FUEL PERFORMANCE CODE BENCHMARK FOR UNCERTAINTY ANALYSIS IN LIGHT WATER

The Pennsylvania State University

The Graduate School

College of Engineering

FUEL PERFORMANCE CODE BENCHMARK FOR UNCERTAINTY ANALYSIS

IN LIGHT WATER REACTOR MODELING

A Thesis in

Nuclear Engineering

by

Taylor S. Blyth

2012 Taylor S. Blyth

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

August 2012

The thesis of Taylor S. Blyth was reviewed and approved* by the following:

Maria Avramova

Assistant Professor of Nuclear Engineering Thesis Advisor

Kostadin Ivanov Distinguished Professor of Nuclear Engineering

Arthur Motta

Professor of Nuclear Engineering and Materials Science and Engineering

Chair of Nuclear Engineering

*Signatures are on file in the Graduate School

iii

ABSTRACT

Fuel performance codes are used in the design and safety analysis of light water reactors.

The differences in the physical models and the numerics of these codes along with input,

manufacturing, and boundary condition uncertainties can lead to more variations in predicting the

target parameter. Because of this, an uncertainty analysis is an important step in code

development and testing. Determining the best estimate values with confidence bounds of

important fuel quantities are becoming a more essential benchmark of the fuel performance

codes. An uncertainty analysis, such as performed in this thesis, targeting the common sources of

variation in the fuel performance codes shows the effects of uncertainty in manufacturing

tolerances and boundary condition variations on the centerline temperature of the fuel. This is

done with an uncertainty analysis code, DAKOTA, driving simulations of randomly sampled

variations in input parameters, as defined by the UAM Benchmark, coupled with the fuel

performance codes FRAPCON and FRAPTRAN. The input parameters with the strongest

influence on the output are also identified. With 100 simulated cases for each test problem, the

overall minimum and maximum calculated output values were within 6% of the calculated

sample mean of the output parameter. The fuel density variations had the largest impact on the

calculated fuel centerline temperature. The results in this study show that the variations of the

input parameters is propagated to the calculated target parameters and best estimate values along

with confidence bounds can be used to define the expected results with 95% confidence. As a

result, a benchmark for fuel performance codes has been designed for these types of cases.

iv

TABLE OF CONTENTS

LIST OF FIGURES ............................................................................................................ vi

LIST OF TABLES .............................................................................................................. viii

NOMENCLATURE ........................................................................................................... xi

ACKNOWLEDGMENTS ................................................................................................... xiii

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

1.1 – Introduction to Uncertainty Analysis in Nuclear Power Plant Safety .................... 1 1.2 – Sources of Uncertainty in Nuclear Fuel Modeling ............................................... 2 1.3 – The UAM Benchmark ......................................................................................... 3 1.4 – Fuel Performance Codes ..................................................................................... 4 1.5 – Analysis Procedure ............................................................................................. 5

Chapter 2 : Exercise II-1 of the UAM Benchmark ............................................................... 9

2.1 – Discussion of Input, Propagated, and Output Uncertainties .................................. 10 2.1.1 – Input Data Uncertainties ........................................................................... 11 2.1.2 – Geometry and Nodalization Uncertainties ................................................. 11 2.1.3 – Modeling and Code Uncertainties ............................................................. 12 2.1.4 – Manufacturing Uncertainties..................................................................... 14

2.2 – Test Problems ..................................................................................................... 18 2.2.1 – Case 1a: Steady-State BWR Numerical Test Problem ............................... 21 2.2.2 – Case 1b: Transient BWR Numerical Test Problem .................................... 28 2.2.3 – Case 2a: Steady-State PWR Numerical Test Problem................................ 32 2.2.4 – Case 2b: Transient PWR Numerical Test Problem .................................... 38 2.2.5 – Case 3a: Steady-State VVER Numerical Test Problem ............................. 41 2.2.6 – Case 3b: Transient VVER Numerical Test Problem .................................. 47 2.2.7 – Case 4a: Steady-State BWR Experimental Test Problem ........................... 49 2.2.8 – Case 4b: Transient BWR Experimental Test Problem ............................... 55 2.2.9 – Case 5a: Steady-State PWR Experimental Test Problem ........................... 59

Chapter 3 : Code Descriptions ............................................................................................. 65

3.1 – FRAPCON ......................................................................................................... 65 3.2 – FRAPTRAN ....................................................................................................... 66 3.3 – DAKOTA ........................................................................................................... 67 3.4 – Other Scripts ....................................................................................................... 67

Chapter 4 : Steady-State Case Results ................................................................................. 70

4.1 – Peach Bottom Unit 2 Steady-State Results ................................................... 70 4.2 – Three Mile Island Unit 1 Steady-State Results ............................................. 74

v

Chapter 5 : Transient Case Results ...................................................................................... 79

5.1 – Peach Bottom Unit 2 Transient Results ........................................................ 79 5.2 – Three Mile Island Unit 1 Transient Results .................................................. 83 5.3 – DAKOTA Seed Study ................................................................................. 87

Chapter 6 : Conclusions ...................................................................................................... 88

Appendix ............................................................................................................................ 90

A.1 – FRAPCON Input File - PB-2 Steady-State Case ................................................. 90 A.2 – FRAPTRAN Input File - PB-2 Transient Case ................................................... 93 A.3 – DAKOTA Input File - PB-2 Transient Case ....................................................... 96 A.4 – DAKOTA Output File nond.out - PB-2 Transient Case (Abridged) .................... 97 A.5 – Batch Script for the PB-2 Transient Case............................................................ 100 A.6 – MATLAB Scripts - PB-2 Transient Case ............................................................ 101

A.6.1 – Driver Script (PB-2 Transient Case) ......................................................... 101 A.6.2 – Input Parameter Reader Script (PB-2 Transient Case) .............................. 102 A.6.3 – DAKOTA Parameter File Reader Script (PB-2 Transient Case) ............... 103 A.6.4 – Parameter Updater Script (PB-2 Transient Case) ...................................... 104 A.6.5 – Fuel Performance Code Input File Creator Script (PB-2 Transient Case) .. 105

A.7 – Python Script - PB-2 Transient Case .................................................................. 108 A.8 – DAKOTA Tabulated Results File (PB-2 Steady-State Case, Abridged) .............. 109 A.9 – DAKOTA Tabulated Results File (PB-2 Transient Case, Abridged) ................... 110

References .......................................................................................................................... 111

vi

LIST OF FIGURES

Figure 1: Effect of Changing Fuel Density on the Fuel Centerline Temperature ................... 18

Figure 2: PB-2 Fuel Pin Image ............................................................................................ 23

Figure 3: PB-2 Power History Plot ...................................................................................... 25

Figure 4: PB-2 Axial Power Profile Plot .............................................................................. 27

Figure 5: PB-2 Transient Power History Plot ....................................................................... 30

Figure 6: TMI-1 Fuel Pin Image .......................................................................................... 33

Figure 7: TMI-1 Power History Plot .................................................................................... 35

Figure 8: TMI-1 Axial Power Profile Plot ........................................................................... 36

Figure 9: TMI-1 Transient Power History Plot .................................................................... 39

Figure 10: VVER-1000 Fuel Pin Image ............................................................................... 42

Figure 11: VVER-1000 Power History Plot ......................................................................... 44

Figure 12: VVER-1000 Axial Power Profile Plot ................................................................ 45

Figure 13: VVER-1000 Transient Power History Plot ......................................................... 48

Figure 14: IFA-432 Fuel Pin Image ..................................................................................... 50

Figure 15: IFA-432 Power History Plot ............................................................................... 52

Figure 16: IFA-432 Axial Power Profile Plot....................................................................... 54

Figure 17: FK-1 Transient Power History Plot ..................................................................... 57

Figure 18: IFA-429 Fuel Rod Image.................................................................................... 60

