Nuclear Science

ISBN 978-92-64-99124-8
NEA/NSC/DOC(2007)21

NEA Nuclear Science Committee
NEA Committee on Safety of Nuclear Installations
US Nuclear Regulatory Commission

NUPEC BWR Full-size Fine-mesh
Bundle Test (BFBT) Benchmark
Volume II: Uncertainty and Sensitivity Analyses ofVoid Distribution and Critical Power – Specification

F. Aydogan, L. Hochreiter, K. Ivanov
The Pennsylvania State University
M. Martin
Commissariat à l’Énergie Atomique
H. Utsuno
Japan Nuclear Energy Safety Organization
E. Sartori
OECD Nuclear Energy Agency© OECD 2010
NEA No. 6343

NUCLEAR ENERGY AGENCY
Organisation for Economic Co-operation and Development

ORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENTThe OECD is a unique forum where the governments of 31 democracies work together to address the
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NUCLEAR ENERGY AGENCYThe OECD Nuclear Energy Agency (NEA) was established on 1st February 1958 under the name of the
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FOREWORD

Foreword

The need to refine models for best-estimate calculations based on good-quality experimental data has
been expressed in many recent meetings concerning nuclear applications. The needs arising in thisrespect should not be limited to the currently available macroscopic methods but should be extended
to next-generation analysis techniques that focus on more microscopic processes. A very valuable
database for thermal-hydraulics modelling was developed by the Nuclear Power Engineering
Corporation (NUPEC), Japan; it includes sub-channel void fraction measurements in a representative
BWR fuel assembly (NEA, 2006). Part of this database has been made available for this international
benchmark activity entitled “NUPEC BWR full-size fine-mesh bundle tests (BFBT) benchmark”. This
international project has been officially approved by the Japanese Ministry of Economy, Trade andIndustry (METI), the US Nuclear Regulatory Commission (NRC) and endorsed by the OECD/NEA. The
benchmark team has been organised based on collaboration between Japan and the United States. A
large number of international experts have agreed to participate in this programme.
The fine-mesh high-quality sub-channel void fraction and critical power data encourages
advancement in understanding and modelling complex, two-phase flow behaviour in real bundles.
Considering that the present theoretical approach is relatively immature, the benchmark specification
is designed so that it will systematically assess and compare the participants’ analytical models on
the prediction of detailed void distributions and critical powers. The development of truly mechanistic
models for critical power prediction is currently under way. These innovative models include processes
such as void distributions, droplet deposition and liquid film entrainment. The benchmark problem
includes both macroscopic and microscopic measurement data. In this context, the sub-channel grade
void fraction data are regarded as the macroscopic data and the digitised computer graphic images
are the microscopic data, which provide void distribution within a sub-channel.
The NUPEC BFBT benchmark consists of two phases. Each phase consists of different exercises:•

Phase I – Void distribution benchmark– Exercise 1 – Steady-state sub-channel grade benchmark– Exercise 2 – Steady-state microscopic grade benchmark– Exercise 3 – Transient macroscopic grade benchmark– Exercise 4 – Uncertainty analysis of the void distribution benchmark•

Phase II – Critical power benchmark– Exercise 0 – Steady-state pressure drop benchmark– Exercise 1 – Steady-state critical power benchmark– Exercise 2 – Transient critical power benchmark– Exercise 3 – Uncertainty analysis of the critical power benchmark

In order to study the basic thermal-hydraulics in a single channel, where the concern regarding
the cross-flow effect modelling could be removed, an elemental task has been proposed. It consists of
two sub-tasks that are placed in each phase of the benchmark scope as follows: Sub-task 1: Void
fraction in the elemental channel benchmark; and Sub-task 2: Critical power in the elemental channel
benchmark.
This report provides the specification for the uncertainty exercises of the international OECD/NEA,
NRC and NUPEC BFBT benchmark problem including the elemental task. The specification was
prepared jointly by Pennsylvania State University (PSU), USA and the Japan Nuclear Energy Safety(JNES) Organisation, in co-operation with the OECD/NEA and the Commissariat à l’énergie atomique(CEA Saclay, France). The work is sponsored by the US NRC, METI-Japan, the OECD/NEA and the
Nuclear Engineering Program (NEP) of Pennsylvania State University. This uncertainty specification
covers the fourth exercise of Phase I (Exercise-I-4), and the third exercise of Phase II (Exercise II-3) aswell as the elemental task.

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3

ACKNOWLEDGEMENTS

Acknowledgements
The authors would like to thank Professor Hideki Nariai, President of the JNES, Japan, whose supportand encouragement in establishing and carrying out this benchmark are invaluable.
This report is the sum of many efforts: those of the participants, the funding agencies and theirstaff – the METI, Japan, US NRC and the Organisation of Economic Co-operation and Development(OECD). Special appreciation goes to the report reviewer, Maria Avramova from the Pennsylvania State
University. Her comments, corrections and suggestions were very valuable and significantly improved
the quality of this report. Further, the authors would like to thank Penn State graduate students
Jeffrey Lane, James Spring and Douglas Miller for their contribution to the development of the
Phenomena Identification Ranking Tables.
The authors wish to express their sincere appreciation for the outstanding support offered by the
JNES personnel in providing the test data and discussing the test characteristics.
The authors would also like to acknowledge the contribution of Daniel Caruge and Eric Royer of
CEA Saclay, France in enriching this benchmark activity for the uncertainty analyses.
Particularly noteworthy are the efforts of Farouk Eltawila and James Han from US NRC. With their
help funding was secured, enabling this project to proceed. We also thank them for their excellent
technical advice and assistance.

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TABLE OF CONTENTS

Table of contents

Foreword ..........................................................................................................................................................

3

Chapter 1 Introduction ................................................................................................................................

7

Chapter 2 Definitions of the UA/SA terms ...............................................................................................

9

2.1 Accuracy ..............................................................................................................................

9

2.2 Probability density function..............................................................................................

9

2.3 Precision ..............................................................................................................................

9

2.4 Uncertainty .........................................................................................................................

10

2.5 Sensitivity analysis ............................................................................................................

11

2.6 Verification and validation ...............................................................................................

11

Chapter 3 Selection and characterisation of uncertainty input parameters for
BFBT benchmark ........................................................................................................................

13

3.1 Uncertainty input parameters for Exercises I-4 and II-3...............................................

13

3.2 Description of the elemental task ....................................................................................

14

Chapter 4 Suggested methods to be used in the uncertainty exercises of BFBT
benchmark ..................................................................................................................................

17

4.1 Step 1: Accuracy analysis ..................................................................................................

17

4.2 Step 2: Uncertainty analysis .............................................................................................

18

Chapter 5 Selection of the exercise cases from the BFBT database ....................................................

21

5.1 Selected cases for accuracy analysis of void distribution.............................................

21

5.2 Selected cases for accuracy analysis of critical power ..................................................

22

5.3 Selected cases for uncertainty analysis of void distribution ........................................

23

5.4 Selected cases for uncertainty analysis of critical power .............................................

24

5.5 Selected cases for elemental task ....................................................................................

25

Chapter 6 Requested output .......................................................................................................................

27

6.1 Introduction ........................................................................................................................

27

6.2 Requested output for Exercise 4 of Phase I .....................................................................

27

6.3 Requested output for Exercise 3 of Phase II ....................................................................

28

6.4 Requested output for elemental task ..............................................................................

29

Chapter 7 Conclusions.................................................................................................................................

31

References .......................................................................................................................................................

33

Appendix 1: Uncertainty and sensitivity analysis methods .....................................................................

35

Appendix 2: PIRT information ......................................................................................................................

39

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LISTS OF FIGURES AND TABLES

List of figures
2.3.1

Precision in measurement of variable x ........................................................................................

10

3.1.1

Distortions of test assembly observed with fine-mesh void measurement.............................

14

3.2.1

Image to quantify the cross-flow ...................................................................................................

16

4.1.1

Example of predicted critical power versus measured critical power ......................................

17

4.1.2

Example of predicted to measured critical power versus quality .............................................

18

6.4.1

Example of relation between equilibrium quality and void fraction ........................................

30

6.4.2

Structure of annular-mist flow.......................................................................................................

30

List of tables
3.1.1 Estimated accuracy and PDF of the main process parameters .....................................................

14

3.2.1 Geometry and power shape of the elemental channel .................................................................

15

4.2.1 Estimated accuracy of void distribution measurements ...............................................................

19

5.1.1 Selected cases for accuracy analysis in Exercise 4 of Phase I .......................................................

21

5.1.2 Test matrix of selected cases for Exercise 4 of Phase I ..................................................................

22

5.1.3 Test numbers of selected cases for Exercise 4 of Phase I ..............................................................

22

5.2.1 Selected cases for Exercise 3 of Phase II...........................................................................................

22

5.2.2 Test matrix of selected cases for Exercise 3 of Phase II .................................................................

23

5.2.3 Experimental conditions of selected cases for Exercise 3 of Phase II .........................................

23

5.3.1 The bundle types used in the BFBT measurements .......................................................................

24

5.3.2 Selected cases for uncertainty analysis of void distribution.........................................................

24

5.4.1 Selected cases for uncertainty analysis of critical power ..............................................................

