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Tutorial 8 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 1 Method

  1. Content of the section Method 0 : Foreword, Definitions M0 : Foreword 1 M1 : Estimator Uncertainty No Error Factor Method 1 : Determination of an Estimator Uncertainty 2 Analysis Error Factor Analysis No Error Factor Analysis Extra Calc Error Factor Analysis M2 : Monte Carlo Method Extra Calculation A0 : Presentation Study Case A1 : Temperature Method 2 : The Monte Carlo Method 3 Uncertainties A2 : IT Uncertainties Application 0 : Presentation of Study Case 4 Application 1 : Temperature Estimation Uncertainties 5 Application 2 : Inverse Technique Uncertainties 6 10

  2. No Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Analysis Error Factor Analysis Extra Calc Raw Data Raw Data Total Raw Data Total Estimation Total M2 : Monte Carlo Errors Error RV Error RV Method A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties 11

  3. No Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Analysis Error Factor Analysis Extra Calc Raw Data Raw Data Total Raw Data Total Estimation Total M2 : Monte Carlo Errors Error RV Error RV Method A0 : Presentation Study Case A1 : Temperature Uncertainties Population of raw data A2 : IT Uncertainties Measurement of the same true value Y true Y true not necessarily known. p = Y true + ε Y p Y 11

  4. No Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Analysis Error Factor Analysis Extra Calc Raw Data Raw Data Total Raw Data Total Estimation Total M2 : Monte Carlo Errors Error RV Error RV Method A0 : Presentation Study Case A1 : Temperature Uncertainties Step 0 : Approximation of the raw data errors A2 : IT Uncertainties Approximation of ε p Y p − Y = ε δ p Y = Y p Y − ε p Y 11

  5. No Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Analysis Error Factor Analysis Extra Calc Raw Data Raw Data Total Raw Data Total Estimation Total M2 : Monte Carlo Errors Error RV Error RV Method A0 : Presentation Study Case A1 : Temperature Uncertainties Step 1 : Determination of ∆ Y A2 : IT Uncertainties N Y = N Y 15 ν Y = N Y − 1 10 � N Y p ( δ ) p = 1 ( δ p Y ) 2 s Y = 2 5 ν Y 0 −0.1 −0.05 0 0.05 0.1 δ 11

  6. No Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Analysis Error Factor Analysis Extra Calc Raw Data Raw Data Total Raw Data Total Estimation Total M2 : Monte Carlo Errors Error RV Error RV Method A0 : Presentation Study Case A1 : Temperature Uncertainties Step 3 : Determination of ∆ Y m A2 : IT Uncertainties N Y m = N Y ε 1 + .. + ε N m ν Y m = ν Y N m Y m = s 2 s 2 Y Decreasing N m 11

  7. No Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Analysis Advantages Error Factor Analysis Extra Calc Quick Method M2 : Monte Carlo Method Fluctuation of the Estimation Value A0 : Presentation Study Case A1 : Temperature Drawbacks Uncertainties A2 : IT No Bias Uncertainties No Interpretation of the Error Structure 12

  8. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties 13

  9. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Population of raw data A2 : IT Uncertainties f k are blocked by controlling environment 1 1 Y f M = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M Y f M = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε 2 2 Y , f M ................... N fM N fM Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M 13

  10. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Population of raw data A2 : IT Uncertainties f k are blocked by controlling environment 1 1 Y f M = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M Y f M = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε 2 2 Y , f M ................... N fM N fM Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M Y f M = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M 13

  11. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Step 0 : Approximation of the raw associated errors A2 : IT Uncertainties Approximation of ε p Y , f M p f M − Y f M = ε p p Y , f M − ε Y , f M δ Y , f M = Y 13

  12. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Associated Raw Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Step 1 : Determination of ∆ Y , f M A2 : IT Uncertainties N Y , f M = N f M 15 ν Y , f M = N f M − 1 10 � N fM p ( δ ) p = 1 ( δ p Y , f M ) 2 s 2 Y , f M = 5 ν Y , f M 0 −0.1 −0.05 0 0.05 0.1 δ 13

  13. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Step 2 : Determination of ∆ Y , f M A2 : IT Uncertainties N Y , f M = N Y , f M ε 1 + .. + ε N m ν Y , f M = ν Y , f M N m Y , f M = s 2 Y , f M s 2 Decreasing N m 13

