g, 0.003 g, 99.998 g, 99.999 g, 100 g can be obtained by selecting appropriate weights from the set. Usually a set will employ the smallest

number of weights possible to cover each decade. A set with a sequence 5, 3, 2, 1 could for instance range from 100 g to 1 mg with the sequence repeated for each

decade (e.g. 500 mg, 300 mg, 200 mg, 100 mg).

Design: This specifies explicitly how and in what order a set of weights is to be

compared (measured). It states in detail what weights of the set have to be used at

a particular step.

ST: Transfer standard weight with true mass (ST)^.

As mentioned under "Set" above, in many cases a number of weights, e.g. from

1 kg to 10 g must be calibrated. These are usually divided into decade groupings;

that is, weights from 1 kg to 100 g are calibrated at the same time using one design

(A standard kilogram is also included in the design). Next the weights from 100 g

to 10 g are cal ibrated--possibly using a different balance. In this procedure, it

is common for the 100 g weight which was calibrated as a result of the first series

of measurements to become the standard for the second series. The 100 g weight thus serves as a "transfer standard" because it is used to transfer our knowledge of a standard kilogram to weights of smaller denomination.

4.1. Surveillance

Two distinct types are outlined below as Type I and Type II. (For more com- plete detailed discussions of many different designs, the reader is referred to ref.

[4]).

41
The basic idea of surveillance testing is to ensure the self-consistency of the

weight set. For example, the 20 g, 30 g, and 50 g weights can be checked against

the 100 g weight to see if the difference is within expected limits. It is also

advantageous (though not essential) to compare one weight of the set (usually the

largest) against an independent standard, S. The latter operation establishes

whether the entire weight set has undergone a change- -even though there may still be

self-consistency within the set. The basic motivations for surveillance testing

are:• to verify the values of newly calibrated weights • to establish the stability of a new weight set • to determine if an accident (such as being dropped on the floor) has damaged

the weights involved.

Buoyancy corrections may not be needed in surveillance testing. One should

check the magnitude of such corrections compared to the surveillance limits (see

below) to see whether it is worthwhile to make the buoyancy corrections.

4.1.1. Type I

The object of this surveillance method is to perform an intercomparison of all

weights in a set using a minimum number of steps. It is preferable to have one standard weight as a member of the set. We will denote it as S for this discussion.

The standard S should always be as large as the largest member of the set; or as large as is convenient.

For the comparison measurement one always starts out with the largest weight

and compares it with a summation of weights next in magnitude such that the sum is

equivalent to the largest. Next, a weight from the first summation is compared with

a lower summation, and the process is continued until all the weights have been

used. If a standard is included in the set, then the S has to be compared first

with the largest mass. An example will help to visualize the procedure.

Example:

The set contains a standard weight S whose nominal value is 100 g and weights

of 100, 50, 30, 20, 10, 5, 3, 2, 1, 0.5, 0.3, 0.2, 0.1, 0.05, 0.03, 0.02, 0.01,

0.005, 0.003, 0.002 and 0.001 g; called 1^ to M

21 , respectively. Such a set is

known as a set with mass ratios of 5, 3, 2, 1 from 100 g to 1 mg. For the measurement sequence, we start out with the standard S versus M-| = 100 g and work down to 1 mg such that all masses are included via a minimum number of steps. The designation

M- refers to the nominal value of the ith weight whereas mT refers to the true mass value of the ith weight.

1st Meas : m{ - S

T

= 6-, , where M

] = 100 g, S = 100 g

2nd Meas : mJ - M^' = 6

2

, where M

2

' = (Mg + M

3 + M4

)

= (50 + 30 + 20) g

= 100 g

42
3rd Meas : NlJ - M3' = 63, where M

3

'= (Mg + Mg + M

? + Mg

)

= (10 + 5 + 3 + 2) g

= 20 g

4th Meas : mJ - mJ' = <5 4

, where M

4

* = (M

g + M

10 + Mn

+

= (1 + 0.5 + 0.3 + 0.2) g

= 2 g

5th Meas : mJ2

- Mg' = 6

g

, where Mg' = (M]3

+ M

]4 + M]5 + M

]6 )

= (0.1 + 0.05 + 0.03 + 0.02) g

= 0.2 g

6th Meas : mJ6

- Mg' = S

g

, where Mg' = (M]7 + M

]g

+ M

]g

+ M

2Q )

= (0.01 + 0.005 + 0.003 + 0.002) g

= 0.02 g

7th Meas : mJ9

- M^' = 67, where M

?

' = (M2Q

+ M

2] )

= (0.002 + 0.001) g

= 0.003 g

Note that the last measurement is amended because only M^ is left to measure.

With a very simple software routine, the differences (tf's) can then be compared

against the known (accepted) values. These new differences should then be plotted

and compared chronologically with previous tests. Together with predetermined

uncertainty limits one can then monitor the particular weight set. Usually one devotes one chart for each 6. For example, fig. 5 shows a survellance chart for 6*.

The horizontal line is the value of <5 4inferred from the most recent calibration

report of the set. The points represent values of 6

4

which were derived from

surveillance testing. As long as the points remain within the upper and lower

horizontal lines, (known as the surveillance limits), we have no evidence that any

of the weights involved in the measurement of 6

4

has changed from its reported

calibration. If an obvious trend is apparent, however, a recal ibration can be done

before the surveillance limit is exceeded. Two questions remain to be answered:

(1) How are the surveillance limits determined?; and (2) If a point lies outside the

surveillance limits, how does one determine which of the weights (that is 2 g, 1 g,

500 mg, 300 mg, or 200 mg in the case of fig. 4) has changed? The answers to these

questions will be deferred until after the description of Type II Surveillance.

4.1.2. Type II

For Type II surveillance, a more sophisticated approach to the mass comparisons

is taken. Again, the procedure is best described by an example.

Suppose we consider the following three weight groupings from the set used in

the example for Type I.

43
CO

00

C/)

LUO

LU>

DC

(/)

CNJ .00

LU

CM

00 o o

CO oCNJ oo oo

CUo

a> 4—O +-> -a 10 o

QJ «o

Q-

<T3 • S- +-> +-> CU S_ > E o O i— CL

cu J- S- <o CDu

4— c o O n3 •I—

+-> fO

SU CD -O

QJ > • 1

-C S- ]

o =300 u

aiu cu +-> x:

cuu

QJ cu

*i

-a SCD> 00 4-> 00 3 o

00

O E

00 cu

+->

— <Q E

4-> o

00 Sa) 4- c o

cu T3 c _c: CU o 4-> 4- rO O CD

JZ. 00 +J (J •M 4- 3 O u

oo

cu

LT)

J- <0

cu

S- =3 03 i—

=3 CU 03

CTi >, >

CU

30N3d3ddlQ SSVIAI

44
Weight Grouping Designation

3 g M

7

2 g + 1 g M

3

= M

8 + M

9

2 g + 500 mg + 300 mg + 200 mg MJ' = M

s + M

10 + M

ll + M

12

For Type II surveillance, the following three mass differences are measured.

1. «}-«]" =6,

2. My - Kj" 1 = 6

2

3. M[" - Mj1 " = 63

(Note that 63 cannot be measured by transposition weighing because Mg is common to

M3" and M

3

'" .)

The three weighing operations shown above form a simple "design". The concept

of weighing designs is crucial to calibration operations and will be discussed at

length in Section 4.2.2. Here we may simply note that we are working with a "3-1 1 s design" (i.e., 3 different weight groupings with 1 nominal mass).

If the weighing process had no scatter (standard deviation equal to zero), then

63 = 6

2 - 6-j . In that case we would derive no additional information from the 63

measurement and its inclusion would, in fact, be a waste of time. Of course, the

process does have some scatter and the 63 which we measure has only a slight probability of being equal to the measured value 6

2

- <5-| . Therefore, far from being

useless, the measurement of the trio <$•, , 6

2 , 5 3 can be use(^ to Q00 ^ advantage in two

ways

First, a statistical technique known as "least squares" fitting [3,8] of the

data provides better estimates of the three mass differences than can be gotten with

a single measurement.

