Uncertainty of Measurements JMSeynhaeve

8/17/2019 Uncertainty of Measurements JMSeynhaeve

1/21

TERM 12/4/2007

Les écoulement de fluide dans l'industrie 1

Measurement and uncertainty of measurement 1

Measurements and uncertainty ofmeasurements

Jean-Marie SEYNHAEVE

Unité TERM – UCL

[email protected]

How to be sure of the uncertainty ?

• Measurement and uncertainty of measurements – Principles and standards

• Applied Regression analysis – Generalised ordinary linear fitting• Measurement device – calibration and use

• Traceability in measurement – Dissipation of the information

• Redundancy in the information – Lagrange multipliers

• References and acknowledgment


Measurement and uncertainties of measurement - Princip les

Reference : GUM - Guide to the expression of th e uncertainty in measurement

Approved by :

BIPM Bureau international des poids et mesures

CEI Commission électrotechnique internationale

FICC Fédération internationale de chimie clinique

ISO International standard organization

OIML Organisation internationale de métrologie légale

UICPA Union internationale de chimie pure et appliquée

UIPPA Union internationale de physique pure et appliquée

Uncertainty of measurement = doubt on the validity of the measuring value

Uncertainty Error


2/21

TERM 12/4/2007



• Let us consider a known random variable x defined by :

Recall : Normal distribution, Student distribution

• Normal distribution if :

1

limn

n i

i

x x n→∞=

⎛ ⎞⎜ ⎟⎝ ⎠∑

( )( )

( )2

0.5 2

1exp

22

x x f x

σ σ π

⎛ ⎞−= ⎜ ⎟

⎜ ⎟−⎝ ⎠

( ) 2 21

limn

n i

i

V x nσ ε →∞=

⎛ ⎞⎜ ⎟⎝ ⎠∑

Mean value

Variance

Mathematical interest !


• Student distribution

• Gamma function : ( )1

0

x qq e x dx

∞− −Γ = ∫

⇒ Gamma function properties :

• Recurrence : ( ) ( )1q q qΓ = Γ − For all q

• Particular case : if q integer then Γ (q) →

• particular values : ( )1 2 π Γ = ( )1 1Γ =

( )

12 2

1

21

2

t f t

ν

ν

ν

ν ν πν

+−

+⎛ ⎞Γ ⎜ ⎟ ⎛ ⎞⎝ ⎠= +⎜ ⎟⎛ ⎞ ⎝ ⎠Γ ⎜ ⎟⎝ ⎠

ν : number of degrees of freedom

If → ∝ : Student distribution → Normal distribution

Factoriel function


3/21

TERM 12/4/2007



Some examples of distribution Height human body…

LDV measurements

Droplets distribution of spray…

Development of QI test…

frequency

0

100

200

300

400

500

600

700

800

-0.60% -0.50% -0.40% -0.30% -0.20% -0.10% 0.00%

frequency

0

100

200

300

400

500

600

700

800

- 0 . 4

5 %

- 0 . 4

0 %

- 0 . 3

5 %

- 0 . 3

0 %

- 0 . 2

5 %

- 0 . 2

0 %

- 0 . 1

5 %

- 0 . 1

0 %

- 0 . 0

5 %

0 . 0

0 %

Example 1

Example 2


• Suppose a q quantity which varies at random : q is a random variable

• Suppose n independent observation q k obtained in same conditions of measurement

Best-estimate of q :

1

1 n

k

k

q qn =

= ∑

Estimate of the standard deviation : ( ) ( )2

1

1

1

n

k k

k

s q q qn =

= −−

∑

Estimate of standard deviation on the “mean value” : ( ) ( )k s q

s qn

=

Extended uncertainty of type A: ( ) ( )U q k s q= ⋅ 2k =

Uncertainty at confidence level of about 95 %

with

1. Measurement uncertainty of type A


4/21

TERM 12/4/2007



Comments

Valeur vraie

σ

Confidence level of 68 %

Number of degrees of freedom :

1nν = −

Student coefficient at 95 % :

( )95k t f ν =

If→ ∝ :

Coefficient de Student correspondant à un niveau de confiance de 95 %

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9 10 1 1 1 2 13 1 4 15 1 6 1 7 18 1 9 2 0

Nombre de degrés de liberté

C o e f f i c i e n t d e S t u d e n t

Limite acceptable

( ) ( )

( )

95

0

1.956

k

q True value

s q and U q

s q

t

σ

→

→

→

→ Conclusions :

Si σ ⇒ n If σ ⇒ ν

GUM recommended value : ν > 10


2. Measurement uncertainty of type B

Evaluation based on all available information :

- Results on previous measurements

- Scientific background

- Technical specification of the manufacturer

- Data from the calibration certificate

- Uncertainties assigned to reference values ...

