Uncertainty Analysis of NIST's 20 Liter Hydrocarbon Liquid

187

Uncertainty Analysis of NIST's 20 Liter Hydrocarbon Liquid Flow Standard

Uncertainty Analysis of NIST's 20 Liter Hydrocarbon LiquidFlow Standard

AARON N. JOHNSON*, CHRIS J. CROWLEY and T.T. YEH

Fluid Metrology Group, National Institute of Standards and Technology (NIST)100 Bureau Drive, Gaithersburg, MD 20899

*e-mail:[email protected]

[Received: 08.04.2011 ; Revised: 15.07.2011 ; Accepted: 21.07.2011]

AbstractThe National Institute of Standards and Technology (NIST) uses a bi-directional 20 L-displacementpiston prover as its primary standard for measuring hydrocarbon liquid flows ranging from 1.86 × 10-5 m3/s (0.3 gpm) to 2.6 × 10-3 m3/s (40 gpm). Our analysis shows that the prover's uncertainty over this flowrange is 0.074 % (k = 2, corresponding to a 95 % confidence interval). Using a dual rotor turbine meteras the transfer standard, we compare the new 20 L piston prover standard with NIST's other hydrocarbonliquid standards and NIST's water flow standard. The results are consistent with the presenteduncertainty analysis.

© Metrology Society of India, All rights reserved 2011.

1. Introduction

NIST measures liquid hydrocarbon flows usingthe four primary standards listed in Table 1. NIST'scustomers establish traceability to the SI unit of flowby calibrating their meter against these U.S. nationalstandards. These standards accommodate all typesof pulse producing flowmeters, although turbinemeters are the most frequently calibrated at NIST. Thefirst two standards listed in Table 1 (i.e., the Smalland Large Cox Bench) work on a gravimetric principlewhereby the flow is determined by weighing the massof liquid collected in a tank during a measured timeinterval. The latter two primary standards are the 2 LHydrocarbon Liquid Flow Standard (2 L HLFS) andthe newly developed 20 L Hydrocarbon Liquid FlowStandard (20 L HLFS). These piston prover standardswork on a volumetric principle whereby the pistondisplaces a known volume of fluid during a measuredtime interval. The first three standards in Table 1 have

MAPAN - Journal of Metrology Society of India, Vol. 26, No. 3, 2011; pp. 187-202ORIGINAL ARTICLE

been in use for several years and their operation anduncertainty analyses are documented in NIST internalrecords and in previous publications [1, 2]. In thismanuscript, we document the operating principle forthe 20 L HLFS, its governing flow equations,calibration procedure, uncertainty analysis, and theresults of comparisons with the three older NIST liquidhydrocarbon flow standards and also with NIST'sWater Flow Calibration Facility [3].

2. NIST's 20 L HLFS

2.1 Description of the 20 L HLFS

The photograph in Fig. 1a shows the majorcomponents of the 20 L HLFS and the diagram inFig. 1b shows the flow path of the calibration fluidthroughout the prover assembly when the piston isstroking to the right. As the piston strokes rightwardthe flow exits the cylinder on the right side and movesthrough the upper section of the four-way valve intothe test section. The meter under test (MUT) is

Aaron N. Johnson, Chris J. Crowley and T.T. Yeh

188

Table 1Capabilities of NIST's four Hydrocarbon Liquid Flow Standards. Flow standards are used with Stoddard

solvent (i.e., MIL-C-7024 B Type II) or fluids with similar kinematic viscositiesi,ii

NIST Primary Flow range Expanded Absolute Temperatureflow standard uncertainty pressure range range

Volumetric Massflow flow

Small Cox 0.05 to 8 lpm 0.12 % 0.12 % 150 kPa to 250 kPa 14ºC to 30ºCBench (0.014 to 2.1 gpm)

Large Cox 5.3 to 918 lpm 0.12 % 0.12 % 150 kPa to 250 kPa 14ºC to 30ºCBench (1.4 to 242 gpm)

2 L HLFS 0.19 to 5.7 lpm 0.04 % 0.05 % 150 kPa to 380 kPa 21ºC to 23ºC(small piston prover) (0.05 to 1.5 gpm)

20 L HLFS 1.1 to 151 lpm 0.074 % 0.08 % 150 kPa to 380 kPa 21ºC to 23ºC(medium piston prover) (0.3 to 40 gpm)

Four wayFluid

Reservoir

Piston

ShaftCylinder

Test Section

Heat Exchanger

Encoder Encoder

MUT CSCheck Standard

valve

Flow Direction

CompressedGas Source

(a) (b)

Fluid Reservoir

cylindershaft

Test Section

encoder

D d

FlowD

irection

Air or N2

StoddardSolvent

Fig. 1. Photograph and drawing of the essential components of the 20 L HLFS

calibrated using the known flow delivered frompiston-cylinder assembly as the reference. The datacollected during a flowmeter calibration is verifiedagainst a turbine meter check standard installeddownstream of the MUT. Fluid leaving the test sectionenters into a fluid reservoir where a compressed gassource applies pressure to the fluid. The appliedpressure helps prevent cavitation on the turbine meterblades of NIST's check standard as well customerturbine meters. As the flow exits the reservoir it isdirected through a heat exchanger and then around

i. Throughout this document the units lpm are actualliters per minute. Divide lpm by 60000 to convert tom3/s and divide lpm by 3.785412 to convert to gallonsper minute which is abbreviated gpm herein.

ii. Uncertainty values at approximately the 95 %confidence interval having a coverage factor of two(k = 2). These uncertainties do not include repeatability,reproducibility, and hysteresis of the flowmeter beingcalibrated. These additional uncertainty componentsare assessed during a flowmeter calibration and aredocumented in a report of calibration.

