NIST Special Publication 250-39 2009 

NIST Calibration Services for 

Pressure Using Piston Gauge 

Standards 

Douglas A. Olson 
NIST Special Publication 250-39 2009 

NIST Calibration Services for 

Pressure Using Piston Gauge 

Standards 

 Douglas A. Olson 

Chemical Science and Technology Laboratory 

Process Measurements Division 

Pressure and Vacuum Group 

July 2009 

U.S. Department of Commerce 

Gary Locke, Secretary 

National Institute of Standards and Technology 

Patrick D. Gallagher, Deputy Director 

 ii
Certain commercial entities, equipment, or materials may be identified in this 

 document in order to describe an experimental procedure or concept adequately. Such 

identification is not intended to imply recommendation or endorsement by the 

National Institute of Standards and Technology, nor is it intended to imply that the 

entities, materials, or equipment are necessarily the best available for the purpose. 

National Institute of Standards and Technology Special Publication 250-39 2009

Natl. Inst. Stand. Technol. Spec. Publ. 250-39 2009, 69 pages (July 2009) 

CODEN: NSPUE2 

 iii
NIST Calibration Services for Pressure Using Piston Gauge Standards 

Table of Contents 

1 Introduction.................................................................................................................... 1 

2 Description of the measurement service ...................................................................... 1 

3 NIST piston gauge pressure standards........................................................................ 3 

3.1 NIST gas primary standards ................................................................................. 6 

3.1.1 Dimensional measurements............................................................................... 6 

3.1.2 Force models...................................................................................................... 9 

3.1.3 Comparisons between standards...................................................................... 10 

3.1.4 Final result for effective area and uncertainty ................................................. 12 

3.2 NIST oil primary standards................................................................................. 12 

3.2.1 Piston area........................................................................................................ 14 

3.2.2 Fall rate measurements for determining pz ...................................................... 15 

3.2.3 Cross-float measurements for determining d................................................... 15 

3.2.4 Summary for NIST oil primary standards ....................................................... 17 

3.3 Other terms in measurement equation ............................................................... 17 

3.3.1 Mass................................................................................................................. 17 

3.3.2 Local acceleration due to gravity..................................................................... 17 

3.3.3 Ambient density............................................................................................... 19 

3.3.4 Density of masses ............................................................................................ 20 

3.3.5 Surface tension and piston circumference ....................................................... 20 

3.3.6 Temperature ..................................................................................................... 20 

3.3.7 Thermal expansion........................................................................................... 21 

3.4 NIST transfer standards....................................................................................... 21 

4 Calibration techniques and procedures..................................................................... 29 

4.1 Calibration of piston gauges using the crossfloat method................................. 29 

4.1.1 Experimental arrangement for a crossfloat...................................................... 30 

4.1.2 Reference levels in a crossfloat........................................................................ 35 

4.1.3 Hydraulic connections ..................................................................................... 36 

4.1.4 Cleaning ........................................................................................................... 36 

4.1.5 Piston gauge rotation........................................................................................ 37 

4.1.6 Measurements and data evaluation .................................................................. 37 

4.2 Calibration of sensing-only pressure instruments............................................. 38 

4.2.1 Experimental arrangement and calibration procedure ..................................... 38 

4.2.2 Measurements and data evaluation .................................................................. 40 

5 Uncertainty analysis..................................................................................................... 40 

5.1 Uncertainties in a crossfloat calibration ............................................................. 41 

5.1.1 Uncertainty due to reference area .................................................................... 45 

5.1.2 Uncertainty due to mass on REF piston gauge ................................................ 45 

5.1.3 Uncertainty due to mass on TEST piston gauge.............................................. 45 

5.1.4 Uncertainty due to ambient density ................................................................. 46 

5.1.5 Uncertainty due to REF gauge mass density ................................................... 46 

5.1.6 Uncertainty due to TEST gauge mass density ................................................. 47 

5.1.7 Uncertainty due to fluid surface tension .......................................................... 48 

5.1.8 Uncertainty due to REF piston circumference................................................. 48 

 iv
 v

5.1.9 Uncertainty due to TEST piston circumference............................................... 48 

5.1.10 Uncertainty due to acceleration of gravity..................................................... 48 

5.1.11 Uncertainty due to thermal expansion of the REF piston and cylinder......... 49 

5.1.12 Uncertainty due to thermal expansion of the TEST piston and cylinder....... 49 

5.1.13 Uncertainty due to temperature of REF piston and cylinder ......................... 49 

5.1.14 Uncertainty due to temperature of TEST piston and cylinder....................... 50 

5.1.15 Uncertainty due to density of pressure fluid.................................................. 50 

5.1.16 Uncertainty due to reference height difference.............................................. 50 

5.1.17 Uncertainty due to pressure difference .......................................................... 51 

5.1.18 Summary of uncertainties in a crossfloat calibration..................................... 51 

5.2 Uncertainty in pressure sensing-only calibration .............................................. 52 

5.2.1 Uncertainty due to reference area .................................................................... 53 

5.2.2 Uncertainty due to mass................................................................................... 53 

5.2.3 Uncertainty due to ambient density ................................................................. 55 

5.2.4 Uncertainty due to mass density ...................................................................... 55 

5.2.5 Uncertainty due to fluid surface tension .......................................................... 56 

5.2.6 Uncertainty due to piston circumference ......................................................... 56 

5.2.7 Uncertainty due to acceleration of gravity....................................................... 56 

5.2.8 Uncertainty due to thermal expansion of piston and cylinder ......................... 56 

5.2.9 Uncertainty due to temperature of piston and cylinder.................................... 57 

5.2.10 Uncertainty due to density of pressure fluid.................................................. 57 

5.2.11 Uncertainty due to reference height difference.............................................. 58 

5.2.12 Uncertainty due to vacuum pressure.............................................................. 58 

6 Quality system.............................................................................................................. 59 

7 References..................................................................................................................... 60 

Appendix A. Recommended formula for the calculation of the density of moist air 

and its uncertainty .......................................................................................................... 61 

Appendix B. Sample calibration report of a customer piston gauge ......................... 64 
1 Introduction 

The National Institute of Standards and Technology (NIST) is responsible for realizing, 

maintaining, and disseminating the derived SI unit of thermodynamic pressure, the pascal 

(Pa). Pressure is an intrinsic property that is equal to the amount of force applied on a 

unit of area. Pressure is a key thermodynamic quantity fundamental to defining the state 

and property of a substance. Currently at NIST, there are no intrinsic methods for 

determining pressure, i.e., no “fixed points” for establishing pressures that can be 

calculated from theoretical considerations. As a derived unit of the SI, the practical 

realization of pressure depends upon measuring other SI units, such as force or length, 

and then relating that measurement through the physical principles of the measurement to 

the pascal. The device which realizes an SI unit is often referred to as a standard.

This special publication describes the calibration services, methods, standards, and 

uncertainties for the calibration of pressure measuring devices using NIST piston gauge 

standards. These standards operate from 10 kPa to 280 MPa (1.4 psi to 40,000 psi). 

NIST provides other calibration services for devices using manometer standards and 

vacuum standards for pressures from 10-7 Pa to 360 kPa, which are not covered in this 

publication. The measurement service is described in Sec. 2. The pressure standards 

used by NIST are described in Sec. 3, and the calibration techniques are discussed in 

Sec. 4. Section 5 is a complete uncertainty analysis of the measurements, and Sec. 6 

describes the NIST quality system. 

2 Description of the measurement service 

The National Institute of Standards and Technology provides calibration services for 

pressure measuring instruments known as piston gauges, pressure transducers, electronic 

barometers, and pressure gauges. Piston gauges are also referred to as pressure balances 

or dead weight gauges. This document will use the term “sensing-only” instrument for 

all pressure transducers, electronic barometers, and pressure gauges. For devices using 

gas as the pressure medium, the pressure range is 10 kPa to 104 MPa (1.4 psi to 

15,000 psi). For devices using oil as the pressure medium, the pressure range is 1 MPa to 

280 MPa (150 psi to 40,000 psi). Routine calibrations are performed using NIST transfer 

standard piston gauges. Special calibrations can be performed against NIST primary 

standard piston gauges, at higher cost and longer turn-around times. The uncertainty in 

pressure using a NIST piston gauge is dominated by the uncertainty in “effective area” 

associated with the NIST piston gauge. The effective area is the “calibration factor” of 

the piston gauge; it is the quantity that when combined with the loaded forces on the 

gauge produces the pressure in the fluid line connected to the gauge. A complete 

discussion of uncertainty is found in Sec. 5. Expanded1

 (k=2) relative uncertainties in 

effective area for the NIST transfer standard piston gauges range from 8x10-6 to 40x10-6

(8 ppm to 40 ppm) for gas, and from 22x10-6 to 37x10-6 (22 ppm to 37 ppm) for oil. 

 

1

 All references to expanded uncertainty in this document shall be interpreted as an uncertainty at the k=2 

level of the quantity. All references to standard uncertainty in this document represent one standard 

deviation (k=1) of the quantity. 

1
Calibrations are performed using a range of piston gauge standards. Typically, the 

customer's instrument is calibrated against the NIST standard that best matches the 

desired pressure range and media. An item can be calibrated against more than one NIST 

standard; however it will be charged as a multiple calibration. Most NIST piston gauges 

operate over a pressure range between 20:1 and 10:1. Although any pressure instrument 

can be calibrated in this service, devices which have a relative expanded uncertainty 

greater than about 1x10-4 are commonly calibrated by independent calibration 

laboratories rather than at NIST. At NIST, all instruments are calibrated using piston 

gauge standards, and there is no cost reduction for a device with higher uncertainties. 

A piston gauge is a differential pressure measurement device measuring the difference in 

pressures applied to the top and bottom of the piston. When the piston gauge is used with 

the top of the piston at ambient pressure, the unit is operating in “gauge” mode. When 

the piston gauge is used with the top of the piston in a vacuum, the unit is operating in 

“absolute” mode. The "gauge" mode is used for all oil calibrations and most gas 

calibrations; certain gas calibrations can also be performed in the "absolute" mode upon 

request. Sensing-only instruments can also be calibrated in gauge or absolute mode. 

Customers should consult the most current version of the NIST Calibration Services 

Users Guide [1], which can be found on the NIST website at 

http://ts.nist.gov/MeasurementServices/Calibrations/upload/feesch-09.pdf, to find up-todate information on calibration services, calibration fees, technical contacts, turn around 

times, and instrument submittal procedures. The NIST calibration service using piston 

gauges follows all procedures set forth in the Users Guide. Chapter 2 gives instructions 

for ordering a calibration for domestic customers, and Chapter 3 gives special 

instructions for foreign customers. Chapter 6 lists fees and technical contacts. 

When a piston gauge is calibrated in the present service, the result that is presented to the 

customer is the effective area of the piston gauge rather than the pressure. The pressure 

generated by a piston gauge depends on the effective area, the force loaded on the piston 

(which depends on the local gravitational constant), and local operating conditions such 

as temperature and pressure. Presenting the result as an effective area allows the piston 

gauge to be used in a variety of conditions without re-calibration as those conditions 

change. For the pressure generated by the calibrated piston gauge to be traceable to the 

SI at the time of use, the masses loaded on the piston, which generates the force, must 

also be traceable. The calibration service using NIST piston gauges does not include 

calibration of the customer’s masses. That service is described in Chapter 5 of [1]. 

A piston gauge is a pressure-generating device, which means that when two of them are 

connected for calibration a different procedure is required than for a sensing-only 

pressure instrument. The procedure for calibrating two piston gauges is often referred to 

as a “crossfloat”. The conceptual difference between a crossfloat calibration and a 

sensing-only calibration is that in a crossfloat some method is used to determine if the 

two piston gauges are in pressure equilibrium, and if not, there is a method to adjust the 

force (mass) on the gauges to bring them into equilibrium. A sensing-only calibration 

2
requires recording the output of the pressure instrument at the pressure points established 

by the piston gauge standard, but no incremental adjustment of the mass on the NIST 

piston gauge. 

3 NIST piston gauge pressure standards 

A piston gauge is a round piston fitted into a matching cylinder filled with fluid, loaded 

with weights of known mass and density. A schematic of a typical piston gauge of the 

“free deformation” type is shown in Fig. 1. The piston is marginally smaller in diameter 

than the cylinder, and fluid fills the gap between the two components. The piston is 

rotated with respect to the cylinder (most common is that the cylinder is stationary and 

the piston rotates). The rotation minimizes the possibility of metal-to-metal contact. The 

term “free deformation” means that the inside of the cylinder is pressurized by the fluid, 

and the outside of the cylinder is exposed to ambient pressure and is constrained by the 

column. The fluid pressure, p, is determined by an equilibrium balance of the downward 

gravitational force due to the weights plus the surface tension of the fluid on the piston, 

against the upward force of the pressure acting on the “effective area”, Ae(T,p), of the 

piston gauge. Or, 

(, ) 1 a

e i

i mi

pA T p mg C ρ γ

ρ

⎛ ⎞

= −+ ⎜ ⎟ ⎝ ⎠ ∑ . (1) 

Here, 

mi are the masses of the piston and all the weights loaded on it; 

g is the local acceleration due to gravity; 

ρa is the density of the ambient gas surrounding the masses (air for gauge mode, vacuum 

for absolute mode); 

ρmi is the density of weight mi; 

γ is the surface tension of the pressurizing fluid; and 

C is the circumference of the piston. 

Ae(T,p) is written as a function of temperature and pressure, for reasons which will be 

explained below. The pressure acts on the “effective area” rather than the piston crosssectional area, since two other force terms act upward on the piston in addition to the 

pressure on the bottom of the piston. The effective area accounts for the sum of the three 

forces. The first additional force is the frictional force of the fluid flowing upward 

through the gap between the cylinder and piston, acting on the piston. The second 

additional force is the vertical component of the normal force from the fluid on the piston 

walls, which is non-zero if there is any slope or “profile” to the piston over its height. In 

the limit when the gap is small compared to the piston diameter, along with both a 

straight and round piston and cylinder, it can be shown [2] that the effective area is equal 

to the average of the piston and cylinder area. 

The effective area is expressed as a function of temperature and pressure, since the piston 

and cylinder will distort under pressure, and both components will expand or contract as 

3
Cylinder

Piston

Column

Masses

Fluid 

pressure

Figure 1. Schematic of free deformation piston gauge. 

4
the temperature changes. Thermal expansion is accounted through the expansion 

coefficients, or 

ATp A e e pc (, ) 1 = ++ − ( (α α )( ) T Tr ) , (2) 

with 

Ae is the effective area at pressure p and reference temperature Tr; 

αp is the linear coefficient of thermal expansion for the piston; 

αc is the linear coefficient of thermal expansion for the cylinder; and 

Tr is the reference temperature. 

The functional dependence of Ae = Ae(p,Tr) is still implied, but the full notation is 

simplified for all equations which follow. NIST uses 23 oC as the reference temperature, 

while 20 oC is also used at other National Metrology Institutes. Substitution of eq. (2) 

into eq. (1) gives the measurement equation for pressure from a piston gauge: 

( ) ( )( )

1

1

a

i

i mi

e pc r

m g C

p

A T

ρ γ

ρ

α α

⎛ ⎞ ⎜ ⎟ − +

⎝ ⎠ = ++ −

∑

T . (3) 

All NIST piston gauge pressure standards use eq. (3) to determine pressure. When using 

a piston gauge, all terms on the right hand side in eq. (3) must be determined: the mass 

and density values of the weights loaded on the gauge, the density of the ambient gas 

surrounding the masses, the circumference of the piston and the surface tension of the 

fluid, the temperature of the gauge, the thermal expansion of the piston and cylinder, and 

the effective area. A primary standard for pressure is an instrument that does not require 

a pressure calibration to measure pressure. It is characterized by measuring the 

fundamental units of mass, time, length and temperature, along with modeling of the 

physical behavior of the standard. In terms of eq. (3), a primary standard piston gauge 

has a determination of effective area at p and Tr that does not depend on calibration 

against another pressure standard. NIST has two gas piston gauge primary standards, 

designated as PG38 and PG39 [3]. These two gauges were characterized by dimensional 

measurements and modeling of the fluid forces on the piston. NIST has three oil piston 

gauge primary standards, designated as PG20, PG27, and PG67 [4]. The primary 

standards are used to calibrate NIST transfer standard piston gauges; the transfer 

standards are therefore traceable to the primary standards. Prior to the introduction of 

PG38 and PG39 into use in 2008, NIST gas piston gauge standards were traceable to a 

mercury manometer, which was described in the first edition of this SP250 [4]. 

The effective area for NIST primary standards will be considered in Secs. 3.1 and 3.2. 

The determination of the remaining terms in eq. (3) follows in 3.3. The effective area of 

the NIST transfer standards used in the calibration service is discussed in Sec. 3.4. 

5
3.1 NIST gas primary standards 

PG38 and PG39 are gas primary piston gauge standards that operate from 20 kPa to 

1 MPa. They were acquired by NIST from Ruska Instrument Corporation in 1989. They 

are deemed primary standards for pressure as their effective area is derived from 

dimensional measurements of the piston and cylinder diameters (which in turn are 

traceable to the wavelength of an atomic transition in a HeNe laser interferometer), along 

with force models of the gas flow in the gap between the piston and cylinder. 

