Analysis of Error Factors for Measurement Data and Inverse - - PowerPoint PPT Presentation

1

Analysis of Error Factors for Measurement Data and Inverse Techniques Application to Temperature Measurements and Heat Flux Estimations

Damien DAVID

Centre de Thermique de Lyon INSA de Lyon - CNRS - UCBL

14th June 2011

Tutorial 8

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

2

1

Method 0 : Foreword, Definitions

2

Method 1 : Determination of an Estimator Uncertainty

3

Method 2 : The Monte Carlo Method

4

Application 0 : Presentation of Study Case

5

Application 1 : Temperature Estimation Uncertainties

6

Application 2 : Inverse Technique Uncertainties

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

3

Content of the section

1

Method 0 : Foreword, Definitions Bibliography Definitions Scope of the Study

2

Method 1 : Determination of an Estimator Uncertainty

3

Method 2 : The Monte Carlo Method

4

Application 0 : Presentation of Study Case

5

Application 1 : Temperature Estimation Uncertainties

6

Application 2 : Inverse Technique Uncertainties

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

4

Bibliography

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

4

Bibliography

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

5

Definitions : Statistics

t-Distribution Random Variable

∆( E , s

2 , ν , N )

−5 5 0.1 0.2 0.3 0.4

t p(t)

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

5

Definitions : Statistics

t-Distribution Random Variable

∆( E , s

2 , ν , N )

Mathematical Expectation

−5 5 0.1 0.2 0.3 0.4

t p(t)

∆(−1, 1, 8, N) ∆( 0, 1, 8, N) ∆( 1, 1, 8, N)

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

5

Definitions : Statistics

t-Distribution Random Variable

∆( E , s

2 , ν , N )

Variance

−5 5 0.1 0.2 0.3 0.4

t p(t)

∆(0, 1, 8, N) ∆(0, 2, 8, N) ∆(0, 3, 8, N)

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

5

Definitions : Statistics

t-Distribution Random Variable

∆( E , s

2 , ν , N )

Degree of Freedom

−5 5 0.1 0.2 0.3 0.4

t p(t)

Normal ∆(0, 1, 8, N) ∆(0, 1, 4, N) ∆(0, 1, 2, N)

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

5

Definitions : Statistics

t-Distribution Random Variable

∆( E , s

2 , ν , N )

Number of associated elements

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

5

Definitions : Statistics

t-Distribution Random Variable

∆( E , s

2 , ν , N )

Interval of confidence : ∆

95% = E ± t 0.975 ν

√ s2

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

5

Definitions : Statistics

t-Distribution Random Variable

∆( E , s

2 , ν , N )

Interval of confidence : ∆

95% = E ± t 0.975 ν

√ s2

−5 5 0.1 0.2 0.3 0.4

t p(t)

∆(0,1,ν,N)

95% 2.5% 2.5%

−t0.975

ν

t0.975

ν

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

6

Definitions

Estimator of Ytrue

Measurement Device Data Treatment

(Thermocouple, Balance...) (Averaging)

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

6

Definitions

Estimator of Ytrue

Measurement Device Data Treatment

(Thermocouple, Balance...) (Averaging)

Raw Data Estimation Y

p m

Y m = Nm

p=1 Y p m

Nm

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

6

Definitions

Estimator of Ytrue

Measurement Device Data Treatment

(Thermocouple, Balance...) (Averaging)

Raw Data Estimation Y

p m

Y m = Nm

p=1 Y p m

Nm Raw Data Total Error Estimation Total Error Y

p m = Ytrue + ε p Ym

Y m = Ytrue + εY m

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

6

Definitions

Estimator of Ytrue

Measurement Device Data Treatment

(Thermocouple, Balance...) (Averaging)

Raw Data Estimation Y

p m

Y m = Nm

p=1 Y p m

Nm Raw Data Total Error Estimation Total Error Y

p m = Ytrue + ε p Ym

Y m = Ytrue + εY m ε

p Ym ∈ NY(0, σY)

εY m ∈ NY m(0, σY m)

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

6

Definitions

Estimator of Ytrue

Measurement Device Data Treatment

(Thermocouple, Balance...) (Averaging)

Raw Data Estimation Y

p m

Y m = Nm

p=1 Y p m

Nm Raw Data Total Error Estimation Total Error Y

p m = Ytrue + ε p Ym

Y m = Ytrue + εY m ε

p Ym ∈ NY(0, σY)