  14. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Population of raw data A2 : IT Uncertainties f M blocked by averaging 1 1 1 Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε 2 Y , f q + ε Y , f q + 1 + . . . + ε 2 2 Y , f M ................... N fq N fq N fq Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M 13

  15. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Population of raw data A2 : IT Uncertainties f M blocked by averaging 1 1 1 Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε 2 2 Y , f q + ε Y , f q + 1 + . . . + ε 2 Y , f M ................... N fq N fq N fq Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + . . . + ε Y , f M Y f q = Y true + ε Y , f 1 + . . . + ε Y , f q − 1 + ε Y , f q + ε Y , f q + 1 + . . . + ε Y , f M 13

  16. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Step 0 : Approximation of the raw associated errors A2 : IT Uncertainties Approximation of ε p Y , f q δ p Y , f q = Y f q − Y f q = ( ε p Y , f q − ε Y , f q ) + ( ε p Y , f M − ε Y , f M ) p 13

  17. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Step 1 : Determination of ∆ Y , f q A2 : IT Uncertainties N Y , f q = N f q 20 ν Y , f q = N f q − 1 15 � N fq p ( δ ) 10 p p = 1 ( δ Y , f q ) 2 − s 2 Y , f M 2 s Y , f q = ν Y , f q N f q 5 Compensation 0 −0.1 −0.05 0 0.05 0.1 δ 13

  18. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Step 2 : Determination of ∆ Y , f q A2 : IT Uncertainties N Y , f q = N Y , f q ε 1 + .. + ε N m ν Y , f q = ν Y , f q N m s 2 Y , f q s 2 Y , f q = Fluctuating Error N m 2 2 s Y , f q = s Y , f q Bias 13

  19. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Estimation Total Analysis Error RV Error Factor Analysis Extra Calc M2 : Monte Carlo Method Raw Associated Raw Associated Raw Associated Estimation Associated A0 : Presentation Data Populations Error populations Error RV Error RV Study Case A1 : Temperature Uncertainties Step 4 : Determination of ∆ Y m A2 : IT Uncertainties N Y m = � q N Y , f q � � �     � 2 s 2 � � ε f 1 + .. + ε f M � Nf Y , fq    � q = 1 � NY , fq ν Y m = � �     s 2 2 � � � Nf  Y , fq  �   1 �   q = 1 NY , fq ν Y , fq Y m = � s 2 q s 2 Y , f q Sum Variances 13

  20. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty No Error Factor Analysis Error Factor Analysis Advantages Extra Calc M2 : Monte Carlo Description of the Error Structure Method A0 : Presentation Bias Taken Into Account Study Case A1 : Temperature Uncertainties Drawbacks A2 : IT Uncertainties Choice of the Predominant Error Factors 14

  21. Extra Calculation M0 : Foreword M1 : Estimator Uncertainty No Error Factor Intrinsic Average Analysis Error Factor Analysis Extra Calc M2 : Monte Carlo Method A0 : Presentation Raw Associated Study Case Error RVs A1 : Temperature Uncertainties Step 5 : Intrinsic average A2 : IT Uncertainties N Y , f k = � i N Y , f k , i ∆ ν Y , f k = � i ν Y , f k , i � i ν Y , f k , i × s 2 Same Error Factor Y , f k , i 2 � ν Y , f k , i s Y , f k = Different Y true 15

  22. Content of the section Method 0 : Foreword, Definitions M0 : Foreword 1 M1 : Estimator Uncertainty Method 1 : Determination of an Estimator Uncertainty M2 : Monte Carlo 2 Method The MC method Generation Inputs Method 2 : The Monte Carlo Method 3 Error Factor Analysis A0 : Presentation The Monte Carlo method Study Case Generation of Noised Input Populations A1 : Temperature Uncertainties Error Factor Analysis A2 : IT Uncertainties Application 0 : Presentation of Study Case 4 Application 1 : Temperature Estimation Uncertainties 5 Application 2 : Inverse Technique Uncertainties 6 16

  23. The Monte Carlo Method M0 : Foreword M1 : Estimator Uncertainty Inverse Technique M2 : Monte Carlo Method The MC method Generation Inputs Error Factor Analysis A0 : Presentation Study Case Inputs Outputs A1 : Temperature Uncertainties Inverse Technique A2 : IT Uncertainties N I Estimation Inputs N O Outputs 17