Example of distributions … :

( )3

as A =

( ) ( )

2

12

A As A

+ −−=

A+ A-

a

A+ A- A+ A-

a

( )6

as A =

No repeated and independent observations available uncertainty of type B

Justification : « Limit central theorem » ...

Rectangular Triangular Rectangular


5/21

TERM 12/4/2007



3. Evaluation of combined expended uncertainty

Suppose a output quantity F depending on n independent input quantities xi ( s(xi ) known)

( )iF F x=

Sensitivity coefficients :( )( )i

i

i

F xc

x

∂

∂

Combined standard deviation : ( ) ( )22

1

n

i i

i

s F c s x=

= ∑

Combined expended uncertainty of F: ( ) ( ) ( )95iF F x t s F ν = ±

NB : Theory based on the Taylor series of the first order of F

⇒ Non-linearity of F ?


4. Example of presentation of results : manometric weighing device

Gas under pressure

Masses

Piston

Cylinder

( )( ) ( )

0, 0,

0,

20

(1 )

1 20 1

a a a

c

m m

p c

M g C

PS t LP

ρ ρ ρ σ

ρ ρ

α α

−− + +

=⎡ ⎤+ + − +⎣ ⎦

Quantity Units Value Standard Sensitivity Contribution toUncertainty Coefficient Standard Uncertainty

M kg 69.5349771 3.81E-05 2.00E+05 7.6

g m/s2 9.81139778 0.00000002 1.42E+06 0.0

ρ 0a kg/m3 1.2 - - -

ρ 0m kg/m3 8000 - - -

ρ a kg/m3 1.16178 2.71E-03 -1.76E+03 -4.8

ρ m kg/m3 7920 39.5 -8.48E-03 -0.3

σ N/m 3.10E-02 3.10E-03 5.06E+02 1.6

C m 0.024821354 5.926E-08 6.32E+02 0.0

S m2 4.902765E-05 7.354E-10 -2.84E+11 -208.7

α p+ α c 1/°C 9.00E-06 4.50E-07 -1.39E+07 -6.3

t °C 21 0.1 -1.25E+02 -12.5

L 1/Pa 1.01E-13 - - -

P Pa 13913168 - - 209

Pressure (bar) 139.1317

Expanded uncertainty 2k (bar) 0.0042

Relative expanded uncertainty 2k 3.01E-05


6/21

TERM 12/4/2007



Simple illustration

P

5 kg ± 5 g

Result of measurement :

P = 5 kg ± 5 g

P

1 kg ± 1 g

Result of measurement :

1 2 3 4 5P P P P P P= + + + +

( ) ( )

52

2

1( ) 5 0.5 1.12i

is P s P g

== = ⋅ =∑

1...5 1ic = =

P = 5 kg ± 2.24 gConsequences :

- Multiply your sources of information

- Estimate the likelihood of the information

- Verify the independence of your sources


Case study : natural gas metering

ValvesT p p T

Adjusting

valve

q1

q 2

q 3

q 4

Flowmeter

Turbines

Flowmeter

Turbines

Main :

Reference

Back-up :

Verification

q 5

q

C C


7/21

TERM 12/4/2007



• Global flowrate : 1 2 3 4 5q q q q q q= + + + +

• Uncertainty on the flowrate :

We have : where n is the number of operating lines

1 2 3 4 5q q q q q≅ ≅ ≅ ≅ ( ) ( ) ( ) ( ) ( ) ( )1 2 3 4 5iU q U q U q U q U q U q≅ ≅ ≅ ≅

( ) ( )iU q n U q= ⋅

• If Main = Back-up : 2 n independent measurements

- Uncertainty on qi : and

- Uncertainty on q : ⇒ decreasing of the uncertainty of +/- 30 %

- More information :

( ) ( )'

2

i

i

U qU q ≅

( ) ( )2i

n U qU q

⋅=

, ,'

