NIST Measurement Services: 

Gas Flowmeter Calibrations with the 26 m3PVTt Standard

NIST Special Publication 1046 

Aaron N. Johnson and John D. Wright 

U. S. Department of Commerce 

Technology Administration 

National Institute of Standards and Technology 
i 

Table of Contents Gas Flowmeter Calibrations with the 26 m3PVTt Standard 

Abstract.............................................................................................................................1 

1 Introduction to Gas Flow Measurement at NIST.........................................................2

2 Description of Gas Flow Calibration Services...............................................................3

3 Procedures for Submitting a Flowmeter for Calibration............................................ 5

4 Overview of Pressure, Volume, Temperature, and time (PVTt) Flow Standards..... 5

4.1 Description of the 26 m3PVTt Flow Standard ........................................................ 5 

4.2 CFV Check Standards.............................................................................................. 6 

4.3 Theoretical Development of the PVTt Mass Flow .................................................. 7 

4.4 PVTt Operating Procedures ..................................................................................... 9 

4.5 Inventory Volume Mass Cancellation Technique ................................................. 11 

5 Uncertainty of PVTt Subsidiary Components............................................................ 14

5.1 Reference Parameters (Molec. Weight, Univ. Gas Const, Comp. Factor) ............ 15 

5.2 Collection Time ..................................................................................................... 16 

5.3 Pressure and Temperature in the Inventory Volume ............................................. 17 

5.4 Inventory Volume.................................................................................................. 22 

5.5 Pressure and Temperature in the Collection Tank................................................. 22 

5.6 Collection Tank Volume........................................................................................ 25 

6 Mass Flow Uncertainty................................................................................................. 32

6.1 Accumulated Mass in Collection Tank.................................................................. 33 

6.2 Accumulated Mass in Inventory Volume .............................................................. 34 

6.3 Effect of Leaks....................................................................................................... 36 

6.4 Uncertainty Attributed to the Steady Flow Assumption........................................ 37 

7 Summary........................................................................................................................ 37

8 References...................................................................................................................... 38

 Appendix: Sample Calibration Report 
1 

Abstract 

This document describes NIST’s 26 m3 pressure, volume, temperature, and time (PVTt) primary 

flow standard. This standard is used to calibrate gas flow meters over a range extending from 

200 L/min to 77000 L/min where the reference temperature and pressure conditions are 

293.15 K and 101.325 kPa respectively, and the working fluid is dry air. This standard measures 

flow by collecting a steady stream of gas into a tank of known volume during a measured time 

interval. The ratio of the mass of gas accumulated in the tank to the collection time is the mass 

flow. 

The PVTt standard measures mass flow with an expanded uncertainty of 0.09 % at the 95 % 

confidence interval (i.e., k = 2) over the full flow range. This document presents a detailed 

uncertainty analysis evaluating and explaining the various components that comprise the 

expanded uncertainty. In addition to the uncertainty analysis, we specify the various components 

of the 26 m3PVTt system, describe its theoretical basis of flow measurement, document the 

calibration procedure used for the standard, provide information about the calibration service 

(i.e., meters that we commonly test, available pipe sizes, sample calibration report, etc.), and give 

details regarding the unique aspects of NIST’s PVTt systems. 

Key words: calibration, uncertainty, flow, flowmeter, gas flow standard, inventory volume, PVTt standard, inventory 

mass cancellation technique, sensor response, correlated uncertainty sources. 
2 

1. Introduction to Gas Flow Measurement at NIST 

Calibrations of gas flow meters are performed with primary standards [1] that are based on 

measurements of more fundamental quantities, such as length, mass, and time. Primary flow 

calibrations are accomplished by collecting a measured mass or volume of a flowing fluid over a 

measured time interval. The ratio of the collected mass to the measured time interval equals the 

time-averaged mass flow at the meter under test (MUT). To ensure that the instantaneous mass 

flow equals the time-averaged value, the flow at the MUT should be maintained under steady 

state conditions of flow, pressure, and temperature. 

Traditionally, primary flow standards have been based on either gravimetric or volumetric 

methods. Gravimetric based primary flow standards measure the mass of collected gas by 

directly weighing the mass of the collection vessel before and after gas accumulation [2]. On the 

other hand, volumetric based primary standards calculate the mass of collected gas by 

multiplying the measured density of the gas by the volume of the collection tank. Common 

volumetric based primary standards include piston provers [3], bell provers [4], and pressurevolume-temperature-time (PVTt) systems [5, 6]. In the Fluid Metrology Group (FMG) at NIST, 

gas flow is measured exclusively with PVTt primary flow standards. 

Table 1. Flow measurement capabilities of the three NIST PVTt primary gas flow standards.

The FMG of the Process Measurements Division (part of the Chemical Science and Technology 

Laboratory) at NIST has three PVTt flow standards that provide gas flow calibration services 

over a range from 1 L/min to 77000 L/min.1 The lowest flows are measured using the 34 L PVTt

system, medium flows using the 677 L PVTt system, and the largest flows using the 26 m3PVTt

system. The various flow ranges, types of gases, pressure ranges, and uncertainty for each PVTt

primary standard are shown in Table 1. (Not included in this table are flows less than 1 L/min, 

which can be calibrated by the NIST Pressure and Vacuum Group.) 

This document discusses the procedures for submitting a flow meter for calibration, gives the 

readily available pipe sizes and flanges suitable for flow meter calibration, documents the format 

of a standard NIST calibration report, and states the normal range of data collected for a 

calibration. In addition, this document describes the theory, principle of operation, and 

 

1

 Reference conditions of 293.15 K and 101.325 kPa are used throughout this document for volumetric flows. 

Flow 

Standard 

Flow 

Range 

(L/min) 

Gas 

Type 

Pressure 

Range 

(kPa) 

Relative Expanded Uncertainty 

(k = 2) 

(%) 

1 to 100 Dry Air 100 to 1700 0.05 

1 to 100 N2100 to 7000 0.03 to 0.04 

1 to 100 CO2100 to 4000 0.05 

1 to 100 Ar 100 to 7000 0.05 

34 L PVTt 

1 to 100 He 100 to 7000 0.05 

10 to 2000 Air 100 to 1700 0.05 

677 L PVTt

10 to 150 N2100 to 800 0.02 to 0.03 

26 m3 PVTt 200 to 77000 Dry Air 200 to 800 0.09 
3 

uncertainty of the 26 m3PVTt primary flow standard covering the flow range from 200 L/min to

77000 L/min. Details concerning the two smaller PVTt flow standards can be found in the 

following reference [6]. 

2. Description of Gas Flow Calibration Services 

NIST offers calibrations of gas flow meters in order to provide traceability to flow meter 

manufacturers, secondary flow calibration laboratories, and flow meter users. For a calibration 

fee, NIST calibrates a customer’s flow meter and delivers a calibration report that documents the 

calibration procedure, the calibration results, and their uncertainty. The flow meter and its 

calibration results may be used in different ways by the customer. The flow meter is often used 

as a transfer standard to perform a comparison of the customer’s primary standards to the NIST 

primary standards so that the customer can establish traceability, validate their uncertainty 

analysis, and demonstrate proficiency. Customers with no primary standards frequently use their 

NIST calibrated flow meters as working standards or reference standards in their laboratory to 

calibrate other flow meters. 

Table 2. Readily available pipe sizes and fittings.

 2

Nominal 

Pipe 

Diameter 

(cm) (in) 

Fittings and/or 

ANSI 

Flange Ratings 

2.54 1 VCO, Swagelok, AN, and NPT 

5.08 2 ANSI Flanges 150 and 300 

7.62 3 ANSI Flange 300 

10.16 4 ANSI Flanges 150 and 300 

15.24 6 ANSI Flange 300 

20.32 8 ANSI Flanges 300 and 600 

Flowmeters can be calibrated in pipe sizes ranging from 2.54 cm (1 in) to 20.32 cm (8 in). The 

standard pipe sizes and flanges used in the 26 m3PVTt flow standard are listed in Table 2. Flow 

meters can be tested if the flow range, gas type, and piping connections are suitable, and if the 

system to be tested has precision appropriate for calibration with the NIST flow measurement 

uncertainty. The vast majority of flow meters calibrated in the gas flow calibration service are 

critical flow venturis (CFVs), also commonly called critical nozzles. To date, this meter type is 

regarded as the best candidate for transfer and working standards by the gas flow metrology 

community [7]. Other meter types that we have tested include laminar flow meters, positive 

displacement meters, roots meters, rotary gas meters, thermal mass flow meters, and turbine 

meters. Meter types with precisions or calibration instabilities that are significantly larger than 

the uncertainty of the primary standard should not be calibrated by NIST for economic reasons. 

For example, a rotameter for which the float position is read by the operator’s eye normally 

 

2

 Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such 

identification does not imply recommendation or endorsement by the National Institute of Standards and

Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the 

purpose.
4 

cannot be read with precision any better than 1 %. It is not practical to pay several thousand 

dollars to obtain a NIST calibration with an expanded uncertainty of 0.09 %. For such a 

flowmeter, a calibration with an expanded uncertainty of 0.5 % would be perfectly adequate and 

is available from other laboratories at significantly lower cost. 

A normal flow calibration performed by the NIST Fluid Flow Group consists of five flows 

spread over the range of the flow meter. For a CFV, typical calibration set points are at 200 kPa, 

300 kPa, 400 kPa, 500 kPa, and 600 kPa. A laminar flow meter is normally calibrated at 10 %, 

25 %, 50 %, 75 %, and 100 % of the meter full scale. At each of these flow set points, three (or 

more) flow measurements are made with the PVTt standard. The same set point flows are tested 

on a second occasion, but the flows are tested in decreasing order instead of the increasing order 

of the first set. Therefore, the final data set consists of six (or more) primary flow measurements 

made at five flow set points (i.e., at least 30 individual flow measurements). The sets of three 

measurements can be used to assess repeatability, while the sets of six can be used to assess 

reproducibility. For further explanation, see the sample calibration report that is included in this 

document as an appendix. Variations on the number of flow set points, spacing of the set points, 

and the number of repeated measurements can be discussed with the NIST technical contacts. 

However, for data quality assurance reasons, we rarely will conduct calibrations involving fewer 

than three flow set points and two sets of three flow measurements at each set point. 

The FMG prefers to present flow meter calibration results in a dimensionless format that takes 

into account the physical model for the flow meter type. The dimensionless approach helps 

facilitate accurate flow measurements by the flow meter user even when the conditions of usage 

(i.e., gas type, temperature, pressure) differ from the conditions during calibration. For example, 

for a CFV calibration, the calibration report will present Reynolds number and discharge 

coefficient, and for a laminar flow meter, a report presents the viscosity coefficient and the flow 

coefficient [8]. However, we point out that there may be additional uncertainties introduced 

when the dimensionless approach is used to extrapolate a NIST calibration to conditions or gases 

that were not specifically tested. Unless special provisions are made between NIST and the 

customer, it is the customer’s responsibility to determine any additional uncertainty when using 

the dimensionless approach to extend a NIST calibration beyond the measured range. 

When a flow meter is calibrated, the uncertainty of its dimensionless calibration factors depend 

on both the uncertainty of the flow standard as well as the uncertainty of the instrumentation 

associated with the MUT (normally absolute pressure, differential pressure, and temperature 

instrumentation). We prefer to connect our own instrumentation (temperature, pressure, etc.) to 

the meter under test since they have established uncertainty values based on calibration records 

that we would not have for the customer’s instrumentation. In some cases, it is impractical to 

install our own instrumentation on the MUT. This situation typically occurs when the MUT 

outputs flow. In this case, we provide a table of flow indicated by the MUT, flow measured by 

the NIST standard, and the uncertainty of the NIST flow value. 

Customers should consult the web address www.nist.gov/fluid_flow to find the most current 

information regarding our calibration services, calibration fees, technical contacts, and flow 

meter submittal procedures. 
5 

3. Procedures for Submitting a Flow meter for Calibration 

The FMG follows the policies and procedures described in Chapters 1, 2, and 3 of the NIST 

Calibration Services Users Guide [9]. These chapters can be found on the internet at the 

following addresses: 

http://ts.nist.gov/ts/htdocs/230/233/calibrations/Policies/policy.htm, 

http://ts.nist.gov/ts/htdocs/230/233/calibrations/Policies/domestic.htm, and 

http://ts.nist.gov/ts/htdocs/230/233/calibrations/Policies/foreign.htm. 

Chapter 2 gives instructions for ordering a calibration for domestic customers and has the subheadings: A.) Customer Inquiries, B.) Pre-arrangements and Scheduling, C.) Purchase Orders, D.) 

Shipping, Insurance, and Risk of Loss, E.) Turnaround Time, and F.) Customer Checklist. 

Chapter 3 gives special instructions for foreign customers. The web address 

www.nist.gov/fluid_flow has information more specific to the gas flow calibration service, 

including the technical contacts in the FMG, fee estimates, and turnaround times. 

4. Overview of Pressure, Volume, Temperature, and time (PVTt) Flow Standards 

NIST has used PVTt systems as a primary gas flow standards for more than 30 years [5, 6]. In 

this section we provide an overview of the NIST PVTt facility, develop its theoretical basis for 

flow measurements, list its operating procedures, and detail its unique features that distinguish it 

from PVTt systems used in other laboratories. 

4.1 Description of NIST 26 m3PVTt System 

The NIST 26 m3PVTt calibration system is the United States primary standard for measuring gas 

flows ranging from 200 L/min to 77000 L/min. The relative expanded uncertainty over this range 

of flows is 0.09 % (k = 2). The working fluid is filtered, dry air supplied by a three stage 

centrifugal compressor in series with a desiccant drier. The compressor delivers airflow at line 

pressures up to 800 kPa at nominally room temperature conditions and at relative humidity levels 

below 3 %. 

Figure 1. Schematic diagram of the NIST 26 m3 PVTt gas flow standard. 

Flow measurements using the 26 m3PVTt flow standard are completely automated using 

LabVIEW2 software. This software controls each facet of the calibration process including 

CFV

Tank

Inlet

Valve

Vacuum

Exhaust

Valve

Collection

Tank

Bypass

Fan

Fan

Duct

Bypass

Valve

Steady Flow

Source

Inventory

Volume

CFV

Tank

Inlet

Valve

Vacuum

Exhaust

Valve

Collection

Tank

Bypass

Fan

Fan

Duct

Bypass

Valve

Steady Flow

Source

Inventory

Volume
6 

setting the nominal flow, actuating the valves, filling and evacuating the collection tank, taking 

the appropriate pressure and temperature data, measuring the collection time interval, and 

reducing the data. The post-processed calibration data that is calculated by the LabVIEW 

program is verified by recalculating the data on a spreadsheet. The PVTt system is equip with 

various safety features that prevent overpressurizing the collection tank during a calibration. This 

allows the PVTt system to safely perform calibrations during non-business hours, thereby 

allowing a faster turnaround time for our customers. 

The main components of the PVTt calibration system include a source of steady flow, a set of 

appropriately sized CFVs to cover the flow range, an inventory volume sized appropriately for 

the flow, the collection tank, a timing mechanism, a data acquisition system, and pressure and 

temperature instrumentation. A schematic of the PVTt system showing some of these 

components is depicted in Fig. 1. The inventory volume functions to divert the flow to either the 

collection tank or bypass. The timing system measures the duration that gas accumulates in the 

collection tank and inventory volume. The collection tank stores the gas, allowing it to thermally 

equilibrate before determining its mass. The CFV plays multiple roles. First, it isolates the steady 

upstream flow at the CFV inlet from downstream pressure fluctuations that occur in the 

inventory volume during actuation of the bypass and tank inlet valves. Second, the sonic line at 

the CFV throat, in conjunction with the bypass and tank inlet valves, provides a definite 

boundary for the inventory volume. Lastly, it serves as a check standard to help ensure that the 

PVTt system performs consistently over time. 