24

5.5.1 Reference test matrix for elemental task.........................................................................................

25

5.5.2 Reference test numbers for the elemental task ..............................................................................

25

5.5.3 Boundary conditions for the elemental task ...................................................................................

25

6.2.1 Calculated void distribution from deterministic input parameters .............................................

27

6.3.1 Calculated critical power from deterministic input parameters ..................................................

28

6.3.2 The calculated critical power uncertainty output format .............................................................

29

A2.1

PIRT-1 for steady-state void distribution .........................................................................................

39

A2.2

PIRT-2 for steady-state void distribution .........................................................................................

39

A2.3

PIRT-3 for steady-state void distribution .........................................................................................

40

A2.4

PIRT-4 for steady-state void distribution .........................................................................................

40

A2.5

PIRT-1 for steady-state critical power .............................................................................................

41

A2.6

PIRT-2 for steady-state critical power ..............................................................................................

41

A2.7

PIRT-3 for steady-state critical power ..............................................................................................

41

A2.8

PIRT-4 for steady-state critical power ..............................................................................................

42

A2.9

Comparison of two independent expert groups’ decisions about the PIRT
tables for the void distribution ..........................................................................................................

42

A2.10 Comparison of two independent expert groups’ decisions about the PIRT
tables for the critical power ...............................................................................................................

42

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INTRODUCTION

Chapter 1: Introduction

The OECD/NRC BFBT benchmark provides a very good opportunity to apply uncertainty analysis (UA)and sensitivity analysis (SA) techniques and to assess the accuracy of thermal-hydraulic models for
two-phase flows in rod bundles. During the previous OECD benchmarks, participants usually carriedout sensitivity analysis on their models for the specification (initial conditions, boundary conditions,etc.) to identify the most sensitive models or/and to improve the computed results. The comprehensive
BFBT experimental database (NEA, 2006) leads us one step further in investigating modelling capabilities
by taking into account the uncertainty analysis in the benchmark. The uncertainties in input data(boundary conditions) and geometry (provided in the benchmark specification) as well as theuncertainties in code models can be accounted for to produce results with calculational uncertainties
and compare them with the measurement uncertainties. Therefore, uncertainty analysis exercises
were defined for the void distribution and critical power phases of the BFBT benchmark.
This specification is intended to provide definitions related to UA/SA methods, sensitivity/
uncertainty parameters, suggested probability distribution functions (PDF) of sensitivity parameters,
and selected experimental cases from the BFBT database for both steady-state void distribution andsteady-state critical power uncertainty analyses.
In order to study the basic thermal-hydraulics in a single channel, where the concern regarding
the cross-flow effect modelling could be removed, an elemental task is proposed, consisting of two
sub-tasks that are placed in each phase of the benchmark scope as follows:•

Sub-task 1: Void fraction in elemental channel benchmark•

Sub-task 2: Critical power in elemental channel benchmark

The first task can also be utilised as an uncertainty analysis exercise for fine computational fluid
dynamics (CFD) models for which the full bundle sensitivity or uncertainty analysis is more difficult.
The task is added to the second volume of the specification as an optional exercise.
Chapter 2 of this document provides the definition of UA/SA terms.
Chapter 3 provides the selection and characterisation of the input uncertain parameters for the
BFBT benchmark and the description of the elemental task.
Chapter 4 describes the suggested approach for UA/SA of the BFBT benchmark.
Chapter 5 provides the selection of data sets for the uncertainty analysis and the elemental task
from the BFBT database.
Chapter 6 specifies the requested output for void distribution and critical power uncertainty
analyses (Exercises I-4 and II-3) as well as for the elemental task.
Chapter 7 provides conclusions.
Appendix 1 discusses the UA/SA methods.
Appendix 2 presents the Phenomena Identification Ranking Tables (PIRT) developed at PSU forvoid distribution and critical power predictions in order to assist participants in selecting the mostsensitive/uncertain code model parameters.

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DEFINITIONS OF THE UA/SA TERMS

Chapter 2: Definitions of the UA/SA terms

Definitions of commonly used uncertainty analysis terms are given below.

2.1 Accuracy
Accuracy is the closeness of agreement between the measured value and the true value (Wheeler, 1996).

2.2 Probability density function
The f(x)dx is the probability that the random variable (x) will assume a value in the range of x andx + dx. Note that f(x) is defined such that the probability of getting some value x in the range [a,b] isequal to 1. This function, f(x) is called probability density function (PDF). The cumulative distributionfunction (CDF) is defined as:x∫F(x ) = dx ′ f (x ′)a

The PDF is normalised such that its corresponding CDF varies in a range of [0,1]:
0 ≤ F(x) ≤ 1
for all values of x and:F(x ) =b∫ f (x )dx = 1
a

For the input variable (x) within the range a ≤ x ≤ b, the following classical probability density
functions f(x) can be defined:
PDF
Uniform
Triangular
Exponential
NormalLognormal

Features
Shape parameter c with a ≤ c ≤ b
Location parameter γ
Scale parameter λMean value μStandard deviation σ
Mean value of logarithms μStandard deviation of logarithms σ

Definition
f(x) = 1/(b – a)f(x) = 2(x – a)/(b – a)(c – a)f(x) = 2(b – x)/(b – a)(b – c)

if a ≤ x ≤ c
if c ≤ x ≤ bf(x) = λ exp[-λ (x – γ)]f(x) = exp[-(x – μ)2/2σ2]/σ(2π)1/2
f(x) = exp[-(ln(x) – μ)2/2σ2]/(xσ(2π)1/2)

2.3 Precision
Precision characterises the degree of mutual agreement among a series of individual measurements,
values, or results (Coleman, 1995). An example of the precision in measurement of variable x is given
in Figure 2.1 (Coleman, 1995).

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DEFINITIONS OF THE UA/SA TERMS

Figure 2.3.1: Precision in measurement of variable x

Source: Coleman (1995).

For a series of N individual assessments by a code of a variable x(x ncode ) , the following criteria canbe defined to perform a statistical comparison with an experimental result:1) Mean value: x code =∑1
x ncodeN n≤N2) Individual relative bias error: δ ncode =3) Mean relative bias error: δ code =x ncode − x exp
x exp

∑

1
δ ncodeN n≤N4) Maximum bias error: ε code = max δ coden5) Standard deviation: σ code =1(xncode − x code )2
N n≤N

∑

6) Coverage ratio: the proportion of individual assessments among the series with a relative bias
error lower than the experimental uncertainty εexp:
R code =card{δ ncode with δncode ≤ ε exp }N

2.4 Uncertainty
Uncertainty is an estimation of scatter in a measurement or in a result, usually determined with acertain level of confidence (often 95%) (Wheeler, 1996; Hochreiter, 2006). Considering a model producing
output results as a function of “uncertain” input parameters, propagation of uncertainty (or propagation

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DEFINITIONS OF THE UA/SA TERMS

of error) is the effect of the input uncertainties on the output results. In other words, this quantifies
the variation of the results for a given variation (range and PDF) of input parameters. Mainly, thevariables are measured in an experiment, and have uncertainties due to measurement limitations(e.g. instrument precision), which propagate to the results.

2.5 Sensitivity analysis
Sensitivity analysis (SA) is a study of how the response of a model (numerical or otherwise) canqualitatively or quantitatively be apportioned to different sources of input variations (Wheeler, 1996;
Saltelli, 2000, 2004). SA is thus closely linked to uncertainty analysis (UA), which aims to quantify theoverall uncertainty associated with the response as a result of uncertainties in the model input.
The mathematical model of a given phenomenon is defined by a series of equations, input
factors, parameters and variables aimed to characterise the process being investigated. Input issubject to many sources of uncertainty including errors of measurement, absence of information andpoor or partial understanding of the driving forces and mechanisms. The uncertainties impose a limiton the confidence in the response or output of the model. Further, models may have to cope with thenatural intrinsic variability of the system, such as the occurrence of stochastic events. Good modelling
practice requires that the modeller provide an evaluation of the confidence in the model, possibly
assessing the uncertainties associated with the modelling process and with the outcome of the model
itself. Therefore, the uncertainty and sensitivity analyses offer valid tools for characterising theuncertainties associated with the model and can be used to determine:•

the model resemblance with the process under study;•

the quality of model definition;•

the factors that mostly contribute to the output variability;•

the region in the space of input factors for which the model variation is maximum;•

the optimal or instability regions within the space of factors for use in a subsequentcalibration study;•

the interactions between the factors.

The sensitivity analysis is popular in financial applications, risk analysis, signal processing andany area where models are developed. Sensitivity analysis can also be used in model-based policy
assessment studies.
There are several possible procedures to perform uncertainty and sensitivity analyses. The mostcommon sensitivity analysis is the sampling-based sensitivity. In the sampling-based sensitivity
analysis the model is executed repeatedly for combinations of values sampled from the distribution(assumed known) of the input factors. In general, UA and SA are performed jointly by executing themodel repeatedly for combination of factor values sampled with some probability distribution. The
following steps can be listed:•specify the target function and select the input of interest;•

assign a distribution function to the selected factors;•

generate a matrix of inputs with that distribution(s) through an appropriate design;•evaluate the model and compute the distribution of the target function;•

select a method for assessing the influence or relative importance of each input factor on thetarget function.