εY m ∈ NY m(0, σY m) Raw Data Total Error Random Variable Estimation Total Error Random Variable ∆Y(0, s

2 Y, νY, NY)

∆Y m(0, s

2 Y m, νY m, NY m)

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

7

Definitions

Estimator Uncertainty

Ytrue = Y m ± ∆

95% Y m is true with a level of confidence 95%

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

7

Definitions

Estimator Uncertainty

Ytrue = Y m ± ∆

95% Y m is true with a level of confidence 95%

0.1 0.2 0.3 0.4

ξYm p(ξYm)

∆Ym

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

7

Definitions

Estimator Uncertainty

Ytrue = Y m ± ∆

95% Y m is true with a level of confidence 95%

0.1 0.2 0.3 0.4

ξYm p(ξYm)

∆Ym

95%

−t0.975

νY

• s2

Y

t0.975

νY

• s2

Y

∆

95% Y m = t 0.975 νYm

• s2

Y m

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

8

Definitions

Error Factors fq

Physical Phenomena which deteriorate the quality of the

• measurement. They must be independent !

Fixed Error Factors Fluctuating Error Factor

Temperature, Pollutant Concentra- tion, Air speed, measurement tool... Electromagnetic noise...

f1..fM−1 fM

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

8

Definitions

Error Factors fq

Physical Phenomena which deteriorate the quality of the

• measurement. They must be independent !

Fixed Error Factors Fluctuating Error Factor

Temperature, Pollutant Concentra- tion, Air speed, measurement tool... Electromagnetic noise...

f1..fM−1 fM Raw Data Total Error Estimation Total Error Y

p m = Ytrue + ε p Ym

Y m = Ytrue + εY m Raw Data Associated Errors Estimation Associated Errors Y

p m = Ytrue + ε p Ym,f1 + .. + ε p Ym,fM

Y m = Ytrue + ε

p Y m,f1 + .. + ε p Y m,fM

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

9

Scope

Conservation of ∆Y m

Preliminary Measurments Actual Use Estimator

M0 : Foreword

Bibliography Definitions Scope

M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

9

Scope

Conservation of ∆Y m

Preliminary Measurments Actual Use Estimator

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

10

Content of the section

1

Method 0 : Foreword, Definitions

2

Method 1 : Determination of an Estimator Uncertainty No Error Factor Analysis Error Factor Analysis Extra Calculation

3

Method 2 : The Monte Carlo Method

4

Application 0 : Presentation of Study Case

5

Application 1 : Temperature Estimation Uncertainties

6

Application 2 : Inverse Technique Uncertainties

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

11

No Error Factor Analysis

Raw Data Total Error RV Estimation Total Error RV Raw Data Total Errors Raw Data

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

11

No Error Factor Analysis

Raw Data Total Error RV Estimation Total Error RV Raw Data Total Errors Raw Data

Population of raw data

Measurement of the same true value Ytrue Ytrue not necessarily known. Y

p = Ytrue + ε p Y

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

11

No Error Factor Analysis

Raw Data Total Error RV Estimation Total Error RV Raw Data Total Errors Raw Data

Step 0 : Approximation of the raw data errors

Approximation of ε

p Y

δ

p Y = Y p − Y = ε p Y − ε p Y

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

11

No Error Factor Analysis

Raw Data Total Error RV Estimation Total Error RV Raw Data Total Errors Raw Data

Step 1 : Determination of ∆Y

−0.1 −0.05 0.05 0.1 5 10 15

δ p(δ)

NY = NY νY = NY − 1 s

2 Y =

NY

p=1(δ p Y)2

νY

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

11

No Error Factor Analysis

Raw Data Total Error RV Estimation Total Error RV Raw Data Total Errors Raw Data

Step 3 : Determination of ∆Y m

ε1 + .. + εNm Nm NY m = NY νY m = νY s

2 Y m = s2 Y

Nm Decreasing

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

12

No Error Factor Analysis

Advantages

Quick Method Fluctuation of the Estimation Value

Drawbacks

No Bias No Interpretation of the Error Structure

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Data Populations Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV Raw Associated Data Populations