2

i main i back up

i

q qq

−+=

, ,i main i back upq q −=

If

⇒ Estimation under constraints with Lagrange multiplier method


Case study : calibration installation of gas counters


8/21

TERM 12/4/2007



• Determination of the flowrate through one nozzle

*

, 20, ,1000,(1 0.169 ) 1 ( 100000)293.15

iv i v i i dry i

T Q x c P QÈ ˘= + + -Î ˚

Nozzle

Number

Diameter of Nozzle

(mm)

Q 20,dry,1000,i (m

3 /h)

Correction coefficient

ci* (Pa

-1)

1 2.0021 2.1909 2.3 10-7

2 2.7249 4.0884 1.4 10-7

3 3.8504 8.1860 1.3 10-7

4 5.3881 16.0543 1.0 10-7

5 7.6216 32.218 1.7 10-7

6 10.7107 63.780 1.4 10-7

7 15.1357 127.474 9.1 10-8

• Determination of the uncertainty on the flowrate through one nozzle

( )( ) ( ) ( )

( ) ( )

12 2 2 2

2 2 2, , ,2 *

*

, 22

22, ,20 , ,1 00 0,

20 , ,100 0,

v i v i v i

v i i

v i i

v i

v i v ii dry i

i dry i

Q Q QU x U T U c

x T cU Q

Q QU P U QP Q

∂ ∂ ∂

∂ ∂ ∂

∂ ∂

∂ ∂

È ˘Ê ˆ Ê ˆ Ê ˆ Í ˙+ +Á ˜ Á ˜ Á ˜ Ë ¯ Ë ¯ Ë ¯ Í ˙

= Í ˙

Ê ˆ Í Ê ˆ ˙+ +Í ˙Á ˜ Á ˜ Ë ¯ Ë ¯ Í ˙Î ˚

• Total flowrate and global expanded uncertainty

,

1

n

v v i

i

Q Q=

= Â ( ) ( )2,1

n

v v i

i

U Q U Q=

= Â


Relative uncertainty

for a 95 % confidence level

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0 20 40 60 80 100 120 140 160 180 200 220 240 260

Volume flowrate ( m3/h )

R e l a t i v e u n c e r t a i n t y ( % )


9/21

TERM 12/4/2007



Generalized linear ordinary fitt ing

General methodology

• Suppose n independent observations of a set of variables (physical quantities)

⇒ k predictive variables : independent and known (true value)

⇒ 1 (or more) dependent variable :

- function of xi (only…)

- having a random component for each set of xi

1 2, ,..., k x x x

y

⇒ Observation i (1 à n) :1 2, ,..., k i i i i x x x et y

Examples :

⇒ QI = f(weight) :

2 observations of QI for the same person or 1 observation of QI for two person ?

⇒ Heat transfer correlation : Nu (y) → Re (x1) et Pr (x2)


• « fitting » model

⇒ Good choice of a function : ( )1 1ˆ ,..., , ,...,k m y f x x C C =

With m constants C i to be dertermined Example : ( )( )Re ,Pr, i Nu f C λ =⇒ Good choice of the uncertainty model on y : ( ) ( )1 ˆ,..., k U y k f x x ky ou Cte= ⋅ = =

Difficult choice ! - Importance to know very well the phenomenon ...

• Least squares method

( )

2

1

ˆni i

i i

y yF

U y=

⎛ ⎞−= ⎜ ⎟⎜ ⎟

⎝ ⎠∑ Function to minimize : U(yi) are the weighting coefficients

⇒ How to determine the m constants C i : m equations 0i

F

C

∂=

∂

⇒ Number de degrees of freedom: n mν = − ⇒ Statistical validity : 0 10ν ν > → ≥

⇒ Residuals : standard deviation

⇒ Uncertainties on the m constants

:( ) ( ) ( )95i i is C et U C t s C

Bias of the model ?

Examine the residues

2

95r i r r s U t sε ν =∑


10/21

TERM 12/4/2007



Polynomial regression – Linear regression for multiple variables

• Model of «fitting»

2

0 1 2ˆ ... mm y A A x A x A x= + + + +Choice of the function :

⇒ Model for residuals : ( )U y k =

• Least squares method

⇒ Function to minimize : ( )2

1

ˆn

i i

i

F y y=

= −∑

⇒ Calculation of the m+1 constants Ai :

2

02 3

1

2 3 4 22

...

...

...