Figure 19: IFA-429 Power History Plot ............................................................................... 62

Figure 20: IFA-429 Axial Power Profile Plot....................................................................... 63

Figure 21: Code Flowchart .................................................................................................. 69

Figure 22: PB-2 Mean Fuel Centerline Temperatures at each Axial Node with 95% CIs ...... 73

Figure 23: PB-2 Standard Deviation of Fuel Centerline Temperatures at each Axial Node

with 95% CIs............................................................................................................... 74

Figure 24: TMI-1 Mean Fuel Centerline Temperatures at each Axial Node with 95% CIs .... 77

vii

Figure 25: TMI-1 Standard Deviation of Fuel Centerline Temperatures at each Axial

Node with 95% CIs ..................................................................................................... 78

Figure 26: PB-2 Transient Standard Deviation Confidence Bound Plot ................................ 81

Figure 27: TMI-1 Transient Standard Deviation Confidence Bound Plot ............................. 85

viii

LIST OF TABLES

Table 1: Effect of Uncertainties – Manufacturing, BWR...................................................... 15

Table 2: Effect of Uncertainties – Manufacturing, PWR ...................................................... 15

Table 3: Effect of Uncertainties – Modeling, BWR ............................................................. 16

Table 4: Effect of Uncertainties – Modeling, PWR .............................................................. 17

Table 5: Exercise II-1 Core Boundary Condition Variations ................................................ 20

Table 6: Exercise II-1 Code Parameter Variations [14] ........................................................ 20

Table 7: PB-2 Fuel Rod Geometry ...................................................................................... 22

Table 8: PB-2 Power History .............................................................................................. 24

Table 9: PB-2 Axial Power Profile ...................................................................................... 26

Table 10: Case 1 Manufacturing Uncertainties .................................................................... 27

Table 11: PB-2 Transient Power History ............................................................................. 29

Table 12: PB-2 Transient Time Step Sizes........................................................................... 30

Table 13: PB-2 Transient Coolant Temperature History ...................................................... 31

Table 14: TMI-1 Fuel Rod Geometry .................................................................................. 32

Table 15: TMI-1 Power History .......................................................................................... 34

Table 16: TMI-1 Axial Power Profile .................................................................................. 36


Table 18: TMI-1 Transient Power History ........................................................................... 38

Table 19: TMI-1 Transient Time Step Sizes ........................................................................ 39

Table 20: TMI-1 Transient Coolant Temperature History .................................................... 40

Table 21: VVER-1000 Fuel Rod Geometry ......................................................................... 41

Table 22: VVER-1000 Power History ................................................................................. 43

Table 23: VVER-1000 Axial Power Profile ......................................................................... 45


ix

Table 25: VVER-1000 Transient Power History .................................................................. 47

Table 26: VVER-1000 Transient Time Step Sizes ............................................................... 48

Table 27: IFA-432 Fuel Rod Geometry ............................................................................... 49

Table 28: IFA-432 Power History ....................................................................................... 51

Table 29: IFA-432 Axial Power Profile ............................................................................... 53


Table 31: FK-1 Fuel Rod Geometry .................................................................................... 55

Table 32: FK-1 Transient Power History ............................................................................. 56

Table 33: FK-1 Transient Coolant Temperature History ...................................................... 58

Table 34: FK-1 Transient Time Step Sizes .......................................................................... 58

Table 35: IFA-429 Fuel Rod Geometry ............................................................................... 59

Table 36: IFA-429 Power History ....................................................................................... 61

Table 37: IFA-429 Axial Power Profile ............................................................................... 62

Table 38: Case 5 Manufacturing Tolerances ........................................................................ 64

Table 39: PB-2 Steady-State Case Summary ....................................................................... 71

Table 40: PB-2 Steady-State Confidence Intervals............................................................... 71

Table 41: PB-2 Steady-State Maximum and Minimum Values ............................................ 71

Table 42: PB-2 Steady-State Partial Correlation Matrix ....................................................... 72

Table 43: TMI-1 Steady-State Case Summary ..................................................................... 75

Table 44: TMI-1 Steady-State Confidence Intervals ............................................................ 75

Table 45: TMI-1 Steady-State Maximum and Minimum Values .......................................... 75

Table 46: TMI-1 Steady-State Partial Correlation Matrix .................................................... 76

Table 47: PB-2 Transient Case Summary ............................................................................ 80

Table 48: PB-2 Transient Confidence Intervals ................................................................... 81

Table 49: PB-2 Transient Maximum and Minimum Values ................................................. 82

x

Table 50: PB-2 Transient Partial Correlation Matrix ............................................................ 83

Table 51: TMI-1 Transient Case Summary .......................................................................... 84

Table 52: TMI-1 Transient Confidence Intervals ................................................................. 84

Table 53: TMI-1 Transient Maximum and Minimum Values ............................................... 86

Table 54: TMI-1 Transient Partial Correlation Matrix ......................................................... 86

Table 55: Random Seed Variation Results ........................................................................... 87

xi

NOMENCLATURE

BWR Boiling Water Reactor

CI Confidence Interval

CPU Central Processing Unit

FA Fuel Assembly

GWd Gigawatt Days

ID Inner Diameter

LOCA Loss Of Coolant Accident

LWR Light Water Reactor

MTU Metric Ton Uranium

NEA Nuclear Energy Agency

NPP Nuclear Power Plant

NRC United States Nuclear Regulatory Commission

NSC Nuclear Science Committee

OD Outer Diameter

OECD Organization for Economic Co-operation and Development

PB-2 Peach Bottom Unit 2

PDF Probability Distribution Function

PNL Pacific Northwest National Laboratory

PWR Pressurized Water Reactor

RIA Reactivity Insertion Accident

SA Sensitivity Analysis

TD Theoretical Density

TMI-1 Three Mile Island Unit 1

xii

UA Uncertainty Analysis

UAM Uncertainty Analysis in Modeling

VVER Russian-Type Water-Water Energetic Reactor

xiii

ACKNOWLEDGMENTS

I would like to thank my thesis adviser, Dr. Maria Avramova, for taking me into her

research group, for her patience while I adjusted to performing work at the research level, and

also for challenging me with interesting projects and class work. I would also like to acknowledge

Dr. Kostadin Ivanov for his guidance, which has been crucial to my learning experience while I

have studied here. I also need to express my gratitude to Dr. Samuel Levine for explaining to me

almost everything I know about nuclear power as well as providing many of his own fascinating

stories to supplement these lessons. Thank you to The Pennsylvania State University for its

amiable research setting and to the United States Nuclear Regulatory Commission for the

financial support.

For my parents, siblings, and friends, who also deserve recognition for their support and

encouragement, thank you.

Chapter 1 : Introduction

1.1 – Introduction to Uncertainty Analysis in Nuclear Power Plant Safety

Uncertainties are present in virtually all calculations which are modeled off of actual

processes. These processes can be simple or complex. A nuclear power plant, for example,

contains many subsystems and parts that contain uncertainties, whether in dimensional variations,

sensor accuracy, or material compositions. When simulating portions of a nuclear power plant

(NPP), there are several sources of uncertainty that can be analyzed separately or altogether.

An uncertainty analysis differs from a sensitivity analysis in several ways. The sensitivity

analysis often incorporates changing parameters by small increments, one at a time, in order to

perceive the effect on the target parameter caused by these small changes. This is performed in

order to measure how proportional the relation between input and output is. An uncertainty

analysis often utilizes random sampling over a number of cases in order to determine lower and

upper predictions for certain parameters.

The purpose of analyzing these uncertainties is to quantify the effects of certain input

variations on the target parameter. If the actual variations of the input are precisely known, then

this gives a strong indication of the actual variations in the process that should be observed during

actual experiments. Often, these uncertainties are not well-defined and best-estimates are needed

in order to perform an uncertainty analysis of the process.

