NIST Measurement Services: 

NIST Calibration Services for 

Water Flowmeters 

Water Flow Calibration Facility

NIST Special Publication 250

 Iosif I. Shinder, Iryna V. Marfenko
NIST Special Publication 250 

NIST Measurement Services: 

Iosif I. Shinder, Iryna V. Marfenko 

Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratory 

National Institute of Standards and Technology 

Gaithersburg, MD 20899 

August 2006 

NIST Calibration Services for Water Flowmeters 

Water Flow Calibration Facility

2

National Institute of Standards and Technology 

 William Jeffrey, Director 

Robert Cresanti, Under Secretary of Commerce for Technology 

Technology Administration 

U.S. Department of Commerce 

Carlos M. Gutierrez, Secretary 
Certain commercial entities, equipment, or materials may be identified in 

this document in order to describe an experimental procedure or concept 

adequately. Such identification is not intended to imply recommendation 

or endorsement by the National Institute of Standards and Technology, 

nor is it intended to imply that the entities, materials, or equipment are 

necessarily the best available for the purpose. 

National Institute of Standards and Technology Special Publication 

3
NIST Measurement Services: 

NIST Calibration Services for 

Water Flowmeters 

Water Flow Calibration Facility

NIST Special Publication 250 

Iosif I. Shinder, Iryna V. Marfenko 

Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratory 

National Institute of Standards and Technology 

Gaithersburg, MD 20899 

4
TABLE OF CONTENTS 

NIST Special Publication 250 ...........................................................................................4

Abstract...............................................................................................................................6

1 Introduction...................................................................................................................6

2 Description of Measurement Services.........................................................................8

3 Procedures for Submitting a Flowmeter for Calibration........................................10

4 The NIST Water Flow Primary Standard................................................................10

4.1 Fundamentals of the Static Gravimetric Method................................ 10 

4.2 Equipment Arrangement, Schematic and Operation Sequence ........ 11 

4.3 Error-Free Uni-directional Diverter: Theory and Design.................. 14 

5 Uncertainty Analysis for NIST’s Water Flow Calibration Facility .......................17

5.1 Techniques for Uncertainty Analysis ................................................... 19 

5.2 Volumetric Flow ..................................................................................... 19 

5.2.1 Collected Mass Uncertainty.................................................................20 

5.2.1.1 Scale Calibration, Mass Standards Calibration, Long-term 

Stability, and Sensitivity ........................................................................... 20 

5.2.1.2 Buoyancy Correction .................................................................. 24 

5.2.1.3 Splashes and Leaks..................................................................... 24 

5.2.1.4 Storage effects............................................................................. 24 

5.2.1.5 Water Evaporation ...................................................................... 24 

5.2.2 Collection Time Uncertainty ...............................................................24 

5.2.2.1 Uni-directional Diverter Tests.................................................... 24 

5.2.2.2 Counters and Timers................................................................... 29 

5.2.3 Density ...................................................................................................29 

5.2.4 Viscosity.................................................................................................34 

5.2.5 Temperature Measurement.................................................................36 

6 Data Acquisition and Control System.......................................................................37

7 Summary......................................................................................................................38

8 References....................................................................................................................39

Appendix: Sample Calibration Report..........................................................................41

5
Abstract 

This document describes the Water Flow Measurement Standards at the National Institute 

of Standards and Technology (NIST). These primary standards are disseminated using 

calibration services offered by NIST’s Fluid Metrology Group using its Water Flow 

Calibration Facility (WFCF). This facility has three parallel pipelines with diameters of 

100, 200 and 400 mm and three weighing systems with capacities of 1100 kg, 3700 kg 

and 22500 kg. Part of the WFCF which includes a 3700 kg collection tank and a 100 mm 

pipeline (for the sake of simplicity hereafter referred to as WFCF 3700/100) is now 

complete and operated by the Fluid Metrology Group. The WFCF 3700/100 is used to 

provide water flowmeter calibration services as reported in [1] for the Calibration Service 

ID Number 18020C. The WFCF 3700/100 uses the static gravimetric method [2] and an 

error-free uni-directional diverter with a collection/bypass unit (CB unit) to perform 

water flowmeter calibrations between 40 L/min and 1600 L/min. The WFCF measures 

flow using the static gravimetric method that employs weighing the water collected in the 

3700 kg tank during a measured time interval [3]. The expanded uncertainty of the flow 

measurement is 0.033 % when a full tank (3700 kg) is weighed (k = 2 or approximately 

95 % confidence level). 

Key words: calibration, correlated uncertainty, flow, flowmeter, water flow standard, 

diverter, mass calibration, meter, uncertainty. 

1 Introduction 

We provide an overview of the water flow calibration service and the procedures for 

customers to submit their flowmeters to NIST for calibration. We describe the significant 

and novel features of the standard, in particular, the uni-diverter with CB unit, and 

analyze its uncertainty. We demonstrate theoretically and experimentally that the newly 

developed uni-directional diverter with CB unit leads to virtually zero time correction and 

low uncertainty contributions from the diverter performance. The largest source of 

uncertainty is the measurement of the mass of water collected. Procedures and 

periodicities for calibration of reference masses, the weigh scale, density, temperature, 

time, and humidity sensors, are described together with their uncertainty and stability. 

Calibrations of liquid flowmeters are performed with primary standards [2-4] that are 

based on measurements of the more fundamental quantities: length, mass, time. Primary 

flow calibrations are accomplished by timed collection of the measured mass of water 

flowing through the meter under test (MUT) during approximately steady conditions of 

flow, pressure, and temperature. All of the quantities measured in connection with the 

calibration standard (i.e., temperature, mass, time, etc.) are traceable to established U.S. 

national standards. The flow measured by the primary standard is computed along with 

the average of the flow indicated by the MUT during the collection interval. An 

additional flowmeter is normally used to set the test flows and to monitor the flow 

stability. 

6
NIST’s Water Flow Calibration Facility consists of the 3 fundamental component parts: 

• flow generation system: comprised of storage tank, pumping system, and a flow 

control system which actuates the control valves. The flow generation system

produces the water flow through the test section at the constant rate required for 

test point series necessary for a calibration. 

• test section: piping system producing the required flow conditions for the MUT. 

The main purpose of this piping system is to implement ideal flow for flowmeter 

operation, i.e. the appropriate, fully developed pipe flow for the conditions. 

Therefore, an axisymmetric filter and two flow conditioners (tube-bundle and 

perforated plate) are installed upstream of the flowmeter 15 meters (or 150 

diameters of the pipe) away the MUT to avoid any flow disturbance that might 

affect the meter performance. 

• gravimetric reference system: weigh system with collection tank and a flow 

diverting device. This device is the part of the calibration system that directs the 

flowing water into the collection tank while triggering a clock to determine the 

collection time. The water collected can be determined in terms of volumetric or 

gravimetric units. 

NIST offers calibration services for water flowmeters in order to provide traceability for 

flowmeter manufacturers, secondary flow calibration laboratories, and flowmeter users. 

For a calibration fee, NIST calibrates a customer’s flowmeter and delivers a calibration 

report that documents the calibration procedure and the calibration results, with their 

uncertainty. The flowmeter and its calibration results may be used in different ways by 

the customer. The flowmeter is often used as a transfer standard to perform a comparison 

of the customer’s primary water flow standards to NIST’s primary water flow standards 

to establish the customer’s traceability, to validate their uncertainty analysis, and to 

demonstrate the proficiency of their testing process. Customers without primary 

standards can use their NIST calibrated flowmeters as working or reference standards in 

their laboratory to calibrate other flowmeters. 

