Author... Department of Nuclear Science and Engineering August 11, 2016

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1 Methodology for characterization of representativeness uncertainty in performance indicator measurements of thermal and nuclear power plants by Uuganbayar Otgonbaatar Submitted to the Department of Nuclear Science and Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology September 2016 c Massachusetts Institute of Technology All rights reserved. Author Department of Nuclear Science and Engineering August 11, 2016 Certified by Neil.E.Todreas, Sc.D. KEPCO Professor of Nuclear Science and Engineering, and Professor of Mechanical Engineering (Emeritus) Thesis Supervisor Certified by Emilio Baglietto, Ph.D. Norman C. Rasmussen Associate Professor of Nuclear Science and Engineering Thesis Supervisor Certified by Michael Golay, Ph.D. Professor of Nuclear Science and Engineering Thesis Reader Accepted by Ju Li, Ph.D. Battelle Energy Alliance Professor of Nuclear Science and Engineering Professor of Materials Science and Engineering Chairman, Department Committee on Graduate Theses

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3 Methodology for characterization of representativeness uncertainty in performance indicator measurements of thermal and nuclear power plants by Uuganbayar Otgonbaatar Submitted to the Department of Nuclear Science and Engineering on August 11, 2016, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis, a general Methodology framework to characterize, assess and quantify the representativeness uncertainty in performance indicator measurements in thermal and nuclear plants is presented. The representativeness uncertainty arises from the inherent heterogeneity or the variability of the quantity being measured or from the inadequacy of the physical models used to simulate the measurement. The main objective of the Methodology is to gain a deeper physical understanding of the Representativeness uncertainty of the measurement by using numerical simulation tools such as Computational Fluid Dynamics (CFD) and to quantify various sources of representativeness uncertainty. First, the components of the Methodology are expressed using the normal probability distribution for the uncertainty sources. Second, a non-parametric formulation of the Methodology framework is developed and demonstrated. The use of non-parametric techniques allows the quantification and integration of uncertainties that are not expressed by the normal probability distribution. The Methodology is developed based on the analysis of four industrial Case Studies involving uncertainties in performance indicator measurements to structure the analysis. They are: Mass flow rate measurement by an orifice plate (Case Study 1), Steam Generator recirculation ratio measurement using chemical tracers (Case Study 2), The simulation of cooling tower deformation using a Photomodeler (Case Study 3) and the NO x emission measurement from a Combined Cycle Gas Turbine (Case Study 4). In Case Study 1, the non-parametric bootstrap method was used to quantify sampling, iterative and discretization uncertainties thus demonstrating its applicability to CFD uncertainty analysis. In Case Studies 2,3 and 4, the parametric formulation of the Methodology is used to structure the technical analysis. Thesis Supervisor: Neil.E.Todreas, Sc.D. Title: KEPCO Professor of Nuclear Science and Engineering, and Professor of Mechanical Engineering (Emeritus) Thesis Supervisor: Emilio Baglietto, Ph.D. Title: Norman C. Rasmussen Associate Professor of Nuclear Science and Engineering 3

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5 Acknowledgments I am grateful to Professor Todreas for giving me the opportunity to work on this project, and for his guidance. I learned a lot from Prof.Todreas over the course of my Ph.D. that helped me develop not only academically but also personally. I am also grateful to Professor Baglietto for everything I learned about CFD and his help in my learning process. I had fruitful discussions with Prof.Golay that lead to an important piece of my thesis. For that I am thankful. Dr. Yvan Caffari of EDF has been critical to the success of the project. I thank him for his dedication from the inception to the conclusion of this Ph.D. thesis. I also thank fellow graduates of the Nuclear Science and Engineering department Giancarlo Lenci and Etienne Demarly for their help in various Case Studies. This project is supported by EDF R&D in general, the STEP and SEPTEN departments specifically. I want to thank all those involved in this project. 5

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7 This doctoral thesis has been examined by a Committee of the Department of Nuclear Science and Engineering and Électricité de France as follows: Neil.E.Todreas KEPCO Professor of Nuclear Science and Engineering, and Professor of Mechanical Engineering (Emeritus) Emilio Baglietto Norman C. Rasmussen Associate Professor of Nuclear Science and Engineering Michael Golay Professor of Nuclear Science and Engineering Yvan Caffari Member, Thesis Committee Team Manager of Power Generation Group R&D Center EDF China Holding

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9 Contents 1 Introduction Motivation Contributions of the thesis Literature review Literature review of general uncertainty quantification methodologies Literature review of the uncertainty quantification methodologies in CFD Terminology Review of terminologies of uncertainty Terminology adopted for the proposed Methodology Representativeness uncertainty Uncertainty sources Variable Input uncertainty Fixed Input uncertainty Round-off error Iterative uncertainty

10 1.5.5 Discretization uncertainty Sampling uncertainty Systematic model sensitivities Validation uncertainty Outline of the thesis Methodology Formulation using normal distribution Characterization of an existing measurement Simulation model construction Quantification of Random modeling uncertainties Updating the Measurement Value, M C Testing the quality of the simulation Formulation using non-parametric methods Implementation of the Methodology Defining the Reference True Value Defining Representativeness uncertainty Defining two categories of Representativeness uncertainty (Temporal and Spatial) Simulation model building Calculating Random modeling uncertainty Assessing the systematic Validation uncertainty Creating combined Methodology schematic

11 2.3.8 Creating a workflow chart Case Study 1: Mass flow measurement by means of an orifice plate Introduction Flow rate measurement by means of an orifice plate Temporal Representativeness uncertainty Analysis of raw test data LESimulation of SG feedwater line Spatial Representativeness uncertainty Experimental tests RANS simulation Uncertainties of the RANS CFD simulation Factors introducing systematic sensitivity/uncertainty Random modeling uncertainties Application of the Methodology framework to Temporal Representativeness uncertainty Defining the Reference True Value Defining Temporal Representativeness uncertainty Defining two categories of Representativeness uncertainty (Temporal and Spatial) Simulation model building and Calculating Random modeling uncertainty Assessing the systematic Validation uncertainty Creating combined Methodology schematic

12 3.6.7 Creating a workflow chart Application of the Methodology framework to Spatial Representativeness uncertainty Defining the Reference True Value Defining Spatial Representativeness uncertainty Defining two categories of Representativeness uncertainty (Temporal and Spatial) Simulation model building and Calculating Random modeling uncertainty Assessing the systematic Validation uncertainty Creating combined Methodology schematic Creating a workflow chart Combining Spatial and Temporal Representativeness Application of the Methodology using the non-parametric bootstrap method to Spatial Representativeness problem Simulation model building and Calculating Random modeling uncertainty Assessing the systematic Validation uncertainty Creating combined Methodology schematic Conclusion Case Studies 2,3 and Case Study 2: Steam Generator recirculation ratio measurement by means of radioactive or chemical tracer Introduction and main findings

13 4.1.2 Application of the Methodology framework Case Study 3: Study of cooling tower deformation using a Photomodeler Introduction and main findings Application of the Methodology framework Case Study 4: Representativeness uncertainty in measurement of NO x emission from a Combined Cycle Gas Turbine Introduction and main findings Application of the Methodology framework Summary and Conclusion Conclusions and Future work Introduction Case Studies performed The Methodology developed Summary of contributions Lessons Learned in executing this work Difficulty encountered in assuming uncertainty sources are normally distributed Solutions to the analysis of the non-normal distributions offered by the bootstrap approach Assumptions and limitations of the bootstrap approach Justifications of the tools used in the uncertainty analysis of Case Studies 2,3 and Future Work

14 A Case Study 2: Steam Generator recirculation ratio measurement by means of chemical tracer 193 A.1 Introduction A.2 Description of the flow in a Steam Generator A.3 Experimental tests A.3.1 The tracer method A.3.2 The experimental results A.3.3 Estimation of the Measurement uncertainty of the experiment 200 A.4 CFD simulation A.4.1 The Model A.4.2 Grid convergence and the discretization uncertainty A.4.3 Validation of the simulation using the experiment A.4.4 Random modeling uncertainty estimation A.4.5 Optimization of tracer measurement using CFD A.5 Conclusion B Case Study 3: Study of cooling tower deformation using the Photomodeler 231 B.1 Introduction B.2 Laser measurement data B.3 The Photomodeler B.4 The simulation of the cooling tower using the Photomodeler B.4.1 Random input uncertainty propagation through the Photomodeler

15 B.4.2 Validation uncertainty of the Photomodeler B.5 Conclusion C Case Study 4: Representativeness uncertainty in measurement of the NO x emission from a Combined Cycle Gas Turbine 253 C.1 Introduction C.2 Experimental data, and Representativeness uncertainty C.3 Theory of natural gas combustion C.4 Statistical modeling of relationship between AMS parameters using the stepwise regression C.5 Development of a Statistical predictive model for the NO x (VERITAS) 267 C.5.1 Linear regression using all explanatory variables C.5.2 Variable selection using the stepwise regression C.6 Conclusion C.A Statistical models for empirical relationship between AMS parameters 272 C.A.1 Simple linear regression C.A.2 Robust linear regression and partial least squares C.A.3 Robust non-linear model with interaction terms C.A.4 Alternating conditional expectation algorithm for the optimal transformation of variables C.A.5 Simple linear regression using the transformed variables C.A.6 Robust linear regression using the transformed variables C.A.7 Comparison of models

16 D CFD methods 287 D.A Theoretical description of RANS simulation D.A.1 Reynolds averaging D.A.2 Standard k-ε model D.A.3 Quadratic k- ε model D.A.4 Cubic k- ε model D.B Theoretical description of LES (Large Eddy Simulation) D.B.1 Filtering operation D.B.2 The filtered N-S equations D.B.3 WALE model E Random input uncertainty propagation techniques in CFD 295 E.A Markov Chain Monte Carlo (MCMC) method E.B Sensitivity coefficient method E.C Latin hypercube method E.D Polynomial Chaos Expansion method E.E Smolyak quadrature rule E.F Stochastic collocation method E.G Comparison of spectral uncertainty propagation techniques F Discretization uncertainty estimation methods 309 F.A Richardson extrapolation F.B Eça and Hoekstra methodology

17 G Statistical methods employed in the methodology 313 G.A Quantification of sampling uncertainty G.B Non-parametric bootstrap resampling method for estimating sampling uncertainty G.C Bootstrap method for comparing two populations G.D Permutation Test G.E Adding random variables G.F Gaussian process regression (Kriging)

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19 List of Figures 1-1 Overview of the Case Studies used in the development of the Methodology framework Performance indicators considered in this thesis Schematic of the uncertainty terms used in the Methodology Validation experiment to validate a computational prediction Schematic for the True Value, Reference True Value, and Measurement Value The schematic for the Measurement, Simulated Measurement and the Reference True Value for both experiment and simulation A schematic of the Methodology framework to quantify Representativeness uncertainty using a CFD simulation. The shape of random uncertainty distribution in each value, in general, does not have to be Gaussian. Gaussian is chosen for illustration purposes Illustration of two cases of the corrected Measurement Value M c. Figure A shows M c sufficiently separated from M and will pass the test for the null hypothesis H 0. Figure B shows a corrected Measurement Value that is not sufficiently separated from the original Measurement Value M due to large modeling and/or Validation uncertainty The Reference True Value is defined in this step

20 2-6 The Representativeness uncertainty is defined in terms of the difference between the Reference True Value and the Measurement Value The Representativeness uncertainty can be spatial or temporal or both can be present simultaneously A simulation model of the measurement is built in this step Random modeling uncertainties are quantified and combined in this step. Simulated Representativeness uncertainty is the difference between M S and TS R Systematic Validation uncertainty is calculated in this step by subtracting the Measurement Value (M) from the Simulated Measurement Value (M S ) In this step, the information from measurement and simulation are combined in a Methodology schematic with all associated random and systematic uncertainties to achieve the specific objective of the Case Study Global workflow of the methodology SG feedwater lines of an operating nuclear plant PWR Schematic of flow rate measurement by orifice plate Test data at 100, 80, 50% power levels Illustration of change in signal shape at a lower sampling frequency. Blue line represent high-frequency test data whereas red and green lines represent hypothetical signal shapes resulting from lower sampling rates with a phase shift on starting point Average signal value with deviation due to starting point phase shift as a function of the sampling frequency

21 3-6 RANS simulation result of integral turbulent length scale given by Equation 3.3 for the orifice plate geometry at 100% power level A cross section of the LES mesh. The base size mesh is used in 2 hydraulic diameters in both upstream and downstream directions and extruded additionally for 3 hydraulic diameters An instantaneous velocity magnitude imposed at the inlet boundary condition. A fully developed flow profile and Synthetic Eddy Method is used to generate the required mass flow rate and sustained turbulence respectively LES calculated velocity magnitude field. The black arrow shows the location of the downstream pressure tap LES result of pressure field at 100% power. The black arrow shows the location of downstream pressure. The red region upstream of the orifice is indicative of minimal friction pressure drop The point probes upstream and downstream the orifice correspond to the locations of the pressure taps Discharge coefficient C calculated from LES at 100% power level is compared with the ISO standard. The time-averaged, LES calculated C coefficient is less than 0.1% higher than the ISO standard prescribed value. A simulation time step = 2.5e 4 second is chosen to keep the Convective Courant number of the simulation close to Discharge coefficient C calculated from LES at 80% power level is compared with the ISO standard. The time-averaged, LES calculated C coefficient is less than 0.1% higher than the ISO standard prescribed value. A simulation time step = 2.5e 4 second is chosen to keep the Convective Courant number of the simulation close to

22 3-14 Discharge coefficient C calculated from LES at 50% power level is compared with the ISO standard. The time-averaged, LES calculated C coefficient is within 0.1% of the ISO prescribed value of the discharge coefficient. A simulation time step = 2.5e 4 second is chosen to keep the Convective Courant number of the simulation close to 1. The artifact around 6 second is due to an unexpected break in the simulation Fluctuation of test pressure signal and LES result at 100% power level Second order polynomial fit to standard deviation in wall pressure fluctuation as a function of flow rate The convergence of standard deviations of the test signal and the LES at the three power levels. This plot is produced as a supplement to Figure 3-16, and does not indicate a computation-related convergence PSD of test and LES simulated pressure drop data. The test signal is the raw signal with no filtering. Vertical axis is non-dimensionalized using Equation 3.7 for comparison. The peaks at frequencies 50Hz(2n+1), n=0,1,2... are indicative of a measurement related noise (possibly due to antenna effect) and should be ignored. Another possible source of the peaks is turbines and pumps in the secondary circuit The effect of impulse line length on the PSD spectrum of LES calculated pressure drop signal. The different colored lines correspond to various impulse line lengths expressed in meters (m) Experimental configuration of XYZ028KD geometry Experimental result expressed in terms of the discharge coefficient along with ISO. Error bars of both the standard and experiment are shown. Systematic downward shift of the experimental result from the ISO standard shows the systematic uncertainty due to non-straight upstream piping geometry in XYZ028KD flow configuration

23 3-22 Computation mesh showing the orifice region (note regional mesh refinement) and perpendicular to flow direction Locations of points with velocity monitors for convergence determination A) Velocity as a function of time at 5 monitors, flow rate 5, for the unsteady simulation using the cubic k-ε turbulence model. The vertical axis shows velocity magnitude in (m/s). B) Convergence of the pressure drop at flow rate 5, for the unsteady simulation using the cubic k-ε turbulence model Results of steady RANS simulation expressed in terms of % deviation from the discharge coefficient prescribed by the ISO standard Angular location of pressure tap (left), piping geometry of the experimental test (right) Dependence of CFD calculated discharge coefficient on angular location of pressure taps. Gaussian Process (GP) regression was used to interpolate the function between discrete computational mesh points. Details on the implementation of GP regression are given in Appendix G Sensitivity to different constitutive relations in k- ε model Cross sections of unstructured and the block structured meshes Sensitivity to different meshing topologies Input uncertainty calculation using a combination of sensitivity coefficient method and bootstrap sampling The objective function S given in Appendix F in Equation F.7 plotted as a function of the order of convergence p. For each value of p, S is minimized with respect to α and φ

24 3-33 Grid convergence and least square fit according to methodology of Hoekstra et al. Discharge coefficient C is calculated for 7 different meshes at flow rate 1. The finest 5 mesh results were selected for fitting, as coarser mesh results do not exhibit monotonic behavior Two examples of discretization uncertainty confidence interval. Histogram A) was constructed assuming the normal distribution for Hoekstra method, histogram B) was constructed by resampling from the data given in Figure Result for the pressure drop across an orifice plate of a URANS simulation using the cubic k-ε turbulence model. The histogram represents the distribution of time-dependent simulation result given in red. The green line is the time-average of the simulation result Histogram representing the distribution of the time-average value constructed using bootstrap resampling procedure with replacement Convergence plot of a steady-state RANS simulation using a linear k- ε turbulence model. The histogram represents the distribution of time-dependent simulation result given in red. The green line is the time-average of the simulation result The distribution of the iteration-averaged value constructed using bootstrap resampling procedure with replacement Combined discretization and sampling uncertainties. The histogram in A) shows the resulting distribution assuming the normal distribution for discretization. B) shows the bootstrap resampled result The Reference True Value of the orifice plate measurement method is defined using the ISO standard

25 3-41 Representativeness uncertainty arises from the difference between the Reference True Value and the Measurement Value given by the orifice method In Case Study 1a, only Temporal component of Representativeness uncertainty is considered The LES simulation result has a significant sampling uncertainty Systematic shift exists between test Measurement Value and Simulated Measurement Value All uncertainties considered are comprehensively combined in this figure A) Summary of the application of the Methodology schematic. B) Workflow in Case Study 1a. Steps 3a and 3b represent the same analysis The Reference True Value of the orifice plate measurement method is defined in terms of the velocity field internal to the pipe Representativeness uncertainty arises from the difference between the Reference True Value and Measurement Value given by the orifice method In Case Study 1b, only Spatial component of Representativeness uncertainty is considered CFD model of orifice plate method has significant discretization uncertainty Systematic shift exists between experimental Measurement Value and Simulated Measurement Value All uncertainties considered are comprehensively combined in this figure A) Summary of the application of the Methodology schematic. B) Workflow in Case Study 1b. Steps 3a and 3b represent the same analysis

26 3-54 CFD model of orifice plate method in XYZ028KD configuration. Input, discretization and sampling uncertainties are each quantified using the boostrap method in Section Systematic shift exists between experimental Measurement Value and Simulated Measurement Value Combined Methodology schematic using non-parametric bootstrap method Histogram of the local recirculation ratio, 6cm base size mesh result. The measurement results in the hot side of the SG are illustrated in red and in the cold side are illustrated in blue The Reference True Value of the tracer measurement method is defined in terms of the feedwater and recirculation flow rates Representativeness uncertainty arises from the difference between the True Value and Measurement Value given by the tracer method In Case Study 2, only the Spatial component of Representativeness uncertainty is considered CFD model of tracer method has significant iterative and discretization random uncertainties Systematic modeling uncertainty exists between the Measurement Value and the Simulated Measurement Value All uncertainties considered are comprehensively combined in this figure A) Summary of the application of the Methodology schematic. B) Workflow in Case Study 2. The steps 3a and 3b are parts of the same analysis The Reference True Value of the PM simulation of a cooling tower is defined as the instantaneous value of a coordinate of a target on a cooling tower

27 4-10 Representativeness uncertainty arises from the difference between the Reference True Value and Measurement Value given by the laser method In Case Study 3, both Spatial and Temporal Representativeness uncertainties are considered The Simulated Measurement Value of the height of a target predicted by the PM. The Random modeling uncertainty is shown as an error bar The systematic modeling uncertainty is the difference between the Measurement Value and the Simulated Measurement Value All uncertainties considered in Case Study 3 are plotted. The green bar takes into account the model and the input uncertainties. Visibly the lines are the same length, but the calculations are conducted according to Equation A) Summary of the application of the Methodology schematic. B) Work-flow in Case Study 3. The steps 2a and 2b belong to the same analysis and 3a, 3b and 3c are parts of a separate analysis/ The Reference True Value of the concentration of NO x in the exhaust stream is an average over flow rate Representativeness uncertainty arises from the difference between the Reference True Value and Measurement Value In Case Study 4, only the Spatial component of Representativeness uncertainty is considered The statistical predictive model has sampling and input uncertainty components Systematic modeling uncertainty exists between the Reference True Value and the model prediction

28 4-21 In the last step, all uncertainties are collected together to present a complete picture A) Summary of the application of the Methodology schematic. B) Workflow in Case Study Schematic of the Methodology framework A-1 A schematic of the coolant flow inside a SG. The chemical tracer mixing is given by different colors. The red arrows indicate the feedwater flow, and the blue arrows indicate the recirculation flow. The purple arrows indicate the mixed feedwater and recirculation flows in the downcomer. 195 A-2 The tracer distribution in the secondary circuit of a PWR. Different colors correspond to different tracer concentrations A-3 The bootstrap distribution of r based on the experimentally obtained values. The horizontal axis is re-scaled due to the sensitivity of the data A-4 CFD simulation geometry. The outlet boundary is indicated by orange and the inlet boundary is indicated by red. The blue boundary indicates a symmetric boundary condition. A) General overview of the geometry B) Feedwater ring region, C) Tubesheet region A-5 Different mesh sizes used in the simulation. A) 2cm mesh, B) 6cm mesh C) 12cm mesh A-6 Temperature-dependent water properties implemented in the CFD model209 A-7 Tube simplification. Left: original tube size. Right: simplified tube size 212 A-8 Wall Y+ distribution for the 4cm Base Size Mesh A-9 The 6cm base-size mesh temperature as a function of the number of iterations at 3 locations

29 A-10 The grid convergence and the least square fit according to the methodology of Hoekstra et al. The passive scalar is calculated for 6 different meshes at a probe location in the tube bundle region A-11 The results of the experimental test conducted on a Model #1 Steam Generator. The theoretical value of the SG recirculation ratio is supplied by the vendor. The vertical axis is re-scaled due to the sensitivity of the data A-12 The results of the CFD simulation on an approximate geometry A-13 A histogram of the passive scalar at h=0.68m, for 6cm and 8cm base size meshes A-14 A histogram of the local recirculation ratio using a 6cm base size mesh result A-15 A histogram of the local recirculation ratio using a 8cm base size mesh result A-16 A histogram of the means of 1000 randomly resampled populations. 223 A-17 The effect of varying the number of probes used to sample the chemical tracer concentration in the periphery of the downcomer A-18 The local recirculation ratio as a function of the angular coordinate at different heights and radial coordinates A-19 The boxplots of the local recirculation ratio at different heights. Red + indicate points considered outliers. The central blue box represents the central 50% of the data with the red line indicating the median of the data. Two vertical lines extending from the central box indicate the remaining data outside the central box that are not regarded as outliers A-20 The sensitivity of the recirculation ratio measurement result to a small angular rotation of the probes

30 B-1 The scope of Case Study 3. A laser-based system is used to measure the coordinates of targets on the cooling tower surface. A Photomodeler (PM) is used to reconstruct a 3D image of a cooling tower based on photos of the cooling tower and a certain number of bundle adjustment points 1. The PM output is analyzed using statistical methods B-2 Laser measurement system B-3 A photograph of a cooling tower used as an input to the PM. The targets on the surface of the cooling tower are used either as bundle adjustment points or as validation points B-4 3D visualization of the target point coordinates as measured by the laser device which includes the bundle adjustment set and the validation set. The axes are removed due to the sensitivity of the data B-5 3D data can be obtained from multiple photographs of the same object. 237 B-6 Physical measurements need to be fed into the PM to bring the resulting 3D model into absolute units B-7 A 3D reconstruction of the cooling tower using the PM. The blue dots represent the simulated target points. The red dots represent the bundle adjustment set used to construct the PM model of the cooling tower. The axes are removed due to the sensitivity of the data B-8 The input uncertainty of the PM model in the XY plane calculated by the MCMC method B-9 Validation set of laser measurements given in blue along with PM predictions of the same targets in red. The axes are removed due to the sensitivity of the data

31 B-10 A) Z as a function of the polar angle, B) Z as a function of Z, C) Z as a function of the radial coordinate, D) Added variable plot for Z using stepwise regression, E) Normal probability plot of Z before and after correction using regression, F) Histograms of Z before and after correction using regression B-11 A) X as a function of the polar angle, B) X as a function of Z, C) X as a function of the radial coordinate, D) Added variable plot for X using stepwise regression, E) Normal probability plot of X before and after correction using regression, F) Histograms of X before and after correction using the regression B-12 A) Y as a function of the polar angle, B) Y as a function of Z, C) Y as a function of the radial coordinate, D) Added variable plot for Y using stepwise regression, E) Normal probability plot of Y before and after correction using regression, F) Histograms of Y before and after correction using the regression C-1 The Representativeness uncertainty in the NO x emission rate from a CCGT in AMS, and VERITAS audit data C-2 Matrix plot of 6 operational parameters in AMS. Grey-background subplots show significantly correlated parameters. The axes of the matrix plots are re-scaled due to the sensitivity of the data C-3 A comparison between the AMS and the VERITAS NO x values C-4 A typical arrangement of a combined cycle Steam and gas turbine(a). T-S diagram (B) [60] C-5 Control volume around a gas turbine

32 C-6 Mass fractions of difference gas species in an exhaust stream, and the combustion Temperature as a function of the inlet fuel equivalence ratio φ. The blue line at φ = 0.93 shows an example of how the mass fractions of different exhaust gases and the combustion Temperature are calculated for a given equivalence ratio. The exhaust gas mass fraction is directly proportional to the flow average concentration of the NO x emission C-7 Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot of residuals C-8 Selected diagnostic plots of the stepwise regression model. A) Model fit plotted with the data, B) model prediction and NO x (AMS) benchmarked with NO x (VERITAS), C)The Cook s distance for each model prediction point (data points above the horizontal line are considered outliers), D) The normal probability plot of residuals C-9 Selected diagnostic plots of the stepwise regression model. A) The model fit plotted with the data, B) The model prediction and NO x (AMS) benchmarked with NO x (VERITAS), C)The Cook s distance for each model prediction point (data points above the horizontal line are considered outliers), D) The normal probability plot of residuals C-10 Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot C-11 Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot of residuals

33 C-12 Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot C-13 Optimal transformation functions found by ACE for the NO x vs Turbine Feed gas flow rate C-14 Optimal transformation functions found by ACE for the NO x vs Turbine Temperature C-15 Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot C-16 Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot C-17 Comparison of selected fits to the NO x data produced using different regression models. A) Simple linear regression, B) Robust linear regression with interaction terms, C) Simple linear regression with the transformed variables, D) Robust linear regression with the interaction terms

34 D-1 One dimensional energy spectrum (solid line), and filtered spectrum (dashed line) for a certain flow condition. The horizontal axis is the non-dimensional wave number associated with Fourier decomposition of the turbulent energy at a point. The filtered spectrum is generated by a filter of size = 1/6L 11. L 11 is one dimensional integral turbulent length scale. K c denotes the wave number associated with filter width and is equal to K c = π/. The filtered spectrum covers 92% of the turbulent energies in the 1D case, and 80% in the full 3D solution[38]. The remaining portion of the spectrum is modeled by sub-grid scale methods E-1 Example of Latin hypercube sampling to generate a sample of size n S = 5 from U, V with U uniform on [0, 10] and V triangular on [0, 10].298 E-2 Gauss-Hermite quadrature nodes for N=2 random input variables that are uncorrelated and given by normal probability distributions. The level parameter l is the same as nq defined in Equation E.11. Note that random input parameters are normalized by their standard deviations. The contour lines shown correspond to the underlying 2 dimensional joint Gaussian probability distribution. Quadrature notes were constructed using the MATLAB function provided in MIT class taught in Fall2015 by Prof. Marzouk E-4 Construction of Smolyak sparse grid. A) shows the lower order grids used (in gray) for constructing the Smolyak sparse grid in B) for l=3. C) shows the lower order grids used (in gray) for constructing the Smolyak sparse grid in D) for l=4. The quadrature notes were constructed using the MATLAB function provided in MIT class taught in the Fall of 2015 by Prof. Marzouk

35 G-1 An illustrated example of an application of Gaussian process regression. The green sinusoidal curve was used to generate a sample of blue data points, which include an added Gaussian noise. The red curve is the mean of Gaussian process predictive distribution [73]

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37 List of Tables 1.1 Comparison of CFD uncertainty quantification methodologies and uncertainty elements considered by different organizations and authors Summary of uncertainty terminology from different authors Terminology used in the Methodology proposed in this thesis The proposed Methodology framework to evaluate Representativeness uncertainties Summary of Temporal Representativeness systematic uncertainty due to low sampling rate LES mesh characteristics Summary of experimental dimensions of XYZ028KD geometry Simulation setup used for RANS simulations Characteristics of the mesh used for the RANS simulation Mass flow rates used for experiment and simulation Sensitivity to turbulence models using the structured mesh General Methodology terminology adopted for the specific quantities in Case Study

38 4.2 General Methodology terminology adopted for the specific quantities in Case Study General Methodology terms adopted for the specific quantities of Case Study A.1 Summary of the experimental data A.2 Summary of the uncertainties of C R and C. The standard deviation assuming the normal probability distribution is expressed as a percentage of the mean A.3 Summary of the number of cells of the meshes used for grid convergence study A.4 Summary of initial conditions used A.5 The boundary conditions that represent a realistic operating condition based on the open literature and the thermal-hydraulic data shared by EDF A.6 The boundary conditions in the U-tube heat transfer region B.1 Summary of the available data B.2 Model coefficients of linear regression with the selected variables for the Z coordinate B.3 Model summary statistics of simple linear regression with selected variables for the Validation uncertainty in the Z coordinate B.4 Model coefficients of linear regression with the selected variables for the X coordinate B.5 Model summary statistics of simple linear regression with selected variables for the Validation uncertainty in the X coordinate

39 B.6 Model coefficients of linear regression with the selected variables for the Y coordinate B.7 Model summary statistics of simple linear regression with selected variables for the Validation uncertainty in the Y coordinate C.1 Availability of AMS and VERITAS data C.2 Abbreviated notation of AMS parameters C.3 The variable selection order using the stepwise regression C.4 The coefficients of the stepwise regression model C.5 Summary statistics of the model using the stepwise regression C.6 The results of linear regression estimate of regression coefficients along with the t-statistics, and the p-values C.7 Linear regression model summary statistics C.8 The variable selection process for NO x (VERITAS) C.9 The results of the stepwise regression estimate of the coefficients along with the t-statistics, and the p-values C.10 The summary statistics of the stepwise regression model C.11 The model coefficients of the simple linear regression C.12 Summary statistics of the simple linear regression C.13 The model coefficients of the robust linear regression C.14 The model summary statistics of the robust linear regression C.15 The model coefficients of the robust regression with interaction terms 277 C.16 The model coefficients of the robust regression with interaction terms

40 C.17 The model coefficients of a simple linear regression with the transformed variables C.18 The model summary statistics of simple linear regression with the transformed variables C.19 The model coefficients of robust regression with the transformed variables282 C.20 The model summary statistics of robust linear regression with the transformed variables C.21 Comparison of the statistical models developed D.1 Coefficients of the quadratic k-ε model D.2 Coefficients of the cubic k-ε model D.3 Selected filter functions in the coordinate and k space E.1 Hermite polynomials up to order E.2 Summary of spectral uncertainty propagation techniques G.1 Common probability distributions functions and their moment generating functions

41 Chapter 1 Introduction 1.1 Motivation The uncertainty in thermal power measurement affects the power level at which a Pressurized Water Reactor is allowed to operate. Reducing the uncertainty margin enables the plant operation at higher power, without compromising safety and reliability. An increase in power output translates into an increased revenue stream and improved economic competitiveness for the plant. The uncertainties in the measurements of the operating parameters of PWRs present untapped potential for increased power output, and are therefore of great interest. Since the early 1990s, utilities in the U.S. have applied for Measurement Uncertainty Recapture (MUR) uprates and have gained between an additional 1% and 2% in the rated thermal power level 2 for PWRs providing a motivation for understanding and managing uncertainty better. Significant advances have been made in the recent decades in numerical modeling of complex systems. For example, computational fluid dynamics (CFD) is used extensively in many industries, including the aerospace and automotive. In the design and operation of nuclear reactors, CFD application is still limited by the incomplete inte

42 $ 90# gration in the overall analysis process and by the complexity related to quantifying its uncertainty. The Methodology presented in this thesis demonstrates the use of CFD and other models. The scope covered is the operation of nuclear and thermal plants. This Methodology was developed in part to facilitate the adoption of CFD by the utility. However, the developed Methodology framework is applicable to other engineering problems that involve both experimental data and computational simulation and there is a need to link the findings of the both. Simulation tools other than CFD can be used depending on the application. Also, new types of uncertainties can be integrated into the framework. This work is part of an industrial project at Électricité de France (EDF), which aims at characterizing and assessing uncertainty due to Representativeness in performance indicators of thermal power plants. Representativeness is defined in this thesis as the difference between a Measurement Value of a performance indicator and its Reference True Value 3. It arises from the inherent heterogeneity or variability of the quantity being measured or from the inadequacy of the physical model used to simulate the measurement process. This thesis presents the development of the general Methodology framework and illustrates its use in 4 different Case Studies illustrated in Figure 1-1. Case study 1 Case study 2 Case study 3 Case study 4 Orifice plate mass flow rate measurement Steam generator Recirculation ratio measurement Uncertainty in cooling tower simulation with Photomodeler Uncertainty in NOx emission from a CCGT Reference True Value (T R ) Reference True Value (T R ) Reference True Value (T R ) Reference True Value (T R ) Simulated Reference True Value (T R S) Measurement Value (M) 4 3 Simulated Reference True Value (T R S ) Measurement Value (M) 3 Measurement Value (M) Simulated Reference True Value (T R S) Measurement Value (M) 1* 2* 3* 4* Simulated Measurement Value (M S) 1 2 Simulated Measurement Value (M S ) 1 2 Simulated Measurement Value (M S ) 1* 2* 3* Simulated Measurement Value (M S) Measurement Simulation Measurement Simulation Measurement* Simula'on* Measurement* Simula'on* Figure 1-1: Overview of the Case Studies used in the development of the Methodology framework 3 An approximate definition for the True Value of the quantity being measured 42

43 Each Case Study is given in a separate chapter and affects a specific measurement chain at a PWR or a Combined Cycle Gas Turbine (CCGT) operation as shown in Figure 1-2. CONTAINMENT BUILDING REACTOR COOLANT SYSTEM SG moisture carryover rate Cooling tower deformation P Z R S/G HP MSR LP ELECTRIC GENERATOR MAIN TURBINE COOLING TOWER MAIN CONDENSER RHR PUMP RHR HX CORE RCP MAIN FEED PUMP FW HTR CONDENSATE PUMP CIRC. WATER PUMP AUXILIARY BUILDING CONTAINMENT SUMP TURBINE BUILDING SG feedwater flow rate NO x emission rate from CCGT Figure 1-2: Performance indicators considered in this thesis 1.2 Contributions of the thesis In this thesis, a Methodology that achieves three high-level tasks is proposed. The first task is to integrate both experimental and simulation insights of a relevant physical process to measure a performance indicator that is as close to its True Value as possible. Second, it aims to identify, quantify, and combine relevant random and systematic uncertainties of the simulation and experimental data, further identifying the relationship between the uncertainties of the simulation model and experiment in a systematic way. The last objective is to provide a road-map to identify and quantify Representativeness uncertainty in measurement processes of power generation systems. 43

44 In achieving these goals, the following specific contributions are made A general Methodology framework is developed to characterize Representativeness uncertainty using experimental data and simulation tools. It allows integration of the insight from simulation to be added to the physical measurement process by assessing relevant uncertainties. The concepts are expressed assuming the normal probability distribution for the uncertainty sources. The Methodology is applied to 4 industrial Case Studies involving uncertainties in performance indicator measurements to structure the analysis. A non-parametric formulation of the Methodology framework is developed and demonstrated. The use of non-parametric techniques allows quantification and integration of uncertainties that are not expressed by normal probability distribution. In Case Study 1, the bootstrap method was used to quantify sampling, iterative and discretization uncertainties thus demonstrating its applicability to CFD uncertainty analysis. 1.3 Literature review Literature review of general uncertainty quantification methodologies A number of general uncertainty quantification methodologies have been produced by EDF, and are available for use in modeling and simulation of industrial processes. Two of the 4 Case Studies on which the current Methodology is based involve statistical modeling and random uncertainty quantification. Parts of the methodology developed by Camy of EDF [1] and the methodology for performance indicator uncertainty quantification by Bailly also of EDF [2] in analyzing such Case Studies are utilized in this thesis. 44