When applied to reactor safety, the results of the analysis are based on how well the input

uncertainties are known. Some values, such as manufacturing tolerances, are more frequently

measured and are therefore more confidently known. Other variations, such as those found within

2

newly-developed computer codes for nuclear engineering applications, may not be presently

known and require benchmarking in order to determine the effects on the output values.

As nuclear reactor modeling codes become more powerful and accurate, the ability to

predict reactor parameters improves. This improvement has the potential to decrease the

discrepancies between current reactor design limits and the actual limits of certain processes. For

example, if nuclear fuel were to have its upper limit on exposure lifetime extended this would

improve the efficiency and utilization of fuel within the plant. This extension would be based on

simulations that must be capable of proving the new fuel lifetime is indeed safe by all current

standards. These codes used in the simulation must also be trusted by a process of benchmarking

with steps similar to those found later in this study. Less conservative design limits, as could be

recommended after sufficient trusted simulations, would improve efficiency while maintaining at

least the current levels of safety.

1.2 – Sources of Uncertainty in Nuclear Fuel Modeling

Nuclear power plants such as the current light water reactors (LWRs) use a large number

of fuel rods which contain stacks of fuel pellets made of uranium oxide. In order to make these

fuel pellets to a certain specification, the uranium is mined and enriched and formed into small

cylindrical pellets which are placed inside of the fuel rod’s cladding layer. This cladding is

typically a zirconium alloy such as Zircaloy which is able to withstand the pressure between the

moderator and the fill gas, typically helium, as well as the high temperature and long-term

exposure to irradiation while the reactor operates. These fuel rods are arranged in batches called

fuel assemblies. Each of the processes, which go into the creation of the fuel, introduces potential

for uncertainty due to allowable levels of variation in parameters such as the density of the fuel.

The fuel rods are manufactured to strict dimensions but there is still some manufacturing error

3

which is represented as the tolerance on a certain measurement. The manufacturers provide these

tolerances based on several factors, including the known precision of their equipment used while

producing the fuel rods.

Nuclear power plant simulations contain input data uncertainties, geometry/nodalization

uncertainties, modeling uncertainties, code uncertainties, and manufacturing uncertainties. The

complexity of NPPs causes much potential for variations due to the large number of components

and processes. Since these components come from different manufacturers, the dimensional

tolerances will vary depending on the vendor. The type of reactor also has an effect on the

associated uncertainties. There are differences in the layout of the core and fuel assemblies, the

operational boundary conditions, and the material composition of the fuel pins. Chapter 2

contains detailed information on each of these uncertainty sources.

1.3 – The UAM Benchmark

An expert group under the Nuclear Science Committee (NSC) of the Nuclear Energy

Agency (NEA) within the Organization for Economic Cooperation and Development (OECD)

convened in 2006 to discuss “Uncertainty Analysis in Modeling” (UAM) with workshops and

decided to proceed with the development of a benchmark. This benchmark contains several

exercises specific to a certain region of reactor modeling [1, 2, 3, 4]. These exercises are designed

to provide uncertainty analysis methodologies for multi-physics (coupled) and multi-scale

simulations [16, 30]. The UAM group aims to emphasize the needs of sensitivity and uncertainty

analysis (SA/UA) with reactor simulations. Modeling LWR scenarios is a primary focus, with a

focus on neutronics, thermal hydraulics, and fuel behavior. Best-estimate calculations for the

exercises can be compared with experimental data from several nuclear power plants and research

facilities. This provides a means for comparison between the calculated and experimental results

4

for certain scenarios. Each exercise within the phase benchmarks has its own set of experimental

data specific to the type of scenario analyzed within that certain exercise.

Phase II of the benchmark was recently released in its draft form, and it contains three

exercises which are Exercise II-1: Fuel Physics, Exercise II-2: Time-Dependent Neutronics, and

Exercise II-3: Bundle Thermal-Hydraulics. Specifically in this study, Exercise II-1 will be

examined and analyzed using sensitivity and uncertainty analysis. Several reactors were chosen

as the representative reactors for which the numerical cases in this benchmark are modeled. Peach

Bottom Unit 2 is selected for the BWR, Three Mile Island Unit 1 is chosen for the PWR, and

Kozloduy Unit 6 is the VVER-1000 representative reactor. The availability of design data and

familiarity led to their selection. The cases analyzed in this report are fundamental to the design

of Exercise II-1 of the UAM Benchmark.

1.4 – Fuel Performance Codes

There are various fuel performance codes available for assessing the properties and

behavior of nuclear fuel. These evaluations are used during the design phase and in safety

calculations for the LWRs. Some codes are developed specifically for steady-state analysis and

others are adapted to various transient analysis. The purpose of these codes is to accurately

predict several fuel parameters during operating or abnormal conditions. Often the codes contain

capabilities to calculate specialized information such as gap conductivity, creep data, and

deformations. Chapter 3 contains more details on the fuel performance codes utilized in this

study.

Within Exercise II-1 of the UAM Benchmark there are several types of experiments

defined. The types of conditions considered in this study are normal operating conditions for the

long-term irradiation cases, and abnormal conditions for the short-term pulse cases. Accident

5

scenarios are not considered at this time. All models are single fuel pins surrounded by

moderator. The irradiation cases are modeled to be typical of the power levels a fuel rod would

receive during a normal fuel cycle in a reactor core. There are some small fluctuations in the

power levels but the power is mostly constant overall which provides an almost linear burnup

over time. The temperature variations are also on a long-term scale that varies directly with the

power. The fuel centerline temperature, which is the target parameter, will be recorded at its peak

value during the irradiation, whenever that may occur.

The abnormal transient conditions are not as severe as a loss of coolant accident (LOCA)

or other accident types, but they still cause a sharp disruption in the reactor power over a small

duration of time. They are more similar to reactivity-initiated accidents (RIAs) which are more

commonly studied and evaluated during design work to ensure safety of operating plants. RIAs

may also include control rod drop/ejection scenarios where a control rod is unintentionally

removed or ejected from its assembly, causing a local spike in reactivity and power. Another

possible scenario is caused by a loss of coolant flow due to a main coolant pump malfunction.

The ability of the components in the reactor to withstand these brief elevated conditions dictates

some of the temperature and power ratings given to the core. The calculated fuel centerline

temperature will be recorded at the onset of the power pulse, at the peak value of the pulse, and at

the end of the transient scenario. This provides a numerical result other than just the maximum

fuel temperature achieved during the simulation.

1.5 – Analysis Procedure

In this study, the uncertainty analysis is performed by examining several of the test cases

defined in Exercise II-1 of the UAM Benchmark and converting the provided data to code input.

The fuel performance codes provide output that includes the target parameter, which in these

6

cases is the fuel centerline temperature. The codes will be run with variations in the input

parameters in order to simulate an uncertainty analysis of the selected test case. The uncertainties

that are examined are the manufacturing tolerances and the boundary condition variations. These

values are provided in the benchmark. The PDFs are also given, and these are used in conjunction

with a code capable of interfacing with the fuel performance code to perform the analysis. This

study will utilize the FRAPCON [17] and FRAPTRAN [18] fuel performance codes running

within a DAKOTA [28] framework. MATLAB and Python are used to create input files and

process fuel code results, respectively. The interface and implementation of the codes and scripts

is discussed in detail in Chapter 3.

The DAKOTA script will cause random sampling of the input parameters for which there

are uncertainty data. A random value will be assigned to each one corresponding to its provided

statistical distribution. A certain number of cases are required in order to generate a confidence

interval for the target parameter. The purpose is to run enough cases in order to say with 95%

confidence that the resulting lower and upper confidence bounds contain the actual mean value of

the target parameter. Confidence intervals are commonly used in statistical analyses as a way to

show the likelihood that the results of a smaller sample can be extrapolated to estimate the mean

of the actual population. The sample mean usually differs from the population mean and its

location with respect to the actual population mean is important to quantize for a simulation

involving a reduced sample size. The required number of cases is determined using Wilks’

Method [26, 27].