  24. The Monte Carlo Method M0 : Foreword M1 : Estimator Uncertainty Inverse Technique M2 : Monte Carlo Method The MC method Generation Inputs Error Factor Analysis A0 : Presentation Study Case Populations Inputs Populations Outputs A1 : Temperature Uncertainties The Monte Carlo Method A2 : IT Uncertainties Noised population for the inputs Noised outputs 17

  25. Generation of Noised Input Populations M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method The MC method Estimation Total Estimation Total Noised Input Generation Inputs Error RV Error Values Error Factor Analysis A0 : Presentation Study Case A1 : Temperature Uncertainties Generation from ∆ Y m A2 : IT Uncertainties 0.4 ∆ ( 0 , 1 , ν, N ) � 0.3 p = t ν Ym δ s 2 p ( t ) rand Y m 0.2 0.1 Command trnd on 0 MATLAB TM −5 0 5 t 18

  26. Generation of Noised Input Populations M0 : Foreword M1 : Estimator Uncertainty Noised Input M2 : Monte Carlo Method The MC method Generation Inputs Error Factor Analysis Estimation Associated Estimation Associated A0 : Presentation Error RV Error Values Study Case A1 : Temperature Uncertainties Generation from ∆ Y m A2 : IT Uncertainties 0.4 ∆ ( 0 , 1 , ν, N ) � 0.3 ν Y , fq δ p f q = t s 2 p ( t ) rand Y , f q 0.2 0.1 Command trnd on 0 MATLAB TM −5 0 5 t 18

  27. Generation of Noised Input Populations M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Noised Input Estimation Assiociated Estimation Associated Method Error RV Error Values The MC method Generation Inputs Error Factor Analysis A0 : Presentation Autocorrelation Matrix Study Case A1 : Temperature Uncertainties Same estimator Y , different times t i A2 : IT Uncertainties Dependency between : δ ( t i ) δ ( t j ) with j < i 18

  28. Generation of Noised Input Populations M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Noised Input Estimation Assiociated Estimation Associated Method Error RV Error Values The MC method Generation Inputs Error Factor Analysis A0 : Presentation Autocorrelation Matrix Study Case A1 : Temperature Uncertainties Same estimator Y , different times t i A2 : IT Uncertainties � i ) = mvtrnd ( R , ν ) ( δ s 2 2 δ f 0 −2 Fully dependent : R i , j = 1 2 δ f 0 Autocorrelated : R i , j � = 0 −2 Independent : R i , j = 0 2 δ f 0 −2 0 20 40 60 80 100 time 18

  29. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty Inverse Technique M2 : Monte Carlo Method The MC method Generation Inputs Error Factor Analysis A0 : Presentation Study Case Populations Inputs Populations Outputs A1 : Temperature Uncertainties A2 : IT Uncertainties 19

  30. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty Inverse Technique M2 : Monte Carlo Method The MC method Generation Inputs Error Factor Analysis A0 : Presentation Study Case Populations Inputs Populations Outputs A1 : Temperature Uncertainties A2 : IT Uncertainties 19

  31. Error Factor Analysis M0 : Foreword M1 : Estimator Uncertainty Inverse Technique M2 : Monte Carlo Method The MC method Generation Inputs Error Factor Analysis A0 : Presentation Study Case Populations Inputs Populations Outputs A1 : Temperature Uncertainties A2 : IT Uncertainties 19

  32. Content of the section M0 : Foreword Method 0 : Foreword, Definitions 1 M1 : Estimator Uncertainty M2 : Monte Carlo Method 1 : Determination of an Estimator Uncertainty Method 2 A0 : Presentation Study Case The Inverse Technique Method 2 : The Monte Carlo Method 3 A1 : Temperature Uncertainties A2 : IT Application 0 : Presentation of Study Case 4 Uncertainties The Inverse Technique Application 1 : Temperature Estimation Uncertainties 5 Application 2 : Inverse Technique Uncertainties 6 20

  33. 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

  34. 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 Interpolation Model 21

  35. The Inverse Technique M0 : Foreword M1 : Estimator 40 Uncertainty M2 : Monte Carlo Method 30 T k , i A0 : Presentation ¯ Study Case 20 The Inverse Technique A1 : Temperature 10 Uncertainties 0 1.5 A2 : IT 500 Uncertainties 1 1000 0.5 1500 Y [m] 0 t [min] 21

