NIST Measurement Services: 

Natural Gas Flow Calibration Service (NGFCS)

NIST Special Publication 1081

Aaron N. Johnson 

U. S. Department of Commerce 

Technology Administration 

National Institute of Standards and Technology 
i 

Table of Contents for the Natural Gas Flowmeter Calibration Service (NGFCS) 

ABSTRACT........................................................................................................................... 1 

1 INTRODUCTION................................................................................................................. 2 

2 DESCRIPTION OF MEASUREMENT SERVICES ........................................................ 3

 2.1 Flow Capacity and Capabilities of the NGFCS............................................................... 4 

 2.2 Description of the Calibration Facility ............................................................................ 5 

 2.3 Mathematical Formulation of Volumetric and Mass Flow.............................................. 5 

 2.4 General Calibration Procedures....................................................................................... 6 

 2.5 Available Pipeline Sizes and Required Safety Inspections for Flanges .......................... 8 

 2.6 Flowmeter Types Typically Calibrated ........................................................................... 9 

 2.7 Procedures for Submitting a Flowmeter for Calibration ................................................. 9 

3 OVERVIEW OF THE PROCESS USED TO ESTABLISH TRACEABILITY................. 9

3.1 Overview of Five Stage Traceability Process used to Establish NIST Traceability...... 10 

3.2 Description of Critical Flow Venturis (CFVs) and their Calibration Parameters ......... 13 

3.3 Summary of the Uncertainty of the Five Stage Traceability Process............................ 14 

4 CALIBRATION ANALYSIS OF THE FIVE STAGE TRACEABILITY PROCESS 15

 4.1 STAGE 1: Calibration and Uncertainty of LP CFVs .................................................... 16 

 4.2 STAGE 2: Calibration and Uncertainty of the MP CFVs ............................................. 20 

 4.3 STAGE 3: Calibration and Uncertainty of the HP CFVs .............................................. 24 

 4.4 STAGE 4: Calibration and Uncertainty of the nine TMWS.......................................... 26 

 4.5 STAGE 5: Typical Uncertainty of a Flowmeter Calibration......................................... 34 

5 SUMMARY AND CONCLUSIONS ................................................................................. 36

REFERENCES.................................................................................................................... 37

 APPENDIX A: CFV Calculations..................................................................................... 40

 APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components......................... 42

APPENDIX C: Analysis of Selected Stage 5 Uncertainty Components......................... 50 

 APPENDIX D: Sample Calibration Report ..................................................................... 54 
1 

Abstract 

This document describes NIST’s high pressure natural gas flow calibration service (NGFCS). Flow 

calibrations are conducted offsite at the Colorado Experimental Engineering Station Incorporated 

(CEESI) in Garner, Iowa. A parallel array of nine turbine meter working standards (TMWS) are 

used to calibrate customer flowmeters over a flow range from 0.25 m3

/s to 9 m3/s at nominal 

pressures of 7500 kPa and at ambient temperatures. Over this flow range the expanded uncertainty 

varies from 0.25 % to 0.27 % (increasing at lower flows). All flowmeter calibrations are traceable 

to NIST standards. In particular, each of the nine TMWS is traceable to the NIST’s primary flow 

standards via a bootstrap process using critical flow venturis (CFVs), while auxiliary 

measurements (e.g., pressure, temperature, frequency, etc.) are traceable to the appropriate NIST 

standard. This document provides a detailed analysis of the bootstrap procedure used to establish 

NIST traceability and the uncertainty of this procedure. In addition, the document gives 

information about the flow measurement capabilities of the NGFCS (i.e., types of meters calibrated, 

calibration setup, calibration procedures, a sample calibration report, etc.). 
2 

1. INTRODUCTION 

NIST has extended its flow measurement capabilities to include natural gas flows at pressures 

and temperature conditions commensurate with U.S. gas pipeline distribution companies. 

Because NIST does not have natural gas flow measurement facilities of the necessary scale, a 

new approach has been developed that utilizes Cooperative Research and Development 

Agreements (CRADA) and Qualified Manufacturers Listings (QML) to augment NIST primary 

flow measurement standards. This combination is the basis for a new NIST flowmeter calibration 

service for large flowmeters ranging in size from 200 mm to 750 mm (8 inches to 30 inches) 

diameter that operate over a pressure range of approximately 6.2 MPa to 9 MPa (900 psi to 

1300 psi) at ambient temperature in natural gas flows. To realize these capabilities NIST has 

established a QML for private sector suppliers having the capability of providing the necessary 

facilities and operational expertise. Currently one provider, Colorado Engineering Experiment 

Station, Inc. (CEESI), has met this NIST QML requirement. To establish and improve the 

necessary chain of traceability to the SI, NIST engages in collaborative research efforts with 

CEESI under the terms of a Cooperative Research and Development Agreement (CRADA) 

These efforts ensure that improvements in NIST primary flow measurement standards, in 

working standards used in establishing traceability, and in improvements to metrological control 

capabilities are implemented in a timely and efficient manner. 

NIST works with CEESI in two distinct modes to provide this calibration service. In the 

operational mode, NIST uses the QML to purchase services provided by CEESI for individual 

calibration tests for a particular NIST customer. The test data obtained is derived from procedures 

and protocols specified by NIST that are performed at CEESI under NIST metrological control. 

CEESI provides access to its facilities and the expertise necessary to test the metering device of a 

customer at an agreed upon cost. Under the terms of the CRADA NIST and CEESI engage in 

cooperative research efforts to develop and improve the traceability ladder that extends NIST’s onsite, primary, gas flowrate measurement standards to the pipeline conditions attained at CEESI’s 

Garner, Iowa test site. These research efforts investigate flow phenomena affecting the traceability 

ladder with the intent of improving various aspects of flow traceability. The result of these joint 

activities is the Natural Gas Flow Calibration Service (NGFCS). 

A key aspect of the NGFCS is that NIST maintains metrological control of flowmeter 

calibrations performed at CEESI’s Garner, Iowa facility. The following measures are taken by 

NIST to ensure metrological control: 

1) A path of traceability is established and maintained linking the calibration of a meter 

under test (MUT) at CEESI to NIST’s primary flow standards. 

2) The NIST quality system, compliant with ISO 17025 [i], is extended to these calibration 

activities. 

3) Each calibration is monitored via a secure internet connection. 

4) All auxiliary instrumentation (e.g., pressure transducers, temperature sensors, frequency 

counters, etc.) necessary for flow calibrations are traceable to NIST via transfer standards 

provided by and maintained by NIST. 

5) Control charts that validate the performance of both auxiliary instrumentation and 

flowmeter check standards are maintained by NIST. 

6) The calibration process is highly automated to increase reproducibility and decrease the 

risk of human error. 
3 

7) Diagnostics are used to quantify flow stability levels, line pack (or mass storage) effects, 

the impact of changing environmental conditions, and other parameters affecting 

calibration results. 

8) The results of each flowmeter calibration are validated by a check standard installed in 

series with the meter under tests (MUT). 

9) The raw calibration data is analyzed by NIST. 

10) The calibration report is written by NIST. 

11) The facility is periodically compared to other national metrology institutes within the 

framework of the CIPM Mutual Recognition Arrangement (i.e., international key 

comparisons) to detect potential biases in the calibration results. 

The focus of this document is to provide a description of the traceability ladder that extends from 

NIST’s primary flow standards to the working standards used for flow meter calibration at 

pipeline conditions. It is also a reference guide for customers looking for specific information 

about operational aspects of NIST’s natural gas flow calibration service (e.g., types and sizes of 

flowmeters that can be calibrated, flow capacity, how to schedule a calibration, cost of a 

calibration, turnaround time, etc.) as well as a technical resource that provides the underlying 

metrological details of how flow traceability to NIST is established and the uncertainty budget 

resulting from this process. 

2. DESCRIPTION OF MEASURMENT SERVICES 

NIST calibration services are continuously being upgraded. Customers should consult the following 

web address http://www.cstl.nist.gov/div836/Group_02/index_836.02.html to find the most current 

information regarding the calibration services (e.g., calibration fees, turnaround times, technical 

contacts, shipping procedures, etc.). 

Table 2.1. Typical natural gas concentration at CEESI’s Iowa flow facility. 

Component Mole (%)

Methane 94.8 to 96.2 

Ethane 1.5 to 2.3 

Propane 0.055 to 0.3 

iButane 0.0008 to 0.03 

nButane 0.0003 to 0.04 

iPentane 0 to 0.01 

nPentane 0 to 0.006 

C6+ 0 to 0.006 

Nitrogen 1.4 to 1.8 

Carbon Dioxide 0.5 to 0.7 

Hydrogen 0.05 to 0.27 

Helium 0.03 to 0.04 
4 

2.1 Flow Capacity and Capabilities of the NGFCS 

The NGFCS can calibrate both volumetric based and mass based flowmeters. The volumetric 

flow range extends from 0.25 m3/s (1.5 × 104 L/min or 3.2 × 104 acfh) to 9 m3/s (5.4 × 105 L/min 

or 1.14 × 106 acfh) at a nominal pipeline pressure of 7500 kPa ± 1200 kPa (1088 psi ± 174 psi) 

and at a nominal temperature of 292.5 K ± 7.5 K (66.8 °F ± 13.5 °F). 1 The expanded 

uncertainties for volumetric flow calibrations vary from 0.25 % to 0.27 %, increasing at lower 

flows. Mass flow capabilities extend from 5.9 kg/s (13.1 lbm/s) to 533 kg/s (1175.8 lbm/s) with 

uncertainties ranging from 0.26 % to 0.28 % depending on flow. The typical natural gas 

composition for flow calibrations is shown in Table 2.1. 

NIST is working toward extending the calibration flow range to 0.1 m3/s and to reducing the 

uncertainty to 0.2 % over the entire flow range. The following website 

http://www.cstl.nist.gov/div836/Group_02/index_836.02.html should be consulted for current 

capabilities. 

762 mm

914mm

Flow Regulating Station

Flow Inlet

1067 mm

P T

TMWS1

f

MUT

P T

508 mm

762 mm

305 mm

305 mm

305 mm

Outlets

A B

C

D

E

MUT

P T

610 mm

MUT

P T

762 mm

Test Sections

Flow

P T

TMWS2

f

P T

TMWS8

f

305 mm

P T

TMWS9

f

Turbine Meter

Working Standards

(TMWS) 1 to 9

762 mm

914mm

Flow Regulating Station

Flow Inlet

1067 mm

P T

TMWS1

f

MUT

P T

508 mm

762 mm

305 mm

305 mm

305 mm

Outlets

A B

C

D

E

MUT

P T

610 mm

MUT

P T

762 mm

Test Sections

Flow

P T

TMWS2

f

P T

TMWS8

f

305 mm

P T

TMWS9

f

Turbine Meter

Working Standards

(TMWS) 1 to 9

Figure 2.1. CEESI Iowa Flowmetering Facility 

 

1

 Note that the variation in pressure and temperature occur seasonally, and not during a flowmeter calibration. 
5 

2.2 Description of the Calibration Facility 

NIST flow calibrations are conducted at CEESI’s Iowa flow facility. This facility is shown in Fig. 2.1. 

The dry pipeline quality natural gas that enters the custody transfer junction in pipeline A is 

transported throughout the northwest region of the United States in pipelines B, C, and D. 

During a flowmeter calibration a fraction of the gas from pipeline A is diverted to pipeline E 

where it is measured by a parallel array of up to nine turbine meter working standards (TMWS). 

The flow measured by the TMWS is used to determine the flow at the MUT installed 

downstream in one of three appropriately sized test sections. Flow exiting the MUT is returned to 

pipelines B, C, and/or D as appropriate. 

A gas chromatograph (not shown in the figure) located upstream of the TMWS array on 

pipeline E is used to measure the gas composition. Each of the nine TMWS is instrumented with 

a pair of pressure transducers, a pair of temperature sensors, and a pair of frequency counters. 

The redundant measurements guard against erroneous instrumentation readings. The MUT is 

also instrumented with redundant pressure and temperature instrumentation (and frequency when 

necessary). 

2.3 Mathematical Formulation of Volumetric and Mass Flow 

Application of the principle of conservation mass shows that the mass flow at the MUT equals 

n st leak

9

n 1

,n ∑ - Δ - m& MUT m& TMWS δ m& m&

=

= (2.1) 

where mTMWS,n & is the mass flow through the nth TMWS; δ n is the TMWS selector function, 

which equals zero when the nth TMWS shutoff valve is closed (i.e., no flow) and unity when it is 

open; Δmst & is the rate of mass storage (or line pack) in the connecting pipe volume between the 

TMWS and the MUT; and mleak & is the net rate of mass leakage out of the connecting volume. 

Leak check procedures ensure that mleak & is negligible relative to the measured flow. Similarly, 

stable flow conditions are maintained so that Δmst & is small relative to mMUT & . Consequently, both 

mleak & and Δmst & are taken to be zero when computing mMUT & , but are included in the uncertainty 

budget. With this simplification the mass flow at the MUT is 

n

n 1

MUT ∑ρ TMWS,n TMWS,nδ

N

m q

=

& = . (2.2) 

where the product of density ( ρ TMWS,n ) and volumetric flow ( qTMWS,n ) is substituted for mass 

flow. 

The volumetric flow measured by the nth TMWS is proportional to the rotational frequency of 

the respective turbine blade ( TMWS,n f ) 
6 

,n

,n ,n

TMWS

TMWS

TMWS K

f

q = (2.3) 

where the inverse of the meter factor or K-factor ( KTMWS,n ) is the constant of proportionality. 

The K-factor of each TMWS is traceable to NIST primary flow standards via an array of critical 

flow venturis (CFVs). The measurement results and uncertainties of this traceability chain are 

outlined in Section 3 and discussed in detail in Section 4. Substitution of Eqn. 2.3 into Eqn. 2.2 

yields an expression for mass flow entirely in terms of parameters associated with the turbine 

meter 

∑

n 1 ,n

,n n ,n

TMWS

TMWS

MUT TMWS

N

K

f

m

= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ & = δ ρ . (2.4) 

The expression for volumetric flow is derived by dividing by the density at the MUT ( MUT ρ ) 

n

n 1 ,n

,n ,n ∑

TMWS

TMWS

MUT

TMWS

MUT δ

ρ

ρ N

K

f

q

= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = . (2.5) 

Alternative formulations for mass flow (Eqn. 2.4) and for volumetric flow (Eqn. 2.5) can be 

derived by using the equation of state for gas density 

Z R T

P

u

M

ρ = (2.6) 

where P and T are respectively, the measured pressure and temperature, Ru is the universal 

gas constant, Z is the compressibility factor as determined from REFPROP 8 Thermodynamic 

Database [ii], and NG ∑ k k M = M x is the molar mass of the natural gas mixture - a linear sum of 

Mk (the molar mass of the kth component) multiplied by kx (the mole fraction of the kth

component) summed over all the mixture components. Combining Eqn. 2.6 with Eqns. 2.4 and 

2.5 respectively yields the following formulation for mass flow 

∑

1 n u n n

n n n

TMWS

NG

MUT

N

n Z R T K

P

m f

=

⎥

⎦

⎤ ⎢

⎣

⎡ = M & δ (2.7) 

and volumetric flow 

∑

1 ,n

,n

,n ,n

,n n

TMWS

TMWS

TMWS

MUT

TMWS

MUT

MUT

TMWS

MUT

N

n K

f

Z

Z

T

T

P

P

q

= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = δ , (2.8) 

respectively. 

2.4 General Calibration Procedures 

The general procedures for calibrating a MUT are divided into four parts including A) installation 

procedures, B) pre-flow calibration checks and procedures, C) flow calibration procedures, and 

D) post processing calibration procedures. Here, we summarize these procedures for a typical 

calibration. The full lists of procedures are included in the NIST NGFCS Quality Manual. 
7 

A. Installation Procedures 

1. Install MUT in the appropriately sized test section: 

a. Upstream and downstream piping will be a minimum of 10 D and 5 D respectively unless 

otherwise specified where D is the pipe diameter. 

b. Check to ensure that inside diameter of the upstream piping is within 1 % of the meter 

bore or the manufacturer’s specifications. Make sure that there is not a observable 

discontinuity in diameter between upstream diameter and meter bore diameter. 

c. Ensure flange faces at the inlet and exit of the meter match up with inlet and exit piping. 

d. Upstream flanges and gaskets shall not protrude into the flow stream by more than 1 % of 

the internal pipe diameter. 

2. Install thermal well(s)/temperature transmitter(s) between 1 D and 5 D downstream of 

the meter. 

3. Install pressure transmitters. 

4. Document (sketch), upstream and downstream piping, the location of pressure and 

temperature taps, and other fittings relative to MUT. 

5. Photograph calibration setup including MUT and instrumentation. 

B. Pre-Flow Calibration Checks and Procedures 

1. Remove any air left in the pipeline from meter installation. 

2. Perform leak checks at 2 MPa, 4 MPa, 6 MPa, and at operating line pressures 

3. Check proper operation of data acquisition system including readouts of pressure, 

temperature, and frequency. 

Table 2.2. Sequence of flow set points for a typical calibration. (Second half of dataset is used to 

assess hysteresis effects and reproducibility) 

Set Point 

Number 

Flow 

Set Point 

No. of Repeats 

at Set Point 

1 min q (Typically 55 m/min or 3 ft/s) 3 

2 10 % of max q 3 

3 25 % of max q 3 

4 40 % of max q 3 

5 70 % of max q 3 

6 100 % of max q 3 

Interrupt and reestablish nominal flow 

7 100 % of max q 3 

8 70 % of max q 3 

9 40 % of max q 3 

10 25 % of max q 3 

11 10 % of max q 3 

12 min q (Typically 55 m/min or 3 ft/s) 3 

13 

1 Verification Point taken 

(between 40 % and 70 % of max q ) 1 
8 

C. Calibration Procedures 

1. Determine the minimum ( min q ) and maximum ( qmax ) flows for the calibration. 

2. Determine the appropriate flow set points (see Table 2.2). 

3. Determine how many of the nine TMWS are needed to achieve the flow set points. 

4. Establish steady flow through the MUT at the desired flow set points. 

5. Data Collection and Calculations: 

a. Collect data (i.e., temperature, pressure, and frequency necessary for flow determination) 

for approximately 120 s. (The software data rate is about 1 Hz so that approximately 120 

points are collected). 

b. The software calculates the desired flow at all 120 points using Eqn. 2.7 for mass flow or 

Eqn. 2.8 for volumetric flow. 

c. The time averaged value of volumetric flow (or mass flow) is defined as the arithmetic 

average of the 120 flow points. 

d. The flow set point is repeated as specified in Table 2.2. 

6. Follow the protocol in Table 2.2. 

a. If the current set point number is not equal to either 6 or 12 then increment the set point 

number by one and return to step 2. 

b. If the current set point number equals 6 then interrupt and reestablish the nominal flow 

before returning to step 2 and starting set point number 7. 

c. If set point number 12 has just been completed proceed to step 7. 

7. If an ultrasonic flowmeter is being calibrated, then calculate the Flow Weighted Mean 

Error (FWME) per AGA 9 [iii]. 

a. Electronically install the FWME adjustment into the ultrasonic flowmeter. 

b. If directed by customer and suitable for the meter design multi-point linearization 

techniques will be used to electronically install adjustment. 

b. Once the meter factor(s) is electronically installed, 1 verification point shall be taken as 

denoted in Table 2.2 (Note that verification points are performed only if electronic 

adjustments are made to the flowmeters). 

c. Upon completion of the Verification point, if available, the meter shall be put into a 

“Read Only” mode. 