4.2 CFV Check Standards

Figure 2. Flow ranges covered by various sized CFVs calibrated on the NIST 26 m3 PVTt gas flow standard.

Critical flow venturis are considered to be among the best transfer standards by the flow 

metering community and are commonly used as transfer standards for international comparisons 

between National Metrology Institutes [7]. Since these devices are inherently part of a PVTt

system, the FMG maintains a calibrated set of variously sized CFVs that span the flow range of 

its three PVTt flow standards (i.e., 34 L, 677 L, and 26 m3PVTt standards). These CFVs are used 

as working standards in another calibration facility called the Working Gas Flow Standard 

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

100 1000 10000 100000

Throat Diameter (cm)

Flow Range (L/min)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

100 1000 10000 100000

Throat Diameter (cm)

Flow Range (L/min)
7 

(WGFS). The WGFS provides calibrations, particularly for laminar flow meters, in which the 

reference flow is measured with a relative expanded uncertainty no greater than 0.16 % (k = 2). 

Figure 2 shows the subset of WGFS CFVs that are calibrated on the 26 m3PVTt flow standard. 

The sizes of these CFVs were selected so that the lower and upper limits of their flow ranges 

overlap. The smallest three CFVs in Fig. 2 have a portion of their flow ranges that can be 

calibrated on both the 677 L and the 26 m3PVTt. Agreement between these independent systems 

adds confidence to the validity of each system’s calibration results. 

4.3 Theoretical Development of the PVTt Mass Flow

PVTt systems measure the CFV mass flow3 using timed-collection techniques based on the 

principle of conservation of mass. For fluid flow into an arbitrary control volume (i.e., region of 

interest), this principle requires that the rate of mass accumulation in the control volume equals 

the net influx of mass through its boundaries. In Fig. 1 we take the control volume to include 

both the collection tank and the inventory volume so that the statement of mass conservation is 

dt

dM

m&

net = (1) 

where the total mass in the control volume includes both MT, the mass in the collection tank, 

and M I, the mass in the inventory volume,

M = MT + M I (2) 

and the net influx of mass into the control volume is

mnet m mleak & = & + &

(3) 

the summation of the CFV mass flow, m& , and the leakage of mass flow into the control volume 

from the environment surrounding the tank, mleak & . Although PVTt systems are designed to 

measure m& , they do not distinguish between the CFV mass flow and flow from other sources 

(i.e., leaks), and therefore the flow that is actually measured is mnet & . Consequently, mleak & must 

be either known or negligibly small relative to m& . 

The effects of leaks can be understood by Eqn. (3), which shows that leakage into the control 

volume will result in overpredicting the actual mass flow (i.e., m&

net > m& ). Conversely, the actual 

mass flow will be underpredicted (i.e., m&

net < m& ) for leakage out of the control volume. During a 

calibration cycle, the gas pressures inside the collection tank and inventory volume 4 are 

maintained at or below atmospheric pressures so that leaks tend to flow into the control volume, 

causing the flow standard to overpredict the actual mass flow. The FMG regularly inspects its 

flow standard for leaks to ensure the quality of calibration data. In cases where leaks cannot be 

completely eliminated their effects are included as part of the uncertainty analysis (see 

section 6.3). 

 

3 PVTt systems can also be used to measure the mass flow of a MUT located upstream of the CFV. In this case the 

uncertainty analysis presented in this document should be modified to include the mass storage effects that occur in 

the piping volume between the MUT and CFV. 

4

 The pressure in the inventory volume briefly exceeds one atmosphere during flow diversion, but are subatmospheric during the majority of the collection interval. 
8 

The expression for mass flow as given by Eqn. (1) is not useful in its present form since the rate 

of mass accumulation in the control volume (i.e., the derivative term), in general cannot be 

directly measured at low levels of uncertainty. This difficulty is circumvented by maintaining 

steady state conditions of pressure and temperature in the piping section upstream of the CFV 

inlet. As long as the appropriate pressure ratio is maintained across the CFV, the mass flow ( m& ) 

remains constant throughout the collection period. If the leak rate is negligible, then the 

instantaneous rate of mass accumulation, dt

dM , is constant, and equals 

t

M

dt

dM

∆

∆

= (4) 

the average rate where f i ∆t = t − t is the collection period, and ∆M = M f − M i is the mass 

accumulated during this period. Here, the initial and final masses in the control volume, M i and 

f M , correspond to the times coinciding with the start and end of the collection period, i

t and 

f

t , respectively. The total accumulated mass in the control volume consist of ∆MT, mass 

accumulated in the collection tank, and ∆MI, the mass accumulated in the inventory volume 

T I

i

I

f

I

i

T

f ∆M = (MT

− M ) + (M − M ) = ∆M + ∆M (5) 

Each of the four of the masses in Eqn. (5) are determined by multiplying the appropriate volume 

(either the collection tank or inventory volume) by the average gas density at the time of interest. 

Both the collection tank and inventory volumes are determined prior to a calibration cycle. They 

are measured as described in sections 5.4 and 5.6 respectively. If both volumes are assumed to 

remain fixed over the range of temperatures and pressures they experience, the mass 

accumulation in the collection tank and inventory volumes are5

T

i

T

f

T T ∆M = (ρ − ρ )V (6a) 

I

i

I

f

I I ∆M = (ρ − ρ )V (6b) 

where VT and VI are the respective collection tank and inventory volumes. Applying the 

equation of state for gas density, ρ = PM ZRuT , the accumulated masses in the collection tank 

and in the inventory volume are 

( ) T i

T

i

T

i

T

f

T

f

T

f

T

T u V

Z T

P

Z T

P

M R

















∆ = M − (7a) 

( ) I

i

I

i

I

i

I

f

I

f

I

f

I

I u V

Z T

P

Z T

P

M R

















∆ = M − (7b) 

where M is the molecular weight of the dry air [10], Ru

 is the universal gas constant [11], Z is 

the compressibility factor for dry air [10], and P and T are the average pressure and 

temperature, respectively. By combining Eqns. (4) and (5) and substituting the result into 

Eqn. (1) the governing expression for mass flow is 

 

5

 The change in the collection tank volume due its elasticity and thermal expansion between its evacuated and filled 

conditions makes a negligible contribution to the uncertainty in mass flow and is neglected. 
9 

t

M M

m

T I

∆

∆ + ∆

& = (8) 

where the effect of leaks is omitted in calculating the CFV mass flow, but accounted for in the 

mass flow uncertainty in section 6.3. Furthermore, by substituting the definitions of ∆MT and 

∆M I given in Eqns. (7a) and (7b) into Eqn. (8), the CFV mass flow is also given by 

































+ −

















 −











= I

i

I

i

I

i

I

f

I

f

I

f

I

T i

T

i

T

i

T

f

T

f

T

f

u T V

Z T

P

Z T

P

V

Z T

P

Z T

P

t

R

m

∆

M & . (9) 

4.4 PVTt Operating Procedures 

The typical process for measuring mass flow with the 26 m3PVTt flow standard entails the 

following procedure: 

1. With the tank valve closed, open the bypass valve and establish a stable flow through the 

CFV at the desired stagnation pressure (see Fig. 1). 

2. Evacuate the collection tank volume (VT

) to a prescribed lower pressure using the vacuum 

pump. (Steps 1 and 2 can begin simultaneously.) 

3. Wait for pressure and temperature conditions in the tank to stabilize and then acquire their 

initial values ( i PT

 and iTT). These values will be used to calculate the initial gas density in the 

tank ( i

T

ρ ) and subsequently the initial mass of gas in the tank ( i MT). With the tank under 

vacuum conditions, reasonable pressure and temperature stability is attained in 300 s or less. 

4. With the tank valve still closed, close the bypass valve. After the bypass is fully closed, the 

flow exhausting from the CFV will dead-end in the inventory volume for a brief interval (i.e., 

100 ms or less) called the first dead-end interval. The time history of the pressure and 

temperature in the inventory volume is measured during the dead-end interval. The start of 

the collection time, ( it ), is selected within this interval. The initial pressure and temperature 

in the inventory volume ( i PI and iTI) correspond to the selected start time. These values of 

pressure and temperature are used with an equation of state to determine the initial 

compressibility factor ( i

I

Z ), and subsequently the initial density ( iρ I), which when 

multiplied by the inventory volume (VI) equals the initial mass in the inventory volume 

(

i M I

). Immediately following the dead-end interval, the tank valve is opened. 

5. Wait for the tank to fill to a prescribed upper pressure (i.e., near atmospheric pressure) and 

close the tank valve. 

6. When the tank valve is fully closed (with the bypass valve still closed) there is a brief time 

interval where the flow emanating from the CFV is again dead-ended in the inventory 

volume, the second dead-end interval. The time history of both the pressure and temperature 

in the inventory volume are again measured during this period. The pressure and temperature 

data are used with the equation of state to calculate the time history of the gas density. A stop 

time, ( ft ), is selected within the second dead-end time so that the final inventory gas density 

equals its initial density (i.e., 

i

I

f

I

ρ = ρ ), and hence the final mass in the inventory volume 
10 

(

f MI

) is the same as the initial mass ( i M I). Immediately following the second dead-end 

interval, open the bypass valve. 

7. Turn the fan on inside the collection tank and wait for temperature stability before acquiring 

the final pressure and temperature ( fPT and f

T

T ). These values will be used with an equation 

of state for dry air to determine the final compressibility factor ( f

T

Z ), and subsequently the 

final density ( f

T

ρ ). The volume of the collection tank (VT) is multiplied by the final density 

to determine the final mass of gas in the collection tank ( f MT). The usual waiting period for 

pressure and temperature stability is 2700 s. (Steps 6 and 7 can begin simultaneously.) 

8. Equation (9) is used to determine the CFV mass flow ( m& ). 

9. Return to step 1 for next calibration point or end calibration. 

Table 3. Nominal values of the parameters and measured variables used in Eqn. (9). 

System 

Components 

and 

Parameters 

Quantity Nominal Value Instrumentation 

or Reference 

Universal Gas Constant, Ru 8134.472 J/(kg⋅K) Reference [11] 

Molecular. Mass (dry-air), M 28.9647 g/mol Reference [10] Reference 

Parameters 

Compressibility Factor (dry-air), Z Z = Z( T,P ) Reference [10] 

Initial Pressure, i PT

0.08 kPa to 0.1 kPa Vacuum Gauge 

Final Pressure, f PT

93 kPa to 103 kPa Abs. Pressure Gauge 

Initial Temperature, iTT

292 K to 297 K 

Final Temperature, f

T

T 292 K to 297 K 

37 Thermistors 

Collection 

Tank 

Volume, VT 25.8969 m3

 see section 5.6 

Initial Pressure, i PI

100 kPa to 450 kPa

Final Pressure, f PI

100 kPa to 450 kPa

2 Fast Pressure 

Transducers 

Initial Temperature, iTI

293 K to 320 K 

Final Temperature, f

I

T 293 K to 320 K 

2 Thermocouples 

Inventory 

Volume 

Volume, VI 0.025 m3

 to 0.1 m3 see section 5.4 

Base time, ∆τ 20 s to 8300 s 2 Universal Counters 

1

st Dead-End Interval, ∆ 1

t 0.03 to 0.1 sec 

Timing 

System 

(see Eqn. 11) 

2

nd Dead-End Interval, 2 ∆t 0.03 to 0.1 sec 

Data acquisition card 

sampling at 3000 Hz 

Table 3 list the instrumentation used to make the pressure, temperature and time measurements 

as well as their normal range of values during a calibration. The table also gives the values of the 
11 

reference parameters Ru, M , and Z . The measurement of the collection tank volume and the 

inventory volume are discussed later in sections 5.4 and 5.6 respectively. 

4.5 Inventory Volume Mass Cancellation Technique

Many of the operating procedures used by the FMG are standard to all blow-down PVTt systems. 

However, the inventory mass cancellation technique outlined in steps 4 and 6 of the PVTt

operating procedures (see section 4.4) is unique to NIST. During the dead-end periods, both the 

pressures and temperatures in the inventory volume increase. The start and stop times, it and ft , 

are selected so that the initial and final densities in the inventory volume are equal. Since the size

of inventory volume remains fixed for both dead-end intervals, matching the densities ensures 

that the accumulated mass in the inventory volume is identically zero (i.e., ∆M I = 0 ). 

Figure 3. The time history of the mass in the inventory volume ( M I), and the mass in the collection tank 

( MT ) for a typical calibration cycle. (The mass plots are based on a semi-empirical model.) 

The inventory volume mass cancellation technique is an extension of the pressure-matching 

scheme described in [6]. In the pressure-matching scheme, the initial and final pressures in the 

inventory volume are matched so that the accumulated mass in the inventory volume is nearly 

zero. The mass cancellation technique, introduced here, further develops this strategy, by 

matching the initial and final densities. By matching density instead of the pressure, the initial 

and final masses are made to completely cancel. The advantages of these matching schemes are 

two fold. First, the correlated uncertainty sources between the initial and final densities will 

completely cancel. Second, because the uncertainty in the size of inventory volume does not 

significantly contribute to the mass flow uncertainty, a highly accurate measurement technique is 

not necessary to determine the size of the inventory volume. In practice, the size of the inventory 

volume is rudimentarily measured to within 25 % of its actual size using a simple tape measure. 

This straightforward approach for measuring the size of the inventory volume is especially 

convenient when calibrating customer CFVs requiring modifications to the normal piping 

configuration of the inventory volume. 

t it f

0

15

30

45

0

15

30

45

60

75

MT(kg)

time

∆t

Region 1 Region 2 Region 3 Region 4 Region 5

a b c a b c ∆τ

density overlap 

region

MI

(g)

∆t

2

∆t

1

t it f

0

15

30

45

0

15

30

45

60

75

MT(kg)

time

∆t

Region 1 Region 2 Region 3 Region 4 Region 5

a b c a b c ∆τ

density overlap 

region

MI

(g)

∆t

2

∆t

1
12 

Figure 3 illustrates how the inventory mass cancellation technique is applied during a calibration 

cycle. The figure shows time histories for the mass in collection tank, MT (left), and the mass in 

the inventory volume, M I (right), during a typical calibration cycle. The values of M I and MT

are obtained from a semi-empirical model based on mass conservation. The results of the model 

agree reasonably well with measured results, and are used here to explain the inventory matching 

technique. 

The time histories of M I and MT are divided into five regions. Region 1 corresponds to steps 1 

and 2 in the PVTt operating procedures. In this region, M I is constant since the mass flow 

entering the inventory volume through the CFV equals the mass flow exiting via the bypass 

valve (see Fig. 1). Simultaneously, MT decreases as the collection tank is evacuated via the 

vacuum exhaust valve. Region 2 corresponds to step 4 where the flow is diverted from the 

bypass into the collection tank. Region 3 includes the first part of step 5 where flow accumulates 

in the collection tank through the tank inlet valve (bypass is closed). The latter part of step 5, and

step 6 correspond with Region 4 where the flow is diverted from the tank back to the bypass. 

Finally, Region 5 corresponds to the end of the calibration cycle as explained in step 9. 