45 The Open Turns methodology developed by EDF 4 is also a comprehensive framework methodology to frame and propagate uncertainty in simulations of industrial processes. It has three steps: specifying the Case Study, quantifying uncertainty sources, and propagating uncertainty through the model. The framework is implemented in a python code available for download. The Dakota code produced by the Sandia National Laboratory in the U.S provides a comprehensive approach to uncertainty quantification in CFD and other simulations 5. The Dakota code provides capabilities in uncertainty propagation including Monte Carlo sampling, reliability methods, and Polynomial Chaos Expansion methods. In addition to uncertainty propagation, it includes tools to perform sensitivity analysis, optimization and calibration. More recently, the UQLab framework developed by Prof. B.Sudret, and Dr. S.Marelli at ETH Zurich provides a comprehensive overview of uncertainty quantification methods 6. The MATLAB based tool posted on their website provides tools for generating random input generation methods such as Monte Carlo, Latin hypercube samples (LHS)and Polynomial chaos expansion, tools to connect input descriptions to the simulation model, sensitivity analysis tools and implementation of Gaussian process regression. The tools posted on the website are included in our Methodology and described in Appendix E and G. Spectral methods such as Polynomial Chaos Expansion and Stochastic Collocation method are adopted in this thesis to propagate input variable uncertainty through a simulation model. The book by Omar Knio and O.P.Le Maître [3] provides a comprehensive overview of various models. Uncertainty quantification methods are an active area of research and the groups of Marzouk 7 and Iaccarino 8 are two examples of cutting edge research

46 In the 2nd Case Study related to the moisture carryover rate measurement in a steam generator, the bootstrap method is applied for sampling random uncertainty quantification. An introduction to the use of the non-parametric bootstrap method is also presented by Bailly in [2] in addition to the excellent guides provided by Chernick [4], Zoubir [5] and Rapacchi [6]. However, the 1st and 2nd Case Studies employ computational fluid dynamics tools for model building, and the methodologies listed in this section fail to address CFD uncertainty comprehensively. Therefore, a separate literature review presented in Section is dedicated to the approaches to uncertainty quantification in CFD Literature review of the uncertainty quantification methodologies in CFD Two out of 4 Case Studies considered in the development of the Methodology utilize CFD to model the physical measurement process of a performance indicator. Uncertainty quantification in CFD codes alone is an active area of research and many groups and institutions propose different frameworks to approach this problem. In this section, relevant literature on uncertainty quantification in CFD in the context of the Methodology development needs of this thesis is reviewed. It is important to note however that the Methodology proposed in this thesis is not purely based on CFD, but is much broader in that it integrates experiment and simulation with the main objective of addressing Representativeness uncertainty. The following is a review of approaches taken by different organizations in the area of uncertainty quantification in CFD. The frameworks proposed by different organizations and authors are summarized in Table 1.1 along with the present proposed Methodology. A review of terminologies related to Representativeness has been given in Section 1.4. Note that Table 1.1 does not include Representativeness uncertainties which is introduced in Section 1.4. The review provided in this subsection is only on the uncertainty quantification in CFD. 46

47 Representativeness is a type of uncertainty in measurement processes that we aim to address using tools like CFD. EDF proposed two approaches to CFD uncertainty quantification [1]. The approach involves analyzing errors arising from physical chaotic behaviors, discretization error, convergence error, round-off error, error in geometry definition, error arising from uncertain input physical parameters, domain limitation error and the oversight of important physical phenomena. NRG [7] suggests the following approach for use in reactor safety. First, numerical errors including code verification, and solution verification and quantification of discretization error are addressed. Second, input parameters uncertainty is assessed for the CFD model. The use of simplex stochastic collocation method [8] is suggested in this step. Lastly, experimental uncertainties are addressed. A probabilistic method based on propagating uncertain input parameters is proposed by CEA [9]. It treats all model inputs including meshing, numerical schemes selection, physical modeling, physical properties, and initial, boundary conditions as uncertain parameters, and performs one at a time sensitivity study to quantify the overall model uncertainty. The approach taken by Oberkampf and Roy [10] identifies and treats model uncertainties such as round-off error, iterative uncertainty, validation error and extrapolation from the validation domain. The framework provided by Oberkampf et al is one of the most robust and applied frameworks in the nuclear industry. 47

48 Present work Variable Input uncertainty Fixed input uncertainty Discretization uncertainty Iterative uncertainty Round-off error Sampling uncertainty Validation error Turbulence model Meshing topology EDF [1] NRG [7] CEA[9] Areva [11] Oberkampf [10] Uncertain Input parameters Physical Initial and Input uncertainty input physical parameters uncertainty properties boundary conditions, Physical properties Error in geometry definition Discretization Discretization Meshing Meshing Discretization error error uncertainty Convergence Solution verification Numerical Numerical Iterative error schemes schemes uncertainty Round-off Solution verification Numerical Numerical Round-off er- error schemes schemes ror Experimental uncertainty Validation metric Physical Physical modeling modeling Meshing Physical Chaotic behavior Table 1.1: Comparison of CFD uncertainty quantification methodologies and uncertainty elements considered by different organizations and authors 48

49 1.4 Terminology In this section, terminologies from various authors and institutions relating to the Representativeness are reviewed, and the terms used throughout the thesis are defined. Note that the word "Methodology" and the terms given in Table 1.3 are capitalized throughout the thesis to emphasize their fundamental importance in the thesis Review of terminologies of uncertainty In this subsection, a high-level nomenclature for uncertainty is reviewed. Table 1.2 summarizes uncertainty terminologies used by two main authors that were adopted in this Methodology. As shown in the table, there are two main classes of uncertainty in a measurement process: Epistemic and Aleatory. Epistemic or systematic uncertainties arise from lack of knowledge of the physical process or from lack of data. It is possible to reduce epistemic uncertainty with the introduction of more accurate modeling and data. Aleatory or random uncertainties, on the other hand, arise from the inherent randomness or variability of a quantity and cannot be reduced multiple runs of an experiment. The Methodology proposed in this thesis considers the Representativeness uncertainty as described in Section to be epistemic and attempts to identify, quantify and reduce it by using modeling and simulation. In addition, the Methodology provides procedures to quantify the random and validation uncertainties of the adopted models or simulations. 49

50 Authors Current work Meaning Oberkampf [10] DeRocquigny [12] Epistemic Epistemic uncer- Representativeness Uncertainty that uncertainty uncertainty, tainty sys- is reducible from tematic modeling injection of more uncertainty data Aleatory uncer- Aleatory irre- Random modeling Arises due to in- tainty or vari- ducible, intrinsic, uncertainty, ran- herent random- ability variability dom measurement ness or variabil- uncertainty ity of a quantity Table 1.2: Summary of uncertainty terminology from different authors Terminology adopted for the proposed Methodology Table 1.3 shows the terminology used in the Methodology proposed in this thesis. In this table, the nomenclature of the terms are adopted from the International vocabulary of metrology [13] and ASME V&V 20 standard [14]. More detailed definitions of the Representativeness uncertainty, and the systematic modeling uncertainty are given later in this section. Figure 1-3 shows a schematic of the uncertainty terms used in the Methodology. The schematic is explained in detail in Section 2.1. Table 1.4 is presented to serve as a collection of methods to calculate various uncertainties plotted in Figure 1-3. The definitions of terms plotted are given in Table 1.3. The steps of the Methodology are given in Section 2.3 and applied to the industrial/experimental Case Studies in Chapters 3 through C 50

51 Current work 9 JCGM definitions [13] 1 True Value True quantity value, True Value of a quantity 2 Reference True Value 3 Measurement Value 4 Measurement uncertainty 5 Simulated Reference True Value 6 Simulated Measurement Value 7 Representativeness uncertainty 8 Simulated Representativeness uncertainty 9 Validation uncertainty of Reference True Value 10 Validation uncertainty of Measurement Value 11 Random modeling uncertainty Measured quantity value, measured value of a quantity, measured value un- Definitional certainty Notation T Meaning True but unknown value of the Measurand 10 T R An approximate definition of the True Value M Quantity value representing a measurement result with aleatory uncertainty. σ M Random uncertainty representing the accuracy of a measurement chain TS R Reference True Value calculated using simulation M S Measurement Value calculated using simulation M T R Measurement uncertainty resulting from limited detail in the definition of a Measurand (M S - TS R ) 11 Representativeness uncertainty calculated using simulation T R T R S M S M σ MS Random uncertainty representing the accuracy of a simulation of the Measurement Value Table 1.3: Terminology used in the Methodology proposed in this thesis 9 Based on ASME V&V standard[14] 10 The quantity intended to be measured 11 Uncertainty is defined to be the deviation from the Reference True Value 51

52 True Value (T) Reference True Value (T R ) Simulated Reference True Value (T R S) Measurement Value (M) Simulated Measurement Value (M S ) Representativeness uncertainty Measurement uncertainty Validation uncertainty of Reference True Value Simulated Representativeness uncertainty Validation uncertainty of Measurement Value Random modeling uncertainty Measurement Simulation Figure 1-3: Schematic of the uncertainty terms used in the Methodology. Category Sub-categories Methodology Examples Physical representativeness Modeling Random Modeling uncertainties Systematic Modeling sensitivities Input uncertainty Spatial Temporal Variable (physical properties) Fixed (geometry) Discretization uncertainty Estimated from simulation or experiment LES, statistical methods Sensitivity coefficient method, PC expansion Sensitivity study using CFD Richardson extrapolation, ASME, Hoekstra et al C dependence on angular location (Case Study 1) Finite sampling frequency Physical properties Geometrical uncertainty Finite computational cell size Round-off error Oberkampf et al Number approximations Iterative uncertainty Sampling uncertainty Validation error Turbulence model Meshing topology Quantified from model results Statistical methods Validation metrics Oscillatory convergence Sampling Recirculation ratio (Case Study 2) Difference between model and experimental results Constitutive relation in turbulence model Structured, unstructured, 46 hexahedral, tetrahedral Table 1.4: The proposed Methodology framework to evaluate Representativeness uncertainties 52

53 1.4.3 Representativeness uncertainty Representativeness uncertainty is a type of uncertainty that is central to the thesis of this work. In this section, representativeness uncertainty is defined and the origins of representativeness are explained. Representativeness or Representativeness uncertainty arises due to the inherent spatial or temporal variations of the Measurand or inadequacy of a simulation model. It is the difference between the Reference True Value (T R ) of the desired quantity and Measurement Value (M) 12 [10]. The above definition of the Representativeness is consistently used throughout this thesis. In each Case Study, a Representativeness uncertainty present in performance indicator measurement process at a power plant is considered. In this thesis, integration of CFD uncertainty quantification Methodology with the processes of performance indicator measurements to address the Representativeness uncertainty is demonstrated. The overarching objective of studying the Representativeness uncertainty is to improve the economics, efficiency and safety of the power plants under consideration. There are two types of Representativeness uncertainties considered in this thesis: Spatial Representativeness uncertainty arises due to the inherent variations of the Measurand. It is the difference between the intended Reference True Value (T R ) and the Measurement Value (M) due to spatial and geometrical considerations adjusted for temporal Representativeness uncertainty. Examples of spatial Representativeness uncertainty are the non-fully developed nature of secondary coolant entering a flow restriction device. Case Studies 1, 2, 3, 4 each have a form of spatial Representativeness uncertainties. 12 Adopted from Roy and Oberkampf s definition of aleatory uncertainty 53

54 Temporal Representativeness uncertainty arises due to the inherent temporal variations of the Measurand. It is the difference between the continuous time averaged value and the discrete time averaged value at a certain sampling frequency. Temporal Representativeness uncertainty may arise in situations when the quantity to be measured is varying in time, and the temporal resolution of the device measuring the quantity is insufficient to resolve the timescale of the inherent fluctuation. Case Studies 1 and 3 partly involve the temporal Representativeness component. A simulation model used to model the measurement introduces modeling Representativeness uncertainty. In this thesis, this type of uncertainty is divided into two categories: Random modeling uncertainty and systematic Validation uncertainty of the model. Therefore, all model related Representativeness uncertainty sources are treated as either one of these two categories. 1.5 Uncertainty sources Random uncertainty can arise from a number of sources both in physical measurements and numerical simulations. As discussed in Section 1.4.1, random uncertainties can not be reduced by multiple repeats of an experiment or better performance of simulation since they relate to inherent randomness or numerical uncertainties. In this section, various random uncertainty sources are explained, and methods to quantify and combine them are introduced. The discussion of the proposed Methodology framework in Chapter 2 makes use of the definitions given in this section. In each Case Study, a set of criteria is used to judge whether a random uncertainty should be included or neglected as part of the Methodology. The following is a list of guidelines that can be used in doing so. Numerical uncertainties specific to a simulation tool are addressed only for that tool. For example, discretization, iterative and round-off errors are specific to 54

55 CFD models. For other tools such as the Photomodeler, these are not relevant. Quantification of input and sampling uncertainties are subject to availability of the information from experiments. For example, in order to quantify input uncertainty, the conditions of the experiment with a quantified level of uncertainty in physical properties is necessary. Similarly with sampling uncertainty, a sufficient number of experimental data points are required for quantification. An independent judgment has to be made regarding the inclusion of fixed input uncertainty. If the computation domain such as the conduit shape does not change significantly compared to other dominant sources of uncertainty such as turbulence induced fluctuation and the associated sampling error, fixed input can be neglected. Next, within the scope of this thesis, a random uncertainty can either be treated as a normal random variable or non-parametrically using the bootstrap method. The following criteria are used in judging when to adopt one approach over another. If an experimental result is reported only using a confidence interval at a given significance level, there is inherent assumption of normality in the error distribution and the normal distribution has to be adopted in the absence of more insight. In quantifying numerical uncertainties such as iterative and discretization that are described by highly skewed distribution, the use of bootstrap method provides robustness. In calculations that combine non-parametric distributions with normally distributed uncertainty terms, non-parametric methods need to be employed. 55

56 1.5.1 Variable Input uncertainty Input uncertainty arises from uncertainty in input variables to a model. A simulation model is intended to model the physical measurements process of interest. Examples of variable input parameters include density, viscosity and uncertain boundary conditions. It is important to note that the input uncertainty refers not only to input variables to a CFD model but also for uncertain input data for the statistical models given in Case Studies 3 and 5. Four methods to propagate the variable input uncertainty through a model are proposed. While the Markov Chain Monte Carlo (MCMC) method is used for the unbiased, most accurate results, the complexity of large models makes the large number of runs required by the MCMC method computationally too heavy. This created the interest in the following alternatives: Sensitivity coefficient method, Polynomial Chaos method, Latin Hypercube method, and Stochastic Collocation Method. More details on the mechanics of implementing these methods are given in Appendix E Fixed Input uncertainty The fixed input uncertainty arises from the definition of the computational domain or the geometry of the problem. The fixed input uncertainty needs to be treated separately from the variable input uncertainty for the following reasons. First, for each different value of the input geometry, the computational domain has to be re-meshed making it a challenging problem since the computational domain becomes random. Therefore, a rigorous treatment of a random computational boundary requires the application of Karhunen-Loeve decomposition [3]. Several different methods exist in the literature on how to address the geometric uncertainty. As described in Appendix G, the stochastic collocation method based on sparse grid first proposed by Smolyak in 1963 [15] can be used to propagate the parametrized geometric input uncertainty through a CFD model. Compared to the tensorial-expanded chaos collocation method [16], sparse grid enables computational cost saving and 56

57 makes the use of the method feasible in industry. Application of this method requires first parametrizing the input geometry, and secondly minimizing the effect of grid size sensitivity. The first objective can be achieved for example in a pipe geometry by choosing the radius and length of the pipe as parameters. The second objective can be achieved by the deformation approach by the adjoint method as illustrated in [17]. An example of propagating parametrized geometrical uncertainty in a turbo-machineryrelated application is given by Montomoli et al in [18]. The effect of roughness in supersonic flows past a wedge is considered by Karniadakis et al in [19] where the stochastic collocation method is combined with random domain decomposition. The stochastic collocation method applied to geometric uncertainty in the context of robust design to estimate the variance of the design performance can be found in [20]. Given the recent maturity of the approach, it is adopted in the thesis Round-off error According to Roy and Oberkampf [10] Round-off errors occur due to the fact that only a finite number of significant figures can be used to store floating point numbers in a digital computer. They are usually small and can be reduced by increasing the number of significant figures carried in computation Iterative uncertainty The iterative uncertainty arises from an oscillatory convergence of an output quantity of a model. The iterative uncertainty of the solution is defined as the difference between the exact and the iterative solutions of the discretized equations [10]. In treatment of the iterative uncertainty several guidelines are given. According to Oberkampf, the iterative uncertainty should be 100 times smaller than the discretization uncertainty on the finest computational mesh[10]. For most engineering applications, the relative accuracy of 3-4 significant digits is considered sufficient. 57

58 1.5.5 Discretization uncertainty The discretization uncertainty arises from the fact that computations are done using a computational mesh that represents the real geometry of a physical process. Since an infinitely fine and theoretically exact mesh is not achievable, there is a need for quantifying the uncertainty due to the discrete mesh size. Two commonly used approaches for quantifying the discretization uncertainty are that of Richardson [14] and Eça and Hoekstra [21]. These are described in more detail in Chapter F Sampling uncertainty According to the central limit theorem, the mean and standard deviation of the underlying distribution can be estimated precisely with an infinite number of samples [22]. The sampling uncertainty arises due to a finite number of data available to estimate a certain quantity. Sampling uncertainty is statistical in nature and can be reduced by increasing the data used to estimate a quantity. Traditional techniques involve an assumption of normal distribution to estimate the sampling uncertainty [22]. However, they are not suitable in cases when the underlying probability distribution is non-gaussian. Non-parametric techniques such as bootstrap sampling [4] should be used in those cases. Case Study 2 illustrates a use of the bootstrap method and a summary of the principles of the bootstrap method is given in Appendix G Systematic model sensitivities The systematic model sensitivities that introduce systematic change in model prediction as shown in Table 1.4 are examples of systematic sensitivities. They lead to a systematic shift in model prediction as opposed to a random contribution to uncertainty. In the absence of a separate and dedicated validation process, it is impossible to quantify these systematic uncertainties. 58

59 1.5.8 Validation uncertainty The uncertainties described in Section lead to a systematic shift in model prediction as opposed to a random contribution to uncertainty. In the absence of a validation experiment, it is impossible to quantify these systematic uncertainties. Validation experiments are performed to validate the predictions of computational tools such as CFD and are subject to more stringent and comprehensive set of criteria compared to typical experiments [23, 24]. A metric to compare the closeness of the prediction of a computational model to the results of the experiment is therefore needed in estimating the combined systematic uncertainties. Here, the validation metric approach developed by Oberkampf et al [24] is adopted and integrated into the Methodology. The validation error is defined as Ẽ = M M S (1.1) where M is the result of a validation experiment and M S is the prediction of the same quantity using a computational tool. The validation experiment is subject to a well-quantified uncertainty as shown in Figure 1-4. The 90% confidence interval of the validation error is defined as ( ) s Ẽ t 0.05,ν, Ẽ + t s 0.05,ν (1.2) n n where s is the validation experiment sampling standard deviation and n is the number of samples. The term s n represents the standard deviation of the error distribution given by a Gaussian in Figure 1-4. Validation error that is of the same magnitude as the experimental error will correspond to a simulation model that represents the physical reality within the experimental error. If the validation error is large compared to the uncertainty of the validation experiment, the physical model implemented by the code should be improved or calibrated to have higher fidelity simulation of the physical reality. Validation experiments are used to calibrate various physics models such as turbulence and multiphase flow closure relations in CFD. In the context of 59

60 Systematic & Random uncertainty Experiment value (M) Simulated value (M S ) Measurement uncertainty (s) u Validation error u Validation experiment Simulation Figure 1-4: Validation experiment to validate a computational prediction the current Methodology framework, the significance of validation experiment is as follows. Random uncertainties described in Sections i, ii and iii can each be quantified and combined. However, a validation experiment is needed if one wants to quantify the systematic uncertainties of Section It presents a challenge if the Methodology is being used for prediction of a certain measurement process. The need for a validation experiment means that a similar or more precise experiment than the current measurement process needs to have been done before in order to quantify the simulation uncertainty of that process fully. This concludes the discussion of methods employed in the Methodology. 60

61 1.6 Outline of the thesis In this thesis, we proposed a Methodology to characterize, assess and quantify representativeness uncertainty in power plant measurements. The representativeness uncertainty arises from the inherent heterogeneity or variability of the quantity being measured or from the inadequacy of the physical model used to simulate the measurement process. The Methodology aims also to serve as a link between the general CFD community and the engineering divisions at the utilities in order to facilitate the use of CFD for practical applications in the design operation of nuclear/thermal power plants. The Methodology is applied to two Case Studies involving statistical modeling and two Case Studies involving CFD simulation of power plant performance indicator measurements to demonstrate the flexibility and robustness. The rest of the thesis is organized as follows. The terminology used in the Methodology is defined in Chapter 1 based on a review of relevant literature of high-level uncertainty terminology and methodologies published by various organizations. The proposed Methodology is presented next in Chapter 2. Chapter 3 is dedicated entirely to the Case Study 1. Chapter 4 is dedicated to a condensed presentation of Case Studies 2,3 and 4. Chapter 5 is a conclusion of the thesis. Full details of the technical work completed in Case Studies 2,3 and 4 are given in Appendices A, B and C respectively. CFD and Statistical methods employed in this thesis are given Appendices D through G. 61

62 62

63 Chapter 2 Methodology The Methodology presented in this work allows us to achieve greater knowledge about the Representativeness uncertainty by gathering insights from both conducting experiment and performing modeling simulation of the measurement process of interest. In this chapter, the origin of Representativeness uncertainty is introduced and a Methodology framework is proposed to obtain better knowledge about it using a numerical simulation and data analysis. 2.1 Formulation using normal distribution The Methodology developed in this thesis is based on the results of 4 Case Studies each involving a different performance indicator measurement. The Methodology is intended to be applicable to a wide range of measurements. The Representativeness uncertainty derived from the Methodology is specific to the choice of the measurement method. 63

64 2.1.1 Characterization of an existing measurement The Representativeness uncertainty is defined as δ rep = M T R (2.1) where M and T R are the Measurement Value and the Reference True Value of the measurand respectively. The user defines this value by including physics details that the computational tool to be employed is often able to resolve. Here, there is a distinction between the True Value used in the ASME V&V 20 standard on CFD [14] and the Reference True Value defined in this thesis. According to the standard, the True Value is an abstraction and any attempt to define it gives rise to additional uncertainty, which is indicated in Figure 2-1 by the vertical separation between the True Value and the Reference True Value. The Reference True Value differs from the Measurement Value in that it is a more accurate definition of the quantity being measured. The Measurement Value, according to the JCGM terminology [13], is the Quantity value representing a measurement result with sampling uncertainty. The sampling uncertainty arises from the availability of only a finite number of data to estimate a certain quantity as discussed in Appendix G. The existence of a sampling uncertainty makes the Measurement Value M a random variable. It is considered in this Section to be Gaussian and given by M = N ( µ M, σ 2 M ) (2.2) where µ M and σ M are the mean and the standard deviation of the Measurement Value M. Figure 2-1 shows a schematic that illustrates the relationship between True Value, Reference True Value and Measurement Value. 64

65 True Value (T) Reference True Value (T R ) Representativeness uncertainty Measurement Value (M) Experiment Figure 2-1: Schematic for the True Value, Reference True Value, and Measurement Value Simulation model construction In order to obtain insight into the Representativeness uncertainty of a measurement, a simulation model can be used. This step requires an appropriate definition of the problem. For example in CFD, the initial and boundary conditions of the measurement need to be specified. For a verified and validated CFD code, the result of the simulation is expected to be accurate within a code validation error in the corresponding validation domain. The validation of scientific computational tools is discussed by Oberkampf et al [10]. The validation of a code can be quantified using the validation metric approach suggested by these authors. The validation error of a code in a given domain based on a validation experiment in that same domain is given as a Gaussian random variable: δ val N ( µ val = M val.exp M val.sim, σ 2 val ) (2.3) 65

66 where M val.exp is the validation experiment result of the specific measurement process and M S is the simulation result using the validation experiment configurations such as initial and boundary conditions. The terms µ val and σ val are the mean and the standard deviations of the normal distribution that represent the Validation uncertainty. The validation error δ val in Equation 2.3 is a random variable with a mean and variance determined from a validation experiment using a specific set of settings in a code such as the turbulence model and meshing topology. The assumption of normal distribution for the Validation uncertainty given in Equation 2.3 is not generally applicable in all validation experiments. Like numerical uncertainties related to the simulation model such as iterative, discretization uncertainty and input uncertainty, the Validation uncertainty is assumed to be Gaussian 13 in this section for ease of illustrating the calculation of combining random uncertainties. The main effect of relaxing this assumption will be in the calculation of combined probability distributions. The Validation uncertainty enters into the calculation of the corrected Measurement Value in Equation 2.7 and its origin is explained in more detail in Section Quantification of Random modeling uncertainties After the simulation model is built, the Random modeling uncertainties are quantified. In this step, 4 types of Random modeling uncertainties are considered. They are round off error, iterative uncertainty, discretization uncertainty and input uncertainty. The methods to quantify these model random uncertainties are discussed in Appendix G. The sampling uncertainty is also introduced in Appendix G, and is part of the measurement uncertainty illustrated in Figure 2-2. The method to combine the random uncertainties is given by the ASME V&V 20 standard [14] discussed in Appendix G. The Simulated Measurement Value M S is then given by M S N ( µ MS, σ 2 M S = σ 2 it + σ 2 d + σ 2 in ) (2.4) 13 The assumption of normal distribution is relaxed in Section

67 Reference True Value (T R ) Measurement Value (M) Simulated Representativeness uncertainty Simulated Reference True Value (T R S) Simulated Measurement Value (M S ) Experiment Simulation Figure 2-2: The schematic for the Measurement, Simulated Measurement and the Reference True Value for both experiment and simulation. where σ it, σ d, σ in are the standard deviations of random uncertainties of the model introduced due to iterative, discretization and input uncertainty sources. The round off error is not included in Equation 2.4 because it is usually negligible. The variances of the random model uncertainties can be added as given in Equation 2.4 if they are uncorrelated and Gaussian. In general, this assumption is not valid. When the probability distributions of the simultaneous uncertainties are not normal and if they are expressed by continuous distributions, the convolution integral should be employed to calculate the mean and the variance of combined probability distribution [25]. Figure 2-2 shows the schematic of the Measurement and Reference True Value in both experiment and simulation along with the illustrations for the probability distributions. Note that the Simulated Measurement Value M S has uncertainties due to modeling and validation. The Validation uncertainty has also a systematic component µ val which introduces a vertical shift from M. Once the total model random uncertainty is quantified, the Representativeness un- 67

68 certainty can be estimated from the simulation as shown in Figure 2-2 by the vertical distance between the Simulated Measurement Value M S and the Simulated Reference True Value TS R. It is important to note that the Simulated Reference True Value will also have random model uncertainty, and the ability to resolve the Representativeness uncertainty using the simulation depends on the magnitude of model random uncertainties. The total model random uncertainty of the Simulated Reference True Value TS R can be quantified in the same way as the Simulated Measurement Value M S as: T R S N ( µ T R S, σ 2 T R S = σ 2 it + σ 2 d + σ 2 in ) (2.5) Updating the Measurement Value, M C Lastly, the value M obtained by a given measurement method can be updated using the simulation result. In performing this update, as described below, the following cautions need to be observed. First, the CFD simulations need to be validated by validation experimental data in a relevant validation domain as defined by Oberkampf[24]. Second, the model or simulation can be used to "correct" the measurement, only if the measurement is within the same validation domain of the validation experiment. This quantity was introduced to extend the use of the methodology as much as possible. The corrected measurement value M c is not used in the Case Studies analyzed in this Ph.D. project due to the lack of validation experimental data. The corrected Measurement Value M C is calculated as follows. M C N ( µ M + µ T R S µ MS, σ 2 M C = σ 2 M + σ 2 M S + σ 2 T R S ) (2.6) Here, the random uncertainties of M, M S and TS R are assumed to be uncorrelated. Assuming that the measurement process is within the validation domain, the Validation 68

69 True Value (T) Reference True Value (T R ) Corrected Measurement Value (M C ) Simulated Reference True Value (T R S) Measurement Value (M) Simulated Measurement Value (M S ) Experiment Simulation Figure 2-3: A schematic of the Methodology framework to quantify Representativeness uncertainty using a CFD simulation. The shape of random uncertainty distribution in each value, in general, does not have to be Gaussian. Gaussian is chosen for illustration purposes. uncertainty of Equation 2.3 when added to Equation 2.6 yields M C N ( µ T R S + µ δval, σ 2 M C = σ 2 M + σ 2 M S + σ 2 T R S + σ 2 δ val ) (2.7) where µ δval, σ δval are determined from a validation experiment and µ T R S is obtained from a simulation making sure the simulation is within the validation domain. Further, both σ MS and σ T R S are quantified using methods described in Appendix G. The Measurement uncertainty σ M arises from the accuracy of the physical measurement process. Figure 2-3 shows the Methodology schematic with the corrected Measurement Value added. Note that the corrected Measurement Value is vertically shifted 14 from the Simulated Reference True Value TS R due to the Validation uncertainty. The symbols used in this Methodology schematic are summarized in Table 1.3. In this work, it is not suggested that the measurements taken at power plants be 14 This shift is to a lower value since µ δval is assumed for illustration to be positive 69

70 updated using the results of simulation models such as CFD. However, the framework specifies the uncertainties that need to be considered if there is no experimental way of determining the Representativeness uncertainty, and if a simulation model is to be relied on for quantifying the Representativeness uncertainty. As shown in Figure 2-3, the application of this Methodology can lead to a more accurate knowledge of the Reference True Value only if the model is accurate enough to avoid introducing much wider random uncertainties than the ones previously available from the measurement. If the model random uncertainty were large compared to the systematic uncertainty it is aiming to resolve, this Methodology would not provide significant benefit Testing the quality of the simulation The decision to update the Measurement Value based on the result of the simulation can be quantitatively made within the hypothesis-testing framework involving random variables. A discussion of hypothesis-testing can be found in [22]. Let us start with the null hypothesis H 0 : M C M 0 which means that the corrected Measurement Value M C is statistically different from the original value M. The subtraction term can then be calculated from Equations 2.2 and 2.7 as M C M N ( µ T R S + µ δval, σ 2 M C = σ 2 M + σ 2 M S + σ 2 T R S + σ 2 δ val ) N ( µm, σ 2 M ) (2.8) or M C M N ( µ T R S µ M + µ δval, σ 2 M C = σ 2 M S + σ 2 T R S + σ 2 δ val ) 15 (2.9) The following test is proposed for testing the null hypothesis H 0 - accept the null hypothesis H 0 with significance level α if Equation 2.10 is satisfied. µ T R S µ M + µ δval z(α/2) σm 2 S + σ 2 TS R + σ 2 δ val (2.10) 15 It can be shown by calculating the variance of M c M that σ 2 M term is eliminated 70

71 True Value (T) Reference True Value (T R ) Corrected Measurement Value (M C ) Measurement Value (M) Figure 2-4: Illustration of two cases of the corrected Measurement Value M c. Figure A shows M c sufficiently separated from M and will pass the test for the null hypothesis H 0. Figure B shows a corrected Measurement Value that is not sufficiently separated from the original Measurement Value M due to large modeling and/or Validation uncertainty. The significance level is a preference of the user and the commonly used values are α = 1% or α = 5%. Here, the standard normal cumulative probability distribution function(cdf) z(α) is used. The function is readily available in tabulated forms and implemented in software packages such as MATLAB. In practice, the test for H 0 means that the corrected Measurement Value M C has to be sufficiently separated from M in comparison with the total standard deviation due to various uncertainties in order to be statistically significant. Figure 2-4 shows two cases illustrating the usefulness of the test for H 0. Figure 2-4A represents a high-quality simulation result with small modeling and/or Validation uncertainty compared with the Representativeness uncertainty that is likely to pass the test given in Equation Figure 2-4B is an illustration of a simulation result with large uncertainties and likely to fail the test. 71

72 2.2 Formulation using non-parametric methods The calculations given by Equations 2.4, 2.6, 2.10 rely on the assumption of normal distribution for the uncertainties being added. The calculation prescribed by the ASME standard makes the same assumption[14]. However, the sources of uncertainty may not be considered as normal in all cases thus limiting the applicability of the method. Different types of numerical uncertainties are taken into account in the Methodology framework. The formulation given in Section 2.1 requires the assumption of normal distribution for numerical uncertainties. It is not evident that these are strong assumptions. For example, the methodology for discretization uncertainty estimation proposed by Hoekstra et al[21] does not provide a second-moment measure for the uncertainty, but rather a confidence interval due to discretization uncertainty. This hinders the process of combining discretization uncertainty with other sources of uncertainty. Non-parametric methods have the potential to make a contribution to the process of combining uncertainties that are not normal. In addition to numerical uncertainties which may not be treated as normal, measurement chains of a performance indicators may be influenced by a number of factors that introduce uncertainty. They include the uncertainty of the measurement of environmental temperature which can influence the measurement chain and the physical properties of the working fluid and inherently chaotic physical phenomena such as turbulence induced fluctuation of physical quantities. A literature review was conducted on the probability distribution of velocity fluctuation in turbulent flows in various flow conditions. The third and fourth moments of turbulence statistics corresponding to the skewness (lopsidedness of the distribution) and kurtosis (tailedness) were examined. The results of both experiments and direct numerical simulations (DNS) of turbulent flows indicate that while the distributions of velocity fluctuation converge to normal in the bulk flow, the distribution is highly skewed and non-normal near the wall [26, 27, 28, 29, 30, 31, 32]. 72

73 Parametric methods such as the one described in Section 2.1 rely on normal probability distribution of uncertainty, which reduces the dimension of the data and results in loss of information. The move towards the adoption of non-parametric bootstrap methods are advocated by different authors including Tim Hesterberg of Google[33]. Prescriptions for the implementation of the non-parametric bootstrap method can be found in [5, 4, 6] and given by Hesterberg 16. The following calculations need to be performed non-parametrically in order to translate the Methodology framework given in Section 2.1 into full non-parametric Methods. Sampling uncertainty estimation. Determining the confidence interval for a random variable is necessary in quantifying the total numerical uncertainty of a CFD model. The bootstrap method can be used for confidence interval calculation[34] as discussed in Appendix G. Adding random variables is a necessary step in combining numerical uncertainties. Non-parametric bootstrap method can provide flexibility in integrating uncertainty sources given by non-gaussian continuous probability distributions or a data-set not specified by a known parametric probability distribution. Comparing two populations of random variables is necessary in determining the quality of a simulation model and in making the decision to update a Measurement Value. Both the bootstrap method and permutation test can be adopted to accomplish this task as explained in Appendix G. An example application of the Methodology using non-parametric methods is given in the Case Study 1 presented in Chapter tibs/stat315a/supplements/bootstrap.pdf 73

74 2.3 Implementation of the Methodology In this subsection, a road-map of implementing the methodology framework given in Section 2.1 is given. The implementation is divided into the following steps which are introduced in Sections Defining the Reference True Value 2. Defining Representativeness uncertainty 3. Defining two categories of Representativeness uncertainty (Temporal and Spatial) 4. Simulation model building 5. Calculating Random modeling uncertainty 6. Assessing the systematic Validation uncertainty 7. Creating a combined Methodology schematic 8. Creating a workflow chart 74

75 2.3.1 Defining the Reference True Value First, the Reference True Value is defined. Figure 2-5 illustrates the The Reference True Value plotted. Each Case Study included in this thesis features a definition of the Reference True Value of the relevant performance indicators. 1. Defining Reference True Value Reference True Value (T R ) Experiment Figure 2-5: The Reference True Value is defined in this step. 75

76 2.3.2 Defining Representativeness uncertainty In this step, the Measurement Value of the performance indicator is defined. The Representativeness uncertainty is the difference between the Measurement Value and the Reference True Value defined in Section 1.4. Figure 2-6 shows the Methodology schematic and the Representativeness uncertainty. 2. Defining Representativeness uncertainty Reference True Value (T R ) Representativeness uncertainty Measurement Value (M) Experiment Figure 2-6: The Representativeness uncertainty is defined in terms of the difference between the Reference True Value and the Measurement Value 76

77 2.3.3 Defining two categories of Representativeness uncertainty (Temporal and Spatial) In this step, the Representativeness uncertainty is further specified. As shown in Figure 2-7, in this Methodology, the Representativeness uncertainty is considered to be either spatial or temporal or both simultaneously. If both are available in a given Case Study, they need to be defined separately. 3. Defining two categories of Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) Spatial Representativeness uncertainty Temporal Representativeness uncertainty Experiment Figure 2-7: The Representativeness uncertainty can be spatial or temporal or both can be present simultaneously. 77

