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SOP 26

Standard Operating Procedure

For

Gravimetric Calibration of Dynamic Volumetric Systems used as Standards1

1 Introduction

Purpose of Test

This procedure describes the gravimetric calibration of volumetric systems that 

may be used as volumetric measuring standards. (Gravimetric calibration of fixed 

volume graduated-neck or slicker-plate type standards should be completed using 

SOP 14). This procedure uses gravimetric calibration principles with mass values 

and pure water sources to obtain calibration uncertainties that are generally less 

than those obtained during volume transfer procedures of similar standards. 

Alternative test liquids are not considered in this procedure. Accordingly, the 

procedure is especially useful for high accuracy calibrations. The procedure uses 

replicate testing to incorporate measurement assurance steps to ensure the validity 

of the measurement process; additional analyses must be performed to assess for 

bias (e.g., gravimetric calibration of other similar sized standards using the same 

methods, or interlaboratory comparisons with similar standards or provers). The 

procedure makes use of an electronic balance and is suitable for all sizes of 

gravimetric calibrations as limited by the capacity and resolution of the balance,

available water source, mass standards, and handling capacity of the laboratory. 

Detailed measurement ranges, standards, equipment, and uncertainties for this SOP 

are generally compiled in a separate document in the laboratory.

This procedure can be used for single direction and bi-directional calibration of 

small volume provers, for calibration of volumes delivered through a master meter, 

for fixed volume delivery from alternative volumetric measuring systems, or other 

alternative test methods and standards where the gravimetric calibration provides a 

useful calibration result and measurement uncertainty. All steps in the procedure 

and calculations must be thoroughly reviewed and evaluated for appropriateness 

for the specific standards under test to ensure accurate measurement results are 

reported to the user.

Conformity Assessment

Standards that are calibrated for use in legal weights and measures applications 

should be evaluated for conformance to the appropriate specifications and 

tolerances that apply. Where compliance is required by law, conformity evaluations 

should be conducted prior to performing calibrations. See Section 6.2 for reporting 

results.

 1 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 are used to reflect the practical needs of the laboratories 

performing these measurements when appropriate.
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Prerequisites

Verify that valid calibration certificates with appropriate values and 

uncertainties are available for all standards, instruments and sensors used in 

this procedure. Calibration values must have demonstrated metrological 

traceability to the International System of Units (SI), which may be to the 

SI, through a National Metrology Institute such as NIST.

Verify that all standards and instruments (e.g., mass standards, 

thermometer, barometer, and hygrometer) have sufficiently small standard 

uncertainties for the level of calibration. Reference standards that need 

special care and protection should not be used for gravimetric calibration

where there is risk for contamination or damage from water.

Verify that the balance has sufficient capacity and is in good operating 

condition with sufficiently small process standard deviation. Process 

standard deviations need to be verified by a valid control chart or 

preliminary experiments to ascertain performance quality. The accuracy of 

the balance and weighing procedures should also be evaluated to minimize 

potential bias in the measurement process.

Verify that the operator is experienced and has demonstrated proficiency in 

precision weighing techniques and volumetric measurements and has had 

specific training in NISTIR 6969 SOP 2, SOP 4, and NISTIR 7383 SOP 19, 

SOP 29, GMP 10, and especially in gravimetric calibrations such as those 

used in SOP 14.

Verify that an adequate quality and supply of distilled or de-ionized water 

(see GLP 10) is available. Do not use tap or other impure water sources for 

this procedure! The laboratory must have a continuous flow of water 

available and may thermally equilibrate water in a large volumetric prover 

or a water tank provided purity is ensured. Note: Calibration of some 

systems may require other liquids due to viscosity or lubricity requirements 

and alternative procedures should be considered. Use of calibration liquids 

other than water is not considered in this SOP.

Verify that the laboratory facility is free of air currents and/or temperature 

fluctuations where the mass measurements will be performed. The 

volumetric proving system must also be calibrated in a thermally stable 

environment. Where the volume and weighing areas are not the same, 

movement of standards and transfer vessels must not negatively affect 

environmental stability. Stable temperatures, and thermal equilibration of 

water, are especially important to minimize possible convection currents on 

precision balances. Humidity control is essential for minimizing possible 

evaporation and/or condensation during the measurement process. 
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Verify that the laboratory facilities meet the following minimum conditions 

to meet the expected uncertainty possible with this procedure:

Table 1. Laboratory environmental conditions.