The time durations of Regions 2 and 4, corresponding to flow diversion into and away from the 

collection tank, have been expanded relative to the other regions in Fig 3. These brief intervals 

play an important role in the mass cancellation technique. By expanding these regions, the 

behavior of M I can be clearly identified. Region 2 and 4 each last approximately 0.3 s, in 

contrast to Region 3, which can last from 20 s to 5500 s depending on flow, and Regions 1 and 5 

which together, last approximately 4000 s. Regions 2 and 4 are both divided into three distinct 

subdivisions labeled “a”, “b”, and “c”. In Region 2 these three subdivisions denote the following: 

subdivision “a” shows the slight increase in M I during the closing of the bypass valve; 

subdivision “b” shows the nearly linear increase in M I during the first dead-end interval where 

both the bypass and tank valves are closed; and subdivision “c” shows the initial increase in M I

as the tank valve just begins to open followed by its rapid decrease as the inventory volume gas 

is sucked into the nearly evacuated collection tank through the fully opened tank valve. The three 

subdivisions in Region 4 are similar to those in Region 2 and denote the following: subdivision 

“a” shows the slight increase in M I

 as the tank valve is closing; subdivision “b” shows the 

increase in M I during the second dead-end interval; and subdivision “c” shows the initial 

increase in M I

 followed by its rapid drop off to match the atmospheric pressure condition when 

the bypass is fully opened. 

For the lowest uncertainty, the collection time measurement should begin in the first dead-end 

interval (i.e., Region 2b) and end in the second dead-end interval (i.e., Region 4b). If the 

collection time began or ended in any other region, the uncertainty in mass flow could be 

substantially larger. For example, if the collection time began while the bypass valve was closing 

(Region 2a), the gas emanating from the CFV could escape into the room through the partially 

opened bypass valve. The uncertainty attributed to airflow leaking into or out of the bypass is 

difficult to quantify, and thereby increases the mass flow uncertainty. 

An increase in the mass flow uncertainty also occurs if collection time begins in Region 2c while 

the tank valve is opening. In this case, the initial mass in the collection tank, i MT

, must be 

measured dynamically (i.e., while mass is accumulating in the tank) rather than statically. 
13 

Figure 3 shows the increase in MT attributed to mass flow through the partially opened tank 

valve in Region 2c. Since dynamic mass determinations have larger uncertainties than static 

determinations, it is not advantageous to begin the collection time in this region. By default, 

Region 2b is the best choice to begin the collection time. Similar arguments can be made to show 

that Region 4b is the best choice to stop the collection time. 

Figure 4. Time adjustment factor versus the percent density overlap parameter. 

Unlike MT, the nature of the diversion process necessitates that M I be measured dynamically. 

Because M I

 must be measured dynamically, the initial and final inventory mass measurements 

must coincide with it (i.e., the start of collection time) and ft (i.e., the end of collection time). 

On the other hand, the initial and final mass measurements in collection tank do not need to 

coincide with it and ft . For example, the initial mass of the gas in the tank, i MT, can be 

measured any time starting from the closing of the vacuum exhaust valve (at the latter part of 

Region 1) until just before the tank valve starts to open (at the beginning of Region 2c). Likewise, 

the final mass, f MT, can be measured any time in Region 4b, 4c, or 5. During either of these 

time intervals the collection tank is isolated so that the mass of gas in its interior remains 

constant as shown in the Fig. 3. In practice, however, the time traces of these mass measurements 

are not constant, but asymptote toward a constant value as the spatial pressure and temperature 

gradients in the gas dissipate. If there are no leaks, any non-uniformities in the time traces of the 

initial mass (in Regions 1, 2a, and 2b) or the final mass (in Regions 4b, 4c, and 5) are a result the 

method used for measuring the mass (i.e., via pressure, temperature, volume, and an equation of 

state) and not an actual change in the mass. During a calibration, 

i MT

 and f MT are determined 

only after sufficient time is allotted to allow the gas to equilibrate as discussed in steps 3 and 7 of 

the operating procedures in section 4.4. 

In Fig. 3 the duration of the collection time period is shown by the horizontal line that extends 

between ∆t1 and ∆t2 . The shaded region, called the density overlap region, denotes all of the 

∆t12 (s)

ζρ

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0 20 40 60 80 100

80000 L/min

14317 L/min

∆t12 (s)

ζρ

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0 20 40 60 80 100

80000 L/min

14317 L/min
14 

plausible collection times consistent with the inventory mass cancellation technique. In practice, 

the percent density overlap parameter













−

−

=

max min

match min 100

ρ ρ

ρ ρ

ζ ρ (10) 

determines which of the manifold of possible collection times is used to calculate the mass flow. 

Here, ρ min and ρ max are the lower and upper limits of the density overlap region, and ρ match is 

the matched density. From the geometry in the figure, the collection time is 

∆t = ∆τ + ∆t12 (11) 

where the base time period, ∆τ , extends from the start of the first dead-end period to the start of 

the second dead-end period, and the time adjustment factor, ∆t12 ≡ ∆t2 − ∆t1 , is defined as the 

difference between the two time intervals, ∆t1 and ∆t2 , where the subscripts “1” and “2” 

indicate which of the two dead-end periods the time intervals occur. As shown in Fig 3., these 

time intervals persist for a fraction of their respective dead-end intervals. Although the duration 

of both ∆t1 and ∆t2 depend on ζ ρ, the time adjustment factor ( 12 ∆t ) should ideally have no 

dependence on the percent density overlap parameter. The almost uniform distribution of the 

time adjust factor shown in Fig. 4 confirms that it is nearly independent of ζ ρ

. In this figure the 

time adjust factor is measured at two flows, 14317 L/min and 80000 L/min, corresponding to 

collection times of approximately 109 s and 20 s respectively. The relative uncertainty of ∆t12

due to its dependence on ζ ρ is defined as the ratio of its standard deviation (

12 σ ∆t

) to the 

collection time (∆t ). For longer collection times (i.e., lower flows), ∆t increases while 12 σ ∆t

remains nearly fixed so that its relative uncertainty decreases. Thus, we selected two relatively 

large flows to determine an upper uncertainty bound. At the largest flow the relative uncertainty 

is t σ ∆t ∆ 12 = 5 × 10-6. This value is one of the contributing components for the collection time 

uncertainty discussed in section 5.2.2. 

5. Uncertainty of PVTt Subsidiary Components 

The mass flow determinations of a PVTt flow standard rely on accurate measurements of 

pressure, volume, temperature, and time and on the reference parameters Ru, M , and Z . In 

general, the largest uncertainties in mass flow can be attributed to the measurements of volume, 

temperature, and pressure. However, timing measurements can also play an important role near 

the maximum flow capacity of a PVTt system when collection times are shortest. For these short 

collections, the largest contribution from timing uncertainties is typically associated with timing 

errors introduced by the flow diversion processes. On the other hand, at the lower flow capacity 

the collection times are longer and timing measurements typically play only a minor role in the 

mass flow uncertainty budget. The reference parameters, M and Z , are well known for common 

gases (e.g., air, N2, CO2, Ar, He, etc), and contribute little to the mass flow uncertainty. 

In this section, the uncertainty of the various reference parameters and measured quantities are 

assessed. We begin with the reference parameters, followed by the timing system, the pressure 

and temperature measurements in the inventory volume, the size of the inventory volume, the 

pressure and temperature measurements in the collection tank, and the size of collection tank. 
15 

Throughout the document all of the uncertainty components are categorized as being either 

Type A (i.e., those which are evaluated by statistical methods) or Type B (i.e., those which are 

evaluated by other means) as described in [12]. Uncertainties having subcomponents belonging 

to both Type A and Type B are categorized as (A, B) as specified in [12]. 

5.1 Reference Parameters (M , Ru

, and Z )

5.1.1 Universal Gas Constant

The universal gas constant has a value of Ru = 8314.472 J/(kg·K) with a Type B relative 

standard uncertainty of [ ( ) ] u Ru Ru = 1.7 × 10-6 [11]. 

Table 4. Composition of dry air. 

Species Mole Fraction

(xk)

Nitrogen 0.780849 

Oxygen 0.209478 

Argon 0.00934 

Carbon Dioxide 0.000314 

Neon 1.82 × 10-5

5.1.2 Molecular Mass

In this work the value used for the molecular mass of dry air is M = 28.9647 kg/kmol. It is 

computed using the Refprop Thermodynamic Database [10] for the composition shown in 

Table 4. The relative molecular mass has two sources of uncertainty: 1) a Type B uncertainty 

attributed to the air moisture level, and 2) a Type A uncertainty resulting from the variation in 

the composition of dry air. The air moisture level is maintained below 3 % relative humidity 

(RH). Since the RH measurement is made under room temperature conditions at a nominal 

pressure of P = 800 kPa, the mole fraction of water vapor is 9.82 × 10-5 resulting in a relative 

standard uncertainty attributed to air moisture level of 37 × 10-6. 

Various references list slight difference in the composition of dry air at sea level [13-15]. We 

estimated that the relative standard uncertainty attributed to the variation in composition is 

35 × 10-6. Thus, propagation of these two uncertainty components yields a combined relative 

standard uncertainty of [ ( ) ]air u M M = 51 × 10-6. 

5.1.3 Compressibility Factor 

The compressibility factor is determined using the Refprop Thermodynamic Database [10] in 

conjunction with the corresponding measurements of pressure and temperature. The ranges of 

the pressures and temperatures in the collection tank differ from those in the inventory volume so 

that the uncertainties of the compressibility factors corresponding to these ranges also differ. In 

the collection tank the temperature ranges from 292.5 K to 298.5 K and the pressure ranges from 

0 kPa to 110 kPa. For this range of conditions the relative standard uncertainty of the 

compressibility factor is estimated to be no more than [ ( ) ] u ZT ZT =50 × 10-6 [10] for both the 
16 

initial and final conditions.. In the inventory volume the temperature ranges from 290 K to 340 K 

and the pressure ranges from 100 kPa to 450 kPa. For this range of conditions the relative 

standard uncertainty of the compressibility factor is conservatively estimated to be no more than 

[ ( ) ] u ZI ZI = 100 × 10-6. Both of these uncertainty components are Type B. 

5.2 Collection Time 

The collection time, defined previously in Eqn. (11), consist of the base time, ∆τ , and the time 

adjustment factor, ∆t12 . Applying the method of propagation of uncertainty [16] the 

corresponding collection time uncertainty is6

( ) ( ) ( ) 2

12

2 2 2

t

u u t

t t

u t













+ 























= 











∆

∆

∆τ

∆τ

∆

∆τ

∆

∆

(12) 

where the relative uncertainty of the time adjustment factor (the second term) is normalized by 

the collection time, ∆t , instead of ∆t12 . Moreover, since t t ∆ 12 << ∆ (as explained in section 4.5) 

the ratio ∆τ ∆t in the first term is close to unity. The total relative standard uncertainty for the 

collection time is [u(∆t) ∆t]=15 × 10-6. It is comprised of the following two components: 1) the 

base time measurement (2 × 10-6), and 2) the time adjustment factor (14 × 10-6). These 

components are itemized in Table 5 for a 20 s collection period (i.e., the shortest collection 

period used) and the uncertainty value of each is discussed here. The abbreviations in the table 

have the following meanings: Abs. Unc. is the Absolute Uncertainty, Rel. Std. Unc. is the 

relative standard uncertainty, Sens. Coeff. is the dimensionless sensitivity coefficient, Unc. Type 

is the uncertainty type, and Perc. Contrib. is the percent contribution to the combined uncertainty 

in collection time. 

Table 5. Collection time uncertainty for a 20 s collection. 

 Collection Time Uncertainty. Abs. 

Unc.

Rel. 

Std. 

Unc. 

(k=1)

Sens. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

 Collection time, ∆t = 20 s (ms) (× 10-6) (-----) (%) (-----) 

 Base time, ∆τ 0.04 2 ≈1 1.9 A Calib. of HP Counters 

 Time adjustment factor, 12 ∆t 0.29 14 1 98.1 B See section 5.2.2. 

 Combined Uncertainty 0.29 15 100 

5.2.1 Base Time Measurement

The base time spans the time interval from the beginning of the first dead-end interval to the 

beginning of the second dead-end interval. It is measured with a redundant pair of HP counters 

each having a relative standard uncertainty of 2 × 10-6. The redundancy provided by two counters 

helps prevent against erroneous time measurements should one of them malfunction. The HP 

counters are triggered by the voltage output of an electric circuit. A photodiode sensor aligned 

with the closed position of the bypass valve activates the electric circuit and starts the time 

measurement during the first flow diversion. In a similar manner, the time measurement is 

 

6

 For convenience all equations symbolically expressing uncertainty are given as the variances rather than standard 

uncertainties unless otherwise noted. 
17 

terminated during the second flow diversion by another photodiode sensor that produces a 

voltage signal when the tank valve reaches its fully closed position. Timing errors associated 

with misalignment of the triggering signal and the valve fully closed positions of either valve are 

inherently accounted for by the inventory mass cancellation technique. For example, if during 

the first flow diversion the triggering signal is set off prematurely before the bypass valve is fully

closed, the measured base time, ∆τ , will be slightly longer than its actual value. However, the 

measured time interval ∆t1 will be extended by the same amount so that the collection time as 

calculated by Eqn. (11) is invariant. Consequently, misalignment of the triggering signal does not 

contribute to the uncertainty. Nevertheless, proper mass accounting requires that the tank valve 

remain closed until the bypass valve is fully closed. 

5.2.2 Time Adjustment Factor

The time adjustment factor is a small correction that adjusts the time measurement to ensure 

mass cancellation in the inventory volume. The time adjustment factor is evaluated by taking the 

difference between the time intervals ∆t1 and ∆t2 . The first interval, ∆t1 , begins during the first 

diversion period when a photodiode is activated by the closing of the bypass valve. The 

photodiode triggers an electric circuit that in turn outputs a voltage signal that starts the time 

measurement. Similarly, the measurement of ∆t2 starts during the second flow diversion when 

the photodiode on the tank valve is activated by its closing. The duration of both ∆t1 and ∆t2

are based on the percent density overlap parameter, ζ ρ. In particular, measurements of pressure 

and temperature are used to calculate the density time histories during the 1st and 2nd dead-end 

intervals, and ζ ρ

 selects the particular matched density from the region of density overlap. Since 

the voltage, pressure, and temperature measurements used to determine the duration of ∆t1 and 

∆t2 are acquired by a data acquisition card sampling at 3000 Hz, the resolution of the calculated 

time intervals is limited to 0.33 ms. If a rectangular distribution is assumed, the standard 

uncertainties for both ∆t1 and ∆t2 equal 0.19 ms, so that for a 20 s collection the corresponding 

relative uncertainties are 10 × 10-6. 

The total uncertainty in ∆t12 consists of three components. These include the uncertainty 

attributed to ∆t1 (10 × 10-6), the uncertainty attributed to ∆t2 (10 × 10-6), and the uncertainty 

attributed to the uniformity of ∆t12 with ζ ρ

 (5 × 10-6) discussed previously in section 4.5. The 

first two are Type B uncertainties while the third is a Type A uncertainty. Propagation of these 

three components yields a total relative standard uncertainty for the time adjustment factor equal 

to 14 × 10-6.