A cross-section of the piston/cylinder assemblies is shown in Fig. 2, and a picture of the 

assemblies is shown in Fig. 3. The assemblies are “twins’ in the sense that they were 

made from the same casting of tungsten carbide and have the same nominal dimensions. 

The pistons are hollow, with the hollow end pointed downward in normal operation as 

shown in Fig. 2 on the left-hand side. Their nominal diameters are approximately 

35.8 mm and their length is 75 mm. The radial clearance between pistons and cylinders 

is about 600 nm. The construction of the pistons is such that they can be inserted into 

their cylinders either upright or inverted. When operated in the inverted configuration, a 

special cap with a spherical pivot is placed onto the hollow end to allow the loading of 

masses. That cap is not sealed to the piston. In the upright position, the interior of the 

piston is subjected to the system pressure, whereas in the inverted position the piston 

interior is subjected to ambient pressure. The two orientations of the piston have a 

different calculable value for the pressure coefficient (relative change in Ae with 

pressure). The effective area of both PG38 and PG39 is given by the linear distortion 

equation: 

A A bp e = + 0 1 (1 ) . (4) 

Here, A0 is the effective area at atmospheric pressure and the reference temperature, 

23 oC, and b1 is the pressure coefficient. 

There are two components in the establishment of PG38 and PG39 as primary standards. 

The first is the dimensional measurements of the piston and cylinder diameters; the 

second is the analysis of that data with force and distortion models to determine the 

effective area. The modeling and dimensional measurements are used to determine both 

A0 and the linear pressure distortion coefficient, b1. An important verification of the 

results is comparisons of the effective area of PG38 and PG39 against each other when 

operated as pressure standards; and comparison of the gauges against the NIST mercury 

manometer, which is an independent primary pressure standard traceable to the density 

and speed of sound of mercury. 

3.1.1 Dimensional measurements 

Both PG38 and PG39 were first dimensioned in 1989 at NIST [5] using a stack of gauge 

blocks, a precision comparator to compare the length of the gauge blocks to the piston 

diameters, and a laser interferometer to measure the length of the gauge blocks. Based on 

those measurements the total relative standard uncertainty of the effective area was 

6
 

Cylinder

36 mm

75 mm

Piston Cap

 

Figure 2. Schematic diagram of the PG38 and PG39 piston/cylinder assembly with the 

piston in upright (left) and inverted (right) orientations. The cap on the right is 

used to support the weight carrier plus weights. 

7
Figure 3. Picture of PG39 cylinder (left), piston (right), and mass set (top). Closed end 

of piston is shown. 

8
estimated as 10x10-6. In 1999, PG39 was dimensioned by Physikalisch Technische 

Bundesanstalt (PTB) using a state-of-the art diameter and form comparator in which a 

calibrating laser interferometer is integral to the apparatus. Absolute diameters were 

measured at four places on the piston and four places on the cylinder, with a standard 

uncertainty of 15 nm. Relative roundness was measured at 5 latitudes and relative 

straightness was measured at 8 longitudes. In 2003, both PG38 and PG39 were measured 

at PTB with the same device as PG39 was measured in 1999. This time, absolute 

diameters were measured at 10 places on the piston and 10 places on the cylinder. Four 

of the locations on PG39 were the same in 2003 as in 1999; the relative difference from 

the 1999 to 2003 ranged from -0.1x10-6 to -0.8x10-6. The standard uncertainty of the 

absolute diameters in 2003 was 12.5 nm and 25 nm for the piston and cylinder, 

respectively. Relative roundness and relative straightness were measured again in 2003 

for both pistons and both cylinders, at 5 latitudes for roundness and 8 longitudes for 

straightness. The standard uncertainty for the roundness and straightness measurements 

was 50 nm. The 2003 data for both PG38 and PG39 showed that the pistons were round 

to within the standard uncertainty of measurement. Changes in diameter with height for 

both pistons and both cylinders were larger than the standard uncertainty of the 

measurement. 

3.1.2 Force models 

The conventional method for determining the uncertainty of the effective area, which is 

based on the uncertainty of the dimensional measurements only, would imply that A0 has 

a relative standard uncertainty of 1.0x10-6. However, the low uncertainty of the 

dimensional measurements requires that we consider the appropriate model for 

converting the measurements into “effective area” when the piston gauge is used for 

generating pressure. The model needs to account for all of the forces on the piston: the 

external mass load, the normal pressure force on the piston base, the shear forces on the 

piston flanks, and the normal forces on the piston flanks. It also needs to account for the 

complete dimensional data which describes the artifacts. In the analysis which 

establishes PG38 and PG39 as primary standards [3], the data from PTB on roundness, 

straightness, and absolute diameters were reconstructed in the form of cylindrical 

“bird-cages” providing longitudinal and latitudinal crevice (piston-cylinder gap) 

variation. Forces were computed assuming two models of nitrogen flow behavior in the 

crevice: (1) viscous flow, and (2) flow of gas that interpolated between molecular flow 

and viscous flow. The effect of operating mode (gauge or absolute) was evaluated for 

both models. The effect of the dimensional uncertainty on the standard uncertainty in A0

was included by increasing or decreasing all piston and cylinder diameters by their 

standard uncertainty. A complete mathematical description of the models is given in [6]. 

The results of the two flow models, including the dimensional uncertainty, gives a 

distribution of A0 values. The accepted value for A0 was taken as the average of the 

maximum and minimum value of the results, and the standard uncertainty as one half of 

the difference between the maximum and minimum. Statistically this means that the 

distribution of A0 from the models is part of a normal distribution, and that the maximum 

and minimum results represent about a 2 out of 3 chance that the true quantity lies 

between those values. The relative standard uncertainty in A0 evaluated in this way is 

9
3.0x10-6 for both PG38 and PG39. The largest A0 occurred for the viscous flow model, 

and the smallest A0 occurred for the interpolated flow model in absolute mode. 

The value for the pressure distortion coefficient, b1, was determined from elasticity 

theory. Two model implementations of elasticity theory were considered. In one, both 

the cylinder and piston were modeled as infinitely long components subjected to radial 

forces due to the pressure on the walls, which allowed using analytical formulae. These 

formulae require a constant pressure (pg) in the piston-cylinder gap, even though the gap 

pressure varied from p at gap entrance to ambient at the top. The formulae were solved 

for pg = 0, p/2, and p. In a second model, finite-element analysis was used to include the 

constraint of the closed-end of the piston (Fig. 2) and vertical loading on both the piston 

and cylinder; these constraints produce two-dimensional stresses. The two models and 

three gap boundary conditions produced a distribution of b1 values from 7.95x10-12 Pa-1

to 10.0x10-12 Pa-1. The accepted value, 8.97x10-12 Pa-1, was chosen as the average of the 

maximum and minimum, and the standard uncertainty was taken as one-half the 

difference. To within the standard uncertainty of the distortion models, both PG38 and 

PG39 have the same b1 and uncertainty in b1 due to modeling. The combined standard 

uncertainty u(b1) includes the standard uncertainty in the Young’s modulus added in 

quadrature. u(b1) equals 1.12x10-12 Pa-1. 

3.1.3 Comparisons between standards 

NIST also realizes pressure with a primary standard mercury manometer known as the 

ultrasonic interferometer manometer (UIM) up to 360 kPa. PG38 and PG39 have been 

compared numerous times since 1989 to the UIM, serving as check standards to confirm 

UIM stability and also to confirm the stability of the piston gauges. The combined 

relative standard uncertainty of the UIM from 20 kPa to 360 kPa is 2.6x10-6. All 

comparisons of the UIM to the piston gauges has shown agreement to within one 

standard deviation of the combined standard uncertainty of the difference, with UIM 

pressures both higher and lower than those of the piston gauges. These comparisons 

show the combined stability of the UIM and the piston gauges, and given the independent 

nature of the realization technique, the likely stability of each method. 

PG38 and PG39 have been compared directly to each other from 20 kPa to 1 MPa, 

utilizing the unique feature mentioned earlier that both can be operated in the upright and 

inverted position. This comparison measures the area ratio A38/A39. Four comparisons 

(PG38-up, PG39-up; PG38-down, PG39-down; PG38-up, PG39-down; PG38-down, 

PG39-up) were performed, which were compared to predictions from the distortion 

models. With both gauges in the same orientation, the distortion coefficients should be 

the same and the slope (Δb1) of A38/A39 should be zero. With the two opposing 

orientations, the models predict (Δb1) = ± 7.2x10-12 Pa-1. Figure 4 shows the results of 

the four comparisons along with the predicted slopes from the analytical models. There 

is good agreement between the experimental result and the modeling. This helps confirm 

the use of elasticity theory to establish b1 and its uncertainty. 

10
0.999990

0.999995

1.000000

1.000005

1.000010

1.000015

0 200 400 600 800 1000 1200

Pressure / kPa

A 38/A 39

Up-Up

Dn-Dn

Dn-Up

Up-Dn

TWF

5 ppm

Figure 4. The ratio of the effective area for PG38 to that of PG39 (A38/A39). Symbols 

indicate ratios from crossfloat measurements of PG38 versus PG39 for 

different combinations of piston orientation (Up-Dn means PG38 upright, 

PG39 inverted). The dashed lines indicate ratios based on thick wall formulae 

from elasticity theory and A0 from dimensional characterization. 

11
3.1.4 Final result for effective area and uncertainty 

The final result for the primary standards for Ae in m2

 with p in Pa at a temperature of 

23 oC is: 

PG38: 

( ) 3 1 1.0079497 10 1 8.97 10 A p e

− − = × +× 2 , (5) 

PG39: 

( ) 3 1 1.0079484 10 1 8.97 10 e

2 A p − − = × +× . (6) 

The uncertainties from A0 and b1 are added in quadrature, giving the relative standard 

uncertainty2

 for both as 

( )( )

1/2 2 6 12 ( ) 3.0 10 1.12 10 e

e

u A

p A

⎡ − − =× +× ⎢⎣

2 ⎤

⎥⎦ . (7) 

The coefficient of thermal expansion for the combined piston and cylinder was measured 

in [6] for PG39, and is assumed the same for PG38. The result is: 

( ) 6 1 αα α p c 8.754 0.03 10 K − − =+= ± × . (8) 

Eq. (2) is then used to find the effective area when the temperature differs from Tr. 

3.2 NIST oil primary standards 

The NIST oil primary standards are of the type known as controlled clearance piston 

gauges (CCPG). In this type of piston gauge, shown schematically in Fig. 5, a pressure 

independent of the system pressure is applied to the outside of the cylinder. This “jacket 

pressure”, pj, minimizes the elastic distortion of the cylinder and controls the annular gap 

between the piston and cylinder. The ability to control the width of the gap (clearance) 

allows one to obtain the best operating conditions of the piston gauge, such as reduced 

fall rate and high mass sensitivity. However, the main advantage of the CCPG is that it 

can be characterized as a primary standard without extrapolation from another pressure 

standard. Heydemann and Welch [7] describe a method for characterizing a CCPG that 

involves dimensional measurement of the piston area, estimation of the piston distortion 

with pressure, and estimates of the piston-cylinder gap. The gap is estimated using 

measurements of fall rate of the piston and changes in system pressure in response to 

changes in jacket pressure. This method, known as the Heydemann and Welch (HW) 

method, is used on NIST’s three oil primary standards, PG27, PG20, and PG67. The 

analysis method presented here is explained in more detail in [7] and [8]. 

The measured pressure at the reference level of a CCPG at equilibrium conditions is 

 

2

 Throughout this document, lower-case u variables refer to standard (k=1) uncertainties; upper-case U

variables refer to expanded (k=2) uncertainties. 

12
Piston

Cylinder

Jacket 

pressure

Column

System 

pressure

Figure 5. Schematic of controlled clearance piston gauge. 

13
 determined by using the following equation: 

0 ( )( ) ( )( ) ( )( )

1

11 1

a

i

i mi

p p z j pc r

mg C

p

A bp d p p T T

ρ γ

ρ

α α

⎛ ⎞ ⎜ ⎟ − +

⎝ ⎠ = + + − ++ −

∑

 . (9) 

The reader should note the similarity to eq. (3); the forces due to mass and surface 

tension form the numerator; and the effective area forms the denominator. 

Characterizing the CCPG as a primary standard involves determining the effective area at 

the reference temperature, T = Tr: 

A A bp d p p e p p zj = + +− 0 (1 1 )( ( )) . (10) 

A0p is the area of the piston at ambient pressure and Tr; bp is the distortion coefficient of 

the piston; pz is a HW modeling parameter, equivalent to the jacket pressure for which the 

clearance between the piston and cylinder is zero at a given measured pressure; and d is a 

HW modeling parameter, equivalent to the relative change of effective area due to a 

change in jacket pressure. The pz and d parameters are determined in the characterization 

and are a function of p; bp is determined from analytical models; αp and αc are properties 

of the material; and T and pj are operating conditions. 

In the HW model, the piston area is estimated from A0p and the (1+bpp) piston distortion 

multiplier. A0p is determined dimensionally. The final term in the brackets in eq. (10) 

approximates the additional area due to the gap. We imagine applying sufficient jacket 

pressure to collapse the cylinder onto the piston, reducing the gap to zero and the 

effective area equal to the piston area only. We then reduce the jacket pressure, opening 

up the gap and increasing the effective area. pz is the jacket pressure that reduces the gap 

to zero, and the amount of area increase per change in pj is determined by the parameter 

d. One limitation of the HW characterization is that operating the CCPG at jacket 

pressures close to pz can potentially damage the piston or cylinder, and the mechanical 

design may not accommodate the high pressure. However, the uncertainties of the HW 

parameters, and hence the effective area, decrease if the jacket pressure approaches pz

during the characterization. A characterization is therefore a compromise between 

constraints of the system and the desire to reduce uncertainties. 

3.2.1 Piston area 

The piston diameter at ambient pressure, A0p, is measured by the NIST Dimensional 

Metrology Group in much the same way as was done for PG38 and PG39. The piston 

area is calculated from the average of the diameter measurements. The standard 

uncertainty in the diameter measurement for PG27, PG20, and PG67 is 26 nm, 51 nm, 

and 32 nm, respectively. The piston distortion coefficient, bp, is computed from elasticity 

theory by modeling the piston as uniformly loaded on the ends at p and pressurized in the 

gap at common pressure p/2 [7]: 

14
(1 3 )

p

p

b

E

− − μ = . (11) 

The Poisson’s ratio, μ, and modulus of elasticity, Ep, are material properties of the piston. 

Typical values for bp are -5x10-13 Pa-1 to -7x10-13 Pa, and the standard uncertainty u(bp) is 

about 0.03bp. For PG67 at its maximum pressure of 280 MPa, the relative change in 

piston area is -157x10-6 (157 ppm smaller) from its value at p = 0 MPa. 

3.2.2 Fall rate measurements for determining pz

Instead of determining pz by operating the CCPG at a jacket pressure that reduces the gap 

to zero, the HW model assumes that the gap will change linearly with applied jacket 

pressure, and extrapolates measurements taken at lower pressures. To determine pz, the 

HW model utilizes viscous flow theory that predicts that the flow rate (Q) of fluid in the 

piston-cylinder gap is proportional to the third power of clearance (hg) between the 

cylinder and piston. Or, 

3 Q h ∝

g . (12) 

The gap flow rate is directly proportional to the fall rate, v, of the piston, assuming no 

fluid leakage and neglecting volume changes of the fluid due to thermal expansion. The 

HW model further assumes that the gap width varies linearly with the jacket pressure at 

each measured pressure, p. The jacket pressure for which the clearance becomes zero, pz, 

is computed by measuring v vs. pj(at constant p), and fitting it to the following function: 

1/3

z z kv p p = − j . (13) 

kz and pz are fitting constants, with pz being the intercept of the fitted function at v = 0. 

Figure 6 shows data of a typical set of measurements of fall rate vs. jacket pressure, with 

constant system pressures designated by similar symbols. The solid and dashed lines are 

the fits of the data to eq. (13), showing the extrapolation of the fits to v = 0. pz values 

obtained for each p are fit to a function of p, usually linear: 

zz z 0 1 p = p p + p . (14) 

The parameter pz0 can be thought of as the jacket pressure that closes the piston-cylinder 

gap at zero measured pressure. The Type A standard uncertainty in pz is estimated from 

the standard uncertainty in the predicted values of the fit of eq. (14) [8]. 

3.2.3 Cross-float measurements for determining d

The HW parameter d is determined by monitoring the change in measured pressure due 

to the change in jacket pressure at each constant load: 

1

j

p d

p p

∂ = ∂ . (15) 

15
0

20

40

60

80

100

120

140

160

180

200

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

v

1/3 / (mm/s)1/3

pj / MPa

200

180

160

140

120

100

80

60

40

20

p / MPa:

Figure 6. Fall rate (v) of a typical CCPG over pj/p = 0 to 0.4, plotted as pj vs. v

1/3. 

Similar symbols are constant pressure (p). Linear fits of data over same range 

are extrapolated to v = 0, giving HW parameter pz. 