Procedure Temperature Relative

Humidity

Gravimetric 18 °C to 23 °C

Stable to ± 1 °C / 1 h during the calibration.

40 % to 60 %

Stable to ± 10 % / 4 h

All necessary maintenance to the unknown volumetric standard must be 

completed by the device owner prior to starting the gravimetric calibration. 

It is advisable to perform a static leak test prior to performing the test on 

systems that allow such testing. Replace any seals that are present on the 

prover if there is any doubt as to their integrity. If “as found” data is 

required, it must be obtained before any seal replacements. A second 

calibration will be required after any maintenance is completed to obtain an 

“as left” calibration value. Note: “As found” data is essential for setting, 

monitoring, or adjusting calibration intervals.

Any standard inlet(s) and outlet(s) that are not used in the calibration 

process must be blocked by the device owner before calibration by using a 

blind flange, double-block and bleed valve, or other appropriate hardware. 

All extra valves, piping must be evaluated to ensure that air is not entrapped 

in the system.

The device owner must provide all necessary manufacturers’ manuals and 

hardware (e.g., including pipe fittings, valves, solenoids, wiring, etc).. as 

well as trained personnel to operate the device during the calibration to 

ensure that the device is calibrated in the same way it will be used.

2 Methodology

Scope, Precision, Accuracy 

The procedure is applicable for the calibration of any size of volumetric measuring 

system that, when filled with water, will not overload the electronic balance used. 

Typical volumetric proving systems ranging in capacity from 5 gal to 120 gal may 

be calibrated. Continuous systems that measure these quantities that are 

controllable in the laboratory may be considered as well (e.g., a metering system 

delivering 40 gal per min to 100 gal per min may be possible). Flow rates that will 

be assessed need to be determined between the laboratory and the standard owner 

prior to calibration and should consider the practical range of use, the laboratory 

capability, and any documentary or legal requirements of applicable specifications 

and tolerances. 

The precision of this calibration depends on strict adherence to the various steps of 

the procedure. The accuracy attainable will depend on the uncertainties of the 

standard weights, the uncertainties associated with the air buoyancy corrections, 
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the accuracy of temperature measurements, the purity and density of the water, and 

the thermal expansion corrections that are made. Another important factor to 

consider is the repeatability of the procedure and the repeatability of the standard 

under test. 

Summary 

The mass of the water delivered by the volumetric proving system is determined by 

weighing the water on an electronic balance and comparing it to mass standards. 

Volume is derived from the mass of the mass standards, the density of pure water, 

appropriate air buoyancy corrections, and thermal and pressure expansion 

corrections, using the equations provided in Section 3.

Standards, Equipment, and Calibration Notes

An electronic balance having sufficient capacity to weigh the loaded vessel

is required. The repeatability of the balance, measurement process, and 

standard under test are limiting factors in the accuracy of the measurement. 

The resolution of the balance and repeatability of the measurement process 

must be smaller than the expected uncertainty of the calibration. Note: 

Standard deviations obtained from precision mass calibrations do not reflect 

the process repeatability of this procedure; therefore, repeatability must be 

assessed using this procedure. 

Mass standards of sufficient quantity and capacity are required. Ordinarily, 

standards of ASTM Class 2 or 3 or OIML Class F1 or F2 weight 

specifications are required. Working standards of other classes are generally 

not designed to maintain adequate stability needed; however, corrections 

obtained within a few days of the volume calibration may be used if the 

uncertainty is sufficiently small. Mass standards are selected so that they 

are slightly larger than the volume of water and container or transfer vessel 

to be weighed. Linearity errors or additional uncertainties may need to be 

considered when the mass standards are slightly less than the volume or 

container being measured. When summations of masses are used, the 

summation mass is used, and the “effective density” must be calculated, 

taking care not to use “average” density values. 

Filter paper and padding for protection of mass standards when placed on 

the balance pan, stacked, or placed on the calibration item may be used. 

When all such materials are used, care must be taken to determine when and 

how the mass of these items are incorporated into the process (as tare 

weight, or as tared/zeroed off the balance readings). 