  36. 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

  37. 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

  38. Content of the section Method 0 : Foreword, Definitions M0 : Foreword 1 M1 : Estimator Uncertainty Method 1 : Determination of an Estimator Uncertainty M2 : Monte Carlo 2 Method A0 : Presentation Study Case Method 2 : The Monte Carlo Method 3 A1 : Temperature Uncertainties Calibration Application 0 : Presentation of Study Case 4 Error Factors Results A2 : IT Uncertainties Application 1 : Temperature Estimation Uncertainties 5 Calibration of the Thermocouples The error factors Results Application 2 : Inverse Technique Uncertainties 6 22

  39. The Calibration of the Thermocouples Reference probe Reference probe Thermocouple Thermocouple M0 : Foreword M1 : Estimator Uncertainty Reflux Reflux M2 : Monte Carlo Method A0 : Presentation Study Case Insulated pipe Insulated pipe A1 : Temperature Uncertainties Calibration Error Factors Refrigerated/heated bath Refrigerated/heated bath Results A2 : IT Uncertainties Experimental Setup 23

  40. The Calibration of the Thermocouples M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Data Acquisition ∆ T : Temperature difference between the two junctions V TC : Voltage between the two junctions 23

  41. The Calibration of the Thermocouples Set k+1 M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Set k Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Raw data N T = 13 Sets of acquisition N A = 15 Acquisitions per set 23

  42. The Calibration of the Thermocouples M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Averaging the data 23

  43. The Calibration of the Thermocouples Calibration curve M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Calibration Curve obtained with [ V TC , i , ∆ T ref , i ] 2 ∆ T c = c 2 V TC + c 1 V TC + c 0 23

  44. The Calibration of the Thermocouples M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method Calibration Curve A0 : Presentation Study Case Error Noise A1 : Temperature Uncertainties Error Calibration Error Factors Results A2 : IT Uncertainties The error factors Noise Error Calibration Curve Error 23

  45. The Noise Error M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case Noise A1 : Temperature Uncertainties Error Calibration Error Factors Results A2 : IT Uncertainties Description Due to the electromagnetic noise Fast fluctuating error 24

  46. The Noise Error M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case Noise A1 : Temperature Uncertainties Error Calibration Error Factors Results A2 : IT Uncertainties Approximation of the noise error V TC , i , 1 = V true + ε i V , c + ε i i , 1 V , n i i i , 2 V TC , i , 2 = V true + ε V , c + ε V , n ................... i i i , N A V TC , i , N A = V true + ε V , c + ε V , n i i i V TC , i = V true + ε V , c + ε V , n 24

  47. The Noise Error M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Approximation of the noise error Turning a voltage error into a temperature error δ T , n = ∆ c ( V TC , i , 1 − V TC , i , j ) = ε p i , p T , n − ε i T , n 24

  48. The Noise Error M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method Estimation Noise Raw Noise Error RV A0 : Presentation Error RV Study Case A1 : Temperature Uncertainties Calibration Raw Noise Errors Raw Noise Error RVs Error Factors Results A2 : IT Uncertainties Calculation steps One noise error random variable per set of acquisition Step 5 : Intrinsic average Step 2’ : Fluctuating Error : the Averaging decreases the Variance 24

  49. The Calibration Curve Error M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Calibration Method Curve A0 : Presentation Error Study Case A1 : Temperature Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Description Discrepancy between the curve and the real behavior of the Thermocouple Bias during a set of measurement 25

  50. The Calibration Curve Error M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Calibration Method Curve A0 : Presentation Error Study Case A1 : Temperature Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Approximation of the calibration curve error ∆ c ( V TC , 1 ) = T true + ε 1 1 T , c + ε 1 T , n ∆ c ( V TC , 2 ) = T true + ε 2 T , c + ε 2 2 T , n ................... N T N T N T ∆ c ( V TC , NT ) = T true + ε T , c + ε T , n 25

  51. The 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 Approximation of the errors i We suppose T true ≈ ∆ T ref , i 3 Parameters c 1 , c 2 , c 3 to estimate the errors : p T , c = ∆ T c ( V TC , i ) − ∆ T ref , i δ 25