1. - M^" = 1/3(26-, + 6

2

- 63) = 6\ \

2. My - M^"' = 1/3(6

1 + 26

2 + 63) = 6£

3. M3" - M3 11

= l/3(-6-, + 6

2 + 26

3

) = <5^ .

The least-squares estimates 6-j, <5 2

, and 63 can be checked against the calibration

report as in Type I surveillance (see refs. [3] and [8]).

Second, the differences between the least-squares estimates and the observations, |<5]- |5

1

» IV"^! and

l

<5 3

, " 6

3l ' are related t0 scatter i n tne measurement

process. 6 If one of these differences is markedly higher (say three times higher)

It is characteristic of the least squares solution to a 3-Ts design that |<5-j-<5-j|

l 6 2-6

2 | l 6 r

6

3

This feature cannot be generalized to other designs.

45
than the measured standard deviation of the balance, one should redo the measurements

of 6p 6

2

, and 6^- If trouble persists, it is a good indication that the precision

of the balance (eq. (21)) has deteriorated.

As another example we will use a set similar to the one discussed under Type I,

i.e. mass ratios 5, 3, 2, 1 from 100 g to 1 mg with S = 100 g.

The first 3 - I's weighing series consists of

la. S

T

- m| = 6|, where M

] = 100 g

lb. S

T

- mJ' = 6

2

, where M

2

' = (M

£ + M

3 + M4

)

= (50 + 30 + 20)g

t t

= 100 g

1c. M] - Mg 1 = 6

3

The next 3 - I's weighing series is

2a. M3 - M3 1 = 6 4

, where M3 = 30 g

M

3

' = (M4 + M

5

)

= (20 + 10)g

= 30 g

2b. M3 - mJ' = 6

5

, where M

4

' = (M

4 + Mg + M

? + Mg)

= (20 + 5 + 3 + 2)g

= 30 g

2c. M3 1 - <' = 6

6

This is followed by the third 3 - I's series as

3a. M7 - mJ' = 6

7

, where M

? = 3 g

Mg' = (Mg + Mg)

= (2 + 1) g

= 3 g

3b. - Mg' = 6

8

, where Mg' = (Mg + M

]0

+ Mn+ M

12 )

= (2 + 0.5 + 0.3 + 0.2) g

= 3 g

3c. Mj' - Mj« = 6 g

46
For the fourth sequence we have:

4a. m|

1 - = 6

1Q , where M-j, = 0.3 g

M

7

' = (M12 + M

13 )

= (0.2 + 0.1) g

= 0.3 g

4b. - Mj' = 6 n

, where Mg' = (M]2 + M

]4 + M

]5 + Mlg )

= (0.2 + 0.05 + 0.03 + 0.02) g

= 0.3 g

4c. M7' - Mg' = 6

]2

.

For the fifth series we measure:

5a. m|

5

- Mg* = 6

13 , where M

15

= 0.03 g

M

9

' = (M16

+ M

1? )

= (0.02 + 0.01) g

= 0.03 g

5b. Mj

5

- M

j

' = 6 U. where M{

Q

= (M

lfi + M]8

+ M

]g + M

2Q )

= (0.02 + 0.005 + 0.003 + 0.002) g

= 0.03 g

5C. Mj' - M{ ' =

6, 5

.

Finally, we arrive at the last series with

6a. Mj

g

- Nljj = 6

16

where M

19

= 0.003 g ^

Mn' = ( M

20

+ M

2l)

= (0.002 + 0.001) g

= 0.003 g

6b. m|

q

" m{

2

' = 6

1?

where M

]2

' = (M2Q + M

22 )

= (0.002 + 0.001*) g = 0.003 g

6c. Mjj - = 6 18

.

Note that in the last series a known (standard) 0.001-g weight was added in order to complete the design.

47
Once again the least squares fitting technique provides better estimates such

that eqs. (la), (lb), and (lc) yield

la') 1/3(26

1 + <5 2

-6

3

) = 6-,

2b') l/3(6

] + 26

2 + 63) = fi£

3c') l/3(-6, + 6

2 + 2 63) = S'y

If for any reason a suitable standard, S, is unavailable, then any uncalibrated

weight or summation of weights of the correct nominal value can be used in eqs. (la)

and (lb). However, in this case only 61 can be checked against the reported (known)

calibration and only internal consistency of the entire set can be determined.

We also have

and

2a') l/3(26

4 + 6 5

- «6>- V

2b') l/3(6

4 + 26

5 + 6 6

) = V

2C) l/3(-6

4

+ 6

5 + 26

6

) = 6

6

3a') l/3(26

? + 6g

- 5g) = *7

3b') l/3(6

? + 26g + 6g) = 6 8

3c') l/3(-6

7 + 6g + 26g) = 5 9 as well as

4a') 1/3(26

1Q + fi n

6

12 ) - 6

io

4b') 1/3(6 1Q+ 26 n

+ 6 12 ) =

4c') l/3(-6

10 + 6 n

+ 26

12 ) - s\2

Finally

5a'

)

1/3(26

13

+ 6 U

6 15>

=

5b') 1/3(6

13

+ 26

14

+ «16>

=

5c') l/3(-fi 13 + fi 14

+ 2S15 ) - s \s

and

6a') l/3(26

]6 + fi 17

6

18 ) =

6b') 1/3(6

16

+ 26

1? + 5 18 J = 617

6c') l/3(-6

16 + 6

17 + 2S 18' = S

18

The differences obtained for the first two results of each 3-1' s series (i.e. a and b) should be compared to accepted values derived from the most recent calibration

report.

48
The above examples of surveillance measurements for Type I and Type II present

a simplified picture of the equations. All 6's must contain any necessary buoyancy

corrections for true mass differences. Different 6's are obtained if apparent

masses are used. From the simplified eq. (19a), one finds

M

A

- S

A

= kA6 + (pA- pQ ) (V QM- V

QS ) = 6

where we assumed that the mass of the standard is given in apparent mass units, S .

Deviations of the measurements from values inferred from the most recent calibration report are evident from the previously mentioned surveillance charts.

4.1.3. Surveillance Limits

Upon carrying out a surveillance test of either Type I or II it will become

evident that, after any necessary buoyancy corrections have been made, the measured

values 6

i (Type I) or the predicted values 6^ (Type II) do not exactly agree with

those values, 6^, calculated from the calibration certificate accompanying the set.

To judge how serious the disagreement is, one must also calculate surveillance

limits. The surveillance limits associated with each 6.. or 6j define the limits of

credibility that the difference 6^ - 6. or 6^ - 6.1 could be due to a combination

of calibration uncertainties of the weight set and the random error of the measurement which is associated with the balance used in the surveillance measurements.

Let us designate to random and systematic error limits by SL;

(SL) = U + E (42)

where

U = systematic error as determined from the calibration report

E = limit to random error = 3a

and a = standard deviation of the measurement (refer to Appendix D for

information on estimating this number).

Then any value 6. or 61 should fall between 6?±SL, as shown in fig. 5. The value of

3a gives a 99.7 percent confidence level for the random error (i.e. if the measurements were repeated a great many times there is a 99.7 percent chance the average

would be within ± E of the result of a single surveillance test).

For surveillance, U is the root-sum-square of the individually reported calibration uncertainties such that

U = (ZU?) 1/2

(43)

Combining the uncertainties in this way is strictly valid only if they are uncorre- cted. It is the nature of calibration designs, however, that the uncertainties of

weights within the set generally are correlated. Thus one should view eq. (43) as an approximation which is adequate for surveillance limits. The reader is referred to

the discussion of calibration uncertainties below for a fuller explanation.

The standard deviation used is known from many previous measurements (eq.(21)).

It is a measure of the random errors in the balance being used.

At this stage it is worthwhile to quote a couple of examples and step through

the process. From Type I surveillance, we have

49
S = 10 g

M

1 = 100 g

= 100 g = 50 g + 30 g + 20 g = M

2 + M3 + M

4

Assume the following previously reported calibration:

Mass Value Uncertainty

S

T

= 100.0010196 g U

$ = 0.015 mg

mJ = 100.0009407 g U

M

= 0.020 mg

Nl]> = 50.0004628 g UM= 0.011 mg

M3 = 30.0002926 g U

M

= 0.012 mg

mJ = 20.0001578 g U

M

= 0.010 mg

Also assume that o = 0.026 mg.