...... ... ... ... ...

i i i

i i i i i

i i i i i

An x x y A x x x x y

A x x x x y

Ê ˆ Ê ˆ Ê ˆ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ =Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Ë ¯ Ë ¯ Ë ¯

Â Â ÂÂ Â Â ÂÂ Â Â Â

⇒

0 00 01 02

1 10 11 12

22 20 21 22

...

...

...

... ... ... ... ... ...

i

i i

i i

A C C C y A C C C x y

A C C C x y

Ê ˆ Ê ˆ Ê ˆ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ =Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Ë ¯ Ë ¯ Ë ¯

ÂÂÂ

[C ij] is the inverse matrix


⇒ Number of degrees of freedom : ( )1n mν = − +

⇒ Residuals - standard deviation : ( )2

1

ˆn

r i i

i

s y y n =

= -Â ( )95r r U t sn =

⇒ Incertainty on the coefficients Ai :

( )

( )

( )

2

0 00

2

1 11

2

2 22

...

r

r

r

s A s C

s A s C

s A s C

=

=

=

⇒ Test of statistical significance of the coefficient Ai :( ) 95

m

m

At

s A≥

( )0 0 00 0

ˆm m

j k

r jk

j k

s y s C x x= =

= ÂÂ

⇒ Uncertainty on a predicted value : 20 0 0 0

ˆ ( , ,..., )m y f x x x=

⇒ Coefficient R 2 :( )

( )

2

2 1

2

1

ˆn

i

i

n

i

i

y y

R

y y

=

=

-=

-

Â

Â


11/21

TERM 12/4/2007



Example : Pressure transducer calibration

No X (I) U(X) Y(Pref) U(Y) Y est. U(Yest.) U global Y est. U(Yest.) U global

mA mA bar bar bar bar bar bar bar bar

1 7.996617 8.00E-05 20.0004 0.0007 19.9980 0.0025 0.0026 19.9995 0.0032 0.0033

2 8.996359 9.00E-05 24.9986 0.0009 24.9966 0.0022 0.0024 24.9974 0.0024 0.0026

3 9.997176 1.00E-04 30.0006 0.0011 30.0006 0.0020 0.0023 30.0007 0.0018 0.0022

4 10.99765 1.10E-04 35.0007 0.0012 35.0028 0.0018 0.0022 35.0025 0.0017 0.0022

5 11.99811 1.20E-04 40.0008 0.0014 40.0050 0.0016 0.0022 40.0043 0.0017 0.0023

6 12.99706 1.30E-04 44.9988 0.0016 44.9996 0.0015 0.0022 44.9987 0.0018 0.0025

7 13.99723 1.40E-04 49.9989 0.0017 50.0003 0.0014 0.0024 49.9994 0.0019 0.0027

8 14.99632 1.50E-04 54.9990 0.0019 54.9957 0.0015 0.0025 54.9948 0.0018 0.0028

9 15.99681 1.60E-04 59.9971 0.0021 59.9980 0.0016 0.0027 59.9973 0.0017 0.0028

10 16.99686 1.70E-04 64.9992 0.0023 64.9981 0.0018 0.0030 64.9978 0.0017 0.0030

11 17.99729 1.80E-04 69.9993 0.0024 70.0001 0.0020 0.0033 70.0003 0.0018 0.0032

12 18.99682 1.90E-04 74.9992 0.0026 74.9977 0.0022 0.0036 74.9984 0.0024 0.0037

13 19.99715 2.00E-04 79.9993 0.0028 79.9992 0.0025 0.0039 80.0007 0.0032 0.0044

Pol ynome du 1er de gré Polynome du 2ième de gr é

⇒ Diagnostic :

1.00E-05

3.50E-05

Degree Ai s(Ai) Test Test result

0 -19.9841 0.00228969 8728 +

1 4.99988 0.00015804 31638 +

11

Standard deviation on linear fitting (bar) 0.0021

t coeff ic ient for 95 % confidence level 2 .202

0.999999989

Relative expanded uncertainty on X

Relative expanded uncertainty on Y

Linear regression P (bar) = A0+A1*I

R2 coefficient of regression

Degree of freedom

1.00E-05

3.50E-05

P (bar) = A0+A1*I+A2*I^2

Degree Ai s(Ai) Test test result

0 -1.99718E+01 8.5099E-03 2346.90 +

1 4.99800E+00 1.2750E-03 3920.09 +

2 6.71703E-05 4.5228E-05 1.49 -

Degree of freedom 10

Standard deviation on cur ve fit ting (bar) 0.0020

Student coefficient for 95 % confidence level 2.229

Fitting degree 2

Relative expanded uncertainty on X

Relative expanded uncertainty on Y


Example of Global Uncertainty on the fitting model

0.000

0.001

0.002

0.003

0.004

0.005

6 8 10 12 14 16 18 20 22

Current (mA)