16
The definition of d follows from taking the partial derivative of both sides of eq. (9) with 

respect to pj, holding the load (mass) constant and neglecting higher order terms. p is 

measured by cross-float calibration of the CCPG against another piston gauge standard, 

at each mass load, over a range of jacket pressures. The p versus pj data are fitted to a 

linear function, and d is the fitted slope divided by the average value of p. Using an 

average p is justified since the relative change in pressure produced by changes in pj is 

typically less than 5x10-4, and this amount of change contributes to an error in the relative 

effective area of less than 0.2x10-6 (0.2 ppm). The results for d at each nominal pressure 

are then fit to a function of p or load force W (W is the numerator in eq. (9)). A linear 

function for d is most common: 

0 1 d d dp = + . (16) 

The Type A standard uncertainty in d is estimated from the standard uncertainty in the 

predicted values of the fit [8]. The fitting parameters for both pz and d can be dependent 

on the pressure transmitting fluid. The NIST CCPGs were characterized using Spinesstic 

oil. 

3.2.4 Summary for NIST oil primary standards 

A summary of the characterization parameters for NIST CCPGs is listed in Table 1. 

More details on the design of the gauges and their complete uncertainty analysis is given 

in [4]. The full-scale pressures for PG27, PG20, and PG67 are 28 MPa, 140 MPa, and 

280 MPa, respectively. Values for bp and d are larger for PG20 because it is made of 

stainless steel, which has a lower Young’s modulus (less rigid) than the tungsten carbide 

of PG27 and PG67. 

3.3 Other terms in measurement equation 

What follows is the NIST treatment of the remaining terms in the measurement equation 

for pressure, eq. (3). Details of how these terms contribute to the uncertainty of a 

customer calibration are discussed in Sec. 5. 

3.3.1 Mass

The values of the masses of all weights used on NIST piston gauges are traceable to the 

NIST Mass and Force Group. An identifying number inscribed onto each weight is used 

to recall the appropriate mass value and density. Mass values must be known for all parts 

of the piston gauge supported by the pressure fluid, including the piston, weight table, 

and bell. 

3.3.2 Local acceleration due to gravity 

The best value of g for a given laboratory site may be obtained by having on-site 

measurements made by the Office of National Geodetic Survey, with relative standard 

uncertainties on the order of 1x10-7 (0.1 ppm) or better. The next best value of g can be 

obtained by an interpolation from a grid of measurements prepared by the same 

organization, with typical relative uncertainties of 1x10-6 to 5x10-6 depending on 

geographical location. The website that provides this interpolation is 

17
Table 1. Characteristics of NIST controlled clearance piston gauge primary standards, in 

terms of HW model. Pressure p in Pa. Relative standard uncertainty in pressure given at 

maximum operating pressure. A A bp d p p e p p zj = + +− 0 (1 1 )( ( )) . 

PG27 PG20 PG67

Max p / Pa 28x106 140x106 280x106

A 0p / m2 4.902139x10-5 3.218871x10-5 1.4219412x10-5

b p / Pa-1 -5.49x10-13 -7.23x10-13 -5.61x10-13

d / Pa-1 3.425x10-12 - 1.458x10-20p 8.662x10-12e

4744210/p 3.691x10-12 - 3.512x10-21

p

p z / Pa 40.13x106

 + 0.734p

9.31x106

 + 0.843p - 

8.714x10-10

p

2 39.52x106

 + 0.422p

α p / oC-1 4.5x10-6 9.41x10-6 4.5x10-6

α c / oC-1 4.5x10-6 9.41x10-6 4.5x10-6

u (A 0p )/A 0p 6.5x10-6 15.9x10-6 14.8x10-6

u (b p ) / Pa-1 1.8x10-14 2.2x10-14 1.8x10-14

u (d ) / Pa-1 7.4x10-14 3.0x10-13 1.6x10-13

u (p z ) / Pa 1.5x106 3.0x106 0.57x106

u (p )/p 9x10-6 20.7x10-6 24.4x10-6

NIST Controlled Clearance Piston Gauges Parameter

18
http://www.ngs.noaa.gov/cgi-bin/grav_pdx.prl. For further information on both of these 

services, contact National Oceanic and Atmospheric Administration, National Ocean 

Survey, Office of the National Geodetic Survey, Geodetic Information Center, 

Washington Science Center, Rockville, MD 20852. 

The value of g used for all piston gauge pressure standards at NIST since 1983 is 

9.801011 m/s2

 with a standard uncertainty of 0.000002 m/s2. It was measured at that 

time in Room A46 of the Metrology Building, which is adjacent to the calibration 

laboratories (Rooms B43 and B55). 

3.3.3 Ambient density 

The ambient density is the density of the gas surrounding the weights loaded on the 

piston. In gauge mode with no cover on the piston gauge, it is the air surrounding the 

masses. If there is a tightly fitting cover over the masses, the density will depend on the 

gas used (often nitrogen) and the pressure within the cover. In absolute mode, the gas 

within the cover is pumped out and the pressure is near zero, and the atmospheric density 

can be calculated with the perfect gas equation of state using the measured residual 

pressure, temperature, and molecular weight of the gas. NIST does not use tightly fitting 

mass covers with routine customer calibrations except in absolute mode. The term which 

combines the ambient density with the density of the masses (1 / ) − ρa m ρ is called the 

buoyancy correction. 

For gauge mode, the internationally accepted formula for the density of moist air is given 

in [9], and is referred to as the CIPM-2007 equation for the determination of the density 

of moist air. The relative standard uncertainty from this equation is 2.2x10-5. The moist 

air density depends on the air temperature, barometric pressure, mole fraction of water 

vapor, and to a much lesser degree the mole fraction of carbon dioxide. The CIPM-2007 

equation for ρa in kg/m3

 is: 

( ) ( ) 2

3

CO 3.483740 1.4446 0.0004 1 0.3780 10 . a v

p x x

ZT ρ − = + ⋅− ⋅ − × ⎡ ⎤ ⎣ ⎦ (17) 

Where p is the air pressure in Pa, T is the thermodynamic temperature in K, Z is the 

compressibility factor, xv is the mole fraction of water vapor, and CO2 x is the mole fraction 

of carbon dioxide. Details of the derivation of CIPM-2007 and calculation of Z and xv

(through relative humidity, h, or dew point temperature, td) are given in Appendix A. In 

practice, the uncertainties in the air temperature, pressure, and humidity contribute larger 

components to the air density uncertainty than does the CIPM-2007 equation. When 

calibrating one piston gauge against another in gauge mode, the masses of both are 

subjected to the same air density and approximately the same buoyancy correction; a 3 % 

relative standard uncertainty in air density contributes 3x10-7 in effective area relative 

standard uncertainty, even if the difference in mass densities is large (8400 kg/m3

 on one 

gauge and 7800 kg/m3

 on the other). When calibrating a pressure transducer in gauge 

mode, a 3 % relative standard uncertainty in air density of would contribute to a relative 

standard uncertainty in pressure of about 4x10-6. 

19
3.3.4 Density of masses

The density of the masses is required for the buoyancy correction. When the piston 

gauge is used in the gauge mode, the values of the densities of the masses used in eq. (3) 

must be identical to the values used during the mass calibration. For this case, the values 

can be arbitrary in the sense that they need not be correct for the metal in question, but 

they must be identical to the values used during the mass determination. The situation is 

different for absolute mode, where the air buoyancy correction is reduced to near zero. 

An incorrect mass density value used in determining the mass will result in a mass error 

and therefore a pressure error. For example, a 1 % standard uncertainty in mass density 

translates into a relative pressure standard uncertainty of 1.3x10-6. If masses are used in 

absolute mode it may be necessary to determine density values by hydrostatic weighing, 

or by determining the volume through dimensional measurement and iterating on the 

assumed density. 

3.3.5 Surface tension and piston circumference 

As the piston emerges from the fluid, a force is generated by the fluid surface tension, γ, 

acting on the circumference, C, of the piston. For oils, γ = 3x10-2 N/m. For a 10 mm 

diameter piston in oil, the surface tension produces a force equivalent to about 95 mg of 

additional mass, which represents about 11x10-6 (11 ppm) additional pressure at 1.1 MPa. 

At higher pressures, the relative magnitude of the force is less. For smaller diameter 

pistons, the force from the surface tension is also less. It is important to account for 

surface tension in oil piston gauges but it rarely contributes to the uncertainty of the 

pressure. For gases, the fluid surface tension is zero and the surface tension force is zero. 

3.3.6 Temperature 

The temperature of a piston gauge is measured with platinum resistance thermometers, 

thermocouples, or thermistors. A reliable and consistent temperature measurement is 

important since the area expands or contracts with temperature, and area changes produce 

pressure changes. For example, every 1 oC change in temperature in a tungsten carbide 

piston gauge changes the relative pressure by 9x10-6. The optimum location for the 

temperature sensor would be the working area of the piston and cylinder, but this is 

seldom possible due to practical considerations. Some manufacturers provide an integral 

sensor mounted in the housing containing the cylinder. Normally, the operating 

temperature is determined either on the base supporting the cylinder or on the lower end 

of the cylinder. It is important to keep sources of heat, such as electronics or computers, 

away from the piston gauge to prevent temperature gradients. If that is not possible, all 

electronics should be turned on several hours (typically NIST leaves them on overnight) 

prior to a calibration for temperatures to stabilize. One US piston gauge manufacturer3

has electronics mounted in the base below the piston gauge column, which can generate 

heat. Such a device should be turned on several hours before a calibration is performed. 

All the temperature sensing elements used at NIST are traceable to the Kelvin as realized 

by the NIST Thermometry Group. 

 

3

 DH Instruments, a Fluke Company 

20
3.3.7 Thermal expansion 

Coefficients of thermal expansion are material properties for the piston and cylinder. The 

reference temperature for piston gauge measurements at NIST is 23 oC, and the 

coefficients of thermal expansion along with the operating temperature determine the 

change in effective area compared to the area at the reference temperature. Tungsten 

carbide is the preferred material for modern piston gauges. Stainless steel is used by 

some manufacturers for certain designs, such as very low pressure piston gauges (the 

density of stainless steel is less) or very small diameter pistons (stainless steel tends to be 

more forgiving to mis-handling than tungsten carbide). 

3.4 NIST transfer standards 

NIST primary standard piston gauges are used to calibrate NIST transfer standard piston 

gauges, which are then used to calibrate customer pressure gauges and sensing-only 

instruments. The NIST transfer standard piston gauges area all unmodified, 

commercially available piston gauges. Use of the transfer standards saves wear on the 

primary standards, and saves time in the calibration. The primary standards are used only 

for international comparisons, to calibrate NIST transfer standards, and in rare instances 

to calibrate customer piston gauges. Such a customer calibration would be considered a 

special test, and would not follow the same fee schedule as routine calibrations. 

As of May 1, 2009, NIST uses eleven transfer standards for gas calibrations and three 

transfer standards for oil calibrations. Their NIST designations, pressure ranges, and 

pressure media are listed in Table 2. Also listed are their coefficients for effective area 

(details to follow). Five of the gas ranges use two nominally identical transfer standards, 

or “twins”. Use of the twin gauges provides redundancy if a gauge is temporarily taken 

out of service, and also provides a means for checking the relative performance of the 

gauges without re-calibrating them against the primary standards. 

A transfer standard is calibrated against a primary standard or another transfer standard 

using the cross-floating technique. A calibration means the determination of its effective 

area as a function of pressure. Since both piston gauges are pressure generators, the 

calibration is done by connecting both gauges to a common pressure line, floating both 

pistons at their respective reference levels, and determining if the pressures are equal 

(pressure equilibrium). The process is repeated for several pressures. The effective area 

of the transfer standard is calculated from the known masses on both gauges, the known 

effective area of the primary standard, measured temperatures of each gauge, and the 

pressure difference due the difference in reference levels. The equation of pressure 

equilibrium is: 

pT R fa = p g − − +Δ ( ρ ρ ) h P . (18) 

where pR is the pressure generated by the piston gauge of known area (REF), pT is the 

pressure generated by the transfer standard being calibrated (TEST), h is the height 

difference between the gauge reference levels (positive if TEST is higher than REF), ρf is 

the density of the pressurizing fluid, and ΔP is the residual pressure difference between 

the two piston gauge (its value is usually zero but the uncertainty is not). With p for each 

21
Table 2. Pressure range, medium, and effective area coefficients of NIST transfer 

standard piston gauges. 

A 0 / m2 b 1 / Pa-1 b 2 / Pa-2

PG22 10 kPa to 150 kPa gas 336 3.357224x10-4 0 0

PG36 10 kPa to 150 kPa gas 336 3.357388x10-4 0 0

PG28 20 kPa to 300 kPa gas 336 3.358209x10-4 0 0

PG29 20 kPa to 300 kPa gas 336 3.357227x10-4 0 0

PG34 35 kPa to 1.4 MPa gas 84 8.397281x10-5 5.903x10-12 0

PG37 35 kPa to 1.4 MPa gas 84 8.398156x10-5 8.319x10-12 0

PG13 360 kPa to 7 MPa gas 8.4 8.389145x10-6 2.661x10-12 0

PG35 360 kPa to 7 MPa gas 8.4 8.388724x10-6 4.267x10-12 0

PG23 700 kPa to 17 MPa gas 8.4 8.390295x10-6 -7.968x10-13 0

PG32 700 kPa to 17 MPa gas 8.4 8.389404x10-6 -7.968x10-13 0

PG87 9 MPa to 104 MPa gas 8.4 8.378298x10-6 -2.120x10-12 6.39x10-21

PG42 1 MPa to 26 MPa oil 84 8.402026x10-5 -2.086x10-12 0

PG41 7 MPa to 140 MPa oil 16.8 1.680257x10-5 -2.516x10-12 0

PG21 14 MPa to 280 MPa oil 8.4 8.402894x10-6 -2.744x10-12 0

Effective area coefficients

Standard Pressure range Medium

Nominal 

Area / mm2

22
standard given by eq. (3), substituted into eq. (18) and rearranged, the unknown is the 

effective area of the transfer standard, Ae,T: 

( ) ( )( )

( ) ( )( )

( ) ,

, , ,

, ,

, ,

,

,

1 1 23

1 .

1 23 1

a

i T T

mi T pR cR R f a

eT eR

pT cT T T T a

i R R

mi R

m gC T gh P A A

T p p m gC

ρ γ

ρ α α ρ ρ

ρ α α

γ

ρ

⎛ ⎞ ⎜ ⎟ − +

⎝ ⎠ ++ − ⎛ ⎞ − Δ = ⋅ ⋅ ⋅+ − ⎜ ⎟ ⎛ ⎞ ++ −

− + ⎝ ⎠ ⎜ ⎟

⎝ ⎠

∑

∑

 (19) 

All subscripts R refer to the REF piston gauge (primary standard or previously calibrated 

transfer standard), and all T subscripts refer to the TEST piston gauge4

. This is the 

measurement equation for effective area. The calibration process for determining the 

effective area of a NIST transfer standard against a primary standard is the same process 

as the calibration of a customer piston gauge against a NIST transfer standard. The 

details of this procedure are discussed in Sec. 4. 

The data for the effective area of a transfer standard piston gauge is fit to a function of the 

pressure. This function is then used as the expression for effective area when the NIST 

transfer standard is used for the calibration of customer piston gauges. The general form 

of the equation is: 

 . (20) 2

, 012 (1 ) A A bp bp e f = ++

The subscript T is dropped since the transfer standard will be used as the REF piston 

gauge when calibrating a customer’s pressure instrument. The parameter Ae,f designates 

the fitted function rather than the measured data of eq. (19). The coefficients A0, b1, and 

b2 are determined by least squares fitting of the measured data. The method of least 

squares minimizes the residuals of area at the measured pressures. For most transfer 

standards, b2 is fixed at zero. b1 is also fixed at zero for the low pressure gas gauges 

since the contribution to the effective area by distortion is less than the standard 

uncertainty in the area. 

The three oil transfer standards are calibrated against the CCPG primary standards over 

the full range of operation of the transfer standards. The collection of the transfer 

standards, the primary standards, and their interconnecting calibrations is shown in Fig. 7 

and is referred to as the oil pressure scale. Each circle on the diagram is a piston gauge 

standard, and each line connecting the circles is a calibration. PG87, the 104 MPa gas 

transfer standard, is calibrated against oil transfer standard PG41. These four transfer 

standards are therefore traceable to the CCPG primary standards. 

 

4

 Note that TR is the operating temperature of the REF piston gauge, whereas Tr is the reference temperature 

for both gauges. Tr = 23 oC. 

23
PG42

26 

PG41

140 

PG21

280 

PG20

140 

PG27

28 

PG67

280 

Primary 

Standards 

Figure 7. NIST pressure scale for oil primary and transfer standard piston gauges. 

Circles represent piston gauge standards; the number in a circle is maximum 

pressure in MPa. Lines between circles represent comparisons between piston 

gauges. 

The remaining ten gas transfer standards (five ranges, two gauges each) are traceable to 

the gas piston gauges PG38 and PG39. The gas pressure scale is shown in Fig. 8. It is 

important to note the maximum pressure of three ranges is higher than the maximum 

pressure of the gas primary standards; hence those transfer standards can not be directly 

compared to a primary standard over their full pressure range. Due to the large mismatch 

in effective areas of PG13, PG35, PG23, and PG32 with PG38 and PG39 (nominal area 

ratio of 1:120), these transfer standards are not calibrated against the primary standards. 

As shown in the figure, the 7 MPa gauges (PG13 and PG35) are calibrated against the 

1.4 MPa gauges (PG34 and PG37), and the 17 MPa gauges (PG23 and PG32) are 

calibrated against the 7 MPa gauges. When there is a large mismatch in areas between 

piston gauges, force uncertainties on the smaller gauge become large on a relative basis, 

and the effective area uncertainty resulting from the calibration becomes large. The gas 

pressure scale therefore requires extrapolation beyond the range of direct calibration 

against the primary standard. This is the main reason that uncertainties become higher 

for the higher-range gas transfer standards. Extrapolation is done by considering several 

factors, such as the theoretical values of distortion coefficients, measured differences in 

distortion coefficients between piston gauges of different design or range, and distortion 

determinations from capacitance measurements. 