Since the first measurement is

m| - S

T

= 6^ 6^ = -0.0000789 g = -0.079 mg

and we have

= V Us + uM

J

= ^(0.015)

2 + (0.020)2 = 0.025 mg

and (SL) = U + 3a

= 0.025 + 3 x 0.026

= 0.103 mg.

Hence (mI - S

T

) values should fall in the range of (-0.079 ± 0.103)

For the second measurement we have

so that

m{ - Mg' = 6

2

; 6

2

= 0.028 mg

= 0.019 mg

UM^=V U

M

2 + U

M

3

+ U

M

4

=V5 + U

M

2 "l

= y (0.019)

2 + (0.020)

2

= 0.028 mg

50
and (SL) = 0.028 + 3x0.026

= 0.106 mg

and mJ - ' should fall in the range

0.028 ± 0.106 mg.

Next, we consider the example from Type II. The first series of the design consists

of S, M-j, and M£ with M2

= M

2 + M

3 + M4* us assume tne same calibration report

as above, with the scale standard deviation at a = 0.026 mg. In this case we have

three measurements

S

T

- Mj = 6

] S

T

- Mj' = 6

2

Mj - = « 3

Because we have used a 3 - Ts design, the standard deviation of our result is not

simply the standard deviation of a single measurement, a, but is instead /2/3 a. A

proof of this conclusion is beyond the scope of this text but is explained fully in

ref. [3],

There are several surveillance limits that can be calculated. For the three

differences, we follow the calculations in Type I (but we have to remember that we

are in Type II so that the standard deviation is »^273 a).

Limits for S

T

- mI

We then have

so that

Limits for S

T

- M^'

U = ^ uj + ujjj = ^0.01^f + (o.Q2^ = 0.025 mg

(SL) = U + 3 x (v£73 a) = 0.025 + 3^2/3 x 0.026

= 0.089 mg

(S

T

- m{) = (0.079 ± 0.089) mg.

where U

M

' = ^ U

2

+U

M

2 + ll/ = ^{0.0U) Z + (0.012)

2 + (0.010)2

so that

Hence

'2 1 "2

0.019 mg

= y (0.015)

2 + (0.019)

2 = 0. 024 mg

(SL) = U + 3*^73 a = 0.024 + 3^2/3 0.026 = 0.088 mg

and (S

T

- Mg') = (0.106 ± 0.088) mg.

51
4.1.4. Identifying Weights Which Have Changed

If a measurement falls outside the surveillance limits, it becomes necessary to

determine which individual weights are responsible. Simple deductive reasoning is

all that is required, although a few extra weighing combinations may also be needed

if Type I surveillance was used.

As an example, suppose that in the Type I measurements shown earlier (4.1.1) the

value for 64 was outside but the value for 63 was inside the surveillance limits.

All we know at this point is that there is some change in the subset = 1 g + 0.5 g + 0.3 g + 0.2 g. Three additional weighings are now required to pinpoint the cause of the discrepancy. Recall that MQ = 1 g, M1n = 0.5 g, NU, = 0.3 g, M10 = 0.2 g,

M

]3

= 0.1 g.

y lu 11 u

We then measure

4' ) Mj - Mj" = 6J,

where M][ = M

1Q + Mn

+ M

]2

4" ) m|

q

- mJ" = 6g, where M£ = + M

]2

4'") m|

] - Mg" = 6£, where M£ = M

]2 + M13

After the surveillance limits for 64, 6£, 6£ are calculated one investigates the

results:

1. ) If 6^ lies outside the surveillance limits but 6£ and 6£ do not, then it is

probable that Mg has changed since its last calibration.

2. ) If 6J[ and 6£ lie outside the surveillance limits by opposite amounts and 6£ is

inside, then it is probable that the M-|q (0.5 g) weight has changed.

3. ) If 6)[ and 6£ lie outside the limits by about the same amount and 6jj is outside

also by the same amount but in the opposite direction, then it is probable that

the M-j-j (0.3 g) weight has changed.

4. ) If

6J 9 6^, and 6£ all lie outside the limits by about the same amount in the same direction, then it is probable the M-| 2 (0.2 g) has changed.

If none of the above conditions are met, then it is probable that more than one weight has changed. The reader should then consult ref. [4] for a more thorough

analysis of surveillance methods.

4.2. Calibration

The process of "calibration" assigns mass values to weights by comparing the

unknown weights to recognized standards. An uncertainty limit--3 times the standard

deviation of the measurement process, plus estimated uncertainties systematic to the

measurement process—accompanies each calibrated value. To an even greater degree

than in Type II surveillance, redundant information is gathered in order to determine

whether the measurement scatter is acceptable. A powerful self-consistency check of

the calibration process is also included.

52
Most of the following information can be found in much greater detail in refs.

[1] and [3]. It is repeated here mainly for explaining the overall approach of the

program.

Weight sets come in various denominations. The following groupings are frequently calibrated:

1. Nominally Equal Groups

These sets consist of weights, all of which have the same nominal value. The

number of weights can go from 3 to as high as 13 or more. (Sets of up to 50

members have been utilized.) For large sets, subsets are usually analyzed.

Otherwise the number of weighing operations would become impractically large.

2. Groups for 2, 2, 1,1.. sets.

Several combinations are in use such as 2, 1, 1, 1, or 2, 2, 1, 1, 1, or 2, 1,

1, 1, or 2, 2, 1, 1, 1, etc. Here 2 and 1 imply the same decade of mass such

as 2 kg and 1 kg.

3. Binary and Miscellaneous Groups

These sets are usually comprised of pound units. Many combinations are available

the use of which is slowly diminishing. A few examples provided in ref. [3] are 4, 3, 2, 1, 1, or 10, 5, 2, 2, 1, 1, or 6, 5, 4, 3, 2, 1, etc. Note that a combination of the lower masses always sums to the value of a higher mass unit.

4. The 5, 3, 2, 1 and 5, 2, 2, 1 Groups

These are the sets most widely used. In general a set of weights in this group

can span many decades with each decade comprising, for example, the 5, 3, 2, 1 sequence. In many cases standards or "check standards" (see below) are added

to the sets. Examples of such groups could be 5, 5, 3, 2, 1, 1, or 5, 3, 2, 1,

1 , etc.

4.2.1. Trend Elimination in Direct Reading Balances

If a single-pan, direct reading balance has truly constant sensitivity and is

subject to only a slight linear drift, the following simplification is possible:

Suppose there are four nominally equal weights, designated A, B, C, and D to be

compared in a particular calibration scheme, the following eight "direct" weighings

are done:

(1

(2

(3

(4

(5

(6

(7

(8

(9

(10

P

1 (12

(13

Place A on balance and read

Remove A

Place B on balance and read

Remove B Place C on balance and read

Remove C

Place D on balance and read 6,

Remove D

Replace D on balance and read e 5

Remove D Place C on balance and read

6

Remove C

Place B on balance and read 6^

53
(14) Remove B (15) Place A on balance and read 6

We have taken the weights in the following order: A,B,C,D,D,C,B,A. We then

make the following calculations:

e A

= i/2(e

1 + e 8

)

e B = i/2(e

2 + e 7

)

e c

= l/2(6

3 + e 6>

e D

= i/2(e

4 + e 5

)

Then the estimated difference in mass between any two weights (say C and D) is

C

T

- DT= pA

(v

c

- v

D

) + k(e

c

- e D

)

where we assume k is well-known and constant. The symmetry of the weighing sequence

removes problems caused by any linear drifts in the balance or weighing conditions.

The above example is an instance of "trend elimination." (Double substitution

is another instance.) When weighings are subject to large air buoyancy corrections,

drifts in temperature or barometric pressure may lead to errors if those quantities

are only read once during all the measurements required by the design. Some calibration schemes, however, have the property of "trend elimination" even for this

problem [3].