2.6 Verification and validation
Verification and validation are defined as primary means to assess the accuracy and reliability ofsimulations (Roache, 1998; Mahaffy, 2010; Oberkampf, 2006). Oberkampf (2006) separates verification
into two groups. The first is the code verification and the second is the solution verification. The code

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DEFINITIONS OF THE UA/SA TERMS

verification was defined as an assessment of the reliability of the software coding. The solution
verification deals with the numerical accuracy of the computational model. In the same paper,
Oberkampf defined validation as a physical modelling accuracy of a computational simulation bycomparing with experimental data. Briefly, verification is the process of ensuring that the controlling
physical equations have been correctly translated into computer code or in the case of handcalculations, correctly incorporated into the calculation procedure. Validation is defined as the evidence
that demonstrates that the selected code or calculation method is suitable for the specific analysis
purpose. This includes the confirmation that the results from the verified model agree with thebenchmarks (Ravenswaay, 2006).
Because verification and validation terms have elements similar to uncertainty and sensitivity
analysis, it is possible to confuse the verification and validation with the uncertainty analysis. The
BFBT benchmark team defined uncertainty analyses for void distribution and critical power, notverification and validation analyses. The participants are free to choose the model for theiruncertainty and sensitivity analyses.
A review of UA and SA methodologies is provided in Appendix 1.

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SELECTION AND CHARACTERISATION OF UNCERTAINTY INPUT PARAMETERS FOR BFBT BENCHMARK

Chapter 3: Selection and characterisation of
uncertainty input parameters for BFBT benchmark

3.1 Uncertainty input parameters for Exercises I-4 and II-3
Some of the possible input parameters for the uncertainty analysis are classified in the following
groups:•••Boundary condition effects – hydraulic and thermal input variables as:–

mass flow rate;–

inlet fluid temperature (or inlet sub-cooling);–

system pressure;–

power (or outlet quality).

Geometry effects – dimensions of the bundle components as:–

diameter of the heated rod;–

thickness of the cladding;–flow area of the sub-channel;–

wetted perimeter of the sub-channel, etc.

Modelling parameters used in the code as:–

friction factors (single-phase factor, two-phase multiplier, heating corrector);–

single-phase and two-phase heat transfer coefficients;–

boiling/condensation and interfacial mass transfer factors;–

turbulence and mixing coefficients;–

void drift model parameters;–

spacer loss coefficient.

Each of the hydraulic and thermal input parameters is characterised by a range of experimental
uncertainties that were determined by the experimental team (Table 3.1.1). These uncertainties must
be related to a probability density function (PDF) (Glaeser, 2000). In order to extend and enrich theproposed sensitivity analysis, BFBT participants are encouraged to specify their own additional
uncertainty parameters, especially modelling parameters (with accuracy range and related PDF) andpresent them to the benchmark team while submitting their results.
Table 3.1.1 provides the accuracies and PDF of measured variables (inlet temperature, mass flow
rate, system pressure and bundle power).
As can be seen in Figure 3.1.1 the assembly walls are not quite horizontal or vertical. The
deviation for the horizontal (W-E) or the vertical (N-S) is 4 pixels in each edge. The average deviationof rod centres from the design (in pixels) is shown to be about 1.5 pixels for both the x and y errors.

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SELECTION AND CHARACTERISATION OF UNCERTAINTY INPUT PARAMETERS FOR BFBT BENCHMARK

Table 3.1.1: Estimated accuracy and PDF of the main process parameters
Quantity
System pressure
Mass flow rate
Power
Inlet fluid temperature

Accuracy range
±1%
±1%
±1.5%
±1.5°CPDF
NormalNormalNormalFlat

Figure 3.1.1: Distortions of test assembly observed with fine-mesh void measurementxyBased on the observation of the X-ray CT scan data (see Figure 3.1.1), one can make assumptions
about the distortion for the heater pin and channel box, which can quantify the accuracy of the rod
diameter due to manufacturing (1.5 pixels), the accuracy of the rod pitch (1.5 pixels) and the accuracyof the bundle inner width (4 pixels).
The US NRC developed the Phenomena Identification Ranking Table (PIRT) concept for assessing
the relative importance of individual phenomena (models or correlations) involved in determining
safety margins in operating reactors (Boyack, 1995; Holowach, 2006; Martin, 2005; US NRC, 1989, 1989a).
The PIRT help to determine which phenomena are the most important for a given scenario. A group of
experts at PSU developed PIRT tables to rank the most important phenomena that need to be accurately
simulated for each uncertainty exercise – void distribution and critical power. Another independentgroup of experts reviewed these PIRT tables and the ranking of the phenomena and concurred or
disagreed with the first group. The first group then resolved the differences. The PIRT-related
information is provided in Appendix 2. The components of interest for the void distribution uncertainty
exercise are the sub-channel void distribution and bundle average void fraction at the bundle exit.
Tables A2.1 through A2.4 in Appendix 2 provide the PIRT tables for the void distribution. The
components of interest for the critical power uncertainty exercise are the steady-state critical power
and the dry-out elevation. Tables A2.5 through A2.8 in Appendix 2 provide the PIRT tables for thecritical power.

3.2 Description of the elemental task
The elemental channel is a single sub-channel defined by four adjusting rods in a fuel assembly. The
cross-section of the elemental channel is a square with each side equal to the rod pitch. The referenceof the elemental channel is assembly type 0-1. Table 3.2.1 provides the geometry and power shapedata for the elemental channel.

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SELECTION AND CHARACTERISATION OF UNCERTAINTY INPUT PARAMETERS FOR BFBT BENCHMARK

Table 3.2.1: Geometry and power shape of the elemental channel
Item

Data
132.5mm

Assembly

0-1
Simulated fuel assembly type
Number of heated rods
Number of water rods
Heated rods outer diameter (mm)Heated rods pitch (mm)Axial heated length (mm)Water rods outer diameter (mm)
Channel box inner width (mm)
Channel box corner radius (mm)Inchannel flow area (mm2)Spacer type
Number of spacers
Spacer pressure loss coefficients
Spacer location (mm)Radial power shape
Axial power shape

Elemental channel
8×8

62
2

1
0
12.3
16.2
3 708

15.0
132.5
8.0
9 781
Grid

–
–
–
143.6
Grid and ferrule

71.2455, 967, 1 479, 1 991, 2 503, 3 015, 3 527(distance from bottom of heated length to spacer bottom face)
Uniform
Uniform

The void fraction in a sub-channel ij at the bundle quality xe can be separated as:
α ij (x e ) = α(x e ) + α cf (x e )
ij

(3.1)

where α ij (x e ) is the void fraction in sub-channel ij at the bundle quality xe, α(x e ) is the void fractionin the elemental channel at quality xe and α ijcf (x e ) is the void fraction due to cross-flow into sub-channel
ij at the bundle quality xe.
Then, we obtain:
α ijcf (x e ) = α ij (x e ) − α(x e )(3.2)

Actually, the value of α ijcf (x e ) shall be defined for the deviation of the quality xe. There are threedifferent bundle quality cases in the benchmark exercise. By taking the deviation of Eq. (3.2), the valueof α ijcf (x e ) will be quantified. An image of this process is shown in Figure 3.2.1.

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SELECTION AND CHARACTERISATION OF UNCERTAINTY INPUT PARAMETERS FOR BFBT BENCHMARK

Figure 3.2.1: Image to quantify the cross-flow

Xe (%)
25

α cfij (Δxe )12

α cfij (Δxe )5

16

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SUGGESTED METHODS TO BE USED IN THE UNCERTAINTY EXERCISES OF BFBT BENCHMARK

Chapter 4: Suggested methods to be used in
the uncertainty exercises of BFBT benchmark

Exercise 4 of Phase I and Exercise 3 of Phase II of the BFBT benchmark will be performed in two steps.
The first step is a standard type of accuracy analysis or global sensitivity analysis, where the total
overall code predictions are compared to the experimental data.
The second step is an in-depth uncertainty analysis or local propagation of uncertainty of theboundary conditions, geometry effects and models/correlations of a given computer code by using theexperimental data.

4.1 Step 1: Accuracy analysis
In Step 1 comparisons will be performed between the computer code prediction of the given quantity(void fraction/critical power) and its measured value.
An approach similar to the accuracy analyses of the Duke Power Company thermal-hydraulicstatistical core design methodology is proposed (Duke, 1996). In this approach, the predicted parameteris compared with the measured one as shown in Figure 4.1.1 and Figure 4.1.2. In Figure 4.1.2, theparameter to be compared is the ratio of the predicted (P) void fraction to the measured (M) void
fraction on a sub-channel basis. The P/M ratio can be used to assess the accuracy and degree ofagreement of the analysis model with the data over a range of independent variables as pressure,mass flow, quality and inlet sub-cooling.