5.3 Pressure and Temperature in the Inventory Volume 

The initial pressure measurement in the inventory volume is obtained by averaging the results of 

two fast pressure transducers during the first flow diversion. The final pressure is measured by 

averaging the readings of the same two transducers during the second flow diversion. The first 

transducer is positioned adjacent to the bypass valve and the second is located next to the tank 

inlet valve. The initial and final temperatures are determined by averaging the results of two 

type T thermocouples of 0.025 mm nominal diameter. The thermocouples are positioned 

adjacent to the pressure sensors, one next to the tank valve and the other next to the bypass valve. 
18 

Figure 5 shows the time histories for the pressure (left) and the temperature (right) during the 

first and second flow diversions for a nominal flow of 0.4 kg/s. This data is acquired using a data 

acquisition card sampling at 3000 Hz. The beginning of both the first and second dead–end 

intervals starts at t = 0 s. The pressure and temperature time traces begin at near ambient 

conditions, increase as mass accumulates into the inventory volume, and then sharply decrease as 

the accumulated mass is exhausted either into the nearly evacuated collection tank (i.e., 1st flow 

diversion) or to the bypass at ambient conditions (i.e., 2nd flow diversion). 

To capture the rapidly changing conditions in the inventory volume during flow diversions, both 

the pressure and temperature sensors must have a fast time response. The reading indicated by a 

slow sensor will lag behind the actual value. The error associated with a slow sensor can be 

predicted if the transducer time constant is known. The typical manufacturer specified time 

constant for the pressure transducer is Pτ = 3 ms. The thermocouple time constant depends on 

flow. In a previous work, the thermocouple time constant was measured to be 20 ms at a flow of 

1 g/s [17]. This value agreed to within 70 % of the theoretical value that was predicted using an 

empirical heat transfer coefficient corresponding to flow over a small diameter cylinder [18]. No 

attempt was made to obtain better agreement between the measured and predicted time constant 

since the net effect of the sensor time response has little impact on the mass flow uncertainty. In 

fact, we assumed that the thermocouple time constant remained fixed at 20 ms, instead of 

decreasing at larger flows as indicated by experimental and theoretical evidence [17, 18]. This 

assumption is also justified by the small impact of this parameter on flow uncertainty. 

Figure 5. Time histories of the inventory volume pressure (left) and temperature (right) during the first and second 

flow diversions for a nominal flow of 0.4 kg/s. (Data collected using fast pressure transducers and type T 

thermocouples.) 

Figure 5 shows the complete pressure and temperature time histories during diversion processes. 

However, only a small fraction of this time history is critical for computing mass flow. The 

inventory mass cancellation technique only requires the pressure and temperature data occurring 

within the density overlap region (see section 4.5). Given that the pressure and temperature time 

traces in this region are almost identical (i.e., a symmetric diversion process), and that the same

transducers are used to make both pairs of measurements, several of the sources of uncertainty 

are correlated. Moreover, the correlated quantities cancel almost completely when the inventory 

mass cancellation technique is implemented as part of the flow calibration process. If the 

275

285

295

305

315

-0.4 -0.2 0 0.2 0.4

T (K)

Density

Overlap

Region

t (s)

1

st Flow

Diversion

2

nd Flow

Diversion

0

50

100

150

200

250

300

-0.4 -0.2 0 0.2 0.4

P (kPa)

2

nd Flow

Diversion

Density

Overlap

Region

t (s)

1

st Flow

Diversion

275

285

295

305

315

-0.4 -0.2 0 0.2 0.4

T (K)

Density

Overlap

Region

t (s)

1

st Flow

Diversion

2

nd Flow

Diversion

0

50

100

150

200

250

300

-0.4 -0.2 0 0.2 0.4

P (kPa)

2

nd Flow

Diversion

Density

Overlap

Region

t (s)

1

st Flow

Diversion
19 

diversion process was asymmetric, these correlated uncertainties would not cancel, and the 

corresponding uncertainties from these components could increase significantly. 

Below we assess the uncertainty for pressure and temperature measurements in the inventory 

volume. Since the greatest inventory volume uncertainties occur at the largest flows, the analysis 

gives the uncertainties at the largest flow (77000 L/min). 

5.3.1 Initial Pressure in the Inventory Volume

The initial pressure uncertainty components are itemized in Table 6. These six components 

include 1) the calibration fit residuals, 2) the transducer mounting orientation 3) the response 

time of the sensor, 4) the spatial sampling error 5) the ambient temperature effect, and 6) the 

sensor repeatability. The first three sources of uncertainty are perfectly correlated since neither 

their sign nor magnitude change between the initial and final measurement. The remaining three 

sources of uncertainty are treated as uncorrelated. Propagation of the uncorrelated sources yields 

a total relative standard uncertainty of [ ]

i i

u PI PI

u ( ) = 2.3 %, while propagation of the correlated 

sources gives [ ]

i i

c PI PI

u ( ) = 4.0 %. The total relative standard uncertainty is obtained by 

propagating the uncorrelated and correlated sources, thereby yielding [ ]

i i

PI PIu( ) = 4.7 %. An 

evaluation for each of these uncertainty components is provided below, beginning with the 

uncorrelated sources and followed by the correlated sources. 

Table 6. Uncertainty of the initial pressure measurement in the inventory volume. 

 Uncertainty of initial inventory pressure Abs. 

Unc. 

Rel. 

Std. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type 

Comments

 Initial Pressure, P 190.9 kPa i

I = (kPa) (%) (%) A or B 

Uncorrelated Unc. 

 Spatial sampling error 4.41 2.3 24.6 B Meas. pres. Difference 

 Ambient temperature effects 0.12 0.1 0.0 B Manuf. spec. 

 Sensor repeatability 0.17 0.1 0.0 B Manuf. spec. 

Correlated Unc. 

 Calibration fit residuals 0.75 0.4 0.7 A End-to-end calibration to a pres. standard 

 Time response of Heise transducer 7.68 4.0 74.7 B Dead-End Flow Model 

 Transducer mounting orientation 0.0 0.0 0.0 B Always in same mounting position 

Propagation of Uncorrelated Sources 4.41 2.3 24.6 

Propagation of Correlated Sources 7.72 4.0 75.4 

Combined Uncertainty 8.89 4.7 100 

Among the three uncorrelated uncertainty components, the uncertainty attributed to spatial 

sampling errors is by far the largest. We determined this uncertainty experimentally while the 

other two uncertainty components (the ambient temperature effect and the sensor repeatability), 

were obtained via manufacturer specifications. Based on these specifications both of these 

components have relative standard uncertainties equal to 0.1 %. 

The sampling error is defined as the difference between the calculated average pressure (from the 

two Heise transducers) and the actual average pressure in the inventory volume. Sampling errors 

are caused by pressure gradients formed within the inventory volume during the dead-ended 

intervals. These pressure gradients are caused by two sources: 1) by the low-pressure jet 
20 

exhausting from the CFV stagnating against the closed tank and bypass valves, and 2) by the 

pressure impulse attributed to closing either the bypass valve (i.e., 1st dead-end interval) or the 

tank valve (i.e., 2nd dead-end interval) just prior to the start of the dead-end periods. Because the 

CFV mass flow, and the initial inventory pressures, and temperatures are nearly the same during 

the first and second dead-end intervals, the size and location of pressure gradients formed during 

these periods are expected to be similar and to some extent correlated. However, no attempt was 

made to assess the degree of correlation between the initial and final pressure fields. Instead, we 

conservatively treated the spatial sampling error as an uncorrelated uncertainty component. To 

this end, the initial and final spatial sampling errors are evaluated independently by two separate 

experiments. In each experiment we measured the pressure at the locations in the inventory 

volume where the largest pressure differences are expected. Pressure measurements are made at 

the exhaust of the CFV where we expect the lowest pressure, and adjacent to the bypass and tank 

inlet valves where the flow stagnates and the largest pressures are expected. At the maximum 

flow, the largest pressure difference between these locations is only 2.3 % of the initial average 

pressure, and the sampling error is defined equal to this pressure difference. 

The correlated uncertainties include the calibration fit residuals, transducer orientation, and the 

sensor response time. Experimental records show the relative standard uncertainty of the 

calibration fit residuals is 0.4 %. There is no uncertainty attributed to transducer orientation since

the sensors are calibrated and used in the same orientation. The time response of the sensor is 

estimated using a semi-empirical mathematical model. The model calculates the pressure 

increase during the dead-end interval assuming the process is isentropic. The isentropic pressure 

response is linearized over the dead-end period and used with the sensor time constant in a first 

order differential model to predict the pressure lag. This model is a simplified version of a more 

complex model given in [17]. Although this model is not as accurate, it gives reasonable results 

that are appropriate for the relatively minor importance of this uncertainty component. The 

predicted pressure lag of this simplified model is 

















− 















 +











= 1 1

V

m

P P

atm I

DE

DE

P

lag atm

γ

ρ

Γ

Γ

τ

∆

&

(13) 

where P

τ is the time constant of the Heise transducer, Γ DE is the effective dead-end interval 

(i.e., the actual dead-end period plus half the time required to close either the tank or bypass 

valve), γ =1.4 is the specific heat ratio for air, Patm ≈ 101.325 kPa is the initial atmospheric 

pressure in the inventory volume just before the start of the diversion process, ρatm ≈ 1.2 kg/m3

is the initial density under ambient conditions, and VI is the size of the inventory volume. From 

Eqn. (13), the pressure lag increases with increasing mass flow, longer dead-end intervals, and 

smaller inventory volume sizes. Experience indicates that the mass flow has the most significant 

effect since it varies significantly over the operating range of the PVTt flow standard. For 

example, at the largest flow (77000 L/min), the predicted pressure lag is 7.68 kPa, but makes 

only a negligible contribution at the lower flows (7000 L/min or below). 

5.3.2 Final Pressure in the Inventory Volume

Both the initial and final pressures are measured with the same transducers. Consequently, the 

final pressure uncertainty has the same six uncertainty components as the initial pressure. 

Moreover, each of the six uncertainty components has same uncertainty type (i.e., Type A or B) 
21 

as shown previously in Table 6. While the absolute values of uncertainty for these six 

components is the same for both the initial and final pressure measurements, the relative values 

can differ slightly attributed to differences between the initial and final pressures. The relative 

standard uncertainty of the correlated components of the final pressure include the calibration fit 

residuals (0.4 %), the transducer mounting orientation (0 %), and the sensor response time 

(4.1 %). The uncorrelated components include the spatial sampling error (2.3 %), the ambient 

temperature effect (0.1 %), and the sensor repeatability (0.1 %). The total uncorrelated 

uncertainty is [ ]

f

I

f

uu(PI) P = 2.3 % while the total correlated uncertainty is 

[ ]

f

I

f

uc(PI) P = 4.1 % so that the total uncertainty is [ ]

f

I

f

u(PI) P = 4.7 %. 

5.3.3 Initial Temperature in the Inventory Volume

The four uncertainty components for the initial temperature are categorized into uncorrelated and 

correlated components and are shown in Table 7. The uncorrelated components include the 

spatial sampling error and the thermocouple repeatability while the correlated uncertainties 

include the sensor time response and the correction for the moving fluid stagnating against the 

thermocouple surface (i.e., static versus stagnation). The total uncorrelated uncertainty is 

[ ( )]

i

TIuu

 = 6.0 K and the total correlated uncertainty is [ ( )]

i

TIuc

 = 33.6 K. The correlated and 

uncorrelated components are propagated to give a total relative standard uncertainty of 

[ ( )]

i

TIu = 34.1 K. 

Table 7. Uncertainty of the initial inventory temperature measurement. 

 Uncertainty of initial inventory temperature Abs. 

Unc. 

(k=1)

Rel. 

Std. 

Unc.

Perc. 

Contrib.

Unc. 

Type 

Comments

 Initial Temperature, T 345.2 K

i

I = (K) (%) (%) (A or B)

Uncorrelated Unc. 

 Temperature spatial sampling error 6.0 1.7 3.1 B Meas. Temp. difference 

 Repeatability 0.2 0.1 0.0 B Manuf. Spec. of thermistor used 

 for cold junction compensation 

Correlated Unc. 

 Thermocouple time response 33.3 9.5 95.0 B Dead-End Flow Model 

 Static vs. stagnation 4.7 1.4 1.9 B See section 5.3.3 

Propagation of Uncorrelated Sources 6.0 1.7 3.1 

Propagation of Correlated Sources 33.6 9.6 96.9 

Combined Uncertainty 34.1 9.7 100 

The spatial sampling error is determined experimentally by measuring the temperatures adjacent 

to the bypass and tank valves at the maximum flow (77000 L/min). The largest measured 

temperature difference is less than 6.0 K. The repeatability of the sensor is conservatively 

estimated to be 0.2 K, and the correction for the static temperature versus measured temperature 

is 4.7 K. The uncertainty attributed to the sensor time response is 33.3 K. It is calculated using 

















− 















 +











=

−

1 1

V

m

T T

1

atm I

DE

DE

T

lag atm

γ

ρ

Γ

Γ

τ

∆

&

(14) 
22 

where Tτ is the temperature time constant of the thermocouple, and Tatm ≈ 293.15 K is the initial 

ambient temperature in the inventory volume. This expression is based on the isentropic model 

discussed previously in section 5.3.1. 

5.3.4 Final Temperature in the Inventory Volume

The final temperature has the same uncertainty components as the initial temperature and the 

corresponding uncertainties types are the same. These include the spatial sampling error (6.0 K), 

the sensor repeatability (0.2 K), the sensor time response (34.0 K), and the dynamic correction 

for the static temperature measurement (4.7 K). The total uncorrelated uncertainty is 

[ ( )]

f

TIuu

 = 6.0 K, and the total correlated uncertainty is [ ( )]

f

TIuc = 34.3 K. The correlated and 

uncorrelated components are propagated to give a total relative standard uncertainty of 

[ ( )]

f

TIu = 34.8 K. 

5.4 Inventory Volume 

The size of the inventory volume is adjusted as necessary to accommodate the quantity of flow. 

Larger flows require larger inventory volumes to prevent the pressure rise during the dead-ended 

periods from unchoking the CFV. If the CFV unchokes, then the corresponding decrease in mass 

flow violates the steady state assumption used in deriving Eqn. (8), and thereby introduces 

additional uncertainty. Fortunately, the uncertainty in the size of the inventory volume does not 

play a significant role in uncertainty analysis. The inventory mass cancellation technique causes 

its corresponding sensitivity coefficient to be zero (see section 6.2), and consequently, the 

uncertainty attributed to the size of the inventory volume is also zero. The size of the inventory 

volume does however have a small effect on the overall mass flow uncertainty through its 

influence on the inventory pressure and temperature sensitivity coefficients. As such, reasonably 

accurate values must be used. We measure the size of inventory volume to within 25 % of its 

actual size using a tape measure as discussed previously in section 4.5. Since its value is obtained 

using only a single measurement it is a Type B uncertainty. 

Table 8. Uncertainty of the initial tank pressure. 

 Uncertainty of Initial Tank Pressure Abs. 

Unc. 

(k=1)

Rel. 

Std. 

Unc.

Perc. 

Contrib.

Unc. 

Type 

Comments

 Initial Tank Pressure, = 0.1 kPa i PT

(Pa) (× 10-6) (%) (-----) 

 Transducer Accuracy 0.125 1250 4.2 B Manuf. Spec. 

 Ambient Temperature Effect 0.160 1600 6.8 B Manuf. Spec. (0.04 % reading/oC from 22oC) 

 Drift from Cal. Records 0.557 5774 89.0 A <1 Pa per year, assume rect. 

 Spatial Gradients in Pressure 0.001 14 0.0 B Based on Hydrostatic Pressure Head 

Combined Uncertainty 0.612 6120 100 

5.5 Pressure and Temperature in the Collection Tank 

5.5.1 Initial Tank Pressure

The initial tank pressure is measured by averaging the result of two 1333.22 Pa (10 Torr) MKS 

capacitance diaphragm gages, each with a manufacturer specified relative uncertainty of 0.25 % 

taken to be at the 95 % confidence level. Additional uncertainties are attributed to ambient 

temperature effects, to zero drift, and to spatial pressure gradients in the tank. All of these 
23 

uncertainty components are shown in Table 8. The total relative standard uncertainty attributed 

to the initial pressure measurement is [ ]

i

T

i

u(PT) P = 6120 × 10-6

. 