4.2.2. Designs

Sets or subsets of weights are calibrated together by means of a weighing design. A design simply prescribes what weighings are to be made. Each weighing is

used to estimate a mass difference between two nominally equal weights or groups of

weights in the set.

A typical design for a group, say 5, 3, 2, 1, 1, 1, could be as follows:

Mass

Observation M

g

M

3

M

2

M

] M

£l

SC

6

1 + - - +

6

2 + +

63 + - - - +

a4 + -

-

6

5 + ....

6

6 + - + - -

6

7 +_-+-

6

8 + - - - +

6

g

f -

-

6 10 * - -

6

11 "

"

54
Here +/- signs indicate the weight starts out on the left/right pan, for transposition weighing, or that the weight is the first/second in substitution weighing. We

notice that in this group there are six masses, which must be determined (K = 6), 11

measurements (N = 11), and suppose 1 restraint (e.g. the mass of the 5+3+2 summation

is known from previous measurements, L = 1). We can then calculate the number of

degrees of freedom (D.F.) for this design as

D.F. = N- K + L = ll-6 + l=6_.

From the 11 observations and the given value of the restraint one can then use the

least squares method to solve all equations to obtain best values of all differences

and therefore all masses, standard deviations, and variances. Such fitting techniques

are well described in literature available from the NBS [3]. (See Appendix D.)

In general the more degrees of freedom provided by a particular design, the

lower the estimates of standard deviation. Recall the previous section on surveillance testing. Each measurement in Type I surveillance had no degrees of freedom.

That is, we needed to determine a quantity 6. and we had only one measurement of that

quantity (K = 1, N = 1, L = 0; D.F. = 0). In Type II surveillance, however, we used

a simple 3-1 design: Three quantities were determined, for example <S.j , 63

(N = 3), three measurements were made (6-j, 6

2 , 63, so that K = 3), and there was one restraint (6^ = <$£ " 6 *p' ^nus D.F. = 3 - 3 + 1 = 1 . The appearance of a degree of

freedom allowed a crude statistical check (e.g., whether - 6-j < balance standard

deviation) and also led to a slightly reduced standard deviation (i/2/3 a instead of

a).

The design illustrated in this section is much more sophisticated than the 3-1

but is motivated by the same twin desires for good statistical analysis and small

standard deviation of the calibrated masses. The design in the above example is

shown in figure 6, where it is designated C.2. This figure is reproduced from ref.

[3]. Note that two different restraints are considered: restraint A is that the

7 mass of the summation of the 5, 3, and 2 weights is known; restraint B is that the

o mass of a single "1" weight is the known standard. We will discuss the results

based on restraint A in detail. That is, we will assume that the true mass of

M

5

+IV"M2

^ s known - Let us designate this mass as R

1".

In the 532111 design shown above, we have labeled\one of the weights as El.

This indicates that we could calibrate the 5+3+2 summation of the next lower decade

of the same weight set which could then serve as a transfer standard for a subsequent

design. By "SC" we designate a weight which is external to the set being calibrated

and which will serve as check standard.

This would be useful, for instance, if the 5+3+2 summation had been calibrated in a previous design. In this case, the 5+3+2 summation would be a transfer standard.

This restraint is used in working down in mass from 1 kg.

8

This would be useful in working up in mass from 1 kg.

55
The least-squares solutions for the five unknown weight are [3]:

fij = (1/920) {100(6

1 +62

+6

3

+6

4 )+606

5

- 20 (

6

6

+6

7

+6

8

+6

g

+6-, q+6-j

] )+460RT

}

M3 = (l/920){-68(6

1 +6

2

+6

3

+6

4

)-4<S

5+124(66

+6

7

+6

8 )-60(6

g

+6

10

+6

11 )+276R

T

}

M

2

= (l/920){-32(6

1 +62

+6

3

+6

4 )-5665 -104(6

6

+6

7

+6

8 )+80(6

9

+6

10

+6

11

)+184R

T

}

fij = (1/920){1 196-,+46

2

-1 116

3

+46

4

-1086

5

+1286

6 -102(6

7

+6

8 )-125(6

9

+(S 10 )-106

11

+92RT

}

= (l/920){-1116

1 +1196

2+4(6

3

+6

4 )-1086

5

-1026

6

+1286

7

-1026

8

-1256

g

-106

10

-1256

11

+92R

T

}

In addition, the least-squares solution provides a value for the check standard:

(SC)

T

= (l/920){46

1 -1116

2

+n96

3

+46

4

-1086

5

-102(6

6

+6

7 )+12868

-106

9

-125(6

10

+6

11

)+92R

T

}

The metrologist must determine the values 6-j , 6

2

, ... 6^ and the value of r"*" by

experiment. The least-squares solution shows how to combine the observations with

the value of the restraint in order to arrive at mass values for the weights used in

the design.

In addition to mass values, the least-squares solution provides fitted values

for the observations* which when compared with the measured values give:

Vs\ - fi lM

5

+M

3

+H2-M

l +MEl

& 2

= «

2-mJ+mJ+mJ-m^:1t+(sc ) T

S

3

= 6 3"M

5

+M

3

+M

2

+M

1 (SC ^ V

/\ »y /\ "T" ^ T

VM

5+M3+M2

V S

5

= fig-Mg+Mg+M^+M^+CSC)

V

• 5

6

=

/\ *p /V /N mm* /N w /N ^ fis-Mj+Mg-M^+M^ + CSC)

1

sr

" S

7

" aT aT /n T /\ T* T

6

7

-M

3

+M

2

+M

1 -M

zl

+(SC)

V 68-M

3

+M

2

+M

l +MH"^SC ^ V

T ~ T ^ T

6 g-M

2

+M

1 +Mzl

10"

" { icf 6

10-mJ+m|+(sc) t

IV

- 5ir

aT ^ T

6 n'M

2

+M

zi + ^ sc ^

56
These deviations are useful for two reasons. First, the estimated standard deviation

of the least-squares fit is equal to

1/2

1 N-K+l

Second, a glance at each of the eleven values of 6 . —6^ can often pinpoint the source of a blunder in entering the raw data into a computer—that is, if a blunder has

been made, the value of 6 - -6 1 affected by the mistake will often appear much larger

than all the other values.

Finally, the least-squares solution also provides estimates of the standard

deviation for each computed mass and each combination of mass. Below the design

in fig. 6 is a table called "Factors for Computing Standard Deviations." Suppose

the standard deviation of the measurement process is a. Then if design C.2 is used

to calibrate the mass of the "5" weight, for example, the table tells us that the

standard deviation assigned to this mass, ov, will be:

0.2331 x a if restraint A were used

1.7846 x a if restraint B were used.

Also from the table we see that the standard deviation of the sum of the 5 and 3

weights, Og, is given by:

0.2638 x a if restraint A were used

2.8284 x a if restraint B were used.

It is typical of least squares results that, for example,

, Ma 2 a

Z

) VZ

In this case,

m n

2 2 1/2

a

8

< ^ a

5 + a

3 ' for restraint A

2 2 1/2 a

8

> ^ a

5 + 0r>- ) ^or res ^raint B.

Thus the metrologist must choose both design and restraint carefully to minimize the

resulting standard deviations.

4.2.3. Statistical Checks

Computer programs, such as those developed by NBS, are routinely utilized in mass calibration laboratories. The user supplies all the measured data for the set,

all environmental conditions, and other necessary data. The program then provides:

1. a detailed listing of data provided

2. the least squares fit, i.e. the desired mass values

3. the "F ratio" and the "t value".

Two crucial assumptions underlie the calibration of unknown weights by leastsquares fitting of design data:

57
OESIGN C.l 5. 3. 2.1.1 K= 5 t DE S I GN C . 3 5.3 ,2,2 .1.1 N= 8 1

( HA YFORD) .F .= 4

\ CESERVAT IONS 5 3 2 1 1 {

1 CESERVAT IONS 5 3 2 2 1 1 V( 11 + - Y< 1 ) + + Y( 2) - + f Y(2> + VC3) + Y(3) + — + Y< 41 - - Y<4 ) Y<5> - Y(S) + — + Y(6» + - Y<6> + ~ — Y«7» + - - 4 Y<7> + vent -

J Y<8) + Y<9> + RESTRAINT A + f Y( 10

)

f rill!