D. Post Processing Calibration Procedures 

1. Raw data sent to NIST electronically. 

2. NIST checks stability criteria to assess the quality of the data. 

3. NIST ensures data is consistent with check standards used in series with the MUT 

during the calibration. 

4. If data quality is acceptable, NIST processes data and compares results to values 

determined by the software at the time of calibration. 

5. NIST checks to ensure post processed calibration factors match those electronically 

installed in ultrasonic flowmeter. 

6. NIST writes and sends out the calibration report. 

2.5 Available Pipeline Sizes and Required Safety Inspection for Flanges 

Flowmeters can be calibrated in pipe sizes ranging from 30.48 cm (12 inches) to 76.2 cm 

(30 inches). When a customer submits upstream and downstream pipe lengths along with the 

flowmeter to be calibrated, the associated flanges should be rated to withstand a minimum 

pressure of 10 MPa at ambient temperatures (i.e., flange ratings must be 600 lb or higher). 
9 

Additionally, for safety reasons, customers must have all flange welds x-rayed and 

hydrostatically tested before shipping their flowmeters for calibration. 

2.6 Flowmeter Types Typically Calibrated 

The vast majority of flowmeters calibrated by the NGFCS are ultrasonic flowmeters that are used 

for the custody transfer of natural gas. However, other flowmeters types can also be calibrated. 

Table 2.3 shows a list of flowmeters types that are calibrated. Some flowmeter types not listed 

here may also be suitable for calibration. Contact NIST staff for details. 

Table 2.3 Typical Flowmeters Calibrated 

Flowmeter Types 

Ultrasonic Flowmeters 

Turbine Flowmeters 

V-Cone Flowmeters 

Coriolis Meters 

Subsonic Venturis/Nozzles 

Sonic Venturis/Nozzles 

Positive Displacement Meters 

Variable Area Flowmeters (Rotameters) 

Vortex Shedding Flowmeters 

2.7 Procedures for Submitting a Flowmeter for Calibration 

The Fluid Metrology Group follows the policies and procedures described in Chapters 1, 2, and 3 of 

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

following addresses: 

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

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

3. 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. 

3. OVERVIEW OF THE PROCESS USED TO ESTABLISH TRACEABILITY

This section gives an overview of the five stage procedure used to establish the traceability chain 

between a flowmeter calibrated at CEESI’s Iowa facility back to NIST’s primary flow standards. A 

key element of the five stage process was the critical flow venturi (CFV) bootstrap process. NIST’s 

low pressure, low flow standards were used to calibrate multiple CFVs, which were used in 

parallel to provide traceability at pipeline conditions for natural gas flows. A description of the 

CFVs and their basic calibration parameters are described, and an overview of the uncertainty 

attributed to this scale-up procedure is presented. 
10 

3.1 Overview of the Five Stage Procedure used to Establish NIST Traceability 

The traceability chain linking flowmeters calibrated at CEESI’s Iowa facility in natural gas to 

NIST’s low pressure, air flow primary standard were accomplished in five stages. A diagram of 

the five stage process is shown in Fig. 3.1. The first column in each row identifies the calibration 

stage, followed by the flow standard, the reference flowmeter being calibrated, the fluid medium, 

the expanded uncertainty (i.e., k = 2) of the reference flowmeter, and the calibrated flow (or 

Reynolds number) range of the reference flowmeter. The nominal pressure conditions for all five 

stages are also specified in the figure. 

Stage Working

Fluid

Filtered 

dry air

Filtered 

dry air

Filtered 

dry air

Natural

Gas

1

2

3

4

5

Natural

Gas

Standard

NIST

PVTt

NIST 26 m3 PVTt

LP Nozzle Bank

MP Nozzle Bank

4x P0

TMWS Array

P ≈ 7500 kPa

9 x

HP Nozzle Bank

P0 ≈ 7500 kPa

P0 ≈ 350 to 700 kPa

Reference

Meter

LP CFVs

P0 ≈ 350 to 700 kPa

4 x

4 x

8 x

9 x

MP CFVs

(Upstream of LP CFVs)

4x P0

HP CFVs

(Upstream of MP CFVs)

16x P0

MUT

P ≈ 7500 kPa

MUT

(Downstream of TMS)

TMWS

P ≈ 7500 kPa

(Upstream of HP CFVs)

Expanded 

Uncertainty

of Reference 

Flowmeter

0.10%

0.13%

0.17%

0.24 % to 0.25 %

0.25 % to 0.27 %

Flow or 

Reynolds 

Number

Range

LP CFV Re Range 

1.1 × 106

to

2.4 × 106

MP CFV Re Range 

3.7 × 106

to

8.6 × 106

HP CFV Re Range

(dry air) 

20 × 106

to

27.5 × 106

HP CFV Re Range

(natural gas) 

24 × 106

to

27.5 × 106

TMWS Flow Range 

0.25 m3/s

to

9.0 m3/s

( ) P ≈ 7500 kPa

and T ≈ ambient 

Figure 3.1. Schematic of the five stage process used to establish traceability of a MUT at CEESI’s Iowa natural gas 

flow facility to NIST 26 m3 PVTt primary flow standard. 

In Stage 1, the NIST 26 m3 PVTt flow standard [vvi-vii] was used to successively calibrate four 

CFVs in air over a pressure range extending from 350 kPa to 700 kPa. For these CFVs this 

pressure range corresponded to a Reynolds numbers range from 1.1 × 106

 to 2.4 × 106. The four 

CFVs are referred to throughout this document as the low pressure (LP) CFVs. The expanded 

uncertainty of each of the LP CFVs was 0.10 % as verified in Section 4.1. 
11 

In Stage 2, the four LP CFVs were combined in parallel and positioned downstream of a single 

CFV. Under choked flow conditions, the stagnation pressure (P0) of the upstream CFV equaled 

four times that of the four downstream LP CFVs. In this way the upstream CFV is calibrated at 

pressures and flows four times greater than any one of the downstream LP CFVs. This 

calibration process was done in dry air. We repeated this procedure four times so that a total of 

four CFVs were calibrated in Stage 2, which are herein referred to as the medium pressure (MP) 

CFVs. The Reynolds number range of each MP CFV extended from 4.7 × 106

 to 8.4 × 106, and 

the expanded uncertainty of each MP CFVs was 0.13 %. The calibration results and uncertainty 

analysis are documented in Section 4.2. 

Stage 3 is analogous to Stage 2. The four MP CFVs were combined in parallel and used to 

calibrate a single upstream CFV in dry air at four times the pressure. A total of eight CFVs were 

calibrated in Stage 3, which are herein referred to as high pressure (HP) CFVs. The bootstrapping approach implemented in Stages 2 and 3 resulted in eight HP CFVs that were traceable 

to the 26 m3 PVTt primary standard, but with a flow capacity nearly sixteen times the Stage 1 

calibration. The calibrated Reynolds number range of each HP CFV was 20 × 106

 to 27.5 × 106

and the expanded uncertainty of each of the HP CFVs was 0.17 %. The calibration results and 

uncertainty analysis are documented in Section 4.3. 

In Stage 4 the eight HP CFVs were combined in parallel and used to calibrate a total of nine 

TMWS, one at a time, in natural gas, at the nominal pressures for which they are used, and 

mounted in the location of pipeline where they are used. Reynolds number matching was used to 

apply the air-based calibration of the HP CFVs to natural gas. At these high Reynolds numbers 

(i.e., above 16 × 106

), theoretical predictions indicated a difference in the CFV discharge 

coefficient between dry air and natural gas of less than 0.01 % [viii]. Each of the TMWS was 

calibrated over a nominal volumetric flow range from 0.25 m3

/s (3.2 × 104 acfh) to 1 m3/s 

(1.3 × 105

 acfh) at nominal pipeline pressures of 7500 kPa and ambient temperatures. The 

volumetric flow was changed by varying the number of HP CFVs. The expanded uncertainty of 

each of the TMWS ranged from 0.24 % to 0.25 % depending on flow. The calibration results and 

uncertainty analysis are documented in Section 4.4. 

In Stage 5, the nine TMWS are combined in a parallel array, and used to calibrate customer 

flowmeters (MUT) in natural gas at ambient temperatures and at nominal pipeline pressures of 

7500 kPa. The flow range of the facility extends from a minimum of 0.25 m3

/s (3.2 × 104 acfh) 

when only a single TMWS is used to a maximum of 9 m3

/s (1.1 × 106 acfh) when all nine 

TMWS are used together at their full capacity. The expanded uncertainty of a MUT (not 

including the repeatability of the MUT) is 0.25 % at the highest flow and increases to 0.27 % at 

the lowest flow. The uncertainty analysis is documented in Section 4.5. 
12 

Figure 3.2 Four Photographs showing the CFVs with end caps and three different size nozzle fixtures

Figure 3.2 shows four photographs that include (A) the single aperture nozzle fixture used in 

Stage 1 to calibrate the LP CFVs; (B) the four LP CFVs with end caps (C) the four aperture 

nozzle fixture used both in Stage 2 to hold the four LP CFVs and again in Stage 3 to hold the 

four MP CFVs, and (D) the twenty-one aperture nozzle fixture used in Stage 4 to calibrate the 

nine TMWS. In Stage 4, the maximum flow was obtained with the HP CFVs mounted in only 

eight of the possible twenty-one apertures. Additional apertures (which allow for higher flows in 

the future) were not used. The unused apertures remain sealed for the entire calibration process. 

The end caps shown in Fig. 3.2 A, B, and D manually screw onto the downstream end of the 

CFVs to prevent flow. The end caps were used in the Stage 4 calibration of the TMWS using the 

eight HP CFVs. Initially, two of the eight HP CFVs were uncapped. The remaining six were 

systematically uncapped to change the volumetric flow through the TMWS being calibrated. 

The apertures on the face of the nozzle fixtures in Fig. 3.2 A, C, and D, were dimensioned so that 

once the CFVs were installed they are both sealed and held in place by friction. When the nozzle 

fixture (with CFVs installed) was mounted in the pipeline, the pressure gradient necessary to 

choke the CFVs also added to the integrity of the seal. An illustration of the single CFV nozzle 

fixture installed in the appropriate sized pipeline is shown in Fig. 3.3. This figure corresponds to 

the Stage 1 calibration setup of the LP CFVs. 
13 

Flow Direction

Pressure

Temperature Single LP CFV installed 

in nozzle fixture

20.32 cm 10.16 cm

Flow Direction

Pressure

Temperature Single LP CFV installed 

in nozzle fixture

20.32 cm 10.16 cm

Figure 3.3. Stage 1 Calibration setup for LP CFVs 

3.2 Description of CFVs and their Calibration Parameters 

The LP, MP, and HP CFVs used to bootstrap between NIST low pressure, low air flow standard 

and pipeline scale flows of natural gas were designed according to the ISO 9300 standard [ix]. 

These flowmeters were selected because of their excellent reproducibility, simple geometric 

design, straightforward application, and well understood physics. The two dimensionless 

calibration parameters relevant for CFVs are the discharge coefficient (Cd ) and the Reynolds 

number ( Re ). The discharge coefficient is a ratio of the actual mass flow ( m& ) to the theoretical 

mass flow ( mth & ) based on one-dimensional isentropic conditions. The theoretical mass flow is [x] 

u 0

0

2

th 4 R T

d P C

m

π s M & = (3.1) 

and the discharge coefficient is 

0 s M 2

u 0

th

d

4

≡ = π d P C

m R T

m

m

C &

&

& (3.2) 

where P0 is the stagnation pressure, T0 is the stagnation temperature, Ru is the universal gas 

constant, M is the molecular weight, d is the throat diameter, and Cs is the critical flow 

function. Appendix A explains how P0 , T0 , and Cs are calculated in this work. The Reynolds 

number definition used herein is 

0 u 0

0 s

0

th = 4

≡

μ R T

P d C

π μ d

m

Re & M , (3.3) 

where μ0 is the dynamic viscosity evaluated at P0 , T0 . All thermodynamic properties are 

computed using the REFPROP 8.0 Thermodynamic Database [ii]. The calibration results for 

Stages 1 through 3 are expressed by plots of Cd versus Re . 

Table 3.1 lists the throat diameters of the four LP CFVs ( LP d ), the four MP CFVs ( MP d ), and 

the eight HP CFVs ( HP d ) referenced by their serial numbers. The throat diameters of the four LP 

CFVs were measured to tolerances better than 0.001 mm at a 95 % confidence level by the 
14 

Precision Engineering Division at NIST. The use of highly accurate d values cause the 

calibration data for the four LP CFVs to collapse onto a single curve that follows trends 

predicted by theoretical CFV models [xixii-xiii]. The throat diameters of the MP and HP CFVs are 

defined so that the MP CFVs and the HP CFVs can be characterized by a single calibration curve 

that is consistent with the LP CFVs and with theory. 

Table 3.1. LP CFVs measured d values and MP and HP CFVs estimated d values. 

Total No. of 

CFVs 

LP 

CFVs 

LP d

(mm)

MP 

CFVs 

MP d

(mm)

HP 

CFVs 

HP d

(mm)

1 #2 25.3932 #10 25.3944 #1 25.4123 

2 #3 25.3910 #11 25.3952 #7 25.3822 

3 #4 25.3935 #12 25.3959 #8 25.3804 

4 #5 25.3883 #13 25.3854 #14 25.3958 

5 N/A N/A N/A N/A #15 25.3870 

6 N/A N/A N/A N/A #17 25.3789 

7 N/A N/A N/A N/A #19 25.3921 

8 N/A N/A N/A N/A #20 25.4006 

3.3 Summary of the Uncertainty of the Five Stage Calibration Process 

The uncertainty of the five stage traceability scheme is determined using the GUM 

procedure [xiv]. The uncertainty for each of the five stages is summarized in the bar graph 

shown in Fig. 3.4. The height of each of the rectangles depicts the standard uncertainty (i.e., 

k = 1) of the stage it represents. The sum of all five rectangles is 100 % and the root-sum-square 

of the standard uncertainties multiplied by a coverage factor of two is the expanded uncertainty 

(i.e., k = 2) of the MUT. For flows ranging from 0.25 m3

/s to 9 m3/s (at a nominal pressure of 

7500 kPa and at ambient temperatures) the expanded uncertainty varies from 0.27 % to 0.25 %, 

decreasing with increasing flow. 

0

10

20

30

40

50

Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

0.049 %

0.041 %

0.051 %

0.082 %

0.051 %

P, T, & misc

NIST Flow Std.

K-factor Reprod.

CFV cross-talk

CFV species effects

% Contribution to Uncertainty

0

10

20

30

40

50

Stage 1 Stage 2 Stage 3 Stage 4 Stage 5

0.049 %

0.041 %

0.051 %

0.082 %

0.051 %

P, T, & misc

NIST Flow Std.

K-factor Reprod.

CFV cross-talk

CFV species effects

% Contribution to Uncertainty

Figure 3.4. The standard uncertainty and the percent contribution to the total uncertainty of each stage for 

MUT q = 2.25 m3

/s. (The height of each rectangle indicates the uncertainty contribution in percent for the 

corresponding stage.) 
15 

The legend in Fig. 3.4 shows the five different shading patterns used in the bar graph. The three 

lined shading patterns (i.e., , , ) denote uncertainty components that NIST plans to reduce in 

the near future. These include the uncertainty introduced in Stage 1 corresponding to the NIST 

flow standard ( ), the uncertainty introduced in Stage 4 attributed to the reproducibility of the 

TMWS K-factors ( ), and the uncertainty introduced in Stages 2, 3, and 4 attributed to cross-talk 

(i.e., interference effects) between the CFVs mounted in a common plenum ( ). The CFV 

interference effects will be reduced by increasing the spacing between the CFVs used in parallel. 

The K-factor reproducibility is currently based on only two calibrations. We anticipate lower 

values in the future as repeated calibrations provide a larger data set to more accurately 

determine the long term random effects and flowmeter drift. Lastly, NIST is currently working to 

reduce the uncertainty of the NIST flow standard used in Stage 1. 

The two solid shading patterns (i.e., and ) in Stages 2, 3, 4, and 5 include multiple 

uncertainty sources that have been grouped together. The first pattern of solid shading ( ) 

includes uncertainty components attributed to pressure and temperature measurements, line pack 

effects, and various other sources. A detailed listing of the individual uncertainty components 

contained in these groupings is included in Section 4 and in references [xv, xvi]. 

The second pattern of solid shading ( ) is attributed to CFV species effects. CFV species effects 

include uncertainty contributions from the following four thermodynamic properties: Cs for air in 

Stage 3, and Cs , Z , and M for natural gas in Stage 4. The cause of this uncertainty is twofold. 

First, the uncertainty attributed to Cs in Stages 3 and 4 is a consequence of calibrating the HP 

CFVs in air, but applying the calibration to natural gas. Second, the Stage 4 uncertainty 

attributed to Z and M results because the density of the natural gas is required to convert from 

the mass flow predicted by the HP CFVs to volumetric flow needed for the TMWS calibration. 

The uncertainties of these parameters are already nearly optimized, and are not likely to be 

reduced in the near future. 

4. CALIBRATION ANALYSIS OF THE FIVE STAGE TRACEABILITY PROCESS 

This section provides a detailed analysis of the calibration results and uncertainty for each of the 

five stages used to establish traceability to NIST. The governing calibration equations are 

developed for each of the five stages. The uncertainty introduced by each stage is determined 

using the method of propagation of uncertainty [xvii]. The GUM procedure [xviii] is followed 

whereby uncertainty sources are categorized as either Type A (i.e., those which are evaluated by 

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

having subcomponents belonging to both Type A and Type B are categorized as (A, B). The 

uncertainty analysis accounts for correlated uncertainty sources within and between each of the 

five stages. 

In all five stages the flow properties (e.g., the critical flow function, density, viscosity, etc.) are 

calculated using the REFPROP 8 Thermodynamic Database [ii]. In the first three stages the 

working fluid is air. The mole fraction of water vapor is maintained below 4.1 × 10-5 so that the 

air can be considered to be dry air. However, effects of water vapor are considered in the 

uncertainty budget. Table 4.1 shows the composition of dry air used to calculate flow properties 

in this work. The air composition is the average value of multiple references [xix- xxxxi]. 
16 

Table 4.1 Composition of air used in Stages 1 through 3 

Species Mole Fraction (%)

Nitrogen 78.0849 

Oxygen 20.9478 

Argon 0.934 

Carbon Dioxide 0.0314 

4.1 STAGE 1: Calibration and Uncertainty of LP CFVs 

Calibration of the LP CFVs

In Stage 1, NIST’s 26 m3 PVTt primary flow standard was used to calibrate the four LP CFVs in 

dry air. The PVTt flow standard uses a timed collection technique to determine the LP CFV mass 

flow. The flow emanating from the LP CFV was diverted from the bypass into the nearly 

evacuated collection tank for a measured time interval. The average gas temperature and pressure 

in the collection tank as well as in the inventory volume were measured before and after the 

filling process. These measurements were used in conjunction with the REFPROP 8 

Thermodynamic Database [ii] to determine the initial and final densities of the gas in the 

collection tank ( i ρT and f ρ T ), and in the inventory volume ( i ρI and f ρI ). In the absence of leaks, 

the time averaged mass flow equals the change in mass within the collection tank (i.e., gas 

density change in the collection tank multiplied by the collection tank volume) plus the change in 

mass in the inventory volume (i.e., gas density change in the inventory volume multiplied by the 

inventory tank volume) divided by the collection time 

t

V V

m

Δ

- - I

i

I

f

T I

i

T

f

T (ρ ρ ) + (ρ ρ ) & = (4.1) 

where VT is the collection tank volume; VI is the inventory volume; and Δt is the collection 

time interval. The standard relative uncertainty of PVTt mass flow measurements is 

[u( ) m& m&]= 0.045 %. A detailed description of the PVTt system and its uncertainty can be found 

in references [v-vii]. 