5.5.2 Final Tank Pressure

The final pressure in the collection tank is measured using a Paroscientific Model pressure 

transducer with a full scale of 200 kPa. This transducer is calibrated at six month intervals using 

a Ruska piston pressure gauge whose piston area is traceable to the NIST Pressure and Vacuum 

Group [6]. The relevant uncertainty components for pressure are itemized in Table 9 including 

the calibration of the pressure transducer, (17 × 10

-6); the measured drift limit from calibration 

records, (60 × 10-6); the calibration fit residuals, hysteresis, and thermal effects, (100 × 10-6), and 

spatial gradients in the tank attributed to the hydrostatic pressure head (0.5 × 10-6). The 

propagation of these components yields a total relative pressure uncertainty of 

[ ]

f

T

f

u(PT) P = 118 × 10-6

. 

Table 9. Uncertainty of the final tank pressure. 

 Uncertainty of Final Tank Pressure Abs. 

Unc. 

Rel. 

Std. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type 

Comments

 Final Tank Pressure, P 59 .25 kPa f

T = (Pa) (× 10-6) (%) (A or B)

 Transfer standard for static pres. 1.6 17 2.1 B Ruska Piston Pres. Gauge 

 Drift from Cal. Records 5.7 60 25.9 A < 0.01 % in 6 months, assume rect. 

 Residual, hystersis, thermal effects 9.5 100 72.0 A From cal. records expts. 

 Spatial pressure gradients in Tank 0.05 0.5 0.0 B Based on hydrostatic pressure head 

Combined Uncertainty 11.2 118 100 

5.5.3 Initial and Final Average Gas Temperature in the Collection Tank

Both the initial and final gas temperatures are measured by averaging 37 thermistors distributed 

throughout the collection tank. Because the collection tank is initially evacuated, the sensitivity 

coefficient corresponding to the initial temperature is significantly lower than the final 

temperature. As a result the initial temperature measurement only requires marginal accuracy 

relative to the final temperature measurement. Therefore, significantly more effort is spent 

obtaining a low uncertainty final temperature measurement. In this analysis the standard 

uncertainty of the initial temperature measurement is [ ( )]

i

TTu = 1206 mK while the standard 

uncertainty for the final temperature is [ ( )]

f

T

u T = 64.6 mK. The various uncertainty components 

comprising the initial and final temperature measurements are evaluated below. 

The standard uncertainty components for the final temperature measurement are shown in 

Table 10. These components include the thermistor calibration transfer standard (1.2 mK), the 

uniformity of the temperature bath used for calibrations (1.0 mK), the standard deviation of the 

calibration fit residuals (7 mK), the manufacturer specified drift (28.9 mK), radiation and selfheating (1.8 mK), the thermistor time response (2.5 mK), and the spatial sampling error 

(57.3 mK). The most significant of these uncertainties are the spatial sampling error and the 

thermistor drift, which together contribute almost 99 % of the uncertainty. The uncertainty 

attributed to thermistor drift (28.9 mK) is obtained by dividing the manufacturer specified drift 

limit of 50 mK (taken to be at the ninety-five percent confidence level) by 3 as prescribed for a 

rectangular distribution. The uncertainty attributed to drift can be decreased, if necessary, by 
24 

calibrating the thermistors more frequently. The 50 mK drift limit is based on a five year 

calibration schedule (i.e., the drift rate is 10 mK/year). The five year interval was selected to 

avoid difficulties associated with retrieving the thermistors inside the collection tank. We may, in 

the future, select to calibrate the thermistors at shorter intervals to further reduce the uncertainty. 

Table 10. Uncertainty of the final tank temperature. 

 Uncertainty of Final Tank Temperature Abs. 

Unc. 

Rel. 

Std. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type 

Comments

 Final tank temperature, = 294 K

f

TT(mK) (× 10-6) (%) (A or B)

 Temperature transfer standard 1.2 4 0.0 B Traceable to NIST temperature group

 Uniformity of temperature bath 1.0 3 0.0 B Expt. varied position of Temp. Std. 

 Fit residuals 7.0 24 1.2 A Based on calibration data 

 Drift (I, R, DMM, thermistors) 28.9 98 20.0 B Manuf. spec < 50 mK/5 year, 

 assume rect distribution. 

 Radiation, self-heating 1.8 6 0.1 A Expt. varied current & calculated 

 Thermistor time response 2.5 8 0.1 B Est. based on theoretical model 

 Temperature spatial sampling error 57.3 195 78.6 A Expt. measured [1] 

Combined Uncertainty 64.6 220 100 

The spatial sampling error was determined experimentally by characterizing the size and decay 

rate of the temperature gradients in the gas after filling. Based on these measurements, a 

mathematical model was developed both to estimate the settling time necessary for the gas in the 

tank to thermally equilibrate and to determine the best arrangement of sensors in the collection 

tank [19]. When the ducted fan (see Fig. 1) is used to mix the gas, the settling time for thermal 

equilibrium is 2700 s. The final average gas temperature is calculated after this period using a 

volume weighted integration of all 37 thermistors. The spatial sampling error is taken to be the 

root-sum-square of the volume weighted temperature differences in three locations of the 

collection tank including 1) temperature differences near the fan motor, 2) temperature 

differences in the thermal boundary layer adjacent to the tank wall, and 3) temperature 

differences in far-field. 

The spatial sampling error has been reduced from 70.3 mK as given in a previous publication [19] 

to its current value of 57.3 mK. In the original calculation of the spatial sampling error, six 

months of temperature measurements indicated that the temperature difference in the region 

close to the fan motor was 500 mK, in the boundary layer region the temperature difference was 

250 mK, and in the far-field it was 62 mK. Given that the region near the fan motor accounts for 

0.6 % of the collection tank volume, the boundary layer region accounts for 20 %, and the farfield accounts for the remaining 79.4 %, the volume weighted temperature differences are 3 mK, 

50 mK, and 49.4 mK, respectively. A root-sum-square of these three components gave a 

sampling error of 70.3 mK. However, additional temperature measurements over the past three 

years have demonstrated that the uncertainty attributed to temperature differences in the 

boundary layer region can be reduced from 50 mK (as given in [19]) to its present value of 

28.9 mK. 

The original temperature characterization indicated that the temperature measurements in the 

boundary layer were sensitive to the degree of stratification in the room enclosing the PVTt flow 
25 

standard.7 Since the degree of stratification changed seasonally (i.e., from the winter to summer 

months), we used the worst case (i.e., 50 mK) as the temperature difference in the boundary layer. 

We conservatively used the largest temperature difference to avoid underestimating the sampling 

error, if for example future measurements showed a larger boundary layer effect due to seasonal 

temperature changes. However, additional temperature measurements suggest that 50 mK is a 

reasonable upper bound. Moreover, an array of 14 thermocouples distributed along the outer 

surface of the collection tank verifies for each calibration that temperature stratification never 

exceeds the worse case. Since 50 mK is the maximum possible value, we assume a rectangular 

distribution so that this value is divided by 3 and the standard boundary layer temperature 

difference is 28.9 mK. 

Several of the uncertainty components for the initial temperature measurement are identical to 

those of the final temperature measurement. These include the thermistor calibration transfer 

standard (1.2 mK), the uniformity of the temperature bath used for calibrations (1.0 mK), the 

calibration fit residuals (7 mK), and the manufacturer specified drift (28.9 mK). The remaining 

uncertainty components, including the thermistor time response (303 mK), the radiation and selfheating (167 mK), and the spatial sampling error (1155 mK), are all significantly larger than the 

final temperature measurement. These components are larger because the heat transfer 

mechanisms that affect the temperature measurements are drastically different between the initial 

and final conditions in the collection tank. In the final condition, the collection tank is at near 

atmospheric conditions and the heat generated by joule heating in the thermistor is dissipated 

mainly by natural convection processes. On the other hand, the tank is initially under vacuum 

conditions so that radiation is the only significant mode of heat transfer from the thermistor. The 

radiation heat transfer dissipates heat less effectively, and has a much lower heat transfer 

coefficient. The lower heat transfer coefficient results in the slower sensor time response and 

higher value of self-heating. In a similar manner, the poor mixing under vacuum conditions 

results in the larger spatial sampling error. 

5.6 Collection Tank Volume

The collection tank volume is determined using a gravimetric weighing procedure whereby a 

measured mass of gas is transferred into the initially evacuated collection tank. The volume of 

the tank is determined by dividing the mass of gas transferred into the tank by the change in gas 

density attributed to filling. When the gas that remains trapped in the volume of tubing 

connecting the supply gas to the tank is considered, the tank volume is

c

i

T

f

T

cyl

T V

M

V -

ρ - ρ

∆

= (15) 

where ∆Mcyl is the mass of gas transferred to the tank, iρT is the initial density measurement in 

the tank, f

T

ρ is the final density measurement in the tank, and Vc is the connecting volume 

between the gas source and the collection tank. 

 

7

 Stratification had little impact on the temperature differences in the far-field or near the fan motor. 
26 

Figure 6. Shows eight measurements (four with argon and four with nitrogen) based on gravimetric weighing 

technique used to determine the collection tank internal volume (VT ), and the standard deviation of 

repeated measurements (σ V ). 

The source of gas used for the volume determination was an array of high pressure gas cylinders. 

The mass of gas displaced into the collection tank was determined by subtracting the initial 

cylinder mass (before filling the tank) with the mass after the filling process. Both the initial and 

final masses are determined using a 600 kg Mettler IDS scale with a resolution of 0.0002 kg. The 

array of cylinders was connected to the tank by nylon tubing of diameter 6.35 mm (0.25 in) that 

served as the connecting volume. Before beginning the experiment, the collection tank was 

purged by repeatedly evacuating its contents and filling it with the source gas (i.e., either argon 

or nitrogen). The experiment began by determining the initial density of the gas in the nearly 

evacuated collection tank via temperature and pressure measurements. Following this, the 

collection tank is filled with the source gas until it reaches atmospheric pressure. The volume of 

the collection tank is determined at atmospheric pressure to match the condition that it is used 

during calibration, and to prevent leakage into or out of the collection tank while waiting for the 

gas to thermally equilibrate. Once equilibrium conditions are reached, the final gas density is 

determined via pressure and temperature measurements. Both the initial and final pressure and 

temperature measurements use the same instrumentation used during an actual calibration cycle 

(see section 5.5). 

As shown in Fig. 6, eight independent measurements were used to determine the collection tank 

volume. Four of the volume determinations used nitrogen as the source gas while the remaining 

four used argon. The tank volume, VT =25.8969 m3

, is the average of these eight measurements. 

The figure shows that this value for the tank volume compares well with the previously used 

value, differing by only 0.0036 %. This level of agreement is not unexpected since the tank 

volume has no reason to change. The standard deviation of repeated measurements is indicated 

by the two dashed lines (σ V VT = 318 × 10-6). The relative standard uncertainty attributed to 

25.880

25.885

25.890

25.895

25.900

25.905

25.910

25.915

1 2 3 4 5 6 7 8

VT

VT

Measurement No.

Tank Volume (m3

)

σV

old

345678

Nitrogen

Argon

25.880

25.885

25.890

25.895

25.900

25.905

25.910

25.915

1 2 3 4 5 6 7 8

VT

VT

Measurement No.

Tank Volume (m3

)

σV

old

345678

Nitrogen

Argon
27 

repeated volume measurements equals to the standard deviation of the mean, 

N

σV VT = 113 × 10-6

, 

where N = 8 are the eight repeated volume measurements. 

Table 11. Uncertainty of the collection tank volume. 

 Tank Volume Uncertainty Rel. 

Std. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

 Tank Volume, VT = 25.8969 m3 (× 10-6) (-----) (%) (A or B)

 Connecting volume, Vc = 128.4 cm3 58940 -5.0 × 10-6 0.0 B See Section 5.6.1 

 Initial tank density, i

ρT

=8.55 × 10-4 kg/m3

570.03

8988 7.4 × 10-4 0.1 A, B See Section 5.6.2 

 Final tank density, f

T

ρ =1.1620 kg/m3

675.2

250 -1.0 82.3 A, B See Section 5.6.3 

 Effects of Leaks 0 1.7 × 10-5 0.0 B See Section 5.6.4 

 Change in cylinder mass, ∆Mcyl = 30.0710 kg 28 1.0 1.0 A, B See Section 5.6.5 

 Std. dev. of the mean for the eight repeated 

 volume meas., V VT σ 8

113 1.0 16.6 A Four volume determinations 

 in Ar and four in N2

 Combined Uncertainty 276 100 

The expression for the relative standard uncertainty of the collection tank volume is determined 

by applying the law of propagation of uncertainty to Eqn. (15). When the uncertainty 

contribution from repeated measurements is included the resulting expression is 

( ) 2 2 2 2 2

( ) ( )

1

-1

1

1

















































+















 −

+





























 = +











f

T

f

T

T

c

f

T

i

cyl T

cyl

T

c

T

T

u

V

V

M

u M

V

V

V

u V

ρ

ρ

∆ ρ ρ

∆

2 2

( )

1

-1 















































+

















+

i

T

i

T

T

c

f

T

i

T

f

T

i

T u

V

V

ρ

ρ

ρ ρ

ρ ρ ( ) 2

c

c

2

T

c

V

u V

V

V

























+

2

8

















+

T

V

V

T

σ

(16) 

where the correlation between the initial and final densities, iρ T and f

T

ρ , due to their common 

dependence on the reference parameters M and Ru is negligible and has been ignored. 

Additionally, the correlation between i

ρ T and f

T

ρ attributed to calibration of the temperature 

sensors is negligible.8 The various uncertainty components and sensitivity coefficients are listed 

in Table 11. When Eqn. (16) is used to combine these uncertainty components the total relative 

standard uncertainty attributed to the collection tank volume is [ ( ) ] u VT VT = 276 × 10-6. The 

documentation for each of the components contributing to the overall uncertainty is discussed 

here. 

5.6.1 Connecting Volume

The volume of the nylon tube used to connect the high pressure gas cylinders to the collection 

tank was geometrically determined by measuring its internal diameter and length. Using this 

 

8

 The initial and final pressures are measured with different transducers so that there is no correlation attributed to 

their calibration. 
28 

method, the size of the connecting volume was 128.4 cm3 and its relative standard uncertainty is 

[ ( ) ] u Vc Vc = 58940 × 10-6. 

5.6.2 Initial Gas Density in the Collection Tank

The initial tank density was determined by using the measured pressure and temperature in 

conjunction with the equation of state for the appropriate gas (i.e., nitrogen or argon). The 

expression of uncertainty for the initial density is 

2 2

2 2 2 2

( ) ( ) ( ) ( ) ( ) ( )













 +











+

















+

















+

















=

















M

u M

R

u R

Z

u Z

T

u T

P

u u P

u

u

i

T

i

T

i

T

i

T

i

T

i

T

i

T

i

T

ρ

ρ (17) 

where the relative standard uncertainties for the universal gas constant (1,7 × 10-6) has been 

documented in previously in section 5.1.1, and the absolute temperature uncertainty (1206 mK) 

has been documented in section 5.5.3. The relative standard pressure (7998 × 10-6) is slightly 

larger than the value given in section 5.5.1 because a lower initial tank pressure was used for the 

volume determination. The relative standard uncertainties attributed to the molecular mass and 

compressibility factor are (11 × 10-6) and (10 × 10-6) respectively. The molecular mass 

uncertainty is attributed to gas impurities in the source gas, and the compressibility uncertainty is 

primarily due to the Refprop thermodynamic database used to evaluate the equation of 

state [6, 20]. The universal gas constant, initial gas pressure in the collection tank, molecular 

mass, and the compressibility factor are all Type B uncertainties. The temperature uncertainty, 

however, has uncertainty subcomponents of both Type A and B. The total relative standard 

uncertainty of the initial density measurement is [ ]

i

T

i

T

u(ρ ) ρ = 8988 × 10-6. Since it is comprised 

of uncertainty subcomponents of both Type A and B, it is categorized as being a Type A, B 

uncertainty. 