K= 6

N = l 1 D.F.= 6

RESTRAINT 8

FACTORS FOR COMPUTI NG ST DEVS WT RESTRAINTS

A B 5 3 2 1

1 .41 75 .0000

1 .41 75 .5345 +

2 .3546 .9258

3 .3338 1.309 3 +

4 .5237 1 .6903

e . 2673 2. 0702

6 .5237 2.4495

7 .3338 2.9277 +

8 .3546 3.3381 +

S .5632 3. 7033 + 10 .0000 4.1748

I

RESTRAI NT A RESTRAINT 8

F ACT CR S FCR COMPUTING ST DEVS WT RESTRAINTS

A B S 3 2 2 1

1 .3592 . 0000

1 .3592 .47 14

2 .4259 .7553

2 .3625 .7553

.3835 1.2536

.6116 1. 1 123

4 .61 30 1 .4254 + £ .2266 1.7412

6 .4790 2. 0893

7 .3835 2.4 199 +

e . 3625 2.9493 + +

9 .4711 3. 2708

10 .0000 3.5922 *

CESIGN C.2 5. 3. 2. 1 • 1.1 K = 6 DESIGN C.4 5.3 .2.2 .1.1 K= 6 N= 1 1 N= 1 2 D.F.= 6 D.F.= 7 CBSEPVAT IONS 5 3 2 1 1 1 CBSERVATICNS 5 3 2 2 1 1 v<u - V< 1 > — Y(2 > - + Y(2) —

YI 3» + - V(3) — Y<4» + — Y(4)

VI 5) Y<5 ) YI6I Y(6>

Y{7) V(7) + Y<8 ) V( 8) VI9J Y<9 ) V< 10} Y< 1 0) Villi V« 11 1 VI 12)

RESTRAINT A + RESTRAINT A RESTRAINT B

1

RESTRAINT 8 FACTORS FOR COMPUTI NG ST DEVS

WT RESTRAINTS FACTORS FCR COMPUTI NG ST DEVS

A B 5 3 2 1 1 1 WT RESTRAINTS

1 .3551 .0000 A 8 5 3 2 2 1 1 1 .3551 .5000 1 .4796 .0000

1 .3551 . 5000 + 1 .4796 .8165 +

2 .2638 .7802 + 2 .40E5 1 . 054 1 3 . 2985 1 .0 885 + 2 .3127 1.0541

4 .4778 1.4781 + a . 2848 1.4530

5 .2331 1 . 7846 3 .6173 1.7638 + 6 .4299 2. 1 64 4 + + 4 .4372 2. 0276 + 7 .2985 2.5216 s . 2887 2.3805

8 .2638 2. 8284 + 6 .5888 3. 1091

9 .46 16 3.2016 + + 7 .5608 3.3 83

1 . 0000 3.5509 + e .3127 3. B006

9 .61 19 4.5216

1 .0000 4. 7958 + +

Figure 6. Least squares information for the design described in the text.

(Reproduced from ref. [3]).

58
1. The scatter in the data just taken is typical of the scatter found in previous measurements.

2. The mass of the standard weight used in the calibration design has not

changed from its accepted value.

The F ratio and t value provide important tests of these two assumptions.

The F ratio essentially monitors the precision of the measuring process. A

detailed discussion is presented in Appendix D. We note that

where a is the variance of the particular balance being utilized (based on a large

collection of previous measurements). Generally, we expect F to be close to 1, but

we must not be surprised if a particular value of F is somewhat larger than 1.

In particular, a simple check for F can be (and has been) established for most

mass metrology laboratories. By comparing F with a fixed ratio F , which could be

defined at the 99 percent confidence level one can then easily monitor whether the

measured ratio F is greater than F^

i.e. F>F

t

.

If this check holds true, then the measurement process is considered "out of control"

and further studies have to be conducted. The quantity Ft depends only on the

2

degrees of freedom (k-|) in s and on the probability level at which we wish to conduct the test. The calculations are outlined in Appendix D.

The second test is called the "t test" which monitors the accuracy of the measurements. For this test, an extra mass, the check standard, is included in the

particular design. A simple check is then performed to see whether the mass value

assigned to the check standard by the calibration agrees with the accepted value.

The t test thus monitors the systematic errors of the measuring process.

In particular one calculates

(SC) = observed mass of the check standard as found by least-squares fitting;

accepted (SC)^ = accepted mass of the check standard;

2 2

F = s Va

(SC)' - accepted (SC)'|/a

c where \

and

to

2 + a

T

2 + (NSC

/N

R )

V

59
Here

Oj = accepted variance of the measurement process between runs (see below)

l = multiplier which is determined by the least squares process,

a = "process" standard deviation of the balance

N = the nominal value of the check standard

N R

= the nominal value of the standard or transfer standard used as the

restraint in the least squares solution

a

R

= the standard deviation of the value of the restraint. a

R

is taken as zero unless the restraint is a transfer standard, whose value was determined as part of the calibration of the complete weight set [1].

For simplicity, we will discuss a case for which a

R

= 0. In general one checks to see that

t < 3. c The probability is less than one in 100 that this inequality will be violated by

chance. Thus t = 3 is taken as the control limit. Any measurement for which t > 3

is considered "out of control" and is repeated.

CAUTION : The t test is one of the best statistical measures of systematic

errors available. Nevertheless, by their very nature, systematic errors are difficult to detect. If, for instance, buoyancy corrections were not important in assignment of mass to the check standard but were important in assigning mass values to

the weights being calibrated, the t test will not detect systematic errors in the

buoyancy correction. Also, the t test cannot detect an identical change in the

standard and check standard. We have taken a very simple example of a t test. The

most general case is shown in ref. [1].

The variance of the measurement process between runs, Gy , is an important

concept [1,8]. Recall that the process standard deviation, a, is a measure of

scatter over the period of time it takes to make a mass measurement: i.e., a few

minutes to a few hours. There may, however, be sources of random uncertainty which

fluctuate more slowly (i.e., over days or months); but still rapidly compared with

intervals between recal ibration of a given weight. How can we estimate the process

standard deviation for this longer period of time? The easiest way is to monitor

the values of mass assigned to the check standard each time it is used in a run. After many runs, over a span of many months, we can estimate the variance, a

2

, of

the observed values of the check standard about their mean value.

If the process standard deviation has no "between-run" component, then

a* = £o

2

(aj * 0)

2 where a is the process variance which was found by pooling variances estimated from.

the least squares fits to many individual runs. That is, a is the within-run pro2 cess variance based on combining calculations of s from many runs.

60
It will generally be true, however, that

2 „ 2 a

c

> to

indicating that there is an additional source of scatter from run to run. We define

this between-run variance for the measurement process as

2 2 2 Oj = a

c

- la

2

If <jy is not negligible, the source of between-run scatter should, of course,

be sought and eliminated if possible? Failing this, the process standard deviation

must be expanded so that the over-all uncertainty assigned in the calibration report

includes the between-run component.

The cautionary statement given above for the t test also applies to the use of

the check standard in estimating between-run scatter. That is, if a weight being

calibrated differs markedly in its construction from the weight(s) used in the check

standard, a Oj component may go undetected if one relies only on check-standard

behavior.

Suppose a series of design data is "in control." True mass values have been

assigned to the unknown weights and uncertainty limits have been given. What do

these uncertainties mean? In a practical sense, the meaning is that the uncertainty

bands assigned to each mass calibrated by the methods of ref. [1] should almost

always overlap the uncertainty bands which would have been obtained had the masses

been calibrated at NBS.

Note that this specification, while satisfying the expectation of the overwhelming

majority of users, is somewhat less stringent than asserting that: the true mass as defined in eq. [1] and reported in S.I. units is almost surely within the uncertainty

band provided by the calibration.

\

oT

has been found to be negligible for the NBS calibration process.