The relative standard uncertainties in the NIST transfer standard piston gauges as of 

September 2008 are summarized in Table 3. They are plotted for the gas gauges in Fig. 9 

and the oil gauges in Fig. 10. For the gas ranges with twin gauges, the gauge with the 

lowest uncertainty is shown in Fig. 9. 

24
Figure 8. NIST pressure scale for gas primary and transfer standard piston gauges. 

Circles represent piston gauge standards; the number in a circle is maximum 

pressure in MPa. Lines between circles represent comparisons between piston 

gauges. 

Primary 

Standards 

PG87

104 Gas/Oil 

Interface 

PG23

17 

PG32

17 

PG41

140 

PG13

7.0 

PG35

7.0 

PG28

0.3 

PG29

0.3 

PG34

1.4 

PG37

1.4 

PG22

0.15 

PG36

0.15 

PG38

1.0 

PG39

1.0 

25
Table 3. Coefficients for Type B relative standard uncertainty in effective area in gauge 

mode of NIST transfer standard piston gauges. For absolute mode, add uncertainty of 

2x10-6 in quadrature. Relative standard uncertainty calculated from: 

( ) ( ) ( )

1/2 2

2 2 1 2

23 4 ,

( )e

ave B

e

u A c

c cp c p p A p

⎛ ⎞ ⎛ ⎞

= ++ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠. 

c 1 / Pa c 2 c 3 / Pa-1

c 4 / Pa-1 p ave ,B / Pa Low High

PG22 0.106 5.11x10-6 1.12x10-12 0 0 5.2x10-6 11.8x10-6

PG36 0.109 5.11x10-6 1.12x10-12 0 0 5.2x10-6 12.0x10-6

PG28 0.073 4.21x10-6 1.12x10-12 0 0 4.2x10-6 5.6x10-6

PG29 0.147 4.22x10-6 1.12x10-12 0 0 4.3x10-6 8.5x10-6

PG34 0.133 4.20x10-6 1.12x10-12 2.33x10-12 520335 4.3x10-6 5.8x10-6

PG37 0.144 4.21x10-6 1.12x10-12 2.36x10-12 530847 4.3x10-6 6.0x10-6

PG13 0.167 5.82x10-6 0 1.12x10-12 828704 5.8x10-6 9.0x10-6

PG35 1.180 6.43x10-6 0 1.14x10-12 828704 6.5x10-6 9.5x10-6

PG23 1.349 6.87x10-6 0 1.16x10-12 828704 7.0x10-6 20.0x10-6

PG32 1.349 6.89x10-6 0 1.16x10-12 828704 7.1x10-6 20.0x10-6

PG87 0 19.5x10-6 000 19.5x10-6 19.5x10-6

PG42 0 11.0x10-6 000 11.0x10-6 11.0x10-6

PG41 0 18.5x10-6 000 18.5x10-6 18.5x10-6

PG21 0 16.0x10-6 000 16.0x10-6 16.0x10-6

Relative standard uncertainty,

coefficient values in gauge mode Range of u (A e Transfer )/A e 

Standard

26
0

5

10

15

20

10 100 1000 10000 100000

Pressure / kPa

u (A e)/A e x 106

PG38

PG22

PG28

PG34

PG13

PG23

PG87

Figure 9. Operating ranges and relative standard uncertainties of NIST gas piston 

gauges. 

27
0

5

10

15

20

0 50 100 150 200 250 300

Pressure / MPa

u (A e)/A e x 106

PG42

PG41

PG21

Figure 10. Operating ranges and relative standard uncertainties of NIST oil transfer 

standard piston gauges. 

28
The relative standard uncertainty for the transfer standard piston gauges is given by the 

function: 

( ) ( ) ( )

1/2 2

2 2 1 2

23 4 ,

( ) B e

ave B

e

u A c

c cp c p p A p

⎛ ⎞ ⎛ ⎞

= ++ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ . (21) 

The coefficients c1, c2, c3, c4, and pave,B are a function of the specific piston gauge, and are 

given in Table 3. For the gas gauges (except PG87) they are determined in the 

calibration against PG38 and PG39, or for the higher pressure range gauges, in the 

calibration against the lower range transfer standards. For the oil gauges, the uncertainty 

coefficients arise from the calibration against the controlled clearance primary standards. 

4 Calibration techniques and procedures 

The procedures used for calibrating customer pressure instruments depend on which of 

two general classes the instruments fall within. The first class consists of piston gauges 

(or ball gauges), which are similar to the NIST transfer standard piston gauges as 

discussed in Sec. 3.4. These gauges generate pressure based on the incremental masses 

loaded on them. The second class consists of pressure transducers, electronic barometers, 

and pressure gauges, which are referred to as sensing-only instruments. As the name 

implies, this class of instrument senses pressure only, but does not generate pressure. A 

piston gauge calibration, or “crossfloat calibration”, uses a method to “balance” or 

equalize the pressure generated by the NIST transfer standard and the customer piston 

gauge. Once that is done, the masses are tallied on each gauge and eq. (19) is used to 

calculate the effective area. Although eq. (19) was derived for calibrating a NIST 

transfer standard against a NIST primary standard, it is equally valid for calibrating a 

customer piston gauge. For a sensing-only instrument, the pressure is established on the 

NIST piston gauge, and that pressure along with the output of the customer’s instrument 

is recorded. 

4.1 Calibration of piston gauges using the crossfloat method 

In the following, all references to piston gauges refer also to ball gauges. For a crossfloat 

calibration, both the NIST transfer standard and the customer (or test) piston gauge are 

connected to a pressure line along with an appropriate pressure generator. The pressure 

generator can be a hand screw pump, a pressure controller, a gas tank with a pressure 

regulator, or a volume changer. NIST usually uses a common line to both gauges; 

however, each gauge can have its own generator. The effective area of the test gauge at 

the operating temperature TT is determined by balancing the mass and surface tension 

forces loaded onto the piston with the upward force produced by the fluid pressure pT at 

the reference level, or rearranging eq. (1): 

29
,

,

,

1

(, )

a

i T T

i mi T

eT T T

T

m g C

A Tp

p

ρ γ

ρ

⎛ ⎞ ⎜ ⎟ − +

⎝ ⎠ =

∑

 (22) 

pT is determined by the NIST piston gauge and the difference in reference levels between 

it and the customer’s gauge. When these expressions are combined with eq. (22), we get 

the measurement equation for effective area (at reference temperature 23 oC) of the 

customer’s gauge, which is identical to eq. (19): 

( ) ( )( )

( ) ( )( )

( ) ,

, , ,

, ,

, ,

,

,

1 1 23

1 .

1 23 1

a

i T T

mi T pR cR R f a

eT eR

pT cT T T T a

i R R

mi R

m gC T gh P A A

T p p m gC

ρ γ

ρ α α ρ ρ

ρ α α

γ

ρ

⎛ ⎞ ⎜ ⎟ − +

⎝ ⎠ ++ − ⎛ ⎞ − Δ = ⋅ ⋅ ⋅+ − ⎜ ⎟ ⎛ ⎞ ++ −

− + ⎝ ⎠ ⎜ ⎟

⎝ ⎠

∑

∑

 (23) 

For what follows, subscript T now refers to the customers gauge (the TEST gauge is the 

one with the undetermined area), and subscript R refers to the NIST transfer standard (it 

is now the REF piston gauge)5

. 

The two gauges are brought into equilibrium by adjusting the masses on the TEST or 

REF piston gauges, and then eq. (23) is used to calculate the effective area of the TEST 

gauge. The measured area data for the TEST gauge are fit to a function very similar to 

that used in characterizing the NIST transfer standard gauges, that is: 

 . (24) 2

, 012 (1 ) / A A bp bp t e f = ++ − p

 

t is a tare coefficient that may indicate an error in the data, such as an unaccounted for 

mass, or a low pressure characteristic of the performance of the TEST gauge. 

4.1.1 Experimental arrangement for a crossfloat 

The schematic of the fluid circuit for a crossfloat calibration is shown in Figs. 11 and 12. 

The only difference between the fluid circuits is the mechanism by which the equilibrium 

in pressure is established. This general circuit is used for both gas and oil calibrations. 

In the case of a gas calibration, the gas tank will supply the pressure and a volume 

changer will adjust the piston heights. For an oil calibration, the reservoir is the source of 

oil to fill the system, and the screw pump sets the pressure. 

It is essential that the pistons be vertical so that the force due to the masses is totally 

supported by the fluid under the piston, and no component of the force is supported by 

the cylinder wall. Manufacturers usually mount levels on the piston gauge base, and 

5

 The normal reference temperature for a piston gauge calibration is 23 oC. Upon request, NIST can 

provide the effective area at a different reference temperature. 

30
Test 

Reference piston gauge 

piston gauge 

MT

MR

Figure 11. Schematic diagram for fluid circuit for crossfloat calibration of two piston 

gauges, when pressure equilibrium is established using a differential pressure 

cell. SV1, SV2, and SV3 are shut-off valves; CVV1 and CVV2 are constant 

volume valves. Screw pump below SV2 is optional. 

Screw pump or 

volume changer 

CVV1

CVV2

DP Cell 

h 

Reservoir or 

gas tank 

TR

TT Piston 

Cylinder 

SV1 SV2

SV3 

Screw pump 

(optional) 

31
Screw pump or 

volume changer 

CVV1

Reference 

piston gauge 

Test 

piston gauge 

CVV2 

Pressure 

transducer 

h 

MR

Reservoir or 

gas tank 

MT

TR

TT

Screw pump 

(optional) 

SV1 SV2

SV3 

Piston 

Cylinder 

Figure 12. Schematic diagram for fluid circuit for crossfloat calibration of two piston 

gauges, when pressure equilibrium is established using the fall rate method or 

the TAC method. SV1, SV2, and SV3 are shut-off valves; CVV1 and CVV2 

are constant volume valves. Screw pump below SV2, pressure transducer, 

and CVV2 are optional for fall rate method. 

32
provide leveling screws to assure that the piston is vertical. With many designs, a level 

can be placed temporarily on the top of the cylinder. 

The test piston gauge and the NIST reference piston gauge are connected through a short 

length of tubing. Shut-off valves SV1, SV2, and SV3 isolate the pumps and fluid source 

from the piston gauges. Constant volume valves CCV1 and CCV2 isolate the piston 

gauges from each other and are used in establishing pressure equilibrium. NIST uses a 

design [10] that is pneumatically operated and limits the movement of fluid, and hence 

piston position, as the valves are actuated. The valves can be computer controlled 

through use of a solid-state relay to actuate a solenoid supplying the air to the CVV. The 

pump/volume changer providing fluid to the test piston gauge through SV2 is only 

required for one of the methods (described below) for establishing pressure equilibrium. 

Prior to a calibration, the screw pumps are filled from the reservoir, or the volume 

changers are loaded from the gas tank. With ambient pressure on the piston gauges, they 

are loaded with nominal masses to achieve the desired pressure and rotated. Normally, 

the first mass placed on the piston is called the “bell”. The bell is a hollow cylinder with 

one closed end, whose inner diameter is large enough to fit over the column containing 

the piston and cylinder. A narrow lip at the open bottom holds the masses as they are 

stacked on the bell. The closed end is designed to mate with the top of the piston. For 

some piston gauge designs, there is an intermediate mass element between the piston and 

the bell called the “table”. NIST prefers to spin the gauges manually and allow them to 

coast to avoid possible vertical force components from continuous motor drives. Contact 

forces between the masses, bell, table, and piston enable spinning the piston by spinning 

the masses. Both pistons are then raised by means of the pumps/volume changers (if 

there is only one pump CVV1 and CVV2 are both open). SV1, SV2, and CVV2 are 

closed and the pistons are left coasting until both gauges have reached temperature 

equilibrium. Capacitive or inductive proximity sensors monitor the height of the mass 

stack, and therefore the vertical position of the piston in the cylinder. Most pistons are 

operated at a vertical position midway between fully up and fully down. 

Once the piston gauges have reached temperature equilibrium, the pressure of the gauges 

is brought into equilibrium by adjusting the masses on one or both gauges. The 

procedure for accomplishing this depends on which technique is used. In the differential 

pressure cell method (DP Cell, Fig. 11), the output of the DP Cell is monitored. A DP 

Cell is a differential pressure transducer which has an electronic output proportional to 

the pressure difference across a diaphragm. With CVV2 closed and CVV1 open, the 

differential pressure across the cell is zero and output of the cell is electrically adjusted to 

read zero. Then, CVV1 is closed and CVV2 is open to apply the pressure difference 

from the piston gauges to the cell, providing a non-zero output. Masses are adjusted to 

bring the DP Cell to the null position again. This procedure of zeroing the DP Cell, 

applying the pressure difference, and adjusting the masses may need to be repeated 

several times due to the coupling of the fluid elements in the circuit. In addition, if the 

pistons fall or rise from their mid-points, SV1 or SV2 may need to be opened to 

add/withdraw more fluid. If the pumps are known to be leak-tight, SV1 and SV2 can 

remain open during the pressure balancing. 

33
Although the DP Cell is convenient in determining pressure equilibrium, it is not 

essential. The fall rate method can be used with the circuit shown in Fig. 12. Although 

the circuit is shown with CVV2, a pressure transducer, and two screw pumps, the 

technique can be used with a single pump, one CVV connecting the two piston gauges, 

and no pressure transducer. After the piston gauges are raised to about mid-stroke, 

CVV1 is closed (and SV1 if the pump is known to leak), and the only loss of fluid from 

the system will be the leakage through the piston and cylinder gap. This leakage for each 

piston is determined by measuring the rate at which the pistons fall in their cylinders, 

designated as the natural fall rate. This is done by monitoring the proximity sensors, 

which can be sensitive to 0.01 mm. Position vs. time can be recorded on the computer, or 

a stop-watch can be used to time the piston fall through a pre-determined distance. 

CCV1 (and CVV2 if it is in the circuit) is then opened, and the fall rate is measured 

again. NIST has extensive data on the fall rate of its transfer standards, so it is often 

sufficient to measure the fall rate of the NIST piston gauge only. If the piston gauges are 

in pressure equilibrium, there will be no fluid flow in the line connecting the piston 

gauges, and the fall rate will equal the natural fall rate. A mismatch in fall rate indicates 

a pressure difference; the masses are adjusted and the fall rate is measured again. 

Both the DP Cell and fall rate methods are in widespread use; however, they require close 

interaction and judgment of a skilled calibration technician to measure a system 

characteristic (DP Cell imbalance or fall rate as appropriate), switch CVVs, and adjust 

masses. The transducer assisted crossfloat (TAC) method has recently been developed 

[11, 12], which lends itself to automation and less operator judgment. The TAC method 

uses the full circuit shown in Fig. 12. A high precision pressure transducer is placed 

between CVV1 and CVV2. The resolution of the transducer should be 1x10-6 of the 

pressure, and it should have an output which is stable to 3x10-6 to 5x10-6 of the pressure 

over 15 minutes. By alternately opening and closing CVV1 and CVV2, the NIST (REF) 

and customer (TEST) test piston gauges are sequentially connected to the pressure 

transducer. The difference in pressure between the readings on the TEST and REF piston 

gauges is used to adjust the mass on the NIST gauge, which can be calculated by the 

computer with the known effective area of that gauge. It is not necessary to have exact 

pressure equilibrium between the gauges; it has been shown [11] that a residual pressure 

difference of up to 1x10-4 of the system pressure can still yield an effective area of the 

test gauge with negligible uncertainty. The entire measurement process, once the 

nominal pressures have been set, can be executed by the computer with computercontrolled CVVs and sampling of the pressure transducer. Current NIST practice is to 

sample the REF gauge over a period of 30 s, wait 60 s, sample the TEST gauge over 30 s, 

then repeat. Averaging the REF gauge reading before and after the TEST gauge reading 

compensates for linear drift errors in the pressure transducer. 5 sets of TEST readings are 

sandwiched between 6 sets of REF readings, and the 5 sets of difference readings are 

averaged to yield the pressure difference between the two piston gauges. In addition to 

eliminating much of the subjective judgment of the calibration technician, the TAC 

method also eliminates the fluid transients that are inherent in the DP Cell and fall rate 

methods. 

34
4.1.2 Reference levels in a crossfloat 

For piston gauges with straight pistons, the reference level is normally defined as the 

lower end of the piston. For pistons with an irregular shape of the submerged part, an 

adjustment of the reference level is made as illustrated in Fig. 13. In this example the 

piston has a flange at the lower end of diameter D and height h1 serving as a stop. The 

reference level is found by determining the mass of the irregular shape and equating that 

to the mass of an additional length of the piston, at its same nominal area and density. 