(1.1)

In Equation 1.1, α corresponds to the percentage of coverage of the intervals, which

would be 0.95 for the case of creating 95% confidence intervals. The value of β is the confidence

7

that the actual population’s mean is contained within the confidence intervals. Using a value for β

of 0.95 implies 95% confidence that the actual mean is within the determined bounds. The

confidence is important when working with smaller sample sizes because it is strongly affected

by the number of cases which are run. The last parameter in Equation 1.1 is n, which is the

number of cases or trials that need to be run during the simulation. In this study, the values of α

and β are set at 0.95 each and the target parameter from Equation 1.1 is n, the number of cases

that must be run in order to assure that the actual mean value has a 95% likelihood to be within

the determined confidence intervals with 95% confidence. Substituting these values for α and β

yields a result of 93 for n. For simplicity and slightly more confidence, this value is rounded up to

100. Using this number of cases instead of n = 93 gives a confidence of 96.3% instead of 95% on

the resulting probability bounds on the mean. Therefore, 100 cases of input parameter variations

will be run for each test case examined from Exercise II-1. When creating a two-sided tolerance

limit such as with Equation 1.1, the number of cases to run is independent of the number of

parameters that are varied. The objective is to determine a 95% confidence interval on the target

parameter for each test case with 95% certainty. Due to the small sample size compared to the

total possible variations, the resulting confidence intervals from various random seed values will

also be examined in order to determine any effect of the seed value.

The reason for using this formula is to determine a reduced sample size that will still

provide reliable results. If each parameter in the simulation were incrementally varied to every

possible value, this would produce many more trials and take much longer. By reducing the

number of cases with Wilks’ formula, the parameters are not varied to as many permutations but

they will still provide reliable data with 95% confidence. For example, if there were to be ten

parameters varied to six different values each, then this would require 610

, or 60466176, total

cases to cover each possible combination of values. This practice is normal in large scale

simulations when it is not practical to cover every value of each parameter. The value used in this

8

study is 100 cases, which is less than 0.001% of the total number of combinations. This allows

the codes to generate results much more quickly at the cost of some accuracy. Increasing the

number of cases would diminish the distance between the confidence interval bounds and the

actual mean of the target parameter. If the simulation requires extensive CPU time then the

practical choice is to implement a limited sample size.

Chapter 2 : Exercise II-1 of the UAM Benchmark

Phase II of the UAM benchmark specification contains Exercise II-1, which is entitled

“Fuel Modeling” and is focused on evaluating uncertainties associated with modeling and

prediction of one of the most important feedback parameters in coupled neutronics/thermal-

hydraulic calculations - fuel temperature (Doppler feedback). Its objective is identifying and

propagating input uncertainties in the standard fuel rod models (whether it is an average rod, a

fuel assembly, a group of assemblies, or the whole core) used in the current thermal-hydraulics

codes for steady-state and transient analysis. More sophisticated simulation tools such as fuel

performance codes (for example FRAPCON [17] and FRAPTRAN [18]) will be utilized for

obtaining reference solutions. Halden in-pile experimental fuel temperature data will be used to

determine the uncertainty in the participants’ fuel rod models [11, 22, 23, 24]. The other relevant

considerations in this exercise include data, models, and recommendations on the most

appropriate fuel properties which reflect burn-up as well as uncertainties in fuel manufacturing.

Modeling fuel behavior in a reactor under certain transient conditions is an important step

in the calculations required to assess the safety and operation of the reactor [10, 15]. The models

in this exercise will be single-pin models. Variations in the Doppler feedback properties of the

fuel can change greatly with smaller variations in power levels. The propagation of these

uncertainties throughout calculations will affect other outputs desired by the participants.

Proper modeling of thermal behavior during normal and transient conditions includes

surface heat transfer, the heat transfer across the fuel-to-cladding gap, thermal conductivity of

fuel and cladding, power generation distribution in the fuel, and determining a solution of the

conduction equation. There are various inputs required for this type of modeling and the

uncertainty inherent in each of them will be examined in this exercise.

10

2.1 – Discussion of Input, Propagated, and Output Uncertainties

Input uncertainties include the fuel pellet nodalization, gas gap composition, as well as

cladding, fuel and gap conductivities. For some of the conductivities the uncertainties of code-

utilized correlations will be taken into account. The fuel conductivity has burnup dependence,

which can be treated either in a simple way or can be left for Phase III. For steady state

simulations, the input uncertainties (such as uncertainties of thermal conductivities) will be

propagated to the uncertainty of Doppler temperature prediction. The output uncertainty

parameter of interest in this Exercise is the nodal fuel (Doppler) temperature.

The input, output, and propagated uncertainties have been identified for this exercise and

include the following:

Input (I) uncertainty parameters: local pin power, local pin exposure, local pressure, local

bulk temperature and local surface heat transfer coefficient.

Output (O) uncertainty parameters: fuel temperature profiles, local flow area reductions,

gap conductance, and axial elongation.

The propagated uncertainty (U) parameters are the same as the output ones.

Certain assumptions need to be made in order to improve the feasibility of the models.

Using single pin modeling makes the input to the codes easier but it leaves out some complexities

which would affect the results if included. These complexities include the exact geometry of the

fuel assemblies within the core along with spacer grid effects and localized flow patterns. These

assumptions will cost some accuracy but make the simulations more manageable. Uncertainty is

introduced from modeling the scenario, the manufactured tolerances and dimensions, and from

the code’s calculation of the desired outputs.

In principle, the sources of Input (I) uncertainties in computer code simulations are

identified as:

11

Input data uncertainties;

Geometry and nodalization uncertainties;

Modeling and code uncertainties;

Manufacturing uncertainties.

2.1.1 – Input Data Uncertainties

These uncertainties are included in parameters such as the cross-sections used by the

codes or the initial and boundary conditions of the scenario. Since there is a small inherent

inaccuracy in every cross-section value it will propagate this error throughout the calculations. It

is difficult to remove these uncertainties since most of them are limited by scientific knowledge

and some error is typically accepted for normal analyses. The boundary conditions such as the

shapes of the applied power transient or the duration of irradiation also affect results. These are

sometimes difficult to model because in real life the shapes are not as smooth as the

approximations that are often used in calculations.

2.1.2 – Geometry and Nodalization Uncertainties

It is difficult to quantize a transient situation that may have been observed experimentally

into a series of inputs that are suitable and realistic to model. It requires some simplifications that

stray from the exact scenario in order to make the input more general. When modeling the

scenario it is very difficult to exactly match all of the inputs such as coolant temperature and

reactor power because these parameters can change very often and would require thousands of

inputs to be modeled accurately. Usually some generalizations are made and some lines become

flattened in order to simplify the input enough for practical use. During this truncation some

uncertainty is introduced and the produced results will be slightly different than the measured

12

ones. Steady-state codes especially ignore this changes that occur over seconds/minutes because

the overall case is likely on the order of hours/days. Transient code input might include

smoothing out a power pulse in order to make it symmetric or even triangular, which is not as it

actually occurs.

The geometry of the fuel rods and assemblies also receive some simplifications when

they are modeled. These may include ignoring certain parameters altogether such as plenum

volumes and springs in order to comply with what the input requests or does not request. Other

shapes are simplified and homogenized in order to make the codes more practical and easy-to-

use. The uncertainty involved with these simplifications is difficult to quantize because of all the

other uncertainties in codes but its accuracy can be increased by utilizing nodes and time steps as

well as geometry of the scenario as well as possible

2.1.3 – Modeling and Code Uncertainties

These uncertainties are described as the ones included in the code used for the analysis.