Population of raw data

fk are blocked by controlling environment Y

1 fM =Ytrue+εY,f1+. . .+εY,fq−1+εY,fq+εY,fq+1+. . .+ε 1 Y,fM

Y

2 fM =Ytrue+εY,f1+. . .+εY,fq−1+εY,fq+εY,fq+1+. . .+ε 2 Y,fM

................... Y

NfM fq =Ytrue+εY,f1+. . .+εY,fq−1+εY,fq+εY,fq+1+. . .+ε NfM Y,fM

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV Raw Associated Data Populations

Population of raw data

fk are blocked by controlling environment Y

1 fM =Ytrue+εY,f1+. . .+εY,fq−1+εY,fq+εY,fq+1+. . .+ε 1 Y,fM

Y

2 fM =Ytrue+εY,f1+. . .+εY,fq−1+εY,fq+εY,fq+1+. . .+ε 2 Y,fM

................... Y

NfM fq =Ytrue+εY,f1+. . .+εY,fq−1+εY,fq+εY,fq+1+. . .+ε NfM Y,fM

Y fM =Ytrue+εY,f1+. . .+εY,fq−1+εY,fq+εY,fq+1+. . .+εY,fM

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Error RV Estimation Associated Error RV Raw Associated Data Populations Raw Associated Error populations

Step 0 : Approximation of the raw associated errors

Approximation of ε

p Y,fM

δ

p Y,fM = Y p fM − Y fM = ε p Y,fM − εY,fM

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Associated Raw Data Populations Estimation Associated Error RV Raw Associated Error populations Raw Associated Error RV

Step 1 : Determination of ∆Y,fM

−0.1 −0.05 0.05 0.1 5 10 15

δ p(δ)

NY,fM = NfM νY,fM = NfM − 1 s

2 Y,fM =

NfM

p=1(δ p Y,fM)2

νY,fM

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Data Populations Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV

Step 2 : Determination of ∆Y,fM

ε1 + .. + εNm Nm NY,fM = NY,fM νY,fM = νY,fM s

2 Y,fM = s2 Y,fM

Nm Decreasing

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV Raw Associated Data Populations

Population of raw data

fM blocked by averaging Y

1 fq =Ytrue+εY,f1+. . .+εY,fq−1+ε 1 Y,fq+εY,fq+1+. . .+ε 1 Y,fM

Y

2 fq =Ytrue+εY,f1+. . .+εY,fq−1+ε 2 Y,fq+εY,fq+1+. . .+ε 2 Y,fM

................... Y

Nfq fq =Ytrue+εY,f1+. . .+εY,fq−1+ε Nfq Y,fq+εY,fq+1+. . .+ε Nfq Y,fM

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV Raw Associated Data Populations

Population of raw data

fM blocked by averaging Y

1 fq =Ytrue+εY,f1+. . .+εY,fq−1+ε 1 Y,fq+εY,fq+1+. . .+ε 1 Y,fM

Y

2 fq =Ytrue+εY,f1+. . .+εY,fq−1+ε 2 Y,fq+εY,fq+1+. . .+ε 2 Y,fM

................... Y

Nfq fq =Ytrue+εY,f1+. . .+εY,fq−1+ε Nfq Y,fq+ εY,fq +. . .+ε Nfq Y,fM

Y fq =Ytrue+εY,f1+. . .+εY,fq−1+εY,fq+εY,fq+1+. . .+εY,fM

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Data Populations Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV

Step 0 : Approximation of the raw associated errors

Approximation of ε

p Y,fq

δ

p Y,fq = Y p fq − Y fq = (ε p Y,fq − εY,fq) + (ε p Y,fM − εY,fM)

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Data Populations Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV

Step 1 : Determination of ∆Y,fq

−0.1 −0.05 0.05 0.1 5 10 15 20

δ p(δ)

NY,fq = Nfq νY,fq = Nfq − 1 s

2 Y,fq =

Nfq

p=1(δ p Y,fq)2

νY,fq −s2

Y,fM

Nfq Compensation

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Data Populations Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV

Step 2 : Determination of ∆Y,fq

ε1 + .. + εNm Nm NY,fq = NY,fq νY,fq = νY,fq s

2 Y,fq =

s2

Y,fq

Nm Fluctuating Error s

2 Y,fq = s 2 Y,fq Bias

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

13

Error Factor Analysis

Estimation Total Error RV Raw Associated Data Populations Raw Associated Error populations Raw Associated Error RV Estimation Associated Error RV

Step 4 : Determination of ∆Y m

εf1 + .. + εfM NY m =

qNY,fq

νY m =

• 

Nf

q=1

 

s2 Y,fq NY,fq

   