Figure 4.1.1: Example of predicted critical power versus measured critical power
12+%10Predicted Critical Power (MW)

11
10
-%10
9
8
7
6
5
4
4

5

6

7

8

9

10

11

12

Measured Critical Power (MW)NUPEC BWR FULL-SIZE FINE-MESH BUNDLE TEST (BFBT) BENCHMARK, VOLUME II – © OECD/NEA 201017

SUGGESTED METHODS TO BE USED IN THE UNCERTAINTY EXERCISES OF BFBT BENCHMARK

Figure 4.1.2: Example of predicted to measured critical power versus quality
1.3

P/M Critical Power

1.2
1.1
1.0
0.9
0.8
0.7
25

35

45

55

65

75

Quality (%)
Several meaningful comparisons can be obtained by comparing the measured void fraction/criticalpower to the predicted void fraction/critical power and plotting the P/M ratios. For example, P/M
sub-channel void fraction histogram plots can be developed for the different sub-channel types. Theseplots will indicate a potential bias in the computer codes’ capability to model different geometricalconfigurations and the related local flow patterns.
The other plots of interest are the P/M plots as a function of the test independent variables aspressure, mass flow, quality and inlet sub-cooling. One example is shown in Figure 4.1.2, where theP/M critical power ratio is plotted as a function of the exit quality. These types of plots will indicate apossible bias with the independent variables and will indicate potential deficiencies of the different
computer code models.
In the Step 1 analyses in order to access the importance of a given input parameter on the code
predictions, one parameter changes while the rest remain constant.

4.2 Step 2: Uncertainty analysis
Step 2 is devoted to uncertainty analysis of the void fraction and critical power predictions. This step
will fulfil two main objectives: to assess the actual accuracy of thermal-hydraulic models for void
fraction/critical power predictions in rod bundles and to compare the available methods for uncertainty
analysis in thermal-hydraulics. There are several stages in the uncertainty analysis:
1) Analysis of experimental data and its sources of uncertainty (boundary conditions, geometry,measurement technique).
2) Selection of the most (or potential) sensitive parameters in the physical models and/or in the
input data (parametric identification).
3) Propagation of the selected uncertainties from Part 2 through the calculations; this may requirea large number of calculations using PDF (sampling) for the selected model parameters usedin the code.
4) Analysis of the computational results (sub-channel data of void fraction for Phase I, Exercise 4
or overall critical power for Phase II, Exercise 3) with the criteria given in Chapter 2:

18–

to assess the code precision with the sample-averaged bias error;–

to assess the code uncertainty with the sample-averaged standard deviation;–

to assess the code reliability with the proportion of specimens among the sample with abias error lower than the experimental uncertainty.

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SUGGESTED METHODS TO BE USED IN THE UNCERTAINTY EXERCISES OF BFBT BENCHMARKThe uncertainty parameters in the experiments with their characteristics are of two types:•

Input uncertainties – related to the geometry (e.g. fuel rod diameter) and the boundary
conditions (temperature, mass flow, pressure and power) of the test section.

•

Output uncertainties – related to the measurements of sub-channel void fractions and criticalpower.

The uncertainties of the input parameters, with their probability density functions, are indicatedin Chapter 3. The participants are requested to propagate these uncertainties through the codes,
possibly combined with uncertain parameters of the physical models as in Stages 2 and 3 above,and perform the void fraction and critical power calculations associated with the uncertainty. It is
recommended to propagate input uncertainties on all parameters at once in order to assess cross-term
effects. Sensitivity analysis on sampling technique or sampling size is possible to assess the effect on
statistical results such as standard deviation.
The uncertainty of measured void fraction on different spatial levels is given in Table 4.2.1.

Table 4.2.1: Estimated accuracy of void distribution measurements
Quantity
X-ray CT scanner
Local void fraction
Sub-channel void fraction
Cross-sectional void fraction
X-ray densitometer chordal void
Fraction

Accuracy
8%
3%
2%
2%

In Table 4.2.1, the provided accuracy values of measured void fraction are the absolute accuracy
in measurement data. According to Inoue (1995), the accuracy of the void fraction measurements with
X-ray CT scanner depends on: i) the photon statistics of the X-ray source; ii) the detector non-linearity;iii) the accuracy of fluid condition (temperature and pressure) measurements. The accuracy on
different levels – pixel, sub-channel, bundle and etc., has been logically estimated as a sum of theeffects on measurements with these error elements.
Further, the cross-sectional accuracy was confirmed by the calibration test. The void pattern inthe bubbly and annular flow was simulated with acrylic and water. The cross-sectional void fraction
for reference was calculated geometrically. The deviation of the measured cross-sectional void fraction
from the reference value was evaluated. The logically estimated accuracy for the cross-sectional void
fraction is 2%. The calibrated value meets the estimated accuracy. In this discussion, the deviation
was dealt with an absolute value.

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SELECTION OF THE EXERCISE CASES FROM THE BFBT DATABASE

Chapter 5: Selection of the exercise cases from the BFBT database

The BFBT benchmark team selected the BFBT data sets for the uncertainty analysis exercises of thevoid distribution and critical power benchmark phases. The selections of the cases for the accuracy
analysis (Step 1) and for the uncertainty analysis (Step 2) were performed separately and therefore,
the number of the selected cases for both steps is different. Another difference is the selected bundle
type for void distribution accuracy/sensitivity analyses: for accuracy analysis (Step 1) bundle type 1
was selected, while for the uncertainty analyses (Step 2) bundle type 4 was selected. The reason forthis difference is the pressure drop measured data collected for bundle type 4 that can be used toperform optimisation/initialisation of the uncertainty analysis.
The selected cases for the accuracy analysis step are discussed in Sections 5.1 and 5.2
respectively.
The selected cases for the uncertainty analysis step are discussed in Sections 5.3 and 5.4
respectively.
The selected cases for the elemental task are given in Section 5.5.

5.1 Selected cases for accuracy analysis of void distribution
To cover a wide range of operating conditions with a limited number of cases from the BFBT database,a sampling was performed by the benchmark team and a list of eleven experimental cases was defined(see Table 5.1.1). Assembly type 1 was selected for this exercise. A detailed description of assembly
type 1 is given in NEA (2006). Briefly, it is a standard 8 × 8 rod bundle that includes two central waterrods and has a non-uniform radial power distribution and a cosine axial power profile.

Table 5.1.1: Selected cases for accuracy analysis in Exercise 4 of Phase I
Analysis
numberPressure

Flow
rate

QualityInlet
sub-cooling

1
2
3
4

Constant
Constant
Constant
Changing

Constant
Constant
Changing
Constant

Constant
Changing
Constant
Constant

Changing
Constant
Constant
ConstantFirst
selectedcase1071-55
1071-53
1071-34
1071-12

Secondselectedcase1071-621071-55
1071-40
1071-27

Third
selectedcase–1071-58
1071-55
1071-55

Fourthselectedcase–1071-611071-65
1071-82

The set of the three test cases 1071-55, 1071-58 and 1071-61 already selected for Exercise 1 ofPhase I was expanded with eight additional test cases to be used for the accuracy analysis step. These
are test cases1071-12, 1071-27, 1071-34, 1071-40, 1071-53, 1071-65, 1071-62 and 1071-82.
The selected cases are summarised in Tables 5.1.2 and 5.1.3. The values of sub-cooling in
Table 5.1.2 are approximate values and the real sub-cooling during the experiments varied aroundthese approximate values.

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SELECTION OF THE EXERCISE CASES FROM THE BFBT DATABASE

Table 5.1.2: Test matrix of selected cases for Exercise 4 of Phase I
Assembly

Pressure
(MPa)

1
1
1
1
1
1
1
1

1.0
3.9
7.2
7.2
7.2
7.2
7.2
8.6

Inlet
sub-cooling(kJ/kg)
050.2
050.2
050.2
050.2
050.2
050.2
126.0
050.2

Flow
rate
(t/h)
55
55
10
20
55
70
55
55

Exit quality (%)
2

5

8

12

18

25

X
X
X
X
E4,W
X
E4
X

E4
E4
E4
E4
E4
E4
X
E4

X
X
X
X
W
X
–
X

X
X
X
X
E4
X
X
X

–
X
X
X
W
X
–
W

–
X
X
X
E4
–
–
–

X: Test case, W: Duplicated test case, E4: Exercise 4 case.

Table 5.1.3: Test numbers of selected cases for Exercise 4 of Phase I
Test no.

Assembly

Pressure(MPa)

Flow rate(t/h)

1071-121071-271071-34
1071-40
1071-53
1071-55
1071-58
1071-611071-621071-65
1071-82

1
1
1
1
1
1
1
1
1
1
1

1.016
3.944
7.164
7.161
7.185
7.190
7.160
7.200
7.147
7.195
8.633

54.92
54.71
10.22
20.00
54.58
54.60
55.10
54.70
54.85
69.49
54.69

Inlet
sub-cooling(kJ/kg)
056.2
050.7
049.5
063.6
052.2
052.8
050.3
051.8
124.6
053.9
051.7

Power
(MW)

Exit quality(%)

Exercise
cases

2.32
2.09
0.36
0.70
1.23
1.92
3.52
6.48
3.07
2.45
1.85

05.0
05.0
05.0
05.0
02.0
04.9
11.9
25.1
05.0
05.0
05.0

E4
E4
E4
E4
E4
E4
E4
E4
E4
E4
E4

5.2 Selected cases for accuracy analysis of critical power
Similar to the approach adopted in the previous section, to cover a wide range of operating conditions
with a limited number of cases from the BFBT database, a sampling was performed by the benchmark
team and a list of nine experimental cases was defined (see Table 5.2.1). Assembly C2A (NEA, 2006)
was selected for this exercise. Assembly C2A is a “high burn-up” rod bundle that includes one central
large water rod and has a non-uniform radial power distribution and a cosine axial power profile.
Tables 5.2.2 and 5.2.3 show, respectively, the test matrix and experimental conditions of selectedcases. The values of sub-cooling in Table 5.2.2 are approximate values and the real sub-cooling during
the experiments was varying around these approximate values.