There is some inaccuracy in the values used in the computer codes for parameters such as fuel

and cladding thermal conductivity. Since many of these parameters and correlations are

temperature-dependent the inaccuracies can propagate through iteration if the initial value is

inaccurate. There is also the fact that the most of the codes use a 1-D scheme for temperature

calculations which means it works outward from the center of the fuel to the coolant. In order to

keep computation time reasonable it is important to find a balance between a small convergence

value for the correlation calculations and the accuracy of the solution. Subroutines in the codes

are also a source of uncertainty; they take one input parameter and perform a calculation with it to

produce another. A common example of this is the calculation of the thermal conductivity given a

temperature of operation. During this calculation some error is introduced as the models used are

13

not completely accurate and are often based on best-fit lines of previously measured data. With

codes that determine fuel rod failure there is importance on the selection of this failure criteria

that the code uses. It is based on several factors such as the yield strength and the melting point of

the material at the experienced temperatures. Also, many codes are capable of running very

detailed calculations with the use of many optional inputs that may not be utilized by the user.

Without these options specified the code is forced to output a more general solution that may lack

some of the intricacies found in the measured data. Other code factors that introduce some error

are machine rounding and varying significant figures kept during code subroutines. It is difficult

to quantize a transient situation that may have been observed experimentally into a series of

inputs that are suitable and realistic to model. It requires some simplifications that stray from the

exact scenario in order to make the input more general. When modeling the scenario it is very

difficult to exactly match all of the inputs such as coolant temperature and reactor power because

these parameters can change very often and would require thousands of inputs to be modeled

accurately. Usually some generalizations are made and some lines become flattened in order to

simplify the input enough for practical use. During this truncation some uncertainty is introduced

and the produced results will be slightly different than the measured ones. Steady-state codes

especially ignore this changes that occur over seconds/minutes because the overall case is likely

on the order of hours/days. Transient code input might include smoothing out a power pulse in

order to make it symmetric or even triangular, which is not as it actually occurs. Modeling a

changing axial power ratio is often necessary when the void fraction changes in a BWR and it can

be done but the transitions will not be as smooth or continuous as they are in experiments.

Differences in measured versus calculated temperatures may arise also from the code’s treatment

of cladding build-up such as oxidation and other corrosion that may appear on the outside of the

fuel rods. The amount of Burnup in the rods at the time of the transient can also add some

14

uncertainty depending on how accurately the code can model this past irradiation and

accommodate for it.

2.1.4 – Manufacturing Uncertainties

Other uncertainties arise from the values as provided by the manufacturers of the fuels

and equipment used in the reactor. There are design tolerances due to machining precision that

are built into each of the pieces of equipment and each fuel rod used in the reactor. Since it is not

possible to measure each fuel pellet accurately the dimensions provided by the manufacturer are

sufficient for most calculations but it is important to understand that these values do carry some

error. The significance of this error can grow depending on the model setup and the types of

equations used. Some of these manufacturing parameters are cladding Inner Diameter (ID)/Outer

Diameter (OD), fuel pellet OD, pellet/cladding roughness, enrichment, yield strength (and several

other material properties), fill gas composition/pressure, as well as the coolant properties. Of

these geometry parameters, it was determined in “Predictive Bias and Sensitivity in NRC Fuel

Performance Codes” that the ones with the most effect on fuel temperature are the fuel pellet

density and the pellet roughness [14]. The roughness is especially important in determining the

effective heat transfer coefficient from the coolant to the cladding and also from the fill gas to the

fuel pellets. Of the material properties, changing the thermal conductivity of the fuel had a very

large effect on the resulting fuel temperature.

The effect of these uncertainties is illustrated below with data from a study on NRC fuel

performance codes [14]. The manufacturing tolerances are varied by specified amounts and the

effect on the resulting peak fuel centerline temperature is noted in Table 1 for the BWR case and

in Table 2 for the PWR case. The columns correspond to changes made in either the positive or

negative direction from the design value, and the results are measured at both 0 GWd/MTU and

15

50 GWd/MTU to show how burnup affects the calculations. These cases were analyzed using

FRAPCON-3. The two columns under the “0 GWd/MTU” heading show the percentage change

in the fuel centerline temperature for a negative deviation in the input parameters (left column

under the heading) and for a positive deviation from the design value of the input parameters

(right column). The same is repeated for the 50 GWd/MTU heading.

Table 1: Effect of Uncertainties – Manufacturing, BWR

Manufacturing Uncertainties % Change in Peak Fuel Centerline Temperature

BWR % Change 0 GWd/MTU 50 GWd/MTU

Clad ID +0.46 -0.46 2.59 0.24 -0.33 0.33

Clad Thickness +6.06 -6.06 0.08 -0.16 0.13 -0.13

Clad Roughness +60.00 -60.00 0.24 -0.32 0.52 -0.52

Fuel Pellet OD +0.15 -0.15 0.00 -0.08 0.13 -0.13

Fuel Pellet Density +0.96 -0.96 -1.30 1.14 -1.24 1.18

Pellet Re-sinter Density +44.44 -44.44 -0.24 0.49 -0.59 0.59

Pellet Roughness +25.00 -25.00 0.49 -0.57 0.85 -0.98

Rod Fill Pressure +10.00 -10.00 -0.32 0.08 -0.33 0.26

Rod Plenum Length +4.49 -4.49 -0.08 0.00 0.00 0.00

Table 2: Effect of Uncertainties – Manufacturing, PWR

Manufacturing Uncertainties % Change in Peak Fuel Centerline Temperature

PWR % Change 0 GWd/MTU 50 GWd/MTU

Clad ID +0.49 -0.49 -0.25 0.25 -0.28 0.28

Clad Thickness +6.56 -6.56 0.06 -0.13 0.06 -0.06

Clad Roughness +60.00 -60.00 0.25 -0.25 0.28 -0.34

Fuel Pellet OD +0.16 -0.16 0.00 -0.06 0.06 -0.06

Fuel Pellet Density +0.96 -0.96 -1.20 1.20 -1.12 1.06

Pellet Re-sinter Density +44.44 -44.44 -0.57 0.57 -0.56 0.56

Pellet Roughness +25.00 -25.00 0.44 -0.44 0.50 -0.56

Pellet Dish

Diameter/Depth +12.47 -12.47 -0.06 0.00 -0.06 0.00

Rod Fill Pressure +2.86 -2.86 -0.06 0.00 -0.06 0.06

Rod Plenum Length +4.49 -4.49 -0.06 0.00 0.00 0.00

16

The modeling uncertainties include the changes in code correlations and parameters used

during the calculation of the scenario. Table 3 below shows the effects of changing some of these

code values on a BWR steady-state scenario and Table 4 contains the information for a PWR.

The target/output parameter is the peak fuel centerline temperature, as it is for the manufacturing

uncertainties.