78 2.3.4 Simulation model building In this step, a numerical simulation is performed with the purpose of identifying and quantifying the Representativeness uncertainty. The simulation model is built using the boundary and initial conditions of the measurement. 4. Simulation model building Reference True Value (T R ) Measurement Value (M) Simulated Reference True Value (T R S) Simulated Measurement Value (M S ) Experiment Simulation Figure 2-8: A simulation model of the measurement is built in this step 78

79 2.3.5 Calculating Random modeling uncertainty A model prediction is not perfect and subject to many uncertainties. In this step, all relevant random modeling uncertainties are quantified and combined to come up with the total random modeling uncertainty. In combining the random modeling uncertainties, the random uncertainties are assumed to be normally distributed and uncorrelated with each other. Figure 2-9 shows the combined Random modeling uncertainty indicated by a Gaussian. 5. Calculating Random modeling uncertainty Reference True Value (T R ) Measurement Value (M) Simulated Reference True Value (T R S) Simulated Measurement Value (M S ) Experiment Simulation Figure 2-9: Random modeling uncertainties are quantified and combined in this step. Simulated Representativeness uncertainty is the difference between M S and T R S 79

80 2.3.6 Assessing the systematic Validation uncertainty Systematic Validation uncertainty is calculated by comparing the Measurement Value (M) with the Simulated Measurement Value (M S ) as shown in Figure It is an indicator of how far the simulation result is from the measurement. 6. Assessing the systematic Validation uncertainty Reference True Value (T R ) Measurement Value (M) Simulated Reference True Value (T R S ) Simulated Measurement Value (M S ) Experiment Simulation Figure 2-10: Systematic Validation uncertainty is calculated in this step by subtracting the Measurement Value (M) from the Simulated Measurement Value (M S ) 80

81 2.3.7 Creating combined Methodology schematic In this step, the information from measurement and simulation are combined in a Methodology schematic with all associated random and systematic uncertainties as shown in Figure Objectives of applying the Methodology can vary depending on the specific Case Study. True Value (T) Reference True Value (T R ) Corrected Measurement Value (M C ) Measurement Value (M) Representativeness uncertainty Measurement uncertainty 7. Creating combined Methodology schematic Validation uncertainty of Reference True Value Simulated Representativeness uncertainty Validation uncertainty of Measurement Value Random modeling uncertainty Simulated Reference True Value (T R S ) Simulated Measurement Value (M S ) Measurement Simulation Figure 2-11: In this step, the information from measurement and simulation are combined in a Methodology schematic with all associated random and systematic uncertainties to achieve the specific objective of the Case Study. 81

82 2.3.8 Creating a workflow chart Figure 2-12 shows a global workflow of this Methodology. Depending on the available data and physical of statistical models employed in a given Case Study, different paths are taken along the workflow diagram. The application of the Methodology in each Case Study is accompanied by a workflow chart. Global work flow Representa/ve ness of model result Analysis of representa/veness of model results using sta/s/cs Physics model Physics model Experimental data Uncertainty of model Analysis of representa/veness using the simula/on model Sta/s/cal model development Sta/s/cal model uncertainty es/ma/on Figure 2-12: Global workflow of the methodology 82

83 Chapter 3 Case Study 1: Mass flow measurement by means of an orifice plate 3.1 Introduction Steam generator (SG) feedwater flow rate measurement, the subject of this Case Study, is a performance indicator used in nominal thermal power calculation (BIL100 in France) in pressurized water reactors (PWR)[35]. Currently in France, the SG feedwater flow rate is measured indirectly from the pressure drop across a flow restriction device. Uncertainty in the SG feedwater flow rate measurement is the largest source of uncertainty in BIL100, and it has a consequential influence on operational margins in the plant and affects the economic performance of the plant. Therefore, better understanding and handling of uncertainty in feedwater flow rate measurement is of importance. This chapter first presents the technical analysis of experimental data and CFD simulations to yield characterization of the uncertainties of the orifice plate mass flow rate measurement method in Sections 3.3, 3.4 and 3.5. Then the Methodology framework 83

84 is used to structure the technical analysis to characterize the Representativeness uncertainty of this orifice plate measurement. Sequentially, the Temporal (Sections 3.3), the Spatial (Section 3.4) and then the combination of these two types of Representativeness are presented. Finally, the results of the application of the non-parametric bootstrap method to the Spatial Representativeness problem is presented (Section 3.5). This chapter combines several experiments, tests, and simulation studies. are: They 1. High-frequency test data collected at an operating nuclear plant on feedwater flow rate measurement 2. Large eddy simulation of the SG feedwater flow rate measurement conducted using CFD code STAR-CCM+. 3. Experimental test conducted at EDF on an experimental flow configuration called XYZ028KD which has orifice plate flow rate measurement section. 4. RANS and URANS simulations conducted using STAR-CCM+ on XYZ028KD configuration. These works are divided into two mutually exclusive sets of analyses. First, LES is performed on the high-frequency test configuration, and the results are compared. This set, called Case Study 1a, involves the Temporal Representativeness uncertainty of the orifice plate mass flow measurement. Second, RANS simulation is performed on an experimental configuration XYZ028KD. The objective of this set is to characterize the Spatial Representativeness uncertainty, and it is called the Case Study 1b. The Methodology framework is applied to each set of analyses. 84

85 3.2 Flow rate measurement by means of an orifice plate Steam generator (SG) feedwater flow rate measurement is used to calculate the nominal thermal power at 100% in the BIL100 procedure in French nuclear plants. The nominal thermal power at 100% of rated power is calculated as given in Equation 3.1[35]. W r = N i=1 [ ( ) Q i ARE H i v He i Q ( ) ] p H i N v Hp i W k (3.1) where W r -thermal power N-number of steam generators Q i ARE-feedwater flow rate of the i-th SG H v -feedwater enthalpy at the steam generator outlet H e -feedwater enthalpy at the steam generator inlet Q p -blowdown flow rate H p -blowdown enthalpy at the steam generator outlet W k -primary power not coming from the core (heaters, primary pumps) 85

86 Figure 3-1: SG feedwater lines of an operating nuclear plant PWR A typical nuclear plant has 4 SGs. The feedwater lines are shown for an operating plant in Figure 3-1 where circled are the pressure taps with which measurements are taken. The pressure signal is then carried by an impulse line connection to where it is processed. The effect of such impulse line connections is discussed in Section vi. Mass flow rate of the feedwater to a SG is measured by a differential pressure device called an orifice plate which is inserted into the feedwater pipe. The orifice plate is the most commonly used flow rate measurement device in France with 80% of all flow rate measurements across the French fleet of nuclear reactors using it. Mass flow rate measurement by the orifice plate method is prescribed by ISO-5167 standard[36]. Figure 3-2 shows a simple schematic of the orifice plate used for flow rate measurement. Flow is restricted by an orifice of smaller diameter, internal to the conduit, creating a pressure differential upstream and downstream of the orifice. Mass flow rate is then calculated as given in Equation

87 d D l 2 l 1 Orifice p 2 p 1 Figure 3-2: Schematic of flow rate measurement by orifice plate q m = π 4 Cε 1 1 β 4 d2 2(p 1 p 2 )ρ (3.2) where q m is the mass flow rate, d is the orifice diameter, D is conduit diameter, p 1 and p 2 are upstream and downstream pressure values, ρ is the density of the fluid and β = d/d. The expansion coefficient ε is equal to 1.0 for water. C is the discharge coefficient prescribed by the standard for given pressure tap configurations. According to the standard, C is calculated as[36] ( 10 C = β β 8 6 ) 0.7 ( β ( A)β Re D Re D + ( e 10L e 7L 1 )(1 0.11A) 1 β 0.031(M (M 2) 1.1 )β 1.3 where L 1 = l 1 D is the coefficient of distance of upstream tapping with l 1 being the distance between upstream pressure tap and the orifice plate. L 2 = l 2 D is the coefficient of distance of downstream tapping with l 2 being the distance between downstream pressure tap and the orifice plate. with M 2 = (2L 2)/(1 β) and A = (19000β/(Re D )) 0.8. C coefficient depends on fluid mass flow rate in the pipe through Reynolds number Re D, and also depends on the upstream and downstream distances of pressure taps from the orifice. The specific 87 β 4 ) 0.3

88 pressure tapping configuration used in flow rate measurement is called D and D/2 pressure tap configuration meaning that pressure measurement is taken upstream at distance D from the orifice and D/2 downstream from the orifice. In the experiment conducted at EDF described in Section corner pressure taps are used. The discharge coefficient for that configuration was calculated according to the standard and compared with CFD simulation. 3.3 Temporal Representativeness uncertainty Temporal Representativeness or sampling uncertainty arises from the inherent temporal variations of the quantity being measured. The Temporal Representativeness is the difference between continuous time averaged value and discrete time averaged value at a certain sampling frequency. In this Section, we first analyze the raw test data and then implement a Large Eddy Simulation (LES) on the same test configuration in order to understand the nature of the coherent turbulent structure. Lastly, the temporal spectrum produced by the LES is benchmarked with the test data spectrum Analysis of raw test data High-quality test data (1000Hz) was obtained to understand the fluctuation of the differential pressure signal used to calculate feedwater mass flow rate. Three sets of available data at 100%, 80%, 50% power level flow rates are shown in Figure 3-3. The figure shows that enhanced high-frequency noise appears at the 100% power level. The analysis given in this Section is purely statistical and does not include any CFD simulation as input. The spectral shape of data in Figure 3-3 is used in later Section vi for comparison with the CFD simulation results. During normal operation of the plant, the pressure signal is sampled with a frequency of 0.2Hz corresponding to a period of 5 seconds. The statistical sampling uncertainty introduced due to 88

89 Figure 3-3: Test data at 100, 80, 50% power levels this relatively low sampling rate (Temporal Representativeness uncertainty) of this high-frequency test data is analyzed in this Section. In order to assess the Temporal Representativeness uncertainty, we re-sampled the high-frequency test data (1000Hz signal) at lower frequencies and calculated the mean and standard deviation of the new low frequency sampled signal. The effect of sampling phase shift, which represents the starting point of a lower frequency sampling on the mean value of the signal, was also studied. To illustrate the concept of sampling at low frequency, Figure E-1 was constructed showing an example of sampling with 100Hz frequency with phase shifts of φ shift = 0 and φ shift = π. Figure E-1 shows that phase shift results in different mean values. The means of the green and red curves corresponding to 0 and π phase shifts are different from the blue which is the original signal. Figure G.1 shows the average value of the signal as a function of the sampling frequency. The error bar of the average values at each sampling frequency are due to phase shift. A conclusion that can be drawn is that low-frequency sampling is subject to large sensitivity to the 89

90 Figure 3-4: Illustration of change in signal shape at a lower sampling frequency. Blue line represent high-frequency test data whereas red and green lines represent hypothetical signal shapes resulting from lower sampling rates with a phase shift on starting point P(mbar) Sampling Frequency(Hz) Figure 3-5: Average signal value with deviation due to starting point phase shift as a function of the sampling frequency 90

91 Sampling Frequency(Hz) Systematic deviation δ of signal average at low sampling rate in mbar, and (as % of mean) 50% level 80% level 100% level CD2 sensor CD3 sensor CD2 sensor CD3 sensor CD3 sensor δ as % of δ as % of δ as % of δ as % of δ as % of mean mean mean mean mean (4.6e (0.001%) (4.6e (3.0e (2.2e- 4%) 04%) 4%) 04%) (0.067%) 0.173(0.12%) 0.117(0.026%) 0.196(0.044%) 0.163(0.02%) (0.066%) 0.169(0.11%) 0.115(0.026%) 0.192(0.04%) 0.161(0.02%) (0.066%) 0.169(0.11%) 0.115(0.026%) 0.192(0.04%) 0.165(0.02%) (0.067%) 0.169(0.11%) 0.121(0.027%) 0.198(0.04%) 0.284(0.035%) (0.076%) 0.180(0.12%) 0.235(0.053%) 0.302(0.07%) 0.637(0.08%) (0.12%) 0.221(0.15%) 0.325(0.07%) 0.388(0.09%) 0.980(0.12%) (1.02%) 1.61(1.07%) 2.42(0.54%) 2.45(0.55%) 3.69(0.46%) Table 3.1: Summary of Temporal Representativeness systematic uncertainty due to low sampling rate. starting phase. The sampling error decreases with increasing sampling frequency. Table 3.1 shows the amount of deviation of the signal mean when sampled at lowfrequency for all available data. The Temporal Representativeness error due to low sampling rate accounts for less than 0.18% at all power levels. Table 3.1 shows the summary of Temporal Representativeness uncertainties at different sampling frequencies. The systematic shift in the mean value of the signal due to low sampling rate is expressed as a percentage of the mean signal value. The highlighted row shows the frequency at which data is collected during normal operation of the plant. Rosemont 3051 CD2 and CD3 sensors are redundant sensors measuring the same data. In conclusion, the Temporal Representativeness uncertainty introduced due to lower sampling rate during a normal operation of the plant is below 0.12% based on the available test data. 91

92 3.3.2 LESimulation of SG feedwater line Several hypotheses exist to explain the fluctuation of the pressure differential signal indicated by the test. They are turbulence near pressure taps, feed water flow control, pipe vibrations and pumps. In order to gain a complete understanding of the mechanisms behind the fluctuations, a Large Eddy Simulation (LES) was performed using the STAR-CCM+ code. i Theoretical description of LES Large Eddy Simulation (LES) is a CFD simulation technique intermediate in physical accuracy between the direct numerical simulation (DNS) and the Reynolds-Averaged Navier-Stokes (RANS) simulations. The LES resolves the contribution of the large, energy-carrying structures to turbulence, and models the effect of the smallest scales of turbulence[37]. More details on the LES is given in Appendix D. ii Preparatory RANS simulation A preliminary RANS simulation was performed in order to obtain the turbulence length scale of the flow which is then used to construct a computational mesh for Large Eddy Simulation. It is assumed that the flow entering the orifice plate is fully developed. Therefore, the fully developed flow profile is used as an inlet boundary condition. Fully developed flow profile is first obtained by RANS simulation in a straight pipe with periodic boundary condition. Then the flow profile is imposed as inlet conditions using table functionality in STAR-CCM+ in the orifice geometry. The rest of the settings used in the steady-state simulation are given in Section Figure 3-6 shows the integral turbulent length scale based on the mixing length calculated by Equation 3.3 where k and ε are turbulent kinetic energy and dissipation rates respectively and C µ = 0.09[38]. L 11 = Cµ 0.75 k 1.5 ε (3.3) 92

93 Figure 3-6: RANS simulation result of integral turbulent length scale given by Equation 3.3 for the orifice plate geometry at 100% power level LES mesh criteria k 1.5 / >ε10 Number of cells 17,166,922 Base cell size 3.0mm Number of prism layers 3 Prism layer thickness 4.75mm Extruder Constant axial growth rate Table 3.2: LES mesh characteristics iii Implementation of LES The turbulent integral length scale calculated by Equation 3.3 gives a quantitative estimate of the size of turbulent eddies and enables the construction of the mesh for LES. Table 3.2 summarizes the characteristics of the LES mesh constructed using the information from the preparatory RANS simulation. The LES mesh is shown in Figure 3-7 zooming into the orifice region and showing the extrusion length. A fully developed flow profile is imposed as the inlet boundary condition. The Synthetic Eddy Method (SEM) is adopted for generating sustained turbulence. SEM is demonstrated to produce a realistic turbulent inflow condition while preserving time, and length scale and coherent structure of turbulence[39]. A representative instantaneous velocity magnitude field at the inlet boundary is given in Figure

94 Extrude d 3D Base size 2D Figure 3-7: A cross section of the LES mesh. The base size mesh is used in 2 hydraulic diameters in both upstream and downstream directions and extruded additionally for 3 hydraulic diameters. Figure 3-8: An instantaneous velocity magnitude imposed at the inlet boundary condition. A fully developed flow profile and Synthetic Eddy Method is used to generate the required mass flow rate and sustained turbulence respectively. 94

95 The modeling of subgrid scales is based on the wall-adapting local eddy-viscosity (WALE) model, leveraging its generality and extensive assessment[40]. In this work, the Near-Wall Modeled (NWM) LES was adopted in order to keep the computational cost low (practicable) in a high Reynolds number case such as this. Validity of using NWM LES is discussed by Baglietto et al.[40]. iv Results of LES, and validation using the ISO standard The LES simulation results are presented in this Section. Figure 3-9 and Figure 3-10 show the velocity magnitude and pressure fields as resolved by LES. One hypothesis for the origin of the fluctuation of pressure differential signals observed during the test is the presence of coherent structures, visible in the alternating high and low-pressure vortices originating at the orifice edge in LES resolved pressure plot in Figure In order to validate LES results, the discharge coefficient C described in Section 3.2, calculated by LES is compared with the discharge coefficient prescribed by the ISO standard. Proving that turbulence is the original of the fluctuation needs further study. Figure 3-9: LES calculated velocity magnitude field. location of the downstream pressure tap The black arrow shows the The discharge coefficient is calculated in LES as given in Equation 3.4. The upstream 95

96 Figure 3-10: LES result of pressure field at 100% power. The black arrow shows the location of downstream pressure. The red region upstream of the orifice is indicative of minimal friction pressure drop. and downstream pressure values are taken from point pressure probes as shown in Figure At each time instant, both upstream and downstream pressures are probed and the discharge coefficient C is calculated as a function of the simulation time. The time-dependent value of the discharge coefficient calculated using the LES result is given as 4 1 C LES (t) = q m 1 β π 4 d 2ρ(p 2 1 (t) p 2 (t)) (3.4) Figures 3-12 through 3-14 show the discharge coefficient calculated using the LES simulation result for a simulation time of 10 seconds at 100%, 80% and 50% power levels respectively (test signal acquisition length is 1200 seconds). At all power levels, the time-average LES simulation result is within 0.1% of the value prescribed by the ISO standard. 96

97 D D/2 Figure 3-11: The point probes upstream and downstream the orifice correspond to the locations of the pressure taps Figure 3-12: Discharge coefficient C calculated from LES at 100% power level is compared with the ISO standard. The time-averaged, LES calculated C coefficient is less than 0.1% higher than the ISO standard prescribed value. A simulation time step = 2.5e 4 second is chosen to keep the Convective Courant number of the simulation close to 1. 97

98 Figure 3-13: Discharge coefficient C calculated from LES at 80% power level is compared with the ISO standard. The time-averaged, LES calculated C coefficient is less than 0.1% higher than the ISO standard prescribed value. A simulation time step = 2.5e 4 second is chosen to keep the Convective Courant number of the simulation close to 1. Figure 3-14: Discharge coefficient C calculated from LES at 50% power level is compared with the ISO standard. The time-averaged, LES calculated C coefficient is within 0.1% of the ISO prescribed value of the discharge coefficient. A simulation time step = 2.5e 4 second is chosen to keep the Convective Courant number of the simulation close to 1. The artifact around 6 second is due to an unexpected break in the simulation. 98

99 v Spectral analysis To assess the capability of LES to reproduce the frequency spectrum of the oscillation of the pressure drop signal across an orifice plate, three LES simulations were performed at mass flow rates corresponding to the test data that was available for comparison. These simulation results are presented in Figures 3-12 through It is important to note that the pressure signal considered in this chapter is the difference between upstream and downstream pressure measurements. In LES, the pressure difference is obtained by simple subtraction of instantaneous downstream pressure at a point probe from an instantaneous upstream pressure value at a point probe. Separate time-dependent signals from upstream and downstream pressure taps are not available from the test measurement, but only the difference is available. The subtraction of point signals in LES could influence the spectrum analysis. Based on a comparative analysis of test and LES results, several observations are made. The magnitude of the pressure fluctuation simulated in LES is higher than that of the test data. Figure 3-15 shows the histogram of both the test data and the LES result at 100% power level along with Gaussian distribution fits. The standard deviation of pressure fluctuation in a turbulent flow has a strong deterministic relationship with the flow rate[41], and the possibility of measuring flow rate based only on the standard deviation of wall pressure fluctuation has been proposed. Figure 3-16 shows a second order polynomial fit to the std of pressure fluctuations as a function of flow rates for both the test data and the LES result. Figure 3-17 shows a good convergence of the standard deviations of both test signal and the LES result at the given length. Next, spectral analysis was carried out on both the test signal and LES result in order to assess the feasibility of using LES to understand the signal spectrum. The available test data is of the duration of 1200 seconds with a sampling frequency of 1000Hz. On the other hand, LES is of 10 seconds physical time and a computational time step of 2.5e-4 sec (an equivalent of 4000Hz sampling rate). Therefore, the lowest 99

100 Figure 3-15: Fluctuation of test pressure signal and LES result at 100% power level Figure 3-16: Second order polynomial fit to standard deviation in wall pressure fluctuation as a function of flow rate 100

101 [A] [B] Figure 3-17: The convergence of standard deviations of the test signal and the LES at the three power levels. This plot is produced as a supplement to Figure 3-16, and does not indicate a computation-related convergence. frequency components of the test signal ( sec) are not achievable by LES due to limited computational resources. The results of the analysis using power spectral density are given in Figure Power Spectral Density (PSD) is used in order to assess the distribution of the power of a time varying signal as a function of frequency[42]. PSD is defined as P xx (f) = T t=0 e 2πift R xx (t)dt (3.5) where the time-averaged auto-correlation of the sequence is defined as R xx (t) = 1 T t T t t=0 p(τ)p(t + τ)dτ for a time-dependent pressure signal p(t) of length T. The PSD (units P a 2 /s) is then non-dimensionalized using the method used by Qing et al.[42] Φ p (f) = P xx(f) 4q 2 d/v (3.6) where q = ρv 2 /2 is the dynamic pressure in the pipe. V is taken to be the cross section averaged velocity of the fluid in the pipe. The PSDs of the raw test data, calculated using the power spectral density function implemented in MATLAB, is given in Figure The peaks at frequencies 50Hz(2n+1), n=0,1,2... are indicative of a measurement related noise. In comparing with the simulation, the measurement noise can be smoothed out using moving average method. 101

102 Figure 3-18 shows the LES spectrum along with raw test data spectrum. The order of magnitude and the shift in the spectrum with flow rate are reproduced using LES. However, close examination of the shape of the spectrum reveals significant differences. While the exact reasons are not understood, the factors contributing to the shape of the spectrum might include the difference in the sampling frequencies of the test signal and the LES simulation. The sampling frequency of the test signal is 1000Hz (data acquisition time of 1200sec) whereas the LES simulation is done with a time step equivalent of 4000Hz (simulation time of 10sec) sampling frequency. The Courant number of the simulation dictated the time step in LES. More information about the hydraulic configuration is necessary in order to understand the damping of pressure signals. 102

103 Figure 3-18: PSD of test and LES simulated pressure drop data. The test signal is the raw signal with no filtering. Vertical axis is non-dimensionalized using Equation 3.7 for comparison. The peaks at frequencies 50Hz(2n+1), n=0,1,2... are indicative of a measurement related noise (possibly due to antenna effect) and should be ignored. Another possible source of the peaks is turbines and pumps in the secondary circuit. Impulse line or gauge line is the tubing or piping which connects the pressure taps on the primary element (orifice pipe) to a secondary element (recording device or transmitter) [43]. The general guideline provided by Harris and McNaught[44] lists the factors that could induce potential pulsation in a pressure measurement. These are impulse line diameter and length, the location of a secondary device, the routing of the line, the location of pressure tapping, ambient temperature, temperature fluctuation, fluid type, valves and connections for venting. Although we do not have access to the detailed configuration of the impulse line of the test, the photographs of the test arrangement suggest that impulse lines are long with multiple bends and turns. This could explain the deviation in frequency spectrums produced by LES and measured spectrum from the test. In future, while challenging, detailed CFD calculation with the impulse line incorporated could clarify the frequency spectrum more precisely. 103

104 The problem encountered with respect to the impulse line effect is similar to the underwater acoustic damping phenomenon. X.Lurton[45] discusses the frequency dependence of acoustic signal damping. Due to the dissipative nature of water, the amplitude of pressure wave is attenuated with respect to distance traveled. According to the François and Garrison model[46], the attenuation coefficient of pure water due to viscosity is given by α = C 3 f 2 (3.7) where C 3 is a model constant that depends on temperature, f is the frequency and α is the attenuation coefficient, which is defined for a planar pressure wave by ( p(x, t) = p 0 exp α ) exp (jω(t x ) 20 log e c ) (3.8) This formula is then applied to the PSD of the LES calculated pressure drop signal to isolate the impact of impulse line. Figure 3-19 shows the PSD for the LES result for various lengths of impulse line. The left shift in the PSD spectrum is observed due to the frequency dependence of the attenuation coefficient of pure water. This information could help explain the shift between the spectrums of the LES result and the test signal given in Figure

105 10Log 10 P xx /(4q 2 d/v) m 2m 20m m 2000m 20000m Frequency (Hz) Figure 3-19: The effect of impulse line length on the PSD spectrum of LES calculated pressure drop signal. The different colored lines correspond to various impulse line lengths expressed in meters (m). vi Conclusion In this section, a LESimulation of the feedwater line of an operating nuclear plant is performed. The objective of the simulation is to gain an understanding of the temporal fluctuation in pressure differential signal obtained at the plant during a test. First, the time-averaged LES result was validated within 0.1% of the mass flow rate correlation prescribed by the ISO standard. Second, the spectral shape of the data obtained at the plant is compared with the LES simulation result. The spectrums are found to be shifted horizontally on the frequency axis. A possible hypothesis associated with frequency dependence of wave attenuation in water is proposed in this work. 105

106 3.4 Spatial Representativeness uncertainty Spatial Representativeness uncertainty arises due to the inherent variations of the Measurand. It is the difference between the Reference True Value and the Measurement Value due to Spatial and geometrical considerations. This Section focuses on CFD simulation effort on XYZ028KD geometry, an experimental test configuration at the Facility#1 in France. In order to use the ISO-5167 standard, two criteria regarding the geometry of the flow configuration have to be met. The upstream length between the orifice plate and the first non-straight pipe section must be greater than a prescribed length (44D 17 for XYZ028KD configuration [36]) or else an additional uncertainty is imposed. The downstream length between the orifice plate and the first non-straight pipe section must be greater than a prescribed length (8D for XYZ028KD configuration) or else an additional uncertainty is imposed. The XYZ028KD configuration deliberately does not meet these criteria in order to assess the effect on the measurement result. The simulation effort in this Section is aimed at replicating the experimental flow conditions by including the geometry of the upstream and downstream piping and the orifice. Then the simulation results are compared with the experiment and the ISO standard, and sources of uncertainty and the quantities affecting the CFD model are identified Experimental tests This experimental test was conducted in order to assess the uncertainty due to using standards in calculations involving the orifice plate for flow rate measurement when 17 D-hydraulic diameter of the pipe 106

107 Figure 3-20: Experimental configuration of XYZ028KD geometry upstream flow development is deviant from that prescribed by the standard[47]. The geometry of the orifice plate is summarized in Table 3.3. Parameter β D(mm) at 20C 0 d(mm) at 20C 0 XYZ028KD value %(2σ) %(2σ) %(2σ) Table 3.3: Summary of experimental dimensions of XYZ028KD geometry A controlled flow rate was induced through an orifice plate, and the pressure difference between upstream and downstream pressure values was measured. The values of reference flow rate and flow rate calculated using the orifice plate method are compared. The experimental configuration is shown in Figure Flow rate was controlled to 5 different reference values, and the pressure drop across 107

108 C& 0.614& 0.612& 0.610& 0.608& 0.606& 0.604& 0.602& 0.600& 0.598& 0.596& ISO&standard& 0.594& 250& 350& 450& 550& 650& 750& 850& 950& Volumetric&flow&rate&(m 3 /h)& Experiment& Figure 3-21: Experimental result expressed in terms of the discharge coefficient along with ISO. Error bars of both the standard and experiment are shown. Systematic downward shift of the experimental result from the ISO standard shows the systematic uncertainty due to non-straight upstream piping geometry in XYZ028KD flow configuration. the orifice plate was measured. The mass flow rate was then reconstructed using the pressure drop and the ISO standard outlined in Section 3.2. The results of the experiment on geometry XYZ028KD are summarized in Figure 3-21 showing the deviation of mass flow rates calculated by the orifice plate method from the Reference True Value as a percentage. The error bars in the experimental result arises from uncertainties in the measurement of physical properties of the fluid such as temperature, density; geometrical dimensions of the loop such as d, and D; and the ISO tolerance levels given by the standard. The systematic shift between the experimental results and the ISO standard is to be expected since the XYZ028KD geometry of the experiment does not respect the upstream flow development length before the first non-straight Section of the pipe. In other words, the systematic shift of about 0.5% shown in Figure 3-21 is the systematic uncertainty due to non-straight upstream piping geometry as identified by experiment. 108

109 Steady RANS Simulation High y+ wall treatment, standard wall function approach Segregated flow, 2nd order convection scheme Standard k-ε, 2nd order convection scheme URANS simulation Implicit unsteady, t=0.001 sec Segregated flow, 2nd order convection scheme Standard k-ε, 2nd order convection scheme High y+ wall treatment, standard wall function approach Table 3.4: Simulation setup used for RANS simulations RANS simulation In this Section, a simulation effort on the exact conditions used at the Facility#1 test facility is presented. First, simulation details are given starting with the mesh generation, preliminary simulations and results analysis followed by grid convergence study and sensitivity studies on turbulence model and model parameters. Finally, a special category of uncertainty called Spatial Representativeness identified and quantified using CFD is presented. Detailed settings in the STAR-CCM+ simulations in this Case Study are summarized in Table E.1. i Mesh generation In this work, both unstructured and block structured meshes were used to discretize the computational domain. The unstructured computational mesh is generated using the STAR-CCM+ built-in meshing algorithms, in particular, the trimmed and prism layer meshes were adopted, which allow producing hexa dominant cells, in combination with boundary fitted cells in the near-wall region. The base size of cells is 5 mm, and two prism layers with the absolute total thickness of 1 mm and stretching factor 1.5 were used. Volumetric mesh refinement in the region near the orifice of 25% of the base size is employed to better resolve the region of important turbulent shedding. This resulted in a total of 4,963,800 cells with 14,727,751 faces. The characteristics of the unstructured mesh are given in Table 3.5. Cross-sections 109

110 Base size 5mm Number of prism layers 2 (stretch factor of 1.5) Thickness of prism layers 1mm Mesh refinement The region near orifice is refined to 25% of the base size Number of cells 4.9M Table 3.5: Characteristics of the mesh used for the RANS simulation of the unstructured and structured meshes near the orifice region and perpendicular to flow direction are shown in Figure [A] [B] Figure 3-22: Computation mesh showing the orifice region (note regional mesh refinement) and perpendicular to flow direction. ii Convergence The convergence of the Unsteady RANS simulation using the cubic k-ε model is determined by monitoring velocity at 5-point probes with locations given in Figure The probes are located in regions in which convergence is expected to be the slowest: in highly deformed flow regions around the orifice and near the outlet. The convergence of velocity at those points as a function of the number of iterations is given in Figure 3-24A for the flow rate 5, which is the highest. 110

111 OUTLET# INLET# Downstream# Pressure#tap# D/2# D# ORIFICE# Upstream# Pressure#tap# L=28D# Figure 3-23: Locations of points with velocity monitors for convergence determination [A] [B] Figure 3-24: A) Velocity as a function of time at 5 monitors, flow rate 5, for the unsteady simulation using the cubic k-ε turbulence model. The vertical axis shows velocity magnitude in (m/s). B) Convergence of the pressure drop at flow rate 5, for the unsteady simulation using the cubic k-ε turbulence model. Oscillatory behavior as a function of time is observed at the 5 monitors considered when using the cubic k-ε turbulence model. This is not unexpected and mainly caused by flow unsteadiness in this regime; fully steady results are not achievable in URANS without excessive numerical diffusion of the solution. Figure 3-24B shows the convergence of the pressure drop across the orifice plate P as a function of time. The unsteadiness of the simulation results in iterative uncertainty and introduces an uncertainty term with standard deviation σ r = 0.5%µ M. The iterative uncertainty is discussed in Section iv. 111

112 Mass flow(m 3 /h) Reynolds number Flow 1 Flow 2 Flow 3 Flow 4 Flow Table 3.6: Mass flow rates used for experiment and simulation iii Post-Processing The results of CFD simulation are compared to the ISO-5167 standard in order to assess the extent of deviation in simulation. The discharge coefficient C is chosen as the parameter to compare simulation results between the standard and experiment. The reason is first, CFD simulation conducted at EDF used C as the benchmark quantity, and second that ISO standard specifies the tolerance bound on discharge coefficient C making it convenient to assess the capability of CFD simulation. C is also used as the quantity to compare different turbulent models and different settings of those models. The CFD computed discharge coefficient is calculated as 4 1 C CF D = q m 1 β π 4 d 2ρ(p 2 1 p 2 ) (3.9) where p 1 and p 2 are upstream and downstream pressure values obtained from point probes. p 1 and p 2 are time averaged pressure values meaning that they are obtained by averaging Unsteady RANS calculation result once they reach oscillatory behavior which was encountered in the RANS simulation. q m is the mass flow rate imposed at the inlet condition. Mass flow rates are chosen as the experimental reference flow rates and there are 5 separate values as given in Table 3.6. The CFD calculated discharge coefficients are then plotted in Figure 3-25 as a percentage of deviation from ISO-5167 prescribed value. C = C C ISO(q m, d, D, ρ) C CF D C ISO (q m, d, D, ρ) (3.10) 112

113 iv Results and validation Results of all RANS simulations performed on the experimental geometry shown in Figure 3-23 are given in Figure The figure includes the simulation results using both unstructured and block-structured meshes with the standard k-ε model and the cubic k-ε nonlinear eddy-viscosity model (NLEVM). The main observation from the results of the CFD simulations is that all RANS results are approximately within 1% of both the experimental and ISO prescribed values for C. The sensitivity to two different mesh structures is found to be small as indicated by linear k- ε model results using two different meshes. Using the same unstructured mesh, the cubic k- ε model is found to be the closer to the experimental result than the linear k- ε model. It is important to note that the ISO standard is not the benchmark for the experimental geometry since the upstream piping geometry does not satisfy the requirements of the standard. Instead, the experimental result should serve as the benchmark for the simulation (C-C ISO )/C ISO (%) Unstructured mesh, linear k-ϵ Block-structured mesh, linear k-ϵ Unstructured mesh,cubic k-ϵ Experiment ISO standard and tolerance bounds Mass flow rates (kg/s) Figure 3-25: Results of steady RANS simulation expressed in terms of % deviation from the discharge coefficient prescribed by the ISO standard. In the experimental test described is Section 3.4.1, the upstream and downstream piping geometries do not allow enough length for the flow to fully develop. The full 113

114 OUTLET# INLET# R Downstream# Pressure#tap# D/2# D# ORIFICE# Upstream# Pressure#tap# L=28D# Figure 3-26: Angular location of pressure tap (left), piping geometry of the experimental test (right) piping geometry used in the experiment and the CFD modeling is shown in Figure 3-26, where inlet T junction and outlet elbows are indicated in pink and orange along with the location of the orifice plate. Cylindrical coordinates were used with the z-axis along the direction of the fluid flow in the straight section of the geometry as shown in left 3-26 in order to understand the heterogeneity of pressure field due to lack of fully developed flow. Results of CFD simulation are post-processed in order to see the dependence of CFD calculated discharge coefficient as a function of an angular location θ of the pressure taps. The results are given in Figure 3-27 for five different flow rates using the structured mesh. Mesh sensitivities are discussed in the following Section. The result indicates that the persistent structure in the CFD calculated C coefficient as a function of an angular tap location possibly indicative of geometry of the experimental circuit. Although the upstream flow development length in the experimental test meets the requirements of the ISO standard, our RANS simulation shows significant heterogeneity in the pressure field as a function of an angular location of the pressure taps used. 114

115 (C CFD -C ISO )/C ISO )(%) Flow rate Polar angle 3(Degrees), x (C CFD -C ISO )/C ISO )(%) Flow rate Polar angle 3(Degrees), x (C CFD -C ISO )/C ISO )(%) Flow rate Polar angle 3(Degrees), x (C CFD -C ISO )/C ISO )(%) Flow rate Polar angle 3(Degrees), x (C CFD -C ISO )/C ISO )(%) Flow rate 5 GP error bound GP mean ISO mean ISO bound Polar angle 3(Degrees), x Figure 3-27: Dependence of CFD calculated discharge coefficient on angular location of pressure taps. Gaussian Process (GP) regression was used to interpolate the function between discrete computational mesh points. Details on the implementation of GP regression are given in Appendix G 115