U g l o b a l ( b a r )

Linear fitting

Degree 2 fitting


12/21

TERM 12/4/2007



Comments on the linear regression with multiple variables

⇒ Model on the uncertainty ? : ( )U y k =

x

y

U(y)

( )ˆU y

U(y) is not a function of x (constant)

( ) ˆ

U y decreases if increases : level of confidence on the model

⇒ Verification of the model of uncertainty : ( ) ( ). ?U y k f x=

Various possible tests (Durbin-Watson…) – Determination of the « bias » of the model


Particular case : linear regression passing through the origin

⇒ Test of statistical significance on the coefficients Ai :

( ) ( )1 95

1

At

s An ≥

( ) ( )1 95

1

i

i

At

s An

π

π

£

• Model 1 : ( )U y k =

11

2

1

n

i i

i

n

i

i

x y

A

x

=

=

=Â

Â

( )2

1

2

1

r

n

i

i

ss A

x=

=

Â

( ) ( ) ( )2

00 0 1

2

1

ˆr n

i

i

xs y s abs x s A

x=

= =

Â

( )2

1

ˆ

1

n

i i

ir

y y

sn

=

-=

-

Â

• Model 2 : ( )U y k x= ⋅

11

ni

i i

y

x A

n

==Â

( )1

i

i

ys

xs A

n

Ê ˆ Á ˜ Ë ¯

=

( ) ( ) ( )0 0 1ˆs y abs x s A=

x

y

x

y

• Generalisation : ( ) ( )U y k f x= ⋅


13/21

TERM 12/4/2007



Example Model 2 – Chromatograph calibration

Standard deviation on Area for n-C4

0

0.4

0.8

1.2

1.6

0 50 100 150 200 250

Area

S t a n d a r d d e v i a t i o n

Expanded Uncertainty on X (%)

C1 3.15E-04 C2 6.11E-05


1 643.730 3.65 176.29 +

14

2.145

0.999315R2 coefficient of regression

U(X)=C1*X+C2

Linear regression Area = A1*[n-C4](%)

Degree of freedom

t coefficient for 95 % confidence level

FID : Uncertainty of n-C4 calibration

Chromatograph HP12000

0.00

0.01

0.01

0.02

0.02

0.03

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Mole fraction (%)

E x p a n d e d U n c e r t a i n t y ( % )


The measurement apparatus – Calibration and use

TransmitterReferenceOutput

MeasuringDevice

ref Q

out Q

ref Q

out Q

Reference physical quantity

Signal of the transducer

• Calibration

⇒ Choice of the model of fitting :

• Qref = f(Qout ) or Qout = f(Qref ) ? ⇒ Compare U(Q ref ) and U(Q mes )

Examples : Calibration of thermometric sensors, calibration of weighing device

• Which model ? ⇒

Choice of the function : linear or other – Statistical relevance

Choice of the model of uncertainty U(y) =k f(x) ? : test Durbin-Watson


14/21

TERM 12/4/2007



⇒ Choice of the calibration range : Range of use of the transducer

⇒ Choice of the calibration points : Experimental plans with multiple levels

The choice depends on what you need …

⇒ Interpretation of the results

Analysis of the residuals… Statistical Relevance…

Is U(y) model well known – Are the results in concordance of the initial model

Random property of the procedure ?


• The use

⇒ Inverse or direct regression ? :