Vacuum

Exhaust

Valve

LP CFV

Tank

Inlet

Valve Collection

Tank

Bypass

Fan

Fan

Duct

Bypass

Valve

Steady Flow

Source

Inventory

Volume

Tm P

Vacuum

Exhaust

Valve

LP CFV

Tank

Inlet

Valve Collection

Tank

Bypass

Fan

Fan

Duct

Bypass

Valve

Steady Flow

Source

Inventory

Volume

Tm P

Figure 4.1. Schematic of NIST 26 m3 PVTt Primary Flow Standard 
17 

The mass flows measured by the PVTt flow standard were used to calculate Cd values for each 

of the four LP CFVs using Eqn. 3.2, while the Reynolds number was computed using Eqn. 3.3. 

Figure 4.2 shows the calibration data for all four LP CFVs plotted on a logarithmic Reynolds 

number axis. Included in the figure are the results for the MP and HP CFVs. These results are 

plotted along with the LP CFV data so that general trends for all three stages of CFV calibration 

data (i.e., Stages 1, 2, and 3) can be easily observed. The MP and HP CFV data, however, are not 

discussed in this section, but are covered in Sections 4.2 and Sections 4.3, respectively. 

The LP CFV data shown in the figure incorporate two different PVTt calibrations, the first in 

2004, and the second two years later in 2006. In both sets of calibration data the four LP CFVs 

are depicted by triangles having four different orientations. Open triangles are used for the 2004 

dataset (i.e., - CFV #2; - CFV #3; - CFV #4; - CFV #5) while closed triangles are 

used for the 2006 data set (i.e., - CFV #2; - CFV #3; - CFV #4; - CFV #5). For clarity, 

this nomenclature is also denoted in the legend of Fig. 4.2. Each data point in the figure is the 

average of a minimum of four repeated PVTt flow measurements at the same nominal flow. In 

general, the standard deviation of the four (or more in some cases) repeated flow measurements 

is 0.006 %. 

0.991

0.992

0.993

0.994

0.995

0.996

0.997

Re

Cd

105 106 107 108

0.1 %

Stage 3

HP CFV

Data

Stage 2

MP CFV

Data

MP & HP CFV

Curve Fit

LP CFV

Curve Fit

Turbulent

Flow

CFV Theory

LP CFVs 2004 2006

#2

#4

#5

#3

Laminar Flow

CFV Theory

0.998

Stage 1

LP CFV

core Re

range

0.991

0.992

0.993

0.994

0.995

0.996

0.997

Re

Cd

105 106 107 108 105 106 107 108

0.1 %

Stage 3

HP CFV

Data

Stage 2

MP CFV

Data

MP & HP CFV

Curve Fit

LP CFV

Curve Fit

Turbulent

Flow

CFV Theory

LP CFVs 2004 2006 LP CFVs 2004 2006

#2

#4

#5

#3

Laminar Flow

CFV Theory

0.998

Stage 1

LP CFV

core Re

range

Figure 4.2. Calibration data for the LP, MP, and HP CFVs. 

The theoretical Cd values for the laminar CFV flow model ( ) [viii] and the turbulent CFV 

flow model ( ) [viii] are plotted with the calibration data in Fig. 4.2 to clearly illustrate that 
18 

the LP CFV results lie almost entirely in the transitional flow regime. As expected, the portion of 

the calibration data below 6 Re < 10 closely follows the laminar flow model. However, at higher 

Reynolds numbers the data falls slightly below the turbulent flow model. The apparent 

discrepancy at higher Reynolds numbers could be the result of the flow not being fully turbulent. 

In any circumstance the difference (i.e., less than 0.1 %) is well within expected capability of the 

turbulent flow model. 

In total, the LP CFV results include 95 data points that span a Reynolds number range from 

6.5 × 105

 to 2.8 ×106. However, only CFV #3 and CFV #5 were calibrated over this extended 

Reynolds number range. The core Reynolds number range for which all four LP CFVs were 

calibrated extends from 1.1 × 106

 to 2.4 × 106, corresponding to stagnation pressures ranging 

from 350 kPa to 700 kPa at ambient temperatures. The data within the core Reynolds number 

range is used in Stage 2 when the array of LP CFVs are used to calibrate the MP CFVs. The 

Reynolds numbers values outside this core region were obtained primarily to ensure that the data 

followed the expected theoretical trends in the laminar and turbulent flow regimes. Within the 

core region, Cd values were measured at no less than 11 equally spaced Re to capture the 

changes in concavity that occur in the transitional flow regime. Considering that the core data 

includes four different LP CFVs, each calibrated twice two years apart, and that the data lies 

entirely in the transitional flow regime, the tight overlap between the data is remarkable. As 

indicated in the figure, the data in the core region for all four LP CFVs can be represented by a 

single calibration curve 

-2 5

2 -1 5

d, = 0 + 1 + LP C FIT b b Re b Re (4.2) 

where the coefficients b0 , 1b , and 2b are given in Table 4.2, and the standard deviation of the 

curve fit residuals is 0.018%. The same polynomial expression is used to fit the calibration data 

for the MP and HP CFVs, and their curve fit coefficients are also included in Table 4.2. Lastly, 

the curve fits shown in Fig. 4.2 agree to better than 0.05 % with data from PTB (flowing air at 

low flows and natural gas at high flows) and with LADG-LNE (flowing air) [xxii ]. This 

comparison was done using the following nozzles, CFV #2, CFV #3, CFV #4, and CFV #5. 

Table 4.2. Calibration Coefficients for LP, MP, and HP CFVs and the Reynolds number range where the fit is valid. 

CFV ReMIN ReMAX b0 b1 b2

(-----) (-----) (-----) (-----) (-----) (-----) 

LP CFVs 1.1 × 106 2.4 × 106 1.101 -3.917 35.683 

MP CFVs 3.69 × 106

 2.74 × 107 1.0003 -0.1323 0 

HP CFVs 3.69 × 106

 2.74 × 107 1.0003 -0.1323 0 
19 

Uncertainty Analysis of the LP CFVs

When the method of propagation of uncertainty is applied to Eqn. 3.2 (i.e., the calibration 

equation used for the LP CFVs in Stage 1) the result is2

( ) ( ) ( ) ( ) 2

0

0

2

0

0

2 2

d

d

LP1 LP1 LP1 4

1 ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥ +

⎦

⎤ ⎢

⎣

⎡

+ ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥ =

⎦

⎤ ⎢

⎣

⎡

T

u T

P

u P

m

u m

C

u C

&

& ( ) 2

LP1

4 ⎥

⎦

⎤ ⎢

⎣

⎡ +

d

u d ( ) 2

LP1 4

1

⎥

⎦

⎤ ⎢

⎣

⎡ +

M

u M

( ) 2

u

u

4

1 ⎥

⎦

⎤ ⎢

⎣

⎡

+

R

u R ( ) 2

s

s

LP1

⎥

⎦

⎤ ⎢

⎣

⎡

+

C

u C ( ) 2

d

d

LP

FIT

FIT

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

C

u C

(4.3) 

where the subscript “LP1” indicates the Stage 1 LP CFVs, and [ ] FIT FIT

d d u(C ) C is the standard 

deviation of the curve fit residuals. In this expression, the correlations of Cs with P0 and T0

have been neglected. In a more exact representation the normalized sensitivity coefficients of P0

and T0 would include the appropriate pressure and temperature derivatives of Cs . However, a 

sensitivity study showed that these dependencies could be omitted with negligible error in both 

Stages 1 and 2 where the pressure is sufficiently low. On the other hand, we include these 

dependencies in the uncertainty expressions in Stages 3, 4, and 5 where the pressure is 

substantially higher. Furthermore, the correlation between P0 and T0 (through their common 

dependence on the specific heat ratio, γ , and on the Mach number, M , [see Eqns. A1 and 

Eqn. A2 in Appendix A]) affects the uncertainty budget by less than 1 × 10-6 and is ignored. 

Table 4.3. Stage 1 Uncertainty Budget for the LP CFV Discharge Coefficient 

Unc. Components for Stage 1 LP CFVs Rel. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

LP CFV Discharge Coeff., Cd = 0.9936 (× 10-6) (-----) (%) (A or B) 

PVTt primary standard, ( m& = 675.2 g/s) 450 1.0 79.9 A, B Based on References [v-vii] 

Stag. Pres.; (P0 = 570.0 kPa) 118 1.0 5.5 A, B Fit residuals, drift, cal std, etc.

Stag. Temp.; (T0 = 296.0 K) 176 0.5 3.1 A, B Spatial Sampling error, drift, 

fit residuals, etc . 

Nominal Throat diameter; (d = 2.54 cm) 0* 2.0 0 B Nom. value is fixed betwn. 

Stages 1 and 2 

Molar Mass; (M = 28.9646 g/mol) 25 0.5 0.1 A, B Variation in air comp. [xix-xxi] 

Univ. gas constant; (Ru = 8314.47 J/kmol K) 2 0.5 0.0 A, B See Reference [xxiii] 

Critical Flow Function; (Cs,LP =0.6865 ) 0* 1.0 0 B Same flow cond. in Stages 1 

and 2 

CFV Curve Fit Residuals 170 1.0 11.4 A Fit Residuals, Hysteresis, and 

Reproducibility 

Combined Uncertainty 503 100 

 

2

 For convenience, all uncertainty formulas give the square of the actual value unless otherwise noted.
20 

Based on Eqn. 4.3, the Stage 1 uncertainty for the discharge coefficient was 0.05 % with a 

coverage factor of k = 1. The various uncertainty sources are itemized in Table 4.3. The 

abbreviated titles in the heading of the table, “Rel. Unc.”, “Sen. Coeff.”, “Perc. Contrib.”, and 

“Unc. Type”, are taken to mean the following: (1) standard relative uncertainty; (2) normalized 

sensitivity coefficient; (3) percent contribution of a single component to the overall uncertainty; 

and (4) the uncertainty type, respectively. By far, the largest component is the relative standard 

uncertainty of the PVTt primary standard (450 × 10-6). Brief explanations of the remaining 

uncertainty components are provided in Table 4.3 under the heading “Comments”. 

Throughout this document an asterisk (*) next to an uncertainty value in the “Rel. Unc.” column 

is used to indicate self-canceling measurement errors between adjacent stages. In Table 4.3 the 

asterisk is used next to the uncertainties for the nominal CFV throat diameter and the critical 

flow function. Both of these uncertainties are self-canceling since biases introduced in Stage 1 

identically cancel with biases of opposite polarity in Stage 2. For example, any bias in the value 

of the throat diameter used in Stage 1 cancels when the same value of d is used in Stage 2. This 

argument also applies to the critical flow function since the LP CFVs are used in Stage 2 for the 

same gas type (i.e., dry air) and at the same nominal conditions (i.e., P0 and T0 ) for which they 

were calibrated in Stage 1. 

4.2 STAGE 2: Calibration and Uncertainty of the MP CFVs 

Calibration of the MP CFVs using the LP CFVs 

In Stage 2, the four LP CFVs calibrated in Stage 1 are combined in a parallel array and used to 

calibrate four MP CFVs, one at a time. The calibration setup is shown in Fig. 4.3. Both the 

downstream LP CFVs and the upstream MP CFV have a CPA 50E flow conditioner3

 installed 

upstream of their respective pressure and temperature instrumentation. A heat exchanger is used 

to bring the cold jet exiting the MP CFV back to room temperature conditions before the flow is 

measured by the array of LP CFVs.

Du = 15.24 cm

Heat 

Exchanger

Tm P

Tm P

CPA 50E Flow

Conditioner

CPA 50E Flow

Conditioner LP CFVs

MP CFV

Direction

Flow

Dd = 30.48 cm

Du = 15.24 cm

Heat 

Exchanger

Tm P

Tm P

CPA 50E Flow

Conditioner

CPA 50E Flow

Conditioner LP CFVs

MP CFV

Direction

Flow

Dd = 30.48 cm

Figure 4.3. Schematic showing the setup for the Stage 2 calibration of the MP CFV using four LP CFVs calibrated 

in Stage 1 (figure not drawn to scale) 

 

3 Throughout this document certain commercial equipment, instruments, or materials are identified 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.
21 

The operating conditions for the downstream LP CFVs are controlled so that P0 and T0 are 

nearly equal to their Stage 1 values. Because the working fluid is the same as that in Stage 1 (i.e.,

dry air), the resulting Reynolds number range corresponds to the values used in the Stage 1 LP 

CFV calibration. By applying the principle of conservation of mass and using the LP CFVs as 

working standards, the mass flow through the upstream MP CFV is

2

n 1

∑ th,n d,n Δ

2

LP2 LP

mMP2 m C m

N

& = & + &

=

(4.4) 

where N2 = 4 is the number of LP CFVs mounted in the downstream nozzle fixture, and LP2

mth,n &

is the Stage 2 theoretical mass flow for the LP CFVs which is calculated using Eqn. 3.1; and 

Δm2& is the rate of mass storage in the connecting volume between the array of LP CFVs and the 

MP CFV. The mass storage (i.e., line pack) term accounts for density transients in the connecting 

volume. Because Δm2& is small relative to mMP2 & , we set Δm2& equal to zero in Eqn. 4.4; however, 

the uncertainty attributed to line pack is included in the uncertainty budget. 

The calibration procedure begins by establishing steady-state flow conditions at the desired 

stagnation pressure. Subsequently, pressure and temperature data is collected for approximately 

360 s. The data acquisition system cycles at 1.5 Hz, resulting in approximately 540 pressure and 

temperature data points. At each data point P0 , T0 , Cs , and Re are computed at both the LP 

CFVs and at the MP CFV using the methods explained in Section 3.2 and in Appendix A. These 

values are used to calculate the MP CFV discharge coefficient 

∑

2

LP

MP

LP

MP2 MP2

LP2 LP2

LP2

MP2

MP

MP

n 1

2 d,n

2

,n

0 s,

0 s,

0

0

th

d ≡

N

,

,

,

, C

d

d

P C

P C

T

T

m

m C

= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥ =

⎦

⎤ ⎢

⎣

⎡

&

& (4.5) 

where the subscripts “LP2” and “MP2” correspond, respectively, to the Stage 2 LP and MP 

CFVs. The CFV throat diameters are given in Table 3.1, and ( ) LP

LP

FIT

LP Cd,n = Cd, Re is the LP CFV 

calibration curve given in Eqn. 4.2. The reported MP Cd values are averaged over the 360 s data 

collection interval. In the worst case, the standard deviation of the mean for this averaging 

process is 0.002 %, which is negligible relative to other sources of uncertainty, and is therefore 

not included in the uncertainty budget. 

The MP Cd values are measured at a minimum of five discrete ReMP values. A minimum of two 

MP Cd measurements are made at each of the five Reynolds numbers so that a total of no less than 

ten measurements are made. A typical calibration begins at the minimum pressure set point, 

P0,LP2 =375 kPa (or P0,MP2 = 1500 kPa). The set point is increased in equal increments until 

reaching its maximum value at the fifth pressure set point, P0,LP2 =630 kPa (or 

P0,MP2 = 2520 kPa). After finishing the fifth data collection, the flow is shutdown (i.e., zero flow) 

and then reestablished at the maximum set point. The sixth data collection is taken at the 

maximum set point. The pressure set point is subsequently decreased in equal increments until 

reaching its minimum value at the tenth set point. This method of collecting data accounts for 

repeatability, short term reproducibility, and hysteresis effects. 
22 

The averaged MP Cd data is depicted in Fig. 4.2 (shown earlier in Section 4.1) where the symbol 

( ) represents the results of all four MP CFVs. This data set includes a total of 77 points and 

spans a Reynolds number range from 3.7 × 106

 to 8.6 × 106. Unlike the LP CFV data, the MP 

CFV data is entirely within the turbulent flow regime. Moreover, the entire MP CFV data set can 

be fit to a single calibration curve 

MP -1 5 Cd,FIT = b0 + b1 Re (4.6) 

where the coefficients b0 and 1b are given in Table 4.2. Considering that the data corresponds to 

five different CFVs, the small degree of scatter in the data is remarkable; moreover, the standard 

deviation of the curve fit residuals is only 0.017 %. Perhaps more remarkable is the fact that the 

same curve also fits the eight HP CFVs in Section 4.3. 

Uncertainty Analysis of the MP CFVs

The uncertainty of the MP CFV discharge coefficients is determined by applying the law of 

propagation of uncertainty to Eqn. 4.5. Because all four LP CFVs are traceable to the same 

calibration standard, many of their common uncertainty sources are correlated. When these 

correlations are taken into account, the resulting expression for uncertainty is 

( ) ( ) ( ) 2

d

d

2

2

2

d

d

LP

LP

MP

1 -1 ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + ⎥ =

⎦

⎤ ⎢

⎣

⎡

C

u C

N

r N

C

u C ( ) ( ) 2

0

0

2

0

0

MP2 LP2

⎥

⎦

⎤ ⎢

⎣

⎡ ⎥ +

⎦

⎤ ⎢

⎣

⎡

+

P

u P

P

u P

( ) 2

0

0

MP2

4

1 ⎥

⎦

⎤ ⎢

⎣

⎡

+

T

u T ( ) 2

0

0

LP2

4

1 ⎥

⎦

⎤ ⎢

⎣

⎡

+

T

u T ( ) 2

MP2

4 ⎥

⎦

⎤ ⎢

⎣

⎡ +

d

u d ( ) 2

LP2

4 ⎥

⎦

⎤ ⎢

⎣

⎡ +

d

u d

( ) 2

s

s

MP2

⎥

⎦

⎤ ⎢

⎣

⎡

+

C

u C ( ) 2

s

s

LP2

⎥

⎦

⎤ ⎢

⎣

⎡

+

C

u C ( ) 2

d

d

MP

FIT

FIT

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

C

u C 2 + uIE,2

2

2

MP2

Δ

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

m

m

&

&

(4.7) 

where d d LP [u(C ) C ] in the first term is the uncertainty for a single LP CFV used by itself. Its 

uncertainty is determined using Eqn. 4.3 and its value is given in Table 4.3. The uncertainty of 

all four LP CFVs used together is given by the entire first term, including d d LP [u(C ) C ] and the 

coefficient in parenthesis to which it is multiplied. The coefficient multiplying d d LP [u(C ) C ]

accounts for the correlation between the four LP CFVs where the correlation coefficient ( LP r ) 

specifies the degree of correlation. In the hypothetical case when = 0 LP r (i.e., no correlated 

sources of uncertainty), the coefficient multiplying d d LP [u(C ) C ] is 2 1 N , indicating that the 

uncertainty of all four LP CFVs used together is less than their individual use. On the other hand, 

if = 1 LP r (i.e., perfectly correlated uncertainty sources), the coefficient multiplying 

d d LP [u(C ) C ] is unity, and there is no difference in uncertainty between using a single LP CFV 

or multiple LP CFVs together. In the scope of the current work we expect the correlation 

coefficient to be close to unity since many of the uncertainty sources are correlated. Using the 

method outlined in reference [xv], we calculated the correlation coefficient to be = 0 92 LP r . . 
23 

The last three terms in Eqn. 4.7 are the uncertainties attributed to the standard deviation of the 

MP CFVs curve fit residuals, MP

FIT FIT [ ] d d u(C ) C ; followed by the uncertainty caused by 

interference effects (i.e., cross-talk) between the four downstream LP CFVs, (uIE,2 ); and lastly 

the uncertainty attributed to the line packing effect, [ MP2 ] Δm& 2 m& . The uncertainty attributed to 

interference effects was measured experimentally by varying the number of open LP CFVs 

(between two and four) in the nozzle fixture, and comparing the performance to an upstream 

reference CFV held at constant mass flow. These tests were done on three different occasions 

and showed that the maximum influence of interference effects was 500 × 10-6. Assuming a 

rectangular distribution the standard uncertainty attributed to interference effects is 289 × 10-6. 