A transfer vessel is required that is made of some inert and stable material 

with a volumetric capacity larger than the volume to be measured and a port 

large enough to fill the vessel on the top surface. The transfer vessel should 

be as small as possible to minimize its influence on the balance, but 

approximately 25 % larger than the volume to be measured to prevent water 
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loss due to splashing out of the top fill port. The transfer vessel may need 

to be suspended with a hoist or mounted to a pallet to be moved with a 

forklift or pallet jack. 

Weight handling equipment is required for this procedure that has adequate 

capacity for handling the filled and empty transfer vessel. It must have the 

ability to be moved slowly and gently to minimize water sloshing and to 

gently set the empty or filled vessel on the balance pan. 

Thermometers, with resolution and uncertainty less than 0.1 °C to determine 

water and volumetric proving system temperatures. Submersible 

thermometer probes and cables must be long enough to measure the 

temperature of the water near the center of the transfer vessel. Therefore, 

liquid-in-glass thermometers are not adequate. 

Barometer with resolution and uncertainty less than 135 Pa (1 mmHg to 

determine barometric pressure (absolute). 

Hygrometer with resolution and uncertainty less than 10 % to determine 

relative humidity.

Pressure gauge(s) with resolution and uncertainty less than 1 psig for 

determining the volumetric proving system pressure (where applicable).

If a proving system does not have operational temperature sensors and 

pressure gauges installed, the initial inspection should reveal that the 

proving system is not adequately prepared for calibration. If the calibration 

relies on the accuracy of the temperature sensors and pressure gauges 

installed on the prover, they must be installed prior to the volume calibration 

and have current calibration reports from an accredited calibration 

laboratory that are readily available for inspection. When laboratory 

pressure and temperature sensors are used during the calibration all 

laboratory sensors must all have current calibration reports. 

Mechanical problems and sources of errors that may be encountered during 

a calibration and affect the ability of the system to repeat may include many 

possibilities, several of which are listed below. Mechanical problems with 

the volumetric standard must be corrected before calibration. Quantification 

of errors that cannot be eliminated must be evaluated and included in the 

uncertainty budget table.

- Proper operation of the system;

- Air in the system;

- Instability of the temperature of water or ambient conditions;

- Pressure fluctuations in the system;

- Error in reading the test measure(s);

- Incorrect drain time for reading test measure(s);

- Contamination of the systems;
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- Damaged measures/system;

- Error in reading temperature or pressure;

- Displacer problems (when present);

- Malfunction of switches (when present);

- Damage, worn, or badly fitted seals;

- Damage, scoring, or pitting on the inside of the proving system;

- Damaged or misaligned detector switches (when present);

- Light interference in optical switches (when present);

- Malfunction of solenoid valve (when present);

- Leaks in prover/system or bypass valves. 

Errors that may be encountered in weighing steps of the calibration may 

include any of the items listed below. Quantification of errors that cannot 

be eliminated must be evaluated and included in the uncertainty budget 

table.

- Balance drift;

- Off center loading of the balance;

- Non-linearity of the balance and improper selection of the mass 

standards to calibrate the weighing instrument in the range of use; 

- Air currents or convection currents when weighing is performed in a 

large mass laboratory or when the temperature of the water, system, or 

laboratory are not adequately equilibrated;

- Care in placing items on the weighing instrument; 

- Inadequate water quality.

Procedure

Inspect the volumetric proving system to verify that it and associated piping 

are drained, clean, and safe for placement in the laboratory. For large 

volumetric proving systems, ensure that there is adequate space in the 

laboratory to safely position the system near the pure water source and that 

a drain is accessible and that the proximity of the mass calibration area is 

convenient. Direct the driver to position to system appropriately, taking 

appropriate precautions to prevent injury to anyone or damage to the 

laboratory and/or equipment and standards. Allow adequate time for the 

system under test to equilibrate with the laboratory environment. The time 

required will be dependent on the difference between inside and outside 

temperature and relative humidity.

Initial inspection. Record the data about the system being calibrated, taking 

care to record all applicable serial numbers, coefficients of cubical 

expansion, reference temperatures, flow rates, presence or lack of the 

calibration status of all auxiliary standards associated with the standard 

(such as pressure gauges and temperature sensors), and means for sealing 

the system after calibration. 
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Install, or have the owner install, volumetric proving system hardware per 

manufacturer’s instructions or have the owner provide this service. 

(Hardware must be provided by device owner).

Connect the water supply to the volumetric proving system.