189


(a)

SectionTo Test

PistonMotion

FromReservoir

(b)

SectionTo Test

PistonMotion

FromReservoir

Fig. 2. Sketch showing the four-way valve positions and the corresponding flow directions for the piston strokingrightward (a) and leftward (b)

the piston-cylinder assembly to the lower section ofthe four-way valve. The flow loop is completed whenthe fluid reenters the left side of the cylinder. Flowcontinues to accumulate in the left side of the cylinderuntil the piston reaches the end of the rightwardstroke, at which time the four-way valve is actuatedand the piston strokes leftward. As illustrated inFig. 2, the four-way valve maintains unidirectionalflow in the test section regardless of the direction ofthe piston stroke.

The piston is driven using a servomotor and drivegear system that is coupled to the piston shaft. During

each stroke a feedback circuit controls the servomotoroutput so that the piston velocity quickly reaches aconstant value corresponding to the desired flow setpoint. Data is collected only while the velocity isconstant. A timing circuit measures the duration ofthe piston stroke (Δt ) while the distance is measuredby two redundant, linear encoders, each having anominal encoder constant of ΔL’e = 20 μm/pulse(or 50 pulses/mm). The length of the piston stroke isthe encoder constant multiplied by the averagenumber of encoder pulses (Ne) from the two encoders.Each encoder gives two pulse counts (correspondingto the leading and trailing edge of a pulse) so that Ne

Table 2Nominal characteristics of NIST's 20 L HLFS during a flow measurement

Volumetric Piston Duration of Length of Encoder Percent offlow velocity piston piston pulses cylinder

stroke (Δt) stroke (Ne ) volume swept[lpm] [gpm] [cm/s] [s] [cm] [pulse] [%]

151.4 40 14.2 4.6 64.8 32410 85113.6 30 10.7 6.1 64.8 32410 85 75.7 20 7.1 9.1 64.8 32410 85 37.9 10 3.5 18.2 64.8 32410 85 18.9 5 1.8 36.4 64.8 32410 85 9.46 2.5 0.9 72.8 64.8 32410 85 3.78 1 0.36 120.0 42.7 21361 56 1.1 0.3 0.11 120.0 12.8 6408 17

D

D

D

( )ref st refa=+-é ùë û1D D T T

( )ref st refa=+-é ùë û1d d T T

( )e e,ref en en refa¢ ¢D=D+ -é ùë û1L L T T

a

a

V cs e¢=DK A L

ref v K

'e,refDL

refvK

2 2A = p(D -d )/4,cs

ref refcyl v e /= DQ K N t

r= D&ref refcyl ref v /em K N t r

=ideal refMUT cylQ Q

=& &ideal refMUT cylm m

idealMUTQ

rrb k=--+-r[1 ( ) ( P )]ref ref efT T P

b k

rb

k

ˆ ˆ ˆ ˆ

= =


192

temperatures and pressures can be used in Eq. (4) tocalculate the density at the cylinder exit (ρcyl) and atthe MUT (ρMUT). The volumetric flow at the MUT (forsteady flow conditions) equals the measured densityratio (ρcyl/ρMUT) multiplied by the volumetric flowexiting the piston-cylinder.

A third type of non-ideality results from temporalvariations in the thermodynamic conditions (i.e.,temperature and pressure) during the flowmeasurement. In this case the mass flow exiting thecylinder is not equal to the mass flow at the MUT. Thedifference in mass flow is attributed to accumulationof mass in the unswept region of the cylinder and inthe volume of piping connecting the cylinder exit andthe MUT. That is, the mass in these volumes at thestart of the piston stroke is not equal to mass at theend of the piston stroke. Although NIST does notcurrently correct for mass storage effects, we estimatethe magnitude of these effects for each calibration andinclude them in the uncertainty budget.

In next section the mass conservation principle isused to derive expressions for the three types ofcorrections considered in this manuscript (i.e.,reference condition corrections, gradient corrections,and storage effect corrections).

3. Formulation of the Governing Flow Equations

Figures 3a and 3b show the location of the pistonat the start of the measured time interval (ti) and at theend of the measured time interval (tf). The whitedashed lines constitute the control volume where massconservation is applied. The control volume includesthe volume of fluid to the right of the piston inside thecylinder (Vcyl), and the fluid in the connecting volumebetween the exit of the cylinder and the MUT (Vcv). Asthe piston strokes rightward the size of the controlvolume decreases such that the time-averaged massflow through the MUT is

if ifcyl cv

kaelTUM

M Mm mt t

∆ ∆= + −

∆ ∆ (5)

where if i fcyl cyl cylM M M∆ = − is the difference between

the initial and final mass in the cylinder,if i fcv cv cvM M M∆ = − is difference between the initial

and final mass in the connecting volume, ∆t=tf-ti is

the measured time interval, and leakm is net time-averaged mass flow leaking out of the control volume.The mass terms in Eq. (5) can be replaced by theproduct of the spatially averaged density multiplied

by volume (e.g., )vi where

for example icylρ̂ is the initial average density in the

cylinder volume ( ), and is the final average

density in the control volume ( ). After thissubstitution time-averaged mass flow at the MUT is

or

(6)