  52. The Calibration Curve Error M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case Raw Calibration Raw Calibration Estimation Calibration Curve Errors Curve Error RV Curve Error RV A1 : Temperature Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Calculation steps Step 1’ : ν T , c = N T − 3, because of 3 parameters Step 1’ : Variance Correction Step 2’ : Bias : the Averaging does not modify the Variance 25

  53. The Calibration Curve Error Calibration Curve 'True' Response M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method A0 : Presentation Study Case A1 : Temperature Uncertainties Calibration Error Factors Results A2 : IT Uncertainties Autocorrelation 25

  54. The 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 Autocorrelation τ T , k = k × 5 ◦ C Temperature delay Autocorrelation : � N A i = 1 δ i T , c δ i − k T , c G T , c ( τ T , k ) = � N A i = 1 ( δ i T , c ) 2 25

  55. The Calibration Curve Error 1 M0 : Foreword M1 : Estimator 0.5 Uncertainty G ( τ T ) M2 : Monte Carlo Method 0 A0 : Presentation Study Case A1 : Temperature Uncertainties −0.5 Calibration −40 −20 0 20 40 Error Factors τ T [ o C ] Results A2 : IT Uncertainties Autocorrelation 25

  56. Temperature Estimation Uncertainties 0.04 � s 2 M0 : Foreword T , n 0.035 Standard Deviation [°C] � M1 : Estimator s 2 T , n / N m 0.03 Uncertainty � 0.025 M2 : Monte Carlo s 2 T , c Method 0.02 A0 : Presentation 0.015 Study Case A1 : Temperature 0.01 Uncertainties 0.005 Calibration Error Factors 0 Results 1 3 5 7 9 11 13 N o Thermocouple A2 : IT Uncertainties Standard Deviations N m = 90 26

  57. Temperature Estimation Uncertainties 0.065 M0 : Foreword 0.06 M1 : Estimator Uncertainty 0.055 [ o C ] M2 : Monte Carlo Method 0.05 ∆ 95 % A0 : Presentation T ¯ Study Case 0.045 A1 : Temperature Uncertainties 0.04 Calibration Error Factors Results 1 3 5 7 9 11 13 N o Thermocouple A2 : IT Uncertainties Uncertainties T = T SF + ∆ T c ( V TC ) Error on cold junction neglected Average : ∆ 95 % T m = 0 . 05 ◦ C 26

  58. Content of the section Method 0 : Foreword, Definitions M0 : Foreword 1 M1 : Estimator Uncertainty Method 1 : Determination of an Estimator Uncertainty M2 : Monte Carlo 2 Method A0 : Presentation Study Case Method 2 : The Monte Carlo Method 3 A1 : Temperature Uncertainties A2 : IT Application 0 : Presentation of Study Case 4 Uncertainties NIP Generation Temperature Uncertainty Application 1 : Temperature Estimation Uncertainties 5 Heat Flux Uncertainty Application 2 : Inverse Technique Uncertainties 6 Generation of the Noised Input Populations Temperature Uncertainty Heat Flux Uncertainty 27

  59. Generation of the Noised Input Populations M0 : Foreword M1 : Estimator Uncertainty M2 : Monte Carlo Method Inverse Technique A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties NIP Generation Temperature Uncertainty Heat Flux Uncertainty 28

  60. Generation of the Noised Input Populations 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 Noised Temperature T k , i = T k ( t i ) 28

  61. Generation of the Noised Input Populations 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 Noise Errors Independent Errors 28

  62. Generation of the Noised Input Populations 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 Calibration Curve Errors Autocorrelated Errors R i , j = G T ( T ( t i ) − T ( t j )) 28

  63. Temperature Uncertainty IM Mean Temperature IM Temperature Uncertainty M0 : Foreword M1 : Estimator 40 0.08 Uncertainty M2 : Monte Carlo [ o C ] 30 0.06 T [ o C ] Method ∆ 95 % A0 : Presentation 20 0.04 � � T Study Case 10 0.02 A1 : Temperature 0 0 Uncertainties 1.5 1.5 500 500 1 1 A2 : IT 1000 1000 0.5 0.5 Uncertainties 1500 Y [m] 1500 Y [m] t [min] t [min] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Temperature Uncertainty 95 % ∆ constant with time � T 29