2

Nf

q=1

    

s2 Y,fq NY,fq

 

2 1

νY,fq   

• s

2 Y m = qs 2 Y,fq Sum Variances

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

14

Error Factor Analysis

Advantages

Description of the Error Structure Bias Taken Into Account

Drawbacks

Choice of the Predominant Error Factors

M0 : Foreword M1 : Estimator Uncertainty

No Error Factor Analysis Error Factor Analysis Extra Calc

M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

15

Extra Calculation

Intrinsic Average Raw Associated Error RVs

Step 5 : Intrinsic average

∆ Same Error Factor Different Ytrue NY,fk =

iNY,fk ,i

νY,fk =

iνY,fk ,i

s

2 Y,fk =

• i νY,fk ,i × s2

Y,fk ,i

νY,fk ,i

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

16

Content of the section

1

Method 0 : Foreword, Definitions

2

Method 1 : Determination of an Estimator Uncertainty

3

Method 2 : The Monte Carlo Method The Monte Carlo method Generation of Noised Input Populations Error Factor Analysis

4

Application 0 : Presentation of Study Case

5

Application 1 : Temperature Estimation Uncertainties

6

Application 2 : Inverse Technique Uncertainties

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

17

The Monte Carlo Method

Inputs Inverse Technique Outputs

Inverse Technique

NI Estimation Inputs NO Outputs

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

17

The Monte Carlo Method

Populations Inputs Inverse Technique Populations Outputs

The Monte Carlo Method

Noised population for the inputs Noised outputs

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

18

Generation of Noised Input Populations

Noised Input Estimation Total Error Values Estimation Total Error RV

Generation from ∆Y m

−5 5 0.1 0.2 0.3 0.4

t p(t)

∆(0, 1, ν, N)

δ

p = t νYm rand

• s2

Y m

Command trnd on MATLABTM

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

18

Generation of Noised Input Populations

Noised Input Estimation Associated Error Values Estimation Associated Error RV

Generation from ∆Y m

−5 5 0.1 0.2 0.3 0.4

t p(t)

∆(0, 1, ν, N)

δ

p fq = t νY,fq rand

• s2

Y,fq

Command trnd on MATLABTM

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

18

Generation of Noised Input Populations

Noised Input Estimation Associated Error Values Estimation Assiociated Error RV Autocorrelation Matrix

Same estimator Y, different times ti

Dependency between : δ(ti) δ(tj) with j < i

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

18

Generation of Noised Input Populations

Noised Input Estimation Associated Error Values Estimation Assiociated Error RV Autocorrelation Matrix

Same estimator Y, different times ti

−2 2

δf

−2 2

δf

20 40 60 80 100 −2 2

δf time

(δ

i) = mvtrnd(R, ν)

• s2

Fully dependent : Ri,j = 1 Autocorrelated : Ri,j = 0 Independent : Ri,j = 0

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

19

Error Factor Analysis

Populations Inputs Inverse Technique Populations Outputs

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

19

Error Factor Analysis

Populations Inputs Inverse Technique Populations Outputs

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method

The MC method Generation Inputs Error Factor Analysis

A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

19

Error Factor Analysis

Populations Inputs Inverse Technique Populations Outputs

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case

The Inverse Technique

A1 : Temperature Uncertainties A2 : IT Uncertainties

20

Content of the section

1

Method 0 : Foreword, Definitions

2

Method 1 : Determination of an Estimator Uncertainty

3

Method 2 : The Monte Carlo Method

4

Application 0 : Presentation of Study Case The Inverse Technique

5

Application 1 : Temperature Estimation Uncertainties

6

Application 2 : Inverse Technique Uncertainties

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case

The Inverse Technique

A1 : Temperature Uncertainties A2 : IT Uncertainties

21

The Inverse Technique

Adiabatic Conditions

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case

The Inverse Technique

A1 : Temperature Uncertainties A2 : IT Uncertainties

21

The Inverse Technique

Adiabatic Conditions

Model Interpolation

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case

The Inverse Technique

A1 : Temperature Uncertainties A2 : IT Uncertainties

21

The Inverse Technique

500 1000 1500 0.5 1 1.5 10 20 30 40

Y [m] t [min] ¯ Tk,i

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case

The Inverse Technique

A1 : Temperature Uncertainties A2 : IT Uncertainties

21

The Inverse Technique

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case

The Inverse Technique

A1 : Temperature Uncertainties A2 : IT Uncertainties

21

The Inverse Technique

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

22

Content of the section

1

Method 0 : Foreword, Definitions

2

Method 1 : Determination of an Estimator Uncertainty

3

Method 2 : The Monte Carlo Method

4

Application 0 : Presentation of Study Case

5

Application 1 : Temperature Estimation Uncertainties Calibration of the Thermocouples The error factors Results