By adding and subtracting the terms icylρ̂ and

to the right side of Eq. (6) the time-averagedmass flow (with no leaks) is

v Currently NIST measures the pressure at the MUT, butnot at the piston-cylinder exit. Consequently, the pressurecorrection factor G2 shown in Table 3 (see Section 3.1) isnot used to correct the volumetric flow, but is estimatedand included in the uncertainty budget

i i i i i icyl cyl cyl cv cv cvˆ ˆ,M V M V

icylV f

cyl̂

fcylV

i i f f i i f fcyl cyl cyl cyl cv cv cv cv

leak

ˆ ˆ ˆ ˆMUT

V V V Vm mt t

− −= + −

∆ ∆

i i f fi i f fcv

MUT leakcyl cv

ˆ ˆˆ ˆ −−= + − ∆ ∆

V VV Vm mt t

fcylV

i fcv cvˆ V

( )i i f f i f i f

MUT

cyl

ˆ ˆ ˆ ˆ- +=

V V V - Vm

t

+ ∆

( )i i f f i f i f

cv

ˆ ˆ ˆ ˆV V V Vt

− + −

∆

( ) ( )

( ) ( )

i i i f i f f f

cyl

i i i f i f f f

cv

− + −= +

∆

− + −

∆

ˆ ˆ ˆ ˆV V V Vt

V V V Vt

∆ = −

∆ ∆ ˆ ˆV V∆ ∆ ˆ ˆV V

ˆ + +

∆ = −

193


(7)

where if i fcyl cyl cylV V V∆ = − is the volume swept by the

piston during the measurement interval indicated bythe dotted lines in Fig. 3b. The terms in the squarebrackets account for mass storage effects in the portion

of the cylinder volume not swept by the piston ( fcylV )

shown in Fig. 3b, and in the connecting volume ( ).

Here, and if i fcv cv cvˆ ˆ ˆρ ρ ρ are the

change in density in the cylinder volume and in theconnecting volume between the start (ti) and stop (tf)of the timing interval. These density changescorrespond to changes in the average temperatureand/or pressure in these volumes between the startand stop of the timing interval (i.e.

i f i f i fcyl cyl cv cv cyl cyl

ˆ ˆ ˆ ˆ ˆ ˆ, , ,T T T T P P≠ ≠ ≠ and i fcv cv

ˆ ˆP P≠ ). Temporalchanges in the average temperature and pressure inthe connecting volume can also alter the connectingvolume size ( i f

cv cvV V≠ ) via thermal expansion andelastic deformation caused by pressure inducedforces. However, changes in the connecting volume

Fig. 3. Sketch showing the orientation of the piston before and after the piston stroke

vi The symbol ~ is placed above a variable to indicate atime-average while the symbol ^ indicates a spatialaverage.

i if if f i if if f

cyl cv

ˆ ˆ ˆ ˆV V V Vt t

∆ + ∆ ∆ + ∆= + ∆ ∆

i if f if i if f ifcyl cyl cyl cyl cv cv cv cv

ˆ ˆ ˆ ˆV V V Vt t t t∆ ∆ ∆ ∆

= + + +∆ ∆ ∆ ∆

f ifcyl cyl

if ii ifcyl cylcyl cyl

f if i ifcv cv cv cv

if i i ifcyl cyl cyl cyl

MUT

∆

∆∆= ∆ +

ˆ1

ˆVVV

mt

if i fcyl cyl cylˆ ˆ ˆ

cvV

ρ ρ ρ ρ

ρ

ρ ρ ρ ρ

ρρρ

ρ ρρ

ρρ ρ

(a) Prover at =t ti

Encoder

(b) Prover at =t tf

Encoder Encoder

Encoder

Mcyli

Vcyli

Vcv

i

Mcvi

Lsi

Mcylf

Mcv

f

Vcv

f

Vcylf∆Vcyl

if

∆Le

Ls

∆ ∆

∆ ∆


194

( if i fcv cv cvV V V∆ = − ) are generally negligible relative to

density related changes.

The time-averaged volumetric flow at the MUT isdetermined by dividing Eq. (7) by the density at theMUT

(8)

As expected, the mass flow in Eq. (7) and the

volumetric flow in Eq. (8) simplify to ideal refMUT cylm m= and

for the ideal operating conditions givenin section 2.2. In general, Eqs. (7) and (8) incorporatethe appropriate corrections to account for non-idealeffects. In the special case where flow conditions donot deviate substantially from ideal conditions, thecorrections for non-ideal effects are small relative to

idealMUTm and , and Eqs. (7) and (8) can be expressed

as a product of near unity correction factors multipliedby the ideal flow.