The Fluid Metrology Group of the Process Measurements Division (part of the Chemical 

Science and Technology Laboratory) at NIST now provides water flow calibration 

services over a range of 40 kg/min to 1600 kg/min. After completion of 200 mm and 

400 mm pipelines with 1100 kg and 22500 kg collection tanks, NIST will be able to 

provide water flow calibration services in the range of 8 kg/min to 38,000 kg/min. 

Table 1 presents the flow ranges covered by the present and planned primary water flow 

standards available from the Fluid Metrology Group. 

7
Table 1. Primary water flow calibration capabilities within the NIST Fluid Metrology 

Group. Green regions represent operational systems, white regions represent those under 

construction. 

Feature Tank 1 Tank 2 Tank 3 

Tank Volume, L 22000 3700 900 

Tank Material Steel Fiberglass Fiberglass 

Scale Type Load Cell Weigh Scale Weigh Scale 

Scale Capacity, kg 22500 4500 1100 

Scale Resolution, kg 2 0.2 0.04 

Pipe Size, mm 200 to 400 100 to 200 25 to 100 

Flow Range, L/min 880 to 38,000 40 - 1,600 (100mm) 

200-4,900 (200mm) 

8-1500 

Working Pressure, kPa 100-1000 100-1000 100-1000 

Expanded Uncertainty 0.086% (projected) 3000 kg 600 kg 

0.033% 0.051% 

0.075% (projected) 

This document describes the theory, methods of operation, and uncertainty of the 

100 mm pipeline with the 3700 kg collection tank primary standard that covers the flow 

range from 40 L/min to 1600 L/min. 

2 Description of Measurement Services 

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

information regarding NIST’s calibration services, fees, technical contacts, and 

flowmeter submittal procedures. 

NIST uses the WFCF 3700/100 primary standard described herein to provide water 

flowmeter calibrations for flows between 40 L/min and 1600 L/min. The facility can be 

used at flows as low as 10 L/min and as high as 1800 L/min, but calibrations below 

40 L/min and above 1600 L/min should be discussed with NIST flow calibration staff 

before a flowmeter is submitted for calibration. 

8
The WFCF does not have a temperature control system and, therefore, only room 

temperature calibrations are available. Since pump heating occurs, the temperature 

increases during the test procedure at a rate of about 0.2 K/hour. Typically, flowmeter 

calibration results will produce unique curves when appropriate compensation is made 

for the viscosity and density of the water and thermal expansion effects in the meter. 

Meters can be calibrated at NIST if the flow range and piping connections are suitable, 

and if the system to be tested is judged to have the precision appropriate for the WFCF 

flow measurement uncertainty. Typical flowmeters calibrated in the WFCF are high 

quality turbine, ultrasonic, Coriolis, and magnetic flowmeters. The precision, 

repeatability, and reproducibility (see [8, 9]) of these flowmeters are generally considered 

appropriate for the WFCF. However, other flowmeters can be tested as well, where 

customers feel they need the cost-benefits of having a NIST calibration of their meter. 

Flowmeter types with significant imprecision (instabilities significantly larger than the 

WFCF uncertainty) should probably not be calibrated in the WFCF for economic reasons. 

A normal flow calibration performed by the NIST Fluid Metrology Group is intended to 

quantify meter performance and its stability or precision. This is done by making multiple 

measurements in the desired test conditions to produce meter factor averages and 

standard deviations. Calibrations generally consist of two sets of measurements, with 

increasing and decreasing flow adjustments, where 5 successive measurements are made 

at each flow set point. The scatter in each set of 5 can produce a standard deviation that 

can be termed a Repeatability. Repeatability can quantify the short-term stability of the 

meter. Longer-term stability is usually quantified by changing the conditions to include 

typical usage patterns for the meter, such as turning it off and then turning it back on. The 

two sets of 5 measurements determined at essentially the same set point, but after the 

system is turned off and then turned back on can produce a standard deviation that can be 

termed a TOTO (Turn-Off-Turn-On) Reproducibility. Typical calibration set points 

might be chosen at nominal rates, such as: 40 L/min, 200 L/min, 500 L/min, 1000 L/min, 

and 1600 L/min. Therefore, the final data set consists of 50 (or more) primary flow 

measurements with corresponding meter outputs made at five flow set points. The sets of 

five measurements can be used to assess meter performance in terms of averages and 

repeatability (short term stability or the closeness of agreement among a number of 

consecutive measurements), while the sets of ten can be used to assess reproducibility 

(long term stability or the closeness of agreement among a number of repeated 

measurements when conditions change). For further explanation, see the sample 

calibration report in Appendix. Variations on the number of flow set points, spacing of 

the set points, and the number of replicated measurements can be discussed with the 

NIST technical contacts. However, for data quality assurance reasons, NIST rarely 

conducts calibrations involving fewer than three flow set points and two sets of three 

flow measurements at each set point. 

Whenever it is possible, the Fluid Metrology Group presents flowmeter calibration results 

in a dimensionless format that takes into account the physical model for the flowmeter 

type. The dimensionless parameter approach usually facilitates accurate flow 

9
measurement performance when the conditions of use (temperature, viscosity, and 

dimensional changes) differ from the conditions used for the calibration. 

Hence, for a turbine flowmeter calibration, the calibration report will present Strouhal 

number versus Reynolds or Roshko number. 

3 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 [7]. These chapters can be found 

on the internet at the following addresses: 

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

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

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

Chapter 2 gives instructions for ordering a calibration for domestic customers and has the 

sub-headings: 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. NIST 

contact information can be found www.nist.gov/fluid_flow . 

4 The NIST Water Flow Primary Standard 

The static gravimetric liquid flow measurement method is used by the WFCF 3700/100. 

The WFCF 3700/100 is designed to have uncertainty levels that are lower than highperformance flowmeters or secondary standards for which it provides flow traceability. 

The main features of the WFCF 3700/100 are a well established theory, a complete and 

detailed uncertainty analysis, and traceability to the NIST mass, time, temperature 

standards. The WFCF is a highly automated facility. It uses the National Instruments 

LabView environment and custom designed virtual instruments to operate mechanical 

and electronic equipment of the standard during calibration and to store data (see section 

6 for details). 

4.1 Fundamentals of the Static Gravimetric Method 

Static gravimetric liquid flow calibration systems are widely used as primary liquid flow 

standards by NIST and other laboratories. Primary static weigh flow calibrations are 

arranged by collecting a measured mass of the fluid flowing through the meter being 

calibrated over a measured time interval under approximately steady conditions of flow, 

pressure, and temperature at the meter under test. All of the quantities measured in 

connection with the calibration standard (i.e., temperature, mass, time, and density) are 

traceable to established national standards. The flow measured by the primary standard is 

computed along with the average of the flow indicated by the meter under test during the 

collection interval. The calibration result, in the form of a meter factor, is the ratio of the 

10
computed flow result from the standard to the averaged meter indication, or the reciprocal 

of this ratio. An additional flowmeter is normally used in the test pipe to set flow, to 

check the flow stability, and possibly to assist in processing final results. 