2.4.4.1 In general, ensure that all air is bled from the system, suitable water 

supply is provided, and the device temperature is in equilibrium with 

that of the water supply. Several runs are generally required prior to 

the calibration to ensure that all air is out of the system and that all 

components are in thermal equilibrium with the water. Set device 

valves as appropriate for the test to be conducted.

2.4.4.2 Follow the operator’s guidance (and per manufacturer’s 

instructions) to become familiar with and to establish correct valve 

positions for each step of the calibration when applicable.

Install and/or use calibrated thermometers with evidence of current 

metrological traceability to record air temperature, water temperature, 

detector bar temperature (Td) (where present), volumetric proving system 

temperatures (Tp). 

Install and/or use calibrated pressure gauge(s) to record prover pressure(s)

(Pp) when present and appropriate.

Analyze the piping and valves to determine if the configuration needs to be 

changed to deliver water into the transfer vessel. 

Record all the prover-specific variables on the data entry form. 

Note: All references in the following weighing options to the empty transfer 

vessel assume that the empty transfer vessel has been placed in a “wetteddown” condition. This wet down condition is achieved when the vessel has

been filled with water and then drained using the same process that will be 

used throughout this calibration. Several runs may be needed on the volumetric 

proving system to ensure proper wetting of all lines and drains before the 

measurement data is recorded. 

Weighings (Option A)

2.4.9.1 Zero the balance. Place a standard mass, Ms1, on the balance 

platform. Ms1 should be slightly larger than the mass of the empty 

transfer vessel. Record the balance indication as O1.

2.4.9.2 Zero the balance and then place the empty transfer vessel on balance 

platform and record the balance indication as O2. Caution: the 

transfer vessel must be dry on the outside for all weighings.
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2.4.9.3 Record air temperature, barometric pressure and relative humidity

at the time of these measurements. Due to timing of this type of 

calibration, environmental conditions must be recorded for both the 

empty and full transfer vessel. 

2.4.9.4 Zero the balance and then place a standard mass, Ms2, on the balance 

platform. Ms2 should be slightly larger than the mass of the filled 

vessel. Record the balance indication as O3. (This observation may 

be made before or after the weighing of the filled transfer vessel, 

depending on laboratory configurations and convenience).

2.4.9.5 Fill the transfer vessel through the volumetric proving system. Do 

not fill the transfer vessel while it is on a laboratory balance. Ensure 

that all the water from the volumetric standard or registered on a 

meter is captured in the transfer vessel. Verify that the water 

temperature in the system is uniform and stable throughout the 

filling process (± 0.2 °C). Read and record all pertinent volumetric 

standard temperatures and pressures during the calibration 

sequence. Place the lid on the transfer vessel to limit evaporation.

2.4.9.6 Zero the balance and then place the filled transfer vessel on the 

balance and record the balance indication as O4. 

2.4.9.7 Read and record the temperature of the water in the transfer vessel 

immediately after weighing. It is critical to measure the temperature 

after weighing to ensure that water is not removed from the transfer 

vessel prior to obtaining the mass value. 

2.4.9.8 Record the air temperature, barometric pressure and relative 

humidity at the time of these measurements.

2.4.9.9 Make at least five replicate runs with at least two runs at a flow rate 

differing from the others by at least 25 % The measured result (V60) 

of each of these calibration runs must agree within 0.02 % of the 

average volume, or the limits on the standard deviation or range 

charts (whichever is smaller) which may be evaluated through the 

use of an F-test, of all volume calibration runs (i.e., the maximum 

result minus the minimum result must be less than 0.02 % of the 

average measured volume).

2.4.9.10 Calculate the volume using Equation 1 in Section 3.1.1., 

(Option A).

Weighings (Option B)

2.4.10.1 Place mass standards that approximate the mass of the empty 

transfer vessel (with a lid) and adequate filter paper or other clean, 

lint-free padding material (to protect the standards being used) on 
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the electronic balance. With these items on the balance, zero the 

balance. Note: The padding materials must be included on all 

measurements or their mass treated as tare.

2.4.10.2 Place additional mass standards approximating the nominal 

mass of the water volume to be measured on the balance. Take care 

to place the weights on filter paper or other appropriate protective 

padding. Record this balance indication as O1.

2.4.10.3 Remove all mass standards. Record the air temperature, 

barometric pressure, and relative humidity readings.

