NBSIR 74-454

The Equivalence of Gravimetric and

Volumetric Test Measure Calibration

Randall M. Schoonover

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

February 1974

Final

Prepared for Optical Physics Division

Institute for Basic Standards

Washington, D C. 20234

NBSIR 74-454

THE EQUIVALENCE OF GRAVIMETRIC AND

VOLUMETRIC TEST MEASURE CALIBRATION

Randall M. Schoonover

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

February 1974

Final

Prepared for

Optical Physics Division

Institute for Basic Standards

Washington, D. C. 20234

1 .

U. S. DEPARTMENT OF COMMERCE, Frederick B. Oent. Secretary

NATIONAL BUREAU OF STANDARDS. Richard W. Roberts. Director

The Equivalence of Gravimetric and Volumetric

Test Measure Calibration

by

Randall M. Schoonover

INTRODUCTION

Recently the Mass and Volume Section began taking steps toward

modernizing the calibration program for volumetric test measures,

devices used to contain or deliver known volumes of water. A

review of the procedures used indicated a need for Improvement in

the following areas.

1. Methods of data reduction

2. Data analysis

3. Closure which we define as follows: when the

values assigned by different calibration methods do

not disagree by an amount larger than can be

explained by measurement uncertainties, closure is

said to exist.

The first two of these concerns have been attacked by improved

models for data reduction and application of statistical

techniques

In this study, we have addressed our attention to the closure of

values assigned to our working standards by different methods.

As reported earlier [1]* there are two practical methods in use

by which test measures are usually calibrated. The gravimetric

method establishes the mass of water either contained or delivered from a test measure and hence the associated volume.

The volumetric method employs a known volume from which water is

transferred to an unknown volume and thereby the contained or delivered volume can be derived.

It can be argued that both methods of calibration are equally

correct or that one is more correct than the other. We shall not

pursue these arguments in this report but will demonstrate

closure for the NBS processes, regardless of the calibration

method, from 1 to 830 gallons.

* The numbers in brackets refer to similarly numbered references

at the end of this paper.

- 1 -
TEST MEASURES

There are several basic types of test measures, many of which

were used to obtain our closure data. Almost all test measures,

regardless of design, have some characteristics in common. The

majority have nominal volumes ranging from 1 to 1,000 gallons in

increments of 5 gallons. Those most frequently encountered are

the 1, 5, 50 and 100 gallon vessels.

Another common feature is that nearly all test measures are fabricated from mild steel or 304 stainless steel. Their respective

thermal coefficients of expansion (cubical) are 18. 6X10-^/°? [2]

and 26.5X10*"^/°F [3]. Various organizations by agreement report

the volumes of test measures at 60"?, although in use the

temperature span may be quite large. The advantage of dealing

with only two materials whose thermal coefficients are well known

is obvious.

All test measures have at least one reference point (end point)

from which volume is defined and in many vessels there are two

points. These may be valves, graduated necks, slicker plates, or various combinations of them. Spirit levels are also an important part of defining some end points as well as drainage

characteristics.

MAINTENANCE AND DISSEMINATION OF VOLUME STANDARDS

NBS calibrates volumetric test measures as a service for the

public. These measures vary in design and size from 1 to 1,000

gallons, the vast majority of which are between 1 and 100

gallons

.

Our working standards (1, 5, 30, 50, 100 gal) are now maintained

by both the gravimetric and volumetric transfer methods of

calibration. Both of these methods are described with adequate

detail in NBSIR 73-287. In essence any working standard or combination thereof can be transferred to the 100 gallon

standard. Thus we can demonstrate what we call within-group

closure, the internal consistency of our standards regardless of

calibration method. However, in practice we must exceed the 100

gallon level to include our expected working range of 1,000

gallons. In doing so, we face a dilemma unless we have the secure foundation of closure, that is we could unknowingly

propagate two volume scales, one based on gravimetric

calibrations and the other based on volumetric calibrations which

might not necessarily agree with each other.

Not only should we provide within-group closure of the standards,

but also we should extend this property to include the range of

their usage in our calibration program. The remainder of this

report describes the measurements and data collected over the

past few years to demonstrate closure up to 830 gallons.