One can derive an equation for the average mass flow through the meter being calibrated 

during the collection time by writing a mass balance for the control volume composed of 

the inventory and tank volumes: 

( VI t

M

m −+ ρρ 12 Δ  = ) , (1) 

where M is collected mass and Δt is collection time. The inventory volume, VI is the 

volume of piping between the meter under test and the standard used, at the end of the 

pipe, to measure the flow. The densities ρ1 and ρ2 are those in the inventory volume at 

the beginning and end of the collection interval. This equation applies to an idealized set 

of conditions: (1) the flow velocity profile exiting the pipe and entering the standard is 

symmetric with respect to the middle of the fishtail, and (2) the motion of diverter in the 

middle of the water jet is horizontal. Under these conditions, the start and stop time 

signals provide the proper collection time. More information about Static Gravimetric 

Method can be found in the Refs. [2, 3]. 

4.2 Equipment Arrangement, Schematic and Operation Sequence 

The layout of the WFCF is shown below (Figure 1). 

11
Figure 1. A Perspective Drawing of NIST’s Water Flow Calibration Facility. 

The NIST WFCF is a closed loop flow system that consists of a flow source (centrifugal 

pumps), flow conditioners, pipe lines, test section for the MUT installation, valves for 

changing the flow, a diverter with CB unit (see below), collection tanks, weigh scales, 

and a timer (see also Figure 2 and 3). The facility is located above a water reservoir that 

has a capacity of approximately 230 m3. Water flow is produced and maintained in the 

system by four constant velocity pumps, three driven by 112 kW electric motors (pumps 

P-1 to P-3, in Fig 2) and one driven by a 75 kW electric motor (pump P-4). A manifold 

splits the flow into three separate test section pipelines and a bypass of 200 mm diameter. 

The three pipelines are coupled to facilitate flow comparisons between the tanks and to 

permit tests with long collection times by collecting low flows in the larger tanks. 

Downstream of the manifold, each pipeline has a flow conditioner that delivers a 

symmetric, fully developed turbulent velocity profile to the flow meter in the test section. 

Upstream of the meter under test, the facility has straight lengths of 24 diameters for the 

400 mm pipeline, 76 diameters for the 200 mm pipeline and 174 diameters for the 

100 mm pipeline. 

Figure 2. A Sketch of the Flow Paths of the Water Flow Calibration Facility. 

Flow at the test section is controlled by two sets of valves. One set is located upstream 

near the pump, a main valve for each pipeline and a bypass throttle valve that controls the 

amount of water returned to the reservoir without passing through the meter test section, 

see Fig 2. The other valves on each pipeline (located downstream of the test section) are 

12
the fine and coarse controls for setting the water flow rate and the pressure in the test 

section of the WFCF. Once the flow passes through the meter under test and the control 

valves, it goes through two valves in series (a leak detection system) and then a fishtail 

which produces the rectangular jet flow for the diverter mechanism, see Fig 3. See also 

Ref. 5 for details. 

The process of making a gravimetric flow measurement normally consists of the 

following steps: 

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

pipeline and the meter under test. 

2. Zero the weigh scale or measure the initial mass of water in the collection tank. 

3. Switch the diverter to the position to collect water in the collection tank and start the 

timer to initiate the collection time measurement. At the same time, record the initial 

water temperature in the pipeline between the MUT and the collection tank and the 

room air temperature, pressure, and humidity (for buoyancy corrections to the 

collected mass measurement.). 

4. Collect the water passing through the MUT during the measured collection time (time 

should be long enough to collect >600 kg of water). 

5. Reverse the diverter position and trigger the stop time. 

6. Measure the final mass of water in the collection tank, and, after this, drain the 

collection tank. 

13
Figure 3. Sketch of the Arrangement of Equipment in the WFCF 3700/100 System. 

4.3 Error-Free Uni-directional Diverter: Theory and Design 

The nozzle and diverter are designed to 1) rapidly switch the flow from the collection 

tank bypass to the collection tank and back again without disturbing the flow conditions 

in the test section and 2) generate the timing signals for accurate measurement of the 

collection time. During a normal calibration cycle, two diverter traverses are required: a 

first traverse to switch flow from the bypass loop into the collection tank (starting a 

timer), and a second one to switch flow back to the bypass (stopping the timer). 

The WFCF 3700/100 uses a uni-directional diverter that is immune to certain sources of 

uncertainty found in traditional diverters. In the traditional diverter design, a dividing 

plate is moved through a rectangular jet flowing out of the fishtail, see Fig 3. At the start 

of the collection, the dividing plate switches the flow from the bypass channel into the 

collection tank. At the end of the collection interval, the diverter traverses the jet in the 

opposite direction returning the flow to the bypass. The traditional diverter usually 

requires a correction time resulting from any asymmetry of the distribution of the flow 

exiting the fishtail and any asymmetry of the diverter motions through the rectangular jet 

14
exiting the fishtail. The procedure for conventional diverter correction, which is usually a 

function of liquid flow rate, is given in many flow measurement standards, see 

Refs. [2, 5, 6]. 

The basic concept of the new diverter system developed for the NIST WFCF by the Fluid 

Metrology Group makes use of repeated unidirectional motions of the diverter valve in order 

to reduce errors associated with asymmetry in the diverter valve motion and in the liquid jet 

velocity profile. A theoretical consideration and supporting experimental data of the unidirectional diverter performance are given in [5, 6]. Here only a brief description will be given. 

The uni-directional diverter exploits the idea of error self cancellation under condition A+D = 

B+C or A=C & B=D or A=B and C=D (see Figure 4). 

Figure 4. Collection Flow Diagram and Error-free Conditions. 

The uni-diverter is made to move in the same direction through the liquid jet both at the 

beginning and the end of the water collection. Its design has two separate active 

elements (See Fig. 5): 1) a traditional divider (which cuts the flow) is operated by a 

pneumatic angular actuator and 2) a Collection/Bypass (CB) unit that directs the flow to 

the bypass or the collection tank regardless of the divider position. The CB unit is 

mounted below the divider on linear bearings and can be moved horizontally under 

computer control. Three proximity sensors detect the location of CB unit and transmit it 

to the computer. The CB unit consists of three separate channels and coordination of the 

position of the CB unit with the position of the divider allows cutting the water jet in the 

same direction for both the start and stop of the collection. The uni-directional travel of 

the divider dramatically reduces errors due to asymmetry in 1) the divider actuated 

15
motion, 2) the liquid jet velocity profile, and 3) the position of the diverter trigger [6]. The 

full operation sequence of the uni-diverter system is shown in Figure 6. 

Diverter 

Nozzle 

CB unit 

Proxy 

sensors 

Figure 5. Photograph of the Uni-diverter 

Thus uncertainties due to jet profile asymmetry and any asymmetries in the divider 

velocity versus time as it moves through the jet are negligible. Hence, time uncertainties 

of the uni-directional diverter arise only from temporal instabilities of the flow, 

instabilities in the water jet profile, and any irreproducibility of the divider motion in a 

single direction. The freedom to choose the locations of the start and stop switches is a 

valuable feature of the uni-directional diverter. Performance tests of the uni-directional 

diverter are given below. 

A comprehensive analysis of the uni-directional diverter can be found in [6]. 