Table 3: Effect of Uncertainties – Modeling, BWR

Modeling Uncertainties % Change in Peak Fuel Centerline

Temperature

BWR Change 0 GWd/MTU 50 GWd/MTU

Fuel Thermal Conductivity

+0.5 W/m-K -0.5 W/m-K -10.54 27.39 -18.24 30.33

Oxide Conductivity +0.5 W/m-K -0.5 W/m-K -0.16 0.16 -0.20 0.33

Cladding Conductivity +5 W/m-K -3 W/m-K -1.22 1.05 -1.18 0.98

Gas Conductivity +0.02 W/m-K -0.02 W/m-K -0.16 0.08 -0.20 0.20

Young's Modulus +5 MPa -10 MPa 0.00 0.00 0.00 0.00

- % Change - - - -

Fuel Thermal Expansion

+15.00 -13.04 -0.16 0.00 -0.20 0.07

Fission Gas Release +200.00 -50.00 1.95 -1.30 0.85 -0.92

Cladding Corrosion +40.00 -28.57 0.24 -0.24 0.39 -0.26

Irradiation Creep +10.00 -28.57 0.00 0.00 0.00 0.07

Yield Strength +30.00 -9.09 0.00 0.00 -0.65 0.13

Axial Growth +50.00 -33.33 0.00 0.00 0.07 -0.07

Cladding Thermal

Expansion +30.00 -23.08 0.00 0.00 0.07 0.00

17

Table 4: Effect of Uncertainties – Modeling, PWR

Modeling Uncertainties % Change in Peak Fuel Centerline

Temperature

PWR Change 0 GWd/MTU 50 GWd/MTU

Fuel Thermal

Conductivity +0.5 W/m-K -0.5 W/m-K -15.41 26.54 -17.00 26.40

Oxide Conductivity +0.5 W/m-K -0.5 W/m-K -0.38 0.69 -0.95 3.58

Cladding Conductivity +5 W/m-K -3 W/m-K -1.07 0.88 -0.78 0.67

Gas Conductivity +0.02 W/m-K -0.02 W/m-K -0.13 0.06 -0.11 0.11

Young's Modulus +5 MPa -10 MPa 0.00 0.00 0.00 0.00

- % Change - - - -

Fuel Thermal

Expansion +15.00 -13.04 0.00 0.00 -0.06 0.00

Fission Gas Release +200.00 -50.00 0.57 -0.44 0.39 -0.39

Cladding Corrosion +40.00 -28.57 0.69 -0.44 3.36 -1.12

Irradiation Creep +10.00 -28.57 0.00 0.00 0.00 0.00

Yield Strength +30.00 -9.09 0.00 0.00 -0.11 0.00

Axial Growth +50.00 -33.33 0.00 0.00 0.00 -0.06

Cladding Thermal

Expansion +30.00 -23.08 0.00 0.00 0.00 0.00

These tables show the possible effect on the target parameter due to relatively small

variations in manufacturing tolerances and modeling correlations. These observed changes in the

measured output provide a purpose for this study. Some of the input parameters have more of an

impact on the output values than others, which is important to note. This value can be studied

independently for each input parameter in a sensitivity study. This would include performing a

case while one parameter is varied by small increments and measuring the corresponding output

parameter for each increment. This can easily show which parameters have the largest effect on

the target values. Figure 1 below shows the large effect on the centerline temperature that changes

in the fuel density will illicit on a PWR fuel pin [29].

18

91.5 92 92.5 93 93.5 94 94.5 95 95.5 961320

1340

1360

1380

1400

1420

1440

1460

Fuel Density (%)

Fu

el C

en

terlin

e T

em

pe

ratu

re (

K)

Figure 1: Effect of Changing Fuel Density on the Fuel Centerline Temperature

2.2 – Test Problems

The test problems included in this exercise cover three different types of reactors (the

BWR, PWR, and VVER-1000) as well as two types of cases (transient and steady-state) for both

experimental and numerical tests. The numerical test cases include different stand-alone

neutronics single pin-cell test problems, designed for the purposes of the Exercise I-1 utilizing

information from the previous OECD coupled code benchmarks for each of the representative

reactors- BWR PB-2, PWR TMI-1, and Kozloduy-6 (Kalinin-3) VVER-1000 [16].

For Exercise II-1 the available experimental data has been identified using the CRISSUE-

S database. The Halden in-pile fuel temperature data have been examined for appropriate

19

transient fuel thermal property data. For the steady-state cases, FRAPCON is utilized to generate

reference results relevant to the studies in this exercise while for transient cases, FRAPTRAN is

used to generate reference solutions and study the problem definition for transient simulations.

The first six cases listed (cases 1a through 3b) are the numerical cases, which means that

they are based off of experiments but the dimensions and other important parameters have been

modified in order to represent the three reactors of interest in this phase (PB-2, TMI-1, and

Kozloduy-6). There is a steady-state irradiation case and a transient case for each of these three

reactors. The steady-state cases provide small changes in the power over a large amount of time,

often on the scale of months.

The transient cases (cases 1b through 6b) are often quick pulses on the order of

milliseconds, and the total duration of the simulation is usually a few seconds or less. Some of the

experimental transient cases occur after a long period of irradiation, such that the fuel rod will

have a specified value of burnup at the onset of the scenario, such as 65 GWd/MTU.

For the steady-state cases, the main parameter of interest is the centerline fuel

temperature at various burnup steps [13]. For some experiments, fuel temperature was not

directly measured so cladding temperature is also relevant. There are several dimensions and

values that the users are asked to modify in order to determine the effect of the changes on the

resulting temperatures. The transient cases also focus on determining the temperature profile at

each time step of the simulation.

The major uncertainty input parameters, their ranges of change and associated PDFs are

summarized below. These parameters include manufacturing uncertainties, uncertainties in the

boundary conditions, and uncertainties in the parameters in the code models.

The manufacturing uncertainties are specified for each test case. Uncertainties of

boundary conditions are the same for all test cases as shown in Table 5. The uncertainties of

20

boundary conditions (these include the coolant flow rate and temperature, as well as the reactor

pressure and power level) have normal distribution around the average value.

Table 5: Exercise II-1 Core Boundary Condition Variations

Parameter BWR PWR VVER

Coolant Flow Rate ±2.0% ±2.0% ±2.0%

Coolant Inlet Temperature ±10 K ±10 K ±10 K

Core Pressure ±3.0% ±3.0% ±3.0%

Power ±5.0% ±5.0% ±5.0%

The users are also asked to modify the inputs to their codes or the source of their codes in

order to represent some code and model uncertainties. The code and model uncertainties are

given in Table 6 and follow the normal distribution also.

Table 6: Exercise II-1 Code Parameter Variations [14]

Parameter BWR PWR VVER

Fuel Thermal Conductivity ±0.5 W/m-K ±0.5 W/m-K ±0.5 W/m-K

Fuel Thermal Expansion ±15% ±15% ±15%

Cladding Thermal Conductivity ±5 W/m-K ±5 W/m-K ±5 W/m-K

Cladding Thermal Expansion ±30% ±30% ±30%

Gas Conductivity ±0.02 W/m-K ±0.02 W/m-K ±0.02 W/m-K

Heat Transfer Coefficient ±5.0% ±5.0% ±5.0%

The uncertainties in nodalization and time steps will not be studied during these this

exercise, but if the user chooses to implement a different nodalization scheme then that should be

noted in the results. The nodalization can have a significant effect on the propagated uncertainty

parameters and its variation could be studied further.

21

For each test case, the geometry and important parameters of the fuel pin are provided, as

well as the core boundary conditions and an axial power profile. This axial power profile is

assumed to remain constant during the transient cases. The steady-state cases have a defined

irradiation history and the transient cases have a defined pulse power history.

2.2.1 – Case 1a: Steady-State BWR Numerical Test Problem

This case is modeled after a PNL assessment case written for the FK-1 reactor [5, 12, 23,

25]. It has been modified for application to the PB-2 reactor, a typical BWR representative [7].

The case should be run with the parameters listed below, and then with the variations provided by

the manufacturing tolerances for a BWR. The participants should apply the code uncertainties in

order to determine their effects as well. The geometry of the PB-2 Fuel rod is defined in Table 7

and shown in Figure 2.