3.1 Mass Flow and Volumetric Flow at the MUT Near Ideal Conditions

When flow conditions are close to ideal, Eqs. (7)and (8) for mass flow and volumetric at the MUTsimplify to

(9a)

and

(9b)

where the near unity correction factors indicated byRi's, Gi's, and Si's account for reference conditioncorrections, gradient corrections, and storage corrections,

respectively. Expressions for these correction factorsare given in Table 3 along with a description of theirphysical meaning. Equations (9a) and (9b) are derivedby making the following modifications to Eqs. (7) and

(8). First, the swept volume ( ifcylV∆ ) in Eqs. (7) and (8)

is replaced by the product of refVK and Ne multiplied

by the appropriate temperature correction factorsspecified in Eqs. (2), (3a), (3b), and (3c)vii. Similarcorrection factors are used for the volume change inthe connecting volume ( ). The density expressionin Eq. (4) is substituted for all of the density terms inEqs. (7) and (8). The result of these substitutions forthe volumetric flow, for example, is expressedas MUTQ = ideal

MUT corrQ y where is afirst order Taylor series consisting of the deviationfunctions δR, δG and δS that correct for non-idealitiescaused by reference condition effects, gradient effects,and storage effects. The deviation functions equalδR,i = Ri-1,δG,j = Gj-1, and δS,k = Sk-1 where Ri, Gj, and Skare the correction factors in Table 3. For example, thefirst two reference condition deviation functions are

R ,1 st en ref( )T Tδ α= − and iR ,2 st cyl ref

ˆ2 ( )T Tδ α= − .Provided that the deviation functions are small relativeto unity (δR<<1, δG<<1 and δS<<1), higher orderproducts of the deviations are negligible so that

corr R G S(1 )(1 )(1 )y δ δ δ= + + + or corr i i iG Sy R= Π is theproduct of the correction factors as indicated in Eqs.(9a) and (9b).

The Ri's in Eqs. (9a) and (9b) correct the fluiddensity and the measured cylinder volume to thereference conditions. The Gi's correct the flow whenthe pressure and/or temperature differs between thepiston-cylinder assembly and the MUT. The Si's arestorage corrections that account for a change in massbetween the start and stop of the timing interval in theunswept region of the cylinder and in the connectingvolume during a flow measurement.

For NIST operating conditions (which closelyapproximate ideal) many of the correction factorslisted in Table 3 are essentially unity and havenegligible impact on flow calculations. For clarity, wevii For completeness elastic deformation of the crosssectional area attributed to forces induced by pressurestresses are included in Eqs (9a) and (9b).

i f ifcyl cyl cyl

ififcylcyl

MUT f if i ifcv cv cv cv

if ifcyl cyl

MUT MUT

MUT MUT

ˆ ˆ

ˆ ˆ

VVV

Qt V V

V V

∆+ +∆ ∆

= ∆ +

ideal refMUT cylQ Q=

idealMUTQ

ref1 2 3 4 5

refV

MUT 1 2 3 4 5 6

K Nm R R R R R S S S S S S

t

= ∆

e

MUT 1 2 3 1 2 3

refV

4 5 6 1 2

K NQ R R R S S S S S S G G

t

= ∆

e

ifcvV∆

corr R G S1 +y = + +

195


specify correction factors that are negligible for NISTcalibration conditions when using Eqs. (9a) and (9b)in the remaining sections of the manuscript.Nevertheless, we have included these correctionfactors in Table 3 to provide guidance for fieldapplications when operating conditions cannot bemaintained as close to the ideal conditions. In thiscase, we point out that besides the list of correctionfactors in Table 3, one must also account for potentialmechanical problems such as seal failure if a pistonprover will be operated under extreme conditions.

Many of the correction factors in Table 3 requiremeasurements of the average temperature andpressure at the start of the timed interval in both thecylinder and in the connecting volume (e.g.

i i icyl cv cyl

ˆ ˆ ˆ, ,T T P and ). Since NIST data acquisitionsystem is not currently equip to provide the timehistory of pressure and temperature measurements,

time-averaged values are used (i.e. and

) instead of the initial values. Temperature andpressure stability is determined by monitoring thetime-averaged temperatures and pressures throughoutthe entire calibration process. During a calibrationthe average temperatures and pressures remain nearlyconstant for repeated measurements at the same setpoint (i.e., temperature stability is typically less than45 mK and pressure stability less than 0.1 %).Corrections attributed to mass storage are not currentlymade at NIST as these require both the initial andfinal temperatures and pressures in the connectingvolume (i.e., Si's are set equal to unity in Eqns. 9a and9b). However, the uncertainty attributed to storageeffects is included in the uncertainty budget. For NISTcalibrations the most significant correction factor forstorage effects is S1 which contributes less than 0.01 %.

4. Uncertainty Analysis

The method of propagation of uncertainty [8] asspecified in the GUM [9] is used to determine theuncertainties of i) the reference volumetric proverK-factor ( ); ii) the volumetric flow at the MUT

(QMUT) and iii) the mass flow at the MUT ( ). In allthree cases the uncertainty sources are taken to beuncorrelated. The combined standard uncertainty (uc)

is determined by root-sum-squaring (RSS) thestandard uncertainties (i.e., 68 % confidence level orunity coverage factor, k = 1) multiplied by theirsensitivity coefficients. The standard combineduncertainty is multiplied by a coverage factor of two(k = 2) to obtain the expanded uncertainty (U = 2 uc) atan approximate 95 % confidence level.