ˆ ( )ref out Q f Q= ˆ ( )out ref Q f Q= direct inverse

⇒ Suppose q independent observations of Q out for an inverse regression

( )ˆ

out s Q

Qout

Qref

( )out s QQout

Uncertainty on the fitting

Uncertainty on the observation

Uncertainty on the output device

Uncertainty on Q ref

Combined uncertainty

( ) ( ) ( )95ˆ ˆout out U Q t s Qν =

( ) ( )95out out U Q t s Q q=

( )mesU Q

( )ref U Q

( ) ( ) ( ) ( ) ( )2 22 22 ˆ

out out mes ref U Q c U Q U Q U Q U QÈ ˘= + + +Í ˙Î ˚

ˆ1 out

ref

dQc

dQ


15/21

TERM 12/4/2007



Traceability – Continuous ramified chain

Example : Determination of PCi o u s of a natural gas

Toutes mesures doivent être traçables vis-à-vis de références primaires

Preparation of the

gravimetric mixtures

ISO 6142

Calibration of the

chromatograph

ISO 6975

Determination of the

calorific value

ISO 6976


Preparation of


ISO 6142

«Parent» gas

Component Mole fraction St. deviation

x 10-6

x 10-6

H2 0.05 0.03

CH4

H2O 0.5 0.29

CO 0.25 0.14

N2 999998.75 0.36

C2H4

C2H6

O2 0.25 0.14

Ar

CO2 0.1 0.06

C3H6

C3H8

C4H10 n

CnHm 0.1 0.06

C4H10 i

Autre 2

Autre 3

Parent gas : Nitrogen N2 - N57

Reference masses

Primary references

Molar masses

Reference Mass Mass (g) U(mass)

10 10 0.000060

20A 20 0.000080

20B 20 0.000080

10 + 20A 30 0.000100

20A + 20B 40 0.000113

50 50 0.000100

50 + 10 60 0.000117

50 + 20A 70 0.000128

50 + 10 + 20A 80 0.000141

50 + 20A + 20B 90 0.000131

100 100 0.000150

Reference masses and uncertainties

Atomic weight Standard Uncertainty

Atome kg/kmole kg/kmole

H 1.00794 0.00007

C 12.011 0.001

N 14.00674 0.00007

O 15.9994 0.0003

Ar 39.948 0.001

IUPAC Commission on Atomic Weights (1993)


16/21

TERM 12/4/2007



Masses of “parent” gas

Weighing

Calibration of the weighing device

Reference masses

Preparation of the “parent” gas

Ramified chain

Determination of molar

fractions of the mixture

Molar mass

Preparation of


ISO 6142

Pure gas


Component Mole fraction Expanded Uncertainty St. Dev. Mol. St. Dev. Mass St. Dev. Compo.

x 10-6

x 10-6

x 10-6

x 10-6

x 10-6

H2 0.05 0.06 0.00 0.00 0.03

CH4 970340.95 5.23 1.87 1.65 0.78

H2O 0.99 1.12 0.00 0.00 0.56

CO 0.01 0.01 0.00 0.00 0.00

N2 29657.63 5.13 1.87 1.65 0.59

C2H4 0.00 0.00 0.00 0.00 0.00

C2H6 0.05 0.06 0.00 0.00 0.03

O2 0.25 0.28 0.00 0.00 0.14

Ar 0.00 0.00 0.00 0.00 0.00

CO2 0.05 0.06 0.00 0.00 0.03

C3H6 0.00 0.00 0.00 0.00 0.00

C3H8 0.00 0.00 0.00 0.00 0.00

n-C4H10 0.00 0.00 0.00 0.00 0.00

CnHm 0.03 0.03 0.00 0.00 0.01

i-C4H10 0.00 0.00 0.00 0.00 0.00

Autre 2 0.00 0.00 0.00 0.00 0.00

Autre 3 0.00 0.00 0.00 0.00 0.00

Results

Preparation of


ISO 6142


17/21

TERM 12/4/2007



Calibration of the

chromatograph

ISO 6975

Gravimetric

mixtures

Chromatograph

calibration

Calibration results

- Fitting model and U fitting- U ( Area)- U (gravimetric mixtures)

Normalisation at Patm

No Gases X (Ref) U(X) Patm Aera Y(Aera st.)