  64. Temperature Uncertainty 0.08 M0 : Foreword M1 : Estimator 0.07 Uncertainty M2 : Monte Carlo [ o C ] 0.06 Method ∆ 95 % A0 : Presentation � T 0.05 Study Case A1 : Temperature 0.04 Uncertainties A2 : IT 0.03 Uncertainties 0 0.5 1 1.5 Y [m] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Time-Averaged Temperature Uncertainty Local Maximums at Thermocouple locations 29

  65. Heat Flux Uncertainty Uncertainty IM Mean Heat Flux M0 : Foreword M1 : Estimator 0.2 8 Uncertainty ) [ W / m 2 ] 6 M2 : Monte Carlo 0.15 φ [ W / m 2 ] Method 4 0.1 A0 : Presentation 2 Study Case ( ∆ 95 % � � φ 0.05 0 A1 : Temperature Uncertainties 1.5 1.5 −2 0 1 0 1 0 A2 : IT 500 0.5 500 0.5 1000 1000 Uncertainties Y [m] 1500 Y [m] 1500 t [min] t [min] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Heat Flux Uncertainty proportional to � 95 % ∆ φ � φ 30

  66. Heat Flux Uncertainty Relative Uncertainty 0.1 M0 : Foreword M1 : Estimator 0.09 0.2 Uncertainty M2 : Monte Carlo 0.15 / � φ / � φ 0.08 Method ∆ 95 % 0.1 ∆ 95 % � φ � φ A0 : Presentation 0.07 0.05 Study Case 0 A1 : Temperature 400 0.06 Uncertainties 1.5 600 1 A2 : IT 800 0.5 0.05 Uncertainties 0 0.5 1 1.5 1000 Y [m] t [min] Y [m] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Relative Heat Flux Uncertainty φ / � 95 % ∆ φ almost constant with time � 30

  67. Heat Flux Uncertainty : Error Factors Interpretation M0 : Foreword M1 : Estimator Uncertainty T i − T i − 1 M2 : Monte Carlo φ ≈ e ρ c p Method ∆ t A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT Uncertainties NIP Generation Temperature Uncertainty Heat Flux Uncertainty 31

  68. Heat Flux Uncertainty : Error Factors Interpretation M0 : Foreword M1 : Estimator Uncertainty T i − T i − 1 M2 : Monte Carlo φ ≈ e ρ c p Method ∆ t A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT φ + ε φ ≈ e ( ρ + ε ρ )( c p + ε c p )( T i + ε i T ) − ( T i − 1 + ε i − 1 T ) Uncertainties NIP Generation ∆ t Temperature Uncertainty Heat Flux Uncertainty 31

  69. Heat Flux Uncertainty : Error Factors Interpretation M0 : Foreword M1 : Estimator Uncertainty T i − T i − 1 M2 : Monte Carlo φ ≈ e ρ c p Method ∆ t A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT φ + ε φ ≈ e ( ρ + ε ρ )( c p + ε c p )( T i + ε i T ) − ( T i − 1 + ε i − 1 T ) Uncertainties NIP Generation ∆ t Temperature Uncertainty φ + ε φ ≈ φ + ε ρ ρ φ + ε c p T ) e ρ c p Heat Flux Uncertainty φ + ( ε i T − ε i − 1 c p ∆ t 31

  70. Heat Flux Uncertainty : Error Factors Interpretation M0 : Foreword M1 : Estimator Uncertainty T i − T i − 1 M2 : Monte Carlo φ ≈ e ρ c p Method ∆ t A0 : Presentation Study Case A1 : Temperature Uncertainties A2 : IT φ + ε φ ≈ e ( ρ + ε ρ )( c p + ε c p )( T i + ε i T ) − ( T i − 1 + ε i − 1 T ) Uncertainties NIP Generation ∆ t Temperature Uncertainty φ + ε φ ≈ φ + ε ρ ρ φ + ε c p T ) e ρ c p Heat Flux Uncertainty φ + ( ε T − ε i i − 1 c p ∆ t � ε ρ � ρ + ε c p T ) e ρ c p ε φ ≈ i T − ε i − 1 φ + ( ε c p ∆ t 31