4.1 Determination of Uncertainty of the Volumetric Prover K-factor ( )

The reference volumetric prover K-factor ( ) isdetermined using a water draw procedure at areference temperature and pressure of Tref = 21 ºC andPref = 101.325 kPa. After temperature equilibrium isestablished in the both the room and the fluid, thepiston is slowly traversed through the cylinder andthe displaced fluid is directed into a weigh tankinstead of through the MUT. Thus, we determine

using Eq. (9a) for mass flow with the exceptionthat the total mass that would have passed through

the MUT (i.e., ) is replaced by the buoyancycorrected and calibrated weigh scale readings. Theresulting expression for is

5 6

or

( )( ) 1f iT T air refref

V ie ref cyl ref

1 /ˆ[1 ( )]

−− −

=− −

M MK

N T T

ρ ρ

ρ β (10)

where iTM is the tare weight of the empty collection

vessel (after zeroing the scale), and is the finalscale reading after filling the collection vessel, and

the quantity is the buoyancy

correction. The air density ( airρ ) is calculated as afunction of the pressure, temperature, and relativehumidity in the room using the correlation developedby Jaeger and Davis [10]. During the measurement of

the room temperature was controlled to within

icvP̂

cyl cv cyl, ,T T P

cvP

refVK

MUTm

refVK

refVK

refVK

MUTm t∆

refVK

( )( ) 1f iT T air refref

V 5 6

e ref n nn 1 n 1

1 /−

= =

− −=

Π Π

M MK

N R S

fTM

( ) 1air ref1 /

−− ρ ρ

refVK


196

Table 3Correction factors for mass flow in Eq. (9a) and volumetric flow in Eq. (9b)

Region of Correction factors Type of Descriptionprover where correctioncorrection applies

Encoder Reference Axial change of encoder scalecondition from the reference condition duecorrection to thermal expansion

Displaced Reference Radial change in the cylinder andvolume condition shaft from reference condition due

correction to thermal expansionDisplaced Reference Radial change in cylinder andvolume condition shaft from the reference condition

correction due to internal fluid pressureviii

Displaced Reference Change in the fluid density fromvolume condition the reference density (ρref) due to

correction thermal expansionDisplaced Reference Change in the fluid density fromvolume condition the reference density (ρref) due to

correction pressure change from PrefDisplaced Temperature Ratio density change betweenvolume/MUT gradient cylinder and MUT attributed to

correction temperature difference between thecylinder and MUT

Displaced Pressure Ratio density change betweenvolume/MUT gradient cylinder and MUT attributed to

correction pressure difference between thecylinder and MUT

Upswept Mass Change in the fluid density in thevolume of storage unswept region of the cylinder duecylinder to a temperature change between

the start and stop of a calibrationUpswept Mass Change in the fluid density in thevolume of storage unswept region of the cylinder duecylinder to a pressure change between the

start and stop of a calibrationUpswept Mass Radial change in the unsweptvolume of storage region of the cylinder and shaftcylinder diameters due to a temperature

change between the start and stopof a calibration

Upswept Mass Radial change in the unsweptvolume of storage region of the cylinder and shaftcylinder diameters due to a pressure

change between the start and stopof a calibration

Connecting Mass Change in the connectingvolume storage volume and the density of fluid in

this region due to a temperaturechange between the start and stopof a calibration

Connecting Mass Mass storage in connectingvolume storage volume attributed to pressure

difference between start and stopof flow measurement

1 en refen( )1R T Tα= + −

2i

st cyl ref( )α= + −ˆ1 2R T T

i3 eff cyl ref( )ε= + −ˆ1 2R P P

4icyl ref( )Tβ= − −ˆ1R T

5icyl ref( )Pκ= + −ˆ1R P

1 MUTicyl( )Tβ= − − ˆ1G T

2 MUTicyl( )Pκ= + − ˆ1G P

1

fcyl i f

cyl cylrefe V

( )β⎛ ⎞⎜ ⎟= − −⎜ ⎟⎝ ⎠

ˆ ˆ1V

S T TN K

2

fcyl i f

cyl cylrefe V

( )κ⎛ ⎞⎜ ⎟= + −⎜ ⎟⎝ ⎠

ˆ ˆ1V

S P PN K

3

fcyl i f

st cyl cylrefe V

( )α⎛ ⎞⎜ ⎟= + −⎜ ⎟⎝ ⎠

ˆ ˆ1 2V

S T TN K

4

fcyl i f

eff cyl cylrefe V

( )ε⎛ ⎞⎜ ⎟= + −⎜ ⎟⎝ ⎠

ˆ ˆ1 2V

S P PN K

( )ref

cv5

i fcv cv cvref

e V( )α β

⎛ ⎞= + − −⎜ ⎟⎜ ⎟

⎝ ⎠

ˆ ˆ1 3VS T TN K

( )ref

cv6

i fcv cv cvref

e V( )ε κ

⎛ ⎞= + + −⎜ ⎟⎜ ⎟

⎝ ⎠

ˆ ˆ1 2VS P PN K

197


± 2 °C of Tref, the fluid temperature was controlled towithin ± 1 °C of Tref, and the fluid pressure was 101kPa ± 1.5 kPa. For these conditions the only significant

correction factor is ( )i4 cyl ref

ˆ1= − −R T Tβ as shown

explicitly in Eq. (10). All of the other corrections forreference conditions and storage effects attribute lessthan 0.003 % to ref

VK and are neglected.