16
Figure 6. Collection-bypass Cycle. 

5 Uncertainty Analysis for NIST’s Water Flow Calibration Facility

In this section, we will analyze summarized in the Table 2 the uncertainties of the WFCF 

3700/100. Firstly, we will briefly describe the subject of uncertainty analysis itself, 

together with the current conventions that apply. Next, we will give the results of the 

uncertainty analysis for mass flow. We will give uncertainties of the sub-components 

such as those for collected mass, the density of the fluid flowing through the meter during 

the collection, and the collection time which must be combined to obtain the mass and 

volume flow rate uncertainties. The comprehensive graphical representation of the 

measurement equation that serves as the basis for the uncertainty analysis and 

explanation of notations is given in Ref.1. An additional uncertainty source (beyond those 

listed in Ref. 1) is the water evaporation ME , shown in the diagram in Figure 7. 

17
Figure 7. Uncertainty Diagram (Ref.1). 

Table 2. Uncertainty Budget of the WFCF 3700/100 for Mass Collections of 3000 kg and 

600 kg. 

Reference Value Uncertainty, % Uncertainty, %

Collected Mass 3000 kg 600 kg 

1. Mass uncertainty

Scale indication, 0.2 kg 5.2.1.1 0.2 kg 0.004 0.02

Scale drift 5.2.1.1 0 0

Scale calibration 5.2.1.1 0.01 0.01

Buoyancy correction 5.2.1.2 0.0005 0.0005

Leaks and splashes 5.2.1.3 0 0

Storage effects 5.2.1.4 0.003 0.003

Evaporation 5.2.1.5 0.004 0.004

Total mass uncertainty 0.012 0.023

2. Collection time uncertainty

Timer calibration 5.2.2.2 0.0001 s 0.0004 0.0004

Timer actuation and diverter 5.2.2.1 0.01 0.01

Total time uncertainty 0.010 0.010

3. Water density uncertainty 5.2.3 0.0053 0.0053

Combined uncertainty for Q 0.016 0.026

Expanded uncertainty for Q 

(95% confidence level) 0.033 0.051

18
5.1 Techniques for Uncertainty Analysis 

The uncertainty of a mass flow measurement with the WFCF 3700/100 is based on the 

techniques described in Refs.[8, 9] The process identifies the equations involved in the 

flow measurement so that the sensitivity of the final result to uncertainties in the input 

quantities can be evaluated. The uncertainty of each of the input quantities is determined, 

weighted by its sensitivity coefficient, and combined with the other uncertainty 

components to arrive at the combined uncertainty, using the root-sum-squared technique. 

As described in [8, 9], consider a process that has an output, y, based on N input 

quantities, xi. For the generic basis equation: 

),,,( 21 N = ... xxxyy , (2) 

if all the uncertainty components are uncorrelated, the standard uncertainties can be 

combined using the root-sum-squares (RSS) technique to give: 

( ) ∑ ( ) = ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ =

N

i

i

i

c xu

x

y yu

1

2

2

, (3) 

where u(xi) is the standard uncertainty for each of the inputs, and uc(y) is the combined 

standard uncertainty of the measurand. The partial derivatives in Eq. 3 represent the 

sensitivity of the measurand to the uncertainty of each input quantity. 

5.2 Volumetric Flow 

Volumetric flow, Q, is derived from the mass flow knowing the average liquid density, 

ρ , at the MUT location during the collection interval, i.e., 

m M Q

ρ tρ = = Δ

 (4) 

The combined uncertainty of the volumetric flow rate is calculated using the propagation 

of component uncertainties and then using the root-sum-square (RSS) method to combine 

the results. The three main uncertainty components are those that arise from the collected 

mass, collection time, and water density, and their combination gives: 

2 2

M t u uuu

2 = ++ Δ ρ (5) 

where Mu , , and t uΔ uρ are the standard uncertainties of the collected mass, collection 

time, and liquid density, respectively. The confidence interval given by this equation is 

68%. A coverage factor of k = 2, will be used to convert the combined standard 

uncertainty to the expanded uncertainty with approximately 95% confidence level. 

19
5.2.1 Collected Mass Uncertainty 

5.2.1.1 Scale Calibration, Mass Standards Calibration, Long-term Stability, and 

Sensitivity 

The first component of mass uncertainty we will consider is the weigh scale resolution. 

For a collection mass M the weigh scale resolution of 0.2 kg leads to an uncertainty of 

0.2

M 3

 or 0.004% for fully filled tank (3000 kg) and for 20% filled tank 0.02% (the 

square root of 3 takes into account Rectangular to Normal Probability Distribution 

conversion). 

Another part of the mass uncertainty is scale calibration. Calibration of the weighing 

scale uses a sequential, incremental loading method based on a set of 45 kg weights 

calibrated at NIST using the 65 kg weight set and 60 kg mass comparator from NIST’s 

Volume calibration service. The traceability of water flow measurements to the national 

standards of mass is through a 65 kg set of stainless steel weights calibrated by the NIST 

Mass and Force Group, http://www.mel.nist.gov/div822/groups.htm. Weights from the 

65 kg set are used to calibrate the 60 kg mass comparator. The weighing system is 

composed of 12 mass standards of 45 kg each and used for calibration and checking of 

the operation of the weighing scale (See Fig. 8). 

The calibration procedure requires replacing the steel weight of the mass standards using 

water and then adding mass standards to sequentially progress across the scale range to 

calibrate the full capacity of the weighing scale. This is done because a sufficient number 

of mass standards are not available to complete the calibration without partial water fills. 

The procedure with using mass standards and water load equivalents leads to a higher 

uncertainty. Figure 9 illustrates the scale calibration procedure, and the following steps 

describe the tasks. 

1) Close the drain valve of the weigh tank. At the initial position of the calibration 

process, the masses lie on a stand attached to the floor (See Fig. 8). When the 

calibration starts, the system sets 0 kg, at this point the balance bears the tare weight 

of residual liquid in the tank. Record the scale indication. 

2) Place the set of 45 kg mass standards on the scale. These masses are raised by a 

pneumatic system of pistons that grab the 3-stacked masses in 4 sets. The load cells 

read the corresponding nominal mass of 540 kg. Record the scale indication. 

3) Remove the set of 45 kg mass standards from the scale and fill the tank until the scale 

indicator value is identical to that obtained in the last iteration of step 2. Care must be 

exercised to match the value indicated when loaded with the mass standards. 

4) Repeat steps 2 and 3. 

Buoyancy corrections are taken into account for both water and steel weights (see 

paragraph 5.2.1.2). 

20
Pneumatic 

system 

Mass 

standards 

Stand 

Figure 8. Weighing System for WFCF 3700/100 and Mass Standards 

The maximum tank load is about 3700 kg, the reference mass is approximately 540 kg so 

the number of mass increments necessary to fully load the scale is 6 or 7. 

21
Figure 9. The Scale Calibration Sequence 

The following system of equations is used to describe the data reduction procedure. 

( ) on off m Mm M i i = + ref , (6) 

where , mi ( ) M mi are scale indications and corrected masses for each step and Mref is 

the mass of the reference weights. Superscripts on and off indicate the position of the 

weights. The task of the fitting procedure is to find a minimum by the least square 

method of the value: 

( ( )

0 2

,

i on

i fit i

n

M Mm

=

∑ − ) , (7) 

where - fitted value. Three different functional forms for the corrected mass 

dependence were chosen: linear dependence, quadratic dependence and linear 

dependence which crosses the origin. It was found that the last form was sufficient to fit 

all data. In the procedure described above the actual parameter which is being calibrated 

is the slope 

Mi fit ,

M K

m

Δ = Δ , for different scale loads. 