The uncertainty attributed to the line packing effect was assessed by multiplying the connecting 

volume by the average rate of density change (in the connecting volume) during the collection. 

An itemized list of all the uncertainty components along with brief explanations of their 

uncertainty values are given in Table 4.4. 

Table 4.4. Stage 2 Uncertainty Budget for the MP CFV Discharge Coefficient. 

Unc. Components for Stage 2 MP CFVs. Rel. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

MP CFV Discharge Coeff., Cd = 0.9936 (× 10-6) (-----) (%) (A or B) 

Discharge Coeff. For array of 4 LP CFVs, 

( LP Cd = 0.9936) 487 1 57.6 A, B Stage 1 calibration (Corr. 

effects included) 

MP Stag. Pres.; (P0 = 2280 kPa) 125* 1 3.7 A, B Uncorr. Unc.: Random Effects, 

Data Acq., Barometric. Pres.

LP Stag. Pres.; (P0 = 570 kPa) 129 1 3.9 A, B Cal. Std., Random Effects, 

Data Acq., Barometric Pres. 

MP Stag. Temp.; (T0 = 295 K) 230 0.5 3.1 A, B Cal. Std., Random Effects., 

Spatial Non-uniformity, etc. 

LP Stag. Temp.; (T0 = 291 K) 241 0.5 3.5 A, B Cal. Std., Random Effects., 

Spatial Non-uniformity, etc. 

MP Critical Flow Function; (Cs,MP = 0.6912) 0* 1 0 A,B Perfectly correlated betwn. 

Stages 2 and 3 

LP Critical Flow Function; (Cs,LP = 0.6865) 0* 1 0 A,B Perfectly correlated betwn. 

Stages 1 and 2 

Nominal Diameter MP CFV, (d = 2.54 cm) 0* 2 0 B Perfectly correlated betwn. 

Stages 2 and 3 

Nominal Diameter LP CFV, (d = 2.54 cm) 49* 2 2.3 B Unc. for thermal expansion 

betwn. Stages 1 and 2 

LP CFV Curve Fit Residuals 170 1 6.9 A Std. Dev. of Curve Fit 

Residuals, (see Section 4.1) 

Interference effects of multiple CFVs in a 

Common Plenum 289 1 19.8 B Exp. varying number of open 

CFVs in nozzle fixture 

Line pack effect 27 1 0.2 B Based on measured Vdρ dt in 

connecting volume during cal.

Combined Uncertainty 648 100 

When the uncertainties listed in Table 4.4 are used in Eqn. 4.7 the relative standard uncertainty 

for MP Cd is 648 ×10-6 (k = 1). As expected, the largest uncertainty contribution derives from the 
24 

Stage 1 calibration of the four LP CFVs. The value for this uncertainty (487 × 10-6) is slightly 

less than that given in Table 4.3 for the Stage 1 analysis because < 1 LP r . As discussed in 

Section 4.1, the asterisk (*) next to selected uncertainty components indicates our assumption 

that the correlated measurement errors between adjacent stages identically cancel. In the cases 

where an asterisk is positioned next to the pressure or temperature uncertainties the correlation is 

caused by the same transducer being used to measure the same nominal conditions in adjacent 

stages. In this case the listed uncertainties consist entirely of the uncorrelated uncertainty 

sources. When there is ambiguity as to whether an uncertainty source is perfectly correlated, we 

conservatively define it to be uncorrelated. 

4.3 STAGE 3: Calibration and Uncertainty of the HP CFVs 

Calibration of the HP CFVs 

In Stage 3 the MP CFVs are configured in parallel and used to calibrate a total of eight HP CFVs, 

one at a time, in dry air. The calibration setup is similar to the Stage 2 setup shown in Fig. 4.3, 

but in this case a single HP CFV is positioned upstream of the four MP CFVs configured in 

parallel. The Stage 3 stagnation pressure at the MP CFVs is controlled so that the Reynolds 

number overlaps values from the previous calibration stage. The resulting Reynolds number 

range for the MP CFVs varies from 5.4 × 106

 to 7.5 × 106, and the Reynolds number range for 

the upstream HP CFV varies from 2.0 × 107

 to 2.75 × 107. The pressures and flows of the HP 

CFVs are approximately 16 times the Stage 1 values for the LP CFVs. 

The calibration procedure is similar to the calibration of the MP CFVs. After establishing steady 

state flow conditions in the pipeline, data is collected for approximately 200 s resulting in 

approximately 270 data points. At each data point we compute P0 , T0 , Cs , and Re at the MP 

CFVs and at the HP CFV using the methods described in Section 3.1. These values are used to 

calculate the HP CFV discharge coefficient 

∑

3

MP

HP3

MP3

HP3 HP3

MP3 P3

MP3

HP3

HP

HP

1

2 d,n

2 ,n

0 s,

0 s,M

0

0

th

d

m

m

≡

N

, n

,

,

, C

d

d

P C

P C

T

T

C

= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥ =

⎦

⎤ ⎢

⎣

⎡

&

& (4.8) 

where the subscripts “MP3” and “HP3” correspond, respectively, to the Stage 3 MP and HP 

CFVs, and 4 N3 = is the number of MP CFVs mounted in the downstream nozzle fixture. The 

throat diameters of the CFVs are obtained from Table 3.1, and ( MP ) MP

FIT

MP Cd,n = Cd, Re is the MP 

CFV calibration curve given by Eqn. 4.6. The reported HP Cd values are averaged over the 200 s 

data collection interval. The averaging process introduces negligible uncertainty since in the 

worst case, the standard deviation of the mean is less than 0.001 %. 

For each HP CFV, HP Cd measurements are made at a minimum of three discrete ReHP values. In 

most cases these measurements were made on two separate occasions so that six measurements 

are made in total. The format for collecting the data is analogous to the procedure described in 

Section 4.2. The entire data set for all eight HP CFVs consists of 43 points. The data is shown in 

Fig. 4.2 (shown earlier in Section 4.1) where the same symbol ( ) is used to represent all eight 

HP CFVs. The eight HP CFVs fit a single calibration curve 
25 

HP -1 5 Cd,FIT = b0 + b1 Re (4.9) 

where the coefficients b0 and 1b are given in Table 4.2, and the standard deviation of the curve 

fit residuals is 0.02%. 

Uncertainty Analysis of the HP CFVs

The uncertainty of the HP CFVs is determined by applying the method of propagation of 

uncertainty to Eqn. 4.8. The resulting expression of uncertainty is analogous to Eqn. 4.7 for the 

MP CFVs 

( )⎥ =

⎦

⎤ ⎢

⎣

⎡ 2

d

d

HP C

u C ( ) ( ) 2

d,n

d,n

3

3

MP

1 MP -1

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ +

C

u C

N

r N ( ) 2

0

0

2

0

s

s

0

HP3 HP3 ∂

∂ 1 ⎥

⎦

⎤ ⎢

⎣

⎡

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ +

P

u P

P

C

C

P

( ) 2

0

0

2

0

s

s

0

HP3 HP3 ∂

∂ -

2

1 ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

T

u T

T

C

C

T ( ) 2

0

0

MP3

⎥

⎦

⎤ ⎢

⎣

⎡

+

P

u P ( ) 2

0

0

MP3 4

1 ⎥

⎦

⎤ ⎢

⎣

⎡

+

T

u T

( ) 2

s

s

HP3

⎥

⎦

⎤ ⎢

⎣

⎡

+

C

u C ( ) 2

s

s

MP3

⎥

⎦

⎤ ⎢

⎣

⎡

+

C

u C ( ) 2

HP3

4 ⎥

⎦

⎤ ⎢

⎣

⎡ +

d

u d ( ) 2

MP3

4 ⎥

⎦

⎤ ⎢

⎣

⎡ +

d

u d

( ) 2

d

d

MP

FIT

FIT

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

C

u C 2 + uIE,3

2

3

HP

Δ ⎥

⎦

⎤ ⎢

⎣

⎡

+

m

m

&

&

(4.10) 

where in this case the MP CFV correlation coefficient is MP r = 0.95. The normalized sensitivity 

coefficients for 0 0 HP [u(P ) P ] and 0 0 HP [u(T ) T ] include the pressure and temperature derivatives 

of Cs , respectively. These additional terms take into account the uncertainty in Cs attributed to 

uncertainties in the stagnation conditions. At the lower pressures in Stages 1 and 2 these 

additional terms made a negligible contribution to the uncertainty and were omitted. However, at 

the elevated pressures in Stages 3 through 5 they are not negligible and are therefore included. 
26 

Table 4.5. Stage 3 Uncertainty Budget for the HP CFV Discharge Coefficient. 

Unc. Components for Stage 3 HP CFVs. Rel. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

HP CFV Discharge Coeff., Cd = 0.9960 (× 10-6) (-----) (%) (A or B) 

Discharge Coeff. For array of 4 MP CFVs, 

( MP Cd = 0.9947) 635 1 56.4 A, B Stage 2 calibration (Corr. 

effects included) 

HP Stag. Pres.; (P0 = 9120 kPa) 279 1.03 11.6 A, B Pres. Cal. Std., Random Effects, 

Data Acq., Barometric. Pres.

MP Stag. Pres.; (P0 = 2280 kPa) 123* 1 2.1 A, B 

Uncorr. Unc: Random Effects, 

Data Acq., Barometric Pres. 

(Corr. Unc. are self-canceling) 

HP Stag. Temp.; (T0 = 295 K) 238 0.64 3.2 A, B See Reference [xv] (Corr. Unc. 

are self-canceling)

MP Stag. Temp.; (T0 = 290 K) 234* 0.5 1.9 A, B See Reference [xv] (Corr. Unc. 

are self-canceling)

HP Critical Flow Function; (Cs,HP = 0.7097) 215 1 5.8 A,B Comp. of four Therm. 

Databases [ii, xxiv-xxvxxvi] 

MP Critical Flow Function; (Cs,MP = 0.6916) 0* 1 0 A,B Perfectly correlated betwn. 

Stages 2 and 3 

Nominal Diameter HP CFV, (d = 2.54 cm) 0* 2 0 B Perfectly correlated betwn. 

Stages 3 and 4 

Nominal Diameter MP CFV, (d = 2.54 cm) 49* 2 1.4 B Unc. for thermal expansion 

betwn. Stages 1 and 2 

MP CFV Curve Fit Residuals 200 1 5.6 A Std. Dev. of Curve Fit 

Residuals, (see Section 4.2) 

Interference effects of multiple CFVs in a 

Common Plenum 289 1 11.7 B See Section 4.2 

Line pack effect 44 1 0.3 B Based on measured Vdρ dt in 

connecting volume during cal.

Combined Uncertainty 849 100 

4.4 STAGE 4: Calibration and Uncertainty of the TMWS 

Calibration of the TMWS

In Stage 4 all nine TMWS were individually calibrated in natural gas at their place of use and at 

their nominal operating conditions. Each TMWS was calibrated against the HP CFVs that were 

calibrated in air in Stage 3. The air based calibration can be applied to nozzles flowing natural 

gas by accounting for real gas effects via the critical flow function, and by matching the 

Reynolds number. At these high Reynolds numbers (i.e., Re > 16 x 106

) turbulent CFV theory 

predicts that the difference between the Cd values for air and natural gas are less than 

0.01 % [xxii]. Because the viscosity of natural gas is less than dry air, matching the Reynolds 

number required that P0 in the natural gas flow be approximately 20 % lower than its value in 

dry air. During the calibration process the HP CFVs were maintained below their measured 

choking pressure ratio (i.e., pressure downstream of the HP CFVs divided by the stagnation 

pressure) of 0.942. 
27 

Du = 30.48 cm

Dd = 76.2 cm

Flow

Conditioner HP CFV

Array

TMWS

P

f

T

Tm1

P Tm2

Ultrasonic

Flowmeter

Body

Ts

Flow Direction

Du = 30.48 cm

Dd = 76.2 cm

Flow

Conditioner HP CFV

Array

TMWS

P

f

T

Tm1

P Tm2

Ultrasonic

Flowmeter

Body

Ts

Flow Direction

Figure 4.4. Schematic showing the setup for the Stage 4 calibration of the TMWS using the array of HP CFVs 

calibrated in Stage 3 (not drawn to scale) 

The TMWS were calibrated twice, once in May 2006, and again in June 2007. A schematic of 

the calibration setup is shown in Fig 4.4 including the auxiliary pressure, temperature, and 

frequency measurements. For both tests, redundant instrumentation was used to measure 

frequency and pressure at the TMWS, and averaged values were used for these two 

measurements. The temperature at the TMWS was measured with a single resistance temperature 

device (RTD) installed in a thermowell. Pre-calibration tests showed the maximum effect of the 

thermowell on the measured temperature was 50.8 mK (0.017 %). 

A flow conditioner was installed downstream of the TMWS just before the array of HP CFVs. 

The conditioner was used to reduce jetting effects as the flow transitions from the smaller 

30.48 cm (12 inch) diameter pipeline to the larger 76.2 cm (30 inch) diameter pipeline. Just 

downstream of the flow conditioner the first temperature measurement (i.e., Tm1 in Fig. 4.4) was 

made by averaging the readings of 10 RTDs spaced at equal distances around the circumference 

of the pipe. The lengths of the RTDs vary so that their penetration depth into the flow stream 

ranged from 5 cm (i.e., near the pipe wall) to 36 cm (i.e., near the pipe centerline). In this way 

the temperature was sampled at multiple radii across the cross section. The uncertainty attributed 

to temperature variation within the cross section was taken to be the standard deviation of the 

10 RTDs. Two pipe diameters ( Dd ) downstream, just before the array of HP CFVs, a second 

temperature measurement was made (i.e., Tm2 in Fig. 4.4) using 10 RTDs evenly distributed 

around the circumference of the pipeline. None of the RTDs at either cross section were installed 

in thermowells. 

Figure 4.5 includes four pictures showing the flow conditioner (A); the calibration setup (B); the 

10 RTDs used to measure Tm2 (C); and the HP CFV array (D). The eight HP CFVs calibrated in 

Stage 3 were installed in the twenty-one aperture nozzle fixture. The nozzle fixture was installed 

in the pipeline that houses an ultrasonic flowmeter. Although the ultrasonic flowmeter was not 

used in the current work it can provide additional temperature information in the future if 

necessary. To this point, all temperature information was determined using the RTDs. For the 

May 2006 test only the 10 RTDs that measure Tm1 were used. The second set of 10 RTDs used 

to measure Tm2 were added for the June 2007 tests to determine the uncertainty attributed to 

axial temperature gradients. The static pressure instrumentation (not shown in Fig. 4.5B) was 

located approximately one diameter upstream of the HP CFVs on the bottom of the pipe. 
28 

Equations 3.1 and 3.2 in Appendix A were used to calculate the stagnation pressure and 

temperature, respectively. 

Flow Direction

HP CFV

Nozzle Fixture

Tm1

Tm2

Location of

Flow Conditioner

(Viewed from Downstream) (Viewed from Downstream) 

Flow Conditioner

10 RTD’s used to Measure Tm2 HP CFV Array Installed in Pipeline 

A B

C D

(Viewed from Upstream) 

HP CFV Array Setup

Flow Direction

HP CFV

Nozzle Fixture

Tm1

Tm2

Location of

Flow Conditioner

(Viewed from Downstream) (Viewed from Downstream) 

Flow Conditioner

10 RTD’s used to Measure Tm2 HP CFV Array Installed in Pipeline 

A B

C D

(Viewed from Upstream) 

HP CFV Array Setup

Flow Direction

HP CFV

Nozzle Fixture

Tm1

Tm2

Location of

Flow Conditioner

(Viewed from Downstream) (Viewed from Downstream) 

Flow Conditioner

10 RTD’s used to Measure Tm2 HP CFV Array Installed in Pipeline 

A B

C D

(Viewed from Upstream) 

HP CFV Array Setup

Figure 4.5. Four pictures showing the flow conditioner (A); the configuration of HP CFV array (B); the 10 RTD’s 

used to measure Tm2 (C); and twenty-one nozzle fixture installed in pipeline (D). 

The protocol for calibrating the TMWS is shown in Table 4.6. At the start of the calibration all 

eight HP CFVs in the nozzle fixture were capped, additional apertures in the nozzle fixture not 

containing CFVs were sealed, and the TMWS shutoff valves were closed. Subsequently, the 

pipeline was pressurized to the operating pressure and leak checks are performed. After 

eliminating any leaks, the end caps were removed from two of the CFVs (i.e., CFV #8 and 

CFV #17). When steady state conditions were obtained each of the nine TMWS was individually 

calibrated at the first flow set point. Data was collected for 120 s at each set point, and flow 

measurements were repeated a minimum of three times at each set point. An additional CFV was 

uncapped (i.e., opened) at each subsequent set point until all eight HP CFVs were opened as 

shown in the table. This procedure resulted in each TMWS being calibrated at seven set points 

over a flow range extending from 0.25 m3

/s (3.2 × 104 acfh) to 1 m3/s (1.3 × 105 acfh) at a 

reference temperature near ambient conditions and a reference pressure of approximately 

7500 kPa. 
29 

Table 4.6. Calibration protocol for the nine TMWS using the eight HP CFVs. Note that the symbol “O” indicates 

an open CFV (i.e., end cap removed) and the symbol “X” indicates a closed CFV (i.e., end cap securely 

fastened to divergent section of the CFV). Reference conditions for volumetric flows in the table are at 

ambient temperature and nominally at 7500 kPa. 

Set 

Point 

No.