6

Application 2 : Inverse Technique Uncertainties

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

23

The Calibration of the Thermocouples

Reference probe Thermocouple Reflux Insulated pipe Refrigerated/heated bath Reference probe Thermocouple Reflux Insulated pipe Refrigerated/heated bath

Experimental Setup

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

23

The Calibration of the Thermocouples

Data Acquisition

∆T : Temperature difference between the two junctions VTC : Voltage between the two junctions

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

23

The Calibration of the Thermocouples

Set k Set k+1

Raw data

NT = 13 Sets of acquisition NA = 15 Acquisitions per set

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

23

The Calibration of the Thermocouples

Averaging the data

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

23

The Calibration of the Thermocouples

Calibration curve

Calibration Curve

• btained with [VTC,i, ∆Tref,i]

∆Tc = c2V

2 TC + c1VTC + c0

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

23

The Calibration of the Thermocouples

Noise Error Calibration Curve Error

The error factors

Noise Error Calibration Curve Error

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

24

The Noise Error

Noise Error

Description

Due to the electromagnetic noise Fast fluctuating error

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

24

The Noise Error

Noise Error

Approximation of the noise error

VTC,i,1 =V

i true+ε i V,c+ε i,1 V,n

VTC,i,2 =V

i true+ε i V,c+ε i,2 V,n

................... VTC,i,NA =V

i true+ε i V,c+ε i,NA V,n

V TC,i =V

i true+ε i V,c+ε i V,n

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

24

The Noise Error

Approximation of the noise error

Turning a voltage error into a temperature error δ

p T,n = ∆c(VTC,i,1 − V TC,i,j) = ε i,p T,n − ε i T,n

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

24

The Noise Error

Raw Noise Error RV Estimation Noise Error RV Raw Noise Errors Raw Noise Error RVs

Calculation steps

One noise error random variable per set of acquisition Step 5 : Intrinsic average Step 2’ : Fluctuating Error : the Averaging decreases the Variance

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

25

The Calibration Curve Error

Calibration Curve Error

Description

Discrepancy between the curve and the real behavior of the Thermocouple Bias during a set of measurement

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

25

The Calibration Curve Error

Calibration Curve Error

Approximation of the calibration curve error

∆c(V TC,1) =T

1 true+ε 1 T,c+ε 1 T,n

∆c(V TC,2) =T

2 true+ε 2 T,c+ε 2 T,n

................... ∆c(V TC,NT )=T

NT true+ε NT T,c+ε NT T,n

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

25

The Calibration Curve Error

Approximation of the errors

We suppose T

i true ≈ ∆Tref,i

3 Parameters c1, c2, c3 to estimate the errors : δ

p T,c = ∆Tc(V TC,i) − ∆Tref,i

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

25

The Calibration Curve Error

Raw Calibration Curve Error RV Estimation Calibration Curve Error RV Raw Calibration Curve Errors

Calculation steps

Step 1’ : νT,c = NT − 3, because of 3 parameters Step 1’ : Variance Correction Step 2’ : Bias : the Averaging does not modify the Variance

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

25

The Calibration Curve Error

Calibration Curve 'True' Response

Autocorrelation

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

25

The Calibration Curve Error

Autocorrelation

τT,k = k × 5

• C Temperature delay

Autocorrelation : GT,c(τT,k) = NA

i=1 δi T,cδi−k T,c

NA

i=1(δi T,c)2

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

25

The Calibration Curve Error

−40 −20 20 40 −0.5 0.5 1

τT [oC] G(τT)

Autocorrelation

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

26

Temperature Estimation Uncertainties

1 3 5 7 9 11 13 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

No Thermocouple Standard Deviation [°C]

• s2

T,n

• s2

T,n/Nm

• s2

T,c

Standard Deviations

Nm = 90

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties

Calibration Error Factors Results

A2 : IT Uncertainties

26

Temperature Estimation Uncertainties

1 3 5 7 9 11 13 0.04 0.045 0.05 0.055 0.06 0.065

No Thermocouple ∆95%

¯ T

[oC]