61
5. CONCLUSION

This document has been less of a complete treatise on mass metrology than a guide for the mass metrologist. We have indicated appropriate references where the

various topics touched on are treated in greater detail.

What we have tried to offer is a coherent development of the fundamental con- cepts of mass metrology as a means of explaining the relevance to the metrologist of

the references found at the end of this publication.

In brief, starting with the basic law of classical mechanics, we derived:

° Relations for the true mass differences between nominally equal weights

(eqs. (10') and (19b))

Relations for apparent mass differences between nominally equal weignts

(eqs (18) and (19a))

We stressed the importance of measuring or calculating air density in order to make

proper buoyancy corrections.

We then provided:

An explanation of how the basic mass relations apply to measurements on several types of commonly used balances.

A demonstration of how these relations are used in two types of surveillance testing.

A brief sketch of how these relations are used in mass calibrations.

We concluded with a discussion of the statistical checks crucial to a calibration.

In a document such as this, there is a danger of losing the basic outlines of a mass measurement program in the many details essential to carrying out a successful

program. Every program must have a goal --for example, the calibration of weights in

a certain range of masses to a desired uncertainty.

One must then acquire balances equal to the task and house them in a room which

will not degrade their performance. Control charts should be used to establish the

longterm reliability of the balances. Calibrated sets of weights to be used as working standards should be acquired. Surveillance testing can establish the stability of these weights to within the surveillance limits.

Anticipated levels of buoyancy correction should be estimated. Auxiliary equipment such as barometers, hygrometers, and thermometers should be acquired, if necessary, to achieve adequate capability in determining the density of air. These instruments must be calibrated periodically.

When calibrating weight sets, weighing designs should be used. These are selected on the basis of providing the necessary accuracy with the least number of

weighing operations. Great care must be exercised in assigning a total uncertainty

to calibration results.

The techniques outlined in this document form the core of such a mass program.

62
We would like to express our appreciation to Dr. J.D. Simmons of NBS for his

support. One of us (K.J.) wishes to thank R. Weber and Dr. D. Cauffman, both of

Lockheed, for their encouragement, help, and overall support for this endeavor.

Also, one of us (R.D.) extends thanks to his colleagues at NBS for their cooperation

and technical guidance. We especially wish to thank Mrs. Betty King for typing the

manuscript.

\

63
6. REFERENCES

[I] Varner, R.N.; Raybold, R.C. National Bureau of Standards mass calibration com- puter software. Nat. Bur. Stand. (U.S.) Tech. Note 1127; 1980 July. 164 p. NTIS PB80203441

[2] Jones, Frank E. The air density equation and the transfer of the mass unit. J.

Res. Nat. Bur. Stand. (U.S.) 83(5): 419-428; 1978 September-October.

[3] Cameron, J. M.; Croarkin, M. C; Raybold, R. C. Designs for the calibration of

standards of mass. Nat. Bur. Stand. (U.S.) Tech. Note 952; 1977 June. 64 p. NTIS PB268499

[4] Aimer, H. E., Keller, Jerry, ed. Surveillance test procedures. Nat. Bur.

Stand. (U.S.) NBSIR 76-999; 1977 May. 77 p. NTIS PB268130

[5] Pontius, Paul E. Mass and mass values. Nat. Bur. Stand. (U.S.) Monogr. 133;

1974 January. 39 p. NTIS C0M7450309

[6] Aimer, H. E. Method of calibrating weights for piston gages. Nat. Bur. Stand.

(U.S.) Tech. Note. 577; 1971 May. 54 p. NTIS C0M7150264

[7] Precision measurement and calibration statistical concepts and procedures,

Special Publication 300, Volume 1, Pontius, P.E., Cameron, J.M.; pp. 2-20. GPO

Stock No. 003-003-00072-8

[8] Pontius, P.E. Measurement philosophy of the pilot program for mass calibration.

Nat. Bur. Stand. (U.S.) Tech. Note 288; 1966 May. 39 p. (Reprinted January,

1968).

[9] Schoonover, Randall M. A simple gravimetric method to determine barometer

corrections. J. Res. Nat. Bur. Stand. (U.S.)85 (5) 341-345, 1980-SeptemberOctober.

[10] Schoonover, Randall M. A look at the electronic analytical balance. Anal.

Chem. 52: 973A-980A; 1982 July.

[II] Schoonover, Randall M.; Jones, Frank E. Air buoyancy correction in high-accuracy weighing on analytical balances. Anal. Chem. 53: 900-902; 1981 May.

[12] Cameron, Joseph M. ; Hailes, Geraldine E. Designs for the calibration of small

groups of standards in the presence of drift. Nat. Bur. Stand. (U.S.) Tech.

Note 844; 1974 August. 32 p. NTIS C0M7450762

[13] Natrella, Mary Gibbons. Experimental statistics. Nat. Bur. Stand. (U.S.)

Handb. 91; 1963 August (reprinted 1966 October with corrections). GPO Stock No.

0003-003-00135-0

[14] ASTM E617-81. Standard specifications for laboratory weights and precision mass

standards. 1983 Annual Book of ASTM Standards Vol. 14.02: 455-474.

[15] Measurement Assurance Programs Part II: Development and Implementation, Nat.

Bur. Stand. Special Publication 676II, Croarkin, C; 1984, 124 p.

64
References with NTIS numbers are available from:

The National Technical Information Service

Springfield, VA 22161

References with GPO stock numbers are available from:

The Superintendent of Documents

U.S. Government Printing Office

Washington, DC 20402

65

APPENDIX A

APPARENT MASS OF BUILT-IN BALANCE WEIGHTS

Balance manufacturers almost always adjust the built-in dial weights of singlepan balances so that their apparent masses equal the value of the dial position.

No matter of what density material the manufacturer has actually made his weights,

the user generally assumes that the true masses equal the dial values and the

densities of the built-in weights equal p R

, the basis density.

Let us see why this scheme works and what its limitations are. These considerations can be demonstrated by referring to a single dial -weight (of density p D

) that exactly balances a weight X, which is on the pan.

We begin, since we are discussing a balance of the type shown in fig. 3 with

eq. (22):

ke

1 = (mJ-pa

v

x

) - (zdT-p

a

ev.) + ke ,

where we have renamed M by M

x and VM

by V^. We will assume that 6q was adjusted to zero before X was placed on the balance. We have chosen a special case: after X was placed on the balance, the screen reading 6^ returned to zero with the removal

of a single dial weight. Let us refer to this dial weight as "D" and say that it

has a true mass and a volume V at the balance temperature. In this special

case, with 6q = 6-j = 0, eq. (22) becomes

= (Mj-PA

V

X

) - (D

T

-pA

V) . (Al)

Since p

D

= D^/V, we can rewrite (Al) as

mx"Pav

x

= dT

0-pa/pD

)

or,

Mj = D

T

(l-pA

/ P[) ) + pA

V

x

e M

1 \ (A2)

Now we ask the question, "What if one assumes the mass of D is equal to Nq, its

nominal value, and the density of D is equal to p R

, the basis density of the apparent mass scale to which the balance weights have been adjusted?" This assumption

is not correct because

P R * p D

and

D

T

f Nq.

Al
Nevertheless, the incorrect assumption would tell us that

mJ = n

d O-pa

/p

r

) +

Pav

x em2

. (A3)

We can next ask ourselves "Under what conditions does M-| = M

2

?"

Recalling that Mn is made equal to N n , the nominal dial value, we subtract eq.

(A3) from (A2).