2.4.10.4 Place the empty transfer vessel on the balance, with the 

padding material and zero the balance indication. Remove the 

transfer vessel, lid, and padding material and record the base empty 

vessel reading (B). 

2.4.10.5 Fill the transfer vessel from the volumetric proving system.

Read and record all pertinent volumetric standard temperatures and 

pressures during the calibration sequence. Place the lid on the 

transfer vessel to limit evaporation.

2.4.10.6 Immediately prior to weighing full vessel record balance 

reading as (d1). This reading will be used to calculate any balance

drift that occurred while the transfer vessel was being filled.

2.4.10.7 Place the filled vessel and all padding material on the 

balance. If needed, add known mass standards as tare weights to 

bring the water mass up to the mass of the standards used for O1. 

This balance reading is recorded as O2. Record the mass of all tare 

weights.

2.4.10.8 Remove the filled transfer vessel and record the balance 

indication as (d2). This reading is used to calculate balance drift over 

the entire calibration process.

2.4.10.9 Immediately after removing the filled transfer vessel from 

the balance, record the temperature of water in the vessel.

2.4.10.10 Make at least five replicate runs with at least two runs at a 

flow rate differing from the others by at least 25 % when the system 

allows for varying flow rates. The range measured result (V60) of 

each of these calibration runs must agree within 0.02 % of the 

volume (i.e., the maximum result minus the minimum result must 

be less than 0.02 % of the volume being measured) or be less than 

the limits on the standard deviation or range charts (whichever is 

smaller).
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2.4.10.11 Calculate the volume using Equations 2 and 3 in Section 

3.1.1., (Option B).

3 Calculations 

Compute the volume at the temperature of the measurement, Vt, for each 

determination using the equation:

Option A.

4 2

2 1

3 21 1

O O 1 1 1

O O

a a

t s s

s s wa

VM M ρ ρ

ρ ρ ρρ

       = −− −             −  

Eqn. (1)

Table 2. Variables for volume equation.

Variable Description

Ms, Ms1, Ms2

mass of standards

Ms1, is the mass of the standards used with the empty transfer vessel

Ms2, is the mass of the standards used with the transfer vessel after filling 

with delivered water

ρs density of MS standards

ρw density of water at the temperature of measurement 

ρa density of air at the conditions of calibration – may be different for 

empty and filled weighing of the transfer vessel

Vt Volume at the temperature of the test for the unknown prover/system

Option B.

1

s a a t 2c t

s t wa

M 1

V O = 1 - - 1 - M

O - 

ρ ρ

ρ ρ ρρ

                                  Eqn. (2)

( 1 2 )

2 2 2 c

d d

O O B

      +

=+ −          

Eqn. (3)
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Table 3. Variables for volume equations.

Variable Description

O1 Observation 1, balance reading for mass standard

O2 Observation 2, balance reading for water delivered from prover

O2c Drift-compensated O2 reading

d1 Drift while filling the vessel

d2 Drift over entire process

B Base balance reading drift is based on

MS

Mass of mass standards (true mass, vacuum mass)

Mt Mass of tare weights

ρa Air density 

ρs Density of MS

ρt Density of Mt

ρw Water density (calculated using GLP 10 equations)

Compute V60, the volume at 60 °F, for each run, using the equations noted below:

( )

( ( ))( ( ))

( ( ))

( )

60

(1 or 2)

1

2

where,

1

1 0.0000032

1 1

1

1

t

pp p

p

p

p ap b ld b

p cp b

p

p

V V

CCF

CCF CPL CTS CPS

CPL

P

CTS G T T G T T

CTS G T T

P ID

CPS

E WT

=

=× ×

=   − ×  

=+ − + −

=+ −

  ×

= +     ×  

Eqn. (4)
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Table 4. Additional Variables for Volume Equations.