Or, using the flange example, with the flange of density, ρ1, and mass, m1 given as: 

2 2

1 11 11

( )

4

D d mVh π

ρ ρ − = = . (25) 

The mass of the piston (density ρ2) lengthened by L is: 

2

2 22 2 4

d mVL π = = ρ ρ . (26) 

Equating m1 and m2 gives the value of L: 

( ) 2 2

1

2 1

2

D d

L h

d

ρ

ρ

− = . (27) 

For an irregularly shaped part, m1 represents a sum of the masses of the parts, and the 

volumes and densities are substituted appropriately. The reference level for a piston of 

this shape is defined as being L below the bottom of the piston. 

d 

D 

ρ1

ρ2

L 

h1

Reference 

Level 

Figure 13. Adjustment of reference level for irregularly shaped piston. 

35
The pressure at any other location in the fluid line that connects to the NIST piston gauge 

is given by the standard hydraulic formula (eq. 18), with the convention that the elevation 

change, h, is positive when the level is higher than the NIST piston gauge. The fluid 

density needs to be calculated as a function of operating pressure if ρgh/p is on the order 

of 1x10-6 or greater. If the fluid is a gas, density is calculated from the perfect gas 

equation of state. 

 

4.1.3 Hydraulic connections 

When connections are made in a pressure system, care should be taken to install the 

properly rated tubing, fittings, valves, and other components. Plumbing materials should 

be chosen to be compatible with the pressure fluid. Stainless steels are widely used in 

liquid systems and high pressure gas systems, whereas copper and plastics are commonly 

found in low pressure gas systems. When threading and coning high pressure tubing, it is 

essential that the threads be carefully made and that cones have the correct angle and 

proper finish. 

All plumbing should be carefully cleaned. To obtain the optimum response time, lines 

are usually kept short and internal diameters should be as large as practical. Damping of 

a system, if necessary, may be achieved though use of needle valves or other types of 

restrictions. This is usually done where short or long-term oscillations have been 

detected. In high pressure systems, a needle valve can also be utilized as a volume 

changer to make fine adjustments of piston height. Non-rotating stem valves offer the 

advantage of long valve seat life. 

4.1.4 Cleaning 

As in most high precision set-ups containing mechanical components with moving parts, 

a clean system is necessary. The oil piston gauge will function properly over a longer 

period of time if the instrument, lines, and fittings are carefully cleaned. Damage can 

result if particles of dirt become lodged between the piston and cylinder. Oil piston 

gauges should be cleaned with appropriate solvents whenever a change is made from one 

oil to another. 

Gas piston gauges will not function properly when dirty, and if forced to operate under 

such conditions, damage to the piston and cylinder is likely. The calibration technician 

should be able to determine from the behavior of the instrument whether it is functioning 

properly. Common indicators are the rate of decay of spin time, fall rate, and sensitivity 

to an adjustment in mass. Proper operating specifications should also be available from 

the operator’s manual provided by the manufacturer. 

To obtain the highest performance from a gas piston gauge, a good cleaning technique is 

essential. One method is to use mild soap and water. The piston and cylinder should 

separately be scrubbed thoroughly, rinsed with room-temperature water, air blasted to 

remove water droplets, and finally polished with lens tissue. The lens tissue must be lintfree. Before assembling the piston and cylinder, dry, clean air or nitrogen is used to 

remove any particles. The important points in a good cleaning technique are: 1) remove 

all foreign material, such as dirt, grease, and fingerprints; 2) leave no residue; 3) polish; 

36
4) remove all remaining lint before assembly. It may be necessary to repeat the cleaning 

process to obtain satisfactory performance. The only way to judge how clean is “clean 

enough” is by the performance of the gauge. 

4.1.5 Piston gauge rotation 

Many commercial piston gauges provide an electric motor to rotate either the piston or 

cylinder. In some cases the motor is mounted below the piston gauge column, and in 

other cases it is located remotely and a drive belt rotates the piston. Heat given off by 

some motors may increase the temperature of the piston and cylinder above the nominal 

room temperature. Unless requested by a customer, NIST rotates the piston and mass 

stack by hand when operating in gauge mode. A rate of 15 to 30 rev/min is adequate for 

most gauges. For absolute mode with a bell jar covering the masses, NIST uses a motor 

to rotate the piston and mass stack. 

4.1.6 Measurements and data evaluation 

NIST practice is to situate the customer’s piston gauge, and all masses to be used in the 

calibration, in the calibration laboratory at least 24 h prior to performing the 

measurements to allow all components to come to temperature equilibrium. The NIST 

piston gauge and the customer’s piston gauge are tested for functionality (rotation decay, 

sensitivity, fall rate) prior to performing the measurement cycle. To calibrate a piston 

gauge using the cross-float technique, NIST practice is to make a minimum of 10 

measurements (observations) at seven pressures ranging from about 10 % of the full 

range to the full range. The seven pressures are approximately evenly spaced. The 

lowest pressure is repeated as well as two other pressures to give an estimate of 

repeatability. Pressures are alternated in the ascending and descending direction. A 

typical sequence used is (as percent of full scale pressure): 10, 10, 40, 70, 100, 100, 85, 

55, 55, 25. 

The effective area of the customer’s gauge is calculated from eq. (23). The Ae,T vs. p

calibration data is fit to eight variations of eq. (24) in the least-squares fitting routine. 

The eight equations differ by which coefficients are fixed at zero and which are fitted in 

the regression. With a maximum of 4 coefficients to be fit, the 10 observations ensure 

that the degrees of freedom exceed the number of coefficients by at least a factor of two. 

The eight fitting equations are: 

A A e f , 0 = (fit 1) (28) 

, 0 / Ae f = A tp − (fit 2) (29) 

, 01 (1 ) Ae f = A bp + (fit 3) (30) 

, 01 (1 ) / A A bp t e f = + − p (fit 4) (31) 

 (fit 5) (32) 2

, 012 (1 ) A A bp bp e f = ++

2

, 012 (1 ) / Ae f = ++ − A bp bp t p (fit 6) (33) 

2

,02 (1 ) Ae f = + A bp (fit 7) (34) 

 (fit 8) . (35) 2

,02 (1 ) / A A bp t e f =+ − p

37
For gas piston gauges below 17 MPa, pressure coefficient b2 is usually insignificant, and 

b1 may be insignificant below 1 MPa. For oil piston gauges, b1 is nearly always 

significant. A computer program developed by the NIST Statistical Engineering Division 

is used to fit the data to the eight equations. The program provides the coefficients, the 

standard deviations of the coefficients, the standard deviations of the residuals of the fit, 

and the standard deviation of the predicted values of the fit. A plot of the residuals as a 

function of pressures shows whether any gross errors have been made in recording and 

entering the data, and is a valuable aid in judging which equation is appropriate. The best 

fit is generally one which has no systematic structure in the plot of the residuals, a 

minimum in the standard deviation of the residuals, a minimum in the standard deviation 

of the predicted values, and no coefficient that is smaller than its corresponding doubled 

standard deviation. All other factors being equal, a fit with fewer coefficients and with t

fixed at zero will be chosen. If the residuals are randomly distributed around the fit, then 

the Type A standard uncertainty is given by the standard deviation of the predicted value 

of the fit. A typical calibration report is given in Appendix B. 

4.2 Calibration of sensing-only pressure instruments 

Sensing-only pressure instruments (e.g., pressure transducers, pressure gauges, electronic 

barometers) are also calibrated against NIST piston gauge standards. The procedure is 

simplified from a cross-float calibration as the sensing-only instrument does not generate 

a pressure. Calibration of sensing-only instruments is most commonly done in absolute 

mode using gas. Readings are taken of the customer’s instrument at the pressure points 

established by the NIST piston gauge. The calibration report lists the instrument output 

as a function of pressure, and usually a fit of the data (predicted pressure as a function of 

instrument output) from a linear regressing analysis. The true pressure from the NIST 

piston gauge is calculated from eq. (3), with the ambient density, ρa, equal to the density 

surrounding the masses (near zero for absolute mode). 

4.2.1 Experimental arrangement and calibration procedure 

The experimental arrangement for a sensing-only calibration is shown in Fig. 14. The 

test pressure instrument is shown on the right, and is connected to the pressure produced 

by the NIST piston gauge through shut-off valve SV2. A “bell jar” is shown covering the 

piston gauge and masses, which is evacuated with a vacuum pump. For a gauge mode 

calibration, the bell jar is not required, however it may be placed loosely over the masses 

to eliminate air currents. The same cleaning procedures and considerations for hydraulic 

connections as mention in the context of the cross-float calibration are followed. Most 

pressure instruments have a defined reference level as specified by the manufacturer. 

The difference in levels between the instrument and the NIST piston gauge are recorded. 

If the pressure instrument is electronic, the power is turned on at least 24 hours prior to 

the calibration. 

To make a measurement, calibrated masses are loaded on the piston gauge corresponding 

to a pressure point. The piston is raised to the upper-most position, the bell jar is placed 

over the piston gauge, and the bell jar is evacuated (for absolute mode). The masses are 

38
Reservoir or 

gas tank SV2

Reference 

piston gauge 

h 

M 

T 

Piston 

Cylinder 

SV4 

SV3

SV1

Bell jar, pV

Test 

pressure 

instrument 

Vacuum 

pump 

Screw pump or 

volume changer 

Figure 14. Schematic diagram for fluid circuit for sensing-only pressure instrument 

calibration. Bell-jar and vacuum pump used for absolute mode calibration. 

SV1, SV2, SV3, and SV4 are shut-off valves. 

39
rotated using a drive motor, which is then turned off and the masses allowed to rotate 

freely. The piston is lowered to its reference level. The residual vacuum is measured 

with a vacuum gauge. After about 5 minutes to allow any thermal or hydraulic transients 

to decay, the output from the pressure instrument, R, is recorded. Once the instrument is 

read, the piston is lowered, the vacuum is released, and masses are adjusted 

corresponding to the next pressure point. The process is then repeated. 

4.2.2 Measurements and data evaluation 

NIST practice is to take 20 measurements of ten ascending and ten descending pressures, 

ranging from about 10 % of full scale to full scale. The pressure at the test gauge is given 

by: 

( ) T RV f a p = pp g +− − ρ ρ h . (36) 

Where all variable are as defined previously, and pV is the vacuum pressure in the bell jar. 

The fluid density can be calculated from the perfect gas law (if it is a gas). In absolute 

mode with a gas, the gravitational correction is approximately 1x10-6 (1 ppm) per cm of 

reference level difference. 

The calibration data (pT, R) is usually fit to a polynomial function of the measurement 

reading, R, using a least-squares regression fitting routine which minimizes the residuals 

of (pT minus R). A linear function would be: 

f 0 1 p = c cR + . (37) 

Where pf is the fitted value of pressure (predicted pressure) for the reading R. Many 

instruments are scaled such that the units of R are in pressure. Another common fit is to 

fix the value of c1 at 1.0; in that case c0 is the average offset to correct the pressure 

reading from the instrument to the true pressure. Higher order polynomials can be used if 

the data reveals non-linear dependencies, and then the best fit is chosen using the same 

criteria given in Sec. 4.1.6. In all cases when a fit of the data is reported, the Type A 

uncertainty of the calibration is determined from the statistics of the fit. 

5 Uncertainty analysis 

The uncertainty in the calibration results for the test gauge effective area or the sensingonly instrument pressure is estimated by combining the component standard uncertainties 

using the root-sum-square method. The Type A component (uA) is evaluated by statistical 

means, and the Type B components (uB) are evaluated by other means. The current 

international practice (as well as at NIST) is to report the combined expanded 

uncertainty, UC = kuC, at the two standard deviation level (k=2). When normal statistical 

distributions apply, the expanded uncertainty defines an interval having a level of 

confidence of approximately 95%. The same general procedures for the uncertainty 

analysis are used for both a crossfloat calibration and a sensing-only calibration, as 

described in [13]. The two types of calibration use a different measurement equation, 

resulting in different Type B component uncertainties. In the following section, values of 

40
uncertainty components that are given are typical for routine calibrations at NIST. 

Reduced uncertainty values may be possible for special calibrations. 

5.1 Uncertainties in a crossfloat calibration 

The standard uncertainty of the effective area of the calibration consists of Type A and 

Type B components added in quadrature: 

( ) ( ) ( ) 2 22

C e T Cal A e T B e T , , , , uA uA uA = + . (38) 

The Type A component, uA(Ae,T), is due to random errors of the NIST standard, the test 

piston gauge, and the calibration process. Because the result of the calibration is an 

equation for the effective area, the Type A uncertainty is estimated from the standard 

deviation of the predicted values associated with the chosen least-squares fit equation 

(eqs. 28 to 35). Typical values of the Type A relative standard uncertainties range from 

1x10-6 to 10x10-6, and can be a function of pressure. For a fit 1, the Type A uncertainty is 

constant over the pressure range, and is equal to the standard deviation of the residuals, σ, 

divided by the square root of the number of observations, n; or 

, ( ) A eT u A

n

σ = . (39) 

For a fit 3, the Type A uncertainty has a minimum (given by eq. 39) at the mean pressure 

of the data, and increases as the pressure departs from the mean value. 

The Type B component is estimated from the uncertainty analysis of the measurement 

equation for the effective area of the customer’s piston gauge, given earlier as eq. (23): 

( ) ( )( )

( ) ( )( )

( ) ,

, , ,

, ,

, ,

,

,

1 1 23

1 .

1 23 1

a

i T T

mi T pR cR R f a

eT eR

pT cT T T T a

i R R

mi R

m gC T gh P A A

T p p m gC

ρ γ

ρ α α ρ ρ

ρ α α

γ

ρ

⎛ ⎞ ⎜ ⎟ − +

⎝ ⎠ ++ − ⎛ ⎞ − Δ = ⋅ ⋅ ⋅+ − ⎜ ⎟ ⎛ ⎞ ++ −

− + ⎝ ⎠ ⎜ ⎟

⎝ ⎠

∑

∑

 (40) 

The uncertainty of the Type B estimates of Ae,T are found using the Law of Propagation 

of Uncertainty [13]: 

( )

2 1

2 2

,

1 1 1

()2 ( )( )( , )

N N N

B eT i i j ij

i i j i i i j

f f f u A u x ux ux rx x

x x x

−

= = = +

⎡ ⎤ ∂ ∂ ∂

= + ⎢ ⎥ ∂ ∂ ∂ ⎣ ⎦ ∑ ∑∑ . (41) 

In eq. (41), the measurement equation is represented symbolically as 

41
42

A fxx x e T, 12 = ( , ,..., N ) . (42) 

with x1, x2,...., xN the N variables (called components) on the right hand side of eq. (40). 

i

f

x

∂

∂ is the partial derivative of Ae,T with respect to variable (component) xi, referred to as 

the sensitivity coefficient. The standard uncertainty of each component is u(xi). The 

square of the standard uncertainty, u

2

(xi), is the variance. r(xi,xj) is the correlation 

coefficient between variables xi and xj. For a crossfloat calibration, the correlation 

coefficients are all zero except between the masses and between the densities of the same 

mass set. Determining the Type B uncertainty thus involves estimating the sensitivity 

coefficient and standard uncertainty of each component. The following notation is used 

for the uncertainty in Ae,T due to the uncertainty in component xi: 

, () () e T

xi i

i i

f A

u ux u

x x

i x

∂ ∂ = = ∂ ∂ . (43) 

The combined standard Type B uncertainty is found by summing the individual variances 

and taking the square root: 

( ) ,

2 2 2 2 2 2 2 22 2

,

2 2 2 2 2 22

e R R T a MR MT R T

R T RT f

B eT A M M C C g

TT h P

uA u u u u u u u u u u

u u u u u uu

ρρ ρ γ

α α + Δ ρ

= + + + + + ++ + +

+ + ++ + + + +

2

 . (44) 

The correlation terms are included in the uxi terms as appropriate. The thermal expansion 

terms for the piston and cylinder for each gauge have been combined into a single 

variable (α α += α cp ). 

Tables 4 and 5 list values for the component uncertainties for typical customer 

calibrations using NIST transfer standard piston gauges. Each table contains values at a 

single pressure: Table 4 is for oil at 5 MPa, and Table 5 is for gas at 200 kPa. 

Evaluations of the individual components will follow, including the sensitivity coefficient 

and the standard uncertainty. The tables list the definitions of the sensitivity coefficients 

for the components, values at the conditions of the calibration, values of the standard 

uncertainty of the components, and the product (eq. 43) which shows the effect of each 

component on the overall uncertainty in Ae,T. Components which have a relative standard 

uncertainty on Ae,T of 1.0x10-6 or greater are shown in bold. As can be seen from the 

Tables, many of the component uncertainties are negligible (< 1.0x10-7) when compared 

to the uncertainty of the effective area of the NIST piston gauges, Ae R, u . Uncertainties in 

mass, temperature, thermal expansion, fluid density, height difference, and pressure 

equilibrium should be considered in most cases. What follows are the definitions of the 

sensitivity coefficients and the standard uncertainties for each component, and comments 

about their significance or typical value for each term in eq. (44).
Table 4. Type B component standard uncertainties for gauge mode oil piston gauge calibration at 5 MPa, REF p

TEST piston gauge of nominal area 8.4x10-5 m2, and Spinesstic oil as pressure fluid. All uncertainty terms show

Relative combined standard uncertainty is 1.20x10-5 (12.0 ppm). 

Name Symbol Value Units Definition Abs Value Units Value Unit

REF Area

A e,R 8.402E-05

m2 1/A e,R 11901.89

m

-2 9.24E-10

m2 

REF Mass

M R 42.86 kg -1/M R 2.33E-02 kg-1 1.24E-04 kg 

TEST Mass

M T 42.85 kg 1/M T 2.33E-02 kg-1 1.24E-04 kg 

Ambient density

ρ a 1.18 kg/m3 Δρ

M /(

ρ MT

ρ MR ) -

gh /p T 9.80E-08

m3/kg 0.030 kg/m

REF mass dens.