U U

Mr

M

2

= DT

(1-pa

/pd

) - Mg(l-p

A

/p

R

) . (A4)

Also, by definition, (see eq.( 11) in the main text),

Mq = DT

(1-P /PD0 )/( 1 -P /PR0 ) (A5)

where the subscript zeros refer to the standard conditions specified in the definition of apparent mass. Substituting (A5) into (A4),

M

2

-M

] = D

T

[(1-p /pdo )(1-pa

/p

r )/(1-p / Pro ) - (l-pA

/p )3 •

The first approximation we will make is that p Rq

= p R

and p^q = Pq. That is,

the temperature is sufficiently close to 20 °C that the expansion in volume of the

balance weights is negligible. Thus

M

2

-M

] = DT

[(l-p /pD ) (1-pa

/p r )/(1-p /pr

) - (l-pA

/pD

)] •

Next, we use the relation developed in the main text:

-1 2 3

(l-P /pR

)~ = 1 + Pq/PR + (Pq/Pr) + (Pq/PR

) + •••

so that

M

2

-M

1 = D

T

[(l-p /pD )(l-p

A/pR ) (1+P /PR+ (P /PR

) 2 + (p /Pr)

3 + •••) - (1-PA

/PD

)] •

2 2

Finally, we multiply out but throw away all terms of order (pq/Pr) > ( pA

p(/ pR^'

2 3 3

Pq/p d

P r

, and smaller. Since Pq * pa

<_ 1.2 mg/cm and p D * p R

> 7.5 g/cm (the

balance manufacturer has seen to this), then neglecting these terms leads to errors

of 3 x 10"^ percent or less.

Thus, we now have

VM1 =

°T

(p /p R

p /pD

" pA/pR + pA

/pD

}

or, more simply

Mg-M, = D

T

( P(J -pA ) (l/p

R-l/p

D

) (A6)

A2
The balance manufacturer has selected the metal for his weights from stock

having a density Pp sufficiently close to so that, near sea level, M-j -Mg will not

exceed the balance tolerance.

In recent years, balance manufacturers have been making weights of stainless

steel. Equation (A6) shows why it is, therefore, desirable for these weights to be

adjusted on the 8.0 basis.

The chief virtue of the apparent mass scheme is that, even though different

balance manufacturers may make their weights of different alloys, a user can be

completely unaware of these subtleties and still derive reasonably accurate true mass results on any balance so long as the correct basis density is used.

Example :

A balance has 20 g of built-in dial weights. The density of these weights is

3

actually 7.8 g/cm . The weights have been adjusted so that the dial readings

correspond to apparent mass on the brass basis. What is the value of l^-M-j at

maximum load if p^ = 1.16 mg/cm

3

? What if the laboratory is at high elevation so that pA

= 1 .00 g/cm

3

?

Answer :

a) M

2

-M

] = 20 g (0.00120 - 0.00116) - j^-] = -7 ug.

b) M

2

-M

1 = 20 g (0.00120 - 0.00100) - j^] = -37 ug.

By comparison, the calibration uncertainty for high-quality 20 g-weights as measured

by NBS is about 10 ug; the tolerance for 20-g, Class 1, metric weights is 74 yg

[14].

A3

APPENDIX B

OUTLINE OF THE DERIVATION OF pA>

THE DENSITY OF AIR

In order to find p^, an equation of state involving temperature, humidity, and

barometric pressure has been developed. We will follow the definitive derivation of

F. E. Jones [2] and briefly describe his arguments to arrive at a simplified re- lation sufficiently accurate for our applications. The reader is urged to consult

[2] for full details.

In order to derive the density for a mixture of two gasses, let us start with

the ideal gas law.

pV = nRT

and

where

n = number of moles

m = mass of gas

M = molecular weight of gas

R = universal gas constant

T = temperature in kelvins = (273.15 + t) for t in degrees Celsius

From the above, we can rewrite

pV =

jj

RT so that m = ^~

and

Mm PA . m since pA

RT pV p " V

Specifically, consider air consisting of dry air and water vapor. For the dry air. we have

m

D

=

P D VM

D RT

and for the water vapor, we have

p VM ww

Since the total density is given by

pA

mn + m

D w

Bl
we obtain

, Pdvm

d + PwVMw\ /v PA

= ( + ) /V

RT RT V

Using Dalton's Law for partial pressures

P = P-, + P

2 we can substitute for

PD

= P " Pw

so that

and using

pA

^

(PM

D

" P«M

D + PwM

w'

e = —

M

RT

W

At this point it is important to consider for a moment the correction necessary for

a gas which is not ideal. To do so we rewrite the last relation as

P

=

PA

RT

1

1 3(E -D

P

For an ideal gas the ratio

RT

Mo

1

1 + PwI 1 — \z

P

=.= Z

1)

B2
has to be equal to 1 . If we want to consider non-ideal gas corrections, then we

have to incorporate Z into our formalism. Hence,

P

= PA

RTZ

Mr,

1

1 + ^(e-l)

P

where Z is called the "compressibility factor."

Rewriting this relation for the mixture density, we now have

M - [(P + (e - 1)PW

]

RTZ

Let us now turn to the dependence of this relation on the relative humidity. We

define

PW

U = — x 100

where

U = relative humidity in percent

p^ = effective vapor pressure of water in moist air

e£ = effective saturation vapor pressure of water in moist air.

Furthermore, we know that

where

e

s = the saturation vapor pressure of pure-phase water.

The ratio of the two pressures is called the "enhancement factor" for saturated

water vapor and is given by

f =

so that

and since

e' = e.

U • e'

s_ 100

B3
we have

U • e

s

• f

Pw • W

100

Substitution into the mixture density relation yields

PA

- — [P+ (e - 1) —

]

M

RTZ 100

where U must be given in %.

This relation constitutes the final formula. The parameters P, T, and U must

be measured by the user.

Substituting the best available values for R anf and choosing reasonable

values for M

D>

Z, and f, which are approximated as constant parameters, we have

R = 8.31441 MU§_ = 8,314.4 j° uleS

mole-K kmole-K

M

D

= 28.964 g/mole

Z = 0.9996

f = 1.0042

Mw

= 18.0152 g/mole.

Since

we obtain

pA

where

- = \ = 18.0152

M

D 28.964

pA

= 3.4848

(p . 0.0037960 Ue

s

) x 10" 3

mg_\ 0.0034848

(P , .0037960 UeJ . cm

3 / (t+273.15)

t is in °C

P is in pascals (133.3224 Pa = 1 mm Hg) .

Converting to mm Hg pressure, we have

pA

ms_\ 0.46460

(p _ 0i0037960 Ue ) . 3) (t+273.15)

S cm

B4
The parameter e

g has been determined by fitting measured data between 288.15 K and

301.15 K. The relationship developed is

-5315.56

e

s = (1.3146 x 10

9

) e U+273 ' 15)"mm Hg.

It should also be noted that the value for the enhancement factor used above, i.e.

f = 1.0042 can be approximated more accurately by

f = 1.00070 + 4.150 x 10' 6 x P + 5.4 x 10" 7

t

2

.

In the above final equation, the relative humidity is given in percent

U = % (e.g. 51.2, 43.7 etc.)

and the pressure P in mm Hg.

(and, therefore, z) depends on the mixture of gases, other than water vapor,

which makes up ambient air. The chief variability in this mixture comes from C0

2

and 0^ levels. These are assumed perfectly correlated—that is, COg levels can only

increase locally at the expense of

2

and vice versa—as in processes of combustion,

respiration and photosynthesis.

Fortunately, the variation has little effect on Mp. In eqs. (20a) and (20b)

(main text) we have assumed the ambient level of CO2 typically found in the NBS mass laboratories (0.00042 mol of C^/mol of air). A 100 percent increase in this level

would raise by less than 0.02 percent.

Z is slightly dependent on barometric pressure, temperature and relative

humidity [2]. Between 19 °C and 26 °C; 525 mm Hg and 825 mm Hg; and and 100%

R.H., Z varies by less than 0.03 percent.

\

B5

APPENDIX C

PROPAGATION OF ERROR THROUGH THE AIR DENSITY EQUATION

Using eq. (20b) (developed in Appendix B), we have

=

0- 46460 (p_ .0037960.U.ec ) M

(t+273.15)

s

For ease of writing, we will set

P = PA

a = 0.46460

b = 273.15

c = 0.0037960

e

= e

s . such that

p = ——(P - c-U-e) . (t+b)

We want to determine the uncertainty for p using

dp = ^£ dt + 9£ dP + 9£ du . 9t 8P 8U

3p_

: at

Using (CI ) , we find

a (p . c-U-e) idt at t+b

. (-=!> p

t+b

9£.