22

Table 7: PB-2 Fuel Rod Geometry

Geometry Value

Cladding OD 14.30 mm

Cladding ID 12.42 mm

Cladding wall thickness 0.940 mm

Fuel pellet OD 12.12 mm

Pellet-cladding radial gap thickness 0.15 mm

Total fuel column length 3657.6 mm

Fuel pellet height 10.67 mm

Fuel enrichment (atom percent) 3.0%

% of theoretical density (10.96 g/cc) 95.1%

Pellet surface roughness 2.0 μm

Cladding type Zr-2

Cladding surface roughness 0.5 μm

Fill gas type Helium

Fill gas pressure 0.69 MPa

Fuel rod pitch 18.75 mm

Coolant pressure 7.14 MPa

Coolant inlet temperature 550 K

Coolant mass flux around rod 3460 kg/m2-s

Heat transfer coefficient for coolant 2.0e6 W/m2-

K

# of time steps 50

# of axial nodes in pellet 17

# of equal volume radial rings in pellet 45

23

Figure 2: PB-2 Fuel Pin Image

The irradiation history to be used for this case is fairly flat and includes a brief ramp-up

in the beginning to get to power as given in Table 8 and shown in Figure 3. The power is given as

a linear heat rate, which can be applied in conjunction with the axial power profile. The axial

power profile applied to this BWR test problem is provided in Table 9 and shown in Figure 4.

The axial nodalization consists of 17 axial nodes.

24

Table 8: PB-2 Power History

Time Step

-

Time

(days)

Power

(kW/m)

Time Step

-

Time

(days)

Power

(kW/m)

1 0.0 0.00 26 1050 16.55

2 0.1 3.28 27 1100 16.55

3 0.2 6.56 28 1150 16.55

4 0.3 9.84 29 1200 16.55

5 0.4 13.12 30 1250 13.39

6 50 15.70 31 1300 13.39

7 100 15.70 32 1350 13.39

8 150 15.70 33 1400 13.39

9 200 15.70 34 1450 13.39

10 250 15.70 35 1500 13.39

11 300 17.32 36 1550 13.39

12 350 17.32 37 1600 13.39

13 400 17.32 38 1650 13.39

14 450 17.32 39 1700 13.39

15 500 17.32 40 1750 15.16

16 550 17.32 41 1800 15.16

17 600 17.32 42 1850 15.16

18 650 17.32 43 1900 15.16

19 700 17.32 44 1950 15.16

20 750 16.55 45 2000 15.16

21 800 16.55 46 2050 15.16

22 850 16.55 47 2100 15.16

23 900 16.55 48 2150 15.16

24 950 16.55 49 2200 15.16

25 1000 16.55 50 2230 15.16

25

0 500 1000 1500 20000

2

4

6

8

10

12

14

16

18

20

PB-2 Irradiation History

Time (days)

Pow

er

(kW

/m)

Figure 3: PB-2 Power History Plot

26

Table 9: PB-2 Axial Power Profile

Location

(mm)

Relative Power

-

0.0 0.70

228.6 0.82

457.2 0.91

685.8 1.00

914.4 1.09

1143.0 1.13

1371.6 1.13

1600.2 1.13

1828.8 1.13

2057.4 1.13

2286.0 1.13

2514.6 1.13

2743.2 1.09

2971.8 1.00

3200.4 0.91

3429.0 0.82

3657.6 0.70

27

0 500 1000 1500 2000 2500 3000 3500

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

PB-2 Axial Power Profile

Axial Position (mm)

Rela

tive P

ow

er

Figure 4: PB-2 Axial Power Profile Plot

The manufacturing uncertainties to apply to this case are shown in Table 10. These

values are specific to this BWR configuration.

Table 10: Case 1 Manufacturing Uncertainties

Parameter Lower Limit Upper Limit Std. Dev. Distribution

Cladding ID 12.38 mm 12.46 mm 0.0133 mm Normal

Cladding Thickness 0.936 mm 0.944 mm 0.00133 mm Normal

Cladding Roughness 0.2 μm 0.8 μm 0.10 μm Normal

Fuel Pellet OD 12.10 mm 12.14 mm 0.00667 mm Normal

Fuel Density 94.2% 96.0% 0.30% Normal

Fuel Pellet Roughness 1.5 μm 2.5 μm 0.167 μm Normal

Rod Fill Pressure 0.62 MPa 0.76 MPa 0.0233 MPa Normal

28

2.2.2 – Case 1b: Transient BWR Numerical Test Problem

This case is also modeled off of a FK-1 rod case and modified for the PB-2 reactor. It is a

short-duration pulse which causes a spike in the power level of the reactor. It is simulated in this

case as a single fuel pin. The geometry of the pin is the same as described in the previous case

(Case 1a, Table 7) and the initial burnup of the rod used in this simulation is 45.3 GWd/MTU

(since it is assumed to be previously irradiated). The power history of the transient is provided in

Table 11 and shown in Figure 5. The duration of the scenario is 1.00 seconds.

29

Table 11: PB-2 Transient Power History

Time

(s)

Power

(kW/m)

0.000 0.0

0.195 0.0

0.196 200.0

0.197 400.0

0.198 1000.0

0.199 1500.0

0.200 2500.0

0.201 10000.0

0.202 21000.0

0.203 45000.0

0.204 77500.0

0.205 95000.0

0.206 86000.0

0.207 60000.0

0.208 35000.0

0.209 20000.0

0.210 10000.0

0.211 5000.0

0.212 1500.0

0.213 1000.0

0.214 500.0

0.215 200.0

0.216 100.0

0.217 0.0

1.000 0.0

30

0.19 0.193 0.196 0.199 0.202 0.205 0.208 0.211 0.214 0.217 0.220

2

4

6

8

10

x 104 PB-2 Transient Pulse

Time (s)

Pow

er

(kW

/m)

Figure 5: PB-2 Transient Power History Plot

The time step sizes to use during the simulation are defined in Table 12. These are

important for obtaining consistent results.

Table 12: PB-2 Transient Time Step Sizes

Step Size Time Period

0.001 s 0.00 - 0.15 s

0.00001 s 0.15 - 0.25 s

0.001 s 0.25 - 1.00 s

31

The coolant temperature is also provided for this case as a function of time in Table 13.

These can be applied as boundary conditions as needed. The axial power profile and

manufacturing uncertainties are the same as used in Case 1a.

Table 13: PB-2 Transient Coolant Temperature History

Time

(s)

Coolant T

(K)

0.00 305

0.21 305

0.25 393

0.35 436

0.50 510

0.70 452

0.85 452

1.00 383

32

2.2.3 – Case 2a: Steady-State PWR Numerical Test Problem

This case is based on the Na-3 experiments performed at the CABRI test reactor facility

[6, 8]. This case is modified to fit the parameters of the TMI-1 reactor. This is a single pin model

of an irradiation case over a long time period. The geometry of the TMI-1 fuel pin cell is

provided in Table 14 and shown in Figure 6.

Table 14: TMI-1 Fuel Rod Geometry

Geometry Value


Cladding ID 9.58 mm









Cladding type Zr-4



Fill gas pressure 1207 kPa




Coolant mass flux around rod 3460 kg/m2-s

Heat transfer coefficient for coolant 2.0e6 W/m2-K

# of time steps 31



33

Figure 6: TMI-1 Fuel Pin Image

The irradiation history of this pin model is provided in the following Table 15 and Figure

7.

34

Table 15: TMI-1 Power History

Time Step

-

Time

(days)

Power

(kW/m)

Time

Step

-

Time

(days)

Power

(kW/m)

1 0.0 0.00 17 500 23.39

2 0.1 3.28 18 550 23.39

3 0.2 6.56 19 580 23.39

4 0.3 9.84 20 600 19.69

5 0.4 13.12 21 650 19.69

6 0.5 16.40 22 700 19.69

7 0.6 19.69 23 750 19.69

8 50 23.39 24 800 19.69

9 100 23.39 25 850 19.69

10 150 23.39 26 900 19.69

11 200 23.39 27 950 19.69

12 250 23.39 28 1000 19.69

13 300 23.39 29 1050 19.69

14 350 23.39 30 1100 19.69

15 400 23.39 31 1143 19.69

16 450 23.39 -

35

0 200 400 600 800 10000

5

10

15

20

25

TMI-1 Irradiation History

Time (days)

Pow

er

(kW

/m)

Figure 7: TMI-1 Power History Plot

The axial power profile of the TMI-1 fuel rod is given in Table 16 and is shown in Figure

8 as a 13- node model.