ρ MR 7800 kg/m3 (ρ a,cal -ρ a )/ρ MR

2 4.93E-10

m3/kg 45.0 kg/m

TEST mass dens.

ρ MT 7800 kg/m3 (ρ a -ρ a,cal )/ρ MT

2 4.93E-10

m3/kg 45.0 kg/m

Surface tension

γ 3.06E-02 N/m

CR /(M R g )(D R /D T -1) 9.33E-09 m/N 1.77E-03 N/m 

REF piston circ.

C R 3.25E-02 m -γ /(M R g ) 7.28E-05 1/m 3.25E-05 m 

TEST piston circ.

C T 3.25E-02 m

γ /(M T g ) 7.29E-05 1/m 3.25E-05 m 

Gravity

g 9.801011 m/s2 (ρ f - ρ a )h /p T 8.57E-06 s

2/m 2.00E-06 m/s

REF therm. exp.

α p,R

+

α c,R 9.10E-06

C-1

T R - 23.00 0.50 C 5.25E-07

C-1 

TEST therm. exp.

α p,T

+

α c,T 9.10E-06

C-1 -(

T T - 23.00) 0.50 C 5.25E-07

C-1 

REF temperature

T R 23.50 C

α p,R

+

α c,R 9.10E-06

C-1 0.058 C 

TEST temperature

T T 23.50 C −(

α p,T

+

α c,T ) 9.10E-06

C-1 0.058 C 

Fluid density

ρ f 857.8 kg/m3 gh /p T 9.80E-08

m3/kg 9.91 kg/m

Height difference

h 0.05 m (ρ f - ρ a )g /p T 1.68E-03 1/m 0.001 m 

Press. equilibrium

ΔP 0.00 Pa -1/p T 2.00E-07 1/Pa 5.83 Pa 

Relative combinded standard unc. 

Uncertainty term Sensitivity coefficient divided by 

A e,T Standard uncertain

43
44

Table 5. Type B component standard uncertainties for gauge mode gas piston gauge calibration at 200 kPa, REF PG34, TEST piston gauge of nominal area 8.4x10-5 m2, and nitrogen gas as pressure fluid. All uncertainty terms 

completeness. Relative combined standard uncertainty is 5.27x10-6 (5.27 ppm). 

Name Symbol Value Units Definition Abs value Units Value Unit

REF Area A e,R 8.397E-05 m2 1/A e,R 11909 m-2 3.63E-10 m2 

REF Mass M R 1.714 kg -1/M R 0.584 kg-1 3.43E-06 kg 

TEST Mass M T 1.714 kg 1/M T 0.583 kg-1 3.43E-06 kg 

Ambient density ρ a 1.18 kg/m3 Δρ M /(ρ MT ρ MR ) - gh /p T 9.80E-07 m3/kg 0.03 kg/m

REF mass dens. ρ MR 7800 kg/m3 (ρ a,cal -ρ a )/ρ MR 2 4.93E-10 m3/kg 45.0 kg/m

TEST mass dens. ρ MT 7800 kg/m3 (ρ a -ρ a,cal )/ρ MT 2 4.93E-10 m3/kg 45.0 kg/m

Surface tension γ 0.000 N/m CR /(M R g )(D R /D T -1) 3.11E-07 m/N 0.000 N/m 

REF piston circ. C R 3.25E-02 m -γ /(M R g ) 0.000 1/m 3.25E-05 m 

TEST piston circ. C T 3.25E-02 m γ /(M T g ) 0.000 1/m 3.25E-05 m 

Gravity g 9.801011 m/s2 (ρ f - ρ a )h /p T 2.33E-07 s2/m 2.00E-06 m/s

REF therm. exp. α p,R + α c,R 9.10E-06 C-1 T R - 23.00 0.50 C 5.25E-07 C-1 

TEST therm. exp. α p,T + α c,T 9.10E-06 C-1 -(T T - 23.00) 0.50 C 5.25E-07 C-1 

REF temperature T R 22.50 C α p,R + α c,R 9.10E-06 C-1 0.058 C 

TEST temperature T T 22.50 C −(α p,T + α c,T ) 9.10E-06 C-1 0.058 C 

Fluid density ρ f 3.51 kg/m3 gh /P T 9.80E-07 m3/kg 3.51E-03 kg/m

Height difference h 0.02 m (ρ f - ρ a )g /p T 1.14E-04 1/m 0.001 m 

Press. equilibrium ΔP 0.00 Pa - 1/p T 5.00E-06 1/Pa 0.12 Pa 

Relative combinded standard unc. 

Uncertainty term Sensitivity coefficient divided by A e,T Standard uncertain
5.1.1 Uncertainty due to reference area, Ae R, u

The sensitivity coefficient is given by: 

, ,

, ,

eT eT

eR eR

A A

A A

∂ = ∂ . (45) 

The standard uncertainty, u(Ae,R), is a function of the specific NIST piston gauge used. 

Equations to determine those values are given in Table 3. This term is always one of the 

largest contributors to the combined uncertainty. 

5.1.2 Uncertainty due to mass on REF piston gauge, MR u

The sensitivity coefficient is given by: 

eT eT , ,

R R

A A

M M

∂ = − ∂ . (46) 

where = ∑ . R mM ,Ri

The standard uncertainty, u(MR) is more precisely the uncertainty of all the individual 

masses placed on the NIST piston gauge. The result depends on the degree of correlation 

among the masses that are used. Unless there is strong evidence to the contrary, NIST 

assumes that all the REF masses are correlated together, and that they are uncorrelated 

with the TEST masses. This is the most conservative approach. Hence, correlated 

uncertainties add algebraically, and there is no influence of the TEST masses: 

, () ( R

i

uM um = ) ∑ i R . (47) 

This is the assumption for a standard cross-float calibration. This means that if all 

individual masses have a relative uncertainty of 2x10-6 (2 ppm), the relative uncertainty 

on the effective area will also be 2x10-6. In certain instances where the NIST piston 

gauge and the customer piston gauge are of the same design and same nominal area, the 

masses can be interchanged between gauges. Two observations can be made at each 

nominal pressure, with all masses switched between the observations except for the 

pistons. In that case, the Type B uncertainties in the loaded masses from the NIST and 

customer piston gauges cancel out and are not added to the combined Type B uncertainty. 

5.1.3 Uncertainty due to mass on TEST piston gauge, MT u

The sensitivity coefficient is given by: 

eT eT ,

T T

A A ,

M M

∂ = ∂ . (48) 

45
where MT = ∑mi T, . As in the REF masses, the standard uncertainty, u(MT) is more 

precisely the uncertainty of all the individual masses placed on the NIST piston gauge. 

The same considerations for assumed correlation apply for these masses. 

, () ( T i

i

uM um = ) ∑ T . (49) 

Uncertainties in the REF and TEST masses are usually significant in the overall 

uncertainty in effective area. 

5.1.4 Uncertainty due to ambient density, a uρ

The sensitivity coefficient is given by: 

 

, ,

, ,,

,

i R i T

e T i i mi R mi T

e T

a RT

m m

A gh A

M M pT

ρ ρ

ρ

⎡⎛ ⎞ ⎤ ⎢⎜ ⎟ ⎥ ∂ ⎢ ⎥ = − − ∂ ⎢ ⎥ ⎢ ⎥ ⎣⎝ ⎠ ⎦

∑ ∑ . (50) 

If all the REF masses have the same density ρM,R and all the TEST masses have the same 

density ρM,T then this simplifies to: 

, ,,

,

, ,

e T M T M R

e T

a M T M R

A gh A

T p

ρ ρ

ρ ρρ

∂ − ⎡ ⎤ = − ⎢ ⎥ ∂ ⋅ ⎢⎣ ⎥⎦

 . (51) 

This assumption is made if the masses are taken from the same mass set. The ambient 

density is the same surrounding both the REF and TEST masses. The first term in the 

sensitivity coefficient is the effect of the ambient buoyancy on the masses, and the second 

term is the effect of the ambient density on the height difference between the reference 

levels of the piston gauges. 

Standard uncertainty, u(ρa). If the ambient gas is air (gauge mode calibration), the 

uncertainty in density due to imprecisely knowing the air temperature and pressure is 

larger than the uncertainty in the calculation function for the density (Appendix A). Even 

if u(ρa) = 0.03 kg/m3

 and the difference in mass densities between REF and TEST is 

large, the relative uncertainty in Ae,T due to air density is less than 0.2x10-6 and is not 

significant. For absolute mode operation, the density of the air surrounding the masses 

and its uncertainty is extremely small, and can be taken as zero. 

5.1.5 Uncertainty due to REF gauge mass density, MR uρ

The sensitivity coefficient is gauge mode is: 

46
( , ) ,

, 2

, ,

e T a cal a

e T

M R M R

A

A ρ ρ

ρ ρ

∂ − = ∂ . (52) 

Here, ρa,cal is the density of the air at the time the masses were calibrated. To be 

conservative, the air density difference is taken as 0.03 kg/m3

. We again assume that the 

individual densities of each mass on the REF gauge can be approximated by a common 

density. This expression assumes that the same numerical value of the mass density is 

used during the piston gauge calibration as when the masses were calibrated. If that 

value is changed, then the value of the mass must be changed as well. Because the mass 

calibration is effectively providing a force on the balance when the mass is weighed (and 

the mass value comes from inserting the mass density into the force equation), the density 

and mass must be considered together in the uncertainty of the density. (However, the 

mass uncertainty considered above, due to the mass calibration process, is not dependent 

on the density). 

The sensitivity coefficient in absolute mode is: 

, ,

, 2

, ,

e T a cal

e T

M R M

A

A

R

ρ

ρ ρ

∂ = ∂ . (53) 

Note that the sensitivity coefficient is larger in absolute mode calibrations. 

Standard uncertainty, u(ρM,R): The possible error in the mass density is assumed to have a 

rectangular distribution with ±0.01ρM,R maximum deviation from a nominal value, or a 

standard uncertainty of 0.0058ρM,R. The term is negligible for gauge mode calibrations 

and less than 1x10-6 in absolute mode calibrations. 

5.1.6 Uncertainty due to TEST gauge mass density, MT uρ

The sensitivity coefficients are similar to those for the reference mass density. In gauge 

mode it is: 

( , ) ,

, 2

, ,

e T a a cal

e T

M T M T

A

A ρ ρ

ρ ρ

∂ − = ∂ . (54) 

and in absolute mode it is: 

, ,

, 2

, ,

e T a cal

e T

M T M

A

A

T

ρ

ρ ρ

∂ = − ∂ . (55) 

The same arguments apply for the TEST mass densities as for the REF mass densities. 

47
Standard uncertainty, u(ρM,T): We use similar arguments as for the REF masses, so the 

standard uncertainty is therefore 0.0058ρM,T. The term is negligible for gauge mode and 

less than 1x10-6 in absolute mode calibrations. 

5.1.7 Uncertainty due to fluid surface tension, uγ

This term is zero for gas gauges, and negligible for oil gauges of the same nominal area. 

It needs to be considered only for pressures below 5 MPa and when the nominal areas 

between the NIST and customer piston gauge differ by a factor of 5. In those cases, the 

sensitivity coefficient is (with DR and DT the diameters of the REF and TEST piston 

gauges): 

,

, 1 e T R R

e T

R T

A C D A

γ Mg D

∂ ⎛ ⎞ = ⋅ − ⎜ ⎟ ∂ ⎝ ⎠

 . (56) 

Standard uncertainty, u(γ). A reasonable assumption is that the surface tension 

uncertainty is a rectangular distribution of possible relative errors of 10 % of γ. Hence 

u(γ)=0.058γ.

5.1.8 Uncertainty due to REF piston circumference, CR u

This term is always negligible. For completeness, the sensitivity coefficient is: 

,

,

e T

e T

R R

A

A

C M g

∂ γ =− ⋅ ∂ . (57) 

5.1.9 Uncertainty due to TEST piston circumference, CT u

This term is always negligible. For completeness, the sensitivity coefficient is: 

,

,

e T

e T

T T

A

A

C M g

∂ γ = ⋅ ∂ . (58) 

5.1.10 Uncertainty due to acceleration of gravity, g u

The sensitivity coefficient is: 

 

,

,

( ) e T f a

e T

T

A h

A

g p

∂ ρ − ρ = ∂ . (59) 

In absolute mode, ρa is taken as zero for this term. Because gravity is identical at the 

REF and TEST masses (even if there is uncertainty on what the value is), the uncertainty 

on force at the REF piston gauge due to the uncertainty in gravity will cancel with 

uncertainty on force at the TEST piston gauge due to the uncertainty in gravity. The only 

contribution to the sensitivity coefficient is the pressure difference due to the difference 

in reference levels. 

48
Standard uncertainty, u(g): The standard uncertainty in g, based on measurements in the 

NIST calibration laboratory, is 2x0-6 m/s2

. g u is negligible for all crossfloats. 

5.1.11 Uncertainty due to thermal expansion of the REF piston and cylinder, R uα

The sensitivity coefficient is: 

,

, ( 23 e T

eT R

R

A

A T

α

)

∂ = − ∂ . (60) 

This sensitivity coefficient becomes larger as the temperature of the reference piston 

gauge departs from 23 oC. 

Standard uncertainty, )( R u α : NIST uses the manufacturer’s stated thermal expansion for 

the transfer standard piston gauges. We estimate that the uncertainty of the thermal 

expansion is represented by a rectangular distribution of possible relative errors of 0.1 of 

the stated value. Hence the standard uncertainty is ( ) 0.058 R u α = α R . For a tungsten 

carbide piston and cylinder, 6 9 10 R α x −  m/(mK), 6 ( ) 0.5 10 R u x α − = m/(mK), and the 

relative uncertainty R uα is 0.5x10-6 if TR is within 1 oC of 23 oC. If the temperature is kept 

close to the reference temperature, this term is small compared to the uncertainty in the 

NIST standard, but it should be included. 

5.1.12 Uncertainty due to thermal expansion of the TEST piston and cylinder, T uα

The sensitivity coefficient is: 

,

, ( 23 e T

eT T

T

A

A T

α

)

∂ =− − ∂ . (61) 

This sensitivity coefficient becomes larger as the temperature of the customer’s piston 

gauge departs from the reference temperature (taken as 23 oC unless otherwise 

requested). 

Standard uncertainty, ( ) T u α : NIST uses the manufacturer’s stated thermal expansion for 

the customer’s piston gauges. We estimate that the uncertainty of the thermal expansion 

is represented by a rectangular distribution of possible relative errors of 0.1 of the stated 

value. Hence the standard uncertainty is ( ) 0.058 T u α = α T . As for the REF piston gauge, 

for a tungsten carbide piston and cylinder, T uα is 0.5x10-6 if TT is within 1 oC of 23 oC. If 

the temperature is kept close to the reference temperature, this term is small compared to 

the uncertainty in the NIST standard, but it should be included. 

5.1.13 Uncertainty due to temperature of REF piston and cylinder, TR u

The sensitivity coefficient is: 

49
,

,

e T

eT R

R

A

A

T

α

∂ = ∂ . (62) 

The standard uncertainty, u(TR) is determined assuming a rectangular distribution of 

possible errors of 0.1 oC, so u(TR) = 0.058 oC. A good-quality calibrated thermometer 

will have an uncertainty better than this, however it is difficult to place the thermometer 

close to the piston or cylinder, and this uncertainty allows for spatial or time-dependent 

temperature differences. With tungsten-carbide pistons, the relative uncertainty TR u is 

0.5x10-6. 

5.1.14 Uncertainty due to temperature of TEST piston and cylinder, TT u

The sensitivity coefficient is: 

,

,

e T

eT T

T

A

A

T

α

∂ = − ∂ . (63) 

The standard uncertainty, u(TT) is determined assuming a rectangular distribution of 

possible errors of 0.1 oC, so u(TT) = 0.058 oC. A good-quality calibrated thermometer 

will have an uncertainty better than this, however it is difficult to place the thermometer 

close to the piston or cylinder, and this uncertainty allows for spatial or time-dependent 

temperature differences. With tungsten-carbide pistons, the relative uncertainty is 

0.5x10-6. 

TT u

5.1.15 Uncertainty due to density of pressure fluid, fuρ

The sensitivity coefficient is: 

,

,

e T

e T

f T

A gh A

ρ p

∂ = ∂ . (64) 

Standard uncertainty, u(ρf). This term depends on the fluid used. For oils, NIST assumes 

a rectangular distributions of possible errors of 0.02ρf , hence the standard uncertainty is 

0.012ρf . At atmospheric pressure, common hydraulic oils have densities from 860 kg/m3

to 910 kg/m3

. The uncertainty term can become significant for pressures below about 

10 MPa, especially if h is 0.1 m or greater. 

 

For gases below 1 MPa, the uncertainty in density is due to possible differences in 

temperature along the tubing connecting the gauges. Taking a rectangular distribution of 

possible errors of 0.5 oC, u(ρf) = 0.001ρf. The uncertainty term is negligible for h less 

than 1 m. Above 1 MPa, the density begins to depart from the perfect gas model used in 

our data reduction program. As long as h is less than 0.1 m, the uncertainty term is less 

than 0.2x10-6. 