5.6.3 Final Gas Density in the Collection Tank

The uncertainty of the final gas density is determined using an expression analogous to Eqn. (17) 

with the corresponding uncertainties being of the same type. While the uncertainties in the 

universal gas constant, compressibility factor, and molecular weight are identical to the values 

used for the initial density, the uncertainties for the pressure and temperature measurements 

differ. The pressure and temperature uncertainties correspond to the uncertainties discussed 

previously in sections 5.5.2 and 5.5.3 for the final condition in the collection tank. The relative 

standard uncertainty in pressure is 118 × 10-6 while the standard absolute temperature uncertainty 

is 64.6 mK. The total relative standard uncertainty for the final gas density is 

[ ]

f

T

f

T

u(ρ ) ρ = 250 × 10-6, and its uncertainty type is A, B. 

5.6.4 Effect of Leaks on Collection Tank Volume Determination

During the volume determination, every precaution was taken to minimize the influence of leaks. 

The high pressure gas cylinders were checked for leaks before weighing using a soap solution. If 

no leaks were discovered, the array of cylinders was wiped dry and then allowed to sit over night 

to allow any remaining soap solution to evaporate. The next morning the initial mass of the 

cylinders was determined by averaging at least three separate weighings. If each of the three 

successive weighings decreased in value, additional weighings were performed to ensure that the 

cylinders were not leaking gas. 
29 

Before beginning the mass transfer into the collection volume, the connecting volume of nylon 

tubing was checked for leaks using a soap solution. If no leaks were found, the high pressure 

cylinders were emptied into the collection tank. After filling the tank, both the tank and the 

cylinders were near ambient pressure so that any leakage due to a pressure difference was 

minimal. Because of these precautions, leaks are not expected to make a meaningful 

contrinbution to the uncertainty of the collection tank volume. 

5.6.5 Mass Transferred into the Collection Tank

The mass of gas transferred from the high pressure cylinders into the collection tank is 

determined by weighing the cylinders before and after the collection tank is filled. A double 

substitution method is used to determine the initial and final mass of the cylinders. Based on this 

method, the expression for either the initial or final cylinder mass is 

















+ + −

















= −

sen

air

air cyl DS sen

ref

air Mcyl M ref 1 V O M 1

ρ

ρ

ρ

ρ

ρ

(18) 

where M ref is the total mass of the set of NIST traceable reference masses, M sen is one of the 

reference masses that is used as the sensitivity mass, and ODS is the correction factor 

corresponding to the double substitution method 

( ) ( )

( ) ref sen sen

cyl cyl sen ref ref sen

DS 2 O O

O O O O

O

−

+ − +

≡

+

+ +

(19) 

The correction factor consists of the four mass observations made using the 600 kg Mettler IDS 

scale. These four mass observations include the following: 1) Ocyl , the observed mass of the 

cylinder by itself, 2) Ocyl+sen , the observed mass of the cylinder and sensitivity weight together, 

3) Oref +sen , the observed mass of the reference mass and sensitivity weight together, and 

4) Oref the observed mass of the reference mass by itself. The remaining variables in Eqn. (18) 

account for air buoyancy forces. These variables include the density of the room air ( ρ air ); the 

densities of the stainless steel reference and sensitivity masses ( ρ ref and ρ sen respectively); and 

the external volume of the gas cylinders (Vcyl ). 

The difference between the initial and final cylinder masses equals the amount of gas displaced 

from the cylinders into the collection tank. When the initial and final masses are computed using 

Eqn. (18), the gas displaced from the cylinders into the collection tank is 

[ ] [ ]

















− + −

















= + −

ref

f

air

sen

f

DS

f

ref

ref

i

air

sen

i

DS

i Mcyl M ref O M M O M

ρ

ρ

ρ

ρ

∆ 1 1

( )

f

air

i +Vcyl ρ air − ρ

(20) 

where for these measurements the densities of the stainless steel reference and sensitivity masses 

are equal (i.e., ρref = ρ sen ). The uncertainty expression for the mass of gas transferred to the 

collection tank is
30 

=

















2

( )

cyl

cyl

M

u M

∆

∆

2

2

( )

































−































 +

i

air

i

air

cyl

i

cyl air

ref

i

air

cyl

i

sen

i

DS

i

ref u

M

V

M

M O M

ρ

ρ

∆

ρ

ρ

ρ

∆

2

2

( )

































−



































 +

+

f

air

f

air

cyl

f

cyl air

ref

f

air

cyl

f

sen

f

DS

f

ref u

M

V

M

M O M

ρ

ρ

∆

ρ

ρ

ρ

∆

2 2 2

( )

1 1 











































− −

















−

















+

sen

sen

ref

i

i air

DS

ref

f

f air

DS

cyl

sen

M

u M

O O

M

M

ρ

ρ

ρ

ρ

∆

2

2

( )



































































 +

−































 +

+

ref

ref

ref

i

air

cyl

sen

i

DS

i

ref

ref

f

air

cyl

sen

f

DS

f

ref u

M

M O M

M

M O M

ρ

ρ

ρ

ρ

ρ ∆

ρ

∆

( )

2 2

( )































 −

+

cyl

cyl

cyl

i

air

f

cyl air

V

u V

M

V

∆

ρ ρ

2 2

( )

1

















































−

















+

i

ref

i

ref

ref

i

air

cyl

i

ref

M

u M

M

M

ρ

ρ

∆

2 2

( )

1

















































−





















+

f

ref

f

ref

ref

f

air

cyl

f

ref

M

u M

M

M

ρ

ρ

∆

2

cyl

res

scale

M

M

2

















+

∆

2

∆Mcyl

+σ

(21) 

where the last term is the standard deviation of repeated measurements (consisting of the 

standard deviation of the repeated initial mass measurements divided by the average initial mass 

root-sum-squared with the standard deviation of final mass measurements divided by the average 

final mass) and the second to last term accounts for the resolution of the scale. The square of the 

sensitivity coefficient is two in this term to account for the uncertainty of scale resolution for 

both the initial and final weighings. The various uncertainty components are itemized in 

Table 12. The total relative standard uncertainty for the mass of gas transferred into the 

collection tank is [u(∆Mcyl ) ∆Mcyl]= 28 × 10-6. The value of uncertainty for each component is 

documented below. 

5.6.5.1 Initial and Final Reference Masses The set of NIST traceable reference masses ranges 

in value from 0.001 kg to 45 kg. The uncertainty differs for each mass in the set. For the 

combination of masses used for the initial weighing the relative standard uncertainty is 

[ ]

i

ref

i

u(M ref ) M = 0.69 × 10-6, and for the final weighing it is [ ]

f

ref

f

u(M ref ) M = 0.75 × 10-6. We 

made no effort to account for the decrease in uncertainty attributed to correlations between any 

common reference masses used for both the initial and final weighings. The relatively small 

impact of the mass measurement in the volume uncertainty warrants this simplified approach 

(see Table 11). 

5.6.5.2 Sensitivity Mass The same sensitivity mass is used for both the initial and final 

weighings. The sensitivity mass is NIST traceable and its relative standard uncertainty is 

[u(M sen ) M sen ]= 0.04 × 10-6. 
31 

5.6.5.3 Density of the Reference and Sensitivity Masses Both reference and sensitivity masses 

are made of stainless steel and have identical densities equal to 7950 kg/m3. The relative standard 

uncertainty for density of the reference mass (or sensitivity mass) is conservatively estimated to 

be [u(ρref ) ρref ]= 9200 × 10-6

. 

Table 12. Uncertainty of the mass of gas transferred into the collection tank. 

 Mass Transfer Uncertainty Rel. 

Std. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

 Mass Transfer, ∆Mcyl =30.0710 kg (× 10-6) (-----) (%) 

 Initial ref. mass, i M ref = 345.7532 kg 0.69 11.5 8.1 B See Section 5.6.5.1 

 Final ref. mass, f M ref = 315.6720 kg 0.75 -10.5 8.1 B See Section 5.6.5.1 

 Sensitivity mass, M sen = 1.0000454 kg 0.04 -2.63×10-4 0.0 B See Section 5.6.5.2 

 Density of ref. mass, ρ ref = 7950 kg/m3

9200 -1.63×10-4 0.3 B See Section 5.6.5.3 

 External vol. of cylinder, Vcyl = 0.27 m3

83942 -6.72×10-5 4.1 B See Section 5.6.4.4 

 Initial air density, i

air ρ =1.194 kg/m3

844 -8.79×10-3 7.1 A ,B See Section 5.5.4.5 

 Final air density, f

air ρ =1.186 kg/m3

616 9.26×10-3 4.2 A ,B See Section 5.6.5.5 

 Scale Resolution, res M scale =0.2 g 6.7 1.41 11.4 A Cal. records Expts. 

 Std. dev. of repeated measurements 21 1 56.7 A Three or more meas. 

 Combined Uncertainty 28 100 

5.6.5.4 External Cylinder Volume The measurement of external volume of the array of

cylinders required only marginal accuracy since its sensitivity coefficient is small. The small 

sensitivity coefficient results because the buoyant forces are nearly identical during the initial 

and final cylinder weighings. As observed in Eqn. (20), when the air density is the same during 

the initial and final cylinder weighings, the buoyancy forces completely cancel. Since only 

marginal accuracy is necessary, this volume is measured using a tape measure. The external 

cylinder volume is estimated to be 0.27 m3, and the relative standard uncertainty is 

[u(Vcyl ) Vcyl]= 83942 × 10-6 or 8.4 %. 

5.6.5.5 Initial and Final Air Density for Buoyancy Correction The air density was determined 

by measuring the pressure, temperature, and relative humidity in conjunction with the curve 

fit [21] 

( ) 2 sat

1

air P b RH P

T

b

 −











ρ = (22) 

where Psat 1.7526 10 exp( ) 5315.56 T

11

= × − is the saturation pressure, and the values of the 

coefficients are b1 = 0.0034848 K·Pa-1kg/m3 and b2 = 0.003796 respectively. For relative 

humidities between 0 percent and 100 percent, at ambient pressures, and temperatures ranging 

from 290 K to 300 K, the relative difference between the densities calculated using Eqn. (22) and 

the Refprop thermodynamic database was less than 300 × 10-6. Assuming a rectangular 

distribution, the relative standard uncertainty of Eqn. (22) is 173 × 10-6. The additional 

uncertainty components for the initial air density are shown in Table 13, and the total relative 
32 

standard uncertainty is [ ]

i

air

i

air u(ρ ) ρ = 844 × 10-6. The major sources of uncertainty are 

attributed to the measurement of temperature and relative humidity. The temperature uncertainty 

is primarily due to temperature drift in the room during the weighing procedure, and the relative 

humidity uncertainty results from uncertainty in its calibration as well as instrument drift during 

the measurement. The uncertainty components for the final air density have values comparable to 

the corresponding initial values, and corresponding components have the same uncertainty type. 

The total relative standard uncertainty for the final air density is [ ]

f

air

f

air u(ρ ) ρ = 616 × 10-6

. 

Table 13. Uncertainty of the initial air density for buoyancy correction. 

 Initial Air Density Uncertainty Rel. 

Std. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

Initial air density, i

ρ air = 1.194 kg/m3

 (× 10-6) (-----) (%) (A, B) 

 Equation of state for moist air 173 1 4.2 B Comparison w/ Refprop [10] 

 Relative humidity, i RHair = 20.9 percent 169750 -1.8 × 10-3 14.0 B Cal. residuals of RH sensor 

 Initial room temperature, i

air T = 293.24 K 712 -1.0 76.2 A, B Mainly temp. drift during meas.

 Initial room pressure, i Pair = 100.654 kPa 200 1.0 5.6 A, B Mainly pres. drift during meas. 

 Combined Uncertainty 844 100 

6. PVTt Mass Flow Uncertainty

At the most basic level, a PVTt flow measurement system involves measuring mass and time. In 

particular, the accumulated mass in the inventory volume and in the collection tank is measured 

over the collection period. For a general PVTt system, the time-averaged CFV mass flow was 

given previously in Eqn.(8), but repeated here for convenience 

t

M M

m

T I

∆

∆ + ∆

& = (8) 

where the uncertainty attributed to leaks is omitted in the calculation of mass flow but considered 

in the uncertainty analysis. The expression for the mass flow uncertainty is determined by 

applying the method of propagation of uncertainty to Eqn. (8). The resulting expression of 

uncertainty is 

( ) ( ) ( ) 2 2

T

T

2

T

2

I

I

2

I

2

t

u t

M

u M

M

M

M

u M

M

M

m

(u m )













+ 























+ 























= 











∆

∆

∆

∆

∆

∆

∆

∆

∆

∆

&

&

2

leak

leak

m

(u m )













+

&

&

( ) ( )





























+ 



































+

2

u

u

2

I T

R

u u R

M

M

M

M

2

M

M

∆

∆

∆

∆ 2 +σ ss

(23) 

where the last term is the uncertainty attributed to the steady state assumption, and the second to 

last term accounts for the correlation between u(∆M I ) and u(∆MT ) attributed to the parameters 

M and Ru which are common to both ∆M I and ∆MT (i.e., see Eqns. 7a and 7b). Since the 

inventory mass cancellation technique (section 4.5) ensures that ∆M I is identically zero, the 

relative uncertainty of [u(∆M I ) ∆M I ] is infinite, making it impractical to use. By canceling the 
33 

repeated occurrences of ∆M I in the numerator and denominator of the first term and observing 

that the correlated terms vanish as ∆M I tends to zero we obtain 

( ) ( ) ( ) 2 2

T

T

2

T

I

2

t

u t

M

u M

M

u M

m

(u m )













+ 











+ 











= 











∆

∆

∆

∆

∆

∆

&

&

2

leak

leak

m

(u m )













+

&

&

2 +σ ss (24) 

the appropriate mass flow uncertainty corresponding to the inventory matching technique where 

we note that inventory mass uncertainty is normalized by ∆MT and the sensitivity coefficient 

for the tank mass uncertainty is unity (i.e., ∆MT = ∆M ). 

Table 14. Uncertainty of the CFV mass flow. 

PVTt Mass Flow Uncertainty Rel. 

Std. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

 CFV mass flow, m& =30.0710 kg (× 10-6) (-----) (%) (A, B)

 Tank mass accumulation, ∆MT =29.2054 kg 379 1 73.1 A, B See Section 6.1 

 Inv. Vol. mass accumulation, ∆M I = 0 kg 206 1 21.7 A, B See Section 6.2 

 Collection time, ∆t = 20 s 15 1 0.1 A, B See Section 5.2 

 Steady state assumption 100 1 5.1 B See Section 6.4 

 Leaks 0 1 0.0 B See Section 6.3 

 Combined Uncertainty 443 100 

The five uncertainty components shown in Eqn. (24) are itemized in Table 14. Although the 

inventory mass cancellation technique ensures that ∆M I = 0, as observed in the table, its 

uncertainty is not zero. The most significant contribution to the mass flow uncertainty (i.e., more 

than 73 %) stems from measuring the mass accumulated in the collection tank. When Eqn. (24) 

is used to combine all of the uncertainty components shown in the table, the relative standard 

uncertainty for mass flow is [u(m& ) m&] = 443 × 10-6, and the expanded relative uncertainty is 

[Uexp (m& ) m&] = 0.09 % (i.e., k = 2). A detailed explanation of each uncertainty component is 

given below. 