116 3.5 Uncertainties of the RANS CFD simulation The uncertainty sources relevant to Case Study 1 are given in Table 1.4. This Section is organized as follows. First, the systematic sensitivities applicable to Case Study 1 are discussed. Second, the sources of Random Modeling uncertainties of the CFD simulation relevant to the Case Study 1 are discussed, and the method to combine them is presented Factors introducing systematic sensitivity/uncertainty The systematic sensitivities that introduce systematic change in the prediction of the simulation as shown in Table 1.4 are discussed in this Section. A CFD code has a number of parameters or model settings that can impact the results of a simulation. In this Section, the types of parameters of model settings that can have the largest influence on the model output are identified and quantified where possible. i Sensitivity to turbulence model The turbulence model used in simulations using STAR-CCM+ is the standard k- ε model. The results of the model are compared against the results obtained by Code_Saturne using other models in Table 3.7. Two constitutive relations employed in the standard k-ε model for this case are:the linear and the cubic constitutive relations [6, 13]. The sensitivity to using different constitutive relations in the standard k-ε model is given in Figure

117 Deviation from ISO C CFD C ISO Turbulence model Mass flow rate(kg/s) Average C ISO (%) k-ε C CFD C ISO C ISO (%) R ij -ε Table 3.7: Sensitivity to turbulence models using the structured mesh. Figure 3-28: Sensitivity to different constitutive relations in k- ε model 117

118 ii Sensitivities to meshing topology In order to determine the effect of meshing algorithms, results of CFD simulation for two different meshes are compared. A block-structured mesh generated by ANSYS- ICEM and an unstructured mesh generated by the STAR-CCM+ mesh generator. The cross-sections of the orifice region and perpendicular to the flow for both the block-structured and unstructured meshes are shown in Figure The base cell sizes of the two meshes are comparable as is the total number of cells. The result of the two meshes using the same linear k-ε turbulence model is given in Figure The result indicates a minimal effect of using different meshing algorithms in the case under consideration. The linear k-ε turbulence model is chosen since it was judged to be the appropriate turbulence model of choice. Figure 3-29: Cross sections of unstructured and the block structured meshes. 118

119 Figure 3-30: Sensitivity to different meshing topologies The sensitivity of the mesh generation method is found to be minimal and use of the cubic k-ε model results in a value closer to experimental measurement. This is expected since the cubic constitutive relation is more appropriate for the present flow case compared to the linear k-ε model. It is important to note that the ISO standard is not the benchmark in XYZ028KD geometry since upstream piping geometry does not satisfy the requirements of the standard. Instead, the experimental result should serve as the benchmark for computation Random modeling uncertainties Random modeling uncertainties, given in Table 1.4, are assumed not to introduce any systematic shift in model prediction, and only increase the variance of the prediction. When multiple Random modeling uncertainties are present, methods to combine them are discussed in Section iv. 119

120 i Input uncertainty Input uncertainty arises from uncertainty in input variables to a simulation model and introduced in Section In Case Study 1, a CFD model of a specific experimental flow configuration was created. The input variables to the simulation are subject to random uncertainties due to factors including the fluctuation in environmental temperature during the experiment and is reported in[48]. In this Section, we quantify the uncertainty due to uncertain density ρ input on the simulation result. The sensitivity coefficient method is combined with the bootstrap method given in detail in Appendices E and G respectively. Figure 3-31 shows a sensitivity coefficient of the pressure drop predicted by the model to the changes in density. Then the uncertainty due to random density is estimated using the linear regression fit. The result of input uncertainty estimation is integrated into the Methodology framework in Section Pressure drop (Pa) Simulation result Linear sensitivity coefficient fit Input variable histogram Response variable histogram Input Density (kg/m 3 ) Figure 3-31: Input uncertainty calculation using a combination of sensitivity coefficient method and bootstrap sampling 120

121 ii Discretization uncertainty The methodology proposed by Eça and Hoekstra[21] which is given in Appendix F is adopted to quantify the discretization uncertainty. Figure 3-33 shows the data points corresponding to the meshes of certain base sizes, along with the polynomial curve found from minimizing the error function S given in Equation F.7 for the Case Study discussed in this chapter. The grid sizes taken for grid convergence study have to preserve the same increment ratio for robustness. The finest grid size in Figure 3-33 is 0.4 cm. The following grid sizes were chosen to approximately preserve the increment ratio of 1.2 so that 0.49cm, 0.6cm, 0.72cm, 0.9cm, 1.1cm, and 1.35cm are the following sizes. Moreover, Hoekstra et al. method requires having at least 6 points in order to perform grid convergence study. Figure 3-32: The objective function S given in Appendix F in Equation F.7 plotted as a function of the order of convergence p. For each value of p, S is minimized with respect to α and φ 0. Carrying out calculations according to the prescriptions of the methodology, we 121

122 Figure 3-33: Grid convergence and least square fit according to methodology of Hoekstra et al. Discharge coefficient C is calculated for 7 different meshes at flow rate 1. The finest 5 mesh results were selected for fitting, as coarser mesh results do not exhibit monotonic behavior. get U φ = max (1.25δ RE + U S, 1.25 M) = (3.11) Further analysis on grid convergence of XYZ028KD is necessary for more confidence. It is important to note that the methodology provided by Hoekstra et al. does provide a reasonable order of magnitude estimate for the discretization uncertainty. On the other hand, Hoekstra et al. s method presented in this Section has several limitations in application to Case Study 1. First, the method provides 95% confidence interval for discretization error bounds but does not give guidance to calculate second statistical moment or the standard deviation. This hinders the combination of discretization uncertainty with other numerical uncertainties by Equation 3.12, which is prescribed by the ASME V&V 20 standard. The second limitation identified is the conservative safety factor in cases of a non- 122

123 monotonic grid convergence. In CFD applications aimed at understanding small measurement uncertainties such as the orifice plate mass flow measurement discussed in this chapter, the conservative safety factor results in a CFD result with large discretization uncertainty that constrains the use of the computational result with a high level of confidence. Recognizing that the problem of discretization error estimation in CFD can not be resolved within the scope of this Ph.D. thesis, two assumption-based examples are drawn to demonstrate the flexibility of the overall Methodology framework. First, let us assume that the 95% confidence interval provided by the Hoekstra s method corresponds to a normal distribution. Then, the non-parametric bootstrap method is applied to combine the discretization uncertainty with other types of uncertainties. Figure 3-34A shows a histogram of 10,000 samples drawn from the assumed normal distribution. The discretization uncertainty can then be combined with other numerical uncertainties as demonstrated in Section v. Second, the bootstrap resampling method can be applied to the observed grid convergence data given in Figure The resulting histogram of 10,000 samples drawn from the data is given in Figure 3-34B. Note that the asymmetry in the histogram due to coarse grid results included in the convergence plot given in Figure In Section v, the asymmetric histogram is combined with other numerical uncertainties by using the bootstrap method for demonstration [A] 1.22 P(Pa) [B] 1.23 P(Pa) Figure 3-34: Two examples of discretization uncertainty confidence interval. Histogram A) was constructed assuming the normal distribution for Hoekstra method, histogram B) was constructed by resampling from the data given in Figure

124 iii Sampling uncertainty Sampling uncertainty arises due to a finite number of samples drawn from a probability distribution as introduced in Section Details of parametric and nonparametric methods to quantify the sampling uncertainty is given in Appendix G. In this Section, an application of the bootstrap resampling method to quantify sampling uncertainty in Unsteady RANS simulation of Section is illustrated. Figure 3-35 shows the pressure drop across the orifice calculated from a URANS simulation with the cubic k- turbulence model. The green line represents the time-averaged value of the URANS simulation result, which is then used for further calculations. The purpose of the bootstrap resampling procedure is to quantify the uncertainty of the time-averaged value URANS result Time average Histogram of URANS result P(Pa) Time(sec) Figure 3-35: Result for the pressure drop across an orifice plate of a URANS simulation using the cubic k-ε turbulence model. The histogram represents the distribution of time-dependent simulation result given in red. The green line is the time-average of the simulation result. The histogram given in Figure 3-35 is an alternative representation of the simulation result given in red. The parametric methods to quantify the uncertainty of the timeaverage value relies on the underlying distribution to be normal. In practice, the 124

125 P(Pa) 10 4 Figure 3-36: Histogram representing the distribution of the time-average value constructed using bootstrap resampling procedure with replacement. observed distributions can have long tails and asymmetry as shown in Figure The bootstrap resampling method can be applied in this example to calculate the uncertainty of the time-average pressure drop value given by a green line in Figure Figure 3-36 shows the resulting distribution of the time-averaged mean of the pressure drop given in Figure The histogram is constructed by uniformly resampling times from the empirical distribution given in Figure 3-35 with replacement. This distribution can be used in quantifying the total Random modeling uncertainty as given in Equation iv Iterative uncertainty Similar to the sampling uncertainty discussed in Section iii, iteration uncertainty needs to be quantified in steady RANS simulation. The origin of the iterative uncertainty is introduced in Section The convergence plot, which shows the quantity of interest as a function of iteration for the Case Study 1 CFD model is given in Figure Based on the convergence plot, iterative uncertainty can be quantified using the non-parametric bootstrap resampling method. Two separate cases can be 125

126 Steady RANS result Average over iterations Histogram of RANS result P(Pa) Iteration Figure 3-37: Convergence plot of a steady-state RANS simulation using a linear k- ε turbulence model. The histogram represents the distribution of time-dependent simulation result given in red. The green line is the time-average of the simulation result. identified in estimating the iterative uncertainty. If the quantity of interest of the simulation is not averaged over the number of iterations, and the last value is used then the histogram given in blue in Figure 3-37 is used as the probability distribution of the iterative uncertainty. It can then be combined the other random uncertainties as given in Equation Otherwise, if the result of the steady-state RANS simulation is averaged over the number of iterations as given by the green line in Figure 3-37, then the bootstrap method can be employed. Figure 3-38 shows an empirical probability distribution for the iteration-averaged value, which is constructed based on resampling from the empirical distribution given in blue in Figure This distribution can be used in quantifying the total Random modeling uncertainty as given in Equation

127 P(Pa) 10 4 Figure 3-38: The distribution of the iteration-averaged value constructed using bootstrap resampling procedure with replacement. v Combining numerical uncertainties ASME standard on V&V of CFD[14] prescribes a method to combine uncertainties in a validation experiment used to validate a CFD code. In the ASME standard, numerical error, experimental error, and input uncertainties are considered. total validation uncertainty is calculated as u val = The u 2 num + u 2 input + u 2 D (3.12) where u D,u input,u num are the standard deviations of probability distributions of the validation experiment uncertainty, CFD input uncertainty and the numerical uncertainty of the CFD model. The standard deviation of a total uncertainty can be calculated as in Equation 3.12 only if the probability distributions of the simultaneous uncertainties are normal[25] and uncorrelated. When the probability distributions of the simultaneous uncertainties are not all normal and if they are expressed by continuous distributions, the convolution integral should be employed to calculate the combined probability distribution[25]. 127

128 Alternatively, non-parametric bootstrap method can be applied in combining distributions given expressed by the observed data. Here, the application of a non-parametric method to combining discretization uncertainty with sampling uncertainty is demonstrated. This situation arises when performing time-dependent CFD simulation and the result of the simulation is averaged over simulation time. Two cases of treating the probability distribution of the discretization uncertainty are considered as discussed in Section ii. The resulting combined uncertainties are given in Figure Combined uncertainty Discretization uncertainty Sampling uncertainty Combined uncertainty Discretiation uncertainty Sampling uncertainty [A] P(Pa) [B] P(Pa) Figure 3-39: Combined discretization and sampling uncertainties. The histogram in A) shows the resulting distribution assuming the normal distribution for discretization. B) shows the bootstrap resampled result. 3.6 Application of the Methodology framework to Temporal Representativeness uncertainty This Section is concerned with the Temporal Representativeness uncertainty. Here, the Methodology framework is applied to the technical analysis presented in Section 3.3. The technical analysis of Section 3.3 is referred to as Case Study 1a. In each step, a general schematic figure and a specific schematic to Case Study 1a are given. Steps involving a detailed complex schematic specific to Case Study 1a, starting with Figure 3-43, do not include the general schematic to simplify the flow of discussion. 128

129 3.6.1 Defining the Reference True Value The Reference True Value of the mass flow rate across a cross section of a pipe is an integral of the velocity as a function of space as given in Equation 3.13 q T = 1 τ τ t=0 ρv(x) ndadt (3.13) A pipe Since the Reference True Value of the velocity field internal to the pipe is not available, the Reference True Value is defined by q R = π 4 CEεd2 2ρ 1 τ τ t=0 [p 1 (t) p 2 (t)] dt (3.14) The objective of this Section is to address the Temporal Representativeness uncertainty only. Thus the ISO expression for mass flow rate is used. Figure 3-40A shows first the general Methodology framework step defining the Reference True Value and Figure 3-40B shows the specific application to Case Study 1a. 1. Defining Reference True Value 1. Defining Reference True Value Reference True Value (T R ) Reference True Value (T R ) q R = π 4 CEεd 2 2ρ 1 τ p 1(t) p t=0 2(t) τ [ ]dt [A] Experiment [B] Experiment Figure 3-40: The Reference True Value of the orifice plate measurement method is defined using the ISO standard Defining Temporal Representativeness uncertainty The Reference True Value given in Equation 3.14 requires continues time measurement of the pressure drop in order to carry out the integral. However, in practice, the pressure drop is sampled at a given finite frequency due to the constraints of the equipment installed on site. So the Measurement Value of the flow rate is given in 129

130 Equation 3.15 as q M = π 4 CEεd2 2ρ 1 N N [p 1 (t k ) p 2 (t k )] (3.15) k=1 where the main difference from the reference flow rate is the discrete sampling of pressure drop as opposed to continuous time integration. Figure 3-41B shows the Reference True Value and the Measurement Values of the mass flow rate. The difference between the Reference True Value and the Measurement Value constitutes a Temporal Representativeness uncertainty. 2. Defining Representativeness uncertainty 2. Defining Representativeness uncertainty Reference True Value (T R ) Representativeness uncertainty Reference True Value (T R ) q R = π 4 CEεd 2 2ρ 1 τ p 1(t) p t=0 2(t) τ [ ]dt Representativeness uncertainty Measurement Value (M) Measurement Value (M) q M = π 4 CEεd 2 2ρ 1 N [ p 1(t k ) p 2(t k )] N k=1 [A] Experiment [B] Experiment Figure 3-41: Representativeness uncertainty arises from the difference between the Reference True Value and the Measurement Value given by the orifice method Defining two categories of Representativeness uncertainty (Temporal and Spatial) In Case Study 1a, Spatial aspects of the measurement of mass flow rate by an orifice plate are not considered, and Representativeness uncertainty is purely temporal as given in Figure 3-42B. 130

131 3. Defining two categories of Representativeness uncertainty 3. Defining two categories of Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) Spatial Representativeness uncertainty Temporal Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) q R = π 4 CEεd 2 2ρ 1 τ p 1(t) p t=0 2(t) τ [ ]dt q M = π 4 CEεd 2 2ρ 1 N [ p 1(t k ) p 2(t k )] N k=1 Temporal Representativeness uncertainty [A] Experiment [B] Experiment Figure 3-42: In Case Study 1a, only Temporal component of Representativeness uncertainty is considered Simulation model building and Calculating Random modeling uncertainty In this step, a simulation model of piping and orifice plate system is built. LES is carried out to characterize the Temporal Representativeness uncertainty as described in Section 3.3. The relevant Random modeling uncertainty considered in Case Study 1a is sampling uncertainty. The total Random modeling uncertainty is then given as σ MS = σ 2 r (3.16) where σ r is the sampling uncertainty of the simulation. Figure 3-43 shows the pressure drop which is used to calculate mass flow rate as given in Equation 3.15 for the LES result. The value shown by the red triangle is the pressure drop value sampled at 0.5Hz. This frequency is chosen to match the frequency at which the pressure signal is sampled at a power plant. The black diamond is the time-average value for the entire length of LES simulation which serves as the Simulated Reference True Value. The error bars indicate the Random modeling uncertainty associated with the both values. The Random modeling uncertainty is shown to overwhelm the difference 131

132 between the Simulated Reference True Value (T R S ) and the Simulated Measurement Value (M S ) i.e. the Simulated Temporal Representativeness uncertainty. The implication of this is that the LES simulation can not distinguish Temporal Representativeness due to low signal to noise ratio indicated by the ratio between Temporal Representativeness and the standard deviation of the Random modeling uncertainty. The random sampling uncertainty can be reduced by a higher sampling frequency, longer signal length or the combination of the two. The signal length is chosen in Figure 3-43 is associated with the computational cost limitation in LES. 4. Simulation model building 5. Calculating Random modeling uncertainty Simulated Representativeness uncertainty LES Reference True True Value(T Value R S) LES Measurement Value(M Value S ) Figure 3-43: The LES simulation result has a significant sampling uncertainty 132

133 3.6.5 Assessing the systematic Validation uncertainty The systematic Validation uncertainty is calculated by comparing the Measurement Value (M) of the test with the Simulated Measurement Value (M s ). It is an indicator of how far the simulation result is from the measurement. Figure 3-44 shows the validation uncertainty between the Simulated Measurement Value and the test Measurement Value. The sampling uncertainty due to turbulent fluctuation dominates any systematic uncertainty of interest. 6. Assessing the systematic Validation uncertainty Systematic Validation uncertainty Test Reference True Value (T R ) Test Measurement Value (M) LES Reference True Value (T R S ) LES Measurement Value (M S ) Figure 3-44: Systematic shift exists between test Measurement Value and Simulated Measurement Value. 133

134 3.6.6 Creating combined Methodology schematic 7. Creating combined Methodology schematic Representativeness uncertainty Test Reference value Test Measured value LES Reference value Simulated LES Measured value Representativeness uncertainty Test Reference True Value (T R ) Test Measurement Value (M) LES Reference True Value (T R S ) LES Measurement Value (M S ) 136 Figure 3-45: All uncertainties considered are comprehensively combined in this figure. Uncertainty due to sampling from a fluctuating pressure drop signal to measure the mass flow rate by using the orifice plate is considered in this Section. Figure 3-45 shows all uncertainties considered in Case Study 1a combined in one figure. In Case Study 1a, a LES simulation was created to replicate the test measurement, and Random modeling uncertainty which is purely due to sampling uncertainty is calculated. The Simulated Temporal Representativeness uncertainty (M S TS R ) is found to be too small to be determined from a highly fluctuating pressure drop signal in the LES as shown in Figure

135 3.6.7 Creating a workflow chart Reference True Value (T R ) Simulated Reference True Value (T R S) Measurement Value (M) Simulated Measurement Value (M S ) 4 Case 1a Case 1a Experimental data 1 Simulation model 4 2 Uncertainty of model 3a 3b Analysis of representativeness using the model [A] Measurement Simulation [B] 70 Figure 3-46: A) Summary of the application of the Methodology schematic. B) Workflow in Case Study 1a. Steps 3a and 3b represent the same analysis. As shown in Figure 3-46A, the analysis started with the test Measurement Value and used it as an input to the LESimulation model in step 1. Then, the Random modeling uncertainties of the simulation model are quantified in step 2. In step 3, the simulation model was used to quantify the Simulated Temporal Representativeness uncertainty and the Temporal Representativeness uncertainty. The Simulated Representativeness uncertainty is then compared with the Representativeness uncertainty from the test in Step 4. They are not expected to be the same regardless of the quality of the simulation since they are statistical in nature. This concept is shown in Figure 3-46A by not having a connecting arrow between them. The workflow of Case Study 1a illustrated in Figure 3-46B shows the general steps of the analysis in Case Study 1a. The steps 3a and 3b are part of the same analysis of the Simulated Representativeness uncertainty. 3.7 Application of the Methodology framework to Spatial Representativeness uncertainty This Section is concerned with the Spatial Representativeness uncertainty. Here, the Methodology framework is applied to the technical analysis presented in Section

136 The technical work of Section 3.4 is referred to as Case Study 1b. In the following subsections, the steps of the Methodology framework, given in Section 2.3 in the general the form, are applied to Case Study 1b. In each step, a general schematic figure and a specific schematic to Case Study 1b are given. Steps involving a detailed complex schematic specific to Case Study 1b do not include the general schematic to simplify the flow of discussion Defining the Reference True Value The mass flow rate across a cross section of a pipe is an integral of the velocity as a function of space as given in Equation 3.17 q T = ρ v nda (3.17) A pipe The strict definition given above is the Reference True Value of mass flow rate in Case Study 1b which deals with the Spatial Representativeness uncertainty. It is different from Equation 3.13 since the temporal variation is not considered in Case Study 1b. In the absence of accurate experimental techniques such as PIV, the velocity field is hard to obtain, and the integral cannot be carried out. Therefore, approximate measurement techniques such as the orifice plate method are employed for their practicality. Figure 3-47A shows the general Methodology framework step and Figure 3-47B shows the specific application to the Case Study 1b. 136

137 1. Defining Reference True Value 1. Defining Reference True Value Reference True Value (T R ) Reference True Value (T R ) q T = A pipe ρ! υ(x)! n da [A] Experiment [B] Experiment Figure 3-47: The Reference True Value of the orifice plate measurement method is defined in terms of the velocity field internal to the pipe 137

138 3.7.2 Defining Spatial Representativeness uncertainty The principle of the orifice plate method is described in Section 3.2. In the absence of any consideration of the temporal fluctuations, the mass flow rate using an orifice plate is measured based on the pressure drop across the plate as q M = π 4 CEεd2 2ρ(p 1 p 2 ) (3.18) where the discharge coefficient C is prescribed by the ISO standard[36]. A difficulty in applying the ISO standard is in flow configurations with a non-straight upstream piping geometry. The penalty imposed for applying the standard on non-straight geometry is large and results in a conservative margin. Figure 3-48B shows the Reference True Value and the Measurement Value of the mass flow rate. The difference between the two constitutes a Spatial Representativeness uncertainty. 2. Defining Representativeness uncertainty 2. Defining Representativeness uncertainty Reference True Value (T R ) Representativeness uncertainty Reference True Value (T R ) q T = A pipe ρ! υ(x)! n da Representativeness uncertainty Measurement Value (M) Measurement Value (M) q M = π 4 CEεd 2 2ρ ( p 1 p 2 ) [A] Experiment [B] Experiment Figure 3-48: Representativeness uncertainty arises from the difference between the Reference True Value and Measurement Value given by the orifice method Defining two categories of Representativeness uncertainty (Temporal and Spatial) In Case Study 1b, the temporal aspects of the measurement of mass flow rate by an orifice plate are not considered, and Representativeness uncertainty is purely Spatial as given in Figure 3-49B. 138

139 3. Defining two categories of Representativeness uncertainty 3. Defining two categories of Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) Spatial Representativeness uncertainty Temporal Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) q T = A pipe ρ! υ(x)! n da q M = π 4 CEεd 2 2ρ ( p 1 p 2 ) Spatial Representativeness uncertainty [A] Experiment [B] Experiment Figure 3-49: In Case Study 1b, only Spatial component of Representativeness uncertainty is considered Simulation model building and Calculating Random modeling uncertainty In this step, RANS CFD simulation models of the experimental flow configuration XYZ028KD is constructed. Then, the Random modeling uncertainties of the simulation models are quantified and combined. Two turbulence models are employed. First, linear k-ε model was used to conduct a steady RANS simulation. In this simulation, input, discretization and iterative uncertainties are applicable. Therefore, the total Random modeling uncertainty of a RANS simulation of the experimental flow configuration is calculated as σ MS (linear k ε) = σd 2 + σ2 in + σit 2 σd 2 + σ2 in (3.19) where σ d is the discretization uncertainty, σ in is the input uncertainty and σ it is the iterative uncertainty. The discretization uncertainty is quantified using the methods proposed by Hoekstra et al.[21]. A Gaussian probability distribution is assumed for the probability distribution of the discretization uncertainty and for the input uncertainty. The iterative uncertainty was found to be negligible and excluded. The resulting Simulated Measurement Value using linear k-ε turbulence model is given in Figure 3-50 by red. 139

140 Second, the cubic k-ε model was used to conduct Unsteady RANS simulation. In this simulation, input, discretization and sampling uncertainties are applicable. Therefore, the total Random modeling uncertainty of a RANS simulation of the experimental flow configuration is calculated as σ MS (cubic k ε) = σd 2 + σ2 in + σr 2 σd 2 (3.20) where σ d is the discretization uncertainty, σ in is the input uncertainty and σ r is the sampling uncertainty. For the cubic k-ε model, the discretization uncertainty was found to be large calculated from the Hoekstra method. Therefore, the blue error bar for the cubic k-ε model only includes the discretization uncertainty in Figure The Representativeness uncertainty predicted by the simulation is the difference between the Simulated Reference True Value and the Simulated Measurement Value using the linear and the cubic k-ε models. In Figure 3-50, the linear k-ε model is shown to predict the experimentally observed trend of Figure 3-21 qualitatively incorrectly whereas the cubic k-ε model correctly predicts that the use of the orifice plate method will over-predict the true mass flow rate when the non-straight upstream geometry is present. 140

141 4. Simulation model building 5. Calculating Random modeling uncertainty 306 Simulated Reference True Value (T R S ) 304 Simulated Measurement Value (M S, cubic k-ϵ) Mass flow rate(m 3 /h) Simulated Measurement Value (M S, linear k-ϵ) Experiment Simulation Figure 3-50: CFD model of orifice plate method has significant discretization uncertainty Assessing the systematic Validation uncertainty Systematic Validation uncertainty is calculated by comparing the experimental Measurement Value (M) with the Simulated Measurement Value (M s ). It is an indicator of how far the simulation result is from the measurement. Figure 3-51 shows the validation uncertainty between the Simulated Measurement Value and the experimental Measurement Value. 141

142 6. Assessing the systematic Validation uncertainty 306 Reference True value of Experiment (T R ) Measured value of Experiment (M) Simulated Reference True Value (T R S ) Mass flow rate(m 3 /h) Simulated measured Value (M S, cubic k-ϵ) Simulated measured Value (M S, linear k-ϵ) Systematic Validation uncertainty using Cubic k-ε model Systematic Validation uncertainty using Linear k-ε model 296 Experiment Simulation Figure 3-51: Systematic shift exists between experimental Measurement Value and Simulated Measurement Value Creating combined Methodology schematic Figure 3-52 shows all uncertainties considered in Case Study 1b combined in one figure. In this figure, the Spatial Representativeness uncertainty is addressed using a RANS CFD simulation model. The Simulated Representativeness uncertainty is shown for the simulation using the cubic k-ε since it performs better in this flow configuration. The Random modeling uncertainties are shown by error bars in Figure

143 7. Creating combined Methodology schematic 306 Reference True value of Experiment (T R ) Measured value of Experiment (M) Simulated Reference True Value (T R S ) 304 Simulated measured Value (M S, cubic k-ϵ) Simulated measured Value (M S, linear k-ϵ) Mass flow rate(m 3 /h) Representativeness uncertainty Simulated Representativeness uncertainty using Linear k-ε model Simulated Representativeness uncertainty using Cubic k-ε model Experiment Simulation Figure 3-52: All uncertainties considered are comprehensively combined in this figure Creating a workflow chart As shown in Figure 3-53A, the analysis started with the experimental Measurement Value and used it as input to RANS CFD model in step 1. Then, the Random modeling uncertainties of the simulation model are quantified in step 2. In step 3, the simulation model was used to quantify the Spatial Representativeness uncertainty. The Representativeness uncertainty in the simulation was then compared with the Representativeness uncertainty from the experiment in step 4. Unlike Case Study 1a, they are expected to be the same if the quality of the simulation is high. This concept is shown in Figure 3-53A by having a black connecting arrow between them. The workflow of Case Study 1b illustrated in Figure 3-53B shows the general steps of the analysis in Case Study 1b. The steps 3a and 3b are part of the same analysis of the Simulated Representativeness uncertainty. 143

144 Reference True Value (T R ) Simulated Reference True Value (T R S) Measurement Value (M) Simulated Measurement Value (M S ) 4 Case 1b Case 1b Experimental data 1 Simulation model 4 2 Uncertainty of model 3a 3b Analysis of representativeness using the model [A] Measurement Simulation [B] 71 Figure 3-53: A) Summary of the application of the Methodology schematic. B) Workflow in Case Study 1b. Steps 3a and 3b represent the same analysis. 3.8 Combining Spatial and Temporal Representativeness Two types of Representativeness uncertainties are considered in Case Study 1. Spatial and Temporal. The Spatial Representativeness was studied using experimental data from Facility#1 and RANS simulation as given in Sections 3.4. The Temporal Representativeness was studied using test data from an operating nuclear plant and a Large Eddy Simulation as given in Section 3.3. Since the Spatial and Temporal aspects are studied in separate experiments and simulations, it is not necessary to combine them in a unified Methodology framework application. However, it is possible that both exist simultaneously at a power plant for example when piping connections have non-straight junctions, and there is an oscillation in the flow. In such situations, it is necessary to have a simulation that can capture both Spatial and Temporal effects. The large eddy simulation in this chapter was conducted on a straight geometry to match the test geometry. The mesh size requirements in LES that are needed to resolve large turbulent eddies are computationally prohibitively expensive in industrial geometries with high Reynolds number. 144

145 3.9 Application of the Methodology using the nonparametric bootstrap method to Spatial Representativeness problem Methodology implementation given in Sections 3.7 and 3.6 rely on the assumption of normal probability distribution of the uncertainty terms, which can not be justified completely for reasons discussed in Section 2.2. Various uncertainty sources of Case Study 1b are treated non-parametrically in Section In this Section, we combine them into a combined Methodology schematic. The non-parametric bootstrap method was applied to Case Study 1b for the case of the cubic k-ε model simulation. The application of the cubic k-ε model requires the consideration of input, discretization and sampling uncertainties. In the nonparametric implementation, the steps from "Defining the Reference True Value" to "Defining two categories of Representativeness uncertainty (Temporal and Spatial)" are identical to Sections through Therefore they are not repeated in this section Simulation model building and Calculating Random modeling uncertainty In this step, the URANS CFD simulation model of the experimental flow configuration XYZ028KD is constructed using the cubic k-ε model. Then, the Random modeling uncertainties of the simulation models are quantified and combined using the nonparametric bootstrap method. The relevant uncertainty sources are input, discretization and sampling uncertainties. The total Random modeling uncertainty of a RANS simulation of the experimental 145

146 flow configuration is calculated as σ MS (cubic k ε) = σ 2 d + σ2 in + σ 2 r (3.21) where σ d is the discretization uncertainty, σ in is the input uncertainty and σ r is the sampling uncertainty. The operation to add these uncertainties is performed nonparametrically in this section using the bootstrap method implemented in Section v. Figure 3-54 shows the input, discretization and sampling uncertainty sources quantified in Section 3.5 by histograms. The total Random modeling uncertainty for the Simulated Measurement Value is given by a blue histogram. 4. Simulation model building 5. Calculating Random modeling uncertainty 306 Measurement Value of Experiment(M) 304 Simulated Reference True Value (T R S ) Simulated Measurement Value (M S, cubic k-ϵ) M S Mass flow rate(m 3 /h) T R S σ disc σ input 298 σ M 296 Experiment Simulation Figure 3-54: CFD model of orifice plate method in XYZ028KD configuration. Input, discretization and sampling uncertainties are each quantified using the boostrap method in Section

147 3.9.2 Assessing the systematic Validation uncertainty Systematic Validation uncertainty is calculated by comparing the experimental Measurement Value (M) with the Simulated Measurement Value (M s ). It is an indicator of how far the simulation result is from the measurement. Figure 3-55 shows the validation uncertainty between the Simulated Measurement Value and the experimental Measurement Value. 6. Assessing the systematic Validation uncertainty 306 Measurement Value of Experiment (M) Simulated Reference True Value(T R S ) M S Mass flow rate(m 3 /h) Simulated Measurement Value (M S, cubic k-ϵ) M Systematic Validation uncertainty T R S Experiment Simulation Figure 3-55: Systematic shift exists between experimental Measurement Value and Simulated Measurement Value Creating combined Methodology schematic Figure 3-56 shows the combined Methodology schematic with random uncertainties treated non-parametrically. In this figure, the Measurement Value M is used as a boundary condition to a URANS simulation performed using the cubic k-ε turbulence model and the Random modeling uncertainties of the simulation are quantified using the non-parametric bootstrap method. The corrected Measurement Value M c is then 147

148 obtained by adding the information gained from the simulation to the Measurement Value M. 7. Creating combined Methodology schematic 306 Measurement Value of Experiment (M) Simulated Reference True Value (T R S ) M S 304 Simulated Measurement Value (M S, cubic k-ϵ) M Mass flow rate(m 3 /h) T R S M C Experiment Simulation Figure 3-56: method Combined Methodology schematic using non-parametric bootstrap The steps to create a work flow chart is the same as discussed in Section except for the Random modeling uncertainty quantification. In Section 3.7, the normal probability distribution was used to quantify the Random modeling uncertainties. In this section, non-parametric bootstrap method is used. The addition of Random modeling uncertainty was also performed differently Conclusion Flow rate measurement by means of the orifice plate is considered in this chapter. Several experimental and simulation approaches are analyzed and uncertainties as- 148

149 sociated with the experiment and the simulation are evaluated for development of a Methodology for the treatment of Representativeness uncertainty. The CFD tool STAR-CCM+ was used to 1. Identify and qualitatively explain the frequency spectra of pressure drop signals observed during tests at an operating nuclear plant. 2. Identify and quantify deviation from using the ISO standard to calculate mass flow rate from the pressure drop across an orifice plate when upstream and downstream piping geometries are deviant from the prescription in the standard. Also, the following uncertainties associated with the RANS CFD model are quantified as follows Input uncertainty is quantified using a combination of sensitivity coefficient method and the bootstrap method. Discretization uncertainty is first quantified by following the Hoekstra et al. methods. Then it was quantified using the bootstrap method for comparison. The value quantified by the bootstrap method is then combined with other uncertainties. Sampling uncertainty of URANS simulation is quantified using the bootstrap method. The iterative uncertainty of a steady RANS simulation is quantified using the bootstrap method. Lastly, the Methodology is applied to characterize the Temporal and the Spatial Representativeness uncertainties. In Case Study 1a, the Methodology is applied to quantify the Temporal Representativeness uncertainty using the test data signal and LES performed under the same flow conditions as the test. Both the test data and the LES result indicate that the Temporal Representativeness uncertainty is too small to be resolved due to the sampling uncertainty. The sampling uncertainty is quantified 149

150 assuming the normal probability distribution. In Case Study 1b, RANS simulations were performed for the experimental flow configuration XYZ028KD in order to quantify the Spatial Representativeness uncertainty. The Simulated Spatial Representativeness uncertainty obtained using the cubic k-ε model was qualitatively consistent with the experimental results. The Random modeling uncertainties of the RANS simulation model using the cubic k-ε model is quantified using two methods. The method assuming the normal probability distribution method presented difficulty in quantifying and combining different Random modeling uncertainty sources while the non-parametric bootstrap method was shown to be flexible in such tasks. 150

151 Chapter 4 Case Studies 2,3 and 4 This chapter presents the condensed versions of Case Studies 2, 3 and 4. Full details of the technical analyses of these Case Studies are given in Appendices A, B and C respectively. The condensed presentation is done in order to highlight the important technical outcomes of each Case Study and to illustrate how the Methodology developed in Chapter 2 is applied to each Case Study. The parametric version of the Methodology which assumes the normal distribution for uncertainty sources is applied in this chapter. In the future, the non-parametric implementation should be applied to the Case Study 2, 3 and 4. In presenting each Case Study, the following structure is adopted. First, the introduction and a summary of main technical findings are given for each Case Study. Second, the Methodology framework is applied to structure the technical analysis of each Case Study. 151

152 4.1 Case Study 2: Steam Generator recirculation ratio measurement by means of radioactive or chemical tracer Introduction and main findings The Steam Generator (SG) is the main heat exchanger between the primary and secondary sides in a PWR and its performance is critical to the safe and efficient operation of the plant. In this section, the process of measuring the SG recirculation ratio using radioactive or chemical tracer is investigated using both experimental data and computational simulation. The recirculation ratio is an important parameter for the commissioning and the operation of the SG. It is shown experimentally that the measurement result of the recirculation ratio using tracers depends on the location of sampling within the SG. Specifically, the angular location at which the tracer concentration measurement is taken is shown to play a significant role in the measurement result. The computational fluid dynamics (CFD) code STAR-CCM+ is used to perform a simulation to investigate the phenomenon. Note that the geometry used to produce the CFD model is an approximation to the real geometry of the SG. Therefore, it is not possible to correct the experimentally obtained recirculation ratio using the result of the CFD model by producing a Corrected Measurement Value M c. Figure 4-1 shows a result of the CFD simulation showing the recirculation ratio in a histogram for both the hot and cold side of the SG illustrating the phenomenon. The results of the CFD simulation indicate a systematic overprediction of recirculation ratio in the hot-side sampling and a systematic underprediction in the cold-side. The finding was validated by the experimental data consistently. In the analysis of the experimental data, the sampling uncertainty of the recirculation ratio measurement was treated using the normal probability distribution. Figure