Variable Description

Tb Reference (Base) temperature

Td Detector bar temperature (Temperature, detector) – where present

Tp Prover temperature (Temperature, prover)

Pp Prover pressure (Pressure, prover) in psig

E Modulus of elasticity (flow tube)

WT Wall Thickness of flow tube

ID Inside Diameter of flow tube

Gl

Coefficient of linear thermal expansion (switch bar) – where present

(either linear and area values are used in combination or the cubical value is used)

Ga

Coefficient of flow tube area thermal expansion (either linear and area values are used 

in combination or the cubical value is used)

Gc

Coefficient of cubical thermal expansion (either linear and area values are used in 

combination or the cubical value is used)

CPL Correction for the pressure on the liquid

CTS Correction for the temperature on the standard prover CTSp1, based on linear and area 

coefficients of expansion or CTSp2, based on cubical coefficients of expansion

CPS Correction for the pressure on the standard prover

CCF Combined correction factor

Calculate water density for air-saturated water, using the equations provided in 

GLP 10.

Calculate the air density per NISTIR 6969, Selected Mass Calibration Procedures, 

SOP 2, Option B. 

Calculate the within process standard deviation, sw, for the replicate runs and 

determine the applicable degrees of freedom (number of replicates minus one; if 

five runs are made, there are four degrees of freedom). The repeatability data is 

plotted on a standard deviation chart.

An F-test may be incorporated to compare observed standard deviations from the 

current calibration with the accepted standard deviations of the process for similar 

type proving systems used as standards. Where used, calculate the F statistic to 

compare the observed within process standard deviation, sw, to the accepted 

(pooled) within process standard deviation for the measurement process. (See 

NISTIR 6969, Sections 8.4 and 8.9.2, for more information on pooling standard 

deviations and F-tests).

2

Accepted

2

Observed

w

w

s

s F = Eqn. (5)

The calculated F statistic must be less than the F value obtained from an F table at 

95 % confidence level (Table 9.12, NISTIR 6969) to be acceptable. The F value is 

obtained from the F table for numerator degrees of freedom and denominator 
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degrees of freedom equal to the number of degrees of freedom in the pooled within 

process standard deviation. If the data fails the F-test and the source of the error 

cannot be determined conclusively and corrected, the measurement must be 

repeated.

Calculate and report the mean volume of the volumetric standard at its applicable 

reference temperature and reference pressure.

If adjustments were made during replicate runs, report the “as found” volume or 

the mean of “as found” volumes and the “as left” volume or mean of “as left” 

volumes, as applicable, at the appropriate reference temperature. (I.e., do not 

calculate a mean value by combining “as found” and “as left” values when 

adjustments are made). If the standard has bi-directional volumes, report the mean 

volume for each direction at its applicable reference temperature and reference 

pressure. 

Calculate the uncertainty for the calibration using Section 5. 

4 Measurement Assurance

Record and plot the values of the five runs and calculate the observed standard 

deviation to determine short-term repeatability of the measurement process, sw

following SOP 20. The range of the five runs must repeat to within 0.02 % of the 

volume, or the standard deviation must be less than applicable control/action limits 

on standard deviation charts, or the observed standard deviation must pass the Ftest statistics (as applicable).

For standards that may be and are adjusted, do not combine an “as found” value 

with an “as left” value for the replicate runs when entered a standard deviation 

chart; use the adjusted value from those runs with the as left value from remaining 

runs, all at the applicable reference temperature, when entering values in a standard 

deviation or range chart. A minimum of 12 replicate measurements are required to 

establish initial process limits and 25 to 30 points are required for reporting valid 

uncertainty values.

Perform applicable calibrations of check standards where available. 

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, and the additional items 

noted below and in the uncertainty budget table, Table 4, at approximately a 95 % level of 

confidence. See SOP 29 for the complete standard operating procedure for calculating the 

uncertainty. 
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The standard uncertainty for the mass standards, us, is obtained from the calibration 

certificates for the standards used. The combined standard uncertainty, uc, is used 

and not the expanded uncertainty, U, therefore the reported uncertainty for the 

standard is usually divided by the coverage factor k.

The standard deviation of the measurement process is obtained from replicate 

measurements and standard deviation chart performance data. See SOP 17 or 20.

The value for sp is obtained from the standard deviation chart data and results for 

the current run must be compared to the standard deviations observed from replicate 

measurements over time. The standard deviation of the process determined from 

this approach incorporates a repeatability factor related to the precision of all 

weighings and need not be duplicated.

Other standard uncertainties usually included for this type of calibration level 

include uncertainties associated with water temperature measurements, 

thermometer accuracy, calculation of air density, pressure measurements and 

standard uncertainties associated with the density of the standards used or the lack 

of internal cleanliness.