6.1 Accumulated Mass in the Collection Tank 

The mass accumulation in the collection tank is determined volumetrically using Eqn. (7a) 

derived in section 4.2 and repeated here for convenience 

( ) T i

T

i

T

i

T

f

T

f

T

f

T

T u V

Z T

P

Z T

P

M R

















∆ = M − .

(7a) 

The corresponding uncertainty is 

( )

= 











2

T

T

M

u M

∆

∆ ( ) ( ) 2 2













+ 











u

u

R

u u R

M

M

































+

















+

































+

2 2 2 2

( ) ( ) ( )

i

T

i

T

i

T

i

T

i

T

i

T

T

i

T

Z

u

T

u T

P

u P

M

M Z

∆

(25) 
34 





































+

















+

































+

2 2 2 2

( ) ( ) ( )

f

T

f

T

f

T

f

T

f

T

f

T

T

f

T

Z

u Z

T

u T

P

u P

M

M

∆

( ) 2

T

T

V

u V













+

and each component is itemized in Table 15. The uncertainties in the initial pressure, temperature, 

and compressibility factor play a reduced role in the uncertainty analysis since their 

corresponding sensitivity coefficients are much less than unity. More than 86 % of the 

uncertainty results from determining the tank volume and the average final tank temperature. 

Details for the uncertainty of each of these components are located in sections 5.1 to 5.5. The 

total relative standard uncertainty for the mass accumulated in the collection tank is 

[u(∆MT ) ∆MT ]= 379 × 10-6. 

Table 15. Uncertainty of the mass accumulation in the collection tank.

 Uncertainty of mass accumulation in the tank Rel. 

Std. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

 Tank mass accumulation, ∆MT = 29.2054 kg (× 10-6) (-----) (%) (A, B) 

 Tank volume, VT =25.8969 m3 276 1 53.0 A, B See Table 11 

 Tank initial temp., i

T

T =293 K 4115 0.001 0.0 A, B See Section 5.5 

 Tank final temp., f

T

T =294 K 220 -1.001 33.7 A, B See Table 10 

 Tank Initial pres., i

T

P =0.1 kPa 6120 0.001 0.0 B See Table 8 

 Tank final pres., f

T

P =95.246 kPa 118 -1.001 9.7 A, B See Table 9 

 Tank initial compressibility., i

T

Z =1 50 0.001 0.0 B See Section 5.1.3 

Tank final compressibility., f

T

Z =1 50 -1.001 1.8 B See Section 5.1.3 

 Molecular mass, Mair =28.9647 g/mol 51 1 1.8 A, B See Section 5.1.2 

 Univ. gas const., Runiv =8314.472 J/kmol·K 1.7 -1 0.0 B See Section 5.1.1 

 Combined Uncertainty 379 100 

6.2 Accumulated Mass in the Inventory Volume 

As previously defined in Eqn. (6b), and repeated here for convenience, the mass accumulated in 

the inventory volume equals the density change in this volume multiplied by the size of the 

inventory volume 

∆M I = ∆ρ I VI.

(6b) 

where i

I

f

I ∆ρ I ≡ ρ − ρ is the density change between the initial and final density. Using the 

method of propagation of uncertainty, the uncertainty in the mass accumulated in the inventory 

volume is 

( ) ( ) ( ) 2

I

I

2

T

I

2

T

I

2

T

I

2

T

I

V

u V

M

u M

V

V

M

u M

























+ 























= 











∆

∆

∆ρ

∆ρ

∆

∆ (26) 

where the inventory mass cancellation technique ensures that the second term is identically zero 

(i.e., ∆M I = 0). The uncertainty attributed to the density change in the inventory volume is 
35 

normalized by the density change in the collection tank, ∆ρT , instead of ∆ρ I =0, to avoid the 

singularity that would result from dividing by zero. Table 16 itemizes these components and 

shows that the relative standard uncertainty for the inventory volume mass accumulation is 

[u(∆M I ) ∆MT ]= 206 × 10-6. Since the size of the inventory volume contributes no uncertainty, 

all of the uncertainty derives from the density change. 

Table 16. Uncertainty components for the inventory volume mass accumulation.

 Uncertainty of Inv. Vol. Density Difference Rel. 

Std. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

 Mass accumulation in inv. vol., ∆MI =0 kg (%) (-----) (%) (A, B) 

 Density diff. in Inv. Vol., I∆ρ =0 kg/m3

68162 0.003 100 A, B See Section 6.2 

 Inv vol. size, IV =0.078 m3

250000 0 0 B See Section 5.4 

 Combined Uncertainty 206 100 

The uncertainty of the density difference in the inventory volume, ∆ρ I, is less than the 

uncertainty of either i

I

ρ or 

f

I

ρ individually. The lower uncertainty results from the cancellation 

of correlated sources of uncertainty between the initial and final inventory volume densities. The 

correlated sources between i

I

ρ or 

f

I

ρ can be evaluated in a straight forward manner if the 

density difference is expressed in terms of pressures, temperatures, etc. as given previously in 

Eqn.(7b)

( ) 















= −

i

I

i

I

i

I

f

I

f

I

f

I

I u

Z T

P

Z T

P

∆ρ M R . (7b) 

By applying the propagation of uncertainty the uncertainty of the density difference is 

( ) [ ] ( ) [ ] ( ) [ ] ( ) 2

tot Inv

2

tot Inv

2

tot Inv

2

T

I

u P u T u Z

u

= + + 











∆ρ

∆ρ ( ) ( )





























+ 



























+

2 2 2

u

u

T

I

R

u u R

M

M

∆ρ

∆ρ

(27)

where the inventory mass cancellation technique ensures that the last term is zero. The first three 

terms are the total uncertainty for the initial and final pressure measurements, the total 

uncertainty of the temperature measurements, and the total uncertainty of the compressibility 

factor, respectively. The uncertainty of each of these terms is discussed here. 

6.2.1 Total Pressure Uncertainties in the Inventory Volume

The total pressure uncertainty in the inventory volume is 

[ ] ( )





































+

















+

















−

















=

2 2 2 2

2 ( ) ( ) ( ) ( )

f

I

f

u I

i

I

i

u I

f

I

f

c I

i

I

i

c I

T

i

I

tot Inv P

u P

P

u P

P

u P

P

u P

u P

∆ρ

ρ

(28)

where the initial correlated and uncorrelated uncertainties are given in section 5.3.1 as 

[ ]

f

I

f

uc

(PI) P = 4.0 % and [ ]

i

I

i

uu(PI) P = 2.3 % respectively, and the final correlated and 

uncorrelated uncertainties are given in section 5.3.2. as [ ]

f

I

f

uc

(PI) P = 4.1 % and 

[ ]

f

I

f

uu(PI) P = 2.3 % respectively. The total pressure uncertainty is [ ( )] tot P Inv u = 5.5 %. 
36 

Table 17. Uncertainty components for the inventory volume density difference.

 Uncertainty of Density Difference Rel. 

Std. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type

Comments

 Density Difference, ∆ρI =0 kg/m3

(%) (-----) (%) (A, B)

 Total Inv. Vol. pressure uncertainty 4.1 1 35.5 A, B See section 6.2.1 

 Total Inv. Vol. temperature uncertainty 5.5 1 64.5 A, B See section 6.2.2 

 Total Inv. Vol. compressibility uncertainty 0 1 0.0 B See section 6.2.3 

 Molecular mass, Mair =28.9647 g/mol 0.0051 0 0.0 A, B See section 5.1.2 

 Univ. gas const., Runiv =8314.472 J/(kg·K) 0.0002 0 0.0 B See section 5.1.1 

 Combined Uncertainty 6.8 100 

6.2.2. Total Temperature Uncertainties in the Inventory Volume

The total temperature uncertainty in the inventory volume is 

[ ] ( )





































+

















+

















−

















=

2 2 2 2

2 ( ) ( ) ( ) ( )

f

I

f

u I

i

I

i

u I

f

I

f

c I

i

I

i

c I

T

i

I

tot Inv T

u T

T

u T

T

u T

T

u T

u T

∆ρ

ρ

(29)

where the initial correlated and uncorrelated temperature uncertainties are given in section 5.3.3. 

as [ ]

i

I

i

uc(TI) T = 9.6 % and [ ]

i

I

i

uu(TI) T = 1.7 % respectively, and the final correlated and 

uncorrelated uncertainties are given in section 5.3.4. as [ ]

f

I

f

uc(TI) T = 9.7 % and 

[ ]

f

I

f

uu(TI) T = 1.7 % respectively. The total relative temperature uncertainty is 

[ ( )] tot Inv u T = 4.1 %. 

6.2.3 Total Compressibility Factor Uncertainties in the Inventory Volume

The uncertainty in the compressibility factor consists only of correlated uncertainty components. 

Any uncorrelated uncertainty components are fossilized as correlated components by the curve 

fit used to represent the experimental data [16]. Since the initial and final inventory 

thermodynamic conditions are nearly identical, these correlated components completely cancel 

so that the net uncertainty is zero. 

6.3 Effect of Leaks 

The influences of leaks on PVTt flow measurements are most pronounced at the lowest flows 

(200 L/min). At low flows, the collection time is longer so that the sub-atmospheric pressures in 

the collection tank and inventory volume persist for a longer duration, and leaks makeup a larger 

fraction of the accumulated mass. To avoid this situation, the FMG inspects its flow standard for 

leaks prior to each calibration. If the source of a leak cannot be identified, the size of the leak is

estimated by multiplying the rate of density increase from an initially evacuated collection tank 

and/or inventory volume to the appropriate volume. The measured leak rate is then included in 

reported uncertainty of a given calibration. For the purposes of this document the uncertainty 

attributed to leaks is assumed to be zero. 
37 

6.4 Uncertainty Attributed to the Steady Flow Assumption

In developing the expression for the measured mass flow (i.e., Eqn. 8 or 9) we assumed that the 

flow entering the CFV remained steady for the entire collection period. However, in practice 

steady flow conditions at the CFV inlet are never perfectly attained. Instead, the PID controller 

used to set the flow maintains pseudo steady state conditions, whereby the flow fluctuates about 

a fixed baseline. Here we propose a conservative method for estimating the uncertainty 

associated with these fluctuations. 

The mass flow through a choked CFV under steady flow conditions is [22] 

u o

o t s d

R T

P A C C

mCFV

M & = (30) 

where Po is the stagnation pressure, To is the stagnation temperature, At is the throat area, Cs

is the critical flow function, and Cd is the discharge coefficient. Steady flow conditions are 

obtained by maintaining both Po and To constant throughout the collection period. However, 

Eqn. (30) can still be used under pseudo steady state conditions (i.e., small fluctuations in Po

and To) if mCFV & is time-averaged over the collection period. In this case, an estimate of the 

uncertainty attributed to unsteady effects is taken to be the standard deviation of the mass flow 

over the collection period ( mCFV

σ &). A typical value for the relative standard uncertainty 

attributed to unsteady effects is [ mCFV mCFV

& σ &

] = 100 × 10-6. 

In calculating both the average mass flow and its standard deviation we assume that Cd is 

unaffected by small fluctuations in Po and To. Physically, the discharge coefficient corrects for 

boundary layer effects along the CFV wall and for curvature of the sonic line near the CFV 

throat [23]. A small change in either Po or To does not significantly alter the thickness of the 

boundary layer or change the shape of the sonic line so that the changes in Cd are of second 

order. These second order effects can be neglected when assessing the uncertainty attributed to 

steady state assumption. 

7. Summary

This document addresses the flow measurement capabilities of the 26 m3PVTt system, the 

United States primary standard for measuring the flow of dry air. Flow measurements are 

conducted at ambient temperature and at pressures up to 800 kPa, and the flow range extends 

from 200 standard L/min to 77000 standard L/min. This document explains the function of the 

various components comprising the PVTt system, develops the theoretical basis for PVTt mass 

flow measurements, explains the underlying principles for its operation, spells out the operating 

procedures used for flowmeter calibrations, provides details necessary for customers wanting to 

submit a meter for calibration (i.e., pipeline sizes and available fittings, cost, turnaround time, 

etc.), and gives a detailed analysis of the uncertainty of mass flow measurements. 

The uncertainty for mass flow is assessed using the method of propagation of uncertainty [16]. 

The analysis shows that the expanded uncertainty of mass flow is 0.09 % with a coverage factor 

of two. The various uncertainty components are itemized in sections 5 and 6. For convenience a 

summary of the primary uncertainty components is given in Table 18. The largest components of 
38 

uncertainty are attributed measuring the collection tank volume and the final (i.e., after filling) 

temperature of the gas in the collection tank. Together these contribute more than 60 % of the 

overall uncertainty. Any future uncertainty reductions are likely to focus on improving the 

accuracy of these measurements. 

Table 18. Uncertainty of the PVTt mass flow.

PVTt Mass Flow Uncertainty Rel. 

Std. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

 CFV mass flow, m& =30.0710 kg (× 10-6) (-----) (%) (A, B)

Collection Tank Uncertainties 

 Tank volume, VT = 25.8969 m3

276 1 38.7 A, B See Table 11 

 Tank initial temp., i

T

T = 293 K 4115 0.001 0.0 A, B See Section 5.5 

 Tank final temp., f

T

T = 294 K 220 -1.001 24.7 A, B See Table 10 

 Tank initial pres., i

T

P = 0.1 kPa 6120 -0.001 0.0 A, B See Table 8 

 Tank final pres., f

T

P = 95.246 kPa 118 1.001 7.1 A, B See Table 9 

 Tank initial compressibility, i

T

Z = 1 50 0.001 0.0 B See Section 5.1.3 

 Tank final compressibility, f

T

Z = 1 50 -1.001 1.3 B See Section 5.1.3 

Inventory Volume Uncertainties 

 Total Inv. Vol. Pres. Unc., i

PI

 & f

PI

166 1 14.0 A, B See section 6.2.1 

 Total Inv. Vol. Temp. Unc., i

I

T & f

I

T 123 1 7.7 A, B See section 6.2.2 

 Total Inv. Vol. Comp. Unc., i

I

Z & f

I

Z 0 1 0.0 B See section 6.2.3 

 Inv vol. size., IV =0.078 m3

250000 0 0.0 B See Section 5.4 

Reference Properties 

 Molecular mass, Mair =28.9647 g/mol 51 1 1.3 A, B See Section 5.1.2 

 Univ. gas const., Runiv =8314.472 J/kmol·K 1.7 -1 0.0 B See Section 5.1.1 

Timing Uncertainties 

 Collection time, ∆t = 20 s 15 1 0.1 A, B See Section 5.2 

Unsteady Effect and Leaks 

 Steady state assumption 100 1 5.1 B See Section 6.4 

 Leaks 0 1 0.0 B See Section 6.3 

 Combined Uncertainty 443 100 

The 26 m3PVTt system and both of its smaller counterparts (i.e., the 34 L and the 677 L PVTt

flow standards) all implement an inventory mass cancellation technique to reduce the uncertainty 

of the dynamic pressure and temperature measurements made in the inventory volume. The 

technique works by ensuring pre-filling and after-filling thermodynamic conditions in the 

inventory volume are nearly identical. In this way many of the correlated uncertainties associated 

with using the same instrumentation to measure nearly the same conditions cancel each other, 

thereby making little or no contribution to the overall uncertainty. Finally, all three PVTt systems 
39 

are completely automated and able to perform calibrations overnight and on weekends, thereby 

expediting turnaround time for our customers. 
40 

REFERENCES 

[1] International Organization for Standardization, International Vocabulary of Basic and 

General Terms in Metrology, 2nd edition, 1993. 