5.1.16 Uncertainty due to reference height difference, uh

50
The sensitivity coefficient is: 

,

,

( ) e T f a

e T

T

A g A

h p

∂ ρ − ρ = ∂ . (65) 

Standard uncertainty, u(h): The height difference between the gauges is the sum of the 

difference in the reference levels, hT0-hR0, and the difference between the bottom of each 

piston position and its reference level (hT-hT0, hR-hR0). Or, 

 . (66) 00 0 0 ( )( )( TR T R TT RR hh h h h h h h h =−= − + − − − )

The difference in reference levels is measured prior to the calibration. For its uncertainty 

we assume rectangular distribution with a maximum possible error of 1x10-3 m. The 

other two terms are nominally equal to zero, however the uncertainty is not. The pistons 

fall during normal operation, and are intermittently raised above the nominal position to 

allow operational time while establishing equal pressures. For each of the two terms, we 

take a rectangular distribution of errors with a maximum of 1x10-3 m. The combined 

standard uncertainty is u(h) = 1x10-3 m. 

The term uh is always insignificant for gas gauges. For oil gauges, the relative 

uncertainty is 1x10-6 at 10 MPa, and increases as the pressure decreases. 

5.1.17 Uncertainty due to pressure difference, uΔP 

The sensitivity coefficient is: 

,

,

e T 1

e T

T

A

A

P P

∂ = − ∂Δ . (67) 

Standard uncertainty, u(ΔP), depends on the method used for determining pressure 

equilibrium. For the fall rate or DP cell methods, the operator must make a judgment of 

the standard uncertainty. One method is to estimate the amount of mass, ΔM, that can be 

added to one of the piston gauges to unequivocally disturb the balance from equilibrium. 

This can be converted to a pressure using the approximation: 

() ( )

e

g uP uM

A

Δ= Δ . (68) 

Here, Ae refers to whichever piston gauge was “trimmed” with ΔM. Typical values for 

gas gauges are u(ΔM) = 1 to 5 mg; typical values for hydraulic gauges are 25 mg to 

100 mg. These uncertainties become more significant at the low-pressure end of each 

calibration. If equilibrium is determined by the TAC method, then u(ΔP) is given by the 

scatter of the measured pressure differences. 

5.1.18 Summary of uncertainties in a crossfloat calibration 

51
The Type B combined relative standard uncertainties for typical crossfloat calibrations of 

a customer’s piston gauge are shown in Tables 4 and 5. For the oil calibration at 5 MPa, 

the Type B relative standard uncertainty is 12x10-6, with the dominant component being 

the standard uncertainty in the effective area of the NIST piston gauge. If the Type A 

relative standard uncertainty were 5x10-6, then the combined relative standard uncertainty 

would be 13x10-6, with an expanded relative uncertainty (k=2) of 26x10-6 (26 ppm). For 

the gas piston gauge calibration at 200 kPa, the Type B relative standard uncertainty is 

5.3x10-6, again dominated by the standard uncertainty in effective area of the NIST piston 

gauge (4.3x10-6). An uncertainty analysis including all these components is conducted 

over the pressure range of the calibration, and is summarized in the calibration report to 

the customer (Appendix B). 

5.2 Uncertainty in pressure sensing-only calibration 

For a pressure-sensing only calibration, the measurand is pressure produced by the NIST 

transfer standard at the level of the customer’s instrument. The method for the 

uncertainty analysis is the same as for the crossfloat calibration, i.e., the uncertainty of 

the pressure of the calibration consists of Type A and Type B components added in 

quadrature: 

( ) ( ) ( ) 222

c T AT BT up up up = + . (69) 

The Type A component is due to random errors of the NIST standard, the customer’s 

instrument, and the calibration process. The Type A uncertainty is estimated from the 

statistics of the fit provided to the customer. The Type B component is estimated from 

the uncertainty analysis of the measurement equation for the pressure of the customer’s 

instrument, which is found by substituting eq. (3) for the pressure from the NIST standard 

piston gauge into eq. (36) for the pressure at the customer’s instrument: 

( ) ( )( ) ( )

1

1 23

a

i

i mi

T V

e pc

m g C

f a p p g

A T

ρ γ

ρ

ρ ρ

α α

⎛ ⎞ ⎜ ⎟ − +

⎝ ⎠ = + − ++ −

∑ − h . (70) 

The notation of “REF” for the NIST piston gauge standard has been dropped since it is 

the only piston gauge in the measurement equation. The Type B uncertainty follows 

from estimating the sensitivity coefficient of each variable, ∂pT/∂xi, and the standard 

uncertainty of the variable, u(xi). The following notation is used for the uncertainty in pT

due to the uncertainty in variable xi that is uncorrelated with other variables: 

( ) T

xi i

i

p u u

x

x

∂ = ∂ . (71) 

52
53

Hence the Type B combined standard uncertainty can be written as6

: 

( ) 2 2 2 2 2 222222 2

BT A M e aM C g T h p f up u u u u u u u u u u u u = + + + ++++++ ++ ρρ γ α ρ

2

V . (72) 

Table 6 summarizes the Type B component uncertainties for a pressure instrument 

calibration using PG22 as the NIST piston gauge in absolute mode. The calibration gas is 

nitrogen, and the pressure is 100 kPa. Shown are the definitions of each sensitivity 

coefficient, their magnitude, the magnitude of the standard uncertainties of the 

components, and the relative contribution of each component to the uncertainty in pT. In 

this example, the relative Type B standard uncertainty in pressure is about 6.1x10-6

(6.1 ppm), and it is dominated by the uncertainty in effective area of PG22. Components 

of decreasing importance to the overall uncertainty are mass, density of the masses, 

piston gauge temperature, piston gauge thermal expansion, and gravity. Details for each 

component follow. 

5.2.1 Uncertainty due to reference area, Ae u

The sensitivity coefficient is given by: 

 

T T

e e

p p

A A

∂ = − ∂ . (73) 

The standard uncertainty, u(Ae), is a function of the specific NIST piston gauge used. 

Equations to determine those values are given in Table 3. This term is always one of the 

largest contributors to the combined uncertainty. 

5.2.2 Uncertainty due to mass, Mu

The sensitivity coefficient is given by: 

T T p p

M M

∂ = ∂ , (74) 

where M = ∑mi . 

The standard uncertainty, u(M) is more precisely the uncertainty of all the individual 

masses placed on the NIST piston gauge. The result depends on the degree of correlation 

among the masses that are used. Unless there is strong evidence to the contrary, NIST 

assumes that all the masses are correlated, and that the correlated uncertainties add 

algebraically. 

() ()i uM um = ∑ . (75) 

 

6

 If the customer’s pressure instrument is read with an electronic instrument, such as a voltmeter, the Type 

B uncertainty of that instrument should be added in quadrature with the other terms of eq. (72). 
Table 6. Type B component standard uncertainties for pressure-sensing instrument calibration, absolute mode w

NIST piston gauge is PG22 at 100 kPa. All uncertainty terms shown for completeness. Relative combined stand

6.08x10-6 (6.08 ppm).

Name Symbol Value Units Definition Abs value Units Value Un

PG Area

A

E 3.357E-04

m2 -1/A

E 2979

m

-2 1.88E-09

m

PG Mass

M 3.425 kg 1/M 0.292 kg-1 6.85E-06 k

Ambient density

ρ a 0.000 kg/m3 -1/ρ

M + gh/p T 1.09E-04

m3/kg 1.14E-07 kg

Mass density

ρ

M 7800 kg/m3 (ρ a -ρ a,cal ) /ρ

M

2 1.94E-08

m3/kg 45.03 kg

Surface tension

γ 0.000 N/m C/(Mg ) 1.93E-03 m/N 0.000 N

PG circum.

C 6.50E-02 m

γ /(M g ) 0.000 1/m 6.50E-05 m 

Gravity

g 9.801011 m/s2 1/g 0.102 s

2/m 2.00E-06 m

PG therm. exp.

α p +

α c 1.46E-05

C-1 -(T - 23.00) 0.50 C 8.40E-07

C

PG temperature

T 22.50 C − (α p +

α c ) 1.46E-05

C-1 0.0577 

Fluid density

ρ f 1.14 kg/m3 -gh /p T 1.96E-05

m3/kg 0.0011 kg

Height difference

h 0.20 m -(

ρ f - ρ a )g /p T 1.12E-04 1/m 8.20E-04 m 

Bell jar pressure

p V 2.00 Pa 1/p T 1.00E-05 1/Pa 0.010 P

Relative combinded standard un

Uncertainty term Sensitivity coefficient divided by 

p T Standard uncerta

54
5.2.3 Uncertainty due to ambient density, a uρ

The sensitivity coefficient is: 

T T

a m

p p gh

ρ ρ

∂ = − +

∂ . (76) 

The first term in the sensitivity coefficient is the effect of the ambient density on the 

masses, and the second term is the effect of the ambient density on the height difference 

between the reference level of the customer’s instrument and the NIST piston gauge. For 

absolute mode operation, the ambient density surrounding the masses is extremely small, 

and its standard uncertainty, u(ρa), can be taken as zero. In that case the uncertainty 

contribution due to ambient density is also zero. Table 6 displays a value for u(ρa) that is 

determined by the uncertainty in residual pressure in the space surrounding the masses. 

For gauge mode operation, the ambient gas is air and the uncertainty in density due to 

imprecisely knowing the air temperature and pressure is typically larger than the 

uncertainty in the calculation function (Appendix A). If u(ρa) = 0.03 kg/m3

, the relative 

uncertainty in pT due to the uncertainty in the air density is about 4x10-6. 

The uncertainty in ambient density has a larger effect on the combined uncertainty in a 

sensing-only calibration than in a piston gauge calibration for gauge mode. In the 

crossfloat, both sets of masses are exposed to the same buoyancy correction, so the 

uncertainty due to air density cancels out. Here, only one set of masses is exposed to air, 

and more care should be taken in identifying the uncertainty contribution of the ambient 

density. 

5.2.4 Uncertainty due to mass density, M uρ

The sensitivity coefficient in gauge mode is given by: 

( , )

2

a a cal T

T

M M

p p

ρ ρ

ρ ρ

∂ − = ∂ . (77) 

Here, ρa,cal is the density of the air at the time the masses were calibrated. To be 

conservative, the air density difference is taken as 0.03 kg/m3

. It is assumed that the 

individual densities of each mass on the piston gauge can be approximated by a common 

density. This expression assumes that the same numerical value of the mass density is 

used during the pressure calibration as when the masses were calibrated. If that value is 

changed, then the value of the mass must be changed as well. 

In absolute mode the ambient density is negligible, and the sensitivity coefficient is given 

by: 

,

2

T a cal

T

M M

p p

ρ

ρ ρ

∂ = − ∂ . (78) 

55
The same mass density uncertainty has a larger effect on absolute mode calibrations than 

it does on gauge mode calibrations. 

Standard uncertainty, u(ρM): The possible error in the mass density is assumed to have a 

rectangular distribution with ±0.01ρM maximum deviation from a nominal value, or a 

standard uncertainty of 0.0058ρM. The term is negligible for gauge mode calibrations and 

less than 1x10-6 in absolute mode calibrations. 

5.2.5 Uncertainty due to fluid surface tension, uγ

The sensitivity coefficient is given by: 

T

T

p C

p

γ Mg

∂ = ∂ . (79) 

If the calibration is done in gas, the surface tension and standard uncertainty are zero. If 

the calibration is in oil, a reasonable assumption is that the surface tension uncertainty is 

a rectangular distribution of possible relative errors of 10 % of γ. Hence u(γ)=0.058γ. 

The term is negligible above 5 MPa, and even at 1 MPa the relative contribution is less 

than 1.0x10-6. 

5.2.6 Uncertainty due to piston circumference, C u

This term is always negligible, whether gas or oil is used for the NIST piston gauge. For 

completeness, the sensitivity coefficient is: 

T

T

p p C Mg∂ γ = ∂ . (80) 

5.2.7 Uncertainty due to acceleration of gravity, g u

The sensitivity coefficient is: 

T T p p

g g

∂ = ∂ . (81) 

The term due to elevation change has been neglected, as it is always much smaller than 

the term due to mass on the piston gauge. The standard uncertainty, u(g), at NIST is 

2.0x10-6 m/s2

, making the relative contribution of g u equal to 0.2x10-6. Gravity has a 

larger effect on the uncertainty of the measurand in a sensing-only calibration than in a 

piston gauge crossfloat calibration, but it is still small compared to the uncertainty in the 

NIST piston gauge effective area. 

5.2.8 Uncertainty due to thermal expansion of piston and cylinder, uα

The sensitivity coefficient is: 

( 23 T

T

p p T

α

) ∂ =− − ∂ . (82) 

56
This sensitivity coefficient becomes larger as the temperature of the NIST piston gauge 

departs from 23 oC. 

Standard uncertainty, u( ) α : NIST uses the manufacturer’s stated thermal expansion for 

the transfer standard piston gauges. We estimate that the uncertainty of the thermal 

expansion is represented by a rectangular distribution of possible relative errors of 0.1 of 

the stated value. Hence the standard uncertainty is u( ) 0.058 α = α . For the piston gauge 

used near 100 kPa, the piston is stainless steel and the cylinder is tungsten carbide, so 

5 α 1.5 10−  × m/(mK), 6

u( ) 0.5 10 α − = × m/(mK). The relative uncertainty uα is 0.5x10-6

if T is within 1 oC of 23 oC. If the temperature is kept close to the reference temperature, 

this term is small compared to the uncertainty in the NIST standard, but it should be 

included. 

5.2.9 Uncertainty due to temperature of piston and cylinder, T u

The sensitivity coefficient is: 

T

T

p p T

α

∂ = − ∂ . (83) 

The standard uncertainty, u(T) is determined assuming a rectangular distribution of 

possible errors of 0.1 oC, so u(T) = 0.058 oC. A good-quality calibrated thermometer will 

have an uncertainty better than this, however it is difficult to place the thermometer close 

to the piston or cylinder, and this uncertainty allows for spatial or time-dependent 

temperature differences. With stainless steel pistons/tungsten carbide cylinders used at 

100 kPa, the relative uncertainty is 0.8x10-6. T u

5.2.10 Uncertainty due to density of pressure fluid, f

uρ

The sensitivity coefficient is: 

T

f

p gh

ρ

∂ = − ∂ . (84) 

Standard uncertainty, u(ρf). This term depends on the fluid used. For oils, NIST assumes 

a rectangular distributions of possible errors of 0.02ρf , hence the standard uncertainty is 

0.012ρf . At atmospheric pressure, common hydraulic oils have densities from 860 kg/m3

to 910 kg/m3

. The uncertainty term can become significant for pressures below about 

10 MPa, especially if h is 0.1 m or greater. 

 

For gases below 1 MPa, the uncertainty in density is due to possible differences in 

temperature along the tubing connecting the gauges. Taking a rectangular distribution of 

possible errors of 0.5 oC, u(ρf) = 0.001ρf. The uncertainty term is negligible for h less 

than 1 m. Above 1 MPa, the density begins to depart from the perfect gas model used in 

57
our data reduction program. As long as h is less than 0.1 m, the uncertainty term is less 

than 0.2x10-6. 

5.2.11 Uncertainty due to reference height difference, uh

The sensitivity coefficient is: 

( ) T

f a

p g h ρ ρ∂ =− − ∂ . (85) 

Standard uncertainty, u(h): The height difference between the piston gauge and 

customer’s instrument is the sum of the difference in the reference levels, hT-hR0, and the 

difference between the bottom of the piston position and its reference level (hR-hR0). Or, 

 . (86) 0 0 ( )( TR TR RR hh h h h h h =−= − − − )

The difference in reference levels is measured prior to the data-taking. For its 

uncertainty, NIST takes a rectangular distribution with a maximum possible error of 

1x10-3 m. The other term is nominally equal to zero, however the uncertainty is not. The 

piston falls during normal operation, and is intermittently raised above the nominal 

position to allow operational time during the drop. NIST takes a rectangular distribution 

of errors with a maximum of 1x10-3 m. The combined standard uncertainty is 

u(h) = 0.8x10-3 m. 

The contribution of uh to the uncertainty in pT is always negligible when the NIST piston 

gauge uses gas. When oil is used, the relative magnitude is 1x10-6 at 8 MPa, and 

increases as 1/pT as the pressure decreases. 

5.2.12 Uncertainty due to vacuum pressure, (absolute mode only) Vp u

The sensitivity coefficient is: 

1 T

V

p

p

∂ = ∂ . (87) 

The standard uncertainty, u(pV), depends on the vacuum gauge used to measure the 

pressure in the bell jar. NIST uses a calibrated capacitance diaphragm gauge. When 

calibrated, these gauges have a standard uncertainty of 0.05 % of reading. The long-term 

stability uncertainty is typically 0.5 % of reading. A typical vacuum pressure in an 

absolute mode calibration is 2 Pa (this depends on the gas flow in the piston-cylinder gap 

and the capacity of the vacuum pump to evacuate the bell jar). Using these conditions, 

the standard uncertainty is 0.01 Pa. The relative contribution to the uncertainty in 

pressure is 0.1x10-6 at 100 kPa, and 1x10-6 at 10 kPa. 