An example uncertainty budget table is shown in Table 4. Additional components 

may need to be considered depending on the technology being calibrated.
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Table 5. Example uncertainty budget table.

Uncertainty Component 

Description Symbol Source Typical Distribution

Mass Standards (filled; empty) us Calibration certificate(s) Expanded divided 

by k

Mass Density uρs

Assumed reference densities, 

or OIML R111 Rectangular

Standard deviation of the process sp Standard deviation chart Normal, 1s

Prover Temperature utp

Ability to measure and 

possibility of gradients Rectangular

Water Temperature utw

Ability to measure and 

possibility of gradients Rectangular

Water Density Equation uρw GLP 10 Rectangular

Water Compressibility uK Kell, CIPM Reference Rectangular

Cubical Coefficient of Thermal 

Expansion (CCE) uGc

5 % to 10 % of the CCE

(EURAMET CG-19) Rectangular

Area Coefficient of Thermal 

Expansion uGa

5 % to 10 % of the CCE

(EURAMET CG-19) Rectangular

Linear Coefficient of Thermal 

Expansion uGl

5 % to 10 % of the CCE

(EURAMET CG-19) Rectangular

Modulus of Elasticity (E) uE Estimate 1 % of the E Rectangular

Prover Pressure upsig Calibration certificate Rectangular

Flow Tube Internal Diameter (ID) uID

Estimate by one division in 

last decimal place Rectangular

Tube Wall Thickness (WT) uWT

Estimate by one division in 

last decimal place Rectangular

Air Temperature uta

NISTIR 6969, SOP 2, 

Calibration certificate Rectangular

Barometric pressure (absolute) up

NISTIR 6969, SOP 2, 

Calibration certificate Rectangular

Relative Humidity uRH

NISTIR 6969, SOP 2

Calibration certificate Rectangular

Air Density Equation uρa eqn

NISTIR 6969, SOP 2, 

Calibration certificate Rectangular

Viscosity uV NIST SP 250-72 Rectangular

Uncertainty Evaluation

Where applicable, uncertainties for volume calibrations that are assessed for 

conformity must meet decision rule criteria in the applicable documentary 

standards.
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6 Calibration Certificates

Report results as described in NISTIR 6969, SOP 1, Preparation of Calibration

Certificates, with the addition of the following:

Volume, reference temperature, uncertainty, material, thermal coefficient 

of expansion (assumed or measured), construction, any identifying 

markings, tolerances (if appropriate), laboratory temperature, water 

temperature(s) at time of test, barometric pressure, relative humidity, and 

any out–of-tolerance conditions. Volume may need to be reported as “upstream” and “down-stream” volumes for bi-directional provers. 

The status of items that are not calibrated by the laboratory but that may be 

used to perform measurements must be noted on the calibration report (e.g., 

temperature sensors, pressure gauges, or full flow rates that may be used in 

the field but that are not calibrated by the laboratory). A statement such as 

the following must be included on calibration reports: “Only items 

presented on this calibration report were tested under the conditions 

shown”. Any sensors or gauges that were not calibrated by the laboratory 

should be clearly identified on the report (e.g., “Calibration certificates for 

the system included temperature sensors and pressure gauges that were 

provided to the laboratory for evaluation prior to calibration and were found 

to be acceptable.”) 

Conformity Assessment

Evaluate compliance to applicable tolerances as needed or required by the customer 

or by legal metrology requirements. Compliance assessments must note the 

applicable documentary standard and which portions of the standard were or were 

not evaluated. The uncertainty for volume calibrations must be less than the 

tolerances published in the applicable documentary standards. For volume 

calibrations where the unknown standard can be adjusted, it is standard practice to 

adjust the standard or leave the scale plate in a position close enough to its nominal 

volume to ensure that the absolute value of the measurement result plus the 

uncertainty is less than the applicable tolerance. Where the unknown standard 

cannot be adjusted, and a portion of the uncertainty band from the error exceeds 

tolerance limits, it is not appropriate to state compliance with the tolerances unless 

additional decision rules are communicated with and agreed to by the end user. 

Correction values (measurement results) may need to be used by the end user in 

such cases.