Nominal 

Flow 

No. of 

Open 

CFVs

HP CFVs

(-----) (m3/s) (acfh) (-----) CFV 

#8 

CFV 

#17 

CFV 

#7 

CFV 

#14 

CFV 

#20 

CFV 

#15 

CFV 

#1 

CFV 

#19 

1 0.25 3.2 × 104 2 O O X X X X X X

2 0.375 4.8 × 104 3 O O O X X X X X

3 0.5 6.4 × 104 4 O O O O X X X X

4 0.625 7.9 × 104 5 O O O O O X X X

5 0.75 9.5 × 104 6 O O O O O O X X

6 0.875 1.1 × 105 7 O O O O O O O X

7 1 1.3 × 105 8 O O O O O O O O

Mass conservation can be used to show that the flow at the TMWS being calibrated is 

m C m

q ρ ρ

N

TMWS

4

TMWS

HP HP

TMWS

4

n=1

th,n d,n Δ = ∑ + & & (4.11) 

where N4 is the number of open HP CFVs which varies between two and eight; Δm4 & is the rate 

of mass storage in the connecting volume between the TMWS and the nozzle bank; and 

TMWS [ ] ρTMWS = PMgas ZRuT is the density of natural gas at the TMWS where Z Z(P,T,x ) = k

is the compressibility factor calculated using the REFPROP 8 thermodynamic database [ii] and 

gas = ∑ kM k M x , is the mixture’s molar mass. The gas composition ( xk ) was measured during 

the testing with an industrial grade gas chromatograph (GC) that was calibrated prior to the 

testing using a certified gas standard. The theoretical mass flow of the nth HP CFV ( HP

mth,n & ) was 

determined using Eqn. 3.1. The HP CFV discharge coefficient was determined via the calibration 

curve fit, = ( ) HP

HP

FIT

HP Cd,n Cd, Re , given by Eqn. 4.9 where the throat Reynolds number, ( R HP e ) 

was calculated using Eqn. 3.3. 

The calibration performance of each of the nine TMWS was determined by the K-factor, 

( ≡ TMWS K f q ) a ratio of the turbine blade frequency ( f ) divided by volumetric flow ( qTMWS ). 

Combining Eqn. 4.11 with the K-factor definition yields 

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞ ⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

=

∑

4

HP

HP

TMWS HP 1

d,n 2 ,n

0

0 s

gas

u

4

N

n

d C

P

Z T

T

R P C

f K

M

π (4.12) 

where Δm4& from Eqn. 4.11 is set to zero when determining the K-factor, but accounted for in the 

uncertainty budget. Figure 4.6 shows the K-factors for all nine TMWS plotted versus the 
30 

volumetric flow for the May 2006 data ( ) and the June 2007 data ( ). The solid line ( ) is a 

polynomial curve fit of the data 

3

3

2

FIT = 0 + 1 TMWS + 2 TMWS + TMWS K c c q c q c q (4.13) 

where the curve fit coefficients 0c , 1c , 2 c , and 3c are given in Table 4.7. 

210.2

210.4

210.6

210.8

211.0

211.2

211.4

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #1

0.1 %

209.4

209.6

209.8

210.0

210.2

210.4

0.2 0.4 0.6 0.8 1.0

TMWS #2

0.1 %

TMWS q

210.2

210.4

210.6

210.8

211.0

211.2

211.4

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #1

0.1 %

210.2

210.4

210.6

210.8

211.0

211.2

211.4

0.2 0.4 0.6 0.8 1.0

210.2

210.4

210.6

210.8

211.0

211.2

211.4

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #1

0.1 %

209.4

209.6

209.8

210.0

210.2

210.4

0.2 0.4 0.6 0.8 1.0

TMWS #2

0.1 %

TMWS q

209.4

209.6

209.8

210.0

210.2

210.4

0.2 0.4 0.6 0.8 1.0

TMWS #2

0.1 %

TMWS q

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #3

0.1 %

209.4

209.6

209.8

210.0

210.2

0.2 0.4 0.6 0.8 1.0

TMWS #4

TMWS q

0.1 %

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #3

0.1 %

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #3

0.1 %

209.4

209.6

209.8

210.0

210.2

0.2 0.4 0.6 0.8 1.0

TMWS #4

TMWS q

0.1 %

209.4

209.6

209.8

210.0

210.2

0.2 0.4 0.6 0.8 1.0

TMWS #4

TMWS q

0.1 %

209.6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS #5

TMWS q

0.1 %

TMWS #6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8

TMWS q

1.0

0.1 %

209.6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS #5

TMWS q

0.1 %

209.6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS #5

TMWS q

0.1 %

TMWS #6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8

TMWS q

1.0

0.1 %

TMWS #6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8

TMWS q

1.0

0.1 %
31 

208.2

208.6

209.0

209.4

209.8

210.2

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #7

0.2 %

TMWS q

207.2

207.4

207.6

207.8

208.0

208.2

208.4

208.6

0.2 0.4 0.6 0.8 1.0

TMWS #8

0.1 %

208.2

208.6

209.0

209.4

209.8

210.2

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #7

0.2 %

208.2

208.6

209.0

209.4

209.8

210.2

0.2 0.4 0.6 0.8 1.0

TMWS q

TMWS #7

0.2 %

TMWS q

207.2

207.4

207.6

207.8

208.0

208.2

208.4

208.6

0.2 0.4 0.6 0.8 1.0

TMWS #8

0.1 %

TMWS q

207.2

207.4

207.6

207.8

208.0

208.2

208.4

208.6

0.2 0.4 0.6 0.8 1.0

TMWS #8

0.1 %

209.4

209.6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS #9

TMWS q

0.1 %

209.4

209.6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS #9

TMWS q

0.1 %

209.4

209.6

209.8

210.0

210.2

210.4

210.6

0.2 0.4 0.6 0.8 1.0

TMWS #9

TMWS q

0.1 %

Figure 4.6. K-factor (pulse/m3) calibration curves ( ) for all nine TMWS plotted versus volumetric flow 

(m3

/s) with the squares ( ) denoting the May 2006 data and the triangles ( ) signifying the June 2007 

data. (The picture at the lower right shows the nine TMWS.) 

Table 4.7. Fit coefficients for the K-factors of the nine TMWS given in Eqn. 4.13 along with the standard 

deviation of the fit residuals. 

Position Curve Fit Coefficients 

Number 

TMWS 

Serial 

Number 0c 1c 2c 3c

Std. Dev. 

of Fit 

Residuals

(-----) (-----) (pulse/m3) (pulse·s/m6) (pulse·s3/m9) (pulse·s3/m12) (%) 

1 97500360 210.911 -0.380 0 0 0.067 

2 95380241 210.459 -2.103 1.20 0 0.05 

3 97500361 210.306 -0.205 -0.001 0 0.043 

4 97500359 209.982 -0.392 0 0 0.067 

5 95070034 210.57 -1.99 1.265 0 0.059 

6 97310070 210.64 -1.559 0.973 0 0.048 

7 94200263 210.692 -4.404 2.233 0 0.056 

8 95380242 209.579 -6.208 6.872 -2.753 0.053 

9 97470201 210.678 -1.926 1.029 0 0.051 
32 

The May 2006 and June 2007 data sets generally showed good agreement, and were well within 

the K-factor measurement uncertainty of 0.24 % to 0.25 %. Excluding TMWS #4, the two sets of 

data overlap at the majority of flows. In general, the reproducibility between the two data sets 

(i.e., percent difference of the average K-factor’s of the May and June data sets) was better at 

high flows than at lower flows. The larger reproducibility at lower flows could be attributed to 

problems measuring the lower frequencies in the May data. The larger reproducibility could also 

in part be caused by temperature stratification in the 70.62 diameter pipeline at the HP CFV 

array or by line pack (i.e., mass storage effects) resulting from temperature transients in the 

connecting volume between the TMWS and the HP CFVs. Both of these phenomena are more 

prevalent at lower flows. In this work we incorporated the uncertainty associated with 

reproducibility into the calibration curve fit given in Eqn. 4.13. This was done by using both the 

May 2006 and the June 2007 data in the curve fitting process. On average, the standard deviation 

of the fit residuals is 0.056 %. The individual values for each TMWS are given in Table 4.7. 

The K-factors curve fits of the nine TMWS are approximately the same magnitude and generally 

slope downward with increasing flow. A few of TMWS (i.e., #2, #5, and #6) have K-factors that 

reach a minimum value near 0.8 m3

/s before increasing slightly over the remainder of the 

calibrated flow range. For both the May 2006 and June 2007 data sets the repeatability (i.e., 

standard deviation of three or more repeated points at the same nominal flow taken sequentially) 

generally improved at higher flows. In the May 2006 data set the repeatability was generally 

0.02 % at high flows and 0.035 % at low flows. The repeatability of the June 2007 data was 

generally better than the May 2006 data set, being less than 0.01 % at high flow and 

approximately 0.02 % at low flows. 

Uncertainty Analysis for the Calibration of the TMWS

The uncertainty in the K-factor is determined by applying the law of propagation of uncertainty 

to Eqn. 4.12. The resulting expression for relative uncertainty is 

( ) = ⎥

⎦

⎤ ⎢

⎣

⎡ 2

K

u K ( ) 2

d

d

4

4

HP

HP 1 -1 ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ +

C

u(C )

N

r N ( ) 2

s

s

HP4

⎥

⎦

⎤ ⎢

⎣

⎡

+

C

u C ( ) 2

TMWS4

⎥

⎦

⎤ ⎢

⎣

⎡ +

Z

u Z ( ) 2

⎥

⎦

⎤ ⎢

⎣

⎡

+

f

u f

( ) 2

u

u

4

1 ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎠

⎞ ⎜

⎝

⎛ +

R

u R ( ) 2

p

p

HP

4

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

d

u d ( ) k

2 + utot M ( ) k2 + tot u x u ( ) P 2 + tot

u ( ) T 2 + tot

( ) 2

FIT

FIT ⎥

⎦

⎤ ⎢

⎣

⎡

+

K

u K 2 + uIE,4

2 + unchoke u

2 + species u

2

4

TMWS TMWS

Δ ⎥

⎦

⎤ ⎢

⎣

⎡

+

q

m

ρ

&

(4.14) 

where the terms ) ( k

2

tot u x , u ( ) P 2

tot , u ( ) T 2tot , and ) ( k

2

utot M are interim variables, that group 

together like uncertainty terms. For example, u ( ) P 2

tot is the combined pressure uncertainty for 

both the TMWS and the HP CFVs. Detailed explanations of these four terms are discussed in 

Appendix B. The last five terms in Eqn. 4.14 are the following: (1) [u(KFIT ) KFIT] is the standard 

deviation of the curve fit residuals; (2) uIE,4 is the uncertainty attributed to interference (or crosstalk) effects caused by the spacing between the CFVs in the nozzle fixture; (3) uunchoke is the 

uncertainty attributed to CFV unchoking effects; (4) uspecies is the uncertainty in the CFV 
33 

discharge coefficient attributed to applying the air calibrated CFVs in natural gas at the same 

Reynolds number; and (5) Δm4 & is the uncertainty attributed to the line packing effect. 

The K-factor uncertainty decreases slightly with increasing volumetric flow. At the lowest flow 

(i.e., N4 = 2), the uncertainty is 1 231 × 10-6 for k = 1 while at the highest flow (i.e., N4 = 8) 

the uncertainty decreases to 1 190 × 10-6 for k = 1. The decrease in uncertainty at higher flows is 

a result of several factors including 1) reduced temperature transients in the connecting volume 

leading to less line pack, 2) reduced spatial temperature gradients in the 76.2 cm diameter pipe 

immediately upstream of the array of HP CFVs, and 3) a lower uncertainty contribution from the 

HP CFVs (i.e., the coefficient multiplying d d HP [u(C ) C ] in Eqn. 4.14 decreases as N4

increases). 

An itemized list of all of the uncertainty components is provided in Table 4.8 for the highest flow 

where qTMWS = 1 m3

/s and N4 = 8. As expected, the largest source of uncertainty (812 × 10-6) 

derives from the Stage 3 calibration of the HP CFVs. The uncertainty value given in Table 4.8 

for the discharge coefficient of eight CFVs used together is less than the uncertainty given 

previously in Table 4.5 for a single HP CFV used by itself because HP r < 1. In particular, the 

correlation coefficient was calculated to be HP r = 0.91. A brief description of the uncertainty 

components are discussed in the table under comments. 
34 

Table 4.8. Uncertainty of the TMWS K-factor. 

 K-Factor Uncertainty for TMWS Rel. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type 

Comments

TMWS q = 1 m3

/s (8 CFVs opened) (× 10-6) (-----) (%) A or B 

Discharge Coefficient, ( HP

d C = 0.9936) 815 -1 46.8 A, B From Stage 3 unc. anal. (corr. 

effects incl.) 

Critical Flow Function, (Cs,NG = 0.7386) 302 -1 6.4 A, B See Appendix B 

Compressibility Factor, ( TMWS Z = 0.8143) 240 -1 4.1 A, B See Appendix B 

Frequency, (f = 45.9254 Hz) 47 1 0.2 B Manuf. Spec. 

Univ. Gas Constant, 

(Ru = 8314.472 J/kmol·K) 2 -0.75 0 B See Reference [xxiii] 

Throat diameter, (d = 2.5396 cm) 49* -2 0.7 B Unc. for Thermal Expansion 

between Stages 3 and 4 

Total Species Molar Mass Unc. 4.5 1 0 B 

Assumed the Std. Unc. of molar 

mass for various gas comp. is 

[ ] k k u(M ) M =10× 10-6

Total Species Composition Unc. 150 1 1.6 A, B See Appendix B 

Total Pressure Uncertainty 286 1 5.7 A, B See Appendix B 

Total Temperature Uncertainty 336 1 8.0 A, B See Appendix B 

K-factor fit Residuals 560 1 22.1 A Std. Dev. of Curve Fit 

Residuals 

Interference Effects of multiple CFVs in a 

common plenum 214 1 3.2 B 

Exp. varied spacing of CFVs in 

nozzle fixture with 21 apertures; 

(Rect. Dist.) 

Unchoking Effects 116 1 0.9 B Based on unchoking tests done 

in low pressure air (Rect. Dist.)

CFV Species Effects 58 1 0.2 B Based on CFV Theory at high 

Reynolds Number [xxii] 

Line Packing Effect 40 1 0.1 B Based on measured Vdρ dt in 

connecting volume during cal. 

Combined Uncertainty 1191 100 

4.5 STAGE 5: Typical Uncertainty of a Flowmeter Calibration 

A description of the Iowa flow facility, the methodology used for measuring volumetric flow, 

and the flowmeter calibration procedures were described previously in Sections 2.2 through 2.4. 

This section documents the uncertainty for a typical volumetric flow calibration. Repeatability 

and reproducibility of the MUT is not included in the analysis. We begin with the equation used 

for determining the volumetric flow at the MUT 

∑

5

TMWS

TMWS

TMWS

MUT

TMWS

MUT

MUT

TMWS

MUT

TMWS

MUT

1 ,n

,n

,n ,n

,n

N

n K

f

Z

Z

T

T

P

P

q

= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = M

M

MUT

Δ 5

ρ

m& − (4.15) 
35 

given previously by Eqn. 2.8, but repeated here for convenience. In this version of the equation 

the mass storage effects are included ) (Δm5& , and the ratio of the TMWS to MUT molar masses 

are not assumed to cancel. Temporal fluctuations in the gas composition entering the Iowa 

facility result in a spatial distribution of the mole fraction ( kx ) in the piping between the TMWS 

and the MUT. The difference in kx at the TMWS and MUT cause MTMWS and MMUT to differ. 

Since the fluctuations are small, this difference can be neglected when calculating qMUT , but 

should be considered in the uncertainty budget. The uncertainty caused by these fluctuations are 

typically largest at low flows when the residence time is largest between the TMWS and the 

MUT. 

By applying the method of propagation of uncertainty to Eqn. (4.15) the uncertainty for 

volumetric flow of a MUT is 

( ) ( ) ( ) ( ) 2 2

5

5

2

TMWS TMWS

TMWS

MUT

1 -1

⎥

⎦

⎤ ⎢

⎣

⎡

+ ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + = ⎥

⎦

⎤ ⎢

⎣

⎡

f

u f

K

u K

N

r N

q

u q

u (P) () u T 2

tot

2 + tot +

u ( ) Z 2 + tot ( ) k

2 + utot M ( ) k2

tot + u x

2

5

MUT MUT

Δ ⎥

⎦

⎤ ⎢

⎣

⎡

+

q

m

ρ

&

(4.16)

where the correlation coefficient is TMWS r = 0.92, and for brevity the total uncertainty of 

pressure, ) ( utot P ; temperature, ) ( utot T ; the compressibility factor, ) ( utot Z ; species molar 

masses, ) ( utot M k ; and gas composition, ) ( tot k u x ; have been grouped together into like terms. 

The total uncertainties for P , T , and Z include the uncertainties from measurements made both 

at the MUT and at the array of TMWS, and the total uncertainties for kx and Mk include the 

respective uncertainties from all of the constituents in the natural gas mixture. 

Table 4.9 provides an itemized list of all of the uncertainty components along with a brief 

explanation of how it was obtained. For selected components a more detailed explanation is 

given in Appendix C. The relative uncertainty in the MUT volumetric flow ranges from 

1246 × 10-6 at the highest flow (10.7 m3

/s) to 1355 × 10-6 at the lowest flow (0.25 m3/s). The 

increased uncertainty at low flows is caused by the increased effect of line packing and by the 

increased value of the coefficient that multiplies [ ( ) ] K K TMWS u due to a smaller value of N5 .
36 

Table 4.9. Uncertainty of a typical volumetric MUT.

MUT Volumetric Flow Unc. Rel. 

Unc. 

(k=1)

Sen. 

Coeff. 

Perc. 

Contrib.

Unc. 

Type Comments

MUT q = 2.5 m3

/s (3 TMWS opened) (× 10-6) (-----) (%) A or B 

K-factor, (K=211.7 pulse/m3) 1169 1 86.3 A, B From Stage 4 unc. anal. (corr. 

effects incl.) 

Total Pressure Uncertainty 263 1 4.4 A, B Unc. of Pres. Meas. at MUT 

and TMWS. (see Appendix C) 

Total Temperature Uncertainty 351 1 7.7 A, B Unc. of Temp. Meas. at MUT 

and TMWS. (see Appendix C) 

Frequency, (f = 91.851 Hz) 47 1 0.1 A, B Calibration Records 

Total Species Composition Unc. 118 1 0.9 A Based on Typical Fluctuations in 

gas composition (see Appendix C)

Total Species Molar. Mass Unc. 0 1 0 B Perfect correlation betwn. Mk at 

TMWS & MUT (see Appendix C)

Total Compressibility Factor Unc. 0 1 0 B Perfect correlation betwn. ZMUT

& ZTMWS (see Appendix C) 

Line Packing Effect 85 1 0.5 B Based on Historical Vdρ dt in 

connecting volume during cal. 

Combined Uncertainty 1262 100 

5. SUMMARY AND CONCLUSIONS 

This document describes NIST’s Natural Gas Flowmeter Calibration Service (NGFCS). Flow 

calibrations are conducted at the CEESI Iowa natural gas flow facility in Garner, Iowa. However, 

all flow calibrations are traceable to NIST’s low pressure air primary flow standard. Traceability 

is accomplished in five stages. Figure 3.1 shown earlier in Section 3.1 summarizes the five stage 

process and list the uncertainty introduced at each stage. The detailed results of this process 

including flow calibration results and a GUM compliant uncertainty analysis are given in 

Section 4 of this document. 