- 2 -
CLOSURE MEASUREMENTS

To simplify following the procedure used to demonstrate closure,

the reader is referred to figure 1. For convenience we have assumed the calibration process begins with 5 gallon vessels whose

volumes are well defined by gravimetric calibration. This volumetric information is transferred to other vessels which in turn

are also calibrated gravimetrically and the two outcomes are then

compared. The blocks indicate a gravimetric calibration, and the arrows are volume transfers from that vessel. Dotted blocks indicate areas of calibration where both methods are routinely used

for control chart surveillance of the calibration process.

Blocks marked as 5 gallon summations are separated from the 5

gallon blocks only to aid in clarity.

The 5 gallon vessels have not been transferred to all vessels

calibrated gravimetrically, the reason being the conservation of

time and energy as anyone familiar with the process will

recognize.

In comparing the performance of one measurement system to

another, we must know the uncertainty associated with the

measurements produced by each process. For the gravimetric

process the random errors of the process are associated with the

operation of filling and subsequent weighing, so that a standard

deviation based on a sequence of independent fillings provides a measure of the effect of random errors. The error arising from

the weighing operation is negligible so the discrepancies between

successive fillings arise from the following: varying amounts of

liquid retained on vessel walls when drained, erratic meniscus

behavior detrimental to end point determination and failure to

achieve an isothermal condition before measuring the liquid

temperature, etc.

If there are n independent measurements, , X2 , . . . ._X of

the same vessel, the standard deviation of the average, X, is

given by s//n where s is the standard deviation of a single value

from the mean computed from the formula

This value of s is a measure of the process standard deviation,

a^, associated with the gravimetric method for the volume and

type of vessel being measured.

DATA

- 3 -
The uncertainty of the gravimetric value is then ±U^ where

Ui = 3si//n Eq. (1)

Assuming that errors from the weighing process are negligible,

the value of o will be dependent on the nominal size and type of

measure being calibrated.

For a test measure calibrated by volume transfer using a reference test measure calibrated by the gravimetric method,

there will be two sources of error in the value determined for

the larger vessel. The first of these arises from random errors

in the filling operation introduced at each of the K transfers

required to fill the larger vessel. The second source depends on similar errors associated with the larger vessel such as reading

the end point, drainage, temperature, etc. The errors from these

two sources will combine in quadrature to give a standard

deviation of o for the random errors^ associated with a single

filling of tl^e larger vessel. For ni determinations, the

standard deviation of the average will be a / /ni . The

uncertainty, U;^ , assigned to the value of the reference measure

will be repeated K times, therefore the total uncertainty, +U2

,

for the average of n^ determinations for the larger vessel is

U2 = ^s^/y^i + KUi Eq. (2)

The second term on the right-hand side of the above equation is

usually called the systematic component because it cannot be

reduced in magnitude unless the reference measure is recalibrated

or more than one standard is used. In our work this term is much

smaller than the other term because of repeated calibrations of

the reference measure. The first term can, of course, be reduced

by additional work (i.e. by increasing n-^).

The difference between a value X based on the average of M

gravimetric determinations and a value Y based on ni volume

transfer measurements, each consisting of K transfers from a reference measure with n gravimetric determinations will have a standard deviation of the difference as follows:

/S^2 Sj^2 + "TT

^1 M

where s is the standard deviation of the gravimetric values at a volume level of K times the volume of the reference measure.

^ The standard deviation, a , has a component from the filling

operation of size o^t/K plus Yhe component contributed by the

larger vessel which results from variation of the end point

determination and other errors previously mentioned. If the same

type of end point determination is used in both vessels, then a would be nearly equal to a^vK.

- 4 -
The difference (Y - X) will have a possible systematic error of

One regards the measurement system as having the property of

closure if the following inequality test is satisfied:

|y-x| - K(3si//H")

Z = —===============- < 3

' « sgj^ ^ Eq. (3)

7 M

This formula assumes ' that the standard deviations are known,

which means that they are based on a large number of

observations

The data presented in the tables, except for the 1 gallon vessel,

are for delivered volume reduced to 60 °F and to a zero scale

reading where applicable. The closure test is inserted in the

tables for ease in comparing pertinent values and is summarized

in table 11.

Figures 2 and 3 are control charts produced by routine calibrations. Each point on the chart is a measurement produced by

transferring a test measure calibrated gravimetrically to a check

standard of known volume. The check standard is another test

measure of the same nominal volume that is well defined. This

one-to-one transfer recalibrates the check standard each time a gravimetric calibration is performed. Closure is demonstrated

because each new value is within predicted limits.