  71. Heat Flux Uncertainty : Error Factors Uncertainty associated to ρ IM Mean Heat Flux M0 : Foreword M1 : Estimator 0.02 8 Uncertainty ) ρ [ W / m 2 ] 6 M2 : Monte Carlo 0.015 φ [ W / m 2 ] Method 4 0.01 A0 : Presentation 2 ( ∆ 95 % Study Case � � φ 0.005 0 A1 : Temperature Uncertainties 1.5 −2 1.5 0 1 0 1 0 A2 : IT 500 0.5 500 0.5 1000 1000 Uncertainties Y [m] 1500 1500 Y [m] t [min] t [min] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Heat Flux Uncertainty associated to ρ proportional to � φ ε φ ≈ ε ρ ρ φ ∝ φ 32

  72. Heat Flux Uncertainty : Error Factors Uncertainty associated to c p IM Mean Heat Flux M0 : Foreword M1 : Estimator 8 Uncertainty 0.2 ) c p [ W / m 2 ] 6 M2 : Monte Carlo 0.15 φ [ W / m 2 ] Method 4 0.1 A0 : Presentation 2 Study Case ( ∆ 95 % � 0.05 � φ 0 A1 : Temperature Uncertainties 1.5 0 1.5 −2 1 0 0 1 500 A2 : IT 500 0.5 0.5 1000 1000 Uncertainties Y [m] 1500 1500 Y [m] t [min] t [min] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Heat Flux Uncertainty associated to c p proportional to � φ ε φ ≈ ε c p φ ∝ φ c p 32

  73. Heat Flux Uncertainty : Error Factors Uncertainty associated to T IM Mean Heat Flux M0 : Foreword M1 : Estimator 8 0.5 Uncertainty ) T [ W / m 2 ] 6 M2 : Monte Carlo 0.4 φ [ W / m 2 ] Method 4 0.3 A0 : Presentation 2 0.2 Study Case ( ∆ 95 % � � φ 0 0.1 A1 : Temperature Uncertainties 1.5 1.5 −2 0 1 1 0 0 A2 : IT 500 0.5 500 0.5 1000 1000 Uncertainties Y [m] Y [m] 1500 1500 t [min] t [min] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Heat Flux Uncertainty associated to T proportional to � φ T ) e ρ c p ε φ ≈ ( ε i T − ε i − 1 ∝ φ ∆ t 32

  74. Heat Flux Uncertainty : Error Factors 35 1.2 M0 : Foreword 1 30 M1 : Estimator 0.8 Uncertainty G T , c ( τ T ) 25 0.6 M2 : Monte Carlo Method T 0.4 20 A0 : Presentation Study Case 0.2 ∆ t 1 ∆ t 2 A1 : Temperature ∆ T 1 ∆ T 2 15 0 Uncertainties −0.2 A2 : IT 10 0 1 2 3 4 5 Uncertainties 0 200 400 600 800 1000 1200 τ T [ o C ] t [min] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Orders of Magnitude ∆ T 1 < ∆ T 2 33

  75. Heat Flux Uncertainty : Error Factors 35 1.2 M0 : Foreword 1 30 M1 : Estimator 0.8 Uncertainty G T , c ( τ T ) 25 0.6 M2 : Monte Carlo Method T 0.4 20 A0 : Presentation Study Case 0.2 ∆ t 1 ∆ t 2 A1 : Temperature ∆ T 1 ∆ T 2 15 0 Uncertainties −0.2 A2 : IT 10 0 1 2 3 4 5 Uncertainties 0 200 400 600 800 1000 1200 τ T [ o C ] t [min] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Orders of Magnitude ∆ T 1 < ∆ T 2 φ 1 < φ 2 33

  76. Heat Flux Uncertainty : Error Factors 35 1.2 M0 : Foreword 1 30 M1 : Estimator 0.8 Uncertainty G T , c ( τ T ) 25 0.6 M2 : Monte Carlo Method T 0.4 20 A0 : Presentation Study Case 0.2 ∆ t 1 ∆ t 2 A1 : Temperature ∆ T 1 ∆ T 2 15 0 Uncertainties −0.2 A2 : IT 10 0 1 2 3 4 5 Uncertainties 0 200 400 600 800 1000 1200 τ T [ o C ] t [min] NIP Generation Temperature Uncertainty Heat Flux Uncertainty Orders of Magnitude ∆ T 1 < ∆ T 2 G (∆ T 1 ) > G (∆ T 2 ) φ 1 < φ 2 ( ε T − ε i i − 1 T ) 1 < ( ε i T − ε i − 1 T ) 2 ( ε φ ) 1 < ( ε φ ) 2 33

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