22
The value of the scale coefficient averaged over the two year period (2004-2005, see 

Fig.10) is 0.99881 0.00007. In 2006 the scale coefficient was found to be 

0.99884 ± 0.00010 (or ± 0.01%) and this value is included in the uncertainty budget, 

Table 2. 

±

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

April-04 February-05 December-05 October-06

(K /K ave-1)*100%

Figure 10. Residuals from Linear Fit 

The calibration results for the weigh scale over the past two years show that the pressure 

and temperature dependencies of the weigh scale and any other sources of drift in the 

scale calibration are much smaller than the uncertainty in scale calibration, 0.01%, and, 

therefore can be neglected. 

The pneumatic weight handling system has improved the efficiency of the scale 

calibration technique to enable a scale calibration prior to each flowmeter calibration if 

necessary. 

The scale performance is checked periodically by placing a weight stack of known mass 

from NIST’s Volume calibration service on the scale. 

The set of 45 kg weights is used for calibration and checking the operation of the 

weighing scale. These 45 kg weights are calibrated by a double substitution weighing 

method using the 65 kg set and 60 kg mass comparator from NIST’s Volume calibration 

service. The uncertainty of NIST’s Mass Standards calibrations are less than 3 parts per 

23
million and are negligible in comparison with the other types of scale uncertainties 

mentioned here. 

5.2.1.2 Buoyancy Correction 

The mass values observed by the weigh scales in the static weighing system are subject to 

buoyancy forces. According to Archimedes’ Law, to obtain the true mass of the water 

collected, the air and water densities are required. The true mass can be found using the 

following relationship: 

1

M

T

A

W

M M

ρ

ρ

= ⎛ ⎞ ⎜ ⎟ −

⎝ ⎠

 (8) 

where MM is the apparent mass indicated by the scale, ρ A the air density, and ρW is the 

water density. The buoyancy uncertainty for the WFCF was estimated in [1] and it is less 

than 0.0005%. 

5.2.1.3 Splashes and Leaks 

Liquid splashing from the diverter and weigh system and minor leaks from the pipework 

are usually apparent and are eliminated before calibrations start. Therefore, uncertainties 

for splashes or leaks are neglected. 

5.2.1.4 Storage effects 

Storage effects due to changes in the density of the water and dimensions of the pipes for 

the volume between the MUT and the outlet of the fishtail (the inventory volume) must 

be considered. For the WFCF, this effect is estimated to be less than 0.003% [1]. 

5.2.1.5 Water Evaporation 

In the case of low flows, the collection time required to obtain an acceptably large mass 

change in the collection time can be quite long. For example, a calibration of a flowmeter 

at the rate 40 L/min takes about 15 min to fill the tank completely. Tests show that for the 

normal air conditions of 100 kPa, 22°C, and 50% humidity the uncertainty due to the 

evaporation is about 0.0004% (k=1). 

5.2.2 Collection Time Uncertainty 

5.2.2.1 Uni-directional Diverter Tests 

The diverter is one the of critical elements in the design of a liquid flow primary 

standard. A complete flow diverter system provides four important functions: 

24
1. It changes the direction of the flow stream without splashing or leaking, 

2. It channels the desired flow direction to the collection tank, 

3. It starts the timer, and 

4. It stops the timer. 

Diverting the flow and starting the timer can cause a timing error related to the specific 

transition point for the flow into the weighing tank. Similarly, the return diversion of the 

flow to the by-pass channel and stopping the timer can cause timing error. Therefore, the 

determination of the uncertainty of the collection period is crucial. 

Uncertainties in the diversion process originate from several sources: 

1. The finite time required for the dividing plate to traverse through the 

rectangular jet, 

2. Asymmetries in the velocity profile of the rectangular jet flow from the fishtail, 

3. Differences in the divider motions in the two directions, and 

4. The positioning of the timer triggers for the dividing plate positions in the 

rectangular jet. 

Tests designed to evaluate uncertainties related to the diverter vary the collection time at 

nominally constant flow. Timing errors due to the diverter will be more significant for 

smaller collection times, thus leading to differences in flow results (or flowmeter 

calibration factors) that are dependent on collection time. Three validation tests of this 

type have been applied to the uni-directional diverter and are described here. 

1. The uni-directional diverter was compared with a traditional diverter. 

To compare performances of the traditional diverter to the uni-diverter over a range of 

flows, we gathered data in the following sequence: 

1. Traditional diverter (single step, long diversion) 

2. Traditional diverter (n-steps, short diversions) 

3. Uni-diverter (single step, long diversion). 

The derivation of the diverter timing correction assumes constant flow (or compensates 

for flow differences) and therefore to avoid having to compensate, the test flow was 

maintained as stable as possible while gathering data from the three methods. To 

minimize the effects of small flow changes during the course of the test, flow indications 

from a turbine meter in the 100 mm test section were used to monitor the constancy of 

the flow. The measured values of collected mass for the multiple diversions and unidiverter tests were adjusted based on the ratio of the turbine meter frequency, using the 

traditional long diversion frequency as the reference. The data acquisition program was 

automated to perform all three test runs in sequence and without interruption. 

Figure 11 shows the performance of the traditional diverter for 30 and 50 sec. collection 

times if no correction was taken into account. The linear dependence of the differences 

between uni-directional diverter and traditional diverter suggests that time correction for 

traditional diverter must be included to calculate true flow rate. Using the theoretical 

approach the correction to the collection time can be calculated. See Ref [5, 6] for 

detailed information 

25
-0.50

-0.40

-0.30

-0.20

-0.10

0.00

600 700 800 900 1000 1100 1200

Q (L/min)

100 (

Q t / Q -1)

30s

50s

Figure 11. Differences between Traditional and Uni-directional Diverters, where No 

Correction is Made 

Figure 12 shows a comparison between corrected collection time for the traditional 

diverter and uni-directional diverter. Each point is the average of 5 measurements. For 

the tested flow the difference is smaller than 0.005%. 

26
-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

600 700 800 900 1000 1100 1200

Q (L/min)

100 (

Q tc / 

Q -1)

30 s

50 s

Figure12. Difference between Traditional and Uni-directional Diverter, where Time 

Correction is Included 

2. The uni- directional diverter was also tested based purely on statistical analysis. In this 

experiment, two different collection times for each flow were chosen to measure the flow 

over a large number of sequential collections. In order to increase the sensitivity of the 

experiment, a maximum flow of 1800 L/min and a minimum collection time of 20 sec. 

were chosen (instead of the maximum 1100 L/min and of 30 sec. in the previously 

described experiment). Maximum collection time for the 1800 L/min flow rate was 

100 sec. These two collection times were used alternately to measure flow during the 

experiment, which ran continuously for 10 hours. 

It can be seen from the Figure 13 that the 20 s and 100 s curves almost coincide. The 

difference in flow measurement between the 20 s and 100 s collection times can be 

estimated as a ratio of the areas under the curves. It was found that this ratio is 1.00016; 

therefore the differences between 20 s and 100 s collection times is about 0.016%. 

27
28.2

28.4

28.6

28.8

29

29.2

29.4

24400 34400 44400 54400 64400

Time (s)

Q (L/s)

20 s run

100 s run

Figure 13. Ten Hour Run with 20 and 100 s Collection Times 

3. Performance of the uni-directional diverter was assessed by calibrating a very stable 

dual rotor flowmeter which has a repeatability of better than 0.02%. The calibration result 

for this meter (Strouhal number) for 5 collection times spanning the range from 40 s to 

200 s (See Fig. 14) was measured at a flow rate of 15 L/s (tank was filled from 600 kg to 

3000 kg). 