153 shows that the distribution of the recirculation ratio cannot be considered a normal random variable due to the functional dependence on the angular location of the sampling location. 153

154 100 Probe height h=0.68, r=1.59m 6cm mesh Measured recirculation ratio SG hot Side SG cold side Local recirculation ratio Figure 4-1: Histogram of the local recirculation ratio, 6cm base size mesh result. The measurement results in the hot side of the SG are illustrated in red and in the cold side are illustrated in blue. The results of the experiments and the simulation are integrated using the Methodology framework presented in Chapter 2 to characterize the Representativeness uncertainty of the recirculation measurement using a tracer Application of the Methodology framework In this section, the Methodology framework is applied to the experimental test and the simulation model to characterize the Representativeness uncertainty of the SG recirculation ratio considered in Case Study 2. In the following subsections, the steps of the Methodology framework, given in Section 2.3 in general form, are applied to Case Study 2. The steps involving definitions and workflow charts are illustrated in a general schematic figure and a specific schematic applicable to Case Study 2. The steps involving detailed complex schematics specific to Case Study 2 i.e. Figures 4-5, 4-6 and 4-7 do not include the general schematic to clarify the discussion. Table 4.1 shows the relevant general Methodology terminology adopted for the specific quantities in Case Study

155 General Methodology terminology Reference True Value (T R ) Measurement Value (M) Simulated Reference True Value (TS R ) Simulated Measurement Value (M S ) Case Study 2 quantities Vendor-specified value of the recirculation ratio The recirculation ratio measured by the tracer method during an experiment Recirculation ratio imposed as a boundary condition in a CFD simulation Recirculation ratio calculated using the tracer method in a CFD simulation Table 4.1: General Methodology terminology adopted for the specific quantities in Case Study 2 i Defining the Reference True Value The SG recirculation ratio is defined as r T Q R + Q A Q A (4.1) where Q R is the SG recirculation flow rate and Q A is the SG feedwater flowrate. The definition given above is considered to be the Reference True Value of recirculation ratio and is plotted in Figure 4-2B. Typically, the vendor company supplying the SG provides the value of the recirculation ratio to the utility owner. However, after commissioning the SG, the recirculation ratio is not measured continuously by the operators. Therefore, approximate measurement techniques such as the tracer method are employed to determine the recirculation ratio. 155

156 1. Defining Reference True Value 1. Defining Reference True Value Reference True Value (T R ) Reference True Value (T R ) r T = Q A +Q R Q A [A] Experiment [B] Experiment Figure 4-2: The Reference True Value of the tracer measurement method is defined in terms of the feedwater and recirculation flow rates ii Defining Representativeness uncertainty As described in Section A.3.1, the tracer method involves dissolving a chemical or radioactive tracer in the secondary side coolant of a PWR during operation. The concentration of the tracer is measured in multiple locations to indirectly measure the recirculation ratio. If there are N locations at the same height around the downcomer where the tracer concentration is measured, the Measurement Value of the recirculation ratio is given as r M = 1 N N i=1 C R C A C R C i (4.2) where C R and C A are the average tracer concentrations in the recirculation flow and the feedwater flow respectively. C i is the local value of the tracer concentration at the i-th measurement location. As shown in Figure 4-3B, the difference between r T and r M is the Spatial Representativeness uncertainty in the SG recirculation ratio measurement by tracer method. 156

157 2. Defining Representativeness uncertainty 2. Defining Representativeness uncertainty Reference True Value (T R ) Representativeness uncertainty Reference True Value (T R ) r T = Q A +Q R Q A Representativeness uncertainty Measurement Value (M) Measurement Value (M) r M = 1 N N i=1 C R C A C R C i [A] Experiment [B] Experiment Figure 4-3: Representativeness uncertainty arises from the difference between the True Value and Measurement Value given by the tracer method. iii Defining two categories of Representativeness uncertainty (Temporal and Spatial) In Case Study 2, the temporal aspects of the measurement of recirculation ratio measurement are not considered, and the Representativeness uncertainty is purely Spatial as given in Figure 4-4B. 3. Defining two categories of Representativeness uncertainty 3. Defining two categories of Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) Spatial Representativeness uncertainty Temporal Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) r T = Q A +Q R Q A r M = 1 N N i=1 C R C A C R C i Spatial Representativeness uncertainty [A] Experiment [B] Experiment Figure 4-4: In Case Study 2, only the Spatial component of Representativeness uncertainty is considered. iv Simulation model building and Calculating Random modeling uncertainty In this step, a RANS CFD model of the SG is built and the SG recirculation ratio measurement is simulated. A non-dimensional concentration is used to simulate the 157

158 tracer dispersion in the CFD model of the SG operation. The results of the CFD simulation of the flow in the SG are presented in Section A.4.3. In Case Study 2, the iterative, discretization, and sampling uncertainties are the relevant Random modeling uncertainties. These uncertainties are discussed in Sections viii and A.4.2. The input uncertainty is omitted. The reason is the lack of information regarding the experimental test condition such as the temperature of the secondary coolant fluid and its uncertainty. The total Random modeling uncertainty of the Simulated Measurement Value of the recirculation ratio is σ MS = σ 2 it + σ 2 d + σ2 r (4.3) where σ it is the standard deviation due to iterative uncertainty, σ d is the discretization uncertainty and σ r is the sampling uncertainty. The Simulated Measurement Value is indicated by the filled red diamond in Figure 4-5 and is an average of the 4 CFD calculations of tracer concentration. The Simulated Reference True Value in Figure 4-5 is imposed as a boundary condition in a CFD simulation. The Random modeling uncertainty is large compared to the Simulated Representativeness uncertainty suggesting that the uncertainty due to sampling from a chaotically mixing tracer in the SG dominates any systematic effect that the simulation is trying to isolate. 158

159 4. Simulation model building 5. Calculating Random modeling uncertainty Simulated Measurement Value (M S ) Simulated Reference True Value (T R S ) Random modeling uncertainty Simulated Representativeness uncertainty Figure 4-5: CFD model of tracer method has significant iterative and discretization random uncertainties. v Assessing the systematic Validation uncertainty Ideally, the Simulated Reference True Value should be chosen so that the resulting Simulated Measurement Value (M S ) matches the Measurement Value. Due to the chaotic nature of the tracer mixing, an arbitrary value of 4.0 is chosen. This introduces a systematic Validation uncertainty. The systematic Validation uncertainty is calculated by comparing the Measurement Value (M) with the Simulated Measurement Value (M s ). It is an indicator of how far the simulation result is from the measurement. Figure 4-6 shows the systematic Validation uncertainty. 159

160 6. Assessing the systematic Validation uncertainty Simulated Measurement Value (M S ) Measurement Value (M) Systematic Validation uncertainty 101 Figure 4-6: Systematic modeling uncertainty exists between the Measurement Value and the Simulated Measurement Value. vi Creating combined Methodology schematic In Case Study 2, the SG recirculation ratio measurement by means of a tracer is considered. A CFD simulation was created to examine the mixing of a chemical tracer in the single-phase region of the SG, and to assess and quantify the Representativeness uncertainty of the measurement of recirculation ratio. The recirculation ratio is shown to be overpredicted using the tracer method. The Simulated Representativeness uncertainty indicates that the recirculation ratio is underpredicted by the tracer method. This shows a qualitative inconsistency between the test result and the simulation result. Figure 4-7 shows all uncertainties considered in Case Study 2 combined in one figure. 160

161 7. Creating combined Methodology schematic Simulated Measurement Value (M S ) Simulated Reference True Value (T R S ) Measurement Value (M) Reference True Value (T R ) Representativeness Uncertainty Simulated Representativeness Uncertainty Figure 4-7: All uncertainties considered are comprehensively combined in this figure. vii Creating a workflow chart As shown in Figure 4-8A, the analysis in Case Study 2 started with the Measurement Value. In step 1, the experimental conditions were used as inputs to create a RANS CFD simulation model. The Random modeling uncertainties of the model were quantified in step 2. In step 3, the model was used to quantify the Spatial Representativeness uncertainty. The Representativeness uncertainty in the simulation was then compared with the Representativeness uncertainty from the experiment as shown by the black diagonal arrow. The workflow of Case Study 2 is given in Figure 4-8B. 161

162 Reference True Value (T R ) Simulated Reference True Value (T R S) Measurement Value (M) Simulated Measurement Value (M S ) Case Case 2 Experimental data Physics model development 1 2 Uncertainty of model 3a 3b Analysis of representativeness using the model [A] Measurement Simulation 111 [B] 112 Figure 4-8: A) Summary of the application of the Methodology schematic. B) Workflow in Case Study 2. The steps 3a and 3b are parts of the same analysis. 162

163 4.2 Case Study 3: Study of cooling tower deformation using a Photomodeler Introduction and main findings Deformation of the cooling tower of a thermal and nuclear power plant is an important performance indicator and monitored by EDF. Once in every decade, a laser-based measurement system is used to characterize the shape of the cooling tower and compare it with the previous data to estimate the rate of the structural deformation of the cooling tower. Laser measurements of all targets on the cooling tower can take up to 2 weeks during which, weather-related changes in the cooling tower shape occur and Representativeness uncertainty is introduced. To help overcome this problem, a Photomodeler is used. The use of a Photomodeler enables a 3D reconstruction of the cooling tower with a smaller number of laser measurement data (bundle points) compared to full laser scanning of the cooling tower. The smaller number of scans enables a shorter measuring period and hence less cooling tower shape change within the scope of the smaller data set. The Photomodeler is software that extracts 3D data from photographs of an object. The PM has applications in civil, mechanical and chemical engineering 18. In Case Study 3, a full historical data set of a laser measurement campaign and a set of photographs of the cooling tower is available. The laser data set is split into two groups: a smaller bundle adjustment set and a larger validation set. The bundle adjustment set is used to construct the PM model along with photographs and the validation set is used to validate the performance of the PM model. The performance and the accuracy of the PM method have been assessed by Moreau et al. [49]. Specifically, the input uncertainty of the PM model is quantified using the Markov Chain Monte Carlo (MCMC) method. However, the Validation uncertainty

164 of the method was not addressed by the authors. In Case Study 3, a statistical analysis of the Validation uncertainty of the Photomodeler is analyzed based on the set of laser measurement data. The results of the analysis show that the PM overpredicts the height of an object in a statistically significant way. A linear regression model was built to correct for the overprediction and reduce the systematic bias in the Validation uncertainty. The predictions of the X and Y horizontal oriented coordinates were not found to have significant systematic biases Application of the Methodology framework In this section, the Methodology framework is applied to the Case Study 3. In the following subsections, the steps of the Methodology framework, given in Section 2.3 in general form, are applied to Case Study 3. The steps involving definitions and workflow charts give a general schematic figure and a specific schematic to Case Study 3. The steps involving detailed complex schematics specific to Case Study 3 do not include the general schematic to clarify the discussion. Table 4.2 shows the relevant general Methodology terminology adopted for the specific quantities in Case Study 3. Note that the corrected measurement value M c is not applicable in Case Study 3 since this Case Study only deals with the Validation uncertainty of the simulation. General Methodology terminology Reference True Value (T R ) Measurement Value (M) Simulated Measurement Value (M S ) Case Study 3 quantities True height of a particular target on a cooling tower The height of a particular target as measured by the laser system The height of a particular target as predicted by the PM simulation 164

165 Table 4.2: General Methodology terminology adopted for the specific quantities in Case Study 3 i Defining the Reference True Value In Case Study 3, the Reference True Value is the instantaneous coordinate of a target on the outer surface of a cooling tower. If the height of a certain target is the focus of the analysis, then the Reference True Value is defined as Z R = Z(t k ) (4.4) where t k is the time of the laser beam hitting the target. The Reference True Value of the performance indicator is shown in the Methodology schematic in Figure 4-9B. 1. Defining Reference True Value 1. Defining Reference True Value Reference True Value (T R ) Reference True Value (T R ) Z(t k ) [A] Experiment [B] Experiment Figure 4-9: The Reference True Value of the PM simulation of a cooling tower is defined as the instantaneous value of a coordinate of a target on a cooling tower 165

166 ii Defining Representativeness uncertainty In Case Study 3, the Measurement Value is the value that the laser system measures or Z M = Z laser (t k ) (4.5) The Representativeness uncertainty i.e. Z laser (t k ) Z(t k ) given in Figure 4-10B is small since the laser system is highly accurate. The main focus of the application of the Methodology in Case Study 3 is to address the systematic Validation uncertainty of the Photomoder. 2. Defining Representativeness uncertainty 2. Defining Representativeness uncertainty Reference True Value (T R ) Representativeness uncertainty Reference True Value (T R ) Z(t k ) Representativeness uncertainty Measurement Value (M) Measurement Value (M) Z Laser (t k ) [A] Experiment [B] Experiment Figure 4-10: Representativeness uncertainty arises from the difference between the Reference True Value and Measurement Value given by the laser method. iii Defining two categories of Representativeness uncertainty (Temporal and Spatial) Figure 4-11A shows a schematic that illustrates the two types of Representativeness uncertainties. In Case Study 3, only the Spatial Representativeness uncertainty is considered. It is the difference between the Reference True Value of the target height Z R in Equation 4.4 and the Measurement Value obtained using the laser system given Z M in Equation 4.5 as shown in Figure 4-11B. 166

167 3. Defining two categories of Representativeness uncertainty 3. Defining two categories of Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) Spatial Representativeness uncertainty Temporal Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) Z(t k ) Z Laser (t k ) Spatial Representativeness uncertainty [A] Experiment [B] Experiment Figure 4-11: In Case Study 3, both Spatial and Temporal Representativeness uncertainties are considered. iv Simulation model building and Calculating Random modeling uncertainty In Case Study 3, a 3D model of the cooling tower is constructed using the Photomodeler based on a set of photographs of the cooling tower. The Random modeling uncertainties are quantified using the MCMC method as discussed in Section B.4.1 [49]. The Random modeling uncertainty is purely due to input uncertainty arising from the uncertainties of the bundle points. The Simulated Measurement Value (M S ) which is the result of the PM simulation shown in Figure 4-12 is for a particular target on the surface of the cooling tower chosen for illustration. The vertical axis in Figure 4-12 shows the height of the target. σ MS = σ 2 input (4.6) 167

168 Simulation model building 5. Calculating Random modeling uncertainty Z (m) Simulated Measurement Value (M S ) Laser measurement Photomodeler Figure 4-12: The Simulated Measurement Value of the height of a target predicted by the PM. The Random modeling uncertainty is shown as an error bar v Assessing the systematic Validation uncertainty The systematic Validation uncertainty in Case Study 3 is calculated by comparing the Measurement Value (M) with the Simulated Measurement Value (M S ) since the data available enables the assessment of the Measurement Value not the Reference True Value. Note that the general Methodology in Figure 1-3 allows for the Validation uncertainty to be expressed in terms of either the Measurement Value or the Reference True Value. The Validation uncertainty is an indicator of how far the simulation result is from the experiment. The comparison between a PM prediction of the height of a particular target and the height of the same target as measured by the laser system is given in Figure

169 6. Assessing the systematic Validation uncertainty Measurement Value (M) Simulated Measurement Value (M S ) Z (m) Systematic Validation uncertainty Laser measurement Photomodeler Figure 4-13: The systematic modeling uncertainty is the difference between the Measurement Value and the Simulated Measurement Value. vi Creating combined Methodology schematic In Case Study 3, a statistical analysis is used to detect and correct the residual systematic uncertainty of the model prediction. Figure 4-14 shows the prediction of the PM before and after the correction. The systematic Validation uncertainty can be reduced using the linear regression model developed in Section B.4.2. It is important to note that the Random modeling uncertainty indicated by the green error bar in Figure 4-14 is larger than the uncertainty indicated by the red error bar due to uncertainty of the linear regression model prediction since corrected σ M S = σinput 2 + σmodel 2 (4.7) Although the systematic uncertainty was reduced (corrected Simulated Measurement Value is closer to the Measurement Value), the Random modeling uncertainty is increased as a result of applying the regression model. Since the input uncertainty 169

170 is a dominant source of uncertainty, the added uncertainty due to regression model building in the green error bar is imperceptible 170

171 Measurement Value (M) Simulated Measurement Value (M S ) Corrected Simulated Measurement Value 7. Creating combined Methodology schematic Z (m) Systematic Validation uncertainty Laser measurement Photomodeler Figure 4-14: All uncertainties considered in Case Study 3 are plotted. The green bar takes into account the model and the input uncertainties. Visibly the lines are the same length, but the calculations are conducted according to Equation 4.7. vii Creating a workflow chart Reference True Value (T R ) Case 3 Case 3 Valida5on uncertainty of model result 1 2b 3a Regression model development to reduce systema5c uncertainty 3b Physics model Measurement Value (M) Simulated Measurement Value (M S ) a Random Uncertainty of model 3c Combining random uncertainty quan5fica5on and systema5c uncertainty correc5on [A] Measurement Simulation 129 [B] 130 Figure 4-15: A) Summary of the application of the Methodology schematic. B) Workflow in Case Study 3. The steps 2a and 2b belong to the same analysis and 3a, 3b and 3c are parts of a separate analysis/ As shown in Figure 4-15A, the analysis of the Case Study 3 started with a set of laser measurement data considered to be the Reference True Value. In step 1, a set 171

172 of photographs of the cooling tower is used as an input to the Photomodeler, and a 3D model of the cooling tower is constructed. The Random modeling uncertainty of the simulation model is quantified in step 2 using MCMC method. In step 3, the laser measurement data set was used to validate the model prediction. The Validation uncertainty is quantified and a linear regression model is applied to for this uncertainty. Figure 4-15B illustrates the workflow of Case Study 3 in the Methodology framework. 172

173 4.3 Case Study 4: Representativeness uncertainty in measurement of NO x emission from a Combined Cycle Gas Turbine Introduction and main findings The emissions of atmospheric pollutant gasses such as NO x from fossil fuel plants are regulated by the environmental agencies in Europe. At a specific Combined Cycle Gas Turbine plant that was studied, during normal operation, the Automatic Measurement System (AMS) measures the pollutant gas emissions. The Measurement Values read by the AMS sensor are subject to Spatial Representativeness uncertainty because the sensor does not cover the entire cross section of the smoke-stack. Routine auditing tests are conducted at the CCGT plant to estimate the Representativeness uncertainty. During an auditing test called VERITAS, a wire-mesh sensor is inserted into the smokestack, and more accurate reading of the NO x emission is obtained. In order to correct the AMS values so that their use yields results that reflect the auditing values, models need to be developed to supplement the NO x concentration values measured by the AMS. In Case Study 4, first, a statistical model was developed to predict the VERITAS value from two parameters measured by the AMS 19. This model was built using the VERITAS data that was available. Two parameters out of 6 in AMS were chosen purely on the basis of their explanatory powers: feed gas flow rate, and Turbine Temperature. The two selected parameters are shown to have a theoretical basis to influence the NO x emission rate as explained in Section C.3. Second, it was shown in Section C.A through statistical analysis that the AMS Measurement Value of NO x is influenced by the local conditions in the smokestack such 19 If there is a need, the value produced from the statistical model can be used to update the measurement by producing a corrected measurement value M c 173

174 as the temperature in the smokestack rather than the fundamental parameters of natural gas combustion as discussed in Section C.A. Thirds, a linear regression model is built to correct the Representativeness uncertainty based on the available VERITAS and AMS data sets. Lastly, the Methodology framework is applied to structure the analyses to characterize the Spatial Representativeness uncertainty Application of the Methodology framework In this section, the Methodology framework is used to characterize the Representativeness uncertainty of Case Study 4. In the following subsections, the steps of the Methodology framework, given in Section 2.3 in general form, are applied to Case Study 4. The steps involving definitions and workflow charts give a general schematic figure and a specific schematic to Case Study 4. The steps involving detailed complex schematics specific to Case Study 4 do not include the general schematic to clarify the discussion. Table 4.3 shows the relevant general Methodology terminology adopted for the specific quantities in Case Study 4. General Methodology terminology Reference True Value (T R ) Measurement Value (M) Simulated Reference True Value (TS R ) Case Study 4 quantities VERITAS NO x measurement value AMS NO x measurement value Prediction of a linear regression model Table 4.3: General Methodology terms adopted for the specific quantities of Case Study

175 i Defining the Reference True Value In Case Study 4, a set of auditing data called VERITAS is used as the Reference True Value. Since the auditing measurement relies on a mesh sensor, the Reference True Value is defined as a weighted average over all sensors on the mesh C R NO x = mesh w i C i NO x mesh w i (4.8) The Reference True Value is plotted in the Methodology schematic in Figure 4-16B. 1. Defining Reference True Value 1. Defining Reference True Value Reference True Value (T R ) Reference True Value (T R ) R C NOX [A] Experiment [B] Experiment Figure 4-16: The Reference True Value of the concentration of NO x in the exhaust stream is an average over flow rate. ii Defining Representativeness uncertainty During operation, the concentration of NO x emission is measurement by the AMS system. Therefore, the Measurement Value is defined to be the AMS measurement result for NO x C M NO x = C NOx (AMS) (4.9) The AMS relies on a Measurement Value of a single point sensor placed inside the smokestack and is subject to Spatial Representativeness uncertainty as shown in Figure 4-17B. 175

176 2. Defining Representativeness uncertainty 2. Defining Representativeness uncertainty Reference True Value (T R ) Representativeness uncertainty Reference True Value (T R ) R C NOX Representativeness uncertainty Measurement Value (M) Measurement Value (M) M C NOX [A] Experiment [B] Experiment Figure 4-17: Representativeness uncertainty arises from the difference between the Reference True Value and Measurement Value. iii Defining two categories of Representativeness uncertainty (Temporal and Spatial) In Case Study 4, the temporal aspects of the measurement of NO x emission are not considered due to the limited number of auditing VERITAS data. The Representativeness uncertainty is purely Spatial as shown in Figure 4-18B. 3. Defining two categories of Representativeness uncertainty 3. Defining two categories of Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) Spatial Representativeness uncertainty Temporal Representativeness uncertainty Reference True Value (T R ) Measurement Value (M) VERITAS C NOX AMS C NOX Spatial Representativeness uncertainty [A] Experiment [B] Experiment Figure 4-18: In Case Study 4, only the Spatial component of Representativeness uncertainty is considered. iv Simulation model building and Calculating Random modeling uncertainty In Case Study 4, a linear regression model was built to predict the Reference True Value based on two other parameters (feed gas flow rate, and Turbine Temperature) 176

177 measured by the AMS system as discussed in Section C.A. The result of the regression model is considered to be the Simulated Reference True Value in Case Study 4. The only relevant Random modeling uncertainty to the regression model of Case Study 4 is the input uncertainty as discussed in Section C.A. It is quantified from the linear regression analysis, and the resulting Random modeling uncertainty is shown in Figure 4-19 by a red error bar. The calculation of the Random modeling uncertainty is given in Section C.A. With the collection of more auditing data, it is possible to decrease the Random modeling uncertainty. 4. Simulation model building 5. Calculating Random modeling uncertainty NO x concentration (mg/nm 3 ) Simulated Reference True Value (T R S ) 24.5 Experiment Statistical model Figure 4-19: The statistical predictive model has sampling and input uncertainty components. v Assessing the systematic Validation uncertainty Systematic Validation uncertainty in Case Study 4 is calculated by comparing the Reference True Value (T R ) with the Simulated Reference True Value (T R S ) as shown 177

178 in Figure The systematic Validation uncertainty is an indication of how well the linear regression model is performing to simulate the Reference True Value. Note that the Validation uncertainty is defined in terms of the Reference True Value in contrast with the Case Study 3, in which the Validation uncertainty is defined by the Measurement Value. The general terminology given in Figure 1-3 allows for both types of definitions of the Validation uncertainty Assessing the systematic Validation uncertainty 27.5 NO x concentration (mg/nm 3 ) Reference True Value (T R ) Systematic Validation uncertainty of T R 25 Simulated Reference True Value (T R S ) 24.5 Experiment Statistical model Figure 4-20: Systematic modeling uncertainty exists between the Reference True Value and the model prediction. vi Creating combined Methodology schematic Figure 4-21 shows a schematic that illustrates all uncertainties considered by the Methodology in Case Study 4. The Representativeness uncertainty arises due to the difference between a point measurement (AMS) and wire-mesh sensor reading (VER- ITAS). A linear regression model is built to predict the Reference True Value. 178

179 28 7. Creating combined Methodology schematic 27.5 NO x concentration (mg/nm 3 ) Representativeness uncertainty Reference True Value (T R ) Systematic Validation uncertainty of T R Simulated Representativeness uncertainty 25 Simulated Reference True Value (T R S ) Measurement Value (M) 24.5 Experiment Statistical model Figure 4-21: In the last step, all uncertainties are collected together to present a complete picture vii Creating a workflow chart Reference True Value (T R ) Case 4 Case 4 Simulated Reference True Value (T R S ) Measurement Value (M) Simulated Measurement Value (M S ) Experimental data 1 Analysis of representa3veness using the model [A] Measurement Simulation 90 [B] Quan3fica3on of experimental representa3veness uncer3anty 2 3 Sta3s3cal 4 Sta3s3cal model model uncertainty development es3ma3on 91 Figure 4-22: A) Summary of the application of the Methodology schematic. B) Workflow in Case Study 4. As shown in Figure 4-22A the analysis starts with the AMS and VERITAS data sets. In step 1, the Representativeness uncertainty is quantified using the two data sets. In step 2, the two sets of data were used as inputs to develop a linear regression 179

180 model to produce a prediction of the Reference True Value. The Random modeling uncertainties of the regression model was then quantified using the regression analysis in step 3. In step 4, the Simulated Representativeness uncertainty using the new statistical model was then compared with the Representativeness uncertainty of the AMS system. The workflow of Case Study 4 is given in Figure 4-22B with steps numbered sequentially. 4.4 Summary and Conclusion In this chapter, the Methodology framework presented in Chapter 2 is used to structure the technical analysis of three Case Studies involving different performance indicators. For full details of the technical analysis in each Case Study is given in Appendices A, B and C. The application of the Methodology to Case Study 2 enables the following conclusions. The recirculation ratio is shown to be overpredicted by 8% in Figure 4-7 using the tracer method in experiment. The Simulated Representativeness uncertainty indicates an underprediction by 6% using the tracer method. This presents an inconsistency between the test result and the simulation result. The Random modeling uncertainty is large (±40% for a 95% confidence interval) compared to the Simulated Representativeness uncertainty suggesting that the uncertainty due to sampling from a chaotically mixing tracer in the SG dominates any systematic effect that the simulation is trying to isolate. The application of the Methodology to Case Study 3 enables the following conclusions. A very small systematic Validation uncertainty (0.002% for the data point chosen for illustration in Figure 4-14) exists between the laser measurement value of target height and the prediction of the PM. The systematic Validation uncertainty can be corrected using a statistical regression model. However, the Random modeling uncertainty of the PM prediction is increased due to the uncertainty associated with the regression model. 180

181 The application of the Methodology to Case Study 4 enables the following conclusions. A significant Representativeness uncertainty (1.3% for the data point chosen for illustration in Figure 4-21) exists between the Measurement Value (AMS) of NO x concentration and the Reference True Value (VERITAS). The linear regression model using feed gas flow rate, and Turbine Temperature as predictor variables is used to produce a prediction (Simulated Reference True Value) from the VERITAS value (Reference True Value). The systematic Validation uncertainty is small (0.6% for the data point chosen for illustration in Figure 4-21) in comparison with the Random modeling uncertainty (±0.6% for 95% confidence interval) of the Simulated Reference True Value. Random modeling uncertainties are treated using normal distribution in this Chapter. As future work, the non-parametric implementation of the Methodology should be applied to Case Studies 2,3 and 4. A significant benefit from using the non-parametric bootstrap method can be expected in Case Study 2 in which the tracer mixing is chaotic and the distribution of the tracer concentration is highly non-normal and bimodal. 181

182 182

183 Chapter 5 Conclusions and Future work 5.1 Introduction In this thesis, a Methodology framework for characterizing the Representativeness uncertainty in performance indicator measurements of power plants is proposed. Power plant performance indicators are important in the operation and the safety of nuclear and thermal power plants and reducing their uncertainties is desirable. The Representativeness uncertainty arises due to the inherent spatial or temporal variations of the measurand or inadequacy of a simulation model. The Methodology framework is developed based on 4 industrial Case Studies supplied by EDF which sponsored this thesis. 5.2 Case Studies performed The Methodology is applied to 4 industrial Case Studies on the Representativeness uncertainty encountered during the operation of plants. In Case Study 1, the Representativeness uncertainty arising from adopting the ISO standard in the measurement of mass flow rate by means of an orifice plate for non- 183

184 straight upstream and downstream piping geometries is studied. Using the Methodology framework, in Case Study 1a, the Temporal Representativeness uncertainty is found to be too small to be resolved either by test data or the LESimulation performed. In Case Study 1b, both linear and cubic k-ε turbulence models were employed and the model-related numerical uncertainties were quantified using the proposed Methodology. This enables the comparison of the experimental result with the CFD model prediction in quantifying the Representativeness uncertainty. The steady simulation using the linear k-ε model resulted in qualitatively inconsistent agreement with the experiment. The unsteady simulation using the cubic k- ε turbulence model resulted in a prediction that is qualitatively consistent with the experimental result with a quantified level of Random modeling uncertainty. In Case Study 2, the SG recirculation ratio measurement by means of a tracer is considered. Using the Methodology framework, it is concluded that the recirculation ratio is overpredicted using the tracer method in an experimental test. However, the Simulated Representativeness uncertainty using CFD indicates an underprediction using the tracer method. This exhibits a qualitative inconsistency between the test result and the simulation result. The Random modeling uncertainty is large compared to the Simulated Representativeness uncertainty suggesting that the uncertainty due to sampling from a chaotically mixing tracer in the SG dominates any systematic effect that the simulation is trying to isolate. In Case Study 3, the simulation of a cooling tower shape using the Photomodeler is considered. Using the Methodology framework, it is concluded that a significant systematic Validation uncertainty exists between the laser measurement value of a target height and the prediction of the PM. In Case Study 3, the Validation uncertainty is defined in terms of the Measurement Value consistent with the general terminology. Then, the Validation uncertainty is corrected using a linear regression model developed in this Case Study. The Random modeling uncertainty of the PM prediction is found to increase due to the uncertainty associated with the regression model. 184

185 In Case Study 4, NO x emission rate measurement Representativeness uncertainty from a CCGT is considered. Using the Methodology framework, it is concluded that a significant Representativeness uncertainty exists between the Measurement value (AMS) of NO x concentration and the Reference True Value (VERITAS). A linear regression model using feed gas flow rate, and Turbine Temperature as predictor variables is used to produce a prediction (Simulated Reference True Value) of the VERITAS value (Reference True Value). The systematic Validation uncertainty is small in comparison with the Random modeling uncertainty of the Simulated True Value. In Case Study 4, the Validation uncertainty is defined in terms of the Reference True Value consistent with the general terminology. 5.3 The Methodology developed Based on the 4 Case Studies, a general Methodology framework is developed. As shown in Figure 5-1, a measurand has a True Value which is never known exactly. In practice, the measurand is estimated by the Measurement Value which is subject to a Representativeness uncertainty as indicated by the vertical separation between the Measurement Value and the Reference True Value. The Reference True value is defined within the framework of the Methodology by including the relevant physics which describes the measurement more accurately than the definition of the Measurement Value. The Methodology aims to reduce the Representativeness uncertainty by incorporating information from a Simulation Model. 185

186 True value (T) Reference true value (T R ) Corrected Measurement value (M C ) M c ~ N ( µ R TS + µ val,σ 2 M +σ 2 M S +σ 2 R +σ 2 T S val) T R s ~ N(µ T R s,σ 2 T R s ) Simulated Reference true value (T R S ) Measurement value (M) M ~ N(µ M,σ 2 M ) M S ~ N(µ Ms + µ val,σ 2 M s +σ 2 val ) Simulated Measurement value (M S ) Experiment Simulation Figure 5-1: Schematic of the Methodology framework A simulation model is built using the conditions of the Measurement to produce the Simulated Measurement Value and the Simulated Reference True Value. The Random modeling uncertainties of the Simulation model is quantified and combined using the methods given in the Methodology. Lastly, depending on the specific objective of the analysis, the Representativeness uncertainty is quantified using the Simulated Measurement Value and the Simulated Reference True Value if no experimental data are available. If experimental data are available, the Simulated Representativeness uncertainty is compared with the experimentally obtained Representativeness uncertainty to assess the performance of the simulation model. The formulation given in Figure 5-1 relies on the assumption that the uncertainty sources are normally distributed. For non-normal distributions such as the numerical uncertainties of the simulation, this formulation is limiting. Therefore, a non- 186

187 parametric is given. Three operations are identified in order to translate the Methodology to fully non-parametric methods. They are the addition of populations of random variables, the calculation of confidence intervals for populations of random variables and the non-parametric hypothesis tests. The non-parametric methods for performing these three functions are adopted from other authors in this thesis. A roadmap for the implementation of the Methodology is given in a step-by-step manner. 5.4 Summary of contributions In this thesis, several contributions are made. A general Methodology framework is developed to characterize Representativeness uncertainty using experimental data and simulation tools. It allows integration of the insight from simulation with the information from the experiment to achieve a better assessment of the Representativeness uncertainty. The concepts contained in the Methodology are expressed using the normal probability distribution for the uncertainty sources. The Methodology is applied to 4 industrial Case Studies involving uncertainties in performance indicator measurements to structure the analysis. In each Case Study, a set of technical analyses is carried out and the Methodology is used to structure and focus the work to characterize the Representativeness uncertainty. A non-parametric formulation of the Methodology framework is developed and demonstrated. The use of non-parametric techniques allows quantification and integration of uncertainties that are not expressed by the normal probability distribution. In Case Study 1, the bootstrap method was used to quantify sampling, iterative and discretization uncertainties thus demonstrating its applicability to CFD uncertainty analysis. 187

188 5.5 Lessons Learned in executing this work Uncertainty quantification is a common problem encountered in many fields. The methods adopted in this methodology are generally applicable to a broad range of uncertainty problems. First, the non-parametric bootstrap method can be applied universally for computing confidence intervals and performing hypothesis tests on uncertainty terms expressed by any parametric or non-parametric distributions. The bootstrap method offers an improvement over the traditional methods that rely on the assumption of normality, central limit theorem and a large sample size. Second, the Random uncertainty propagation techniques described in Appendix E are generally applicable to any complex simulation model with uncertain input parameters. The methods described in Appendix E enables an efficient computation of the uncertainty of the model output with a minimal number of necessary computations Difficulty encountered in assuming uncertainty sources are normally distributed In Section 3.7, the Spatial Representativeness uncertainty of Case 1b was studied using the assumptions of the normal probability distribution. It was shown that a significant difficulty exists both in quantifying and combining various Random modeling uncertainties using the normal distribution. Specifically, the discretization uncertainty which is a dominating numerical uncertainty in Case Study 1b was expressed by a confidence interval rather than a mean and a standard deviation in the methods given by Hoekstra et al. This created a necessity for the further assumptions of the distribution shape of the discretization uncertainty in order to combine it with other types of Random modeling uncertainties. Furthermore, based on the observation in Case Study 1b, the sampling and iterative uncertainties are not given by a normal distribution which invalidates the quantification using the traditional methods involving the normal distribution described in Appendix G. 188

189 5.5.2 Solutions to the analysis of the non-normal distributions offered by the bootstrap approach In Section 3.9, the Spatial Representativeness uncertainty of Case 1b was studied using the non-parametric bootstrap method. First, it was demonstrated that the discretization, iterative and the sampling uncertainties of the CFD simulation are easily quantified using the bootstrap method. The confidence intervals for these uncertainties can be obtained separately allowing an assessment of the dominant source of the Random modeling uncertainty. Secondly, the problem of combining uncertainties, encountered by the assumption of the normal distribution, is avoided by using the bootstrap method. Moreover, the confidence interval of the CFD simulation result can easily be estimated from the bootstrap population Assumptions and limitations of the bootstrap approach However, one should be aware of the assumptions and the limitations of the bootstrap method. First, the bootstrap resampling technique assumes that the underlying population is independent and identically distributed (i.i.d). A justification for this assumption is necessary when applying the bootstrap method to quantify the discretization uncertainty. Second, the number of resamples used in a bootstrap procedure needs to be carefully considered. Various prescriptions for how many resamples should be used in estimating the confidence intervals and standard errors exist in the literature[4, 50, 34].Third, the application of the bootstrap method to time-series data such as the turbulent fluctuation and the iterative uncertainty needs a careful consideration. The big picture principles of the bootstrapping do not change when applied to time-series data[51]. However, different algorithms need to be adopted. Implementations of the bootstrap method in time-series data in R can be found in [50]. Fourth, bootstrapping from more than one sample needs careful implementation. Bootstrapping from multiple samples comes up in combining numerical uncertainties in Case Study 1. The prescriptions and pseudo-codes for this purpose can be found 189