58
6 Quality system 

The calibration services performed by the piston gauge standards which are described in 

this document are support by the NIST quality system. The NIST quality system 

documentation consists of tiered quality manuals, ranging form the highest level (QM-I) 

to the Division level (QM-IIs) to the Service level (QM-IIIs). The NIST quality manual 

(QM-I) is found at http://ts.nist.gov/QualitySystem/

The integrity, reliability, and traceability of the NIST measurement services relies on the 

NIST Quality System for Measurement Services, which is based on the ISO/IEC 17025 

(General requirements for the competence of testing and calibration laboratories) [14] 

and the relevant requirements of ISO/IEC Guide 34 (General requirements for the 

competence of reference material producers) [15]. The scope of the NIST Quality System 

includes the delivery of Calibration Services and the development and certification of 

Standard Reference Materials. 

The Measurement Services Advisory Group (MSAG) at NIST serves as the corporate 

quality manager; they are assisted by staff from the National Voluntary Laboratory 

Accreditation Program for the implementation of the quality system. The NIST quality 

system for measurement services satisfies the requirements of the International 

Committee for Weights and Measures (CIPM) Mutual Recognition Arrangement (MRA) 

[16] for recognition of national measurement standards; and as such, has been recognized 

as conformant to the ISO/IEC 17025 and ISO Guide 34 by the Inter-American Metrology 

System (SIM) Quality System Task Force and the Joint Committee of the Regional 

Metrology Organizations and the BIPM (JCRB). The BIPM is the International Bureau 

of Weights and Measures. 

In order to maintain compliance with the MRA, NIST participates in a large number of 

international comparisons with other NMIs to support our calibration measurement 

capabilities and uncertainty claims. Comparisons relevant to the present calibration 

service can be found at http://kcdb.bipm.org/AppendixB/KCDB_ApB_search.asp, 

searching on Metrology Area = Mass, Branch = pressure, and Country = United States. 

 

59
7 References 

1. NIST Calibration Program, Calibration Services Users Guide SP250 Appendix (2009). 

2. Dadson, R.S., Lewis, S.L., and Peggs, G.N., “The pressure balance: theory and 

practice”, Her Majesty’s Stationery Office, London (1982). 

3. Schmidt., J.W., Jain, K., Miiller, A.P., Bowers, W.J., and Olson, D.A., “Primary 

pressure standards based on dimensionally characterized piston/cylinder assemblies”, 

Metrologia 43 (2006) 53-59. 

4. Bean, V.E., “NIST pressure calibration service”, NIST Special Publication 250-39, 

US Department of Commerce, Technology Administration, National Institute of 

Standards and Technology (1994). 

5. Veale, R.C., “NBS report of calibration M3565”, Precision Engineering Division 

(1989). 

6. Jain, K., Bowers, W., and Schmidt, J.W., “A primary dead-weight tester for pressures 

(0.05-1.0) MPa”, J. Res. Natl. Inst. Stand. Technol. 108 (2003) 135-145. 

7. Heydemann P.L.M., and Welch, B.E., “Piston Gauges”, in Experimental 

Thermodynamics, vol. 2, ed. B. Leneindre and B. Voder, London: Butterworths (1975) 

147-201. 

8. Bandyopadhyay, A.K., and Olson, D.A., “Characterization of a compact 200 MPa 

controlled clearance piston gauge as a primary standard using the Heydemann and Welch 

method”, Metrologia 43 (2006) 573-582. 

9. Picard, A. Davis, R.S., Gläser, M., and Fujii, K., “Revised formula for the density of 

moist air (CIPM-2007), Metrologia 45 (2008) 149-155. 

10. Markus, W., “A constant volume valve”, Review of Scientific Instruments 43 (1) 

(1972) 158-159. 

11. Kobata, T., and Olson, D. A., “Automating the Calibration of Two Piston Gage 

Pressure Balances”, Proceedings of the National Conference of Standards Laboratories 

International, (2002) August. 

12. Kobata, T., and Olson, D.A., “Accurate Determination of Equilibrium State Between 

Two Pressure Balances Using a Pressure Transducer”, Metrologia 42 (2005) S231-S234. 

13. B.N. Taylor and C.E. Kuyatt, “Guidelines for evaluating and expressing the 

uncertainty of NIST measurements and results”, NIST Technical Note 1297, (1994). 

14. ISO/IEC 17025:2005(E), “General requirements for the competence of testing and 

calibration laboratories” (2005). 

15. ISO Guide 34:2000(E), “General requirements for the competence of reference 

material producers” (2000). 

16. CIPM, “Mutual recognition of national measurement standards and of calibration and 

measurement certificates issued by national metrology institutes”, 

www.bipm.org/utils/en/pdf/mra_2003.pdf, (1999, rev. 2003). 

60
Appendix A. Recommended formula for the calculation of the density 

of moist air and its uncertainty 

A.1 The CIPM-2007 equation 

The following is a summary of the method described in [9] for the calculation of the 

density of moist air, known as CIPM-2007. The formulation begins with the equation of 

state for the air density: 

1 1 a v

a v

a

pM M

x

ZRT M ρ

⎡ ⎛ ⎞⎤ = −− ⎢ ⎜ ⎟⎥ ⎢⎣ ⎝ ⎠⎥⎦

 . (88) 

The quantities and units are 

p air pressure in Pa 

t air temperature in oC 

T thermodynamic temperature in K, T = 273.15+t 

xv mole fraction of water vapor 

Ma molar mass of dry air in kg mol-1 

Mv molar mass of water in kg mol-1 

Z compressibility factor 

R molar gas constant in J mol-1 K-1. 

The value of R is given as: 

 1 1 R 8.314472 J mol K − − = . 

The composition of dry air is given in Table 1 of [9], assuming the mole fraction of CO2

is . With those components and their mole fractions, the value of Ma is CO2 x = 0.0004

3 1 28.96546 10 kg mol Ma− − = × . 

If the mole fraction of CO2 x is available, Ma is given by: 

( ) 2

3 1

CO 28.96546 12.011 0.0004 10 kg mol M x a

− − = + ⋅− × ⎡ ⎤ ⎣ ⎦ . (89) 

The molar mass of water vapor is given by: 

3 1 18.01528x10 kg mol Mv− − = . 

To within typical values of CO2 x and uncertainties of other quantities in the formula, 

 

1 0.3780 v

a

M

M− = . 

61
Substituting eq. (89) and values of the quantities into eq. (88), the formula for ρa in kg/m3

becomes: 

( ) ( ) 2

3

CO 3.483740 1.4446 0.0004 1 0.3780 10 a v

p x x

ZT ρ − = + ⋅− ⋅ − × ⎡ ⎤ ⎣ ⎦ . (90) 

Using the formula is: CO2 x = 0.0004

( ) 3 3.483740 1 0.3780 10 a

p x

ZT ρ v

− = ⋅− × . (91) 

Hence to calculate the density of moist air, p and t are measured, along with either the 

relative humidity or the dew point temperature. xv and Z are calculated (see below) from 

the measured quantities; finally ρa is computed from eq. (90) or (91). 

A.2 Calculation of xv from the measurement of relative humidity or dew-point 

temperature 

The mole fraction of water vapor is given by: 

( ) ( ) ( ) ( ) , , sv s

v d

v d p t p x hf p t f p t t

p p = ⋅= ⋅ . (92) 

Here, h is the relative humidity, td is the dew-point temperature, f is the enhancement 

factor (which is a function of p and t or td), and psv is the vapor pressure at saturation. 

Either h or td can be measured to determine xv. psv is given by: 

( ) 2 1 Pa exp / sv p = × + ++ AT BT C D T . (93) 

T is the temperature in K. The constants are given by: 

5 2

2 1

3

1.2378847 10 K ,

1.9121316 10 K ,

33.93711047 ,

6.3431645 10 K .

A

B

C

D

− −

− −

= ×

=− ×

=

= − ×

Finally, the enhancement factor, f, is given by: 

2 f =+ + α β γ p t , (94) 

with the constants given by: 

62
8 1

7 2

1.00062 ,

3.14 10 Pa ,

5.6 10 K .

α

β

γ

− −

− −

=

= ×

= ×

 

t is the temperature in oC. 

A.3 Calculation of compressibility factor, Z 

The equations for Z and its constants are given by: 

( )( ) ( )

2

2 2

0 1 2 01 01 2 1 v v

p p 2 Z v a a t a t b b t x c c t x d ex

T T

=− ⋅ + + + + + + + ⋅ + ⎡ ⎤ ⎣ ⎦ . (95) 

The constants in the equation are given by: 

6 1

0

8 1

1

10 1 1

2

6 1

0

8 1

1

4 1

0

6 1

1

11 2 2

82 2

1.58123 10 K Pa ,

2.9331 10 Pa ,

1.1043 10 K Pa ,

5.707 10 K Pa ,

2.051 10 Pa ,

1.9898 10 K Pa ,

2.376 10 Pa ,

1.83 10 K Pa ,

0.765 10 K Pa 

a

a

a

b

b

c

c

d

e

− −

− −

− −−

− −

− −

− −

− −

− −

− −

= ×

=− ×

= ×

= ×

=− ×

= ×

=− ×

= ×

=− × .

A.4 Uncertainty of the formula and range of application 

The relative standard uncertainty of the formula is given in [9] as 2.2x10-5. The 

recommended ranges in pressure and temperature over which it can be used are: 

60 kPa ≤ p ≤ 110 kPa 

15 oC ≤ t ≤ 27 oC. 

63
64

Appendix B. Sample calibration report of a customer piston gauge 

Report follows on next page
UNITED STATES DEPARTMENT OF COMMERCE 

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY 

GAITHERSBURG, MARYLAND

REPORT OF CALIBRATION

Pressure & Vacuum Group 

Bldg 220, Room B43 

Requester: 

 No One In Particular 

 123 Main Street 

 Your Home Town, MD 20899 

 

Test Instrument Data:

 Manufacturer: ABC Instrument Company 

 Model: 9999 

 Serial Number: None 

 Piston Number: 99-000 

 Cylinder Number: 99-000 

 Maximum Pressure: 28 MPa 

 Cylinder Type: Re-Entrant 

 Thermal Expansion Coefficient of Piston: 4.55x10-6/ oC 

 Thermal Expansion Coefficient of Cylinder: 4.55x10-6/ oC 

 Nominal Piston Area: 8.4x10-5 m2

Test Record Data:

 Purchase Order Number and Date: XXXXX Dated 01/01/09 

 NIST Identification Number: P-9999A 

 NIST Test Folder Number: TN-999999-09 

 Date Instrument was Received: January 1, 2009 

 Date Test was Completed: March 1, 2009 

Test Conditions:

 NIST Standard and Calibration Reference: PG42, Dec 1996 

 Reference Temperature: 23 oC 

 Mode of Operation: Gauge 

 Pressure Fluid: Spinesstic Oil 

 Pressure Range of Calibration: 1.4 to 26 MPa 

 Surface Tension of Fluid: 0.0 

 Rotation of Weights: Manual 

 Test Gauge Weights Provided by: NIST 

 The test gauge was leveled so that the axis of rotation was vertical. 

 Reference level of test piston: The reference level was 0.080 meter below the uppermost surface of 

the piston. The gauge was operated at mid-stroke. 

The suggested fit for the effective area of the test gauge in m2

, is 

Afit = 8.401099x10-5 (1 – 2.269x10-12 P) , with P in Pa. 

The expanded relative uncertainty in the effective area of the test gauge, including the uncertainty 

of the NIST standard, ranges from 24x10-6 (24 ppm) to 30x10-6 (30 ppm). 

65
The test gauge was cross-floated against the NIST standard. The calibration data are given in Table 1. The 

pressures (P) are at the reference level of the test gauge as determined by the NIST standard gauge. The 

temperature corrected forces (F) on the test gauge were calculated using the expression 

1 ( )( ) 23

1

−++

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −

=

∑

T

gm C

F

cp

i mi

a

i

αα

γ

ρ

ρ

, 

where mi are the masses of the piston, weight hanger and weights corresponding to P, 

ρa is the density of the ambient air, 

ρmi is the density of the material from which the weights are made, 

 g is the local acceleration due to gravity, 

γ is the surface tension of the pressurizing fluid, 

 C is the circumference of the piston in the test gauge, 

αp and αc are the linear thermal expansion coefficients of the piston and cylinder, and 

 T is the temperature in degrees C of the test gauge when operating at pressure P. 

Also listed in Table I is the effective area (A) of the test gauge at each pressure calculated using the 

expression: 

A = F P/ . 

 

To obtain an expression for predicting the effective area at any pressure, the P and A data were fitted to the 

following eight equations using the method of least squares: 

 (fit 1) A A fit = 0

0 / Afit = − A tP (fit 2) 

0 1 (1 Afit = + A bP) (fit 3) 

( ) (fit 4) 0 1 1 A A bP t fit = +− / P

)

/

( (fit 5) 2

0 12 1 A A bP bP fit = ++

( ) 2

0 12 1 Afit = ++ − A bP bP t P

)

/ P

 (fit 6) 

( (fit 7) 2

0 2 1 A A bP fit = +

( ) . (fit 8) 2

0 2 1 A A bP t fit =+ −

The regression coefficients are interpreted as follows: A0 is the extrapolated area at zero applied pressure 

and at 23 °C; b1 and b2 are the first and second order pressure coefficients for the area; and t allows for the 

possibility of a “tare”. The suggested fit based on the calibration data is fit 3, yielding for the effective 

area: 

0 1 ( ) 1 Afit = + A bP . 

The uncertainty in the calibration results for the test gauge effective area is estimated by combining the 

component uncertainties using the root-sum-square method. The Type A component (uA) is evaluated by 

statistical means, and the Type B components (uB) are evaluated by other means. The current 

international practice (as well as at NIST) is to report the combined expanded uncertainty, UC = kuc, at the 

66
two standard deviation level (k=2). When normal statistical distributions apply, the expanded uncertainty 

defines an interval having a level of confidence of approximately 95%. The Type A component is due to 

random errors of the NIST standard, the test piston gauge, and the calibration process; it is estimated from 

the standard deviation of the predicted values associated with the least-squares fit. The dominant Type B 

component is the uncertainty in the NIST standard. uB also includes uncertainties in the masses used on 

both the NIST standard and the test gauge, uncertainties in thermal effects on both gauges, and the 

uncertainty in the reference level correction. 

The fitted effective area over the pressure range of the calibration is listed in Table 2. Table 2 also lists 

the Type A and Type B relative standard uncertainties, and the combined relative expanded (k=2) 

uncertainty of the test gauge effective area, over the pressure range of the calibration. The relative 

expanded uncertainty in the effective area of the test gauge ranges from 24x10-6 (24 ppm) to 30x10-6

(30 ppm). Note that this uncertainty is applicable only over the stated range of calibration, and is valid 

only for the specific operating conditions of this calibration, given on page 1. 

 

Note: The mass used for the piston assembly was 126.82212 g, based on a density of 15.8 g/cm3

, 

determined by NIST Mass Group. 

 

 

For the Director, 

National Institute of Standards and Technology 

Dr. Douglas A. Olson 

Leader, Pressure and Vacuum Group 

Process Measurements Division 

Chemical Science and Technology Laboratory 

67
Table 1. Calibration data of the crossfloat. Listed are pressure (P) on the test gauge at its reference 

level, force (F) on the test gauge, and effective area (A) of the test gauge. 

Obs. PFA

No. (MPa) (N) (m2

)

1 1.43275 120.3657 8.401027E-05

2 2.81103 236.1581 8.401117E-05

3 13.83783 1162.4950 8.400847E-05

4 19.35152 1625.6640 8.400703E-05

5 25.72025 2160.6620 8.400625E-05

6 20.95730 1760.5590 8.400695E-05

7 16.59461 1394.0790 8.400791E-05

8 11.08110 930.9101 8.400881E-05

9 6.94604 583.5333 8.400946E-05

10 4.18937 351.9494 8.401015E-05

68
69

Table 2. Effective area of the test gauge calibration fit (Afit) and its uncertainty, over the pressure range 

of the calibration. Listed are Type A relative standard uncertainty (uA/A), Type B relative 

standard uncertainty (uB/A), and combined relative expanded (k=2) uncertainty (2uC/A). 

 

P A fit u A /A u B /A 2u C /A

(MPa) (m2

) x106 x106 x106

1.40 8.401072E-05 2.1 14.6 29.6

2.62 8.401049E-05 2.0 12.6 25.6

3.84 8.401026E-05 1.8 12.2 24.6

5.06 8.401003E-05 1.7 12.0 24.2

6.28 8.400979E-05 1.6 11.9 24.0

7.50 8.400956E-05 1.5 11.9 23.9

8.72 8.400933E-05 1.4 11.8 23.8

9.94 8.400910E-05 1.3 11.8 23.8

11.16 8.400886E-05 1.3 11.8 23.8

12.38 8.400863E-05 1.3 11.8 23.7

13.60 8.400840E-05 1.3 11.8 23.7

14.82 8.400817E-05 1.3 11.8 23.7

16.04 8.400793E-05 1.4 11.8 23.7

17.26 8.400770E-05 1.5 11.8 23.8

18.48 8.400747E-05 1.6 11.8 23.8

19.70 8.400723E-05 1.7 11.8 23.8

20.92 8.400700E-05 1.8 11.8 23.8

22.14 8.400677E-05 2.0 11.8 23.9

23.36 8.400654E-05 2.1 11.8 23.9

24.58 8.400630E-05 2.3 11.8 24.0

25.80 8.400607E-05 2.5 11.8 24.1

End of Calibration Report, Appendix B.