Cylinder Mixture % % bar - -

N4@dz68 nC4+C2+C1 1.82225 0.000635 1.0181 1205.5518 1199.81

N4@dz68 nC4+C2+C1 1.82225 0.000635 1.01805 1210.9127 1205.20

N4@dz68 nC4+C2+C1 1.82225 0.000635 1.01795 1210.9059 1205.31

n4@dz62 nC4+C2+C1 1. 58759 0. 0005 61 1.01817 1044.6101 1039.56

n4@dz62 nC4+C2+C1 1. 58759 0. 0005 61 1.01814 1052.6324 1047.58

n4@dz62 nC4+C2+C1 1.58759 0.000561 1.0181 1053.1123 1048.10

n4@dz57 nC4+C2+C1 0. 75182 0. 0002 98 1.01775 473. 28247 471.19

n4@dz57 nC4+C2+C1 0. 75182 0. 0002 98 1.01768 477. 08435 475.01

n4@dz57 nC4+C2+C1 0. 75182 0. 0002 98 1.01762 480. 84372 478.78

n4@dz46 nC4+C2+C1 0.19478 0.000122 1.01731 122.8237 122.33

n4@dz46 nC4+C2+C1 0. 19478 0. 0001 22 1.01743 125. 28597 124.77

n4@dz46 nC4+C2+C1 0. 19478 0. 0001 22 1.01755 126. 14828 125.62

n4@dz55 nC4+C2+C1 0. 47936 0. 0002 12 1.01712 302. 20477 301.05

n4@dz55 nC4+C2+C1 0. 47936 0. 0002 12 1.01707 302. 22177 301.09

n4@dz55 nC4+C2+C1 0. 47936 0. 0002 12 1.01699 305. 42725 304.30

Expanded Uncertainty on X (%)

C1 3.15E-04 C2 6.11E-05


1 643.730 3.65 176.29 +

14

2.145

U(X)=C1*X+C2

Linear regression Area = A1*[n-C4](%)

Degree of freedom

t coefficient for 95 % confidence level

Ramified Chain Results


Determination of

Calorific Value

ISO 6975

Calibration results

- Fitting model and U fitting- U ( Area)- U (gravimetric mixtures) Natural Gas

Chromatographmeasurement

Normalisation at 100 %

Constants- Calorific value of the components- Compressibility factor Z n- Universal constant : 8314.51 J/kmole/K

Results

Ramified chain


18/21

TERM 12/4/2007



Units Value Uncertainty (2k)

kJ/Nm3 36454 29

kJ/Nm3 32914 29

- 0.9029 0.0005

kg/kmol 18.71 0.02

Nm3/kmol 22.36 0.01

- 0.9976 0.0005

- 0.6473 0.0006

kg/Nm3 0.8369 0.0008

kJ/Nm3 45311 32

Nm3/Nm3

kg/Nm3

Nm3/Nm3

Physical propertiesGross calorific value -Volumetric basis GCV

Low calorific value -Volumetric basis LCV

LCV/GVC

Molar Mass

Molar Volume

Compression factor

Relative density

REMARKS : Combustion at 25 °C - Metering at 0 °C and 101325 Pa

CO2 Total combustion

Density

Wobbe index

Stoichiometric air-to-gas requirement

H2O combustion

Determination of the

Calorific ValueISO 6975

Results


Redundancy in the informationLagrange Multipliers

General methodology

• Suppose n independent observations of a set of quantities

* *

1 ,..., n x x

( ) ( )

* *

1

,...,n

U x U x

⇒ Independent quantities

⇒ Uncertainties on the quantities

• Suppose q constraints (relations) which link these quantities

⇒ Estimated quantities1 ,..., n x x

⇒ Form of constraints ( ) ( )1 0,..., 0i q iC x C x= =

⇒ Number of degrees of freedom n qν = −


19/21

TERM 12/4/2007



• Least squares methodology

⇒ Function to minimize :

⇒ Equations :

1,..., qλ λ q Lagrange multipliers :

⇒ Number of unknowns : n xi et q k

0i

F

x

∂=

∂

( ) ( )( )

2* *

( )

1 1 1

,

k

q qnk i k j i j

k k i jk j k C

C x C x xF C x

wl

π

= = =

È ˘Ê ˆ -Í ˙= +Á ˜ Í ˙Á ˜ Ë ¯ Í ˙Î ˚Â Â Â

n q+

n equations q constraints 0k

F

λ

∂=

∂

⇒ Weighing coefficients ? : ( )*

*

k

j j

j k C j

j x x

C w U x

x=

∂=

∂

Determination of the best-estimate of quantities x j et U(x j )

Principle of the maximum likelihood


Simple example : Flowrate measurement – Metering station

Main :

Reference

Back-up :

Verification

q

⇒ Observed quantities * *1 2,q q

( ) ( )* *1 2,U q U q⇒ Uncertainties on the observed quantities⇒ Best-estimate

1 2,q q

⇒ Constraints1 2 0q q− =

⇒ Function to minimize( ) ( )

( )

2 2* *

1 1 2 21 2* *

1 2

q q q qF q q

U q U qλ

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟= + + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⇒ Results :( ) ( )

( ) ( )