28
9.174

9.176

9.178

9.18

9.182

0 40 80 120 160 200 240

Collection time (s)

St

Figure 14. Measured Strouhal Number for Different Collection Times 

Each point represents 7 measurements. Maximum difference between points is about 

0.001 units and corresponds to 0.01%. The averaged value for all collection times is 

9.17753 ± 0.006%. Based on the evaluation tests described above, we are therefore using 

a conservative value of 0.01% for the uni-directional diverter with zero time correction. 

5.2.2.2 Counters and Timers 

Liquid collection time is measured with a Hewlett Packard Model 53131A counter. The 

uncertainty of this timer is 0.0001 s. During a typical high flow calibration, which has 

20 s to 30 s collection time, this uncertainty leads to a value of 0.0004% or less. 

5.2.3 Density 

The Static Gravimetric Method for measuring liquid flow rate determines the 

(mass)/(time) value, averaged over the collection interval. There are volumetric type flow 

meters and there are mass flow rate meters. For example turbine flowmeters measure the 

flow in (volume)/(time), ultrasonic and magnetic flowmeters measure the flow in terms 

of (averaged fluid velocity*area)/(time), Coriolis flowmeters measure in (mass)/(time). In 

order to perform necessary conversion from mass to volumetric flow the fluid density 

must be known. Density of the liquid flowing through the meter being calibrated is 

required for both types of meters. 

For liquid density measurement, the Fluid Metrology Group uses a DMA 602 External 

Measuring Cell with Density Meter - DMA 60 manufactured by Anton Paar Corp. The 

29
density determination is based on measuring the period of oscillation of a vibrating Ushaped sample tube, which is filled with liquid. This system is equivalent to a springmass system, where the mass contains the sample liquid. The natural frequency of such a 

system will be 

1

2

c f

π M ρV = +

where c is an elastic constant, M is the mass of the body, V is the filled volume, and ρ is 

the density of the sample liquid. Therefore the period, T, can be calculated as 

2 M V T

c

ρ π+ =

or 

2 T A = ρ + B

where A and B are temperature dependent coefficients to be calibrated using reference 

liquids. At the same temperature, the density difference between two fluids is 

2 2

1 2

1 2

T T

A ρ ρ − − = . 

The densimeter manufacturer claims that the fractional uncertainty for the density of the 

water at room temperature is 6 5 10− ⋅ , provided that the temperature instability is less than 

0.01 K. This uncertainty includes 6 2 10− ⋅ due to the temperature instability and 6 3 10− ⋅

due to deviation of the oscillator. 

In order to estimate the performance of the Anton Paar densimeter and to find 

coefficients A and B, four fluids were used: distilled water, Standard Reference Material 

211d (Toluene), Standard Reference Material 2214 (Isooctane) and air. Certificates for 

SRM 221d and SRM 2214 can be found via: https://srmors.nist.gov/pricerpt.cfm. 

Measurements were performed at 20°C. Water density was calculated using the Patterson 

and Morris polynomial [10]. Density of the air was calculated using the temperature, 

atmospheric pressure, relative humidity and corresponding equation [11]. Relative 

humidity of the air inside the vibrating tube was re-calculated for the tube temperature 

assuming that the partial pressure of the water vapor is the same for the room and tube. 

First, the water vapor pressure was calculated for the room temperature and after this 

value was divided by the saturated water pressure at the tube temperature to get the 

relative humidity of the air in the tube. 

30
Figure 15 shows that experimental data can be fitted with a linear dependence between 

square of the period of vibrating tube versus the density. Fitted values for A and B are 

0.024029(4) ± 0.000002 m3

s

2

/kg and 25.994(0) ± 0.001(5) s2 respectively. 

25

35

45

55

0 250 500 750 1000

ρ (kg/m3

)

T 2

 (s2)

Figure 15. Densimeter Results Using the Four Fluids 

The standard deviation for four fluids (Figure16) is 0.053 kg/m3 or 0.0053%. This is 

considered a good result when one takes into account that this value is only about twice 

as large as the claimed uncertainties for the Standard Reference Materials (0.025 kg/m3

for Toluene and 0.035 kg/m3 for Isooctane). 

31
-0.100

-0.050

0.000

0.050

0.100

0 200 400 600 800 1000 1200

ρ (kg/m3

)

Residuals (kg/m 3

)

Figure 16. Residuals from the Densimeter Tests. 

As mentioned before, the coefficients A and B are temperature dependent. In order to find 

these dependencies, the Anton Paar Densimeter was calibrated using air and water in the 

temperature range 18 °C to 36 °C in 2 °C steps. 

It was found that both A and B can be approximated with linear dependence with 

 A=2.40900x10-2-2.97018 x10-6 (T/°C) and 

 B=26.06049-3.519921 x10-3 (T/°C), 

where T is the tube temperature in °C (see Figs 17 and 18). 

32
0.02398

0.024

0.02402

0.02404

18 23 28 33 T (o

C)

A (m3

s

2

/kg) 

Figure 17. Temperature Dependence of the Coefficient, A

25.9

25.95

26

26.05

18 23 28 33 T (o

C)

B (s2

)

Figure 18. Temperature Dependence of the Densitometer Coefficient, B

33
The density of the flume water was measured in the same temperature range. It was found 

that the difference between distilled water and flume water density has weak temperature 

dependence (Figure 19). 

0.315

0.32

0.325

0.33

18 23 28 33 T (o

C)

Δρ (kg/m3

)

Figure 19. The density difference between flume and distilled water 

The density of the flume water can be found by adding an additional linear term 

4 3 ρ 0.3321 3.631 10 , / T kg m − Δ = − ⋅ , 

to the distilled water polynomial (where T is in °C) . The maximum uncertainty for the 

density difference is about 0.004 kg/m3 which is negligible compared to the uncertainty 

(0.05 kg/m3

) for the 4 fluids previously described. . The expanded uncertainty is about 

100 parts in 106

. The density of the flume water is checked once a month or after adding 

water to the reservoir. 

5.2.4 Viscosity 

Practice indicates and current experiment confirms that universal calibration curves for 

turbine flowmeters can be constructed if calibration data are expressed dimensionlessly 

as: 

2 fD Ro

ν =

34
3 fD St

Q

π

= 

where Ro is the Roshko number, and St is the Strouhal number, where f is the meter 

frequency, D is the flowmeter diameter, ν is the kinematic viscosity and is the 

volumetric flow rate. The kinematic viscosity of the flume water must be measured to 

calculate the Roshko number. 

Q

The Fluid Metrology Group uses a capillary Micro-Ubbelohne Viscometer with Schott 

Gerate AVS 440 measuring system to measure the kinematic viscosity of liquids. The 

manufacturer claims that the uncertainty of the viscosity measurement is 0.1 % with 

temperature stability 0.01 K. 

The kinematic viscosity can be calculated using expression: 

( ) ν = Kt t ν − Δ HC

where K is the calibration constant of the capillary viscometer, t is the flow time, HC Δt

is the Hagenbach correction. The calibration constant was found using distilled water as a 

reference liquid. The constant K 0.002734 ν = (cm2/s2) was found by the Fluid Metrology 

Group with 0.1% uncertainty. The fractional residuals from fitting the kinematic viscosity 

as a function of temperature is shown in Figure 20. There was no noticeable temperature 

dependence of the constant found. 