Where no specific documentary standards apply, or a laboratory does not perform 

an assessment, a notice should be included on the calibration certificate to the effect 

that a legal metrology assessment was not performed. (For example, “Conformity 

assessment was not completed as a part of this calibration. Accurate use in field 

testing requires assessment of repeatability in field applications as well as proper 

training of the operator, maintenance of the proving system, integrity of all seals 
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and sensors, and any other auxiliary mechanical or electronic components. The 

entire system has not been assessed for suitability as a field standard for weights 

and measures applications.”)

7 References

Chapter 4, Manual of Petroleum Measurement Standards, and the following Sections as 

applicable:

Section 1 – Introduction

Section 2 – Conventional Pipe Provers

Section 3 – Small Volume Provers

Section 4 – Tank Provers

Section 5 – Master-Meter Provers

Section 6 – Pulse Interpolation

Section 7 – Field Standard Test Measures

Section 8 – Operation of Proving Systems

NIST Handbook 105-7: Specifications and Tolerances for Reference Standards and Field 

Standard Weights and Measures, 7. Specifications and Tolerances for Dynamic Small Volume 

Provers, 1996.

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, (2009).

EURAMET Calibration Guide 19, Guidelines on the Determination of Uncertainty in Gravimetric 

Volume Calibration, (Version 2.1, 03/2012). 
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Appendix A

Gravimetric Calibration Data Sheet

Company Name: ________________________________________ Test No: __________________

Address: ________________________________________ Purchase Order No: __________________

City, State, Zip: ________________________________________ Date Scheduled: __________________

Representative: ________________________________________ Date Received: __________________

Phone: __________________ Fax: ________________ Date Tested: __________________

Email: ________________________________________ Date Returned: __________________

Company URL: ________________________________________ Next Appointment: __________________

General Description ________________________________________

Manufacturer: ________________________________________

Model Number: _______________ Serial Number: ___________

Condition: ________________________________________

Compressibility factor for water 3.2000E-06 Cubical Thermal Expansion Coefficient (Gc)

Reference Temperature (Tb) (oF) Area Thermal Expansion Coefficient (Ga)

Reference Temperature (oC) Detector Thermal Expansion Coefficient (Gl)

Reference Pressure (psig) Modulus of Elasticity (flow tube) (E)

Flow Tube Inside Diameter (inches) (ID)

Flow Tube Wall Thickness (inches) (WT)

Mass standard(s) data:

Identification (ID) 

(Note ID and whether for 

Filled or Empty Transfer 

Vessel)

Nominal Mass Mass CorrectionA Expanded Unc:

From Cal. Cert.

Unc:

k factor

Density

g/cm3

S

S

S

S

S

S

S

S

A Mass Correction = Mass values are required for this procedure (i.e., Conventional Mass must not be used).
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Observations (Option A):

Fill Number Fill # 1 Fill # 2 Fill # 3 Fill # 4 Fill # 5

Flow Rate Fast Fast Fast Slow Slow

Run Time (sec)

Balance Reading O

1 (Standards for Empty 

Transfer Vessel)

Balance Reading O

2 (Empty Transfer Vessel)

Balance Reading O

3 (Standards for Filled 

Transfer Vessel)

Balance Reading O

4 (Filled Transfer Vessel)

Air Temperature (

o

C)

Barometric Pressure (mmHg)

Relative Humidity (% RH)

Water Temperature (

o

C) – record all values

Detector Bar (where present) 

Temperature (Td) (

o

C)

Prover Temperature (Tp) (

o

C)

Prover pressure (Pp) (psig)

Observations (Option B):

Fill Number Fill # 1 Fill # 2 Fill # 3 Fill # 4 Fill # 5

Flow Rate Fast Fast Fast Slow Slow

Run Time (sec)

Balance Reading O

1 (standards)

Balance Reading Empty Transfer Vessel Tared 

and Removed (B))

Balance Reading Immediately Before O

2

Reading (d

1

)

Balance Reading O

2 (water)

Balance Reading Full Transfer Vessel 

Removed (d2)

Validity Check (Compares d

1

-B, and d2-d

1

)

Drift Compensated O

2

(O2c)

Air Temperature (°C)

Barometric Pressure (mmHg)

Relative Humidity (% RH)

Water Temperature (

o

C) – record all values

Water Temperature (°F)

Detector Bar (where present) 

Temperature (Td) (°C)

Detector Bar (where present) 

Temperature (Td) (°F)

Prover Temperature (Tp) (°C)

Prover Temperature (Tp) (°F)

Prover pressure (Pp) (psig)
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