190 in [51] Justifications of the tools used in the uncertainty analysis of Case Studies 2,3 and 4 In Case Study 2, the Representativeness uncertainty of the measurement of the SG recirculation ratio using chemical tracers was studied using experimental data and CFD simulation. The mixing of chemical tracers within a SG was found to be chaotic and result in large Random uncertainties. In order to quantify the Random uncertainty of the experimental data and the simulation result, the bootstrap method was adopted because of its flexibility in dealing with small sample size. In Case Study 3, the performance of the PM reconstruction of the cooling tower is assessed. The validation uncertainty of the PM was assessed using a subset of the laser test data called the validation set. The regression analysis was adopted to diagnose the bias in PM prediction by analyzing the validation uncertainty. In Case Study 4, the Representativeness uncertainty of the measurement of NO x emission from a CCGT is studied. A regression model is developed to predict a more accurate value for the NO x emission based on the available auditing data. Statistical regression was adopted in order to correct for Representativeness uncertainty in the absence of more detailed information about the local conditions of the plant. 5.6 Future Work The following list of work is recommended as next steps in the developments of the Methodology and for the analysis of the Case Studies. 1. Further development of the non-parametric bootstrap method in quantifying 190

191 discretization uncertainty of CFD simulations. 2. Assessment of the correlation between numerical uncertainty sources such as the sampling uncertainty and the discretization uncertainty. 3. Implementation of the version of the bootstrap method for time-series in quantifying the sampling uncertainty. 4. Application of the non-parametric implementation of the Methodology framework to Case Studies 2,3 and In Case Study 1, a detailed CFD simulation performed with the exact impulse line geometry to replicate the shift between the spectrums of the CFD result and the test data. 6. More sophisticated analyses using Proper Orthogonal Decomposition (POD) and the Dynamic Mode Decomposition (DMD) applied to understand the pressure drop signal spectrum in Case Study 1a. 191

192 [utf8]inputenc [toc]glossaries 192

193 Appendix A Case Study 2: Steam Generator recirculation ratio measurement by means of chemical tracer A.1 Introduction The Steam Generator (SG) is the main heat exchanger between the primary and secondary circuits in a PWR and its performance is critical to the safe and efficient operation of the plant. The recirculation ratio is an important performance indicator for the commissioning and the operation of the SG. In this Appendix, the measurement of the SG recirculation ratio using chemical tracers is investigated. Both experimental data and CFD simulations are employed in the analysis. It is shown experimentally that the measurement of the recirculation ratio using tracers is influenced by the measurement locations within the SG. Specifically, the angular location of the measurement is shown to play a significant role in the measurement result. The CFD code STAR-CCM+ is used to create simulation models to reproduce the trend that was observed by the experiment. 193

194 Using the result of the CFD simulation, optimization and sensitivity studies were performed to investigate the effects of the measurement height, the number of probes, and the angular rotation of the probes on the measurement result. This Appendix is organized as follows. First, a general description of a SG operation is given and the recirculation flow ratio measurement using a tracer is explained. Second, the experimental data and the data analysis is presented. Third, a CFD simulation model is presented with emphasis on the model inputs, assumptions, simplifications, post processing and the uncertainty assessments. A.2 Description of the flow in a Steam Generator The Steam Generator is the main heat exchanger between the primary and the secondary circuits in a PWR. A schematic of a generic U-tube SG design is given in Figure A-1. The primary coolant enters the SG from the bottom of the SG and goes through the U-tubes and exits from the SG through the primary coolant exit nozzle. The secondary coolant enters the SG through the feedwater inlet nozzle and gets distributed around the periphery of the SG by the feedwater ring. The function of the feedwater ring is to distribute the secondary coolant to balance the heat transfer load in the SG. A higher fraction of the secondary coolant feedwater is partitioned to the hot side of the SG i.e. the half of SG where the primary coolant enters, and rises in the U-tubes region. 194

195 Figure A-1: A schematic of the coolant flow inside a SG. The chemical tracer mixing is given by different colors. The red arrows indicate the feedwater flow, and the blue arrows indicate the recirculation flow. The purple arrows indicate the mixed feedwater and recirculation flows in the downcomer. The secondary coolant feedwater is then directed down the downcomer where it mixes with the recirculation water. The mix of the feedwater and the recirculation water then makes an 180-degree turn at the tube plate level and rises in between the U- tubes where boiling is allowed to occur. The moisture separators are located above the U-tubes and separate the vapor generated in the SG from the liquid droplets. The liquid is then recirculated back to the SG and gets mixed with the feedwater, and a high-quality vapor flow exits the SG and gets sent to the turbine to produce electric power. In Case Study 2, the objective is to investigate the measurement process of the SG recirculation ratio by chemical tracers. The basis of the tracer method is explained in Section A.3.1. The recirculation ratio of a SG is defined as r = Q A + Q R Q A (A.1) where Q A and Q R are the total flow rates of the feedwater, and the recirculation flows 195

196 respectively. The SG recirculation ratio is a global quantity for a given SG. A.3 Experimental tests In this Section, the experimental tests conducted using the tracer method are presented. First, an overview of the experimental findings is given in Section A.3.2. Then the Measurement uncertainty estimation using an assumed probability distribution is then given in Section A.3.3. Lastly, a new analysis using the bootstrap method to assess the uncertainty of the same experimental data is presented in Section ii. A.3.1 The tracer method This Section describes the tracer method to measure the SG recirculation ratio and the moisture carryover rate. The Steam Generator moisture carryover rate is an important quantity that has a direct impact on the operation of the turbine and the assessment of the thermal balance of a plant. However, in Case Study 2, a part of the method involving the measurement of the SG recirculation ratio is investigated since it presents a good example of a Representativeness uncertainty. SG Moisture: Q M, C M Recirculation water: Q R, C R Turbine Condenser Heating zone: Q R +Q A, C Feedwater: Q A, C A Feedwater pump Figure A-2: The tracer distribution in the secondary circuit of a PWR. Different colors correspond to different tracer concentrations. 196

197 A chemical tracer that is water-soluble and nonvolatile (is not soluble in water vapor) is injected outside the SG in the secondary circuit of a PWR as illustrated in Figure A-2. Following the injection of the tracer, the system reaches a steady state. Based on the distribution of the tracer in steady state, the SG recirculation ratio can be calculated based on the sampling of tracer concentrations at different points inside a SG [52]. Upon entering the SG, the feedwater flow is mixed with the recirculation flow in the downcomer before boiling. The mass balance of the tracer inside a SG is given as Q A C A + Q R C R = (Q A + Q R )C (A.2) where C A denotes the concentration of the tracer in the feedwater flow in steady state, and C R denotes the concentration of the tracer in the recirculation flow. Q A and Q R are the total flow rates of the feedwater, and the recirculation flow respectively. C denotes the concentration of the tracer in the flow entering the heating zone (the downcomer and the external flow along the SG tubes) in a SG. Since the moisture carryover rate is usually much smaller than 1%, the concentration of the tracer in the feedwater flow is small compared to the tracer concentration in the recirculation flow i.e. C A << C R. Therefore, from Equation A.2, we have Q R C R (Q A + Q R )C (A.3) using Equation A.1, from Equation A.3, we obtain C R C Q A + Q R Q R = r r 1 (A.4) From Equation A.4, the recirculation ratio can be expressed as r = C R C R C (A.5) Therefore, it is sufficient to sample the concentration of chemical tracer in the recir- 197

198 culation flow and the downcomer region to measure the recirculation ratio. To measure the moisture carryover rate of a SG, the following calculation can be performed. Since the tracer is non-volatile, the concentration of the tracer in the vapor flow Q V exiting the SG can be assumed to be zero. In steady state, the mass flow of the tracer entering the SG has to be equal to the mass flow exiting the SG. Q V 0 + Q M C M = (Q V + Q M )C A (A.6) where Q M and C M are the mass flow rate and the concentration of the tracer in the moisture of the exit stream from the SG respectively. The moisture carryover rate is defined as τ Q M Q M + Q V (A.7) Using Equation A.6, from EquationA.7, we obtain τ = Q M Q M + Q V = C A C M (A.8) Since the concentration of the tracer in the moisture of the exit steam C M is the same as the concentration in the heating zone C, the moisture carryover rate τ can be expressed in terms of C A and C. τ = C A = C A C M C (A.9) Using Equation A.4, the moisture carryover rate can be expressed in terms of C A,C R and r. τ = C A C = r r 1 C A C R (A.10) However, in the following analysis of Case Study 2, the recirculation ratio, r, of Equation A.5 is used instead of the moisture carryover rate, τ, of Equation A.10. The reason is that the analysis is performed in CFD using a non-dimensional passive scalar 198

199 for modeling the tracer concentration which takes values between 0 and 1. Equation A.10 requires the ratio of the absolute values of C A and C R. These could only be simulated in CFD if the C A and C R were available. In the available experimental reports of this experiment, these values were not reported. A.3.2 The experimental results The 900MW and 1300MW PWRs class reactors constitute a large fraction of the reactors in operation in France today. The Steam Generators for the 900MW and 1300MW class fleet are of different designs. The experimental data is available for both types of SGs. SG model #1 is based on a Westinghouse reference model, and used for 900MW reactors. SG type #2 is based on #1, but is modified for a higher power and a higher pressure of operation. Table A.1 summarizes the experimental data available for these two types of SGs. Model #1 Model #2 Data on downcomer flow sampling (C) Data on recirculation flow sampling (C R ) X Table A.1: Summary of the experimental data From the experimental test result on Model #1, it was shown that the tracer method overestimates the recirculation ratio in the cold side and underpredicts in the hot side. In Case Study 2, the hot and the cold side of the SG are defined by the feedwater split ratio. In Model#1, 80% of the feedwater is discharged in the hot-side and 20% in the cold-side. The over and under-prediction of the recirculation ratio as a function of the angular location of the sampling device in the downcomer indicates that there is significant 199

200 Spatial Representativeness uncertainty present for this method of measurement. Possible causes for the heterogeneity in the concentration of the tracer are an asymmetric rate of discharge from feedwater ring, asymmetry in the concentration of the tracer in the recirculation flow and the asymmetry in the temperature distribution [48]. A.3.3 Estimation of the Measurement uncertainty of the experiment This Section describes the analysis of the experimental Measurement uncertainty using both the parametric and non-parametric methods. In order to estimate the uncertainty in the Measurement Value of the recirculation ratio, it is necessary to estimate the uncertainties of the concentrations measured in recirculation water C R and in the downcomer C. From Equation A.5, assuming the normal probability distribution and assuming the uncertainties are small, the uncertainty of C R and C are propagated as [53] u(r) r = ] C 2 [ ] 2 [ u(cr ) u(c) + (A.11) C R C C R C where u(c R ) is the uncertainty of C R and u(c) is the uncertainty of C. In order to quantify u(c R ) and u(c), statistical approaches have been taken and the small number of available data points presents difficulties in the analysis [48]. Equation A.11 is derived as follows. The sensitivity coefficient method described in Appendix E applied to Equation A.5 is u(r) 2 = ( C R [ C R C R C ] 2 ( [ u(c R )) + C C R C R C ] u(c)) 2 (A.12) where u(c R ) and u(c) are the standard deviations of C R and C respectively. After taking the partial derivatives [ ] 2 [ ] 2 u(r) 2 C = u(c (C R C) 2 R ) 2 C R + u(c) 2 (A.13) (C R C) 2 200

201 after some manipulation, [ ] ] C R C 2 [ ] 2 [ u(cr ) u(c) u(r) = + (A.14) (C R C) (C R C) C R C using the definition of the recirculation ratio in Equation A.5, we obtain u(r) r = ] C 2 [ ] 2 [ u(cr ) u(c) + (A.15) C R C C R C i Uncertainty of the recirculation ratio assuming the normal probability distribution Only 4 data points are available to estimate the uncertainty of C R and 8 data points are available for C. Table A.2 shows a summary of an evaluation of the Measurement uncertainty based on the available data assuming the normal probability distribution. Measured quantity SG model #1 SG model #2 C R 6% of the mean 6% of the mean C 2% of the mean 12% of the mean Table A.2: Summary of the uncertainties of C R and C. The standard deviation assuming the normal probability distribution is expressed as a percentage of the mean. 201

202 Using the values given in Table A.2 [53] and Equation A.11, the uncertainty of the recirculation ratio for the type #1 SG is calculated. The resulting 95% confidence interval is given as 3.1 < r < 5.76 (A.16) ii Uncertainty of the recirculation ratio using the bootstrap resampling method In this Section, an alternative analysis of the experimental data using the bootstrap method is given. The reason for using this method is to obtain an estimate of the uncertainty of the experimentally measured recirculation ratio without assuming any probability distribution. The recirculation ratio determined using the tracer method is defined in Section A.3.1. Assuming a small feedwater flow tracer concentration i.e C A << C R, the expression for the recirculation ratio is given in Equation A.5. The bootstrap estimation of the recirculation ratio can be carried out by sampling randomly 1000 times with replacement from both experimentally obtained values of C and C R. The resulting bootstrap distribution of the recirculation ratio shown in Figure A-3 is obtained by using Equation A.5 202

203 Recirculation Ratio Figure A-3: The bootstrap distribution of r based on the experimentally obtained values. The horizontal axis is re-scaled due to the sensitivity of the data. The 95% confidence interval is obtained by selecting the 25th and the 975th values of the bootstrap distribution for r < r < (A.17) 203

204 A.4 CFD simulation This Section presents the CFD simulation performed in Case Study 2. First, the simulation model building is discussed in Section A.4.1. The results of the simulation are given in Section A.4.3. A.4.1 The Model In this Section, the CFD simulation model of the SG is described. First, the geometry, meshing, physics modeling, and initial and boundary conditions are discussed. Then the iterative and grid convergence analysis of the CFD simulation with the quantification of the discretization uncertainty are given. The tube simplification process used for Case Study 2 is briefly discussed in Section v. i Geometry In Case Study 2, a representative and approximate geometry is used to create a simple simulation model of a SG. Since the specific design details differ for type #1 and #2 SGs, the work presented here is intended to serve as a demonstration of the application of CFD rather than to produce a conclusive final result. First, the geometry of the SG simulation model used for the CFD simulation only covers the parts of the SG that are of interest to the problem of chemical tracer mixing. Since the tracer mixing only happens in the secondary side of the SG; modeling the primary coolant was not necessary. Second, the modeling of the moisture separators and the steam dryers of the SG was not attempted in this work as they are challenging CFD problems by themselves. The model assumes that the tracer concentration of the recirculating water coming out of those elements is homogeneous. It is a reasonable assumption since the tracer is expected to be well mixed by the time it reaches the recirculation water level above the feedwater ring[52, 48, 53]. 204

205 Third, due to the large number of U-tubes, the tube support plates and the antivibration bars, the real geometry of a SG is vastly complex. Moreover, the physics of two-phase flow is beyond the current capabilities of CFD. Therefore, the simulation was limited to the single-phase region in the riser. For the reasons given above, the riser geometry was limited to a relatively small tube length (18 inches from the tube sheet) and the outlet was placed at this level. In doing so, the two-phase flow region and its associated modeling challenges are kept out of this simulation. Cutting the geometry short at 18 inches also ensures that the first tube support plate which is at a height of 21 inches from the tube sheet is avoided. The simulation geometry includes a portion of the Steam Generator tubes. The number of U-tubes in a SG had to be reduced in order to make the meshing process feasible. A summary of the approach is described in Section v. A description of the method used and its implementation in STAR-CCM+ is given in [54]. Lastly, the outlet of the geometry was moved from its initial 18 inches level to a higher one by adding a funnel at the top of the tube bundle so that the outlet boundary is 30 inches above the tube sheet level as given in Figure A-4. This change of geometry is done for numerical convergence reasons in CFD. In the absence of this change in geometry, significant backflow was generated preventing the convergence of the computation. The addition of the funnel was to help remove the effect of the backflow. It was assumed that this change in the geometry is not of concern for two reasons. Firstly, the region of interest in the mixing of the chemical tracer is the downcomer region. Since the outlet boundary is sufficiently separated from the downcomer, significant change in the mixing of tracer is not expected due to the geometry change at the outlet boundary. Secondly, the geometry used in this simulation is an approximate representative geometry chosen in order to understand the chemical tracer mixing process in a Steam Generator. 205

206 Figure A-4: CFD simulation geometry. The outlet boundary is indicated by orange and the inlet boundary is indicated by red. The blue boundary indicates a symmetric boundary condition. A) General overview of the geometry B) Feedwater ring region, C) Tubesheet region ii Mesh The built-in mesher in STAR-CCM+ is used for the discretization of the computational domain. The built-in mesher in STAR-CCM+ mesher is capable of many automatic optimization tools and the user has the choice to leave these refinement options activated or to disable them to gain access to a finer control of the meshing process. In Case Study 2, a total of 6 high-quality meshes of varying base sizes (2cm, 4cm, 6cm, 8cm, 10cm and 12cm) were created. The reason for creating 6 different meshes is to perform a systematic grid convergence analysis given in Section A.4.2. Figure 206

207 A-5 shows a close-up of three of the six meshes near the tube bundle, which is the most challenging region to mesh, and a major contributor to the total cell count. Table A.3 summarizes the total number of cells for all six different meshes. Figure A-5: Different mesh sizes used in the simulation. A) 2cm mesh, B) 6cm mesh C) 12cm mesh Mesh Base Size (cm) Cell count 64M 13.0M 5.4M 3.8M 2.5M 1.9M Table A.3: Summary of the number of cells of the meshes used for grid convergence study iii Physics models The physics models used in the simulation are as follows: 1. Only the single phase flow of the secondary coolant is modeled. The boiling part of the tube is avoided by cutting boiling region in the U-tube bundle. 2. Steady state RANS simulation is performed due to the complexity of geometry, and to keep the computational time tractable. 207

208 3. Turbulence model: The standard, linear k-ε model is chosen. The quadratic and the cubic formulations were investigated initially and demonstrated, as expected, a higher level of turbulence. In the case of the cubic model, the simulation had to be run unsteady contrary to the linear and quadratic cases. The linear formulation was chosen for its robustness. The choice of turbulence model, in this case, is not unique and other turbulence models such as R ij -ε should be investigated to see the sensitivity of the final result. 4. Wall treatment: High Wall Y+ wall function was implemented. The flow in all regions of the geometry is highly turbulent (low-velocity regions in the downcomer yield Reynolds number of more than 10 6 ). The computational meshes were created with the same size for the first layer of cells next to the wall. This was done to maintain the same wall Y+ across various meshes. The value for wall Y+ is well above 30 in the entire region of interest. The validity of the logarithmic Law of the Wall is also confirmed for Wall Y+ higher than 200 in the case of highly turbulent flows [38] 5. Temperature dependent physical properties of water are implemented. The temperature dependence laws are manually entered using the NIST database in the relevant temperature range. Figure A-4 shows the polynomial fits to water density, dynamic viscosity, heat capacity and thermal conductivity. 208

209 Figure A-6: Temperature-dependent water properties implemented in the CFD model iv Tracer diffusion modeling in CFD The tracer used in the experiment to measure the recirculation ratio of the SG is modeled by a passive scalar in the CFD simulation. A passive scalar is a scalar variable assigned to a fluid phase or an individual particle. It is called passive because it does not affect the physical properties of the simulation. An intuitive way to think of passive scalars is as tracer dye in a fluid, but with numerical values instead of colors, and with no appreciable mass or volume STAR-CCM+ user guide 209

210 In our CFD model, the passive scalar p is defined as in Equation A.18. p = C C A C R C A (A.18) where C A and C R are the concentrations of chemical tracers in the feedwater, and recirculation flows respectively. Therefore, a passive scalar value of 0 corresponds to a tracer concentration equal to the concentration in the feedwater flow. A passive scalar value of 1 corresponds to the concentration in the recirculation flow. It is possible to express the local SG recirculation ratio in terms of the passive scalar. Starting with the mass balance of chemical tracer given in Equation A.2, then using the definition of the recirculation ratio in Equation A.1 (r 1)C R + C A = rc (A.19) Substituting for C from the definition of p in Equation A.18, we have (r 1)C R + C A = r(c A + p(c R C A )) (A.20) or C A [1 r + rp] = C R [1 r + rp] (A.21) Since Equation A.21 has to hold for all values of the feedwater and the recirculation water concentrations, the expressions in the square parenthesis have to equal to zero i.e. 1 r + rp = 0. The SG recirculation ratio can then be obtained as a function of p as r = 1 1 p (A.22) It is important to note that the passive scalar is a local quantity whereas the SG recirculation ratio is a global quantity. In a world of perfectly uniform mixing of chemical tracer in the SG, the same concentration of the tracer would be measured regardless of the location of measurement giving the same value for r. However, due 210

211 to the heterogeneous mixing of the chemical tracer, the resulting global value for r may be different depending on the location of the the chemical tracer sampling. This constitutes a Spatial Representativeness uncertainty in the SG recirculation ratio measurement. In our CFD simulation, the steady state distribution of the passive scalar is obtained by solving the transport equation. The transport equation for the passive scalar in single-phase flow is given by Equation A where: ρφdv + t V ρφ υ d A = [ ( µ ρ Sc µ ) ] t φ d Sc A + S φ dv t V (A.23) ρ is the fluid density υ is the fluid velocity µ is the fluid viscosity µ t = ρc µ k 2 ε is the turbulent viscosity where k and ε are turbulent kinetic energy and turbulent dissipation rates calculated in k-ε models and C µ = Sc = µ ρd m is the molecular Schmidt number with D m is the molecular diffusivity of the tracer in the fluid. The tracer used in the experiments is Cesium chloride (CsCl). An estimate of the Schmidt number for CsCl was obtained from an external source [55]. The sensitivity of the simulation result to small variations in the molecular Schmidt number was found to be small. Sc t = 0.9 is the turbulent Schmidt number. It is assumed to have a value of 0.9, consistent with the turbulent Prandtl number used for energy. S φ is a source term for the passive scalar. because the chemical tracer is injected outside the SG. This term is 0 in our simulation 21 Single phase and VOF flows, transport equation for passive scalar, STAR-CCM+ v User guide. 211

212 v Tube simplification The number of the Steam Generator tubes was reduced while keeping the flow cross sectional area constant. The total volume of the secondary coolant in a SG is kept constant. In this work, the number of tubes was reduced from 6633 to approximately 370 by using a scaling factor of 16. A cross section of the simplified tube geometry is given in Figure A-7. Figure A-7: Tube simplification. Left: original tube size. Right: simplified tube size vi Initial and boundary conditions The initial conditions are set in order to facilitate the convergence of the simulation and they are summarized in Table A.4. Temperature Equal to the feedwater temperature. T = C. Passive scalar Static pressure 0 (i.e. the Feedwater concentration) everywhere. 58 bar everywhere Table A.4: Summary of initial conditions used The boundary conditions are important in making the simulation model realistic in comparison to a real operating SG. Table A.5 and A.6 show a set of boundary 212

213 conditions used for the simulation. Feedwater inlet Temperature C C Recirculation water inlet Passive scalar 0 1 Geometry The feedwater ring nozzles are distributed so that: for half of a SG, 20 nozzles are on the hot side and 6 on the cold side. The flow split is 77/23% Velocity inlet v feed = 3m/s Assumed homogeneous and uniform v recirc = 0.568m/s corresponding to r=4 Uniform inlet with a homogeneous distribution of tracer concentration and velocity Table A.5: The boundary conditions that represent a realistic operating condition based on the open literature and the thermal-hydraulic data shared by EDF 213

214 Characteristics Implementation in CFD Total thermal power of the SG 856MWth Total surface for heat transfer m 2 Average heat flux at the tube surface φ = 115.4kW/m 2 Shape of the heat flux Assumed to vary linearly with the length of the tubes [54] Maximum heat flux φ max = 192.7kW/m 2 Minimum heat flux φ max = 38.2kW/m 2 Heat transfer implementation in STAR-CCM+ Linear dependence is coded by a field function Correction to heat flux due to simplified Implemented as a volumetric heat source tube geometry Table A.6: The boundary conditions in the U-tube heat transfer region. vii Wall Y+ Figure A-8 shows the Wall Y+ on all the wall surfaces of the geometry. The Wall Y+ is cut between 30 and 3000 for a better visualization. The minimum Wall Y+ observed is 14. A visual inspection of the cells with a Wall Y+ below 30 (regions where the High Wall Y+ treatment of the wall velocity distribution is not valid anymore) reveals that only a few coarse cells in the tube bundle region are affected. 214

215 Figure A-8: Wall Y+ distribution for the 4cm Base Size Mesh viii Iterative convergence and iterative uncertainty The steady state simulation performed in Case Study 2 with various mesh sizes exhibits good convergence in general. However, due to the highly turbulent nature of the flow in some regions (especially in the tube bundle), the steady state result can show an oscillatory convergence behavior even though the simulation is converged. This behavior varies from one mesh to another. Figure A-9 shows examples of oscillatory convergence for 6cm base size mesh at three different locations. 215

216 Temperature (K) Iterations Recirculation Region Point Monitor Downcomer Point Monitor Tube Bundle Monitor Figure A-9: The 6cm base-size mesh temperature as a function of the number of iterations at 3 locations When the result is oscillating, an additional post-convergence averaging of the results is necessary. The result of an oscillatory convergence is averaged over the last thousand iterations for each run and the extent of oscillation gives the iterative uncertainty of the simulation. A.4.2 Grid convergence and the discretization uncertainty A grid convergence analysis was conducted on the CFD models used in this work. The methodology described by Eça and Hoekstra [21] is used in this Section. More detail is given for the method in Appendix F. The grid sizes are selected for the grid convergence study to preserve the same increment ratio for the volume average cell size. The choice of the grid size for convergence study is discussed in Section ii. The finest grid size has a base size of 2 cm. Figure A-10 shows the result of using Hoekstra methodology to fit the grid convergence of passive scalar. It gives the fitting function that minimizes the square error 216

217 along with optimal parameters α and φ 0 corresponding to the convergence order. Using the fitting parameters the discretization error is calculated. Figure A-10: The grid convergence and the least square fit according to the methodology of Hoekstra et al. The passive scalar is calculated for 6 different meshes at a probe location in the tube bundle region. Carrying out the calculations according to the Hoekstra et al[21] methodology prescription for the non-monotonic convergence, the discretization uncertainty is U φ = 3 M = (A.24) The 95% confidence interval for the passive scalar calculated by CFD simulation is then [21] φ i U φ < φ exact < φ i + U φ (A.25) Assuming the normal probability distribution, the discretization uncertainty of r can 217

218 be expressed as r r = 1 r 1 p r = 1 8.7% = 43.5% (A.26) (1 0.8) The analysis using the normal distribution highlights an inherent difficulty with the uncertainty estimation in the tracer method. Even a small perturbation in concentration measurement can result in a large swing in the inferred recirculation ratio at that point. A.4.3 Validation of the simulation using the experiment The results of the experimental tests described in Section A.3.2 are compared to the CFD simulation results in this Section. Figure A-11 shows the experimental test result and Figure A-12 shows the result of the CFD simulation. The angular dependence of the local recirculation ratio is well reproduced by the simulation. The main reason for the overprediction of the recirculation ratio in the cold side of the SG and the underprediction in the hot side is the 20/80 feedwater flow split described in Section A

219 Test result from, SG type #1, h=0.68m Cold leg Hot leg Local recirculation ratio (Rescaled) Measured values Average Theoretical value Angular location (degrees) Figure A-11: The results of the experimental test conducted on a Model #1 Steam Generator. The theoretical value of the SG recirculation ratio is supplied by the vendor. The vertical axis is re-scaled due to the sensitivity of the data. Cold leg Hot leg Figure A-12: The results of the CFD simulation on an approximate geometry 219

220 A.4.4 Random modeling uncertainty estimation In Section A.3.3, the normal probability distribution was used to quantify the experimental Measurement uncertainty. The reason is the lack of understanding of the physics of tracer mixing in the downcomer region. Using the results of the CFD simulation, the validity of such an assumption can be tested. The local recirculation ratio is derived from the concentration of the tracer in the downcomer flow i.e from the passive scalar value as given in Equation A.22. First, the distribution of the passive scalar at the height where the measurement is taken in the downcomer is given in Figure A-13. A bimodal behavior is exhibited for two CFD simulations with different mesh sizes. Second, the local recirculation ratio calculated from the passive scalar as in Equation A.22 is shown in Figure A-14. The histograms in the figure shows the distribution of the local recirculation ratio measured by the sampling of the tracer around the downcomer. The tracer measurement in the cold side (given in blue) tends to overestimate the recirculation ratio due to high recirculation to feedwater flow ratio whereas in the hot side (given in red) tends to underestimate the recirculation ratio as there is higher feedwater flow compared to the recirculation flow. The bimodal behavior exhibited in the tracer concentration distribution is persistent in the local recirculation ratio distribution as shown in Figure A

221 Figure A-13: A histogram of the passive scalar at h=0.68m, for 6cm and 8cm base size meshes 100 Probe height h=0.68, r=1.59m 6cm mesh Measured recirculation ratio SG hot Side SG cold side Local recirculation ratio Figure A-14: A histogram of the local recirculation ratio using a 6cm base size mesh result The result obtained using 6cm base size mesh was consistent with the one obtained 221

222 using 8cm base size mesh as given in Figure A-15. The local recirculation ratios measured in the cold side have higher dispersion than the values in the hot side. The asymmetry of the tracer concentration is predominantly due to the asymmetric discharge of the feedwater in the SG. 100 Probe height h=0.68, r=1.59m 8cm mesh Hot Side Measured recirculation ratio Cold side Local recirculation ratio Figure A-15: A histogram of the local recirculation ratio using a 8cm base size mesh result i Analysis of the uncertainty using the Bootstrap method Next, the Random modeling uncertainty of the CFD model is quantified. In Case Study 2, the Random modeling uncertainty arises from the functional dependence of the passive scalar on the angular location. Although the functional dependence is deterministic in nature, it was treated as random in the uncertainty analysis of the experimental data. Therefore, a consistent analysis of the CFD results is conducted in order to compare the results. 222

223 Figure A-16: A histogram of the means of 1000 randomly resampled populations The bootstrap method for the confidence interval calculation is implemented by sampling randomly from the 212 values of the recirculation ratio given in Figure A-15 with replacement 1000 times. The resulting bootstrap distribution is given in Figure A-16. A 95% confidence interval can then be obtained as 4.6 r 8.1 (A.27) The confidence interval is obtained by first ordering all 1000 values represented in Figure A-16, and then selecting 25-th and 975-th largest values in that list. ii Analysis of the uncertainty using an assumed probability distribution The confidence intervals can attain different values if a certain probability distribution is assumed. The confidence interval for r assuming the normal distribution with 223

224 unknown mean and a standard deviation is [22] µ r t n 1 (α/2) S r n r µ r + t n 1 (α/2) S r n (A.28) where µ r and S r are the sample mean and the standard deviations of data given in Figure A-15. Also, the significant level is α and t n 1 (α/2) is the student t distribution with n-1 degrees of freedom corresponding to α = 5%. In the case of our CFD results, the number of data points is n=212. Carrying out the calculation, the 95% confidence interval is 4.4 < r < 8.1 (A.29) A.4.5 Optimization of tracer measurement using CFD During an experimental test, a certain number of chemical probes are used to sample the coolant to measure the concentration of the tracer in the SG. The experimental data takes 8 point measurements using the probes around the periphery of the SG in the downcomer. Four point measurements are taken to sample in the recirculation water level above the feedwater ring. The values measured by these probes are then averaged to come up with the mean Measurement Value. Using the results of the 6cm base size mesh simulation, the effect of using a different number of probes is evaluated. 224

225 Figure A-17: The effect of varying the number of probes used to sample the chemical tracer concentration in the periphery of the downcomer. Figure A-17 shows the statistical average of the local recirculation ratios as a function of the number of probes. In this plot, all probes are given equal weight in averaging. Therefore, the outliers such as the unmixed jet from the feedwater nozzles picked up by a probe can have a significant influence on the average of values measured by the probes. This explains the deviation of the average from the Reference True Value when a relatively high number of probes are used such as N=10, and N=12. Next, the effect of the measurement height on the measurement result is investigated. A group of plots in Figure A-18 shows the local recirculation ratio as defined in Equation A-14 as a function of the angular location at different heights and radial locations. The radial variance is shown to decrease with height. However, the angular dependence is persistent even at the lowest height where the measurement is taken during an experimental test. Figure A-19 shows the same information in a statistical box plot at different heights. 225

226 [A] [B] [C] [D] [E] [F] [G] [I] Figure A-18: The local recirculation ratio as a function of the angular coordinate at different heights and radial coordinates. 226

227 Figure A-19: The boxplots of the local recirculation ratio at different heights. Red + indicate points considered outliers. The central blue box represents the central 50% of the data with the red line indicating the median of the data. Two vertical lines extending from the central box indicate the remaining data outside the central box that are not regarded as outliers. Finally, the sensitivity of the recirculation ratio measurement result to a small angular rotation of the probes is investigated. Figure A-20 shows the change in the recirculation ratio measurement result corresponding to a small change in the angular location of the probes defined as the simultaneous rotation of all probes by polar angular displacement θ. When a large number of probes used such as N=10 or N=12, the sensitivity was shown to be large predominantly due to the locally unmixed jets from the feedwater nozzle. 227

228 !! ΔΘ ΔΘ+!!! Figure A-20: The sensitivity of the recirculation ratio measurement result to a small angular rotation of the probes A.5 Conclusion In Case Study 2, the SG recirculation ratio measurement by means of a chemical tracer is considered. A CFD simulation was created to examine the mixing of chemical tracer in the single-phase region of the SG, and to assess and quantify the Spatial Representativeness uncertainty of the measurement. Using the results of the CFD simulation, a systematic overprediction of the recirculation ratio based on sampling in the hot side of SG, and underprediction in the cold side are found. The finding was qualitatively consistent with the experimental data. In assessing the random uncertainty of the recirculation ratio measurement, the normal distribution is typically assumed and a confidence interval is computed. First, it was shown that the distribution of the recirculation ratio cannot be considered a random variable given that there is a functional dependence on the angular location of the sampling location. When the uncertainty has to be treated as a random variable, 228

229 the assumption of the normal distribution is not appropriate given the bimodal nature of the distribution due to asymmetric feedwater flow split. The non-parametric bootstrap method was employed to recalculate the confidence interval both for the limited set of experimental data and the CFD simulation resulting in a much wider uncertainty range than previously calculated. The assumption of the normal distribution results in a 95% confidence range of 3.1 < r < 5.76, whereas the boostrap method applied to the same data results in 4.38 < r < Lastly, optimization studies were carried out using the results of the CFD simulation on several parameters. The number of probes placed in the downcomer was not found to correlate with the higher accuracy of the method. Secondly, it was found that the height of the measurement should be minimized for a better result since the tracer mixing process is less chaotic at lower heights. Third, a small angular perturbation to the angular location of the sampling probes was found to affect the measurement results significantly due to the unmixed jets from the feedwater nozzles. 229

230 230

231 Appendix B Case Study 3: Study of cooling tower deformation using the Photomodeler B.1 Introduction Deformation of the cooling tower of a thermal and nuclear power plant is an important performance indicator and monitored by EDF. Once in every decade, a laser-based measurement system is used to characterize the shape of the cooling tower and compare it with the previous data to estimate the rate of the structural deformation of the cooling tower. Laser measurements of all targets on the cooling tower can take up to 2 weeks during which, weather-related changes in the cooling tower shape occur and Representativeness uncertainty is introduced. To help overcome this problem, a Photomodeler is used. The use of a Photomodeler enables a 3D reconstruction of the cooling tower with a smaller number of laser measurement data (bundle adjustment points) compared to full laser scanning of the cooling tower. The smaller number of scans enables a shorter measuring period and hence less cooling tower shape change within the scope 231

232 of the smaller data set. The Photomodeler is software that extracts 3D data from photographs of an object. The PM has applications in civil, mechanical and chemical engineering 22. In Case Study 3, a full historical data set of a laser measurement campaign and a set of photographs of the cooling tower is available. The laser data set is split into two groups: a smaller bundle adjustment set and a larger validation set. The bundle adjustment set is used to construct the PM model along with photographs and the validation set is used to validate the performance of the PM model. The performance and the accuracy of the PM method have been assessed by Moreau et al. [49]. Specifically, the input uncertainty of the PM model is quantified using the Markov Chain Monte Carlo (MCMC) method. However, the Validation uncertainty of the method was not addressed by the authors. In Case Study 3, a statistical analysis of the Validation uncertainty of the Photomodeler is analyzed based on the set of laser measurement data as shown in Figure B-1. The results of the analysis show that the PM overpredicts the height of an object in a statistically significant way. A linear regression model was built to correct for the overprediction and reduce the systematic bias in the Validation uncertainty. The predictions of the X and Y horizontal oriented coordinates were not found to have significant systematic biases