[2] Brain T. J. S., Macdonald, L. M., Evaluation of the Performance of Small Scale Critical 

Flow Venturis Using the NEL Gravimetric Gas Flow Standard, Technical Paper: B-3. 

[3] Wright, J. D., and Mattingly, G. E., NIST Calibration Services For Gas Flow Meters: 

Piston Prover and Bell Prover Gas Flow Facilities, NIST SP 250-49, 1998. 

[4] Todd, D. A., NPSL Method for Calibrating Bell Provers, Technical Report: Navy Primary 

Standards Laboratory Code 4.1.4.5.0, 1996. 

[5] Olsen, L. and Baumgarten, G., Gas Flow Measurement by Collection Time and Density in a 

Constant Volume, Flow: Its Measurement and Control in Science and Industry, ISA, (1971), 

pp. 1287 - 1295. 

[6] Wright, J. D., Johnson, A. N., and Moldover, M. R., Design and Uncertainty Analysis for a 

PVTt Gas Flow Standard, J. Res. Natl. Inst. Stand. Technology 108, 21-47 (2003) 

[7] Wright, J. D., What Is the “Best” Transfer Standard for Gas Flow?”, FLOMEKO, 

Groningen, the Netherlands, May, 2003. 

[8] Wright, J. D., The Long Term Calibration Stability of Critical Flow Nozzles and Laminar 

Flowmeters, National Conference of Standards Laboratories Conference Proceedings, 

Albuquerque, NM, USA, pp. 443-462, 1998. 

[9] Marshall, J. L., NIST Calibration Services Users Guide 1998, NIST Special Publication 250, 

January, 1998. 

[10] Lemmon, E. W., McLinden, M. O., and Huber, M. L., Refprop 23: Reference Fluid 

Thermodynamic and Transport Properties, NIST Standard Reference Database 23, 

Version 7, National Institute of Standards and Technology, Boulder, Colorado, 2002. 

[11] Moldover, M. R., Trusler, J. P. M., Edwards, T. J., Mehl, J. B., and Davis, R. S., 

Measurement of the Universal Gas Constant R Using a Spherical Acoustic Resonator, NIST 

J. of Res., 93, (2), 85–143, 1988. 

[12] B. N. Taylor and C. E. Kuyatt, Guidelines for the Evaluating and Expressing the 

Uncertainty of NIST Measurements Results, NIST TN-1297, Gaithersburg, MD: NIST 

1994. 

[13] J. Hilsenrath, C. W. Beckett, W. S. Benedict, L. Fano, H. J. Hoge, J. F. Masi, R. L. Nuttall, 

Y. S. Touloukian, H. W. Wooley, Tables of Thermal Properties of Gases, U.S. Department 

of Commerce NBS Circular 564, 1955. 

[14] R. C. Weast, CRC Handbook of Chemistry and Physics, CRC Press Inc., 58th edition, Ohio, 

1977. 

[15] F. T. Mackenzie and J. A. Mackenzie, Our changing planet. Prentice-Hall, Upper Saddle 

River, NJ, p 288-307, 1995 (After Warneck, 1988; Anderson, 1989; Wayne, 1991.) 

[16] Coleman, H. W. and W. G. Steele, Experimentation and Uncertainty Analysis for Engineers, 

John Wiley and Sons, 2nd edition, 1999. 
41 

[17] Wright J. D. and Johnson, A. N., Uncertainty in Primary Gas Flow Standards Due to Flow 

Work Phenomena, FLOMEKO, Salvador, Brazil (2000). 

[18] A. Bejan, Convection Heat Transfer, John Wiley and Sons, 1st edition 1984. 

[19] A. N. Johnson, J. D. Wright, M. R. Moldover, P. I. Espina, Temperature Characterization 

in the Collection Tank of the NIST 26 m3 PVTt Gas Flow Standard, Metrologia, 2003, 40, 

211-216. 

[20] E. W. Lemmon, R. T. Jacobsen, S. G. Penoncello, and D. G. Friend, Thermodynamic 

Properties of Air and Mixtures of Nitrogen, Argon, and Oxygen from 60 to 2000 K at 

Pressures to 2000 MPa, J. Phys. Chem. Ref. Data 29, (3), 331-362, (2000). 

[21] Wagner, W. and Pruss, A., The IAPWS Formulation 1995 for the Thermodynamic 

Properties of Ordinary Water Substance for General and Scientific Use, J. Phys. Chem. Ref. 

Data, 31(2): 387-535, 2002. 

[22] ISO 9300: (E)., Measurement of Gas Flow by Means of Critical Flow Venturi Nozzles, 

Geneva Switz, 1990. 

[23] Johnson, A. N., Numerical Characterization of the Discharge Coefficient in Critical 

Nozzles, Ph.D. Dissertation, Pennsylvania State Univ., College Park, PA, 2000. 
NIST Test Number: 836-270542-04-04 Service ID Number: 18010C

Calibrated on July 22, 2004 by Gina Kline, Aaron Johnson, and John Wright Page 1 of 6

SAMPLE CALIBRATION REPORT

FOR 

A CRITICAL FLOW NOZZLE 

July 22, 2005 

Mfg.: CFV Builders, Inc. 

Serial Number: 1234 

Throat Diameter: 0.19009 inch (0.48283 cm) 

submitted by 

Flow Nozzles, Inc. 

Metertown, MD 

Purchase Order No. A123 dated May 24, 2004 

The flow meter identified above was calibrated by flowing filtered dry air at a constant rate 

through it into a volumetric prover (the NIST 26 m3PVTt standard). The PVTt standard 

determines mass flow, m& , by measuring the change in density of gas diverted into a known 

volume for a measured period of time.1 The flow meter was tested at five flows and at each flow, 

three (or more) measurements were gathered on two different occasions and used to produce 

averages at each of these flows. As a result, the tabulated data for this test are averages of six or 

more individual calibration measurements. 

A photograph of the flow meter installation is shown in Figure 1. The nozzle temperature, ) (T1, 

and pressure, ) (P1, were measured with NIST sensors, (Keithley SN 687848, thermistor #26, and 

Paroscientific SN 73965). Stagnation temperature, T0, was calculated from the measured 

temperature via the following equation, using a recovery factor, r , of 0.75: 













⋅ ⋅

−

T = T ⋅ + M r

2

0 1

2

1

1

γ

(1)

and the stagnation pressure, P0 , was calculated via the equation: 

 

1

Johnson, A.N., Wright, J.D., Moldover, M.R., and Espina, P.I., Temperature Characterization in the 

collection tank of the NIST 26 m3 PVTt Gas Flow Standard, Metrologia, 40, 211-216, 2003.
SAMPLE CALIBRATION REPORT Gas Flow Meter, S/N 1234

Flow Nozzles, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18010C

Calibrated on July 22, 2004 by Gina Kline, Aaron Johnson, and John Wright Page 2 of 6

P P M 0 1

2

1

1

1

2

= ⋅ +

−

⋅













γ −

γ

γ

(2)

where γ is the specific heat ratio and M is the Mach number in the approach pipe ( d = 3.48 cm), 

both based on P1 and T1.

2

 The largest of these corrections is 0.05 % for pressure. 

Figure 1. Photograph of the flow meter installation. 

The Reynolds number is included in the tabulated data and it was calculated using the following 

expression: 

π ⋅ ⋅ μ

⋅

=

d

m

Re 4 &

(3)

where m& is the mass flow of gas, d is the nominal nozzle throat diameter, and μ is the gas 

viscosity, all in consistent units so that Re is dimensionless. The gas properties (density and 

viscosity) were calculated using best-fit equations which are based on the NIST gas properties 

 

2 Measurement of Gas Flow by Means of Critical Flow Venturi Nozzles, ISO 9300: 1990 (E), 

International Organization for Standardization, Geneva, Switzerland, 1990.
SAMPLE CALIBRATION REPORT Gas Flow Meter, S/N 1234

Flow Nozzles, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18010C

Calibrated on July 22, 2004 by Gina Kline, Aaron Johnson, and John Wright Page 3 of 6

database.3,4 In January 2003, the correlation for viscosity used by the NIST Fluid Flow Group 

was changed from an older reference to the one used in this report. 

The discharge coefficient Cd was calculated from the expression: 

C

m R T

d P C d =

⋅ ⋅ ⋅

⋅ ⋅ ⋅

∗

4 0

2

0

&

π

 (4)

where R is the gas constant [the universal gas constant, 8.314471 J / (mol K), divided by the gas 

molecular weight, 28.9646 g/mol]. The critical flow factor, C

∗

, was calculated from the 

expression: 

1

1

1

2 γ−

γ+

















γ +

= γ

* C (5)

where γ is the specific heat ratio. 

The calibration results are presented in the following table and figure. The figure shows the 

discharge coefficient as a function of the inverse square-root of the Reynolds number. For many 

ISO standardized nozzles in the laminar flow range 5 this has the effect of linearizing the 

calibration data. 

 

3 Lemmon, E.W., McLinden, M.O., and Huber, M.L., Refprop 23: Reference Fluid Thermodynamic and 

Transport Properties, NIST Standard Reference Database 23, Version 7, National Institute of 

Standards and Technology, Boulder, Colorado, 2002.

4 Wright, J., Gas Properties Equations for the NIST Fluid Flow Group Gas Flow Measurement 

Calibration Services, 2/04.

5 Measurement of Gas Flow by Means of Critical Flow Venturi Nozzles, ISO 9300: 1990 (E), 

International Organization for Standardization, Geneva, Switzerland, 1990.
SAMPLE CALIBRATION REPORT Gas Flow Meter, S/N 1234

Flow Nozzles, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18010C

Calibrated on July 22, 2004 by Gina Kline, Aaron Johnson, and John Wright Page 4 of 6

0.986

0.987

0.988

0.989

0.99

0.991

0.992

0.993

0.994

0.995

0.0015 0.0018 0.0021 0.0024 0.0027 0.0030

Re-1/2

Cd

Figure 2. Calibration results for 0.19009 in (0.48283 cm), NIST xxx nozzle. 

Table 1. Calibration results for 0.19009 in (0.48283 cm), NIST nozzle. 

P0

[kPa] 

T0

[K] 

m&

[g/s] 

∗ C

[] 

Re

[] 

Cd

[] 

U

[%] 

200.44 294.72 8.5482 0.6854 1.229×1050.9885 0.11 

300.48 295.32 12.8340 0.6857 1.841×1050.9905 0.11 

400.53 295.60 17.1264 0.6860 2.452×1050.9917 0.11 

500.58 294.77 21.4613 0.6863 3.077×1050.9925 0.11 

601.63 296.26 25.7546 0.6866 3.684×1050.9930 0.11 

An analysis was performed to assess the uncertainty of the results obtained for the meter under 

test.6, 7, 8 The process involves identifying the equations used in calculating the calibration result 

 

6

 International Organization for Standardization, Guide to the Expression of Uncertainty in 

Measurement, Switzerland, 1996 edition. 
SAMPLE CALIBRATION REPORT Gas Flow Meter, S/N 1234

Flow Nozzles, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18010C

Calibrated on July 22, 2004 by Gina Kline, Aaron Johnson, and John Wright Page 5 of 6

(measurand) so that the sensitivity of the result to uncertainties in the input quantities can be 

evaluated. The approximately 67 % confidence level uncertainty of each of the input quantities is 

determined, weighted by its sensitivity, and combined with the other uncertainty components by 

root-sum-square to arrive at a combined uncertainty ( uc). The combined uncertainty is 

multiplied by a coverage factor of 2.0 to arrive at an expanded uncertainty (U ) of the measurand 

with approximately 95% confidence level. 

As described in the references, if one considers a generic basis equation for the measurement 

process, which has an output, y , based on N input quantities, ix , 

y (y x x, ,K x, N ) = 1 2 (6)

and all uncertainty components are uncorrelated, the normalized expanded uncertainty is given 

by, 

( ) ( ) ( ) ∑=

















= =

N

i i

i

i

e c

x

u x

k s

y

U y

k

y

U y

1

2

2

(7)

In the normalized expanded uncertainty equation, the (u x )' s i are the standard uncertainties of 

each input, and s'si are their associated sensitivity coefficients, given by, 

y

x

x

y

s

i

i

i

∂

∂

= (8)

The normalized expanded uncertainty equation is convenient since it permits the usage of 

relative uncertainties (in fractional or percentage forms) and of dimensionless sensitivity 

coefficients. The dimensionless sensitivity coefficients can often be obtained by inspection since 

for a linear function they have a magnitude of unity. 

For this calibration, the uncertainty of the discharge coefficient has components due to the 

measurement of the mass flow by the primary standard, u(m&) = 0.06%,9 as well as the pressure, 

u(P) = 02.0 % , and temperature, u(T ) = 03.0 % , measurements at the meters under test. The 

sensitivity coefficients for mass flow and pressure are 1, and the sensitivity coefficient for 

temperature is 1⁄2. This uncertainty analysis assumes that the user will use the same values for the 

throat diameter and the critical flow factor given herein and that the measurement errors in these 

quantities are correlated and cancel. 

 

7

 Taylor, B.N. and Kuyatt, C.E., Guidelines for Evaluating and Expressing the Uncertainty of NIST

Measurement Results, NIST TN 1297, 1994 edition. 

8

 Coleman, H.W. and Steele, W.G., Experimentation and Uncertainty Analysis for Engineers, John 

Wiley and Sons, 2nd ed., 1999. 

9

 Johnson, A.N., Wright, J.D., Moldover, M.R., and Espina, P.I., Temperature Characterization in the 

collection tank of the NIST 26 m3 PVTt Gas Flow Standard, Metrologia, 40, 211-216, 2003. 
SAMPLE CALIBRATION REPORT Gas Flow Meter, S/N 1234

Flow Nozzles, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18010C

Calibrated on July 22, 2004 by Gina Kline, Aaron Johnson, and John Wright Page 6 of 6

The present uncertainty analysis does not include uncertainty in the experimental measurements 

of viscosity found in the references, which can amount to 1% or more. To prevent errors due to 

viscosity, the user must use the same gas and viscosity expression used by NIST when using the 

results given in Table 1, or must use calibration coefficients calculated with their preferred 

viscosity relationship. Flow measurements made with this nozzle and a gas other than air 

(including humid air) will have greater uncertainty than that given in the present analysis due to 

uncertainty in the gas viscosity. Given these assumptions, the viscosity uncertainty depends 

primarily on the uncertainty of the gas temperature measurement. 

To measure the reproducibility10 of the test, the standard deviation of the discharge coefficient at 

each of the nominal flows was used to calculate the relative standard uncertainty (the standard 

deviation divided by the mean and expressed as a percentage). The reproducibility was 

propagated along with the other uncertainty components to calculate the combined uncertainty. 

Using the values given above, results in the expanded uncertainties listed in the data table and 

shown as error bars in the figure. 

For the Director, 

National Institute of Standards and Technology 

Project Leader’s Official Signature

John D. Wright 

Project Leader, Fluid Flow Group 

Process Measurements Division 

Chemical Science and Technology Laboratories 

 

10 Reproducibility is herein defined as the closeness of agreement between measurements with the flow 

changed and then returned to the same nominal value.