Table 5.2.1: Selected cases for Exercise 3 of Phase II

22

Analysis
numberPressure

Flow
rate

Inlet
sub-cooling12
3

Constant
Constant
Changing

Constant
Changing
Constant

Changing
Constant
Constant

First
Secondselected
selectedcase
case
SA610503 SA610600
SA612500 SA605500
SA510500 SA610503

Third
selected
case
SA610700
SA607500
SA810501

Fourthselected
case
SA610900
SA610503–

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SELECTION OF THE EXERCISE CASES FROM THE BFBT DATABASE

Table 5.2.2: Test matrix of selected cases for Exercise 3 of Phase II
Assembly

Pressure
(MPa)
5.5

C2A

7.2

8.6

Flow rate
(t/h)
20
45
55
65
10
20
30
45
55
60
65
20
45
55
65

25
X
X
W
X
X
X
X
X
E3, W
X
X
X
X
W
X

Inlet sub-cooling (kJ/kg)
50
84
104
W
X
–
X
X
–
E3, W
W
–
X
X
–
X
X
X
E3, W
X
X
E3
X
X
X
X
X
E3, W
E3
E3, W
X
X
X
E3
X
X
W
X
–
X
X
–
E3, W
W
–
X
X
–

126
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X

No. of
data
20

35

20

X: Test case, W: Duplicated test case, E3: Exercise 3 case.

Table 5.2.3: Experimental conditions of selected cases for Exercise 3 of Phase II
Test no.
SA510500
SA605500
SA607500
SA610503
SA610600
SA610700
SA610900
SA612500
SA810501

Outletpressure
(MPa)
5.48
7.16
7.13
7.17
7.18
7.13
7.27
7.16
8.62

Flow rate
(t/h)
55.06
20.09
30.02
55.20
55.05
55.20
55.10
65.36
55.15

Inlet
sub-cooling(kJ/kg)056.41050.55048.35059.39089.53107.61037.73055.66054.89Experimental cases
Duplicated test, E1, E3
Duplicated test, E1, E3
E1, E3
Duplicated test, E1, E3
E1, E3
Duplicated test, E1, E3
Duplicated test, E1, E3
E1, E3
Duplicated test, E1, E3

5.3 Selected cases for uncertainty analysis of void distribution
NUPEC has used different bundle types for various measurements. The bundle types used for thesteady-state void distribution, steady-state critical power, steady-state single- and two-phase pressure
drop measurements are summarised in Table 5.3.1. An optimisation of the code is necessary before
the uncertainty analyses are carried out. For example, an optimisation can be performed in order todetermine the axial node size. Also, the pressure drop data can be used for the optimisation purposes.
As already discussed, in the BFBT database, the pressure drop data was collected for assembly type
C2A, which is a sub-type of assembly type 4. Therefore, assembly types 4 and C2A were selected,
respectively, for the uncertainty analyses of the steady-state void distribution and for the uncertainty
analyses of the steady-state critical power.
Four experimental cases were selected that are considered covering the BFBT database for the
purpose of uncertainty analyses of the steady-state void distribution. Table 5.3.2 summarises theselected cases.

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SELECTION OF THE EXERCISE CASES FROM THE BFBT DATABASE

Table 5.3.1: The bundle types used in the BFBT measurements
Phase

Exercises

Assembly type
0-1
0-2
0-3
1
2
3
4
0-1
0-2
0-3
1
2
3
4
4
C2A (single-phase)
C2A (for two-phase)
C2A
C2B
C3
C2A

Steady-state sub-channel

Void distribution

Steady-state microscopic

Transient macroscopic
Pressure drop
Steady-state

Critical power

Critical power
Transient

Critical power

Table 5.3.2: Selected cases for uncertainty analysis of void distribution
Test case
Test 4101-02
Test 4101-13
Test 4101-69
Test 4101-86

Void
fraction
57.1
86.8
18.2
69.8

Pressure
(MPa)
0.994
1.224
8.638
8.705

Flow rate
(t/h)
10.12
55.01
10.08
54.59

Inlet sub-cooling(kJ/kg)
53.3
92.5
52.5
54.2

Outlet quality
(%)
0.32
4.46
0.23
4.62

5.4 Selected cases for uncertainty analysis of critical power
Four experimental cases were selected that are considered covering the BFBT database for the
purpose of uncertainty analyses of the steady-state critical power. Table 5.4.1 summarises theselected cases.

Table 5.4.1: Selected cases for uncertainty analysis of critical power

24

Test case

Pressure(MPa)

Flow rate(t/h)

SA603901
SA505900
SA812800
SA512800

7.18
5.49
8.67
05.5

10.01
20.14
65.27
65.52

Inlet
sub-cooling(kJ/kg)025.82026.04135.52133.75

Critical
power
(MW)
03.20
05.98
08.90
11.09

Dry-out rod
numberElevation

Angle

08
04
08
53

A
A
B
A

330
240
330
150

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SELECTION OF THE EXERCISE CASES FROM THE BFBT DATABASE

5.5 Selected cases for elemental task
The reference test matrix and test numbers for the elemental task are shown in Tables 5.5.1 and 5.5.2,
respectively, where the cases are denoted with T1 and T2. Table 5.5.3 provides the boundary
conditions for the Sub-tasks 1 and 2 of the elemental task. The power and the flow rate of thereference cases are divided by the ratio of the number of rods and the flow area between the assembly
type 0-1 and the elemental channel. In Sub-task 2, the power shall be increased until the dry-out occurs.
If the spacer effect is modelled in Sub-task 2, both spacer types, grid and ferrule, have to be considered.

Table 5.5.1: Reference test matrix for elemental task
Assembly

0-1

Pressure
(MPa)

7.2

Inlet
sub-cooling(kJ/kg)Flow rate(t/h)

Exit quality (%)
2

5

8

12

18

25

50.2

10
20
30
55
70

X
X
X
W
X

X
X
X
T1
X

X
X
X
W
X

X
X
X
T1
X

X
X
X
W
X

X
X
X
T1, T2–

X: Test case, W: Duplicated test case, T1: Task 1 reference case, T2: Task 2 reference case.

Table 5.5.2: Reference test numbers for the elemental task
Test no.

Assembly

Pressure(MPa)

Flow rate(t/h)

Inlet sub-cooling(kJ/kg)

Power
(MW)

0011-550011-58
0011-61

0-1
0-1
0-1

7.180
7.172
7.210

54.03
54.90
54.79

52.6
51.0
50.9

1.90
3.51
6.44

Exit
quality
(%)
05.0
12.0
24.9

Task
cases
T1
T1
T1, T2

Table 5.5.3: Boundary conditions for the elemental task
Reference
test no.
0011-550011-58
0011-61

Pressure
(MPa)
7.180
7.172
7.210

Flow rate(kg/h)
772.2
806.0
804.4

Inlet sub-cooling(kJ/kg)
52.6
51.0
50.9

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Power
(kW)
030.65056.61103.87

Task
cases
T1
T1
T1, T225

REQUESTED OUTPUT

Chapter 6: Requested output

6.1 Introduction
Participants should provide the output information with the given requirements:•

discussion of the uncertainty approaches used;•

results should be presented in electronic format;•

all output should be in SI units.

Templates of requested output data for each exercise will be provided by the benchmark team.
The Pennsylvania State University will:•

provide code-to-code comparisons for different submittals;•

provide overall conclusions on current code capabilities.

6.2 Requested output for Exercise 4 of Phase I
The requested output format for Exercise 4 of Phase I is given in this section.

Step 1: Accuracy analysis of steady-state void distribution
Using the deterministic values for input parameters, the “deterministic void distribution” for eachselected case should be calculated and submitted as a 9 × 9 matrix (Table 6.2.1).

Table 6.2.1: Calculated void distribution from deterministic input parameters
y/x

1

2

3

4

5

6

7

8

9

1

xx

xx

xx

xx

xx

xx

xx

xx

xx

2

xx

xx

xx

xx

xx

xx

xx

xx

xx

3

xx

xx

xx

xx

xx

xx

xx

xx

xx

4

xx

xx

xx

xx

xx

xx

xx

xx

xx

5

xx

xx

xx

xx

xx

xx

xx

xx

xx

6

xx

xx

xx

xx

xx

xx

xx

xx

xx

7

xx

xx

xx

xx

xx

xx

xx

xx

xx

8

xx

xx

xx

xx

xx

xx

xx

xx

xx

9

xx

xx

xx

xx

xx

xx

xx

xx

xx

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REQUESTED OUTPUT

Step 2: Uncertainty analysis of steady-state void distribution
For each selected case, the following outputs should be submitted as a 9 × 9 matrix (cf. definitions in
Chapter 2):•

sample-averaged void distribution;•

sample-averaged bias error;•

maximum bias error;•

sample-averaged standard deviation;•

coverage ratio with epsilon εexp = (3%/αexp ) at the sub-channel scale. 1

A questionnaire should be filled in to present the sampling technique and size used for thisuncertainty analysis. Since the participants can propagate their own additional uncertain parameters,
especially modelling parameters, they should clearly indicate the associated range of variation and
the PDF considered for each extra input parameter.