2 2* * * *

1 2 2 1

1 2 2 2* *

1 2

U q q U q qq q q

U q U q

+= = =

+( )

( ) ( ) ( ) ( )

( ) ( )

4 2 4 2* * * *

1 2 2 1

2 2* *

1 2

U q U q U q U qU q

U q U q

+=

+


20/21

TERM 12/4/2007



Illustration of results : Metering station

Best-estimate of the flowrate and uncertainty

49

49.2

49.4

49.6

49.8

50

50.2

50.4

50.6

50.8

51

0 2 4 6 8 10 12 14 16 18 20

Incertitude relative sur Q1 (%)

V a l e u r e s t i m é e d u d é b i t ( k g / s )

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

I n c e

r t i t u d e r e l a t i v e s u r l e d é b i t e s t i m é ( % )

Q1 observed = 49 kg/s

Q2 observed = 51 kg/s

Relative uncertainty on Q2 = 5 %

Constraint : Q1 = Q2 =Q


Case study : Normalisation at 100 % of the molar fractions

*

i x⇒ Observed quantities

( )*iU x⇒ Uncertainty on the observed quantities⇒ Best-estimate

i x

⇒ Constraint 1 0i x − =∑

Functions « least squares » to minimize

( )2

*

1 1

1n n

i i i

i i

F x x xl = =

Ê ˆ = - + -Á ˜ Ë ¯ Â Â

*

* 1

1n

i

ii i

x

x xn

=

-= +

Â

( )2

*

*1 1

11

n n

i i i

i ii

F x x x x

l = =

Ê ˆ = - + -Á ˜ Ë ¯ Â Â

*

*

1

ii n

i

i

x x

x=

=

Â

( )

2*

*1 1

1n n

i ii

i ii

x xF x

U xl

= =

Ê ˆ Ê ˆ -= + -Á ˜ Á ˜ Ë ¯ Ë ¯

Â Â

( )

( )

2* *

1*

2*

1

1n

i i

i

i i n

i

i

U x x

x x

U x

=

=

Ê ˆ -Á ˜ Ë ¯

= +Â

Â

Results


21/21

TERM 12/4/2007


Relative uncertainty on CH4 estimated mole fraction Relative uncerta inties on others co mponents equal to 1 %

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Rel. uncertainty on CH4 measured mole fraction (%)

R e l . U n

c e r t a i n t y o n C H 4 e s t i m a t e d m o l e f r a c t i o n ( % )

CH4 mole fraction deduced from other components (Pure gas model)

Case 1 : unweighted coefficients

Case 2 : weighted coefficients based on mole fractions (ISO-6976)

Case 3 : weighted coefficients based on mole fractions uncertainties

Estimated mole fraction of CH4

Pure gas model : 85.5 %

Case 1 : 85.10 %

Case 2 : 85.43 %Case 3 : ...85.49... %

Case study : Normalisation at 100 % of the molar fractions


References and acknowledgment

“Guide pour l’expression de l’incertitude de mesure “ First edition 1993, corrected and reprinted

1995, ISO 14216

“Assessment of uncertainty in calibration and use of flow measurement devices. Part 1: Linear

calibration relationships - Part 2: Non-linear calibration relationships” ISO 7066, 1988.

N.C. BARFORD “Experimental measurements : precision, error and truth” Second edition, 1987

N. DRAPER, H. SMITH “Applied regression analysis” Third edition, 1998

“Gas analysis – Preparation of calibration gas mixture – Gravimetric method” ISO 6142, 1999.

“Expression of the Uncertainty of Measurement in calibration” EAL – R2 and EAL – R2 –S1,

April 1997.

R. WALPOLE, R. MYERS “Probability and statistics for engineers and scientists” 2d edition, 1989

“Engineering Analysis of Experimental Data” ASHRAE Guideline 2-1986 (RA 96),1996

“Méthode statistiques” Recueil de Normes, ISO 3, 1989

“Natural gas – Extended analysis – Gas-chromatograph method” ISO 6975, 1997

“Natural gas – Calculation of calorific values from composition” ISO 6976, 1995.

“IUPAC Commission on Atomic Weights”.

J.V. NICHOLAS and D.R. WHITE “Traceable temperatures – An introduction to temperature

measurement and calibration” Second Edition, 2001 ISBN 0 471 49291 4

Uncertainty of Measurements JMSeynhaeve

Documents

Transcript of Uncertainty of Measurements JMSeynhaeve