Uncertainties

T = TSF + ∆Tc(VTC) Error on cold junction neglected Average : ∆

95% T m = 0.05

• C

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 27

Content of the section

1

Method 0 : Foreword, Definitions

2

Method 1 : Determination of an Estimator Uncertainty

3

Method 2 : The Monte Carlo Method

4

Application 0 : Presentation of Study Case

5

Application 1 : Temperature Estimation Uncertainties

6

Application 2 : Inverse Technique Uncertainties Generation of the Noised Input Populations Temperature Uncertainty Heat Flux Uncertainty

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 28

Generation of the Noised Input Populations

Inverse Technique

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 28

Generation of the Noised Input Populations

Noised Temperature

T k,i = T k(ti)

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 28

Generation of the Noised Input Populations

Noise Errors

Independent Errors

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 28

Generation of the Noised Input Populations

Calibration Curve Errors

Autocorrelated Errors Ri,j = GT(T(ti) − T(tj))

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 29

Temperature Uncertainty

500 1000 1500 0.5 1 1.5 10 20 30 40

Y [m] IM Mean Temperature t [min]

• T [oC]

500 1000 1500 0.5 1 1.5 0.02 0.04 0.06 0.08

Y [m] IM Temperature Uncertainty t [min] ∆95%

• T

[oC]

Temperature Uncertainty

∆

95%

• T

constant with time

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 29

Temperature Uncertainty

0.5 1 1.5 0.03 0.04 0.05 0.06 0.07 0.08

Y [m] ∆95%

• T

[oC]

Time-Averaged Temperature Uncertainty

Local Maximums at Thermocouple locations

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 30

Heat Flux Uncertainty

500 1000 1500 0.5 1 1.5 −2 2 4 6 8

Y [m] IM Mean Heat Flux t [min]

• φ [W/m2]

500 1000 1500 0.5 1 1.5 0.05 0.1 0.15 0.2

Y [m] Uncertainty t [min] (∆95%

• φ

) [W/m2]

Heat Flux Uncertainty

∆

95%

• φ

proportional to φ

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 30

Heat Flux Uncertainty

400 600 800 1000 0.5 1 1.5 0.05 0.1 0.15 0.2

Y [m] Relative Uncertainty t [min] ∆95%

• φ

/ φ

0.5 1 1.5 0.05 0.06 0.07 0.08 0.09 0.1

Y [m] ∆95%

• φ

/ φ

Relative Heat Flux Uncertainty

∆

95%

• φ /

φ almost constant with time

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 31

Heat Flux Uncertainty : Error Factors

Interpretation

φ ≈ eρcp Ti − Ti−1 ∆t

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 31

Heat Flux Uncertainty : Error Factors

Interpretation

φ ≈ eρcp Ti − Ti−1 ∆t φ + εφ ≈ e(ρ + ερ)(cp + εcp)(Ti + εi

T) − (Ti−1 + εi−1 T )

∆t

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 31

Heat Flux Uncertainty : Error Factors

Interpretation

φ ≈ eρcp Ti − Ti−1 ∆t φ + εφ ≈ e(ρ + ερ)(cp + εcp)(Ti + εi

T) − (Ti−1 + εi−1 T )

∆t φ + εφ ≈ φ + ερ ρ φ + εcp cp φ + (ε

i T − ε i−1 T ) eρcp

∆t

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 31

Heat Flux Uncertainty : Error Factors

Interpretation

φ ≈ eρcp Ti − Ti−1 ∆t φ + εφ ≈ e(ρ + ερ)(cp + εcp)(Ti + εi

T) − (Ti−1 + εi−1 T )

∆t φ + εφ ≈ φ + ερ ρ φ + εcp cp φ + (ε

i T − ε i−1 T ) eρcp

∆t εφ ≈ ερ ρ + εcp cp

• φ + (ε

i T − ε i−1 T )eρcp

∆t

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 32

Heat Flux Uncertainty : Error Factors

500 1000 1500 0.5 1 1.5 −2 2 4 6 8

Y [m] IM Mean Heat Flux t [min]

• φ [W/m2]

500 1000 1500 0.5 1 1.5 0.005 0.01 0.015 0.02

Y [m] Uncertainty associated to ρ t [min] (∆95%

• φ

)ρ [W/m2]