6.3 Requested output for Exercise 3 of Phase II
The requested output format for Exercise 3 of Phase II is given in this section.

Step 1: Accuracy analysis of steady-state critical power
Using the deterministic values for input parameters, the “deterministic critical power” for each selectedcase should be calculated and submitted as a 9 × 9 matrix (Table 6.3.1).

Table 6.3.1: Calculated critical power from deterministic input parameters
y/x

1

2

3

4

5

6

7

8

9

1

xx

xx

xx

xx

xx

xx

xx

xx

xx

2

xx

xx

xx

xx

xx

xx

xx

xx

xx

3

xx

xx

xx

xx

xx

xx

xx

xx

xx

4

xx

xx

xx

xx

xx

xx

xx

xx

xx

5

xx

xx

xx

xx

xx

xx

xx

xx

xx

6

xx

xx

xx

xx

xx

xx

xx

xx

xx

7

xx

xx

xx

xx

xx

xx

xx

xx

xx

8

xx

xx

xx

xx

xx

xx

xx

xx

xx

9

xx

xx

xx

xx

xx

xx

xx

xx

xx

Step 2: Uncertainty analysis of steady-state critical power
In this step, uncertainty analyses of critical power have to be performed for the critical power value,
the dry-out elevation, the dry-out rod number and the dry-out angle. Because of the difficulties tostatistically handle and analyse the localisation aspects, it is decided to focus solely on the criticalpower value.

1.
See Chapter 2: εexp is the relative error while Δαexp = 3% is the absolute error on void fraction data at the
sub-channel scale.

28

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REQUESTED OUTPUT

Participants should submit their results by using the format given in Table 6.3.2.

Table 6.3.2: The calculated critical power uncertainty output format
Test
numberSampleSampleaveraged
averaged bias
critical power
error

xx
xx

xx
xx

Samplemaximum
bias error

xx
xx

xx
xx

Sampleaveraged
standard
deviationxx
xx

Coverage ratio with a bias
error lower than theexperimental uncertainty(εexp = 1.5%)xx
xx

6.4 Requested output for elemental task
The following comparisons will be performed in the elemental task:•code-to-code comparisons on the analytical model for different submittals;•

code-to-code comparisons on the analytical results for different submittals.

Sub-task 1: Void fraction in the elemental channel benchmark
Analytical model:•

basic field thermal-hydraulics;•

sub-cooled boiling model;•

flow regime map;•transfer between phases (mass, momentum and energy);•transfer between fluid and wall (momentum and energy);•

spacer effect on pressure drop;•

nodalisation mesh.

Analytical results:•

axial profile of equilibrium quality (xe);•axial profile of flow quality (x);•axial profile of void fraction (α);•

axial profile of liquid velocity;•

axial profile of vapour velocity;•

axial profile of pressure difference from outlet;•

relation of equilibrium quality and void fraction (see Figure 6.4.1).

Sub-task 2: Critical power in the elemental channel benchmark
Analytical model:•

onset of annular-mist flow;•

entrainment at the onset of annular-mist flow;•

droplet entrainment;•

droplet deposition;

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REQUESTED OUTPUT•

spacer effect on droplets;•

film thickness at dry-out;•

nodalisation mesh.

Analytical results:•

axial profile of liquid film flow rate;•

axial profile of entrained droplet flow rate;•

axial profile of flow quality;•

axial profile of entraining droplet mass flux;•

axial profile of depositing droplet mass flux;•

critical power.

Figure 6.4.1: Example of relation between equilibrium quality and void fractionαxeFigure 6.4.2: Structure of annular-mist flow
DryoutDroplets

Liquid film

Fuel rod
Fuel
rod

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CONCLUSIONS

Chapter 7: Conclusions

The objective of the second volume of the BFBT specification is to provide the participants withinformation on the uncertainty and sensitivity analysis of void distribution and critical power.
In order to reduce the amount of work, to make the analysis more precise and to avoid confusion
when comparing the results of participants, specific test cases of the BFBT database were selected out
for the OECD/NRC BFBT uncertainty benchmark exercises.
Exercise 4 of Phase I is devoted to uncertainty analyses of the void distribution predictions. The
BFBT benchmark provides an opportunity to apply techniques of uncertainty analyses and to assess
the accuracy of two-phase flows models. The steady-state void fraction predictions can be used toanalyse these uncertainties and their propagation in the thermal-hydraulic models. The test cases forthis exercise include some of the test cases of Exercise 1 of Phase I plus some additional test cases.
Exercise 3 of Phase II is devoted to uncertainty analyses of the critical power predictions. The goalof Exercise II-3 is to find the critical power uncertainties of the thermal-hydraulic codes using the
BFBT benchmark database. This exercise provides the possibility of performing sensitivity/uncertainty
analyses of the droplet deposition, liquid film entrainment, and spacer grid models and their impact
on the codes’ capabilities of predicting critical power.

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REFERENCES

References

Adomian, G. (1980), Stochastic System Analysis, pp. 1-17, Academic Press, New York.
Aydogan, F., et al. (2007), “Phenomena Identification Ranking Table for BWR Steady State Critical
Power”, Nureth-12 International Conference (255).
Aydogan, F., et al. (2007a), “Phenomena Identification Ranking Table for BWR Steady State Void
Distribution”, Nureth-12 International Conference (254).
Boyack, B. (1995), AP600 Large-break Loss-of-coolant Accident Phenomena Identification and Ranking
Tabulation, LA-UR-95-2718, Los Alamos National Laboratory.
Coleman, H.W., W.G. Steele (1995), “Engineering Application of Experimental Uncertainty Analysis”,
AIAA Journal, Vol. 33, No. 10, pp. 1888-1896.
Doughtery, E.P., H. Rabitz (1979), “A Computational Algorithm for the Green’s Function Method of
Sensitivity Analysis in Chemical Kinetics”, International Journal of Chemical Kinetics, 11, 1237-1249.
Doughtery, E.P., J.T. Hwang, H. Rabitz (1979a), “Further Developments and Applications of the Green’s
Function Method of Sensitivity Analysis in Chemical Kinetics”, International Journal of Chemical Kinetics,
71 (4), 1794-1808.
Duke Power Company (1996), Duke Power Company Thermal-hydraulic Statistical Core Design Methodology,
BWU-Z CHF Correlation.
Dunker, A.M. (1984), “The Decoupled Direct Method for Calculating Sensitivity Coefficients in Chemical
Kinetics”, Journal of Chemical Physics, 81 (5), 2385-2393.
Glaeser, H.G. (2000), “Uncertainty Evaluation of Thermal-hydraulic Code Results”, International Meetingon “Best-estimate” Methods in Nuclear Installation Safety Analysis.
Hochreiter, L.E. (2006), “Thermal-hydraulics Uncertainties as Applied to DNB Analysis and Bestestimated LOCA Calculations”, ACE Uncertainty Workshop, North Carolina State University.
Holowach, M.J. (2000), The Development of a Reflood Heat Transfer Computational Package For Small
Hydraulic Diameter Geometries, MS Thesis, The Pennsylvania State University.
Inoue, A., et al. (1995), “Void Fraction Distribution in BWR Fuel Assembly and Evaluation of Sub-channel
Code”, Journal of Nuclear Science and Technology, July.
Isukapalli, S.S. (1999), Uncertainty Analysis of Transport-transformation Models, PhD Thesis, Rutgers, The
State University of New Jersey.
Mahaffy, John (2010), Verification and Validation website, accessed 10 February 2010,www.personal.psu.edu/faculty/j/h/jhm/470/lectures/VandV/VandV.html.
Martin, R.P., L.D. O’Dell (2005), “AREVA’s Realistic Large Break LOCA Analysis Methodology”, Nuclear
Engineering and Design, 235, 1713-1725.
Nuclear Energy Agency (NEA) (1997), Report of Uncertainty Analysis Methods Study for Advanced Best
Estimate Thermal Hydraulic Code Applications, NEA/CSNI/R(97)35, OECD/NEA, Paris.
NEA (2006), NUPEC BWR Full-size Fine-mesh Bundle Test (BFBT) Benchmark, Volume I: Specifications,
NEA/NSC/DOC(2005)5, OECD/NEA, Paris.