The expanded uncertainties for volumetric flow calibrations vary from 0.25 % to 0.27 %, 

increasing at lower flows. The current flow range extends from 1.5 × 104

 L/min (3.2 × 104 acfh) 

to 5.4 × 105

 L/min (1.14 × 106 acfh) at a nominal pipeline pressure of 7500 kPa (1088 psi ) and at 

a nominal temperature of 292.5 K (66.8°F). NIST is actively working to both reduce the 

uncertainty and to extend the lower end of the flow range as discussed in Section 3.3. The NIST 

website should be consulted for the current capabilities of the NGFCS 

http://www.cstl.nist.gov/div836/Group_02/index_836.02.html. xxvii, xxviii, xxix, xxx 
37 

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Nozzles, Journal of Basic Engineering, September 1964, pp. 519. 
39 

 

[xxviii] McLinden, M. O., and Cornelia, L., Apparatus for wideranging, high-accuracy fluid (p,ρ,T) measurements based on a compact two-sinker 

densimeter, J. Chem. Thermodynamics 39 (2007) 507–530

[xxix] Boggs, P. T., et. al., User's Reference Guide for ODRPACK Version 2.01 -- 

Software for Weighted Orthogonal Distance Regression, NIST IR 4834, U.S. Government 

Printing Office, 1992. 

[xxx] Boggs, P. T., et. al., Orthogonal Distance Regression, 

Contemporary Mathematics, Volume 112: Statistical Analysis of Measurement Error 

Models and their Applications, P. J. Brown and Wayne A. Fuller, Editors, pp. 183-194, 

American Mathematical Society, Providence, Rhode Island, 1990. 
APPENDIX A: CFV Calculations 

40 

Tangent

Point

CL

minimum

length

Flow

Direction

D D

Distances not

drawn to scale

Throat

Tm P Pb

Tangent

Point

CL

minimum

length

Flow

Direction Tangent

Point

CL

minimum

length

Flow

Direction

D D

Distances not

drawn to scale

Throat

Tm P Pb

Figure A1. Cross sectional cut of a toroidal shaped ISO 9300 CFV [ix]

Figure A.1 shows an cross sectional cut of a toroidal throat CFV. The CFV profile consists of a 

circular arc extending slightly beyond the throat cross section to a point of tangency, followed by 

a conical divergent section with a half angle between 2 and 6 degrees. As shown in the figure the 

static pressure, P , is measured one pipe diameter (D) upstream of the CFV inlet while the 

temperature, Tm , is measured with a either a thermistor or RTD two pipe diameters upstream of 

the inlet. The static pressure is used to calculate the stagnation pressure 

2 -1

0 2

-1 1 γ

γ

γ ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ P = P + M , (A1) 

where Cp Cv γ = is the specific heat ratio evaluated at the measured temperature and pressure, 

and M is the Mach number in the approach pipe, and the stagnation temperature is calculated 

using 

( )⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ T = T + M 1-r

2

-1 1 2

0 m

γ (A2) 

where r = 0.75 is the recovery factor, a parameter accounting for the viscous heating that occurs 

as the gas irreversibly comes to rest on surface of the temperature probe. The accuracies of 

Eqns. A1 and A2 increase at lower Mach numbers. For the largest Mach numbers in this work 

( M < 0.016 ) the difference between P and P0 is less than 0.019 % while for Tm and T0 the 

difference is less than 0.002 %. Consequently, more than 99 % of the uncertainty in P0 and T0

derive from P and Tm , respectively, and uncertainty contributions from γ , M , and r can be 

neglected. 

The freestream Mach number can be calculated using either of the two methods. When the mass 

flow, m& , through the CFV is known (i.e., the CFV is being calibrated), the Mach number can be 

calculated using 
APPENDIX A: CFV Calculations 

41 

D c

m

c

u M 2

4

π ρ

& ≡ = (A3) 

where u is average flow velocity in approach pipe, and ρ and c are the gas density and sound 

speed, respectively. On the other hand, when the CFV is being used to determine m& the Mach 

number can be determined by inverting the following expression 

(γ ) ( γ )

γ

γ

β

1 2 2

2 2

2

1 1

1

2

+ −

⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ ⎟

⎠

⎞ ⎜

⎝

⎛ − + ⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛

+ = M M (A4) 

derived from one-dimensional compressible flow theory [x] where β ≡ d D . To avoid inverting 

Eqn. A4 we used the following low Mach number approximation 

( )( ) ( )

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛

+ − − ⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛

+ =

− − 2 −1

4

3 2 2

2 1

2 1 1 2

1

1 2 γ γ γ

γ

β β γ

M (A5) 

which for β <1 6 (as in this work) introduces a negligible error (i.e., less than 0.000001 %). 

Virial effects in CFV flows are accounted for by the real gas critical flow function 

0 M

t t u 0 s P

a R T

C ρ

= . (A6) 

This thermodynamic property is computed numerically by integrating along an adiabat to 

determine the density ( t ρ ) and sound speed ( at ) at the CFV throat [xxvii]. These integrations 

were conducted using the REFPROP 8 Thermodynamic Database [ii]. 
APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components

42 

Appendix B supports the uncertainty analysis of the nine TMWS in Section 4.4. These TMWS 

were calibrated against an array of HP CFVs shown in Fig. 4.4 and the K-factor was determined 

by Eqn. 4.12. The uncertainty of the K-factor was determined by Eqn. 4.14, derived using the 

GUM procedure [xiv]. Table 4.8 gives numerical values for the uncertainty components in this 

equation. Additional details are provided below for the following five uncertainty components: 

1) ) ( utot P , the total pressure uncertainty; 2) ) ( utot T , the total temperature uncertainty; 

3) ) ( utot M k , the total uncertainty of the species molar masses; 4) TMWS [u(Z) Z] , the uncertainty 

of the compressibility factor; 5) utot (xk ) the total uncertainty attributed to the measured gas 

composition.

Table B1. Uncertainty of the two pressure transducers used to measure the average pressure at the TMWS 

Unc. of TMWS Pressure Abs. 

Unc. 

Rel. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type 

Comments

TMWS Pressure, P = 7500 kPa (Pa) (× 10-6) (%) A or B 

Uncorrelated Unc. 

Short Term Random Uncertainty 270.1 38 2.4 A Calibration Control Charts 

Long Term Random Uncertainty 1556.3 217 78.8 A Calibration Control Charts 

Data Acquisition (Agilent 34970A) 331.3 46 3.6 B Manuf. spec. 

Correlated Unc. 

Dead Weight Pressure Standard (Ametek 

EPC 2000) 414.2 58 5.6 B Based on Manuf. Calibration 

Ambient Temperature Effect 543.8 76 9.6 B Manuf. spec. 

Transducer mounting orientation 0.0 0 0.0 B Calibrated and used in the same 

mounting position 

Propagation of Uncorrelated Sources 1614 225 84.8 

Propagation of Correlated Sources 684 95 15.2 

Combined Uncertainty 1753 244 100 

Total Pressure Uncertainty: utot(P)

Pressure measurements were made both at the TMWS being calibrated and at the array of HP 

CFVs as shown in Fig. 4.4 of Section 4.4. The pressure at the TMWS was measured using two 

transducers, and the recorded pressure was the average of the two readings. Likewise, the 

pressure at the array of HP CFVs was also measured using two transducers. All four transducers 

were calibrated by the same pressure standard. Moreover, the nominal pressures at both the 

TMWS and at the array of HP CFVs were identical. Since the transducers were used to measure 

the same conditions and are traceable to the same standard, some of their uncertainty sources are 

correlated. Tables B1 and B2 itemize both the correlated and uncorrelated uncertainty sources 

corresponding to measurements at the TMWS and at the HP CFVs respectively. The standard 

uncertainties are TMWS [u(P) P] = 244× 10-6 at the TMWS and [u(P0 ) P0]HP = 200× 10-6 at the 

array of HP CFVs. 
APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components

43 

The expression for the total pressure uncertainty, utot ( ) P , is derived using the propagation of 

uncertainty 

u ( ) P 2

tot

( ) 2

0

0

2

0

s

s

0

HP ∂

∂ 1 ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = +

P

u P

P

C

C

P ( ) 2 2

TMWS TMWS ∂

∂ 1- ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ +

P

u P

P

Z

Z

P

( ) ( )

TMWS HP TMWS

c

0

c 0

0

s

s

0

∂

∂ 1- ∂∂ - 2 1 ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡

⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

P

u P

P

u P

P

Z

Z

P

P

C

C

P

(B1) 

where the first two terms include the standard pressure uncertainties at the HP CFV array and the 

TMWS, and the last term accounts for the correlated uncertainties, [uc (P0 ) P0]TMWS = 95 × 10-6

and [uc (P0 ) P0]HP = 95 × 10-6, listed in Tables B1 and B2 respectively. The derivative 

expressions in Eqn. B1 are computed numerically using finite difference methods. Based on 

Eqn. (B1) the total relative pressure uncertainty equals utot ( ) P = 243 × 10-6 for k = 1. 

Table B2. Uncertainty of two pressure transducers used to measure the average pressure at the array of HP CFVs 

Unc. of Pressure at HP CFV Array Abs. 

Unc. 

Rel. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type 

Comments

HP CFV Stag. Pressure, 0P = 7500 kPa (Pa) (× 10-6) (%) A or B 

Uncorrelated Unc. 

Short Term Random Uncertainty 408.9 57 8.1 A Calibration Control Charts 

Long Term Random Uncertainty 1143.0 159 63.5 A Calibration Control Charts 

Data Acquisition (Agilent 34970A) 331.4 46 5.4 B Manuf. spec. 

Dynamic Pressure 84.7 12 0.3 B 

Sensitivity Study: Calc. P0 based on 

Eqn (A1) assuming [u(M)/M]= 10 % 

& [u(γ)/γ ]= 5 %

Correlated Unc. 

Dead Weight Pressure Standard (Ametek 

EPC 2000) 414.2 58 8.3 A, B Based on Manufactures Calib. 

Ambient Temperature Effect 543.8 76 14.4 B Manuf. spec. 

Transducer mounting orientation 0.0 0 0 B Always in same mounting position 

Propagation of Uncorrelated Sources 1261 176 77.3 

Propagation of Correlated Sources 684 95 22.7 

Combined Uncertainty 1435 200 100 

Total Temperature Uncertainty: utot(T)

Temperature measurements were made both at the TMWS and at the array of HP CFVs as shown 

previously in Fig. 4.4 of Section 4.4. At the TMWS temperature was measured using a single 

RTD installed in a thermowell. Prior to starting the flow calibration tests we expereimentally 

characterized the temperature uncertainty attributed to the thermowell. These tests were done at 
APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components

44 

the lowest flow when end conduction effects are largest due to reduced convection heat transfer. 

The standard uncertainty due to the thermowell was 172 × 10-6 (or 50 mK). 

The pipe diameter upstream of the HP CFVs is 2.5 times the size of the pipe diameter at the 

TMWS. Since the gas density is nearly the same in both sections of pipe, the velocity in the 

approach piping of the HP CFV is lower than the velocity at the TMWS by a factor of nearly 

6.25. The low gas velocity upstream of the HP CFVs (i.e., ranging from 0.5 m/s to 2 m/s) causes 

a spatial temperature distribution in the gas due to stratification. To account for spatial 

temperature variations we measured the temperature at two axial locations in the pipe indicated 

by Tm1 and Tm2 in Fig. 4.4. Both Tm1 and Tm2 are the average of 10 RTDs spaced evenly 

around the pipe circumference. The lengths of the RTDs vary so that their insertion depths into 

the flow stream span multiple radii within the cross section as shown in Fig. 4.5C. Another 

temperature measurement indicated by Ts in Fig. 4.4 is made using a single RTD located 8 Dd

upstream of the HP CFV array in the 30.48 cm diameter pipeline. None of the RTDs used to 

measure Tm1, Tm2 , or Ts are installed in thermowells. 

The configuration of RTDs used to measure the temperature upstream of the HP CFV array are 

used to assess the uncertainty attributed to spatial temperature variations in the gas both in the 

cross sectional and axial directions. The uncertainty attributed to cross sectional temperature 

variation is taken to be the standard deviation of the 10 RTDs while the uncertainty in the axial 

direction is determined by comparing Tm1, Tm2 , and Ts . 

Table B3. Uncertainty of temperature (TTMWS ) at the TMWS 

Unc. of TMWS Temperature Abs. 

Unc. 

Rel. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type 

Comments

Temp. at TMWS, T = 295 K (mK) (× 10-6) (%) A or B 

Uncorrelated Unc. 

Fit Residuals/Reproducibility 13.1 44 3.4 A Calibration Control Charts 

Effect of Thermowell 50.8 172 50.6 B RTD Exp. with and w/o Thermowell

Data Acquisition (Agilent 34970A) 17.5 60 6 B Manuf. spec. 

Temp. Non-uniformity 30 102 17.7 B Comparison with nearby RTD 

RTD Self Heating 31.7 107 19.7 B Exp. varied current 

Correlated Unc. 

Temperature Transfer Standard 5.8 20 0.7 B Calibration by Manuf. 

Ambient Temperature Effect 5.3 18 0.5 B Manuf. Spec. 

100 Ohm Standard Resistor 8.6 29 1.4 A Std. Dev. of 3 Calibration 

Propagation of Uncorrelated Sources 70.5 239 97.4 

Propagation of Correlated Sources 11.6 39 2.6 

Combined Uncertainty 71.4 242 100 
APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components

45 

Two sets of TMWS flow calibrations were done, the first in May 2006 and the second in 

June 2007. The second temperature Tm2 was not measured in the May 2006 test. In this test, the 

axial temperature uncertainty was estimated by comparing Tm1 to Ts . On average, the difference 

between these temperatures was less than 12 mK. The small difference indicated that the 

measured temperature, Tm1, is a reliable estimate of the true average temperature, in spite of the 

standard deviation of the 10 RTDs being, on average, 151 mK. 

In the June 2007 tests three temperatures, Tm1 , Tm2 and Ts , were used to assess the axial 

temperature differences. These three temperatures agreed to within 20 mK or better, except at the 

lowest flow (i.e., 0.25 m3

/s) where the difference between Tm1 and Tm2 exceeded 100 mK. The 

two sets of 10 RTDs used to measured Tm1 and Tm2 , respectively, were used assess the size of 

temperature differences at two cross sections upstream of the HP CFV array. At high flows the 

standard deviations of the 10 RTDs used to measure Tm1 and Tm2 were both 15 mK, while at 

low flows the standard deviations were 120 mK and 78 mK respectively. 

Table B4. Uncertainty stagnation temperature (T0,HP ) at HP CFV Array 

Unc. of HP CFV Temperature Abs. 

Unc. 

Rel. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type 

Comments

HP CFV Stag. Temp., T0=295 K (mK) (× 10-6) (%) A or B 

Uncorrelated Unc. 

Fit Residuals/Reproducibility 13.1 44 4 A Calibration Control Charts 

Data Acquisition (Agilent 34970A) 17.5 60 7.1 B Manuf. spec. (Rect. Dist.) 

Axial Temperature Differences 20.2 69 9.4 B Diff. betwn. Tm2 and Tm1 with 8 HP 

CFVs opened 

Cross sectional Temperature Differences 47.9 162 53.1 A Std. Dev. of 10 RTD’s with 8 HP 

CFVs opened 

RTD Self Heating 31.7 107 23.2 B Exp. varied current 

Stag. vs. meas. Temp. 0.7 3 0 B 

Sensitivity Study: Calc. T0 based on 

Eqn (A2) assuming [u(M)/M]= 10 % 

& [u(γ)/γ ]= 5 % 

Correlated Unc. 

Temperature Transfer Standard (Ametek 

EPC 2000) 5.8 20 0.8 A, B Manuf. Calibration 

Ambient Temperature Effect 5.3 18 0.6 B Manuf. spec. (Rect. Dist.) 

100 Ohm Standard Resistor 8.6 29 1.7 A Based on Calibration 

Propagation of Uncorrelated Sources 65.8 223 97.0 

Propagation of Correlated Sources 11.6 39 3.0 

Combined Uncertainty 67 227 100 

The uncertainty components for the single RTD installed at the TMWS, and for both sets of 

10 RTDs installed upstream of the array of HP CFVs are listed in Tables B3 and B4, respectively. 
APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components

46 

The uncertainty components are categorized into correlated and uncorrelated sources. In this case 

the correlated sources of uncertainty result from the RTDs being traceable to the same 

temperature standard, and from the standard resistor in the current loop used to measure 

temperature. Applying the method of propagation of uncertainty, the total temperature 

uncertainty is 

u ( ) T 2

tot

( ) 2

0

0

2

0

s

s

0

HP ∂

∂ -

2

1 ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = T

u T

T

C

C

T ( ) 2 2

TMWS ∂

∂ 1 ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ + +

T

u T

T

Z

Z

T

( ) ( )

HP TMWS

c

0

c 0

0

s

s

0

∂

∂ 1

∂

∂ -

2

1 - 2 ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡

⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎥ +

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

T

u T

T

u T

T

Z

Z

T

T

C

C

T

(B2) 

where the first two terms, TMWS [u(T) T] = 242 × 10-6 and 0 0 HP [u(T ) T ] = 227 × 10-6, are the 

standard temperature uncertainties at the TMWS and HP CFV array, and 

TMWS ( ) c [u T T] = 27 × 10-6 and TMWS ( ) c [u T T] = 27 × 10-6 in the last term are the correlated 

uncertainties. The derivative sensitivity coefficients are calculated numerically using finite 

differences. Using Eqn. B2 with the appropriate uncertainty components in Tables B3 and B4 the 

total standard temperature uncertainty is utot ( ) T = 339 × 10-6 (k = 1). 

Uncertainty of the Compressibility Factor: utot(Z)

The compressibility factor was computed using the REFPROP 8 Thermodynamic Database. In 

general, the uncertainty of the compressibility factor for this equation of state can range from 

0.02 % to 0.1 % (k = 2) depending on the constituents of the natural gas mixture, as well as the 

specific pressure and temperature used to evaluate Z . In the present work we determined the 

uncertainty of the REFPROP 8 compressibility factor ( ZR8 ) for natural gas compositions 

corresponding to the Iowa flow facility (see Table 2.1 of Section 2.1) at typical pipeline 

pressures of 7500 kPa and at ambient temperature conditions. The uncertainty was assessed by 

comparing values of the compressibility factor calculated with the REFPROP 8 Database ( ZR8 ) 

to experimentally measured ( Zmeas) at the same conditions. The comparison was done using gas 

samples extracted from the Iowa flow facility. The compressibility factor of gas samples 

obtained from the Iowa flow facility was measured at two pressures along an isotherm of 

298.15 K. These measurements were made near the minimum (6837 kPa) and maximum 

(8018 kPa) operating pressures of the Iowa facility. 

The compressibility factor measurements were done by NIST Physical and Chemical Properties 

Division using a two sinker densimeter [xxviii]. A three step process was implemented to 

determine Zmeas . In the first step, the density of the gas sample was measured at the two 

pressures of interest (6837 kPa and 8018 kPa) at 298.15 K. Next, the density of the sample gas 

was determined at multiple lower pressures ranging from 600 kPa to 3000 kPa at 298.15 K. The 

low pressure data was fit to the virial equation of state with the molar mass (M ) treated as an 

unknown constant that was determined by the regression analysis, 

Dˆ C ρ ˆ B ρ ˆ P ρR T ρ 2 3

u = 1 M + + + . The virial coefficients (which herein are divided by the 

molar mass, Bˆ ≡ B M , Cˆ ≡ C M , and Dˆ ≡ D M ) were also determined by the regression 
APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components

47 

analysis. The curve fit used density as the independent variable so that the point of zero density 

(i.e., the ideal gas limit) was inversely proportional to M . The random uncertainties associated 

with this fitting process were determined using the ODR regression package [xxix, xxx]. 