The volumetric prover K-factor was measured ontwo occasions, first on April 2008, and a second timeon Jan 2009. These measurements were done both with

the piston moving to the left ( ) as well as with

the piston stroking to the right ( refV,rightK ). On the first

occasion the measurements of refV,leftK were each

repeated 20 times using Stoddard solvent as theworking fluid. This data is denoted by the 's and 's

in Fig. 4. On the second occasion refV,leftK was measured

5 times ( ) while refV,rightK was measured 8 times ( )

using reverse osmosis water as the working fluid. The

average of the twenty refV,leftK and twenty ref

V,rightKmeasurements made in Stoddard solvent are indicatedin the figure by the solid line ( ) and by the dotted

line ( ). Similarly, the averages of the five refV,leftK

and eight refV,leftK measurements made in reverse

osmosis water are indicated by the solid line ( )and dotted line ( ) in the figure. The differencebetween the left and right average K-factor values inStoddard solvent is 0.008%, while the difference inreverse osmosis water is 0.002%. Based on this goodagreement we use the average volumetric K-factor,

refVK = ( + ref

V,rightK )/2 in all flow calculations,regardless if the piston is stroking left or right.

The dashed horizontal line (- --) in Fig. 4 thatconnects the left and right K-factor measurements isthe average for the April 2008 measurements in

Stoddard solvent, and ( ) is the average for

Fig. 4. Plot showing measurements of the volumetric prover K-factor done in April 2008 using Stoddardsolvent and in January 2009 using reverse osmosis water

viiiNote that εeff and εcv are parameters with units of inversepressure to be determined using the appropriate pressurevessel equations in terms of the material modulus ofelasticity, Poisson ratio, and cylinder or pipingdimensions.

ix Note that αcv is the linear coefficient of expansion forthe connecting volume material.

refV,leftK

refV,leftK

refVK

-- --

0.3542

0.3543

0.3544

0.3545

0.3546

LeftAvg.

RightAvg.

RightAvg.

LeftAvg.

0.028 %

0.013 %

April2008(Left)

April 2008

(Right)

January2009

(Right)

January2009(Left)

Stoddard Solvent Reverse Osmosis Water

Avg. of Left and

Right

Avg. of Left and

Right

ref

VK

[cm

/ pu

lse

] 3

refVK

Alph1

Line

Alph1

Line

Alph1

Line

Alph1

Line

Alph1

Line

Alph1

Line

Alph1

Line

Alph1

Line

∂ ∂M K u M K


198

the January 2009 measurements in reverse osmosiswater. The difference between these two ref

VK valuesis only 0.013%, in spite of the January 2009measurements being substantially noisier than thosemade in April 2008. We suspect that the noise is relatedto a leak discovered in the 4-way valve during thetestingx. We calculate the square of the combinedstandard uncertainty of the January 2009 K-factormeasurements by

2ref ref

ref

V,left V,right2 2left right

V,right

σ σ−

+ + +3

K KK

(11)

where σleft and σright are the standard deviations of therepeated measurements for the piston strokingleftward and rightward. The last term is the percentdifference between the piston stroking to the left versusto the right. The expanded uncertainty of the K-factor

is ( )ref refV V[ / ]U K K = 0.073%. Table 4 itemizes the

uncertainty of each component, specifying its overallpercent contribution to the total uncertainty and thenormalized sensitivity coefficients (i.e., the derivativeterms in Eq. 11).

4.2 Uncertainty of the Volumetric and Mass Flow at the MUT

The volumetric flow at the MUT is determinedusing Eq. (9b). The square of the combined standarduncertainty of the volumetric flow at the MUT is

calculated by

(12)

Table 5 itemizes each component of uncertainty.The expanded uncertainty in volumetric flow at theMUT is [U(QMUT)/QMUT] = 0.074%. The uncertainty isvalid over the entire flow range from 1.1 lpm to 151lpm. As expected the majority of the uncertainty isattributed to the measurement of the referencevolumetric prover K-factor. The uncertainty in mass

x The leak is believed to be the result of a rusted filter thatwas corroded by the water. Rust particles were found inthe 4-way valve when it was later replaced.

ref ref

ref ref

i ic V s V s

i iV V s s

∂= ∂

2 2 2u K M K u M

K K M M( ) ( )

ref ref

ref ref

f fs V s V

f fV s s V β

+ + ∂ ∂

2 2 2

K M M K( )

ref

ref

i icyl cyle V

i ie V cyl cyl

∂+ +

∂

2 22 ˆ ˆˆ ˆ

T u Tu N KN K T T

( )( )

( ) ( ) ( ) ( )c MUT

MUT

refV

refV

22 2 2u Ku Q u N u t

N tQ K

∆ = + + ∆

e

e

( )MUTMUTMUT

MUT MUTMUT

22 uQQ

∂ + ∂

TTT T

( )enMUTen

en enMUT

2 2u TQTT TQ

∂+ ∂

( )iicylcyl MUT

i iMUT cyl cyl

22 ˆˆˆ ˆ

u TT QQ T T

∂ + ∂

( )iicylcyl MUT

i iMUT cyl cyl

22 ˆˆˆ ˆ

u PP QQ P P

∂ + ∂

( )MUTMUTMUT

MUT MUTMUT

22 uQQ

∂ + ∂

PPP P

( )MUT

MUT

2 2uQQ

κκκ κ

∂ + ∂

( )MUT

MUT

2 2uQQ

∂+ ∂

( )enMUTen

en enMUT

2 2uQQ

∂+ ∂

( )stMUTst

st stMUT

2 2uQQ

∂+ ∂

∂ ∂ K K

+ +ref ref

ref ref

ref V ref air V air

V ref ref V air air

∂ ∂

2 2 2 2K u K u( ) ( )