3P

Using (CI), we find

9£ = _±

8P t+b

and since

p ^

(t+b)

CI
where we ignored the second term in (CI) since it is only a secondary source of

error, we have

. (C4)

BP P

Bp..

8U

From (CI ) , we obtain

aP aceU

p

t+b t+b

so that

3p _ ace

3U (t+b)

aP and again using p

we have

. (C5)

3U P

Next, we have to estimate the combined uncertainties due to t, P, and U. In Section

2.3.2 (main text) we asserted that the temperature should be known to ±0.4 °C so that

dt = ±0.4 °C .

Similarly,

dP = ±1.1 mm Hg

and

dU = ±16% . 3

All these values assume that we want to know the air density to 0.0017 mg/cm .

Substitution of all terms into (C2) yields

dp = =£- (±0.4) + £ (±1.1) - p (±16) . t+b P P

In the worst case, these uncertainties would add linearly. However, we assume

the errors are uncorrected so that a better estimate of the total uncertainty is

given by

C2
Jp

2

-(0.4)

2

, p 2

(1.1)

2

, cVgjjg

" (t+b)

2

P

2

P

2

1 (f

16 + +

C

2e2

256

(t+b)

2

P

2 P2

At standard conditions we have

t = 20 °C

P = 760 mm Hg .

We now have

V

, 0.16 , 1.21 , C 2e

2

256

6p = Pi/ + + (20+b)

2

(-760)

2

(760)

2

Substituting p = and replacing b and c with their numerical values, we have

6pA= J

~ " ~ J0.0037960)

2

• 256-e

2 + ——— +

PA ' (20+273. 15)

2

(760)

2

(760)

2

Also e

s at 20 °C is 17.54 mm Hg, so that finally

<5p —A

- 0.0024

pA

or

6p—A

* 0.24%

PA

C3

APPENDIX D

SOME STATISTICAL CONCEPTS

We start out with an infinitely large population of a random variable; to be

practical, let us say a very large number of similar balance observations. We assume that this population is normally distributed with a mean of y and a variance

of a

2 .

Let us take a random sample of size n from the population. Then the mean of the

sample is defined as

i n

X = - £ X. (Dl)

n 1=1

1

with an estimated variance for a single observation of

n

- ?

(x rxr

s 2 = ^(D2)

n-1

2 2

As n grows infinitely large X -* y and s * o . (This, in fact, defines what we mean

by u and a).

The treatment of least squares data is more complicated than analysis of repeated measurements of the same quantity. In this case, average values of several different masses are calculated at the same time from data obtained during one weighing

design. If N measurements were necessary to complete the design, the metrologist has

in hand the results of these N mass comparisons: 6-j , <$2 » •• 5 n- Tne least squares

analysis provides a set of "best" estimates: 6-j, ££, ^n* as we^ as a set °f

"best" estimates: NlJ, M^.-.M^, of the K unknown masses calibrated by means of the

design. These "best" estimates are linear combinations of the measured values

6-j, 6

2 ,...6N

with the mass of the known standard used as the restraint. The linear

combinations are uniquely determined by the choice of design and restraint. Least

squares solutions to the most useful weighing designs are tabulated in ref. [3].

The standard deviation of the least squares fit to the design data is estimated

from the formula^

N

2

s 2

= ^

(D3)

N-K+l

where the denominator is given by the degrees of freedom (= number of observations - number of unknowns + number of restraints). If the same measurement design were repeated m times, a better estimate for the standard deviation of the process could

be obtained:

Least squares is so named because the least squares solution: 6-j, d^*.

.6m

minimizes the value of s 2in (D3).

Dl
2 2 2

s 2 . J 2 m

(D4) m

2 th 2 2

where s- is the estimated variance of the i run. As n>*», s -kj . This defines

2

what we mean by a in the case of least squares. Note: the number of degrees of

freedom (D.F.) in s as defined by (D4) is m(N-K+l). Also, this a is the same a discussed in the text, the "process standard deviation."

Besides a, we would also like to know the standard deviation of each individual

mass, M., computed from the least squares analysis. The answer involves matrix

algebraJ and depends both on the design and restraint which were used, as well as on

a. All of the commonly used cases are^tabulated in ref. [3]. In general, the

standard deviation of a measured mass M. is some value l.o, where £. is a number that

J J J depends only on the particular design and restraint. This number can be found in

ref. [3]; a is defined above.

When two or more weights--j and p for example—are used in combination, their

2 2 1/2 combined standard deviation is not simply {i. + i^) but is often somewhat

greater. This is because data taken from a design are usually correlated. Both the

recipe for handling weight summations and tabulated values for many special cases can

be found in ref. [3].

From the above discussion, it should be evident that care must be taken in

choosing the design and restraints for any set of weights to be calibrated. Some

choices will minimize individual l- values but weight summations may have large

uncertainties.

2

Assume that an estimate of a has been computed from m designs by application of

2

eq. (D4). Future designs will produce individual values s- which will also be

2

J estimates of a as long as the measurement process remains stable. An F statistic

F = -i (D5)

s

2 2 can be computed for each new s.. Its purpose is to test the agreement between s.

o J 2

* and s --or more precisely to test whether or not s- comes from the same distribution

2

J of measurements that produced s .

Given a large number of F statistics, each of which is based on the same design,

a histogram of these F statistics will fall very nearly on the universal curve known

as the F distribution. The theoretical calculation of this curve depends only on the

2 2 degrees of freedom in each s., i.e. N-K-l, and on the degrees of freedom in s , i.e.

J m(N-K-l). The distribution is scaled so that the area under the curve is equal to 1,

making it possible, for example, to find the point F^ such that 99 percent of all F values will be less than F^.. As a consequence, the next time that we carry out the

weighing design, we expect that there is a 99 percent chance that the F statistic

computed from that particular design will be less than the percent point Ft- If, in

fact, it turns out that the F statistic is greater than the percent point F^, the

D2
precision for that design is poorer than is expected, and the design should be

repeated.

The percent point Ft

is often represented in tables as F(v-,, v

2 ,l-a) because it

depends on:

2

v.j = the degrees of freedom in s^ which is N-K+l

= the degrees of freedom in s^ which is m(N-K+l)

a = the significance level such as a = 0.01 (i.e. 99 percent)

For mass calibrations at NBS, the significance level is chosen to be a = 0.01,

and because the measurement process has been tracked for a long time resulting in an

2 estimate of a that has a very large number of degrees of freedom, the value that is

used for the test is

Ft

= F(vv

co, 0.99)

In this special case, Ft can be well -approximated by* :

3 + 2.32635

for

and

for

9(N-K+1) %9(N-K+1)

v

1 > 2

Ft

= 6.64

=1

Tables and detailed discussion on the F value and t ratio can be found in ref. [13].

A derivation of this result is well beyond the scope of this work. The interested reader may wish to consult: Paulson, Edward. An approximate normalization of the

lSSfo

S °f variance distribution. Annals of Mathematical Statistics. 13: 223-235;

D3
NBS-114A (rev. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

PUBLICATION OR

REPORT NO.

NBS SP 700-1

2. Performing Organ. Report No 3. Publication Date November 19 84

4. TITLE AND SUBTITLE Industrial Measurement Series

A Primer for Mass Metrology

5. AUTHOR(S)

K. B. Jaeger and R. S. Davis

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as item 6.

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 84-601090.

~~| Document describes a computer program; SF-185, FlPS Software Summary, is attached.

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

bibliography or literature survey, mention it here)

This paper attempts to fill the need for a coherent guide to the

many publications which document the NBS program in mass metrology. The

topics we emphasize are generally those which experience has shown to

present the greatest difficulties for metrologists new to the field of mass measurements. Thus we have included many worked examples and have

retained steps often omitted in more scholarly treatments of the same

subjects. A full bibliography is included so that the reader may also

consult the primary sources of this work.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Air buoyancy; apparent mass; calibration; mass; metrology; surveillance

testing.

13. AVAILABILITY

[X\ Unlimited

| | For Official Distribution. Do Not Release to NTIS

fXI Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

[TJ] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

85

15. Price

USCOMM-DC 6043-P80
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