Systematic uncertainties related to biases in the measurements of density, pressure, and 

temperature were considered separately. The total uncertainty in M was estimated to be 88 × 10-

6

 (k = 2). In the final step M was used in conjunction with the density measurements from the 

first step to determine the compressibility factor, Z = PM ρRuT . 

Following the GUM [xviii] the uncertainty in the measured compressibility factor 

( ) 2

meas

meas ⎥

⎦

⎤ ⎢

⎣

⎡

Z

u Z ( ) ( ) ( ) ( ) ( ) 2

u

u

2 2 2 2

⎥

⎦

⎤ ⎢

⎣

⎡

+ ⎥

⎦

⎤ ⎢

⎣

⎡

+ ⎥

⎦

⎤ ⎢

⎣

⎡ + ⎥

⎦

⎤ ⎢

⎣

⎡ + ⎥

⎦

⎤ ⎢

⎣

⎡ = R

u u R

T

u T

P

u u P

ρ

ρ

M

M (B3) 

included standard uncertainty contributions (i.e., k = 1) from the molar mass (44 × 10-6); 

temperature (7 × 10-6), pressure (31 × 10-6), density (15 × 10-6), and the universal gas constant 

(2 × 10-6). Adding these components in quaditure and taking the square root gave a standard 

uncertainty of [u(Zmeas) Zmeas]= 56 × 10-6 (k = 1). The uncertainty of ZR8 (i.e., the 

compressibility factor predicted by the REFPROP 8 Thermodynamic Database) includes 

uncertainty contributions from (1) the measured value of the compressibility factor; (2) the 

difference between the measured and predicted values of the compressibility factor at the same P, 

T, and xk conditions; and (3) the gas composition values (xk) determined by the GC relative to 

their actual values. An expression for the uncertainty of ZR8 that accounts for these three 

uncertainty sources is 

( ) 2

R8

R8 ⎥

⎦

⎤ ⎢

⎣

⎡

Z

u Z ( ) 2

meas

2

meas

meas

3

1 R8 − ⎥ +

⎦

⎤ ⎢

⎣

⎡ = Z Z

Z

u Z ( ) ∑

k 1

2

k

k

2

k

k

∂

∂ K

x

u x

x

Z

Z

x

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ (B4) 

where the difference between the measured and predicted Z values (i.e., the second term) is 

assumed to follow a rectangular distribution so that its standard uncertainty was 219 × 10-6

for k = 1. to The last term accounts for the uncertainty in ZR8 caused by the uncertainty in mole 

fractions ( kx ) measured by the gas chromatograph (GC). Monte Carlo simulations [xvii] 

predicted this uncertainty to be 80 × 10-6 (k = 1). In the simulation both the pressure and 

temperature are held fixed at 8018 kPa and 298.15 K, respectively, while each component in the 

composition was randomly varied in a Gaussian distribution centered around the measured gas 

composition. The uncertainty in the gas composition ( ) k u x included contributions from the 

uncertainty of the certified gas standard that was used to calibrate the GC, and the historical 

reproducibility of the GC. Based on Eqn. B4 the uncertainty of the [u(ZR8 ) ZR8] = 240 × 10-6

(k = 1). 

As an additional check of the estimated uncertainty we compared the compressibility factor 

calculated by the REFPROP 8 Thermodynamic Database with values computed by three other 

databases including the AGA 8 [xxiv], GERG [xxv], and REFPROP 7 [xxvi] thermodynamic 

databases. Comparisons were made at the same pressure, temperature, and gas composition as 

used in the above analysis. The standard deviation of the four different thermodynamic databases 
APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components

48 

was 0.01 %. The good agreement of the four different equations of state gives a good indication 

that the gas mixture was well characterized at the conditions of interest. This result is not totally 

unexpected since the Iowa gas contains very little higher hydrocarbons as shown in Table 2.1 of 

Section 2.1. 

Total Uncertainty of Species Molar Masses: utot(Mk) 

The total uncertainty of the species molar masses is 

( ) k

2

utot M ( ) ∑

k 1

2

k

k

2

k k

4 NG

1 K

M

x M u M

=

⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = M (B5) 

where Mk is the molar mass of the kth species, andMNG is the molar mass of the natural gas, and 

the uncertainty of the molar mass of each component was estimated to be 

[ ] ( ) Mk Mk u = 10× 10-6. Based on Eqn. B5 the total uncertainty of the species molar masses was 

( ) utot Mk = 4.5× 10-6. 

Total Gas Composition Uncertainty: utot(xk) 

The gas composition is measured using an industrial grade GC located upstream of the TMWS in 

the Iowa facility. The typical composition of gas at the Iowa facility is given in Table 2.1 of 

Section 2.1. Although the gas composition does not explicitly appear in Eqn. 4.12 used to 

determine the K-factor, it is contained in several of the variables. The three most significant 

variables influenced by xk include the molar mass (MNG ), the compressibility factor ( Z TMWS ), 

and the critical flow function (Cs,NG ). The composition is also needed to determine the Reynolds 

number of the HP CFVs ( ReHP ) in Eqn 3.3, and the stagnation pressure ( P0 ) and temperature 

(T0 ) given in Eqns. A1 and A2 of Appendix A. However, the uncertainty from these sources is 

small relative to the other components and are therefore omitted. Considering only contributions 

from the molar mass, the compressibility factor, and the critical flow function, the total 

uncertainty in gas composition is 

( ) k

2

tot u x ( ) ∑

k 1

2

k

k

2

k

k

k

s

s

k k k

∂

∂ - ∂∂ - 2

1 TMWS

NG TMWS

K

x

u x

x

Z

Z

x

x

C

C

x M x

=

⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = M (B6) 

where ( ) k u x is the standard composition uncertainty. The linear sum of the sensitivity 

coefficients in Eqn. B6 accounts for correlated effects between MNG , Z TMWS , and Cs,NG . A 

Monte Carlo simulation calculated the total composition uncertainty to be ) ( tot k u x = 150 × 10-6 

(k = 1). The calculated uncertainty was significantly reduced by correlated effects between MNG , 

Z TMWS , and Cs,NG since separate Monte Carlo analysis on each of these variables indicated their 

individual uncertainties attributed to ( ) k u x to be [ ( ) ] u MNG MNG = 300 × 10-6, 

[ ( ) ] u Z NG Z NG = 100 × 10-6, and [ ( ) ] u Cs,NG Cs,NG = 40 × 10-6, respectively. All of the Monte 

Carlo calculations were done at constant pressure and temperature conditions of 298. 15 K and 

7500 kPa, respectively. 
APPENDIX B: Analysis of Selected Stage 4 Uncertainty Components

49 

In the above analysis the uncertainty of the gas composition ( ) k u x was calculated as the rootsum-square of the uncertainty of the certified gas standard used to calibrate the GC, and the 

historical reproducibility of the GC. As a secondary estimate of ( ) k u x , NIST collected gas 

samples from the Iowa facility and analyzed the composition independently by two different 

methods. 

The first method collected a gas sample and had its composition measured by the Gas Metrology 

and Classical Methods Group at NIST. The difference in the molar mass of the two 

measurements was less than 0.05 %. The second method measured the molar mass using a two 

sinker densimeter (see the uncertainty of the compressibility factor above). The agreement 

between the molar mass computed from the Iowa GC composition measurements and the 

densimeter measurement was 0.023 %. Moreover, the molar mass measurement done using the 

densimeter was independent of gas composition. Therefore, the good agreement between the two 

measurements gives us confidence that the Iowa GC is performing adequately, and perhaps just 

as important is not omitting the measurement of any important constituents. 
APPENDIX C: Analysis of Selected Stage 5 Uncertainty Components

50 

Appendix C supports the uncertainty analysis of a MUT in Section 4.5. The calibration setup is 

shown in Fig. 2.1 of Section 2.2. A MUT is calibrated against a parallel array of up to nine 

TMWS. The TMWS are used to determine the volumetric flow at the MUT based either on 

Eqn. 2.8 of Section 2.3 or an analogous formulation given in Eqn. 4.15 of Section 4.5. The latter 

is more appropriate to use when using the GUM procedure [xviii] to determine uncertainty. 

Based on this procedure the uncertainty of volumetric flow is governed by Eqn. 4.16, and the 

values of components given in this equation are listed in Table 4.9. This appendix explains how 

numerical values were obtained for the following five uncertainty components: (1) ) ( utot P , the 

total pressure uncertainty; (2) ) ( utot T , the total temperature uncertainty; (3) ) ( utot M k , the total 

uncertainty of the species molar masses; (4) ) ( utot Z , the total uncertainty of the compressibility 

factor; and (5) utot (xk ) the total uncertainty attributed to the measured gas composition.

Table C1. Uncertainty of pressure transducers used to measure the average pressure at the array of MUT 

 Unc. of Pressure at MUT Abs. 

Unc. 

Rel. 

Std. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type 

Comments

MUT Pressure, PMUT = 7173.9 kPa (Pa) (× 10-6) (%) A or B 

Uncorrelated Uncertainty 

Short Term Random Uncertainty14 1147.8 160 56.1 A Calibration Control Charts 

Long Term Random Uncertainty1 717.4 100 21.9 A Calibration Control Charts 

Data Acquisition (Agilent 34970A) 219.2 30.6 2.1 B Manuf. spec. 

Correlated Uncertainty 

Calibration Transfer Standard 414.2 57.7 7.3 A, B Based on Manufactures Calib. 

Ambient Temperature Effect 543.8 75.8 12.6 B Manuf. spec. 

Transducer mounting orientation 0 0 0 B Always in same mounting position 

Propagation of Uncorrelated Sources 1371.2 191.1 80.1 

Propagation of Correlated Sources 683.6 95.3 19.9 

Combined Uncertainty 1532 214 100 

Total Pressure Uncertainty: utot(P) 

During a flow calibration pressure measurements are made both at the MUT and at each of the 

TMWS being used. The uncertainty components for pressure measurements at the MUT are 

listed in Table C1. Although the uncertainty components for the TMWS pressure measurements 

are not shown, they are analogous to those shown in Table C1 for the MUT. The components are 

congruent since the nominal pressure being measured, the transducer type, and the standard used 

to calibrate both the MUT and the TMWS pressure transducers are the same. 

 

1

 Short term random effects account for those effects that vary randomly during the time it takes to complete a single 

calibration while long term random effects include random variations between multiple calibrations. 
APPENDIX C: Analysis of Selected Stage 5 Uncertainty Components

51 

The total uncertainty of the pressure measurements includes uncertainty sources from both the 

TMWS and the MUT. Because all of the characterizations of these transducers are traceable to 

the same calibration standard, the bias uncertainty components are assumed to completely cancel 

and the total uncertainty in pressure is given by 

( ) = 2

utot P ( ) ( )

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥

⎦

⎤ ⎢

⎣

⎡

+ ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ 2

u

5

2

u

2

MUT TMWS

1

∂

∂ 1- P

u P

P N

u P

P

Z

Z

P

(C1)

where N5 is the number of TMWS in use, the uncorrelated pressure components, [ ] ( ) u MUT u P P

and [ ] ( ) u TMWS u P P , both equal 191 × 10-6. In addition, each transducer has the same random 

uncertainties. The random uncertainty components are from the short term random effects 

(160 × 10-6), the long term random effects (100 × 10-6), and the data acquisition (31 × 10-6). 

When Eqn. (C1) is used with these values, the total pressure uncertainty equals 

utot ( ) P = 263 × 10-6. 

Table C2. Uncertainty of temperature ( MUT T ) at MUT 

 Unc. of HP CFV Temperature Abs. 

Unc. 

Rel. 

Std. 

Unc. 

(k=1)

Perc. 

Contrib.

Unc. 

Type Comments

HP CFV Stag. Temp., TMUT =295 K (mK) (× 10-6) (%) A or B 

Uncorrelated Unc. 

Digital Accuracy 59 200 37.9 A Manuf. Specs. 

Stability 58.4 198 37.2 Manuf. Specs. 

Data Acquisition (Agilent 34980A) 16.3 55 2.9 B Manuf. spec. (Rect. Dist.) 

RTD Self Heating 14.7 50 2.3 B Exp. varied current 

Correlated Unc. 

Temperature Transfer Standard 20 68 4.4 A, B Manuf. Calibration 

Ambient Temperature Effect 37.5 127 15.3 B Manuf. spec. (Rect. Dist.) 

Propagation of Uncorrelated Sources 85.9 291 80.3 

Propagation of Correlated Sources 42.5 144 19.7 

Combined Uncertainty 95.8 325 100 

Total Temperature Uncertainty: utot(T)

The temperatures at the MUT and at each of the TMWS are measured using a pair of the 

Rosemount 3144 RTDs. The uncertainty components for temperature measurements at the MUT 

are listed in Table C2. The analogous temperature components at the TMWS are not listed, but 

are similar in magnitude since the same transducer types are used, these transducers have been 

calibrated by the same temperature standard, and the transducers are used in application to 

measure the same nominal temperature. 
APPENDIX C: Analysis of Selected Stage 5 Uncertainty Components

52 

The total temperature uncertainty includes contributions from both the MUT and the TMWS. 

Because of the high degree of similarity between the TMWS and the MUT temperature 

measurements, bias uncertainty components completely cancel and the total relative uncertainty 

in temperature is given by 

( ) = 2

utot T ( ) ( )

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥

⎦

⎤ ⎢

⎣

⎡ + ⎥

⎦

⎤ ⎢

⎣

⎡ ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ +

2

u

5

2

u

2

MUT TMWS

1

∂

∂ 1

T

u T

T N

u T

T

Z

Z

T

(C2)

where N5 is the number of TMWS in use, and the uncorrelated temperature uncertainty 

components, ( ) u MUT [u T T] and ( ) u TMWS [u T T] , both equal 313 × 10-6. In this case, the 

uncorrelated uncertainty components are assumed to include the digital accuracy (200 × 10-6), 

the stability (198 × 10-6), the ambient temperature effect (127 × 10-6), the data acquisition 

(6 × 10-6), and probe heat transfer effects (50 × 10-6). Using these values in conjunction with 

Eqn. (C2), the total temperature uncertainty equals utot ( ) T = 340 × 10-6. 

Total Uncertainty of Species Molar Masses: utot(Mk)

The total uncertainty of the species molar masses is 

( ) k

2

utot M ( ) ∑

k 1

2

k

k

2

k k

2

MUT MUT

TMWS 1

K

M

x M u M

=

⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − M M

M . (C3) 

where MMUT and MTMWS are the molar masses at the TMWS and MUT, respectively. Since 

these molar masses are practically equal (i.e., MMUT = MTMWS in Eqn. C3), ) ( utot Mk is 

identically zero. 

Total Uncertainty of Gas Composition: utot(xk)

During a Stage 5 flowmeter calibration the gas composition (or mole fraction; kx ) will differ 

slightly between the MUT and the TMWS. This difference in kx results from small fluctuations 

in the gas composition entering the Iowa flow facility. Since the residence times differ for the 

gas to travel from the measurement location to the MUT versus the TMWS, the instantaneous 

values of composition differ at these two locations. The resulting uncertainty attributed to this 

difference is 

( ) k = 2

tot u x ∑

k 1

2

k

2

k

k

k

k k

∂

∂ - ∂∂

K

x

x x

x

x

Z

Z

x

=

⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ M σ

M

(B3)

where x k x k σ is the standard deviation of typical fluctuations for each species in the natural gas 

mixture. This uncertainty is conservatively estimated to be large, including, the reproducibility of 

the gas chromatograph used to make the composition measurements. Using this expression the 

total uncertainty in gas composition is ( ) tot k u x = 208 × 10-6. 

Total Uncertainty of the Compressibility Factor: utot(Z)

The compressibility factor is determined at the MUT as well as at each TMWS being used. In all 

cases, the REFPROP 8.0 Thermodynamic Database [ii] is used to evaluate the compressibility 
APPENDIX C: Analysis of Selected Stage 5 Uncertainty Components

53 

factor. Since the nominal pressure, temperature, and gas composition are the same at the MUT 

and at the TMWS array, the uncertainty resulting from the thermodynamic database completely 

cancels, yielding a zero uncertainty. 
NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 1 of 11

APPENDIX D: 

SAMPLE CALIBRATION REPORT

FOR 

AN ULTRASONIC FLOWMETER 

July 8, 2007 

Mfg.: Ultrasonic Flowmeter Builders, Inc. 

Serial Number: 1234 

Bore Diameter: 30.322 cm (11.938 inch) 

submitted by 

Advanced Technology Ultrasonics, Inc. 

1829 Flowmeter Rd. 

Metertown, MD 123456 

Purchase Order No. A123 dated June 24, 2007 

The ultrasonic flowmeter identified above was calibrated in dry pipeline quality natural gas using 

a parallel array of three turbine meter working standards. The calibration was completed in 

accordance with the American Gas Association Transmission Report Number 9 5 . 

Thermodynamic properties (e.g., speed of sound, density, etc.) are computed using the 

REFPROP 8.0 Thermodynamic Database 6 in its AGA 8 mode. This database is effectively 

equivalent to the AGA 10 Thermodynamic database, having sound speed and density values that 

agree with the AGA 10 model to better than 0.00001% and 0.000002 %, respectively over the 

range of pressures, temperatures, and gas compositions relevant to this calibration. The 

calibration was performed at the Colorado Engineering Experimental Station Incorporated 

(CEESI) flow facility located at 2365 240th St. in Garner, Iowa 50438 under NIST’s metrological 

control. The traceability of all auxiliary flow measurements for pressure, temperature, frequency, 

etc. are maintained by NIST. In addition, flow determination is based on an array of the turbine 

 

5

 American Gas Association (AGA); Measurement of Gas by Multipath Ultrasonic Meters; Transmission Measurement 

Committee Report No. 9 2nd edition (AGA9), AGA, April 2007. 

6

 Lemmon, E. W., et. al., REFPROP 23: Reference Fluid Thermodynamic and Transport Properties, NIST Standard Reference 

Database 23, Version 8, National Institute of Standards and Technology, Boulder, Colorado, 2007. 
SAMPLE CALIBRATION REPORT Gas Flowmeter, S/N 1234

Advanced Technology Ultrasonics, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 2 of 11

meter working standards that are traceable to NIST’s primary flow standards via a bootstrap 

process using critical flow venturis transfer standards.7,8 

The flowmeter was calibrated at six flows and at each flow, three (or more) measurements were 

gathered on two different occasions. These measurements are used to produce averages at each 

of the six flows so that the plots are averages of six or more individual calibration measurements. 

This report contains tabulations of both the averaged flow results and the individual data points. 

At the conclusion of the calibration, the flow adjustment is electronically installed in the 

flowmeter and single verification point is taken to confirm proper meter operation. 

The ultrasonic flowmeter of bore diameter of D = 30.322 cm (11.938 inch) and serial number 

1234 was installed in a piping test section of approximately 30.48 cm (12 inch) diameter located 

downstream of the turbine meter array. The volumetric flow (QNIST ) at the meter under test was 

determined by summing the flow through all of the open turbine meter working standards and 

applying conservation of mass. A photograph of the flowmeter installation is shown in Figure 1. 