-0.0025

-0.0015

-0.0005

0.0005

0.0015

18 23 28 33 T (o

C)

K av/K (T )-1

Figure 20. Residuals for Calibration Constant Fitting 

35
These results show that the flume water kinematic viscosity is very close to the viscosity 

of distilled water. The difference between viscosities of the flume water and distilled 

water is shown below (Fig. 21). 

-0.002

0

0.002

0.004

0.006

18 23 28 33 T (o

C)

νf-ν d (mm2

/s)

Figure 21. Difference Between Viscosities of the Flume and Distilled Water 

The kinematic viscosity of the distilled water [12] can be fitted with an uncertainty of 

0.001% using a rational polynomial 

2 2 ν =+ + + + ( ) /( a cT eT bT dT 1 )

with coefficients: a=1.794345 cm2/s, b=0.034252 cm2/s/K , c=-0.00172 cm2/s/K, 

d=0.000201cm2/s/K2 and e=2.86.10-5cm

2

/s/K2 . In order to obtain viscosity the difference 

 between flume and distilled water should be added. The 

viscosity of the flume water is measured periodically, once every 3-4 months. 

-4 -2 -3.241 10 + 1.011 10 ⋅ T ⋅

5.2.5 Temperature Measurement 

In order to obtain the true mass of water collected and the desired value of the volumetric 

flow, buoyancy effects are needed. This means that temperatures of the water and the 

pressure and temperature and humidity of the room air is required for density 

computations.

36
Temperature measurements are made of the flowing liquid and of the atmospheric 

conditions surrounding the weighing system. These measurements are made using 

calibrated thermistors placed at various locations along the water flow path and a 

Keithley Model 224 current source, a Keithley Model 7001 switch system, and a Keithley 

Model 2002 Multimeter . One sensor is located immediately upstream of the MUT and 

two are inside of the weigh tank. 

The thermistors are calibrated using an isothermal bath (Hart Scientific Model 5003) by 

comparing their responses to those of a four-wire platinum resistance transfer standard 

thermometer (Thermometrics Model TS8901) that is periodically calibrated by the NIST 

Thermometry Group. The four calibration coefficients and the temperature uncertainty 

for each thermistor are obtained using a linear regression method. All sensors are 

calibrated over the range 15 to 40 oC. This range represents the expected range of 

temperatures during use of the WFCF 3700/100. The calibration of the thermistors is 

performed once per year. 

Each thermistor is physically removed from the indicator by as much as 15 m. Extension 

cables are used to connect the sensors to the indicator. The temperature differences for 

the extension cables do not exceed 0.01 K. 

Uncertainty in Measured Temperature Values 

The components of uncertainty in the temperature values measured by each thermistor 

are: 

- the uncertainty of the transfer standard which is taken as 0.0012 K 

- the effect of the extension cable connection which is taken as 0.01 K 

- the residuals after a best fit of the thermistor calibration results. 

6 Data Acquisition and Control System 

The NIST WFCF is a fully automated system. The automatic control of the measurement 

process, the data collection and the calculation of final results are done by computer. The 

data acquisition and control system consists of a NI terminal block integrated by a (I/O) 

NI PCI-MIO-16XE-10 board, an IEEE-488 interface board and a RS-232 

communication port. 

An IEEE-488 interface is used to interface counters and temperature measurement 

devices to the data acquisition system. This interface functions as defined in ANSI/IEEE 

488-1975 and ANSI/IEEE 488.1-1987. 

An RS-232 communication port is used to acquire readings from the weigh scale, the 

flow check standard and the data for the room conditions via the temperature, pressure 

and humidity sensors. 

 

Figure 22 shows the data acquisition flow diagram. 

37
PC 

1. PCI-MIO-16XE-10 

2. PCI-GPIB 

3. Serial port interface

NI terminal Block GPIB RS232 Scale (Mettler Toledo) Flow Check Standard 

(Controlotron)

(Vaisala PTU 200)

Counter (A

gilent 

Counter (A

gilent 

Counter (A

gilent 

Programmable current 

source 

(Keithley 224) 

Switch system 

(Keithley 7001) 

Multimeter 

(Keithley 2002) 

AIO DIO Main, Control and Bypass Valves, Proximity Sensors 

16 Temperature sensors 

Mass 

Flow 

Temp, Pres, 

Humidity 

Time 

Frequenc

y

Pulses 

Pumps, Pumps Valves, 

CBU Motor, Diverter, 

Weights calibration actuator, 

Drain Valves, Manifold Valves 

Figure 22. Data Acquisition Chart for the Facility 

LabView software is utilized to control the system and to put data into spreadsheet format. 

The data acquisition software is designed to minimize operator intervention, thereby 

reducing or eliminating the number of entry errors. Moreover, the software provides 

specific instructions displayed on the screen to inform the operator of current and next 

calibration steps. The software panel allows the user to visualize and to manipulate 

different functions and devices in the facility. 

7 Summary 

We have given a description of NIST’s Water Flowmeter Calibration Facility which is 

used by the Fluid Metrology Group to offer water flowmeter calibration services in the 

range; 40 to 1600 L/min. This standard has an uncertainty of 0.033% for liquid mass 

collections of 3000 kg, and 0.051% for 600 kg collections. 

The intended audience for this document includes NIST’s water flowmeter calibration 

customers, and the characteristics and uncertainties claimed are intended to reflect the 

concerns of this audience. This document explains the method of operation of the WFCF, 

38
the functions of its various components, and provides details necessary for customers 

wanting to submit meters for calibration (i.e., pipeline sizes, costs, turnaround time, etc.), 

and gives a detailed analysis of the uncertainty of its mass (or volumetric) flow 

measurement results. Additionally, we have given a description of typical flow 

calibration, including a sample calibration report. 

8 References

1. V. Gowda, T. T. Yeh, P. I. Espina, and N. P. Yende, “The New NIST Water Flow 

Calibration Facility,” Proceedings of the 2003 FLOMEKO (Groningen, Netherlands: 

Gasunie, 2003). 

2a. ISO 4185:1980 “Measurement of Liquid Flow in Closed Conduits – Weighing 

Method”. 

2b. ASME/ANSI MFC-9M-1988 “Measurement of Liquid Flow in Closed Conduits by 

Weighing Method”, Reaffirmed 2001. 

3. D.W. Spitzer, Editor. Flow Measurement 2nd Edition. Chapter 27, Laboratory Primary 

Standards, by J.D.Wright. pp. 731-761, 2001. 

4. ISO 9368-1:1990 “Measurement of Liquid Flow in Closed Conduits by Weighing 

Method – Procedures for Checking Installations” . 

5. Iryna Marfenko, T.T. Yeh, John Wright, “Diverter Uncertainty Less Than 0.01% for 

Water Flow Calibrations”, 6th International Symposium on Fluid Flow Measurement, 

May 16-18, 2006, Queretaro, Mexico. 

6. T.T. Yeh, N.P. Yende, A.N. Johnson, P.I. Espina, “Error Free Liquid Diverters For 

Calibration Facilities”, Proceedings of ASME FEDSM’02 2002 ASME Fluids 

Engineering Division Summer Meeting Montreal, Quebec, Canada, July 14-18, 2002. 