NUPEC BWR FULL-SIZE FINE-MESH BUNDLE TEST (BFBT) BENCHMARK, VOLUME II – © OECD/NEA 201033

REFERENCES

Oberkampf, W.L., T.G. Trucano (2006), “Design of and Comparisons with Verification and Validation
Benchmarks”, Proceedings of the Meeting on Benchmarking of CFD Codes for Application to Nuclear Reactor
Safety (CFD4NRS), Garching (Munich), Germany, 5-7 September, accessed 11 February 2009,www.nea.fr/html/nsd/reports/2007/nea6298/invited.html.
Prokopakis, G.J. (1993), “Decoupled Direct Method for the Solution of Ordinary Boundary Value
Problems”, Applied Mathematical Modelling, 17 (9), 499-503.
Ravenswaay, Jan P., et al. (2006), “Verification and Validation of the HTGR Systems CFD Code Flownex”,
Nuclear Engineering Design, Vol. 236, Issues 5-6, March, pp. 491-501.
Roache, P.J. (1998), Verification and Validation in Computational Science and Engineering, Hermosa
Publishers, pp. 403-412.
Roselle, S.J. (1994), “Effects of Biogenic Emission Uncertainties on Regional Photochemical Modeling ofControl Strategies”, Atmospheric Environment, 28 (10), 1757-1772.
Saltelli, A., K. Chan, E.M. Scott (2000), Sensitivity Analysis, Wiley.
Saltelli, A., et al. (2004), Sensitivity Analysis in Practice. A Guide to Assessing Scientific Models, John Wiley &Sons.
Sistla, G., S.T. Rao, J. Godowitch (1991), “Sensitivity Analysis of a Nested Ozone Air Quality Model”,
Proceedings of the AMS/AWMA Joint Conference on Applications of Air Pollution Modeling and its Applications,
New Orleans, LA.
Tatang, M.A. (1992), Combined Stochastic and Deterministic Approach in Solving Stochastic Differential
Equations, MS Thesis, Carnegie Mellon University.
Tomovic,´ R., M. Vukobratovic´ (1972), General Sensitivity Theory, Elsevier, New York.
US Nuclear Regulatory Commission (US NRC) (1989), “Best Estimate Calculations of Emergency Core
Cooling Performance”, Regulatory Guide 1.157 (Task RS 701-4).
US NRC (1989a), Quantifying Reactor Safety Margins, NUREG/CR-5249.
Vanderperk, M. (1997), “Effect of Model Structure on the Accuracy and Uncertainty of Results from
Water Quality Models”, Hydrological Processes, 11 (3), 227-239.
Vieux, B.E., B. Needham (1993), “Nonpoint-pollution Model Sensitivity to Grid-cell Size”, Journal of
Water Resources Planning & Management-ASCE, 119 (2), 141-157.
Villadsen, J., M.L. Michelsen (1978), Solution of Differential Equation Models by Polynomial Approximation,
Prentice-Hall, Englewood Cliffs, New Jersey.
Wheeler, A.J., A.R. Ganji (1996), Introduction to Nuclear Engineering Experimentation, Prentice Hall.
Wikipedia (2010), Sensitivity Analysis webpage, accessed 12 February 2010,
http://en.wikipedia.org/wiki/Sensitivity_analysis.

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UNCERTAINTY AND SENSITIVITY ANALYSIS METHODSAppendix 1: Uncertainty and sensitivity analysis methods

Uncertainty analysis (UA) and sensitivity analysis (SA) can be performed by several possible procedures,such as sampling methods, sensitivity testing, analytical methods and computer algebra-based methods(Isukapalli, 1999). In this chapter, these methods are defined with references given.

A1.1 Sampling methods
Some of the most common sensitivity analysis methods are the sampling-based methods (Wikipedia,
2010). These methods do not require modifications in the model equations.
Some of the sampling methods are listed below:•

Fourier Amplitude Sensitivity Test (FAST) methods;•

Monte Carlo and Latin Hypercube sampling methods;•

reliability-based methods.

Sampling methods are executed repeatedly for combinations of values sampled from the
distribution (assumed known) of the input factors in this method.
For the combination of factor values sampled with some probability distribution, uncertainty andsensitivity analyses are performed jointly by executing the model repeatedly. The following steps canbe listed:•specify the target function (the function on which the uncertainty analysis will be applied)and select the input of interest;•

appoint a PDF to the selected factors;•

generate a matrix of inputs with that PDF;•

evaluate the model and compute the distribution of the target function;•

select and run a method in order to assess the influence or relative importance of each input
factor on the target function.

A1.2 Sensitivity testing methods on models
The sensitivity testing method is run for a set of sample points with straightforward changes in themodel structure. The model structure or parameters of the model are changed in this analysis method.
Measuring the robustness of the model is the target of sensitivity testing methods by testing whether
the model response to changes significantly after changing the model parameters or/and structural
formulation of the model. Roselle (1994), Sistla (1991), Vieux (1993) and Vanderperk (1997) used thisapproach for different applications, such as sensitivity analysis.
Even though these sensitivity methods give information regarding the change of the model or theparameters in the model, detailed uncertainty information cannot be obtained.

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UNCERTAINTY AND SENSITIVITY ANALYSIS METHODS

A1.3 Analytical methods for UA/SAThe analytical methods for UA/SA involve either the reformulation of the original model using
stochastic algebraic/differential equations or differentiation of the model equations and subsequent
solution of a set of auxiliary sensitivity equations.
Some of the analytical methods are listed below:•

coupled and decoupled direct methods;•

spectral-based stochastic finite element method;•

Green’s function method;•

differential analysis methods.

The analytical methods need access to the governing model equations. These methods may
include writing additional computer codes for the solution of the auxiliary equations. This may makeanalytical methods impractical and sometimes impossible. For instance, reformulating an existing
computational model developed by others could require prohibitive amounts of resources.

Coupled/decoupled direct method
The coupled/decoupled direct method (CDDM) contains differentiation of the model equations and thesubsequent solution of the sensitivity equations. In the coupled direct method, the sensitivity
equations are then solved along with the original model equations (Tomovic,´ 1972). In the decoupleddirect method, the sensitivity equations and the original model equations are solved separately(Dunker, 1984; Prokopakis, 1993). The decoupled method has the advantage of computational efficiency
and stability of the solution.

Spectral-based stochastic finite element method
The spectral-based stochastic finite element method relies on the use of representing stochastic
processes in terms of a series expansion. This approach results in a set of linear matrix equations withdeterministic matrices multiplied by random vectors for finite element method problems. The matrix
equations are solved by using either the Galerkin’s or method operator expansions (Villadsen, 1978).

Green’s function method
The sensitivity equations of a model are obtained after differentiating the model equations. The
sensitivity equations are solved with constructing an auxiliary set of Green’s functions. This method
reduces the number of differential equations solved for sensitivity, and replaces them with integralsthat can be easily evaluated (Doughtery, 1979; Villadsen, 1978). Generally, The Green’s function method´ 1972).
is less efficient than the decoupled method (Tomovic,

Differential analysis methods
The differential analysis methods use the Neumann expansion (Adomian, 1980; Doughtery, 1979a)
and perturbation theory (Doughtery, 1979a; Tatang, 1992). Since the Neumann expansion method
includes the inverse of the model, it may have some limitations. The perturbation methods include
the expansion of the model outputs in terms of small random perturbations in model parameters.
Since application of these methods is complex for the non-linear systems and the perturbation
terms must be small, the differential analysis methods have some limitations.

A1.4 Computer algebra-based methods
The computer algebra-based methods involve both the direct manipulation of the computer code andthe estimation of the uncertainty and sensitivity of model outputs with respect to model inputs.
These methods do not need information about the model structure or the model equations.

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UNCERTAINTY AND SENSITIVITY ANALYSIS METHODS

A1.5 UA/SA commonly used in the nuclear industry
Uncertainty methods for advanced best-estimate thermal-hydraulic code applications are addressedin a Nuclear Energy Agency report on uncertainty analysis methods (NEA, 1997). In this report, heprovides detailed information on the uncertainty methods given below:•

AEA Technology method;•

University of Pisa method;•

GRS method;•

IPSN method;•

ENUSA method.

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PIRT INFORMATION

Appendix 2: PIRT information

Two independent expert groups worked at PSU on the Phenomena Identification Ranking Table (PIRT)
tables (Tables A2.1 through A2.8) for void distribution (Aydogan, 2007a) and critical power (Aydogan,2007) predictions. The steps of the development of the PIRT tables are given as:•

The first group developed the PIRT tables and ranked them. Three rankings were used: low,
medium and high. The greater the effect on the uncertainty output parameter expected, thehigher the ranking of the uncertainty parameter.

•

The second group members reviewed the PIRT tables and provided their comments.

•

Finally, the first group finalised the PIRT tables according to the second group’s review.

The two expert groups agreed with each other for most of the parameters except for a few
uncertain parameters ranks. Tables A2.9 and A2.10 demonstrate the disagreement and final decision
about the PIRT tables for the cases of void distribution and critical power, respectively.
The BFBT benchmark participants may use the PIRT tables to obtain the most sensitive uncertain
parameters or they can create their own uncertain parameters tables in order to use in their UA/SA.
If they use PDF for the defined sensitive/uncertain parameters, they need to be presented to thebenchmark team.

Table A2.1: PIRT-1 for steady-state void distribution
ID
1
2
3
4

Boundary condition effectRanking
Basis for ranking
M
This parameter affects the saturation temperature.
H
This parameter is in the energy equation.
H
This parameter is in the energy equation.
M
This parameter changes the inlet boundary condition.