Initially estimates of the standard deviation (s) like those of

tables 1 and 2 are necessary to establish starting limits on the

control charts. The limits become better known as more data is

collected and plotted on the control chart.

The data of tables 1 through 8 is straightforward and no difficulties were encountered in demonstrating closure. However the

calculation of U2 for table 5 requires a modification of Eq. (2).

This results from the use of eight identical reference test

measures, each calibrated once. Each reference measure was then

transferred K times to complete a calibration of the larger

vessel. In a more general sense, when n^ reference standards are

each used K times to calibrate a larger vessel n^ times, the

uncertainty will be

/s^ + Ks^

U2 = ±3/-^ Eq. (2a) n

Likewise a similar modification of Eq. (3) is necessary to test

for closure.

- 5 -
'V

X - Y

s^^ + Ks 2 ~ " 3 Eq. (3a)

^ M

Hj, M

Ordinarily, we can weigh vessels up to 30 gallons on one of

several equal -arm balances at our disposal. Beyond 30 gallons

and up to 100 gallons we use a load cell as a mass comparator for

gravimetric calibration and, beyond this point, a multiple-lever

platform scale. However, most test measures larger than 100

gallons are not structurally sound enough to be lifted when

filled with water. To demonstrate closure between the two

methods of calibration near the 1,000 gallon capacity, we used a rigid steel tank with a useful capacity of 830 gallons.

The tank did not have a reference scale appropriate to this work,

so the data shown below are at the test temperature. We simply

filled the tank by transferring known volumes of water into it

and then measured the accumulated volume gravimetrically

.

Temperature Transfer Gravimetric

tj 829.940 gal 829.863 gal

t2 829.873 829.867

U2 ±0.024 ± 0.060

The uncertainty, U^, is calculated by a different but well known

method not discussed in this report. For such a small data set,

closure is considered to be demonstrated when the error bands

overlap as is the case here.

CONCLUSIONS

Based on our data, we conclude that closure does in fact exist

and that the two methods of test measure calibration may be

regarded as equivalent

.

- 6 -
ACKNOWLEDGMENTS

The author wishes to thank Joseph Cameron and Carol

Croarkin of the Office of Measurement Services and

Harry Ku of the Statistical Engineering Section for

their help and patience in preparing this report. Many

thanks to John Houser, Mass and Volume Section, for his

help in collecting data.

REFERENCES

[1] Houser, John F. , Procedures for the Calibration of

Volumetric Test Measures, Nat. Bur. Stand. (U.S.),

NBSIR 73-287 (Aug. 1973).

[2] Tables for Positive Displacement Meter Prover Tanks,

ASTM D1750-62 (API 2541), ASTM Manual (1962).

[3] American Institute of Physics Handbook, Third Edition

(1972).

- 7 -
MEASUREMENT DIAGRAM

ALL

5 GAL

5 GAL.

CHECK

STD.

L

T

.J

N

I 5 GAL

ui

30 GAL

50 GAL

100 GAL 830 GAL

Blocks indicate gravimetric calibration

*- Volumetric transfer

Check std. control chart operation

FIGURE ]
1 GALLON CHECK STANDARD

^. + . —+ + +

+

" MEAN VALUE = 1.00054 GALLONS

I a = 0.00034

LIMITS = ±3a

^ + + + + .+ . +

FIGURE 2

- 9 -
5 GALLON CHECK STANDARD

i2

b, t'02b<-C0 +

+

MEAN VALUE = 5.00428 GALLONS

a = 0.00058

LIMITS = ±3a

C">"'PL'^TF. FROM '''-7^- TO F-?"^

FIGURE 3

- 10 -
TABLE 1: Values for Two 5 Gallon Graduated Neck Test Measures

Assigned by Repeated Gravimetric Calibration.

A B

5.00078 gallons 5.00450 gallons

5.00154 5.00228

5.00116 5.00242

5.00104 5.00318

5 .00097 5.00295

5.00119 5.00375

5.00035 5.00365

5.00080 X = 5.00323

X = 5.00098 S^= 0.00078

S^= 0.00035 Uj=±0. 00088

Uj=±0. 00037

TABLE 2: Repeated Gravimetric Calibrations of a 5 Gallon Slicker Plate Measure

and a 1 Gallon Graduated Neck Pyrex Check Standard.