199


Table 4Uncertainty budget for volumetric prover K-factor corresponding to Eq. (11) for January 2009

measurements using Reverse Osmosis Water

Vol. prover K-factor Nom. Rel. unc. Norm. Type Perc. Commentsvalue (k = 1) sen. A/B contrib.

coeffrefVK = 0.35442 [cm3/pulse] [SI ] [%] [ ] [ ] [%]

Initial Scale Mass, [kg] 0 0.012 0.002 B 0 Control Charts of 60 kg scale

Final Scale Mass, [kg] 13.8519 0.012 0.998 B 10.5 Control Charts of 60 kg scaleWater Density, ρref [kg/m3] 997.855 0.0002 -1.002 B 0.0 Density of Reverse

Osmosis Water [11]Air Density, ρair [kg/m3] 1.18 2 0.0015 B 0.7 Air Density [10] and

measured T, P, and RHEncoder Pulse, Ne [pulse] 39170 0 -1 B 0.0 Integer number of

pulses (assumes nopulses missed)

Thermal Expan. Coeff. for 0.00022 10 0.0001 B 0.0 Variation with Temp.Water at 21 °C; β C°32 ot C° 91 nwteb]C°/1[

(rect. dist assumed)

Initial Fluid Temp., [°C] 21.63 0.02 0.06 B 0.1 Cal. Records and spatialnon uniformity of 6thermistors

Repeat. Meas. of N/A 0.017 1 A 22.6 Std. dev. of 5 meas.

[cm3/pulse]

Repeat. Meas. of refV,rightK N/A 0.03 1 A 66.0 Std. dev. of 8 meas.

[cm3/pulse]

Diff. between refV,leftK N/A 0.001 1 B 0.1 Rect. Dist. Assumed

and refV,rightK

Combined Std. Unc. (k = 1) = 0.0365 % 100 68 % confidence interval Expanded Unc. (k = 2) = 0.073 % 100 95 % confidence interval

flow at the MUT is determined in an analogousfashion. In this case the measurement of the Stoddardsolvent density slightly increases the mass flowuncertainty above the volumetric flow uncertainty. Thetotal uncertainty of determining the Stoddard solvent

density (including contributions from the referencecorrections R4 and R5 in Table 3, but omitting G1 andG2) is 0.016% (k = 1) so that the expanded mass flowuncertainty at the MUT is [U ( )MUT MUT/m m ] = 0.08 %(k = 2).

iTM

fTM

icylT

refV,leftK


200

Vol. flow; QMUT [lpm] Nom. Rel. unc. Norm. Type Perc. Commentsvalue (k=1) sen. A/B contrib.

coeff. QMUT = 113.56 lpm [SI ] [%] [ ] [ ] [%]

Volumetric prover K-factor, 0.35442 0.036 1 B 96.8 see Table 4

refVK , [cm3/pulse]

Duration of piston stroke; 6.1 0.002 -1 B 0.3 Control Charts fort [s] Freq. Calib.Encoder pules; Ne [pulse] 32410 0.0 1 B 0.0 Integer number of

pulses (assumes nopulses missed)

Temp. encoder; Ten [°C] 23 0.02 0.0002 B 0.0 Env. Sensor Calib.Records

Initial fluid temp., [°C] 22 0.02 0.02 B 0.0 Temp. Cal. Recordsand spatial non-uniformity of 6thermistors

Temp. at the MUT; TMUT [°C] 21.75 0.017 -0.02 B 0.0 Temp. Cal. Recordsand spatial non-uniformity of 6thermistors

Pres. at the MUT; PMUT [kPa] 240 0.2 0.0002 B 0.0 Pres. Calib. Records

Pres. at the cylinder; [kPa] 250 25 0.0002 B 2.5 Estimate of worst casepres. drop; (rect. dist.assumed)

Linear thermal expan. coeff. 1.7E-05 5 3E-05 B 0.0 see Reference [4, 5]of stainless steel; st [1/°C]Isothermal comp. factor for 9.3E-07 7 0.0002 B 0.0 See Reference [7]Stoddard solvent at ; [1/kPa]Thermal expan. coeff. of 0.00097 10 0.0002 B 0.4 Fit residuals from theStoddard solvent, ytisned derusaem]C°/1[

using a vibrating tubedensimeter

Linear Thermal Expan. Coeff.of Encoder; en [1/°C] 0.000008 5 1.6E-05 B 0.0 see Reference [5]

Combined Std. Unc. (k % 86001% 730.0= )1 = confidence interval Expanded Unc. (k % 59001% 470.0= )2 = confidence interval

Table 5Uncertainty budget for volumetric flow at MUT corresponding to Eq. (12)

icylT

icylP

201


5. Comparison of Calibration Data between the 20 L HLFS and Other NIST flow Standards

5.1 Calibration Procedure

To begin a calibration the MUT is installed in thetest section, and the calibration fluid is pressurizedto the desired level. Next, an air bleed is performed toremove any air that entered the system duringinstallation of the MUT. After the fluid and roomtemperatures are brought to equilibrium at Tref, acustom software package is used to control thecalibration process. Typical calibrations measure theflow at six set points (i.e., 5 %, 15 %, 30 %, 50 %, 70 %,and 100 % of the full scale flow). Each of the six flowsis measured on two occasions with 5 repeats on eachoccasion. The final calibration result for each set pointis the average of at least 10 measurements. Thestandard deviation of the 10 measurements is takento be the reproducibility.

For turbine meters the calibration data is plottedin the dimensionless units of Roshko and Strouhalnumbers where the Rosko number is defined as

(13)

Here, f is the blade frequency,

is the temperaturecorrected diameter of the MUT, and

ref MUT ref( )[ ]11ν ν a T T+ −= is a curve fit of the fluidkinematic viscosity determined experimentally usinga capillary viscometer. The Strouhal number isdefined by

(14)

where is determined by Eq. (9b).

5.2 Internal Comparison Results with Other NIST Flow Standards

The 20 L HLFS uncertainty calculations wereverified by comparison to the other three liquidhydrocarbon standards in Table 1 using Stoddardsolvent as the working fluid. In addition the 20 L HLFSwas compared to the water flow facility. The samedual rotor turbine meter was used as the transferstandard for all of the comparisons. The flowcomparisons spanned a flow range from 3.8 lpm to

Fig. 5. Comparison results between five NIST liquid flow standards including the four hydrocarbon flowstandards in Table 1 and NIST's water flow standard

2 fRo

ν=

D

ref MUT MUT ref( )[ ]1 T Tα+ −= D D

MUT

3

4f

StQπ

=D

MUTQ

Log(Ro)

10.24

10.29

10.34

10.39

10.44

4.5 5 5.5 6

St

0.5 %

Stoddard Solvent

20 L HLFS

2 L HLFS

Small Cox

Large Cox

WaterWater Flow Facility20 L HLFS


202

38 lpm. The comparison results are plotted in Fig. 5where the symbols in the legend indicate the measureddata. The solid line ( ) is a curve fit of 13 differentcalibrations done in Stoddard solvent over a six monthperiod using the 20 L HLFS. The two dashed lines( ) are the 95% confidence interval of the curve fit.The results show all five of NIST's primary flowstandards agreed within their uncertainties. Inaddition, the repeated calibrations of the turbine meterover a 6 month period demonstrate the stability of the20 L HLFS.

6. Summary and Conclusion

This paper documents the performance anduncertainty of NIST 20 L HLFS. This primary standardis used to provide flow traceability to NIST customersfor hydrocarbon liquid flows. Based on theuncertainty analysis herein the volumetric flowuncertainty is 0.074 % (k = 2) and the mass flowuncertainty is 0.08 % (k = 2). The comparison of the 20L HLFS with 3 other hydrocarbon liquid flowstandards and a water flow standard are consistentwith the above uncertainty analysis. In addition, thecomparison shows that all of the standards are inagreement within their stated uncertainties. Finally,the paper introduced a set of correction factors thatcan be used to estimate uncertainties of other liquidpiston prover flow standards.

Acknowledgement

The technical contributions from Sherry Sheckelsand John D. Wright are gratefully acknowledged.Special acknowledgement is given to Iosif Shinder forcalibrating the turbine meter transfer standard againstthe NIST water flow standards.

References

[1] T.T. Yeh, J. Aguilera and J.D. Wright,Hydrocarbon Liquid Calibration Service, NISTSpecial Publication, (2009) 250-1039.

[2] T.T. Yeh, J. Aguilera, P.I. Espina, G.E. Mattinglyand N.R. Briggs, An Uncertainty Analysis of

A NIST Hydrocarbon Liquid Flow CalibrationFacility, Proceedings of Heat Transfer/FluidsEngineering Summer Conference, Charlotte,North Carolina, (2004).

[3] I.I. Shinder and M.R. Moldover, DynamicGravitational Standard for Liquid Flow: Modeland Measurements, 7th InternationalSymposium on Fluid Flow Measurement,Anchorage, AK , (2009).

[4] D.R. Lide, CRC Handbook of Chemistry andPhysics 73rd Ed., CRC Press, Inc., Boca Raton,FL, (1992).

[5] W. Benson, J.W. Harris, H. Stocker and H. Lutz,Handbook of Physics, Springer, New York, NY,(2006).

[6] G.E. Mattingly, The Characterization of a PistonDisplacement-Type Flowmeter CalibrationFacility and the Calibration and Use of PulsedOutput Type Flowmeters, Journal of Researchof the National Institute of Standards andTechnology, 97 (1992) p. 509.

[7] API, Manual of Petroleum MeasurementStandards, Chapter 4.3: Proving Systems - SmallVolume Provers, 1st ed., American PetroleumInstitute, Washington, D.C., (1993).

[8] H.W. Coleman and W.G. Steele, Experi-mentation and Uncertainty Analysis forEngineers, John Wiley and Sons, Inc., New York,NY, (1989).

[9] B.N. Taylor and C. E. Kuyatt, Guidelines for theEvaluating and Expressing the Uncertainty ofNIST Measurement Results, NIST TN-1297,MD, Pressure Vessel Equation, (1994).

[10] K.B. Jaeger and R.S. Davis, A Primer for MassMetrology, NBS (U.S.), Spec. Publ. 700-1,Industrial Measurement Series, 79 (1984) p. 5.

[11] J.B. Patterson and E.C. Morris, Measurement ofabsolute water density, 1°C to 40°C, Metro-logia, 31 (1994) p. 277.

Documents

Uncertainty Analysis of NIST's 20 Liter Hydrocarbon Liquid