Parameter
Initial vessel operating pressure
Flow rate in the bundle
Power
Inlet flow temperature

L: Low, M: Medium, H: High.

Table A2.2: PIRT-2 for steady-state void distribution
Geometry effect
ID
1
2
3
4

Parameter
Wetted perimeter
Sub-channel area
Nominal gap width
The distance between thecentres of channels

Ranking
L
H
L
L

5

Fraction of channel area
blocked by gridM

6

Grid perimeter

M

7

Heated perimeter

H

8

Housing wetted perimeterL

Basis for ranking
This perimeter affects the friction factor.
This parameter affects the mass flow rate.
This parameter affects the lateral flow.
This parameter affects the lateral flow.
This parameter affects the pressure drop and the heat transfer.
How effective this parameter affects the void distribution has tobe shown with the sensitivity analysis.
How effectively this parameter affects the void distribution has
to be shown with the sensitivity analysis.
This perimeter affects the heat flux and void distribution but thegeometry of the bundle is well known.
This perimeter affects the friction factor.

L: Low, M: Medium, H: High.

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PIRT INFORMATION

Table A2.3: PIRT-3 for steady-state void distribution
ID
1
2
3
4
5

Parameter
The loss coefficient for agap – lateral
The wall friction factor forthe gap
The grid loss coefficient –axial
The mixing coefficientEquilibrium distribution
weighing factor in void drift

6Interfacial friction factor

7

Drag coefficient for bubble
flow regime
Drag coefficient for drop
flow regime
Interfacial friction factor forfilm flow regime

8
9

10
11

Single-phase friction factor
in wall drag force
Two-phase friction factor inwall drag force

Model parameter effect – hydraulics
Ranking
Basis for ranking
This parameter affects the lateral flow in bundle. The ranking of thisMparameter has to be determined with sensitivity analysis.
L

This affects the pressure drop.

HThis parameter affects the pressure drop in the bundle. The
ranking of this parameter has to be determined with sensitivity
analysis.
This parameter affects the lateral void distribution in the bundle

H

This parameter affects the lateral void distribution in the bundle.

M

Interfacial mass transfer
This parameter affects the pressure drop.
Interfacial drag force
M
The ranking of this parameter has to be determined with sensitivity
analysis.
M
The ranking of this parameter has to be determined with sensitivity
analysis.
H
This parameter affects the mass transfer between film flow andvapour.
Friction factor in wall drag force
L
This affects the pressure drop.
H

M

This affects the pressure drop. Because two-phase pressure drop
models have higher uncertainty, this parameter is ranked as M.
The ranking of this parameter has to be determined with sensitivity
analysis.

L: Low, M: Medium, H: High.

Table A2.4: PIRT-4 for steady-state void distribution
ID

Parameter

1

Single phase liquid2Sub-cooled nucleate boiling

3

Saturated boiling region

Model parameter effect – thermal
Ranking
Basis for ranking
Wall heat transfer coefficient
The ranking of this parameter has to be determined with sensitivity
Manalysis.
The ranking of this parameter has to be determined with sensitivity
Manalysis.
The ranking of this parameter has to be determined with sensitivity
Manalysis.
Entrainment/deposition

Entrainment in film flow
Entrainment rate

M

The ranking of this parameter has to be determined with sensitivity
analysis. This parameter has more importance in the post-CHF
scenario.

De-entrainment in film flow
De-entrainment rate for5film flow
De-entrainment on grid spacers

M

The ranking of this parameter has to be determined with sensitivity
analysis.

4

The ranking of this parameter has to be determined with sensitivity
analysis.
Spacer grid enhancement for entrained phase (to create thicker liquid film)
The ranking of this parameter has to be determined with sensitivity
7SEM
analysis.
6

SDE

M

L: Low, M: Medium, H: High.

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PIRT INFORMATION

Table A2.5: PIRT-1 for steady-state critical power
ID
1
2
3
4

Boundary condition effectRankingParameter
Initial vessel operatingpressure
Flow rate in the bundle
Power
Inlet flow temperature

Basis for ranking

M

This parameter affects the saturation temperature.

H
H
M

This parameter is in the energy equation.
This parameter is in the energy equation.
This parameter changes the inlet boundary condition.

L: Low, M: Medium, H: High.

Table A2.6: PIRT-2 for steady-state critical power
Geometry effect
ID
1
2
3
4

Parameter
Wetted perimeter
Sub-channel area
Nominal gap width
The distance between
the centres of channels

Ranking
L
H
L
L

5

Fraction of channel area
blocked by gridM

6

Grid perimeterM

7
8

Heated perimeter
Housing wetted perimeter

H
L

Basis for ranking
This perimeter affects the friction factor.
This parameter affects the mass flow rate.
This parameter affects the lateral flow.
This parameter affects the lateral flow.
This parameter affects the pressure drop and the heat transfer.
How effectively this parameter affects the void distribution has
to be shown with the sensitivity analysis.
How effectively this parameter affects the void distribution has
to be shown with the sensitivity analysis.
This perimeter affects the heat flux and void distribution.
This perimeter affects the friction.

L: Low, M: Medium, H: High.

Table A2.7: PIRT-3 for steady-state critical power
ID
1
2
3
4
5
6
7
8
9

Parameter
The loss coefficient for agap – lateralThe wall friction factor forthe gap
The grid loss coefficient –axial
The mixing coefficientEquilibrium distribution
weighing factor in void drift
Interfacial friction factor
Drag coefficient for bubble
flow regime
Drag coefficient for drop
flow regime
Interfacial friction factor forfilm flow regime

10

Single-phase friction factor
in wall drag force

11

Two-phase friction factor
in wall drag force

Model parameter effect – hydraulics
Ranking
Basis for ranking
Wall heat transfer coefficientThis parameter affects the lateral flow in bundle. The ranking of thisLparameter has to be determined with sensitivity analysis.
L

This affects the pressure drop.

HThis parameter affects the pressure drop in the bundle. The rankingof this parameter has to be determined with sensitivity analysis.
This parameter affects the lateral void distribution in the bundle.

H

This parameter affects the lateral void distribution in the bundle.

M

Interfacial mass transfer
This parameter affects the pressure drop.
Interfacial drag force

H
L

Drag coefficient affect the friction at the beginning of the bundle.

The ranking of this parameter has to be determined with sensitivity
analysis.
Because this parameter affects the drag force which is near theH
dry-out, it affects the dry-out.
Friction factor in wall drag force
M

L

This affects the pressure drop.

M

This affects the pressure drop. Because two-phase pressure drop
models have higher uncertainty, this parameter is ranked as M. The
ranking of this parameter has to be determined with sensitivity
analysis.

L: Low, M: Medium, H: High.

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PIRT INFORMATION

Table A2.8: PIRT-4 for steady-state critical powerID

Parameter

1

Single phase liquid2Sub-cooled nucleate boiling

3

Saturated boiling region

Model parameter effect – thermal
Ranking
Basis for ranking
Wall heat transfer coefficient
The ranking of this parameter has to be determined with sensitivity
Manalysis.
The ranking of this parameter has to be determined with sensitivity
Manalysis.
The ranking of this parameter has to be determined with sensitivity
Manalysis.
Entrainment/deposition

Entrainment in film flow
4
Entrainment rate
De-entrainment in film flow
De-entrainment rate for5film flow

H

Entrainment affects the liquid film rate.

H

De-entrainment affects the liquid film rate.

L: Low, M: Medium, H: High.

Table A2.9: Comparison of two independent expert groups’
decisions about the PIRT tables for the void distribution
EffectParameter(s)First group’s
decision

Boundary
condition effect

Inlet temperature

L

Geometry effectAll

Agreement

Hydraulic effect

The loss coefficient
for a gap

L

Hydraulic effectInterfacial friction factor
in mass transfer andinterfacial drag force

H

Thermal effectAll

Agreement

Second group’s
decision
M (If the inlet temperature is increased, the exit
temperature of the coolant will change for a fixed
power.)
Agreement
M (If the lateral cross loss coefficient is increased,
there will be no cross-flow. Void drift and mixing
coefficient may not be effective in that case.)
H (Although interfacial drag coefficient is used by
two equations, its effect may be different on the
result. Therefore, the ranking of this parameter
may be different in each equation. The groupsuggests changing the ranking of the interfacial
friction factor in interfacial drag force eq. toMedium.)Agreement

Table A2.10: Comparison of two independent expert groups’
decisions about the PIRT tables for the critical power
EffectParameter(s)First group’s
decision

Boundary
condition effect

Inlet temperature

L

Geometry effectHydraulic effect

All
All

Agreement
Agreement

Hydraulic effectInterfacial friction factor
in mass transfer andinterfacial drag force

H

Hydraulic effectThermal effect42

Drag coefficient for
bubble flow regime
All

L
Agreement

Second group’s
decision
M (If the inlet temperature is increased, the exit
temperature of the coolant will change for a fixed
power.)
Agreement
Agreement
H (Although interfacial drag coefficient is used by
two equations, its effect may be different on the
result. Therefore, the ranking of this parameter
may be different in each equation. The groupsuggests changing the ranking of the interfacial
friction factor in interfacial mass transfer eq. toMedium.)M (Suggested that sensitivity analyses are needed
to understand its effect.)
Agreement

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