7. J.L. Marshall “NIST Calibration Services Users Guide”, NIST Special Publication 250, 

January, 1998 . 

8. Barry N. Taylor and Chris E. Kuyatt, “Guidelines for Evaluating and Expressing the 

Uncertainty of NIST Measurement Results”, NIST Technical Note 1297, 1994 Edition. 

9. International Organization for Standardization, “Guide to the Expression of 

Uncertainty in Measurement”, Geneva, Switzerland, 1996 edition. 

10. J.B. Patterson, E.C. Morris, “Measurement of Absolute Water Density, 1°C to 

40 °C”, Metrologia, 1994, V.31, pp 277-288. 

11. K.B. Jaeger, R.S. Davis, A Primer for Mass Metrology . NBS Special Publication 

700-1 Industrial Measurement Series, November 1984.

12. Kestin, J., Sengers, J.V., Kamgar-Parsi, B. and Levelt Sengers, J.M.H. 

"Thermophysical Properties of Fluid H2O", J. Phys. Chem. Ref. Data, 13(1):175-183, 

1984. 

39
Acknowledgment 

Authors wish to acknowledge and sincerely thank 

Dr. John D. Wright, 

Dr. George E. Mattingly, 

and 

Dr. Michael R. Moldover 

for their interest to this paper, fruitful discussions during 

this Special Publication preparation, all suggestions, 

corrections and comments. 

40
 

Appendix: Sample Calibration Report 

REPORT OF CALIBRATION

FOR 

A TURBINE WATER FLOWMETER

May 01, 2006 

Mfg.: ABCD Corporation 

ABCD Serial No: 1234 

Pipe Diameter: 100 mm (4 inch) 

submitted by 

Waterflow , Inc. 

Metertown, MD 

Purchase Order No. A123 dated April 01, 2006 

The flowmeter identified above was calibrated by water at a constant rates through the 

flow meter and then into Water Flow Calibration Facility (WFCF) Standard. The water 

used for calibration was drawn from municipal water supply. Density and viscosity of the 

of the water were measured at a few temperatures near the room temperature and the 

result of measurement was approximated by corresponding equations. The WFCS 

Standard determines volume flow by measuring collected mass, collection time and 

density of the water.1 The flowmeter was tested two times at five flows and at each flow 

three (or more) measurements were gathered at different flow rates. As a result, the 

tabulated data for this test are averages of six or more individual calibration 

measurements. 

The flowmeter was installed in an assembly that meets the ISO Standard for liquid flow 

measurements and a photograph of the installation is shown in Figure 1. The collected 

mass was measured using Mettler Scale, collection time was measured using HP 15331 

counter. The air properties, such as temperature, pressure, and humidity, were taken with 

PTU 200 at the beginning and the end of each individual test. 

1

 2. D.W. Spitzer, Editor. Flow Measurement 2nd Edition. Chapter 27, Laboratory Primery Standard, 

by J.D.Wright. P. 731-761. 
NIST Test Number: 836-123456-03-01 Page 42

Calibration Date: July 2, 2003

Figure 1. A photograph of the flow meter installation. 

The Reynolds number is included in the tabulated data and it was calculated using the 

following expression: 

⋅⋅ μπ

⋅ = d

m Re 4  (1)

where m is the mass flow of the water, d is the nominal diameter of the pipe, and μ is 

water viscosity, all in consistent units so that is dimensionless. The viscosity of the 

water was measured and approximated with quadratic equation. 

Re

2

0 10 20 μ =+ + a aT aT (2)

 

The calibration results are presented in the following table and figure. 
NIST Test Number: 836-123456-03-01 Page 43

Calibration Date: July 2, 2003

K-factor Calibration Curve

0.3

0.305

0.31

0.315

0.32

0.325

0.33

0.335

0.34

0.345

0 500 1000 1500 2000

Volumetric flow rate, L/min

K-factor, Pulse/L

First test

Second test

Figure 2. Calibration results for ABCD water flowmeter, SN 1234 

1st Test 

Flow Rate K-factor StDev(k=2)StDev %(k=2)

38.90595 0.316952 0.003047 0.961393627

198.5555 0.337156 0.001996 0.592000515

488.1079 0.331855 0.000382 0.115249776

984.11 0.328268 0.000139 0.08441277

1626.398 0.326893 0.000159 0.048694467

2nd Test 

Flow Rate K-factor StDev(k=2)StDev %(k=2)

41.03114 0.311506 0.008526 2.737168322

199.3274 0.336018 0.000575 0.17121386

496.488 0.332102 0.000391 0.117796995

962.2345 0.328583 0.000304 0.092614897

1589.378 0.327057 0.000461 0.140951815

An analysis was performed to assess the uncertainty of the results obtained for the meter 

under test.2, 3, 4

 The process involves identifying the equations used in calculating the 

2

 International Organization for Standardization, Guide to the Expression of Uncertainty in 

Measurement, Switzerland, 1996 edition. 

3

 Taylor, B. N. and Kuyatt, C. E., Guidelines for Evaluating and Expressing the Uncertainty of 

NIST Measurement Results, NIST TN 1297, 1994 edition. 
NIST Test Number: 836-123456-03-01 Page 44

Calibration Date: July 2, 2003

calibration result (measurand) so that the sensitivity of the result to uncertainties in the 

input quantities can be evaluated. The confidence level uncertainty of each of the 

input quantities is determined, weighted by its sensitivity, and combined with the other 

uncertainty components by the root-sum-square method to arrive at a combined 

uncertainty ( ). The combined uncertainty is multiplied by a coverage factor of 2.0 to 

arrive at an expanded uncertainty ( ) of the measurand with approximately 

confidence level. 

67%

Uc

Ue 95%

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

),,,( 21 N = ... xxxyy (3)

and all uncertainty components are uncorrelated, the normalized expanded uncertainty is 

given by, 

( ) ( ) ( ) ∑= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ==

N

i i

i

i

e c

x

xu sk

y

yUk

y

yU

1

2

2 (4)

In the normalized expanded uncertainty equation, the are the standard 

uncertainties of each input, and are their associated sensitivity coefficients, given by, 

i s)'x(u

s'si

y

x

x

y s i

i

i ∂

∂ = (5)

The normalized expanded uncertainty equation is convenient since it permits the usage of 

relative uncertainties (in fractional or percentage forms) and of dimensionless sensitivity 

coefficients. The dimensionless sensitivity coefficients can often be obtained by 

inspection since for a linear function they have a magnitude of unity. 

For this calibration, the uncertainty of the K-factor has components due to the 

measurement of the mass flow by the primary standard, u m(  ) = 0.033%4. This applays to 

NIST’s results. This performance may be expected by customer if ther meter is used in 

similar conditions. 

To measure the reproducibility5

 of the test, the standard deviation of the K-factor 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 

root-sum-squared along with the other uncertainty components to calculate the combined 

 

4

 Water Flow Calibration Facility Standard with 100 mm pipeline and 3.700 kg collection tank 

for Water Flowmeter Calibrations. 5

 Reproducibility is herein defined as the closeness of agreement between measurements with the 

flow changed and then returned to the same nominal value. 
NIST Test Number: 836-123456-03-01 Page 45

Calibration Date: July 2, 2003

uncertainty. Using the values given above results for the expanded uncertainties are listed 

in the data table and shown as error bars in the figure. 

For the Director, 

National Institute of Standards and Technology 

Dr. Mike R. Moldover 

Leader, Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratories