Scientific Uncertainty Analysis in Pressure

Pressure & Vacuum Laboratory TAF 2011 Navigate to Table of Contents

Scientiﬁc Uncertainty Analysis in Pressure Metrology
EURAMET.M.P-K13 as an Illustrative Case Study

Vishal Ramnath
R&D Metrologist: Pressure & Vacuum
vramnath@nmisa.org

Pressure & Vacuum Laboratory

April 19, 2011

1 / 30


Table of contents I
Review of Methods for Pressure Generation/Measurement
Mechanism of Pressure Generation
Mathematical Model of Pressure Balance
Mechanism to Solve Formulation of Mathematical Equation
Solution for Roots of Multivariable Equation
Defining an EoS for the Working Fluid
Solution of the Roots (Pressure) of the Nonlinear Equation
Calculation of Generated Pressure Uncertainty
Specific Issues Encountered in Uncertainty Analysis
Methodology Used in Cross-Floating Analysis
Derivation of Basis Equation used in Cross-Floating I.
Estimation of the Straight Line Uncertainties Corresponding to u (A0 )
and u (λ)
Discussion
Specific Comments on Intercomparison Results
Recommendations Going Foward
2 / 30



Generation of Reference Pressure I.
In general there are different possible approaches in deﬁning
pressure, of which two predominate:

3 / 30



Thermodynamic pressure:

1 ∂ ln Z ∂F
P= =− (1)
β ∂V β ∂V T

3 / 30




1 ∂ ln Z ∂F
P= =− (1)
β ∂V β ∂V T

deﬁned in terms of the Maxwell thermodynamic equations:
∂T ∂P ∂S ∂P
∂V S =− ∂S V , ∂V T = ∂T V
∂S ∂V ∂T ∂V (2)
∂P T =− ∂T P , ∂P S = ∂S P

3 / 30




1 ∂ ln Z ∂F
P= =− (1)
β ∂V β ∂V T

deﬁned in terms of the Maxwell thermodynamic equations:
∂T ∂P ∂S ∂P
∂V S =− ∂S V , ∂V T = ∂T V
∂S ∂V ∂T ∂V (2)
∂P T =− ∂T P , ∂P S = ∂S P

Recall: Partition function Z (T , V , N ) = r exp(−β Er ), r microstate,
1
Er energy for r microstate, temperature paramter β = kT , k
Boltzmann fundamental constant, Helmholtz free energy
1 ∂ S (E ,V ,N )
F (T , V , N ) = −kT ln Z (T , V , N ), temperature T
= ∂E
V ,N

3 / 30



Generation of Reference Pressure II.
Mechanical pressure: The full situation is actually more complicated
since recourse to Hamiltonian theory is required to precisely deﬁne
“force” in terms of the generalized displacements and generalized
momenta:
n ∂H ∂H
H= α=1
˙
pα qα − L, ˙
p α = − ∂ qα , ˙
qα = ∂ pα (3)

4 / 30



momenta:
n ∂H ∂H
H= α=1
˙
pα qα − L, ˙
p α = − ∂ qα , ˙
qα = ∂ pα (3)

This conceptual framework is necessary since ideally for
measurement equivalence
Pthermodynamic = Pmechanical (4)

4 / 30



momenta:
n ∂H ∂H
H= α=1
˙
pα qα − L, ˙
p α = − ∂ qα , ˙
qα = ∂ pα (3)

This conceptual framework is necessary since ideally for
measurement equivalence
Pthermodynamic = Pmechanical (4)

Utilizing the concepts of strain and stress the mechanical pressure
for a ﬂuid is deﬁned in terms of the shear stress tensor as
1 2
¯
p = − (τxx + τyy + τzz ) = p − λ+ µ ·V (5)
3 3
4 / 30



Generation of Reference Pressure III.
Mechanical pressure definition: Without going into the details a
working definition for pressure is to apply Stoke’s hypothesis
2
λ + 3 µ = 0 and couple it with a defining equation of state so that

∂F
P= ⇔ P = NkT i .e. PV = nRT (ideal gas EoS ) (6)
∂ An

5 / 30



Generation of Reference Pressure III.
Mechanical pressure definition: Without going into the details a
working definition for pressure is to apply Stoke’s hypothesis
2
λ + 3 µ = 0 and couple it with a defining equation of state so that

∂F
P= ⇔ P = NkT i .e. PV = nRT (ideal gas EoS ) (6)
∂ An

The above definition for pressure is in accordance to the BIPM
choice of using mechanical pressure in a NMI scale realization

On a Fundamental Scientific SI System of Units
In line with international best practise there is a move to define the SI
in terms of fundamental scientific constants, and although pressure is a
derived unit from the seven base SI units viz. P = F developments such
A
as the Watt balance and Avogadro projects to redefine the kilogram will
logically also impact on the definition of pressure.
5 / 30



Generation of Reference Pressure IV.
In practical terms pressure is then essentially ”p = F ” and presently
A
the most accurate technique of generating a pressure is in terms of
a pressure balance

1 a scalar has zero dimensions and only a magnitude e.g. quantity of matter [moles], a

vector has one dimension and a direction e.g. velocity of wind, a matrix has two dimensions
and direction and orientation e.g. strain
6 / 30



A
a pressure balance
A pressure balance is essentially a precision manufactured piston
that is able to slide in a cylinder where the bottom of the piston is
exposed to a ﬂuid under pressure and the top of the piston is stacked
with mass pieces which exert a downwards weight due to gravity


6 / 30



A
a pressure balance
From the full deﬁnition of “pressure” the quantity p is in fact a tensor1
but for most practical purposes we essentially consider it a scalar
e.g. in hydrostatic hydraulic pressure we assume that the pressure
is equal in all directions (true for Newtonian ﬂuids)


6 / 30



A
a pressure balance
From the full definition of “pressure” the quantity p is in fact a tensor1
but for most practical purposes we essentially consider it a scalar
e.g. in hydrostatic hydraulic pressure we assume that the pressure
is equal in all directions (true for Newtonian fluids)
For a piston-cylinder unit (PCU) there is then a fluid exerted
pressure at the “bottom” and a mechanical loading exerted pressure
at the “top”

6 / 30



Illustrative Examples2 of Pressure Balances

(a) Ruska model 2465A- (b) Ruska model 2485- (c) Desgranges & Huot
U-754 pneumatic PB 930D hydraulic PB model DH 5503S hydraulic
PB

Figure: Low range PB (a), medium range PB (b) and high range PB (c)

2 images courtesy of National Institute of Metrology (Thailand) website

http://www.nimt.or.th/
7 / 30



Schematic Illustration of the Pressure Balance

Figure: Geometry and force analysis of a piston-cylinder pressure balance

8 / 30



By an analysis of the physics of the problem one may derive the following
mathematical model:

n ρa
i =1 mi 1 − ρi − Vs (ρf − ρa ) + H (ρf − ρa )S g + σ C
p−Π =
S
(7)
S = A0 (1 + λP )[1 + α(t − t0 )] (8)
In the above equations the nomenclature used is:

p = absolute pressure ( true pressure ) (9)
Π = ambient pressure ( top pressure ) (10)
P = p − Π = applied pressure ( gauge pressure) (11)
In the case of the EURAMET.M.P-K13 key intercomparison Π is known since this the measured atmospheric pressure and in essence the above

equations are to be solved to determine the unknown applied pressure P (and then by implication the absolute pressure p).
9 / 30



Explanation of Nomenclature Used in Mathematical Model
mi [kg] – mass of the i th weight piece

10 / 30



ρa [kg m−3 ] – density of the ambient ﬂuid above the PCU (air density)

10 / 30



ρi [kg m−3 ] – density of the i th weight piece

10 / 30



Vs [m3 ] – volume of the part of the piston which is submerged into the ﬂuid
which applies the pressure

10 / 30



ρf [kg m−3 ] – density of the ﬂuid which applies the pressure

10 / 30



H [m] – height correction term to account for possible variation of laboratory
standard (LS) and transfer standard (TS)

10 / 30



g [m s−2 ] – local gravitational acceleration

10 / 30



σ [N m−1 ] – surface tension term to account for surface tension effects
between liquid (oil) and gas (air) interface

10 / 30



S [m2 ] – ”effective area” of piston-cylinder

10 / 30



A0 [m2 ] – zero-pressure area of PCU LS

10 / 30



λ [Pa−1 ] – distortion coefﬁcient of PCU (typically measured in ppm/MPa)

10 / 30



α [K−1 ] – thermal expansion coefﬁcient term (α = αp + αc )

10 / 30



α [K−1 ] – thermal expansion coefﬁcient term (α = αp + αc )
C [m] – circumference of piston exposed to oil-air interface (typically set as
C ≈ 4π A0 ) 10 / 30



Formulation of Mathematical Model as Nonlinear Root
Search I.
Substituting the above terms we have
n
A0 ρa
(p − Π)[1 + λ(p − Π)][1 + α(t − 20)] = mi 1− − Vs (ρf − ρa )
g ρi
i =1

+ H (ρf − ρa )A0 [1 + λ(p − Π)]

× [1 + α(t − 20)]
σ −1
+ (4π A0 [1 + α(t − 20)]) 2
g
(12)

which can be rewritten as a multi-variable equation f (X ) = 0 for which
the roots (unknown pressure p) must be solved for.
11 / 30



Search II.
The form of f (X ) is then:

n ρa
f (X ) = i =1 mi 1 − − Vs (ρf − ρa )
ρi
+ H (ρf − ρa )A0 [1 + λ(p − Π)][1 + α(t − 20)]
−1 (13)
+ σ (4π A0 [1 + α(t − 20)]) 2
g
A0
− g
(p − Π)[1 + λ(p − Π)][1 + α(t − 20)]
where the input state variable X is

X = [ m1 , m2 , . . . , mn , (14)
ρ1 , ρ2 , . . . , ρn , (15)
T
A0 , λ, αp , αc , t , Vs , H , Π, σ, g , tair , RHair ] (16)

12 / 30



Search III.

The density of the air may be calculated using the CIPM-2007 formula3
which expresses the air density in terms of
ta – atmospheric temperature (measured with in-house thermometer
traceable to water triple point (WTP) cell)
pa – atmospheric pressure (measured with in-house barometer
traceable to absolute medium gas pressures with Ruska LS)
ha – atmospheric relative humidity expressed as a pure number
such that 0 ha 1 (traceable to in-house hybrid humidity
generator: validated in humidity regional inter-comparison)

3 Metrologia 45 2008 149–155 ”Revised Formula for the Density of Moist Air

(CIPM-2007)”, Picard A et al
13 / 30



Details of CIPM-2007 Formula for Moist Air Density
Brieﬂy the underlying formula is of form

pMa Mv
ρa = 1 − xv 1− (17)
ZRT Ma
where
ρa [kg m−3 ] – air density

p [Pa] – atmospheric pressure (total pressure with implicit assumption of Dalton’s partial pressure law)

t [◦ C] – air temperature

T [K] – thermodynamic absolute temperature

xv [−] – mole fraction of water vapour

Ma [kg mol−1 ] – molar mass of dry air

Mv [kg mol−1 ] – molar mass of water vapour

Z [−] – compressibility factor

R [J mol−1 K−1 ] – universal gas constant (using CODATA 2006 values4 cited in: Reviews of Modern Physics, Vol 80, April–June
2008, Mohr P J et al)

4 http://www.nist.gov/pml/div684/fcdc/upload/rmp2006-2.pdf
14 / 30



Note on Mass Input Values

In the state variable we assume n mass pieces in total

15 / 30



Note on Mass Input Values

In the state variable we assume n mass pieces in total
This choice is incorporated by assigning the mass values to
correspond to a weight set of N weights used (where n > N) and
then appending the piston mass mpiston and bell mass mbell viz.

[m1 , m2 , . . . , mn−2 , mn−1 , mn ]T = [m1 , m2 , . . . , mN , mpiston , mbell ]T
(18)

15 / 30



Varying Meanings of Mass
On the Differences Between True Mass, Conventional Mass and Apparent
Mass
Conventional mass: air density of 1.2 kg m−3 and mass density of 8000 kg m−3
Detailsa Objective is to determine the true mass of a weight piece from calibration
data:

ρa
1− ρs
true mass : m = ms ρa (19)
1− ρm
(c ) (s )
(c ) ρa ρa
calibration cert data : ms 1− (c )
= ms 1− (20)
ρs ρs
(c )
ρa
1− (c ) 1− ρa
(c ) ρs ρs
⇔m = ms (s)
× (21)
ρa
1− ρa 1− ρm
ρs

a The Pressure Balance: Theory and Practice, Dadson R S et al, 1982, National Physical

Laboratory / HMSO 16 / 30


Solution for Roots of Multivariable Equation

Solution of Roots of f (X ) = 0
In the mathematical model of the EURAMET.M.P-K13 key
comparison there will in general be a variable number of inputs since
a varying number of mass pieces n are used to generate a
sequence of (applied) pressures for P ∈ [50, . . . , 500] MPa
It can be shown that for n mass pieces used in
X = [m1 , . . . , mn , ρ1 , . . . , ρn , A0 , λ, αp , αc , t , Vs , H , pa , ta , ha , σ, g ]T
there will be a total of (2n + 12) state variables incorporated into the
measurand model to solve for the unknown generated applied
pressure
Consider the terms in f (X ) which are known and unknown
n ρa
f (X ) = i =1 mi 1 − − Vs (ρf − ρa )
ρi
+ H (ρf − ρa )A0 [1 + λ(p − Π)][1 + α(t − 20)]
−1 (22)
+ σ (4π A0 [1 + α(t − 20)]) 2
g
A0
− g
(p − Π)[1 + λ(p − Π)][1 + α(t − 20)]
17 / 30



EoS for a Pressure Balance Working Fluid I.
During the intercomparison both the transfer standard and the NMISA
Ruska high pressure PCU used di(2-ethylhexgl) sebecate i.e. ‘DHS’ as the
working fluid formulated as
ρf /[kg m−3 ] = [912.7 + 0.752(10−6 p) − 1.65 × 10−3 (10−6 p)2
+ 1.5 × 10−6 (10−6 p)3 ] × [1 − 7.8 × 10−4 (tf − 20)]
(23)
Alternative forms that may be considered are linear extrapolations from a
known reference state ρ f0
1+β(tf −tf 0 )
ρf = p −p
(24)
1 − E f0
f
where tf 0 is the reference oil temperature at which the oil’s properties are
known (say 20 ◦ C), Ef is the bulk modulas fluid elasticity, pf 0 is the referance
oil fluid density pressure, tf is the current oil temperature and p is the current
oil pressure
For pure ideal gases the EoS is
ρRT
p= (25)
M 18 / 30



EoS for a Pressure Balance Working Fluid II.

On the Necessary Distinction Between Absolute and Applied Pressure
The is a natural distinction between the absolute pressure p and the
applied pressure P and equations-of-state are fundamentally deﬁned in
terms of the absolute pressure in order to be unambigous. Consider the
following hypothetical cases:
• case (i) {P (i ) = 200 kPa} ∪ {Π(i ) = 100 kPa} ⇒ p(i ) = 300 kPa
• case (ii) {P (ii ) = 200 kPa} ∪ {Π(ii ) = 80 kPa} ⇒ p(ii ) = 280 kPa
As a result even if the applied pressure is the same this does not neces-
sary imply that the ﬂuid density will be the same

19 / 30



Numerical Solution for Roots of a Nonlinear Equation
From the deﬁning equation f (X ) with X = [x1 , . . . , x2n+12 ]T

n ρa
f (X ) = i =1 mi 1 − − Vs (ρf − ρa )
ρi
+ H (ρf − ρa )A0 [1 + λ(p − Π)][1 + α(t − 20)]
−1 (26)
+ σ (4π A0 [1 + α(t − 20)]) 2
g
A0
− g
(p − Π)[1 + λ(p − Π)][1 + α(t − 20)]

20 / 30




n ρa
f (X ) = i =1 mi 1 − − Vs (ρf − ρa )
ρi
+ H (ρf − ρa )A0 [1 + λ(p − Π)][1 + α(t − 20)]
−1 (26)
+ σ (4π A0 [1 + α(t − 20)]) 2
g
A0
− g
(p − Π)[1 + λ(p − Π)][1 + α(t − 20)]

the original nonlinear equation may be transformed into an equation
in a single unknown Y by setting all the all components such that

20 / 30




n ρa
f (X ) = i =1 mi 1 − − Vs (ρf − ρa )
ρi
+ H (ρf − ρa )A0 [1 + λ(p − Π)][1 + α(t − 20)]
−1 (26)
+ σ (4π A0 [1 + α(t − 20)]) 2
g
A0
− g
(p − Π)[1 + λ(p − Π)][1 + α(t − 20)]

xi = p∀i ∈ [1, . . . , 2n + 12] as parameters i.e. by solving

20 / 30




n ρa
f (X ) = i =1 mi 1 − − Vs (ρf − ρa )
ρi
+ H (ρf − ρa )A0 [1 + λ(p − Π)][1 + α(t − 20)]
−1 (26)
+ σ (4π A0 [1 + α(t − 20)]) 2
g
A0
− g
(p − Π)[1 + λ(p − Π)][1 + α(t − 20)]

xi = p∀i ∈ [1, . . . , 2n + 12] as parameters i.e. by solving

f (Y ; X ) = 0 (27)

The solution Y is of course the absolute pressure

20 / 30



Estimation of Generated Pressure Uncertainty

For the EURAMET.M.P-K13 key comparison V&V in terms of the
uncertainty analysis by the research metrologist was performed
within the framework of the GUM

5 Cox M G and Harris P M, Software Support for Metrology Best Practice Guide No. 6 –

Uncertainty Evaluation, Technical Report, National Physical Laboratory (United Kingdom),
2004
21 / 30



Estimation of Generated Pressure Uncertainty

For the EURAMET.M.P-K13 key comparison V&V in terms of the
uncertainty analysis by the research metrologist was performed
within the framework of the GUM
The underlying matrix based equation for the multi-variable input X
was
2
∂f
u 2 (y ) = ( x f )V x ( x )T (28)
∂y
following the methodology5 of Cox and Harris.
In our initial approach I assumed no correlation effects in the
components of the input variable X for simplicity

5 Cox M G and Harris P M, Software Support for Metrology Best Practice Guide No. 6 –

Uncertainty Evaluation, Technical Report, National Physical Laboratory (United Kingdom),
2004
21 / 30



The Validity of the Assumption of
Cov(xi , xj ) = 0∀i = j : 1 i , j 2n + 12
In general the only possible signiﬁcant non-zero correlation effect
that may occur is for the case Cov(A0 , λ)

6 consultation of the statistical literature such as Metrologia journal articles and IMEKO

conferences states that it is poor practice to simply assume unity correlation in the absence
of information: the recommended option is to assume zero correlation
22 / 30



Cov(xi , xj ) = 0∀i = j : 1 i , j 2n + 12
Correlation effects do occur in the case of the weights mass/density
but in general this has a very minor impact on ﬁnal uncertainties as
u (A0 ) and particularly u (λ) will dominate uncertainty contributions at
elevated pressures


22 / 30



Cov(xi , xj ) = 0∀i = j : 1 i , j 2n + 12
Correlation effects do occur in the case of the weights mass/density
but in general this has a very minor impact on final uncertainties as
u (A0 ) and particularly u (λ) will dominate uncertainty contributions at
elevated pressures
(LS )
Correlation is certain to occur in the present instance as A0 and
λ(LS ) are in fact obtained from an experimental cross-floating
againsts another precision piston-cylinder standard however in the
present instance no accessible information on Cov(A0 , λ) is
available as these data values are sourced from an external
certificate that does not provide any information from which one may
reasonably infer a magnnitude of correlation6

22 / 30



Basis Equation for Cross-Floating
Make the assumption of linear elasticity theory in the PDE’s formulation
of the strain/stress constituent behaviour:
Assume a piston-cylinder A with known characteristics i.e.
(A)
DA = {A0 , λ(A) } is cross-ﬂoated with another piston-cylinder B with
(B )
unknown characteristics i.e. DB = {A0 , λ(B ) } which are to be
determined

23 / 30



Basis Equation for Cross-Floating
Make the assumption of linear elasticity theory in the PDE’s formulation
of the strain/stress constituent behaviour:
Assume a piston-cylinder A with known characteristics i.e.
(A)
DA = {A0 , λ(A) } is cross-ﬂoated with another piston-cylinder B with
(B )
unknown characteristics i.e. DB = {A0 , λ(B ) } which are to be
determined
It follows that
m
1 ρa
PB = × g mB j 1− − Vs B (ρf − ρa )g
SB ρB j
j =1

(B )
+ σB 4π A0 + (HB (ρf − ρa )g ) SB (29)

23 / 30



Basis Equation for Cross-Floating II.
Noting that HB = 0 since the height correction was already
accounted for in PCU A when working out the applied pressure

24 / 30




m
1 ρa (B )
⇒ SB = × g mB j 1− − VB (ρf − ρa )g + σB 4π A0
PA ρm B j
j =1

(30)

24 / 30




m
1 ρa (B )
PA ρm B j
j =1

(30)

As a result there is now a data point k with {P , SB } where
(B )
SB = A0 (1 + λ(B ) P ) (31)

24 / 30




m
1 ρa (B )
PA ρm B j
j =1

(30)

(B )
SB = A0 (1 + λ(B ) P ) (31)

for k ∈ [1, . . . , 10] corresponding to pressures [50, . . . , 500] MPa
from which a straight line may be plotted in an analogous form to

24 / 30




m
1 ρa (B )
PA ρm B j
j =1

(30)

(B )
SB = A0 (1 + λ(B ) P ) (31)

for k ∈ [1, . . . , 10] corresponding to pressures [50, . . . , 500] MPa
from which a straight line may be plotted in an analogous form to

y = ax + b ⇐ D = {Dk = (xk , yk )|k ∈ [1, . . . , 10]} (32)
24 / 30


Estimation of the Straight Line Uncertainties Corresponding to u (A0 ) and u (λ)

Estimation of Zero-Pressure and Distortion Coefﬁcient
Uncertainties I.

Standard linear regression analysis in spreadsheet environments
are not suitable for the provision of adequate parameter
uncertainties

25 / 30



Uncertainties I.

Standard linear regression analysis in spreadsheet environments
are not suitable for the provision of adequate parameter
uncertainties
In general optimization routines are necessary for a χ2 merit
function determination and recently a practical implementation has
been proposed in the scientiﬁc literature by scientists at the PTB
(Germany): A Least Squares Algorithm for Fitting Data Points with
Mutually Correlated Coordinates to a Straight Line, Meas. Sci.
Technol. 22 (2011) doi: 10.1088/0957-0233/22/3/035101, Krystek M
and Anton M

25 / 30



Uncertainties II.
The method ﬁts y = ax + b and provides estimates for u (a) and
u (b) noting that for a pressure balance the deﬁning basis equation is

A(P ) = A0 (1 + λP ) (33)

26 / 30



Uncertainties II.
The method fits y = ax + b and provides estimates for u (a) and
u (b) noting that for a pressure balance the defining basis equation is

A(P ) = A0 (1 + λP ) (33)

It then follows that in a pressure cross-floating experimental
measurement that

u 2 (a) = u 2 (A0 ) (34)
2 2 2
u (b ) = λ u (A0 ) + A2 u 2 (λ)
0 (35)
1
⇒ u 2 (λ) = u 2 (b) − λ2 u 2 (a) (36)
A20

26 / 30


Discussion

Speciﬁc Comments on Intercomparison Participation
The participation of the technical metrologists in
EURAMET.M.P-K13 can be considered a learning experience as
there is a signiﬁcant difference is the level and complexity of key
comparisons when compared to industrial level measurements

27 / 30


Discussion

It is recognized that in terms of data processing, analysis, and
post-processing of results as a NMI there is always room for
improvement and that there are certain inherent fundamental
limitations in spreadsheet utilization, both in terms of what can be
mathematically/statistically achieved and also in terms of validation
and veriﬁcation

27 / 30


Discussion

It is recognized that in terms of data processing, analysis, and
post-processing of results as a NMI there is always room for
improvement and that there are certain inherent fundamental
limitations in spreadsheet utilization, both in terms of what can be
mathematically/statistically achieved and also in terms of validation
and veriﬁcation
In terms of internal comparison of processed results there is good
general agreement in generated pressures and fair/moderate
agreement in computed effective area and distortion coefﬁcient of
applicable transfer standard

27 / 30


Discussion

Recommendations and Future Options Going Forward I.

There is from the research metrologist’s perspective a need to more
fully examine the ratio method of analysis in cross-ﬂoating as this
may present a more numerically stable implementation than a more
direct P vs. A(P ) regression analysis however this must be weighed
up against the implication of directly introducing the LS’s distortion
coefﬁcient into the uncertainty analysis

28 / 30


Discussion

Recommendations and Future Options Going Forward I.

There is from the research metrologist’s perspective a need to more
fully examine the ratio method of analysis in cross-ﬂoating as this
may present a more numerically stable implementation than a more
direct P vs. A(P ) regression analysis however this must be weighed
up against the implication of directly introducing the LS’s distortion
coefﬁcient into the uncertainty analysis
There is a need to more fully investigate the actual correlation
between the parameter estimates for A0 and λ as opposed to a
simplistic unity correlation that merely provides an upper bound

28 / 30


Discussion

Recommendations and Future Options Going Forward II.
FEM numerical simulations should be performed to benchmark the
existing experimental/statistical distortion coefﬁcient however the
challenge posed is not actually in terms of the FEM analysis but
rather in terms of the materials properties of the piston-cylinder:
tungsten-carbide is prone to some variation in the Young’s modulas
of elasticty and Poisson’s ratio – a possibility is to submit a research
proposal for an electrical capacitance / materials characterization
experimental study of the distortion coefﬁcent on the low and
medium pressure Ruska piston-cylinder laboratory standards

29 / 30


Discussion

It may be beneﬁcial to consider alternative statistical formulations to
bootstrapping such as Jackkniﬁng

29 / 30


Discussion

It may be beneﬁcial to consider alternative statistical formulations to
bootstrapping such as Jackkniﬁng
Certain aspects of the Pressure & Vacuum Laboratory R&D on the
Ruska high pressure characterization may be able to (in partnership
with another NMI e.g. PTB / CMI) make a modest contribution on
the Avogadro project that requires precision pressure
measurements at 7 MPa
29 / 30


Discussion

Acknowledgements

This work was performed with ﬁnancial support of the South African
Department of Trade and Industry

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Scientific Uncertainty Analysis in Pressure

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