5 GALLON MEASURE 1 GALLON MEASURE

5.00063 gallons 1.00063 gallons

5.00069 1.00062

5.00081 1.00042

5.00029 X = 1.00056

5.00034 S^= 0.00012

X = 5.00055 Uj=±0. 00021

S^= 0.00023

U^=±0. 00031

Contained Volume

@ 60°F

- 11 -
TABLE 3: Transfer Calibrations of 30 Gallon Graduated Neck Measure Based on the 5 Gallon Slicker Plate Measure Shown in Table 2 .

STARTING STANDARD 30 GALLON VALUES

5.00063 gallons

5.00069

5.00081

5.00029

5.00034

X = 5.00055

b J - U . UUUZ

J

U^=±0. 00031

30.0048 gallons

30.0031

30.0047

30.0007

30.0049

Y = 30.0036

S^= U.UUloO

\J^= ±0.00428

TABLE A: Gravimetric Calibrations of the 30 Gallon Measure.

30 GALLON MEASURE CLOSURE

30.00218 gallons

30.00205

, , 30.00153

30.00023

X = 30.00150

Sj^ 0.00089

U^= ±0.00134

Gravimetric Value

X 30.00150 ±0.00134

Transfer Value

Y 30.00360 ±0.00428

Z = 0.261 < 3

- 12 -
TABLE 5: The 50 Gallon Measure Transfer Calibration Based on 8 Different

Slicker Plate Measures Calibrated Gravimetrically

.

STARTING STANDARDS 50 GALLON MEASURE

5.00109 gallons 50.0098 gallons

5.00078 50.0051

4.99952 50.0033

4.99937 50.0086

5.00073 50.0020

5.00090 50.0100

5.00089 50.0079

4.99903 50.0037

S^ = 0.00023 Y = 50.0063

= ±0.00069

=v

= 0.0032

«2 = ±0.0035

TABLE 6: 50 Gallon Measure Gravimetric Calibration via Load Cell Mass

Comparator.

50 GALLON MEASURE CLOSURE

50.0097 gallons Gravimetric Value

50.0023 X 50.0048 ± 0.0027

50.0025

50.0037 . Transfer Value

50.0072 Y 50.0063 ± 0.0035

50.0073

50.0066 Z = 1.02 < 3 50.0079

50.0030

50.0026

50.0003

X = 50.0048

^M

= 0.0030

= ±0.0027

- 13 -
TABLE 7: 100 Gallon Transfer Calibration Using the 50 Gallon Measure

As the Reference Standard. See Table 6 for 50 Gallon Data.

STARTING STANDARD 100 GALLON MEASURE

100.012 Gallons

100.013

100.013

X = 50.0048 Gallons Y = 100.0127

Sj = 0.0030 S^ = 0.0006

= ±0.0027 U^ = ± 0.0064

TABLE 8: 100 Gallon Gravimetric Values Assigned by Load Cell Weighings100 GALLON GRAVIMETRIC CLOSURE

100.012 Gallons Gravimetric Value

100.010 X 100.0096 ± 0.0028

100.008

100.011

Transfer Value

100.007 Y 100.0127 ± 0.0064

X = 100.0096

Sj^ = 0.0021

Z =2.40 < 3

= ± 0.0028

USCOMM-NHS -nc - 14 -
NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PURLICATIGN OR RKI^ORT NO. NBSIR 74-454

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

The Equivalence of Gravimetric and Volumetric

Test Measure Calibration

5. Publication Date A- ft- 7

A

6. Performing Organization ( ode NBSIR 74-454

7. AUTHOR(S)

ixaTiQaxx ii» ocnoonover

8. Performing Organ. Rept>rt No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Proiect/Task/Work Unit No.

11. Contract /Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

SAME

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less [actual summary of most si^iticant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report discusses the statistical importance of

observed differences between gravimetric and volume

transfer calibrations of volumetric test measures.

The data presented are results from the present NBS

calibration program and conclusively show there are negligible differences between the two methods of

calibration

.

I

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Check standard; closure; gravimetric calibration; standard deviation;

test measure; volumetric transfer calibration

18. AVAILABILITY Unlimited

1

' For Official Distribution. Do Not Release to NTIS

1

^ Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402. SD Cat. No. CH

1

! Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

14

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

22. Price

USCOMM-DC 29042-P74
1,

1