Heat Flux Uncertainty associated to ρ

proportional to φ εφ ≈ ερ ρ φ ∝ φ

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 32

Heat Flux Uncertainty : Error Factors

500 1000 1500 0.5 1 1.5 −2 2 4 6 8

Y [m] IM Mean Heat Flux t [min]

• φ [W/m2]

500 1000 1500 0.5 1 1.5 0.05 0.1 0.15 0.2

Y [m] Uncertainty associated to cp t [min] (∆95%

• φ

)cp [W/m2]

Heat Flux Uncertainty associated to cp

proportional to φ εφ ≈ εcp cp φ ∝ φ

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 32

Heat Flux Uncertainty : Error Factors

500 1000 1500 0.5 1 1.5 −2 2 4 6 8

Y [m] IM Mean Heat Flux t [min]

• φ [W/m2]

500 1000 1500 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5

Y [m] Uncertainty associated to T t [min] (∆95%

• φ

)T [W/m2]

Heat Flux Uncertainty associated to T

proportional to φ εφ ≈ (ε

i T − ε i−1 T )eρcp

∆t ∝ φ

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 33

Heat Flux Uncertainty : Error Factors

200 400 600 800 1000 1200 10 15 20 25 30 35

T t [min]

∆ t2 ∆ t1

1 2 3 4 5 −0.2 0.2 0.4 0.6 0.8 1 1.2

GT,c(τT) τT [oC]

∆ T2 ∆ T1

Orders of Magnitude

∆T1 < ∆T2

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 33

Heat Flux Uncertainty : Error Factors

200 400 600 800 1000 1200 10 15 20 25 30 35

T t [min]

∆ t2 ∆ t1

1 2 3 4 5 −0.2 0.2 0.4 0.6 0.8 1 1.2

GT,c(τT) τT [oC]

∆ T2 ∆ T1

Orders of Magnitude

∆T1 < ∆T2 φ1 < φ2

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 33

Heat Flux Uncertainty : Error Factors

200 400 600 800 1000 1200 10 15 20 25 30 35

T t [min]

∆ t2 ∆ t1

1 2 3 4 5 −0.2 0.2 0.4 0.6 0.8 1 1.2

GT,c(τT) τT [oC]

∆ T2 ∆ T1

Orders of Magnitude

∆T1 < ∆T2 φ1 < φ2 G(∆T1) > G(∆T2) (ε

i T − ε i−1 T )1 < (ε i T − ε i−1 T )2

(εφ)1 < (εφ)2

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 34

Heat Flux Uncertainty : Error Factors

15 20 25 30 −0.1 −0.05 0.05 0.1 0.15

T δc Independent δc,i

p 15 20 25 30 −0.1 −0.05 0.05 0.1 0.15

δc Autocorrelated T

15 20 25 30 −0.1 −0.05 0.05 0.1 0.15

δc Constant T

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 34

Heat Flux Uncertainty : Error Factors

15 20 25 30 −0.1 −0.05 0.05 0.1 0.15

T δc Independent δc,i

p 15 20 25 30 −0.1 −0.05 0.05 0.1 0.15

δc Autocorrelated T

15 20 25 30 −0.1 −0.05 0.05 0.1 0.15

δc Constant T

500 1000 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6

Y [m] δc independent t [min] (∆95%

• φ

) ¯

T 500 1000 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6

Y [m] δc autocorrelated t [min]

500 1000 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6

Y [m] δc constant t [min]

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 35

Heat Flux Uncertainty : Error Factors

500 1000 1500 0.5 1 1.5 −2 2 4 6 8

Y [m] IM Mean Heat Flux t [min]

• φ [W/m2]

500 1000 1500 0.5 1 1.5 1 2 3 x 10

−3

Y [m] Uncertainty associated to k t [min] (∆95%

• φ

)k [W/m2]

Peaks

Heat Flux Uncertainty associated to k

Peaks when φ varies

M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties

NIP Generation Temperature Uncertainty Heat Flux Uncertainty 35

Heat Flux Uncertainty : Error Factors

0.5 1 1.5 0.02 0.04 0.06 0.08 0.1

Y [m] ∆95%

• φ

/ φ

All T cp ρ k

Orders of Magnitude

Error in k negligible Error Mainly due to the Temperature Uncertainties

36

Thanks For Your Attention