Table 1 shows the pressure, temperature, and data acquisition system used in the calibration. The 

nominal temperature and pressure during the test was 295.44 K (72.12 °F) and 8 376.9 kPa 

(1215.0 psia), respectively. The serial numbers of the three turbine meter working standards 

used to determine the volumetric flow are 97500360 (1), 95380241 (2), and 7500361 (3), 

respectively, where the number in parenthesis immediately following the serial number specifies 

the position of the turbine meter in the array at the time of calibration. The nominal composition 

of the natural gas mixture during the calibration and verification is given in Table 3. 

Figure 1. Photograph of the flowmeter installed in the pipeline

 

7

 Johnson, A. N. and Wright, J. D., Gas Flowmeter Calibrations with the 26 m3 PVTt Standard, NIST Special Publication 250-

1046, National Institute of Standards and Technology, Gaithersburg, MD, (2006). 

8

 Johnson, A. N. and Kegel, T. Uncertainty and Traceability for the CEESI Iowa Natural Gas Facility, J. Res. Natl. Inst. Stand. 

Technol., 109, pp. 345-369 (2004).
SAMPLE CALIBRATION REPORT Gas Flowmeter, S/N 1234

Advanced Technology Ultrasonics, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 3 of 11

Table 1. Specification of data acquisition system, the pressure transducers, and the temperature transducers used 

for the calibration. 

Data 

Acquisition 

Barometric 

Pressure 

Gauge 

Pressure Temperature 

Agilent 

34980 A 

(QC #999) 

PCU 300 

(QC # 300) 

2 Rosemount 3051 

Pressure Transmitters 

(QC# 901 & 902) 

2 Rosemount 3144 RTD 

Transmitters 

(QC# 950 & 951) 

A B C D E F G H

Flow Direction

Figure 2. Labeling of uni-directional flowmeter installation configuration (Labels are described in Table 2) This 

piping configuration meets straight-run requirements for AGA9.

Table 2. Itemized list detailing the piping sections (e.g., lengths, orientation, serial numbers); the flowmeter being 

calibrated (e.g., type, size, serial number); the flow conditioner (e.g., type, orientation, serial number); 

and thermowell location during the calibration process in Fig. 2

Components Specified in Fig. 2 Description 

A – Pipe Section (Straight pipe or Elbow) Straight Pipe. 

B – Pipe Section Length = 5.1 D , (SN 01-111345) 

C – Flow Conditioner Flow Profile Fixer (SN -02-12345); Flow direction and mounting 

location verified

D – Pipe Section Length = 5.03 D , (SN 01-111346) 

E – Flowmeter Ultrasonic Flowmeter Builders with diameter (SN 1234) 

F – Pipe Section Length = 5.05 D , (SN 01-111347) 

G – Thermowell Location Distance to closest flowmeter flange face is 2 D

H – Pipe Section (Straight pipe or Elbow) Straight Pipe. 
SAMPLE CALIBRATION REPORT Gas Flowmeter, S/N 1234

Advanced Technology Ultrasonics, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 4 of 11

Table 3. Normalized gas composition measured with gas chromatograph (SN 9007671) at the start and end of the 

calibration, and at the start and end of the verification point. 

Normalized Mole Fraction (%)

Component Start of 

Calibration 

End of 

Calibration 

Start of 

Verification 

Point

End of 

Verification 

Point 

Methane 94.8865 94.9474 94.9355 94.937 

Ethane 2.3172 2.2630 2.2709 2.2712 

Propane 0.2264 0.2176 0.2170 0.2177 

Iso Butane 0.0185 0.0181 0.0179 0.0181 

Butane 0.0228 0.0222 0.0218 0.0223 

Iso Pentane 0.0050 0.0051 0.0052 0.0049 

Pentane 0.0037 0.0037 0.0037 0.0038 

Hexane (C6+) 0.005 0.005 0.0049 0.0049 

Nitrogen 1.6206 1.6422 1.6479 1.6457 

Carbon Dioxide 0.6543 0.6330 0.638 0.6392 

Hydrogen 0.2053 0.2080 0.2019 0.2006 

Helium 0.0347 0.0347 0.0353 0.0346 

Heating Value 1 012.1 1 011.5 1 003.6 1 003.6 

Total Un-Normalized 99.9012 99.9012 99.9012 99.9012 

The velocity in the piping immediately upstream of the ultrasonic flow meter is determined by 

dividing the NIST’s volumetric flow (QNIST ) determined by the turbine meter working standards 

by the piping cross sectional area ( 4 2 Acs = π D ) 

2

NIST

cs

NIST

NIST

4

D

Q

A

Q V

π = = (1)

where D = 30.322 cm (11.938 inch). The calibration factor is 

meter

NIST

meter

NIST

Q

Q

V

V φ = = (2)

the ratio of the NIST measured velocity (VNIST ) and the velocity reported by the flowmeter 

(Vmeter ), or alternatively, the ratio of the NIST’s determined volumetric flow (QNIST ) divided by 

the meter volumetric flow ( meter 4 2 Qmeter = π D V ). The calibration factor is defined so that 

when it is multiplied with the meter velocity (or the meter volumetric flow) the result is the NIST 

calibrated velocity (or NIST volumetric flow). The error in the meter volumetric flow is defined 

as the percent difference between the NIST value and the meter reported value 
SAMPLE CALIBRATION REPORT Gas Flowmeter, S/N 1234

Advanced Technology Ultrasonics, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 5 of 11

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ =⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − φ

φ ε

1- 100 1 100

NIST

meter

Q

Q . (3)

which can also be expressed in terms of the calibration factor. 

The calibration results are presented in Figure 3 and in Table 4. Figure 3 plots the velocity 

determined by the Turbine meter standards versus the error and Table 4 gives the averaged 

calibration results and their expanded uncertainty. In addition, the speed of sound calculated by 

the flowmeter is compared to values computed the REFPROP 8.0 Thermodynamic Database at 

the measured pressure, temperature, and gas composition. Table 5 shows the individual data 

points (i.e., before averaging) for the calibration process. 

VNIST (m/s)

VNIST (ft/s)

% Error (

ε ) -1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30 35 0 5 10 15 20 25 30

0 20 40 60 80 100

Figure 3. Calibration results showing the percent difference in NIST flow ( QNIST ) with the measured flow ( Qmeter ) 

plotted versus the reported meter velocity (VNIST ) for ultrasonic flowmeter serial number 1234. The error 

bars denote the combined expanded uncertainties shown in Table 4. (For convenience the velocity units 

are plotted both in m/s and ft/s.) 
SAMPLE CALIBRATION REPORT Gas Flowmeter, S/N 1234

Advanced Technology Ultrasonics, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 6 of 11

Table 4. Calibration results for ultrasonic flowmeter serial number 1234. (The results are tabulated in both SI and 

English units.) 

Flow NIST 

Velocity 

Meter 

Flow 

NIST 

Flow 

% Flow 

Error 

Calib. 

Factor 

Expand.

Unc. 

REFPROP 8.0 

Sound 

Speed 

Meter Avg 

Sound 

Speed 

% Diff. 

Sound 

Speed 

(VNIST ) ( Qmeter ) ( QNIST ) (ε ) (φ ) (Ue ) 

[% FS] [m/s] [m3/s] [m3/s] [%] [ ] [%] [m/s] [m/s] [%] 

100 30.724 2.2143 2.2186 -0.20 1.0020 0.26 423.31 423.31 0.00 

70 21.508 1.5492 1.5531 -0.25 1.0025 0.26 423.79 423.71 0.02 

40 12.291 0.8853 0.8876 -0.26 1.0026 0.26 424.04 424.02 0.00 

25 7.6828 0.5531 0.55478 -0.30 1.0031 0.27 424.08 423.98 0.02 

10 3.0742 0.22112 0.22199 -0.40 1.0039 0.28 423.84 423.75 0.02 

3 0.9151 0.06566 0.06608 -0.64 1.0064 0.28 423.63 423.47 0.04 

[% FS] [ft/s] [acfh] [acfh] [%] [ ] [%] [ft/s] [ft/s] [%] 

100 100.80 281 506 282 061 -0.20 1.0020 0.26 1 388.82 1 388.80 0.002 

70 70.56 196 954 197 448 -0.25 1.0025 0.26 1 390.40 1 390.11 0.02 

40 40.326 112 546 112 839 -0.26 1.0026 0.26 1 391.20 1 391.15 0.00 

25 25.206 70 317 70 531 -0.30 1.0031 0.27 1 391.35 1 391.01 0.02 

10 10.086 28 111.0 28 222.0 -0.40 1.0039 0.28 1 390.54 1 390.27 0.02 

3 3.002 8 347.3 8 401.7 -0.64 1.0064 0.28 1 389.85 1 389.34 0.04 
SAMPLE CALIBRATION REPORT 

Advanced Technology Ultrasonics, Inc. NIST Test Number: 836-270542-04-04 Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson 

Table 5. Calibration results of point by point data for ultrasonic flowmeter serial number 1234. 

Pt Time 

NIST 

Velocity 

( NIST V ) 

Meter 

Flow 

( Qmeter ) 

NIST 

Flow 

( QNIST ) 

% Flow. 

Error 

(ε )

Calib. 

Factor 

(φ ) Temp. Pres. 

REFPROP 8.0 

Sound 

Speed 

A

[ ] [ ] [m/s] [m3/s] [m3/s] [%] [ ] [K] [kPa] [m/s] 

1 7:01:25 AM 30.722 2.2138 2.2185 -0.21 1.0021 295.21 8 313.0 423.27 2 7:03:40 AM 30.725 2.2141 2.2187 -0.21 1.0021 295.24 8 307.6 423.32 3 7:05:47 AM 30.726 2.2146 2.2188 -0.19 1.0019 295.27 8 306.3 423.35 4 7:11:30 AM 21.504 1.5492 1.5528 -0.23 1.0023 295.57 8 349.8 423.78 5 7:13:47 AM 21.509 1.5491 1.5532 -0.26 1.0026 295.58 8 350.1 423.78 6 7:15:48 AM 21.504 1.5489 1.5528 -0.25 1.0025 295.58 8 350.2 423.82 7 7:21:04 AM 12.292 0.8852 0.8876 -0.27 1.0027 295.65 8 390.2 424.03 8 7:23:07 AM 12.289 0.8852 0.8874 -0.25 1.0025 295.66 8 388.9 424.03 9 7:25:27 AM 12.289 0.885 0.8874 -0.27 1.0027 295.67 8 388.9 424.05 

10 7:30:56 AM 7.6844 0.553 0.5549 -0.34 1.0034 295.69 8 401.5 424.14 11 7:33:19 AM 7.6761 0.5526 0.5543 -0.31 1.0031 295.63 8 401.3 424.08 12 7:35:40 AM 7.6913 0.5536 0.5554 -0.32 1.0033 295.58 8 400.8 424.03 13 7:41:24 AM 3.0695 0.2207 0.22165 -0.43 1.0043 295.49 8 407.6 423.92 14 7:43:28 AM 3.0663 0.22055 0.22142 -0.39 1.0039 295.42 8 405.8 423.84 15 7:45:35 AM 3.0796 0.2216 0.22238 -0.35 1.0035 295.34 8 404.4 423.75 16 7:51:25 AM 0.9147 0.06566 0.06605 -0.59 1.0059 295.21 8 407.7 423.70 17 7:53:48 AM 0.9217 0.06613 0.06656 -0.65 1.0065 295.12 8 406.0 423.62 18 7:56:09 AM 0.9155 0.06568 0.06611 -0.65 1.0065 295.06 8 404.5 423.56 

Shutdown Flow and Reestablish Set point 

19 8:12:49 AM 0.9141 0.06563 0.06601 -0.58 1.0058 295.06 1 219.0 423.56 20 8:14:50 AM 0.9126 0.06544 0.0659 -0.70 1.0070 295.12 1 219.2 423.62 21 8:17:01 AM 0.9118 0.06541 0.06584 -0.65 1.0066 295.21 1 219.4 423.70 22 8:22:10 AM 3.0735 0.2211 0.22194 -0.38 1.0038 295.34 1 219.0 423.75 23 8:24:12 AM 3.0768 0.22136 0.22218 -0.37 1.0037 295.42 1 219.2 423.84 24 8:26:19 AM 3.0793 0.22138 0.22236 -0.44 1.0044 295.49 1 219.4 423.92 25 8:31:48 AM 7.6761 0.5527 0.5543 -0.29 1.0029 295.58 1 218.4 424.03 26 8:34:05 AM 7.6899 0.5539 0.5553 -0.25 1.0025 295.63 1 218.5 424.08 27 8:36:08 AM 7.6788 0.5528 0.5545 -0.31 1.0031 295.69 1 218.5 424.14 28 8:41:47 AM 12.289 0.8851 0.8874 -0.26 1.0026 295.67 1 216.7 424.05 29 8:43:58 AM 12.296 0.8857 0.8879 -0.25 1.0025 295.66 1 216.7 424.03 30 8:46:00 AM 12.293 0.8854 0.8877 -0.26 1.0026 295.65 1 216.9 424.03 31 8:51:30 AM 21.510 1.5493 1.5533 -0.26 1.0026 295.58 1 211.1 423.82 32 8:53:32 AM 21.508 1.5495 1.5531 -0.23 1.0023 295.58 1 211.1 423.78 33 8:55:53 AM 21.510 1.5492 1.5533 -0.26 1.0026 295.57 1 211.0 423.78 34 9:01:34 AM 30.721 2.2142 2.2184 -0.19 1.0019 295.27 1 204.7 423.35 35 9:03:53 AM 30.726 2.2144 2.2188 -0.20 1.0020 295.24 1 204.9 423.32 36 9:05:58 AM 30.724 2.2145 2.2186 -0.19 1.0019 295.21 1 205.7 423.27 
SAMPLE CALIBRATION REPORT Gas Flowmeter, S/N 1234

Advanced Technology Ultrasonics, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 8of 11

Upon Completion of the calibration the coefficients in Table 6 listed below were input into the 

meter and then the meter was put into a Read Only mode. Subsequently a single verification 

point was done as shown in Figure 5 and in Tables 7 and 8, respectively. 

Table 6. Calibration coefficients electronically installed in ultrasonic flowmeter serial number 1234. 

Register Name Value Register Name Value 

328 FwdFlwRt1 281506 340 FwdMtrFctr1 1.0020 

329 FwdFlwRt2 196954 341 FwdMtrFctr2 1.0025 

330 FwdFlwRt3 112546 342 FwdMtrFctr3 1.0026 

331 FwdFlwRt4 70317 343 FwdMtrFctr4 1.0031 

332 FwdFlwRt5 28111.0 344 FwdMtrFctr5 1.0039 

333 FwdFlwRt6 8347.3 345 FwdMtrFctr6 1.0064 

334 FwdFlwRt7 0 346 FwdMtrFctr7 1 

335 FwdFlwRt8 0 347 FwdMtrFctr8 1 

336 FwdFlwRt9 0 348 FwdMtrFctr9 1 

337 FwdFlwRt10 0 349 FwdMtrFctr10 1 

338 FwdFlwRt11 0 350 FwdMtrFctr11 1 

339 FwdFlwRt12 0 351 FwdMtrFctr12 1 % Error (

ε ) -1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30 35 0 5 10 15 20 25 30

0 20 40 60 80 100

VNIST (m/s)

VNIST (ft/s)

Figure 4. Verification results showing the error percent difference in NIST flow ( QNIST ) with the measured flow 

( Qmeter ) plotted versus the reported meter velocity (VNIST ) for ultrasonic flowmeter serial number 1234. 

The error bar represents the combined expanded uncertainty shown in Table 7. 
SAMPLE CALIBRATION REPORT 

Advanced Technology Ultrasonics, Inc. NIST Test Number: 836-270542-04-04 Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Table 7. Single point verification results for ultrasonic flowmeter serial number 1234. (The results are tabulated in both SI and Englis

Flow NIST. Velocity Meter Flow NIST Flow % Flow Error REFPROP 8.0 Sound Speed Meter Avg Sound Speed % Diff. Sound Speed 

( NIST V ) ( Qmeter ) ( QNIST ) (ε )

[% FS] [m/s] [m3/s] [m3/s] [%] [m/s] [m/s] [%] 

55 16.896 1.2200 1.2201 -0.01 423.93 423.92 0.002 

[% FS] [ft/s] [acfh] [acfh] [%] [ft/s] [ft/s] [%] 

55 55.434 155 102 155 119 -0.01 1 390.85 1 390.81 0.002 

Table 8. Verification results of point by point data for ultrasonic flowmeter serial number 1234. 

Pt Time 

NIST 

Velocity 

( NIST V ) 

Meter 

Flow 

( Qmeter ) 

NIST 

Flow 

( QNIST ) 

% Flow. 

Error 

(ε )

Temp. Pres. REFPROP 8.0 Sound Speed Avg. M

Sound S

[ ] [ ] [m/s] [m3/s] [m3/s] [%] [K] [kPa] [m/s] [m/

1 9:19:56 AM 16.893 1.2196 1.2199 -0.02 295.49 8 374.1 423.89 423.2 9:22:01 AM 16.893 1.2199 1.2199 0.00 295.57 8 373.3 423.94 423.3 9:24:14 AM 16.903 1.2205 1.2206 -0.01 295.59 8 371.1 423.96 423.
SAMPLE CALIBRATION REPORT Gas Flowmeter, S/N 1234

Advanced Technology Ultrasonics, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 10 of 11

Updated register files produced by the flowmeter are also provided with each calibration report. 

These files are not included in this sample report since the content and format of these files differ 

substantially among flowmeter manufactures. 

An analysis was performed to assess the uncertainty of the results obtained for the meter under 

test.5,6,7 The process involves identifying the equations used in calculating the calibration result 

(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 k = 2.0 to arrive at an expanded uncertainty (Ue ) 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, i x , 

( , , , ) 1 2 N y = y x x K x (4)

and all uncertainty components are uncorrelated, the normalized expanded uncertainty is given by, 

( ) ( ) ( ) ∑= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = =

N

x

u x k s

y

u y k

y

U y

i 1

2

i

2 i

i

e c (5)

In the normalized expanded uncertainty equation, the u( x )' s i are the standard uncertainties of 

each input, and s 's i are their associated sensitivity coefficients, given by, 

y

x

x

y s i

i

i ∂

∂ = (6)

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 includes the base uncertainty as determined in the SP250 

Supplement for natural gas flow measurements3

 and the reproducibility8 of the measurements. 

The values of uncertainty given in Table 4 include contribution from both these sources. To 

 

5 International Organization for Standardization, Guide to the Expression of Uncertainty in Measurement, Switzerland, 1996 

edition.

6 Taylor, B.N. and Kuyatt, C.E., Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, 

NIST TN 1297, 1994 edition.

7 Coleman, H.W. and Steele, W.G., Experimentation and Uncertainty Analysis for Engineers, John Wiley and Sons, 2nd ed., 1999. 

8 Reproducibility is herein defined as the closeness of agreement between measurements with the flow changed and then 

returned to the same nominal value
SAMPLE CALIBRATION REPORT Gas Flowmeter, S/N 1234

Advanced Technology Ultrasonics, Inc. Purchase Order No. A123

NIST Test Number: 836-270542-04-04 Service ID Number: 18015C

Calibrated on July 8, 2007 by Gina Kline and Aaron Johnson Page 11 of 11

measure the reproducibility 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 Figure 4. 

For the Director, 

National Institute of Standards and Technology 

Project Leader’s Official Signature

Aaron Johnson 

Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratories