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SOP 15 Page 1 of 8 

SOP No. 15 

Recommended Standard Operations Procedure for 

Calibration of Intermediate and Large Volume Standards 

Gravimetric Method1 Using an Equal-Arm Balance2

1 Introduction 

 1.1 Purpose of Calibration 

 This procedure is applicable to calibrations where high accuracy is needed, or 

where the volumetric transfer method is not possible. It is especially useful for 

the capacity range of 5 gal to 100 gal (or larger) using an equal-arm balance and 

the double substitution weighing method as described in SOP 4. It may also be 

used for smaller volumes with an equal-arm balance. If an electronic balance is 

used, see SOP 14. 

1.2 Prerequisites 

1.2.1 Verify that valid calibration certificates are available for the standards 

used in the test. 

1.2.2 Verify that the standards to be used have sufficiently small standard 

uncertainties for the intended level of calibration. Reference standards 

should not be used for gravimetric calibration. 

1.2.3 Verify that the balance used is in good operating condition with 

sufficiently small process standard deviation as verified by a valid control 

chart, or preliminary experiments to ascertain its performance quality 

when a new balance is put into service. The accuracy of the balance and 

weighing procedures should be evaluated to minimize potential bias in the 

measurement process. 

1.2.4 Verify that the operator is experienced in precision weighing techniques 

and has had specific training in SOP 2, SOP 4, SOP 29, and GMP 1, GMP 

3, GMP 10, and gravimetric calibrations. 

1.2.5 Verify that the laboratory facilities meet the minimum conditions shown 

in Table 1 to meet the expected uncertainty achievable with this 

procedure. 

1.2.6 Verify that an adequate supply of distilled or deionized water (see GLP 

10) is available. 

 

1Based on the work of J. Houser "Procedure for the Calibration of Volumetric Test Measures" NISTIR 73-287. 

2 Non-SI units are predominately in common use in State legal metrology laboratories, and/or the petroleum 

industry for many volumetric measurements, therefore non-SI units have been used to reflect the practical needs of 

the laboratories performing these measurements as appropriate. The reference temperature for petroleum products 

is 60 oF and volume values are typically in gallons. 
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Table 1. Laboratory environmental conditions 

Procedure Temperature Relative Humidity 

Gravimetric 20 °C to 23 °C, a set point ± 2 °C, 

maximum change 1 °C/h 

40 % to 60 % 

± 10 % max change / 4 h 

2 Methodology 

 2.1 Summary 

This procedure may be used to determine three calibration values: 

2.1.1 The contained volume, VC, which is the volume of water required to fill 

the vessel at a specified temperature (usually 60 °F for petroleum 

products). 

2.1.2 The delivered volume, VD, which is the volume of water that may be 

poured or drained from the vessel at a specified temperature (usually 60 °F 

for petroleum products) under specified conditions (see note 1). 

2.1.3 The neck scale calibration value, NSCV, which relates the true volume of 

the neck to the value indicated by the neck scale. 

 

 The mass of water required to fill the vessel is measured at an observed 

temperature. The known density of water and coefficient of expansion of the 

vessel are used to calculate the volume of the vessel at a reference temperature. 

 The precision attained will depend on the sensitivity of the balance and on the 

care exercised in carrying out the various operations of the procedure. 

Cleanliness of the vessel and strict adherence to the prescribed drainage 

procedure are critical for high precision. The accuracy of calibration depends on 

all of the above plus the uncertainties of the various weighings, accuracy in 

reading a meniscus (where present), the volumetric temperature equilibration and 

corrections, the proper correction for air buoyancy, and the purity of the water 

used as the standard. The metrologist must critically consider all of the above 

factors when assigning uncertainty limits to the calibrated volumes. 

 

 With careful observance of good measurement practices, an overall relative 

uncertainty of ± 0.02 % should be attainable. 

2.2 Apparatus 

2.3.1 Equal-arm balance of adequate capacity and sensitivity (standard 

deviation of 500 mg or better for 5 gal level). 

2.3.2 Weights (meeting ASTM Class 2 or 3 specifications) as appropriate for 

the accuracy requirements. 

2.3.3 Calibrated thermometer accurate to ± 0.1 °C with recent calibration values 

that are traceable to NIST to determine water temperature.3

 

3 Values from the thermometer, barometer and hygrometer are used to calculate the air density at the time of the 

measurement. The air density is used to make an air buoyancy correction. The accuracies specified are 
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2.3.4 Calibrated thermometer accurate to ± 0.5 °C with recent calibration values 

that are traceable to NIST to determine air temperature. 

2.3.5 Calibrated barometer accurate to ±135 Pa (1 mm Hg) with recent 

calibration values that are traceable to NIST to determine the air pressure. 

2.3.6 Calibrated hygrometer accurate to ± 10 % with recent calibration values 

that are traceable to NIST to determine relative humidity. 

2.3.7 Distilled or deionized water (See GLP 10). 

2.3.8 Stopwatch or other suitable timing device (does not need to be calibrated.) 

2.3.9 Cap to place over neck of the vessel. 

 

2.3 Procedure 

2.3.1 Preliminary Operations 

2.3.1.1 Clean the interior of the vessel with biodegradable low-sudsing 

detergent (see GMP 6) with gentle scrubbing as necessary. Rinse 

several times with clean water to remove all detergent, followed by 

pure water. Check for leaks and for uniform drainage. 

2.3.1.2 Clean and dry outside of container to remove any material which 

might subsequently dislodge and cause an erroneous weighing. 

2.3.1.3 Dry inside by rinsing with small amounts of alcohol or by drawing 

clean, filtered air through the vessel (several hours required). For 

large vessels the heat from a low wattage light bulb suspended 

inside the vessel overnight will aid in drying by slightly heating 

the vessel and creating convection currents through the interior. 

Allow the vessel to come to thermal equilibrium with the room 

before proceeding. 

2.3.2 Empty Weight 

2.3.2.1 Weigh the empty vessel and cap using the double substitution 

procedure (SOP 4) with air buoyancy correction. 

2.3.2.2 Record air temperature, humidity, and barometric pressure as 

needed for calculating air density (SOP 2) and performing 

buoyancy corrections. 

2.3.2.3 The empty weight is determined as shown in the calculations 

(Section 3.1, Eqn 2). 

2.3.3 Filled Weight 

 

recommended for high precision calibration. Less accurate equipment can be used with only a small degradation in 

the overall accuracy of the measurement. 
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2.3.3.1 Fill the vessel with pure water to the zero point on the neck scale. 

Be sure the vessel is level. Bounce the liquid to disturb the 

meniscus to assure it has reached equilibrium level. Record the 

water temperature and readjust level if necessary. Be sure that no 

water remains on the outside of the vessel when it is weighed. 

(See note 2.) The filled weight is determined as shown in the 

calculations (Section 3.1, Eqn 2). 

2.3.3.2 Weigh filled vessel and cap as in 2.3.2. 

2.3.4 Drained Weight 

2.3.4.1 Drain vessel using appropriate discharge/pour and drain times (see 

note 1). Replace cap. The empty weight is determined as shown 

in the calculations (Section 3.1, Eqn 2). 

2.3.4.2 Weigh drained vessel and cap as in 2.3.2. 

2.3.5 Make a duplicate determination of steps 2.3.2 through 2.3.4 (Run 2). 

2.3.6 Neck scale plate verification 

2.3.6.1 Fill vessel with water to a scale division near the bottom of the 

scale range. Record reading (sr1). 

2.3.6.2 Make appropriate successive additions of water from a calibrated 

buret or other suitable volumetric measuring device. Record scale 

readings after each addition . 

2.3.6.3 A plot of scale readings with respect to volume added should be 

linear and will be a gross check of the validity of this calibration. 

2.3.6.4 Calculate the neck scale calibration value, NSCV, using the 

expression 

 Table 1. Variables for neck scale calibration value equation 

Variable Description 

NSCV Neck scale calibration value 

VW Total volume of water added to neck 

sri Scale reading, initial 

srf Scale reading, final 

 

( ) f i

W

sr sr

V NSCV − = Eqn. 1
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3 Calculations 

 3.1 To calculate the filled, empty, or drained mass, use the following equation from 

SOP 4 using Option A. 

Table 2. Variables for Equation 2

Symbol Description 

S standard reference weight 

sw small calibrated weight used to evaluate the sensitivity of the 

balance 

M 

the mass (true mass) of a specific weight. Subscripts s, sw, filled, 

empty, drained are used to identify the weight (equals Nominal 

plus Correction in the case of mass standards) 

ρa density of air at time of calibration 

ρ density of masses; subscripts s, x, ts, tx, sw are used to identify the 

weight. 

O Balance observations, designated with subscripts 1, 2, 3..., for the 

observation number 

3.2 To calculate containment volume, VC, at temperature of measurement, t, 

3.3 To calculate delivered volume, VD, at temperature of measurement, t,

3.4 To correct volumes to reference temperature, tref, (be sure to match units for 

cubical coefficient of expansion, α, with temperature units (e.g., degrees C with 

°C-1). 

( )

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − − + − + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − ( )

1 

2

 1 

3 2

2 1 3 4

( , , ) O O

M

O O O O M M sw

a

sw

S

a

filled empty drained S

ρ

ρ

ρ

ρ

( ) ⎥

⎦

⎤ ⎢

⎣

⎡

− = − 

1 ( ) 

w a

VC t M filled Mempty ρ ρ

( ) ⎥

⎦

⎤ ⎢

⎣

⎡

− = − 

1 ( ) 

w a

VD t M filled M drained

ρ ρ

V V [1 (t t )] t ref = t − α − ref

Eqn.2

Eqn. 3

Eqn. 4

Eqn. 5
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3.5 To convert the volume measured in mL (cm3) to gal: 

 Vgal = (0.000 264 172) VmL (There are 3 785.411 784 mL per gal.) 

 Table 2. Variables for volume equation 

Symbol Description 

ρw density of water at temperature of filling 

ρa density of air 

t temperature at time of test 

tref reference temperature, usually 60 °F (or 20 °C) 

α cubical coefficient of expansion of the vessel 

(must match temperature units in °C-1or °F-1) 

 

 3.6 Calculate the average volume at reference temperature (Vt ref ) 

3.7 Physical Constants 

3.7.1 Density of Air 

Value at environmental conditions at time of weighings calculated from 

the equation given in the Appendix to SOP 2. 

3.7.2 Density of Water 

Calculate the density of water using the formula given in GLP 10 or look 

up the value from appropriate tables. 

 3.7.3 Cubical Coefficient of Expansion 

Values for the cubical coefficient of expansion of various materials are 

given in NISTIR 6969, Table 9.10 and are normally printed on the vessel 

identification plate (for test measures and provers). 

4 Measurement Assurance 

4.1 Duplicate the process with a suitable check standard (See GLP 1, SOP 30, and 

NISTIR 6969, Sec. 7.4) 

4.2 Plot the check standard value and verify that it is within established limits OR a 

t-test may be incorporated to check the observed values against an accepted value. 

4.3 The mean of the check standard is used to evaluate bias and drift over time. 

4.4 Check standard observations are used to calculate the standard deviation of the 

measurement process, sp. 
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5 Assignment of Uncertainties 

The limits of expanded uncertainty, U, include estimates of the standard uncertainty of 

the mass standards used, us, plus the uncertainty of measurement, sp, at the 95 percent 

level of confidence. See SOP 29 for the complete standard operating procedure for 

calculating the uncertainty. 

5.1 The standard uncertainty for the mass standard, us, is obtained from the 

calibration report. The combined standard uncertainty, uc, is used and not the 

expanded uncertainty, U; therefore, the reported uncertainty for the standard will 

usually need to be divided by the coverage factor k. See SOP 29 for proper 

handling of uncertainties when multiple standards are used. 

5.2 Standard deviation of the measurement process from control chart performance 

(See SOP 17 or 20.) The value for sp is obtained from the control chart data for 

check standards or estimated based on the range of duplicate measurements over 

time. This value will incorporate a repeatability factor related to the precision of 

the weighings and the setting of the meniscus, but not related to errors in reading 

the meniscus. 

5.3 Other standard uncertainties usually included at this calibration level include 

uncertainties associated with water temperature measurements, thermometer 

accuracy, the coefficient of expansion, viscosity, reading the meniscus, 

calculation of air density, standard uncertainties associated with the density of the 

standards used, improper observance of drainage times, viscosity or the surface 

effects on the volume of liquid clinging to vessel walls after draining and lack of 

internal cleanliness. 

Additional References: 

Bean, V. E., Espina, P. I., Wright, J. D., Houser, J. F., Sheckels, S. D., and Johnson, A. N., NIST 

Calibration Services for Liquid Volume, NIST Special Publication 250-72, National Institute of 

Standards and Technology, Gaithersburg, MD, (2006). 
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Appendix 

Test Notes 

1 Pour and drain times. 

It is impractical to completely drain a filled container, because some of the contents will 

remain as a film. By strict adherence to a specified procedure, the residual contents can 

be held essentially constant so that, reproducible calibration constants can be obtained. 

The conditions conventionally selected are as follows: 

a For bottom-drain containers: open drain valve fully and allow contents to 

discharge at maximum rate. When flow ceases, wait 30 s, close valve, and touch 

off any drops adhering to spout. 

 b For pour-type containers: pour contents by gradually tilting container to an 85 o 

angle, so that virtually all is delivered in 30 s. Allow to drain for an additional 

10 s, then touch off any drops adhering to the lip. The instructions described above must be precisely followed during calibration and use 

of the calibrated vessels. 

2 Evaporation losses. 

A suitable cap (an empty glass beaker, for example) should be placed on the top of open 

vessels to minimize evaporation losses. If used, the cap must be included in all 

weighings. 

3 Slicker-plate. 

 

 When a slicker-plate standard is calibrated, the plate should be used to fix the water level 

in it. This plate must be weighed with the standard during each such operation.