233 Laser measurement of 400 targets on a cooling tower Photomodeler is used to reconstruct a 3D model Validation uncertainty Exists between laser data and 3D model Model inputs are 50 laser target coordinates, and high resolution photos Dependence of ΔZ on Z Non-central distribution of error indicating modeling representativeness Figure B-1: The scope of Case Study 3. A laser-based system is used to measure the coordinates of targets on the cooling tower surface. A Photomodeler (PM) is used to reconstruct a 3D image of a cooling tower based on photos of the cooling tower and a certain number of bundle adjustment points 23. The PM output is analyzed using statistical methods. B.2 Laser measurement data In order to determine the deformation rate of the cooling tower, the coordinates of the targets on the surface of the cooling tower were measured using lasers in 1990 and in Figure B-2a shows a target on the surface of a cooling tower, and Figure B-2b shows a laser measurement device. Any structural deformation that happened during that period would be revealed by the comparison of the coordinates of the targets. However, the laser measurements of all targets on the cooling tower can take up to 2 weeks during which, weather-related changes in the cooling tower shape occur and a Representativeness uncertainty is introduced. 23 Set of laser measurement data used as input to Photomodeler for bundle adjustment algorithm to translate 3D shape into physical units. 233

234 (a) A target on the surface of a cooling tower (b) A laser device for accurate measurement of the target location Figure B-2: Laser measurement system Figure B-3: A photograph of a cooling tower used as an input to the PM. The targets on the surface of the cooling tower are used either as bundle adjustment points or as validation points. To help overcome this problem, the Photomodeler is used. A simple overview of 234

235 the PM software is given in Section B.3. The use of the Photomodeler enables a 3D reconstruction of the cooling tower with a smaller number of laser measurement dataset (bundle adjustment points) compared to full laser scanning of the cooling tower. However, the use of the PM requires the photographs of the object in addition to the bundle adjustment points to translate the resulting 3D model into absolute units. Figure B-3 shows an example of high-resolution photographs used as inputs to the PM. Figure B-4 shows a 3D visualization of the coordinates of targets on the surface of the cooling tower as measured by the laser method. The target coordinate data set given in Figure B-4 is split into two groups: a bundle adjustment set, and a validation set as summarized in Table B.1. The bundle adjustment set is used to construct the PM model in addition to the photographs, and the validation set is used to benchmark the performance of the PM model. Figure B-4: 3D visualization of the target point coordinates as measured by the laser device which includes the bundle adjustment set and the validation set. The axes are removed due to the sensitivity of the data. 235

236 Year Laser data set used Validation laser data Number of photographs as bundle adjustment set points Table B.1: Summary of the available data B.3 The Photomodeler The Photomodeler (PM) is a software that implements a photogrammetric algorithm to extract a 3D data from the photographs of an object. The PM has applications in civil, mechanical, and chemical engineering 24. The examples of its applications are performing high-density surface modeling as well as measuring and modeling of large installations. Following is a description of the PM technology based on the information on a vendor website 25. As shown in Figure B-5, a photograph of an object is produced when the light hits the object and the reflection is captured by a camera. Using multiple photos, it is possible to construct a 3D image of the object. This requires knowing the location of the camera in each photo as well as camera parameters such as the focal length, distortion, and pointing angles of the camera at each shot

237 Figure B-5: 3D data can be obtained from multiple photographs of the same object. Each of the three pieces of information can be inferred from the photos using a photogrammetry algorithm called Bundle adjustment. To get the 3D model in absolute units, a reference scale is needed. This step requires feeding the physical measurement data into the PM thereby producing a 3D model as shown in Figure B-6. Figure B-6: Physical measurements need to be fed into the PM to bring the resulting 3D model into absolute units. 237

238 B.4 The simulation of the cooling tower using the Photomodeler In this Section, the simulation results using the PM are given and the Random modeling and the Validation uncertainties are quantified. A 3D model of the cooling tower is constructed using the PM based on the input data which includes a set of cooling tower photographs, and a bundle adjustment data set given by the laser method. Figure B-7 shows the result of a PM simulation. The red dots represent the bundle adjustment data set used to construct the model. The blue dots are the PM simulation predictions of the validation set. It is possible to benchmark the PM simulation result using the validation data set that was not used to construct the PM model. In the following subsections, the Random modeling and the Validation uncertainties of the PM model are assessed. Figure B-7: A 3D reconstruction of the cooling tower using the PM. The blue dots represent the simulated target points. The red dots represent the bundle adjustment set used to construct the PM model of the cooling tower. The axes are removed due to the sensitivity of the data. 238

239 B.4.1 Random input uncertainty propagation through the Photomodeler The Photomodeler has two inputs: the photographs and the bundle adjustment dataset. The bundle adjustment points used as an input to the PM in Case Study 3 are the laser measurements that are subject to inherent random uncertainty due to the accuracy of the device. Therefore, the PM simulation is subject to input uncertainty. Moreau et al [49] used the MCMC method to propagate the random input uncertainty through the PM model. The principle of the method is given next in this Section followed by the application to the PM [2]. Let Y = G(X 1,... X n ) be the output of a model G that has inputs X 1,... X n. The objective is to quantify the uncertainty in the model output σ Y when the input variables have known uncertainties σ X1,..., σ Xn. The MCMC method relies on generating random samples of the input vector X 1j,... X nj and calculating the model output corresponding to that sample. To obtain an accurate evaluation of the probability distribution of the output quantity, it is necessary to generate a large number of samples j = (1, N) with N being large enough for convergence of the mean and the standard deviation. For large N, the 99% confidence interval of the mean of the model output is calculated as µ ˆ S 2.58σˆ S < µ Y < µ ˆ S σˆ S N 1 N 1 (B.1) where µ ˆ S = 1 N N Y j j=1 (B.2) and σˆ 2 S = 1 N 1 N j=1 (Y j µ ˆ S ) 2 (B.3) In Case Study 3, the MCMC method was employed to propagate the input uncer- 239

240 tainty through the PM [56]. The input variables are the coordinates of the bundle adjustment data set used to construct the PM model. The uncertainties in the input coordinates arise from the accuracy of the laser system. Figure B-8 shows an example of propagating the input uncertainty through the PM using the MCMC method. The red ellipse represents the 99% confidence interval of the coordinates of a validation data point that is not used to construct the PM model in X-Y plane. The blue dot and the ellipse represent the PM simulation prediction of the same point. The blue ellipse represents the result of a MCMC uncertainty propagation indicating a significant sensitivity of the PM output to input bundle adjustment data set. The distance between the bundle adjustment point and the mean value of the PM prediction gives the Validation uncertainty of the model and is analyzed in Section B PM simulation 99% CI Mean of PM simulation Bundle adjustment data 99% CI Y(m) X(m) Figure B-8: The input uncertainty of the PM model in the XY plane calculated by the MCMC method. B.4.2 Validation uncertainty of the Photomodeler In this Section, the output of the PM is examined and compared with the validation set of laser measurement data. The first aim of the analyses provided in this Section is 240

241 to quantify the Validation uncertainty of the cooling tower model produced by the PM. Second, a rigorous analysis of the Validation uncertainty is given and the dependence of the Validation uncertainty on the coordinates of the target is identified. Figure B-9 shows the PM prediction of the validation set of laser targets superimposed on top of the original laser measurement results. In this Section, the Validation uncertainty that is defined as the difference between the coordinates of PM prediction in red and the validation coordinate given in blue for the validation set is assessed. The PM predictions in the X, Y, Z coordinates are investigated separately. 0.5 Z 0 1 Laser data PM reconstruction Y X Figure B-9: Validation set of laser measurements given in blue along with PM predictions of the same targets in red. The axes are removed due to the sensitivity of the data. i Uncertainty in the prediction of the Z coordinate The Validation uncertainty in the prediction of the Z coordinate is defined as Z = Z P M Z V alidation. The Validation uncertainty Z is plotted as a function of the polar radial coordinate (r ), angle(θ ) and the height(z) respectively in Figures B-10A,B,C by red dots. The histogram of Z is shown in Figure B-10F in red. 241

242 Linear regression fit Original data Linear regression fit Original data Z(m) Z(m) [A] r(m) [B] θ (rad) Linear regression fit Original data Adjusted data Fit: y= *x 95% conf. bounds Z(m) Delta z [C] Z(m) [D] Adjusted whole model Probability Original data After correction Data 10 [E] -3 [F] Figure B-10: A) Z as a function of the polar angle, B) Z as a function of Z, C) Z as a function of the radial coordinate, D) Added variable plot for Z using stepwise regression, E) Normal probability plot of Z before and after correction using regression, F) Histograms of Z before and after correction using regression Next, a statistical analysis was carried out to identify the dependence of the Validation uncertainty in the Z direction on the coordinates. It is hypothesized that the Validation uncertainty in the PM prediction of Z is a linear function of r, θ,z and X,Y as given in Equation B.4. The Cartesian coordinates X, Y are included in order to obtain a form of functional dependence that might be present between Z and cos θ or sin θ which can be expressed as linear combinations of the X,Y coordinates. Z = a 1 r + a 2 θ + a 3 Z + a 4 X + a 5 Y (B.4) 242

243 Then, the stepwise regression was implemented to select the strongest predictor variables among the 5 explanatory variables in Equation B.4. The stepwise regression procedure selects the best explanatory variables according to the P-value and the F statistic first. It then proceeds to sequentially add more variables that improve the model based on the F statistic until no further improvement can be made. The stepwise regression is recommended as the best method to pick the best regression equation [57]. Table B.2 shows the estimates of the coefficients selected by the stepwise regression algorithm along with the Standard error (SE),T-statistic and the P-value. The statistical significance of the selected variables is indicated by the P-value. Therefore, the linear regression model with the predictor variables selected by the stepwise regression is statistically significant since low P-values compared to 1.0 correspond to high statistical significance indicating a strong correlation between the explanatory and response variables. The overall performance of the regression model with the selected variables is given in Table B.3. The R 2 value of the linear regression with the selected variables is 0.384, which is not high. However, the F statistic of the regression model versus a constant model is 33.1 corresponding to a P-value of 9.7e-36 suggesting a highly statistically significant correlation. The usefulness or worthwhileness of the regression in situations with high F-statistic and low R 2 was investigated by Box and Wetz in 1964[58, 59]. Their viewpoint is that in order for the regression to be useful, the F statistic should be at least 4 times greater than the chosen significance level. In this case, with significance level α = 5%, the F statistic with degrees of freedom ν 1 = 6 1, ν 2 = would need to exceed 4F (ν 1 = 5, ν 2 = 374, 1 α = 0.95) = Our regression model with F=33.1 does satisfy this criterion. The corresponding R 2 value that needs to be exceeded in order for the model to be regarded as a satisfactory predictor is R 2 = ν 1 F/(ν 1 F + ν 2 ). The R 2 = also satisfies this criteria. So, based on the arguments provided by Box and Wetz, it is concluded that the linear regression model found by the stepwise regression is satisfactory. The calculation in 243

244 this paragraph is based on a similar analysis demonstrated in [57]. Estimate SE tstat P-value (Intercept) e-05 Z e-08 X e e Y e e ZX e e e-05 ZY e e e-05 Table B.2: Model coefficients of linear regression with the selected variables for the Z coordinate Number of observations: 380 Error degrees of freedom: 372 Root Mean Squared Error: R 2 : Adjusted R F-statistic vs. constant model: 33.1 P-value 9.7e-36 Table B.3: Model summary statistics of simple linear regression with selected variables for the Validation uncertainty in the Z coordinate Figures B-10A,B,C show the results of the linear regression fit superimposed on top of the original data as a function of the radial coordinate (r ), angle(θ ) and the height(z ) respectively showing a reasonable fit of the variation of Z in different coordinates. The added variable plot of the stepwise regression is given Figure B-10D showing that the data is highly random and the variation explained by the regression model is subtle. Figure B-10E and Figure B-10F shows the normal probability plots and the histogram of residuals both before and after applying the regression model respectively. The bias in the Validation uncertainty of the Z coordinate prediction can 244

245 be corrected using the regression model from the fact that the distribution is closer to the Gaussian random variable (indicated by a line in the normal probability plot in Figure B-10E) after the linear dependence is extracted. Moreover, the histogram of residuals is non-central before applying the regression model suggesting a significant bias. The histogram is centered after applying the regression model. ii Uncertainty in the prediction of the X coordinate The Validation uncertainty in the prediction of the X coordinate is defined as X = X P M X V alidation. The Validation uncertainty X is plotted as a function of the polar radial coordinate (r ), angle(θ ) and the height(z ) respectively in Figures B-11A,B,C by red dots. The histogram of X is shown in Figure B-11F in red. Next, a regression analysis was carried out to identify the dependence of the Validation uncertainty in the X direction on the coordinates. It is hypothesized that the Validation uncertainty in the PM prediction of X is a linear function of r, θ,z and X,Y as given in Equation B.5. The Cartesian coordinates X, Y are included in order to obtain a form of functional dependence that might be present between X and cos(θ) or sin(θ) which can be expressed as a linear combination of the X,Y coordinates. X = a 1 r + a 2 θ + a 3 Z + a 4 X + a 5 Y (B.5) Then, the stepwise regression was implemented to select the strongest predictor variables among the 5 explanatory variables in Equation B.5. The stepwise regression procedure selects the best explanatory variables according to the P-value and the F statistic first. It then proceeds to sequentially add more variables that improve the model based on the F statistics until no further improvement can be made. The stepwise regression is recommended as the best method to pick the best regression equation [57]. 245

246 X(m) X(m) [A] Linear regression fit Original data r(m) [B] Linear regression fit Original data θ (rad) X(m) 0 \Delta x Adjusted data Linear regression fit Fit: y= *x [C] Original data Z(m) [D] 95% conf. bounds Adjusted whole model [E] Probability Original data After correction data Data [F] Figure B-11: A) X as a function of the polar angle, B) X as a function of Z, C) X as a function of the radial coordinate, D) Added variable plot for X using stepwise regression, E) Normal probability plot of X before and after correction using regression, F) Histograms of X before and after correction using the regression Table B.4 shows the estimates of the coefficients selected by the stepwise regression algorithm along with the Standard error SE, T-statistics and the P-value. The statistical significance of the variables selected is indicated by the P-values. Therefore, the linear regression model with the predictor variables selected by the stepwise regression is statistically significant since low P-values compared to 1.0 correspond to high statistical significance indicating a strong correlation between the explanatory and response variables. The overall performance of the regression model with the selected variables is given 246

247 in Table B.5. The R 2 value of the linear regression with the selected variables is 0.30 which is not high. However, the F statistic of the regression model versus a constant model is 32.1 corresponding to P-value of 3.19e-27 suggesting a highly statistically significant correlation. Similar to the analysis carried out in Section i, the regression model performance can be analyzed. In this case, with significance level α = 5%, the F statistic with degrees of freedom ν 1 = 6 1, ν 2 = would need to exceed 4F (ν 1 = 5, ν 2 = 374, 1 α = 0.95) = Our regression model with F = 33.1 does satisfy this criterion. The corresponding R 2 value that needs to be exceeded in order for the model to be regarded as a satisfactory predictor is R 2 = ν 1 F/(ν 1 F + ν 2 ) = The R 2 = 0.30 also satisfies this criterion. Following the arguments provided by Box and Wetz, it can be concluded that the linear regression model found by the stepwise regression for X is satisfactory. Estimate SE tstat P-value (Intercept) e-14 θ Z e e X -3.32e e e-14 θz e e θx e e e-05 Table B.4: Model coefficients of linear regression with the selected variables for the X coordinate 247

248 Number of observations: 380 Error degrees of freedom: 377 Root Mean Squared Error: R 2 : 0.3 Adjusted R F-statistic vs. constant model: 32.1 P-value 3.19e-27 Table B.5: Model summary statistics of simple linear regression with selected variables for the Validation uncertainty in the X coordinate Figures B-11A,B,C show the results of the linear regression fit superimposed on top of the original data as a function of the radial coordinate (r ), angle(θ ) and the height(z ) respectively showing a reasonable fit of the variation of X in different coordinates. The added variable plot of the stepwise regression is given Figure B-11D showing that the data is highly random and the variation explained by the regression model is subtle. Figure B-11E and Figure B-11F show the normal probability plots and histogram of residuals both before and after applying the regression model respectively. Compared to linear regression analysis conducted for Z in Section i, the distribution of the Validation uncertainty for X does not change significantly after applying the regression model. This suggests that there is no significant bias in the prediction of the X coordinate using the Photomodeler. iii Uncertainty in the prediction of the Y coordinate The validation uncertainty in the prediction of the Y coordinate is defined as Y = Y P M Y V alidation. The validation uncertainty Y is plotted as a function of the polar radial coordinate (r ), angle(θ ) and the height(z ) respectively in Figures B-12A,B,C by red dots. The histogram of Y is shown in Figure B-12F in red. Next, a statistical analysis was carried out to identify the dependence of the validation uncertainty in the Y direction on the coordinates. It is hypothesized that the 248

249 Validation uncertainty in the PM prediction of Y is a linear function of r, θ,z,x,y and the cross terms rx, ry, θx,θy and ZX, ZY as given in Equation B.6. The Cartesian coordinates X, Y are included in order to obtain a form of functional dependence that might be present between Y and sin(θ) or cos(θ) which can be expressed as a linear combination of the X, Y coordinates. Y = a 1 r + a 2 θ + a 3 Z + a 4 X + a 5 Y + a 6 rx + a 7 ry + a 8 θx + a 9 θx + a 10 ZX + a 11 ZY (B.6) Then, the stepwise regression was implemented to select the strongest predictor variables among the 5 explanatory variables in Equation B.6. The stepwise regression procedure selects the best explanatory variables according to the P-value and the F statistic first. It then proceeds to sequentially add more variables that improve the model based on the F statistic until no further improvement can be made. The stepwise regression is recommended as the best method to pick the best regression equation [57]. Table B.6 shows the estimates of the coefficients selected by the stepwise regression algorithm along with the Standard error SE, t-statistics and the P-value. The statistical significance of the variables selected is indicated by the P-values. Therefore, the linear regression model with the predictor variables selected by the stepwise regression is statistically significant since low P-values compared to 1.0 correspond to high statistical significance indicating a strong correlation between the explanatory and response variables. The overall performance of the regression model with the selected variables is given in Table B.7. The R 2 value of the linear regression with the selected variables is 0.43 which is not high. However, the F statistic of the regression model versus a constant model is 25.5 corresponding to P-value of 4.11e-39 suggesting a highly statistically significant correlation. Similar to the analysis carried out in Section i, the regression model performance can be analyzed. In this case, with significance level α = 5%, the F statistic with degrees 249

250 of freedom ν 1 = 12 1, ν 2 = would need to exceed 4F (ν 1 = 11, ν 2 = 368, 1 α = 0.95) = Our regression model with F=25.5 does satisfy this criteria. The corresponding R 2 value that needs to be exceeded in order for the model to be regarded as a satisfactory predictor is R 2 = ν 1 F/(ν 1 F + ν 2 ) = The R 2 = 0.43 also satisfies this criteria. Following the arguments provided by Box and Wetz, it can be concluded that the linear regression model found by the stepwise regression for Y is satisfactory. Estimate SE tstat P-value (Intercept) e-08 r e-06 θ Z e-12 X e-07 Y e rx e e ry e e θx e e θy e e ZX e e e-12 ZY e e e-05 Table B.6: Model coefficients of linear regression with the selected variables for the Y coordinate 250

251 Number of observations: 380 Error degrees of freedom: 368 Root Mean Squared Error: R 2 : Adjusted R F-statistic vs. constant model: 25.5 P-value 4.11e-39 Table B.7: Model summary statistics of simple linear regression with selected variables for the Validation uncertainty in the Y coordinate Y(m) Linear regression fit Original data Y(m) Linear regression fit Original data [A] r(m) [B] θ(rad) Y(m) Linear regression fit Original data \Delta Y [C] Z(m) [D] -2 Adjusted data -4 Fit: y= *x -6 95% conf. bounds Adjusted whole model Probability Normal Probability Plot Original data After correction Data 10 [E] -3 [F] Figure B-12: A) Y as a function of the polar angle, B) Y as a function of Z, C) Y as a function of the radial coordinate, D) Added variable plot for Y using stepwise regression, E) Normal probability plot of Y before and after correction using regression, F) Histograms of Y before and after correction using the regression 251

252 Figures B-12A,B,C show the results of the linear regression fit superimposed on top of the original data as a function of the radial coordinate (r ), angle(θ ) and the height(z ) respectively showing a reasonable fit of the variation of Y in different coordinate axes. The added variable plot of the stepwise regression is given Figure B-12D showing that the data is highly random and the variation explained by the regression model is subtle. Figure B-12E and Figure B-12F show the normal probability plots and histogram of residuals both before and after applying the regression model respectively. Compared to the linear regression analysis conducted for Z in Section i, the distribution of the validation uncertainty for Y does not change significantly after applying the regression model. Similar to the analysis in the X coordinate, the results of the regression analysis in the Y coordinate suggest that there is no significant bias in the prediction of the Y coordinate using the Photomodeler. B.5 Conclusion In this Appendix, the simulation of the cooling tower using the Photomodeler is evaluated. The Photomodeler is used to shorten the amount of time necessary to create a full 3D model of the cooling tower. First, the random uncertainty of the PM is quantified using the MCMC method. Second, the validation uncertainty of the PM is assessed using the validation laser dataset. The stepwise linear regression is employed to obtain a functional dependence of the Validation uncertainty on the X,Y,Z coordinates. Based on the analysis, the validation uncertainties in X,Y and Z directions were found to have a significant correlation with their coordinates. The Z coordinate was found to be over-predicted by the PM in a statistically significant way. The stepwise regression was then used to correct the systematic bias of the PM prediction in the Z direction. The validation uncertainties in the predictions of the X,Y coordinates were not found to have a significant bias based on the results of the regression analyses. 252

253 Appendix C Case Study 4: Representativeness uncertainty in measurement of the NO x emission from a Combined Cycle Gas Turbine C.1 Introduction The emissions of atmospheric pollutant gasses such as the NO x from fossil fuel plants are regulated by the environmental agencies in Europe. At a specific Combined Cycle Gas Turbine plant that was studied, during normal operation, the Automatic Measurement System (AMS) measures the pollutant gas emissions. The Measurement Values read by the AMS sensor are subject to Spatial Representativeness uncertainty because the sensor does not cover the entire cross section of the smoke-stack. Routine auditing tests are conducted at the CCGT plant to estimate the Representativeness uncertainty. During an auditing test called VERITAS, a wire-mesh sensor is inserted into the Smokestack, and more accurate reading of the NO x emission is obtained. In order to correct the AMS values so that their use yields results that 253

254 reflect the auditing values, models need to be developed to supplement the NO x concentration values measured by the AMS. Figure C-1 shows the available data, and the associated Representativeness uncertainty of the problem studied. In Case Study 4, first, a statistical model was developed to predict the VERITAS value from two parameters measured by the AMS. This model was built using the VERITAS data that were available. Two parameters out of 6 in AMS were chosen purely on the basis of their explanatory powers: Feed gas flow rate and the Turbine Temperature. The two selected parameters are shown to have a theoretical basis to influence the NO x emission rate as explained in Section C.3. Second, it was shown in Section C.A through statistical analysis that the AMS Measurement Value of the NO x is principally influenced by the local conditions in the Smokestack such as the Temperature in the Smokestack rather than the fundamental parameters of natural gas combustion as discussed in Section C.A. Thirds, a linear regression model is built to correct the Representativeness uncertainty based on the available VERITAS and AMS data sets. 254

255 Figure C-1: The Representativeness uncertainty in the NO x emission rate from a CCGT in AMS, and VERITAS audit data. AMS data: 9 min intervals of 6 operational parameters Power level Oxygen flow in chimney Turbine temperature Smokestack T Feed gas flow rate NO x (AMS) concentration NOx(AMS) Auditing data (VERITAS): 1 hour average value for 19 hours Power NO x (VERITAS) concentration NOx(VERITAS) Spatial Representativeness Uncertainty The difference between NOx(VERITAS) and NOx(AMS) is about 10% Chimney C.2 Experimental data, and Representativeness uncertainty During normal operation, the Automatic Measurement System (AMS) measures the atmospheric pollutant gas emissions from the CCGT that was studied. The AMS system measures 6 different operational parameters at 5 min intervals. The parameters measured by the AMS are Power (MW), Oxygen flow rate in the Smokestack (kg/s), Turbine Temperature ( 0 C), Smokestack Temperature ( 0 C), Feed gas flow rate (kg/s), and NO x (AMS) emission concentration (mg/nm 3 ). The relationship between the 6 parameters is summarized in Figure C-2. Significant inter-dependence between the parameters measured by the AMS is observed from the matrix plot summary. For example, the Power is strongly correlated with the Feed gas flow rate and there appears to be dependence between the Turbine Temperature and the Oxygen flow rate in the Smokestack as shown in gray-background subplots in Figure C-2.The regression anal- 255

256 ysis using parameters that are highly dependent can introduce a so-called collinearity and result in an unstable prediction[59]. Tools and techniques are available to detect and prevent collinearity. Figure C-2: Matrix plot of 6 operational parameters in AMS. Grey-background subplots show significantly correlated parameters. The axes of the matrix plots are re-scaled due to the sensitivity of the data. In order to estimate the Representativeness uncertainty of the NO x emissions, a verification test conducted by an auditing office called VERITAS is performed at the CCGT plant. The total duration of the VERITAS test is 18 hours, and the hourly averages of Power (MW) and the NO x (VERITAS) emission concentration (mg/nm 3 ) are measured. Therefore, only 18 data points corresponding to the hourly average values of the VERITAS quantities are available to estimate the Representativeness uncertainty of the NO x concentration measurement using the AMS. A wire-mesh sensor measures the NO x concentration during the VERITAS test, whereas the AMS measurement is taken at a single point location in the Smokestack. The AMS data are recorded during the VERITAS test, which enables the estimation of the Representativeness uncertainty. A summary of the available experimental data are given in Table C

257 Available data Measurement type AMS 5 min intervals Point measurement Test VERITAS 18 hourly averaged data points Mesh measurement Table C.1: Availability of AMS and VERITAS data Figure C-3 shows the comparison between the AMS and the VERITAS NO x measurement. The vertical axis label in Figure C-3 is defined as NO x NO x (V ERIT AS) = NO x(ams) NO x (V ERIT AS) NO x (V ERIT AS) (C.1) Figure C-3: A comparison between the AMS and the VERITAS NO x values The AMS is shown to overpredict the NO x concentration compared to the VERITAS data for 15 out of 18 data points. However, one data point indicates an underpredic- 257

258 tion of about 12%. Additional analysis and a predictive model building process are presented in Section C.5. C.3 Theory of natural gas combustion The combined cycle gas turbine is a gas turbine coupled with a Steam turbine. Feed gas in a CCGT is natural gas and a Steam Rankine cycle is designed to recover the heat from the exhaust stream of the natural gas combustion by boiling water to create steam and drive a turbine. Figure E-5 shows a typical arrangement and the T-S diagram of a CCGT where both the gas cycle and the Steam Rankine cycle are plotted [60]. The combustion of natural gas is indicated in the right Figure E-5 by the states The states correspond to a steam cycle that is subcritical, and the states correspond to a supercritical Steam cycle. The combination of the gas combustion with the Rankine cycle enables achieving a higher cycle efficiency compared to a single combustion cycle. As of August, 2016, the world record thermal efficiency of 62.22% belongs to the General Electric HA turbine at EDF s Bouchain CCGT in France. [A] [B] Figure C-4: A typical arrangement of a combined cycle Steam and gas turbine(a). T-S diagram (B) [60] The natural gas used in power generation consists mainly of methane CH 4, ethane 258

259 C 2 H 6, propane C 3 H 8 and small fractions of butane, higher hydrocarbons, S 2, O 2, and CO 2. The combustion of natural gas in a gas turbine is characterized by a set of governing equations: First and Second laws of thermodynamics, mass and chemical species balance equations. In this Section, each of the governing equations is explained in the context of combustion and their relevance to the emissions of pollutant gases. The course notes from Professor Ahmed Ghoniem s class on the Fundamentals of Advanced Energy Conversion 26 at MIT are used for this discussion. The first law of thermodynamics in a control volume defined around the gas turbine given in Figure C-5 is de dt = Q Ẇ + react n i ĥ i n i ĥ i prod (C.2) Figure C-5: Control volume around a gas turbine where the specific enthalpy of a gas stream has both the thermal and the chemical components ĥ i (T ) = ĥi,thermal + ĥi,chemical = T T 0 c p,i (T )dt + ĥ0 i,chemical (C.3)

260 The second law of thermodynamics is ds dt = k Q k T k + n i ŝ i n i ŝ i + Ṡg react prod (C.4) where Ṡg is the entropy generation rate, and the entropy of species i in a mixture is where ŝ i (T, p, X i ) = ( ŝ 00 i + T T 0 c p,i (T ) T dt ) ( R ln p p 0 + R ln X i ) (C.5) X i = p i p (C.6) corresponds to the fractional partial pressure of each of the gas species in a given mixture. One of the parameters characterizing the natural gas combustion is the equivalence ratio that indicates either lean or rich burning compared to the stoichiometric air-fuel mass flow ratios. The equivalence ratio is defined as φ = AF AF st = ṅair/ṅ fuel ṅ st air/ṅ fuel st ṁair,in ṁ fuel,in (C.7) where ṅ air and ṅ fuel are the air and fuel molar flow rates. ṁ air and ṁ fuel are the mass flow rates of air and fuel flows. The equivalence ratio greater than 1 indicates a rich burn and less than 1 indicates a lean burn. The equivalence ratio combined with the specifications of the feed natural gas composition, the mass conservation law, the first, and the second laws determines the flame Temperature and the product gas mass fractions in the gas turbine. Figure C-6 shows an example calculation result of pure CH 4 combustion at different equivalence ratios. First, the equivalence ratio is determined based on the mass flow rate of the Feed gas and the inlet air flow rate. Based on the equivalence ratio, the exhaust gas species 260

261 and the combustion Temperature can be determined. This suggests that the portion of the gases such as the NO x in the exhaust stream has a direct correlation with the combustion Temperature. The N O mass fraction in the exhaust stream is shown to peak at around equivalence ratio of φ = 0.8, whereas the highest combustion Temperature is achieved at φ = 1.0. In Case Study 4, the performance indicator is defined as a flow averaged concentration of the NO x in the exhaust stream of the gas turbine as given in Section i. The mass fractions of the product gas discussed in this Section and given in Figure C-6 are directly proportional to the average concentration of the NO x. The parameters are identified to be critical in determining the exhaust gas composition are the Feed gas flow rate which enters into the calculation of the equivalence ratio as given in Equation C.7 and the combustion Temperature which determines the exhaust gas composition as given in Figure C.7. The combustion Temperature is measured in the AMS via the Turbine Temperature measurement. 261

262 Figure C-6: Mass fractions of difference gas species in an exhaust stream, and the combustion Temperature as a function of the inlet fuel equivalence ratio φ. The blue line at φ = 0.93 shows an example of how the mass fractions of different exhaust gases and the combustion Temperature are calculated for a given equivalence ratio. The exhaust gas mass fraction is directly proportional to the flow average concentration of the NO x emission. C.4 Statistical modeling of relationship between AMS parameters using the stepwise regression In Section C.3, the theory of natural gas combustion is discussed. It was shown that the key parameters such as the combustion Temperature, N O and other exhaust gas emission rates, and the equivalence ratio of inlet fuel have a non-linear relationship with no closed-form solution as indicated in the example of pure CH 4 combustion given in Figure C-6. In this Section, an empirical relationship between the key oper- 262

263 ational parameters of a gas turbine is obtained based on the AMS data. Specifically, a non-linear stepwise regression was employed to select the parameters that have the highest explanatory powers. The implementation of the non-linear regression is necessary to identify cross correlations among the explanatory variables. In this Section, the AMS parameters are abbreviated as shown in Table C.2. NO x concentration y Oxygen flow rate in the Smokestack x 1 Power x 2 Turbine Temperature x 3 Smokestack Temperature x 4 Feed gas flow rate x 5 Table C.2: Abbreviated notation of AMS parameters The "Stepwise regression is a systematic method for adding and removing terms from a multilinear model based on their statistical significance in a regression 27 ". The stepwise regression method is employed in this section to identify the most significant factors from among the AMS parameters that explain the NO x (AMS). As explained in Section C.3, the Feed gas flow rate (x 5 ), and the Turbine Temperature (x 3 ) are expected to be selected if the NO x (AMS) is a good indicator of the total amount of NO x being generated. As can be seen from Table C.3, the Oxygen flow rate in the Smokestack (x 1 ), the Smokestack Temperature (x 4 ), and the Power (x 2 ) are selected in that order. The results of the stepwise regression suggest that the NO x value read by AMS is the most strongly influenced by the Smokestack Temperature rather than the fundamental parameters that explain natural gas combustion such as the Turbine Temperature and the Feed gas flow rate. The estimates of the coefficients of the model using the stepwise regression are given in Table C.4, and the summary of the model performance is in Table C.5. The diagnostic plots are given in Figure C-7. An improvement over the linear regression and robust linear regression models

264 used in Sections C.A.1 and C.A.2 can be seen from comparing the respective plots in Figure C-7C, and Figure C-7B. However, the normal probability plot in Figure C-7D still indicates a highly non-gaussian tail for the residual distribution. The reason for these tails can be the outliers in the original data. The Cook s distance plot in Figure C-7C supports this hypothesis showing the outliers with high Cook s distance values. 1. Adding x 1 FStat = pvalue = 0 2. Adding x 4 FStat = pvalue = 0 3. Adding x 2 FStat = pvalue = e Adding x 1 :x 4 FStat = pvalue = e Adding x 5 FStat = pvalue = e Adding x 1 :x 5 FStat = pvalue = e Adding x 1 :x 2 FStat = pvalue = 0 8. Adding x 4 :x 5 FStat = pvalue = e Adding x 2 :x 4 FStat = pvalue = e-166 Table C.3: The variable selection order using the stepwise regression. 264

265 Estimate SE tstat pvalue (Intercept) e-95 x e-95 x x e-69 x x 1 :x x 1 :x e-72 x 1 :x x 2 :x e-166 x 4 :x e-240 Table C.4: The coefficients of the stepwise regression model. Number of observations: 6600 Error degrees of freedom: 6590 Root Mean Squared Error: 7.21 R-squared: Adjusted R-Squared F-statistic vs. constant model: 3.66e+03 p-value 0 Table C.5: Summary statistics of the model using the stepwise regression. 265

266 Residual NO X (mg/nm 3 ) [A] NO X (mg/nm 3 ) [B] Residual NO X (mg/nm 3 ) Cook's distance Probability [C] Index [D] Residual NO X (mg/nm 3 ) Figure C-7: Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot of residuals In this Section, the result of the stepwise regression model fit to the operational parameters of the CCGT has been presented. Using the stepwise regression model, the Oxygen flow rate in the Smokestack and the Smokestack Temperature were selected to be the strongest explanatory variables for the NO x (AMS). This result suggests that the AMS readings may be influenced by the local conditions in the Smokestack rather than the fundamental parameters of combustion such as the Turbine Temperature and the Feed gas flow rate. In addition to the stepwise regression model given in this Section, a number of models were experimented to come up with an empirical relationship between the AMS parameters and they are given in Supplemental Section C.A. 266

267 C.5 Development of a Statistical predictive model for the NO x (VERITAS) In this Section, statistical predictive models for the NO x (VERITAS) are built using the AMS and the VERITAS data. The objective of the models is to predict the NO x (VERITAS) auditing data based on the AMS parameters. First, a multiple linear regression model is built using all 6 operational parameters of the AMS in Section C.5.1. A non-linear regression is not employed here due to a small number of auditing (VERITAS) data and to keep the number parameters of the model low. Second, the stepwise regression is used for the variable selection to identify the operational parameters that have the highest explanatory power in Section C.5.2. C.5.1 Linear regression using all explanatory variables In this Section, a linear regression model is built using all explanatory variables available. The response variable is the NO x (VERITAS) and the explanatory variables are Power (MW), Oxygen flow rate in the Smokestack (kg/s), Turbine Temperature ( 0 C), Smokestack Temperature ( 0 C), Feed gas flow rate (kg/s), NO x (AMS) emission concentration (mg/nm 3 ). The model that is fit using the data is given in Equation C.8. NO x (V ERIT AS) = a 1 + a 2 x 5 + a 3 x 3 + a 4 x 2 + a 5 x 1 + a 6 x 4 + a 7 y (C.8) where the abbreviated notations are given in Table C.2. The estimates of the coefficients are given in Table C.6, and the summary statistics of the model result are given in Table C.7. The coefficients a 2,a 3,a 6 result in p-values that are relatively high compared to the others. This suggests possible exclusion of them from the model to improve robustness. The performance of the linear regression model with all explanatory variables included is evaluated by the diagnostic plots given in Figure C

268 Estimate SE tstat pvalue a a a a a a a Table C.6: The results of linear regression estimate of regression coefficients along with the t-statistics, and the p-values. Number of observations 17 Error degrees of freedom 10 Root Mean Squared Error R-squared 0.91 Adjusted R-Squared F-statistic vs. constant model 16.8 p-value Table C.7: Linear regression model summary statistics 268

269 [A] [B] [C] [D] Figure C-8: Selected diagnostic plots of the stepwise regression model. A) Model fit plotted with the data, B) model prediction and NO x (AMS) benchmarked with NO x (VERITAS), C)The Cook s distance for each model prediction point (data points above the horizontal line are considered outliers), D) The normal probability plot of residuals. In the following Section, the stepwise regression is employed to select the variables with the highest explanatory power. C.5.2 Variable selection using the stepwise regression In this Section, the stepwise regression is used to select the variables that have the largest explanatory power. The response variable is NO x (VERITAS) and the explanatory variables are Power (MW), Oxygen flow rate in the Smokestack (kg/s), Turbine Temperature ( 0 C), Smokestack Temperature ( 0 C), Feed gas flow rate (kg/s), NO x (AMS) emission concentration (mg/nm 3 ). The stepwise regression works by forward selecting variables. In the first step, one of the 6 explanatory variables that have the highest correlation with the response variable is selected. Next, the decision to add a second variable is based on whether 269

270 the two variables combined result in a better explanatory power. If a previously added variable does not meet the minimum required criterion to stay in the model, a backward elimination procedure is implemented to identify and exclude the weakest explanatory variable [61]. In this Case Study, the MATLAB built-in tool to implement the stepwise regression is employed. The details of the implementation of the algorithm can be found in the documentation page 28. The output of the variable selection process is given in Table C.8. The backward elimination procedure resulted in only two variables surviving the process the Turbine Feed gas flow rate and the Turbine Temperature. 1. Adding (Turbine Feed gas flow rate) 2. Adding (Turbine Temperature) FStat = pvalue = e-07 FStat = pvalue = Table C.8: The variable selection process for NO x (VERITAS) The resulting model after the variable selection is given in Equation C.9. NO x (V ERIT AS) = a 1 + a 2 x 5 + a 3 x 3 (C.9) The estimates of the coefficients of the linear fit are given in Table C.9. Estimate SE tstat pvalue a a a e-08 Table C.9: The results of the stepwise regression estimate of the coefficients along with the t-statistics, and the p-values. The model summary statistics are given in Table C.10. The performance of the

271 stepwise regression model with the two selected explanatory variables is evaluated by the diagnostic plots given in Figure C-9. Number of observations 18 Error degrees of freedom 15 Root Mean Squared Error R-squared Adjusted R-Squared F-statistic vs. constant model 61.2 p-value 6.13e-08 Table C.10: The summary statistics of the stepwise regression model. [A] [B] [C] [D] Figure C-9: Selected diagnostic plots of the stepwise regression model. A) The model fit plotted with the data, B) The model prediction and NO x (AMS) benchmarked with NO x (VERITAS), C)The Cook s distance for each model prediction point (data points above the horizontal line are considered outliers), D) The normal probability plot of residuals. The result of the variable selection using the stepwise regression suggests that the NO x (VERITAS) auditing value is the most strongly explained by the Feed gas flow 271

272 rate, and the Turbine Temperature. This is expected from the theoretical analysis in Section C.3. It is interesting to note that the NO x (AMS) was not selected as a strong explanatory variable for the NO x (VERITAS) even though they measure the same quantity. It is consistent with the finding from Section C.4 that the NO x (AMS) value is the most strongly explained by the Oxygen flow rate in the Smokestack, Smokestack Temperature, Power which are associated with the local Smokestack conditions. The predictive model obtained in this Section can be improved by the collection and the analysis of more auditing data. C.6 Conclusion In Case Study 4, the Representativeness uncertainty of the measurement of the NO x emission rate from a CCGT is considered. A statistical model was developed to predict the VERITAS value from two parameters measured by the AMS. The parameters, chosen purely on the basis of their explanatory powers, are the Feed gas flow rate and the Turbine Temperature. They are shown to have a theoretical basis as explained in Section C.3. In contrast, it was shown in Section C.A through statistical analysis that the NO x (AMS) value is explained by the local conditions such as the Temperature in the Smokestack rather than the fundamental parameters of natural gas combustion. C.A Statistical models for empirical relationship between AMS parameters In this Supplemental section, the empirical statistical models built to model the relationship between the AMS parameters are presented. They are given in the increasing order of complexity and the comparison between the models is given in Section E.G. In Section C.4, a non-linear stepwise regression was employed to accomplish the same 272

273 task. The analysis of Section C.4 is not repeated here. C.A.1 Simple linear regression As a first attempt to model the relationship between AMS parameters, a simple linear regression with the functional form in Equation C.10 is used. y = a 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 (C.10) The estimates of the coefficients of the model are given in Table C.11, and the summary statistics of the model performance are given in Table C.12. Selected diagnostic plots are given in Figure C-10. Figure C-10A shows residuals after subtracting the fitted model from the original data. Figure C-10B and Figure C-10D show the histogram and the normal probability plots of the residuals respectively. A Gaussian random variable corresponding to a perfect fit would have a normal probability plot indicated by a straight line. Figure C-10D indicates non-gaussian shape at the two ends of the residual as shown by the histogram in Figure C-10B. Figure C-10C shows The Cook s distance plot for all data points. The Cook s distance is a statistical measure used to indicate the outliers in the data. Estimate SE tstat pvalue a e-09 a a e-90 a e-11 a a e-26 Table C.11: The model coefficients of the simple linear regression 273

274 Number of observations: 6600 Error degrees of freedom: 6594 Root Mean Squared Error: 11.8 R-squared: Adjusted R-Squared F-statistic vs. constant model: 1.64e+03 p-value 0 Table C.12: Summary statistics of the simple linear regression 150 Plot of residuals vs. fitted values 0.14 Histogram of residuals [A] [B] Case order plot of Cook's distance Normal probability plot of residuals Cook's distance Probability [C] Row number [D] Residuals Figure C-10: Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot. C.A.2 Robust linear regression and partial least squares Second, a robust linear regression is used to explain the relationship between the AMS parameters. The robust linear regression is a technique that gives less weight to the outliers [62]. The estimates of coefficients of the model are given in Table C.13, and the summary statistics of the model performance are given in Table C.14. The F-statistic given in the table indicates whether the model including all variables has 274

275 good explanatory power. High F value corresponding to low p-values indicates that all explanatory variables combined have a strong predictive power. Selected diagnostic plots are given in Figure C-11. Figure C-11A shows residuals after subtracting the fitted model from the original data. Figure C-11B and Figure C-11D show histogram and the normal probability plot of the residuals respectively. Figure C-11D indicates non-gaussian shape at two ends of the residual as shown by the histogram in Figure C-11B and by the normal probability plot in Figure C-11D. The reason for these tails can be the outliers in the original data. The Cook s distance plot in Figure C-11C supports this hypothesis showing the outliers with high Cook s distance values. Estimate SE tstat pvalue a a a a e-21 a e-39 a Table C.13: The model coefficients of the robust linear regression Number of observations: 6600 Error degrees of freedom: 6594 Root Mean Squared Error: 2.38 R-squared: Adjusted R-Squared F-statistic vs. constant model: 3.5e+04 p-value 0 Table C.14: The model summary statistics of the robust linear regression 275

276 Residual NO X (mg/nm 3 ) [A] NO X (mg/nm 3 ) Residual NO [B] X (mg/nm 3 ) [C] Cook's distance Index Probability Residual NO (mg/nm [D] 3 ) x Figure C-11: Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot of residuals. C.A.3 Robust non-linear model with interaction terms Next, the robust regression with interaction terms is implemented. The estimates of coefficients of the model are given in Table C.15, and the summary statistics of the model performance are given in Table C.16. Selected diagnostic plots are given in Figure C

277 Estimate SE tstat pvalue (Intercept) e-07 x e-06 x x e-177 x e-209 x x 1 :x x 1 :x e-177 x 1 :x e-202 x 1 :x x 2 :x e e-09 x 2 :x x 2 :x e-290 x 3 :x e e-05 x 3 :x e-54 x 4 :x Table C.15: The model coefficients of the robust regression with interaction terms Number of observations: 6600 Error degrees of freedom: 6584 Root Mean Squared Error: 1.95 R-squared: Adjusted R-Squared F-statistic vs. constant model: 2.57e+04 p-value 0 Table C.16: The model coefficients of the robust regression with interaction terms 277

278 Residual NO x (mg/nm 3 ) [A] NO x (mg/nm 3 ) [B] NO (mg/nm3) x [C] Cook's distance Index Probability Residual NO (mg/nm [D] 3 ) X Figure C-12: Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot. C.A.4 Alternating conditional expectation algorithm for the optimal transformation of variables The ACE algorithm developed by Brieman and Friedman [63] enables the estimation of optimal transformation function in both response and explanatory variables. In this Section, the ACE algorithm 29 implemented in MATLAB was employed for two variable combinations: the NO x versus Feed gas flow rate and the NO x versus Turbine Temperature. Figure C-13 and Figure C-14 show the result of the optimal transformations in NO x versus Feed gas flow rate and, for NO x versus Turbine Temperature respectively. The resulting optimal functions were hyperbola, and a linear function in Figure C-13, and were both hyperbolas in Figure C Written by H. Voss and J. Kurths 278

279 [A] [B] Figure C-13: Optimal transformation functions found by ACE for the NO x vs Turbine Feed gas flow rate [A] [B] Figure C-14: Optimal transformation functions found by ACE for the NO x vs Turbine Temperature C.A.5 Simple linear regression using the transformed variables In this Section, a simple linear regression is employed using the optimally transformed variables found in Supplemental Section C.A.4. The estimates of coefficients of the model are given in Table C.17, and the summary statistics of the model performance are given in Table C.18. Selected diagnostic plots are given in Figure C

280 Estimate SE tstat pvalue (Intercept) e-118 x e-38 x e-80 x e e x e-257 x e-79 Table C.17: The model coefficients of a simple linear regression with the transformed variables Number of observations: 6600 Error degrees of freedom: 6594 Root Mean Squared Error: 4.98 R-squared: Adjusted R-Squared F-statistic vs. constant model: p-value 0 3.1e+04 Table C.18: The model summary statistics of simple linear regression with the transformed variables 280

281 Residual φ(no X ) [A] φ(no X ) [B] Residual NO X (mg/nm 3 ) [C] Cook's distance Index Probability Reidual NO [D] X (mg/nm 3 ) Figure C-15: Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot C.A.6 Robust linear regression using the transformed variables In this Section, the robust linear regression is employed using the optimally transformed variables found in Supplemental Section C.A.4. The estimates of coefficients of the model are given in Table C.19, and the summary statistics of the model performance are given in Table C.20. Selected diagnostic plots are given in Figure C

282 Estimate SE tstat pvalue (Intercept) x e-17 x e-147 x e e x x e-159 x 1 :x e-136 x 1 :x e e x 1 :x x 1 :x e-279 x 2 :x e e e-46 x 2 :x e e-155 x 2 :x e-54 x 3 :x e e e-162 x 3 :x e e x 4 :x e-40 Table C.19: The model coefficients of robust regression with the transformed variables 282

283 Number of observations: 6600 Error degrees of freedom: 6584 Root Mean Squared Error: R-squared: Adjusted R-Squared F-statistic vs. model: constant p-value e+05 Table C.20: The model summary statistics of robust linear regression with the transformed variables Residual NO X (mg/nm 3 ) [A] NO X (mg/nm 3 ) [B] 0 50 Residual NO (mg/nm 3 ) X [C] NO X (mg/nm 3 ) Index Probability Residual NO [D] X (mg/nm 3 ) Figure C-16: Selected regression diagnostic plots. A) The residuals plot, B) The residuals histogram, C) The Cook s distance, D) The normal probability plot. C.A.7 Comparison of models Lastly, the performance of different models considered in this Section are compared. Figure C-17 shows the original data in blue with regression fits using different methods 283

284 in red. Table C.21 compares the R 2 values of the models in an increasing order of complexity. In general, increasing the complexity of the regression model used results in higher R 2 values as shown in Table C.21. Specifically, improvement in the model by using the robust regression which gives less weight to the outliers can be seen from Table C.21. The outliers affect the data we used to construct the regression model which can be seen from the Cook s distance plots, and the tails in the normal probability plots. The signs of over-fitting can be observed in the models that use the transformed variables with a large number of interaction terms. A cross validation exercise can be performed to prevent over-fitting in the future NO x 100 NO x [A] #data point [B] #data point NO x 30 NO x [C] #data point [D] #data point Figure C-17: Comparison of selected fits to the NO x data produced using different regression models. A) Simple linear regression, B) Robust linear regression with interaction terms, C) Simple linear regression with the transformed variables, D) Robust linear regression with the interaction terms. 284

285 R 2 # of # of linear terms cross terms Simple linear model Robust linear model Nonlinear model using stepwise regression Robust nonlinear model with interaction terms Simple linear model with transformed variables Robust linear model with transformed variables Robust linear model with transformed variables and cross terms Table C.21: Comparison of the statistical models developed. 285

286 286

287 Appendix D CFD methods D.A Theoretical description of RANS simulation The theoretical foundation for the Reynolds Average Navier-Stokes (RANS) model is given in this section. First, the concept of Reynolds averaging is explained. Next, standard, quadratic, and cubic k-ε models are described in this order. Lastly, the implementation of RANS simulation to identify and quantify the representativeness uncertainty in STAR-CCM+ is presented. D.A.1 Reynolds averaging The Navier Stokes (N-S) equation in Einstein notation for Newtonian fluids of viscosity µ and density ρ excluding the effect of gravity is given by ρ ( ) Uj t + U U j i = µ 2 U j x i x 2 i P x j (D.1) The exact solution of the N-S equation without approximation is implemented in Direct Numerical Simulation (DNS) and is computationally expensive and largely confined to the academic community. Reynolds averaging is implemented by separating velocity and pressure into a time- 287

288 averaged mean and a fluctuating component U j = U j + U j (D.2) where U j = 1 T T 0 pressure term, we have U j(t)dt for a long enough T, and U j = 0. Similarly for the P = P + P (D.3) After, Reynolds averaging, the N-S equations become ρ ( Uj t + U i U ) j = P + (2µS ij τ ij ) x i x j x i (D.4) where the Reynolds stress tensor is τ ij = ρ U iu j, and Sij = 1 2 ( ) Uj x i + U i x j. According to the Boussinesq hypothesis, we relate Reynolds stress to the fluid viscosity as τ ij = ρ U iu j where µ t is the turbulent viscosity tensor. ( Uj = µt + U ) i = 2µ t S ij (D.5) x i x j The N-S is not mathematically closed as given in Equation G.6. It is because µ t is an unknown quantity in the Boussinesq hypothesis. Therefore turbulence modeling is needed in order to solve the Reynolds Averaged N-S (RANS) equation. In the following section, we will describe the k- ε model, which is one of the widely employed turbulence models and is chosen in case studies of this Ph.D thesis. D.A.2 Standard k-ε model The k- ε model is one of the two-equation class of models. Standard k- ε model is characterized by having one equation for the turbulent kinetic energy k, and one for 288

289 the turbulent dissipation rate ε. The turbulent kinetic energy k is defined as k = 1 2 U iu i (D.6) The dissipation rate of turbulent kinetic energy is ε = µ 2ρ ( Uj + U ) 2 i x i x j (D.7) The conservation equation for the turbulent kinetic energy is ρ ( k t + U i k ) = x i x j ( [ µ + µ ] ) t k ρ U σ k x iu j j U i x j ρε (D.8) where σ k = 1.0. The conservation equation for the turbulent dissipation rate is ρ ( ε t + U i ε ) = x i x j ( [ µ + µ ] ) ( t ε U C ε1 σ k x i U j U i j x j ) ε k C ε2ρ ε2 k (D.9) where the parameters from Launder and Sharma [64] are C ε1 = 1.44, C ε2 = 1.92, σ ε = 1.3, µ t = ρc µ k 2 ε and C µ = D.A.3 Quadratic k- ε model Quadratic k- ε model is an example of a non-linear k- ε models. It is an extension of the Boussinesq hypothesis, and the constitutive relation is given in [65]. The quadratic extension has a fundamental effect as it reintroduces in the model the inequality of the normal stresses, referred to as flow anisotropy, which can have noticeable influence in complex shear regions. In the quadratic k-ε model, the Reynolds stress tensor is modeled as τ ij = ρ U iu j = µt S ij ρkδ k ij + C 1 µ t (S ik S jk 1 ) ε 3 S iks jk δ ij + k +C 2 µ t (Ω ik S kj + 1 ) ε 3 Ω k iks kj δ ij + C 3 µ t (Ω ik Ω jk 1 ) (D.10) ε 3 Ω ikω jk δ ij 289

290 where Ω ij = ( U i x j U ) j x i, and Sij = ( U i x j + U ) j x i are defined differently from Section D.A.1. The coefficients are given as C 1 = C NL1 (C NL4 + C NL5 S 3 ) C µ (D.11) C 2 = C NL2 (C NL4 + C NL5 S 3 ) C µ (D.12) with C 3 = C NL3 (C NL4 + C NL5 S 3 ) C µ (D.13) C NL1 C NL2 C NL3 C NL4 C NL Table D.1: Coefficients of the quadratic k-ε model D.A.4 Cubic k- ε model The cubic k- ε model is another example of a non-linear k- ε models. It is an extension of Boussinesq hypothesis, and the constitutive relation is given in [65]. The cubic terms reintroduce in the model the sensitivity to flow streamline curvature and flow rotation. In the cubic k-ε model, the Reynolds stress tensor is modeled as τ ij = µ t S ij kρδ k ij + C 1 µ t (S ik S jk 1 ) ε 3 S k iks jk δ ij + C 2 µ t ε (Ω iks kj + Ω ik S kj ) + k +C 3 µ t (Ω ik Ω jk 1 ) ε 3 Ω k 2 ikω jk δ ij + C 4 µ t ε (S klω 2 lj + S kj Ω li ) S kl + k 2 C 5 µ t (Ω ε 2 il Ω lm + S il Ω lm Ω mj 2 ) 3 S lmω mn Ω nl δ ij C 6 µ t k 2 ε 2 S ijs kl S kl + C 7 µ t k 2 ε 2 S ijω kl Ω kl (D.14)

291 where Ω ij = ( U i x j U j x i ), and Sij = ( U i x j + U j x i ).The coefficients of the cubic k- ε model are given in Table D.2. C NL1 C NL2 C NL3 C NL4 C NL5 C NL6 C NL C 2 µ 0-5C 2 µ 5C 2 µ Table D.2: Coefficients of the cubic k-ε model and C µ = ( max( S, Ω) ) exp exp ( ) 0.75 max( S, Ω) (D.15) with Ω = k ε 1 Ω 2 ijω ij and S = k 1 S ε 2 ijs ij D.B Theoretical description of LES (Large Eddy Simulation) LES is a turbulence model that resolves only large-scale unsteady turbulent motion or eddies but applies a model to represent smaller scale eddies. This is done by a low pass filtering operation based on the size of turbulent eddies. Figure D-1 shows an energy spectrum of filtered and non-filtered solutions of a certain flow configuration. The filtering operation truncates the reciprocal space representation of turbulent energy at the filtering width thereby losing the contribution from a far away point to turbulence at a given location. As illustrated in Figure D-1, filtering is done in order to preserve at least 80% of the turbulent energy [38]. In practice, the filtering width is the mesh size of the simulation, and the correct determination of mesh size is critical in performing LES. Mesh size in LES has to be chosen so that it covers at least 80% of turbulent energy. Therefore one needs to know the turbulent length-scale of the problem before performing LES. This necessitates preliminary RANS simulation in order to determine the turbulent length-scale that will in turn be used to generate 291

292 Figure D-1: One dimensional energy spectrum (solid line), and filtered spectrum (dashed line) for a certain flow condition. The horizontal axis is the non-dimensional wave number associated with Fourier decomposition of the turbulent energy at a point. The filtered spectrum is generated by a filter of size = 1/6L 11. L 11 is one dimensional integral turbulent length scale. K c denotes the wave number associated with filter width and is equal to K c = π/. The filtered spectrum covers 92% of the turbulent energies in the 1D case, and 80% in the full 3D solution[38]. The remaining portion of the spectrum is modeled by sub-grid scale methods. the LES mesh. D.B.1 Filtering operation For illustration, we will consider a one-dimensional case to explain the concept of filtering. We start with the Fourier transform of velocity as a function of coordinates Û(k) = F {U(x)} (D.16) where Û(k) is the Fourier transform of velocity value at x in k space. The filtered velocity is then { 1 Ū(x) = F Û(k)Ĝ(k)} (D.17) 292

293 where Ĝ(k) is the transfer function which is the Fourier transform of the filter function of choice. functions. Table D.3 shows selected filter functions along with their transfer Name Filter function Transfer function General G(r) Ĝ(k) = eikr G(r)dr 1 Box H( 1 r ) 2 ( ) 1/2 ( ) Gaussian 6 π exp 6r Sharp Spectral Cauchy sin πr/ πr a π [(r/ ) 2 +a 2 ], a = π 24 sin k /2 k /2 exp ( k ) H(k c k ), k c = π/ exp ( a k ) Pao exp ( π2/3 24 ( k )4/3) Table D.3: Selected filter functions in the coordinate and k space D.B.2 The filtered N-S equations By applying the filtering operation to the N-S equation, we obtain the filtered N-S equations as where U j ρ ( Uj t + U ) iu j = µ 2 U j x i x 2 i P x j (D.18), and P are the filtered velocity component j and the filtered pressure respectively. This equation can be rewritten where T R ij ρ ( ) Uj t + U U j i = µ 2 U j x i x 2 i ρ T r ij x i p x j (D.19) = U i U j U i U j is the residual stress tensor, and T r ij = T R ij 1 3 δ ijt R kk. The modified filtered pressure is defined as p = P T R ii. Note that the filtered N-S equation is not closed as is the Reynolds averaged N-S equation. Therefore we need models for the T r ij term to be able to solve it. By the eddy viscosity assumption, the sub-grid scale turbulent viscosity tensor is 293

294 defined as [64] T r ij = 2 ρ µ ts ij (D.20) where µ t is the sub-grid scale (SGS) turbulent viscosity, and S ij = ( ) U i x j + U j x i the filtered rate of strain tensor. Various LES models operate by modeling (SGS) turbulent viscosity to close the filtered N-S in Equation D.18. One particular model we have implemented is called the Wall Adapting Local Eddy (WALE) viscosity SGS model. is D.B.3 WALE model In the Wall Adapting Local Eddy (WALE) viscosity SGS model [66], the turbulent viscosity term is modeled as µ t = ρl 2 S ( d S ijs d ) 3 2 ij ( ) 5 Sij S 2 ij + ( S d ijs d ij ) 5 4 (D.21) where L S = C ω V 1/3 and parameter C ω = where g 2 ij = U i x j S d ij = 1 2 ( g 2 ij + g 2 ji) 1 3 δ ijg 2 kk (D.22) 294

295 Appendix E Random input uncertainty propagation techniques in CFD E.A Markov Chain Monte Carlo (MCMC) method The Monte Carlo (MC) method relies on the generation of pseudo-random numbers from the probability distribution of input variables to a model to propagate uncertainty [3]. A good introduction to MC methods can be found in [67]. If [ξ 1, ξ 2... ξ M ] is a set of pseudo random set of realization of the input data, and s i = s(ξ i ) for i = [1, 2,... M] are unique solutions of the model, then the expected value of the model output is calculated as 1 s = lim M M M s i i=1 (E.1) where M is the total number of realizations of the model. The MC method is sufficient to resolve a deterministic model and robust since it does not make any hypothesis of the variance of the input variables. The Markov Chain Monte Carlo algorithm is employed to sample from an arbitrary probability distribution of input variables. The mechanics of the implementation of 295

296 the MCMC can be found in [68, 69] E.B Sensitivity coefficient method The AMSE V&V 2009 standard [14] on verification and validation prescribes a Sensitivity Coefficient method for input variable uncertainty propagation. Using the linear Taylor expansion, the input uncertainty is given by (u input ) 2 = N i=1 ( ) 2 Zj u xi (E.2) x i where Z j is a quantity of interest on which the uncertainty is being calculated, x i is the i-th input variable to the model i.e i-th element of vector x. x i are assumed to be mutually independent of each other. N represents the total number of variable input parameter or the length of vector x. The uncertainty due to inputs is then defined for the output variable of interest as Z j Z j. The sensitivity coefficients Z j x i can be calculated from the model. In conclusion, by knowing the uncertainties on all input parameters u ( x i ), and the corresponding sensitivity coefficients, one can calculate the input uncertainty. The sensitivity coefficients method allows calculation of the output quantity of interest with relatively small number of simulations. E.C Latin hypercube method Latin Hypercube Sampling (LHS) methodology is an efficient method of sampling random input variables to propagate input uncertainty through a complex model [70]. In the regular MCMC sampling process, a random sample is generated from the underlying probability cumulative distribution function (CDF). The LHS is different in that it bins the CDF into equal probability segments so that a random number is generated from low probability high consequence region of interest at each time. In order to illustrate the use of the LHS method, let s consider a model Z(U,V) with 296

297 uncertain inputs U and V. Let us further assume that U has a uniform distribution in range [0,10], and V has a triangular distribution in range [0,10]. We are interested in propagating the uncertainty of both input variables through the model. While the MCMC simulation can be used at this point, the LHS approach will be described here. Figure E-1 illustrates the LHS process for uniform (U) and triangular (V) probability distributions. Let s say the sample size is n S =5 for both distributions. In LHS, the vertical axis of cumulative distribution function is divided into five equal probability bins with nodes at 0.2,0.4,0.6,0.8 and 1.0. Then, random sample values U(1); U(2); U(3); U(4); U(5) and V(1); V(2); V(3); V(4); V(5) are drawn from within each bin for both input variables. The sampling of these random values is implemented by 1. Sampling RU(1) and RV(1) from a uniform distribution on [0, 0.2], RU(2) and RV(2) from a uniform distribution on [0.2, 0.4], and so on. 2. Next, we use the CDFs to sample the U and V values that correspond to the RU(1), RU(2). This identification process is given by the dashed lines in Figure E-1. The random samples U(1) through U(5) and V(1) though V(5) are produced as a result. The LHS is completed by randomly pairing (without replacement) the resulting values for U and V. Since the pairing process is not unique, many possible LHS samples can result. The LHS in Figure E-1c results from the pairings U(1)-V(4); U(2)-V(2); U(3)- V(1); U(4)-V(5); U(5)-V(3) and the LHS in Figure E-1d resulting from the pairings U(1)-V(5); U(2)-V(1); U(3)-V(3); U(4)-V(4); U(5)-V(2). Once we have a chosen pairing, the system output is determined at input values determined by the pairing procedure i.e Z1=Z(U(1),V(5)); Z2=Z(U(2),V(1)); Z3=Z(U(3),V(3)); Z4=Z(U(4),V(4)); Z5=Z(U(5),V(2)) if pairing in Figure E-1d is chosen. The estimation of the properties of the output variable is Z = 1 ns N Z i i=1 (E.3) 297

298 The variance is found from Z = 1 ns N (Z i Z ) 2 (E.4) i=1 In LHS sampling methods, these estimators have been shown to be unbiased, and the variance of the estimator to be reduced compared to MCMC techniques[67]. Figure E-1: Example of Latin hypercube sampling to generate a sample of size n S = 5 from U, V with U uniform on [0, 10] and V triangular on [0, 10]. E.D Polynomial Chaos Expansion method Polynomial chaos (PC) expansion can be used in cases when the model is computationally costly and even a single run takes a long time to complete. PC is based on Wiener s theory of homogeneous chaos [3]. Any stochastic process can be represented 298

299 using orthogonal polynomials as z(ξ) = z j H j (ξ) j=0 (E.5) where z represents the model output treated here as a random variable. ξ represent a vector of standard normal variables corresponding to the variable model inputs x. H j (ξ) is a multivariate Hermite polynomial of order j. z j is the deterministic coefficients of the expansion that will be determined from model simulations. ξ is connected to the vector of variable input variables as ξ i = x i x i σ xi (E.6) where x i and σ xi are the mean and standard deviations of i-th variable input. The multivariate Hermite polynomial is given as N H j (ξ) = H j (ξ i ) i=1 (E.7) where i represents a i-th vector element and N is the number of variable inputs or the length of vector x. H j (ξ i ) is a single variable Hermite polynomial of j-th order. The first 6 Hermite polynomials are given in Table E.1. H 0 (y) = 1 H 1 (y) = y H 2 (y) = (y 2 1) H 3 (y) = (y 3 3y) H 4 (y) = (y 4 6y 2 + 3) H 5 (y) = (y 5 10y y) H 6 (y) = (y 6 15y y 2 15) Table E.1: Hermite polynomials up to order 6 Hermite polynomials are orthogonal with respect to a weighting function i.e, they 299

300 satisfy H m (y)h n (y)e y2 2 dy = n! 2πδmn (E.8) where δ mn is Kronecker delta. Single variable PC for a chaotic system z(ξ) with input random variable ξ is given as p z(ξ) = H j (ξ) j=0 (E.9) where ξ is input random variable and z(ξ) is the model output quantity. P is the order of expansion associated with the highest order Hermite polynomial used in the spectral decomposition. ξ is a random variable following a standard normal distribution with p(ξ)dξ = 1 2π e ξ2 2 dξ = 1 (E.10) and the expectation value of Z is given as z = z(ξ)p(ξ)dξ = 1 z(ξ)e ξ2 nq 2 dξ = 2π n=1 w n z(ξ n ) (E.11) and the standard deviation σ 2 z = z 2 z 2 = nq n=1 w n (z(ξ n )) 2 [ nq n=1 w n z(ξ n )] 2 (E.12) Multiplying both sides of equation by 1 2π H i (ξ)e ( ξ2 /2) and integrating with respect to ξ over all possible values, we obtain 1 2π z(ξ)h i (ξ)e ξ2 /2 dξ = 1 z j H i (ξ)h j (ξ)e ξ2 /2 dξ 2π j=0 (E.13) and from the orthogonality of Hermite polynomials, we get z j = 1 j! z(ξ)h j (ξ)e ξ2 /2 dξ 2π (E.14) In order to carry out the integral given in above expression certain values of ξ have 300

301 to be chosen and the model response to that input needs to be evaluated. In case of a large simulation where a single run can take a significant time to complete, approximating the integral with as few runs as possible is essential. We can employ Gauss-Hermite quadrature in order to obtain accurate approximation to the given integral in order to minimize necessary computational time. The calculation of Gauss Hermite quadrature nodes and weights are given in [3] and a complete tabulation is available from netlib 30. Figure E-2 shows the Gauss-Hermite quadrature nodes for level parameter l or nq=1:6 for two uncorrelated normal random variables. Note that in this Figure, two input variables are normalized with respect to their standard deviations and the contour lines correspond to the 2-dimensional joint normal probability density function. It can be observed that the full tensor grid corresponding to l 2 nodes are not optimal and the same degree of exactness can be achieved by a fewer number of nodes. There is no general method to systematically compute the optimal nodes. The Smolyak sparse grid given in Section E.E is one approach and is applicable to any number of random input variables[3]

302 Figure E-2: Gauss-Hermite quadrature nodes for N=2 random input variables that are uncorrelated and given by normal probability distributions. The level parameter l is the same as nq defined in Equation E.11. Note that random input parameters are normalized by their standard deviations. The contour lines shown correspond to the underlying 2 dimensional joint Gaussian probability distribution. Quadrature notes were constructed using the MATLAB function provided in MIT class taught in Fall2015 by Prof. Marzouk. For normal random variable inputs, the Gauss-Hermite quadrature rule is not unique. Figure E-3 shows the quadrature nodes proposed by Genz and Keister [71]. Unlike the Gauss-Hermite quadrature, the Genz-Keister rule is nested which means it is optimal for sparse grid construction using the Smolyak method. In absence of sparse grid, the full tensor grids constructed using the Gauss-Hermite quadrature is computationally expensive requiring l d nodes for d number of random inputs at level l. 302

303 Figure E-3: Genz-Keister quadrature nodes for N=2 random input variables that are uncorrelated and given by normal probability distributions. The level parameter l is the same as nq defined in Equation E.11. The contour lines shown correspond to the underlying 2 dimensional joint Gaussian probability distribution. Quadrature notes were constructed using the data given in 31 E.E Smolyak quadrature rule The sparse grid generation method proposed by Smolyak [15] is an approach to attempt the computation of full tensor grids given in Figure E-2 and E-3 more optimal by eliminating some points in low probability region. The overall aim of applying the sparse grid is to reduce the number of computations necessary for quantifying the uncertainty of the model output of interest. Here we give a brief overview of the 31 jburkardt/m_src/quadrulequadrule.html 303

304 theory of Smolyak sparse grid and move on to practical implications of employing the method. The Smolyak sparse grid is calculated as A(l, d) = l+1 i l+d ( 1) l+d i ( U i 1 U i d ) d 1 l + d i (E.15) where d is the number of input variables, i is a d-dimensional multi-index with components i 1... i d and i = i i d. U i is a univariate linear operator corresponding to a one dimensional quadrature grid of level i. The tensor product operator produces a tensor grid. For example, Figure E-2 shows the tensor products of two univariates for l=1:6. The level parameter l corresponds to the precision in numerical integration. High level parameter indicates low error in the output. In practice, Smolyak sparse grids are designed to have a more optimal shape compared to the full tensor grid. Figure E-4 show the construction of Smolyak sparse grid in Gauss-Hermite quadrate with levels l=3,4. Figure E-4A shows full tensor grids for various l 1 and l 2 levels corresponding to two random inputs as a matrix. The full tensor grid has 9 nodes corresponding to l 1 = 3 and l 2 = 3. The Smolyak grid of level l given in Figure E-4B is constructed from adding and subtracting the lower level elements highlighted by gray in Figure E-4A. Similarly, Figure E-4C and Figure E-4D show the full tensor grid matrix for level l=4 and the corresponding Smolyak sparse grid respectively. Note that the Smolyak sparse grid has 13 nodes compared to full tensor 9 nodes for l=3 and 29 nodes compared to full tensor 16 for l=4. In this sense, the use of sparse grid has increased the computational cost in case of two random inputs as given in Figure E-5B. However, sparse grid is much more useful in higher number of inputs as given in table in the Figure E-5A. For example in case of 10 random input variables, at level l=2, Smolyak sparse grid would require 221 grid points whereas the full-tensor grid would need 3 10 = points. 304

305 [A] [B] [C] [D] Figure E-4: Construction of Smolyak sparse grid. A) shows the lower order grids used (in gray) for constructing the Smolyak sparse grid in B) for l=3. C) shows the lower order grids used (in gray) for constructing the Smolyak sparse grid in D) for l=4. The quadrature notes were constructed using the MATLAB function provided in MIT class taught in the Fall of 2015 by Prof. Marzouk 305

306 [A] [B] Figure E-5: Computational cost saving using the Smolyak sparse grid. Figure A shows a tabulated number of computations necessary for number of random variable inputs d at level l. The Figure B shows a plot of number of nodes necessary for d=2 input variables for full tensor and Smolyak grids as a function of level parameter l 32. E.F Stochastic collocation method We follow the methodology developed by Loeven in 2010 [55] and the discussion given by Dr.Wunsch and Prof. Hirsch at ERCOFTAC course in Chatou, France in Let u(x, t, ξ) be a solution to a stochastic PDE such as the Navier Stokes equation with uncertainty physical properties with x is coordinate, t is time and ξ representing an uncertain input parameter. Lagrange interpolation polynomials can be used to decompose the solution as N p u(x, t, ξ) = u(x, t)h i (ξ) i=1 (E.16) where h i (ξ) is the Lagrange interpolation polynomial of order I defined as h i (ξ) = Np k=1,k i ξ ξ k ξ i ξ k (E.17) The collocation points ξ j are selected by the appropriate quadrature depending on the probability distribution of the input variable [72] as given in Table E paulcon/slides/oxford_2012.pdf 306