36

Table 16: TMI-1 Axial Power Profile

Location

(mm)

Relative Power

-

0.0 0.63

304.8 0.83

609.6 1.03

914.4 1.08

1219.2 1.08

1524.0 1.08

1828.8 1.08

2133.6 1.08

2438.4 1.08

2743.2 1.08

3048.0 1.03

3352.8 0.83

3657.6 0.63

0 500 1000 1500 2000 2500 3000 3500

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

TMI-1 Axial Power Profile

Axial Position (mm)

Rela

tive P

ow

er

Figure 8: TMI-1 Axial Power Profile Plot

37


values are specific to this PWR numerical case configuration.






Gap Thickness 0.065 mm 0.113 mm 0.0080 mm Normal

U235

Enrichment 4.847 w/o 4.853 w/o 0.0010 w/o Normal

38

2.2.4 – Case 2b: Transient PWR Numerical Test Problem

This is a pulse modeled on the Na-3 pulse test completed at the CABRI test facility. The

actual rod used is shorter than a typical PWR fuel rod found in power reactors so the dimensions

have been scaled up to fit the TMI-1 reactor. The geometry of the fuel pin is the same as given in

Case 2a. The duration of this transient is 0.400 seconds. The transient power history is given in

Table 18 and Figure 9 while the step sizes to use during the calculations are also provided in

Table 19.

Table 18: TMI-1 Transient Power History

Time (s)

Power (kW/m)

0.000 0.0

0.060 0.0

0.065 336.9

0.070 1347.8

0.075 8087.3

0.082 25339.6

0.084 18870.1

0.087 10782.8

0.090 3369.8

0.095 269.7

0.100 82.3

0.400 0.0

39

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.120

0.5

1

1.5

2

2.5

3

x 104 TMI-1 Transient Pulse

Time (s)

Pow

er

(kW

/m)

Figure 9: TMI-1 Transient Power History Plot

Table 19: TMI-1 Transient Time Step Sizes


0.0001 s 0.0000 – 0.0800 s

0.00001 s 0.0800 – 0.0815 s

0.0001 s 0.0815 – 0.1500 s

0.001 s 0.1500 – 0.4000 s

The coolant temperature is also provided for this case as a function of time, for two

locations along the fuel pin as shown in Table 20. The lower location is at 887.5 mm and the

upper measurement is at the top, at 3550 mm. These can be applied as boundary conditions as

needed.

40

Table 20: TMI-1 Transient Coolant Temperature History

Time

(s)

Coolant T

Lower

(K)

Coolant T

Upper

(K)

0.000 553 553

0.075 553 553

0.100 568 563

0.150 628 643

0.200 668 683

0.250 668 703

0.300 658 708

0.350 650 708

0.400 642 703

The axial power profile and manufacturing uncertainties are the same as used in Case 2a.

41

2.2.5 – Case 3a: Steady-State VVER Numerical Test Problem

This irradiation case for a VVER-1000 fuel rod has its geometry defined by Table 21 and

shown in Figure 10 [31, 32, 33, 34]. It is important to note that the VVER fuel pins and

assemblies have hexagonal geometry as well as a central void in the fuel pellet, making the

VVER cases slightly different than the other LWR cases. Some modifications to codes may be

necessary in order to accurately represent these changes.

Table 21: VVER-1000 Fuel Rod Geometry

Geometry Value

Cladding OD 9.10 mm

Cladding ID 7.72 mm



Fuel pellet ID 2.40 mm


Total fuel column length 3550 mm




Pellet surface roughness 2 μm

Cladding type Zr-1% Nb


Fill gas type He

Fill gas pressure 2.5 Mpa


Coolant pressure 15.7 Mpa


# of time steps 41

42

Figure 10: VVER-1000 Fuel Pin Image

The fuel pin model is irradiated according to the data in Table 22 and Figure 11.

43

Table 22: VVER-1000 Power History

Time Step -

Time (days)

Power (kW/m)

Time Step -

Time (days)

Power (kW/m)

1 0 0.0 22 350 12.6

2 0.1 1.5 23 400 12.6

3 0.2 3.0 24 450 12.6

4 0.3 4.5 25 500 12.6

5 0.4 6.0 26 550 12.6

6 0.5 7.5 27 600 12.6

7 0.6 10.0 28 635 12.6

8 5 12.6 29 640 8.0

9 50 12.6 30 685 8.0

10 55 12.6 31 690 5.4

11 58 8.2 32 700 5.4

12 80 8.2 33 745 5.4

13 82 12.6 34 750 4.9

14 100 12.6 35 800 4.9

15 135 12.6 36 805 4.9

16 140 8.2 37 810 12.6

17 180 8.2 38 850 12.6

18 185 12.6 39 900 12.6

19 200 12.6 40 915 12.6

20 250 12.6 41 920 0.0

21 300 12.6 -

44

0 100 200 300 400 500 600 700 800 9000

5

10

15

VVER-1000 Irradiation History

Time (days)

Pow

er

(kW

/m)

Figure 11: VVER-1000 Power History Plot

The axial power profile for the rod in this case is supplied in Table 23 and shown in

Figure 12.

45

Table 23: VVER-1000 Axial Power Profile

Location

(mm)

Relative Power

-

0.0 0.426

355.0 0.735

710.0 1.014

1065.0 1.235

1420.0 1.377

1775.0 1.426

2130.0 1.377

2485.0 1.235

2840.0 1.014

3195.0 0.735

3550.0 0.426

0 500 1000 1500 2000 2500 3000 3500

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

VVER-1000 Axial Power Profile

Axial Position (mm)

Rela

tive P

ow

er

Figure 12: VVER-1000 Axial Power Profile Plot

46


values are specific to this VVER numerical case configuration.


Parameter Lower Limit Upper Limit Std.Dev. Distribution

Fuel Pellet Void ID 2.40 mm 2.70 mm 0.05 mm Normal



Cladding OD 8.95 mm 9.15 mm 0.0333 mm Normal

Fuel Density 92.2 g/cc 97.6% 0.91% Normal

3.3 w/o Enrichment 3.25% 3.35% 0.0167% Normal

3.0 w/o Enrichment 2.95% 3.05% 0.0167% Normal

47

2.2.6 – Case 3b: Transient VVER Numerical Test Problem

This transient case is another short-duration power pulse resulting from a reactivity-

initiated accident scenario which is typically a rod-ejection. The geometry of this rod is the same

as in Case 3a, and the power history of the scenario is provided in Table 25and is shown in Figure

13. The duration of the pulse is 8.00 seconds.

Table 25: VVER-1000 Transient Power History

Time (s)

Power (kW/m)

0.00 0.10

1.20 0.7

1.82 26.2

2.22 52.5

2.87 164.0

3.35 383.9

3.67 213.3

4.06 75.5

4.50 26.2

4.92 13.1

6.22 4.9

8.00 1.6

48

0 1 2 3 4 5 6 7 80

50

100

150

200

250

300

350

400

450

VVER-1000 Transient Pulse

Time (s)

Pow

er

(kW

/m)

Figure 13: VVER-1000 Transient Power History Plot

Table 26: VVER-1000 Transient Time Step Sizes


0.002 s 0.00 - 1.50 s

0.0002 s 1.50 - 3.45 s

0.00005 s 3.45 - 4.20 s

0.0005 s 4.20 - 6.00 s

0.001 s 6.00 - 8.00 s

The transient time steps are given in Table 26. The coolant temperature is not available

for this case. The axial power profile and manufacturing uncertainties are the same as the one

used in Case 3a.

49

2.2.7 – Case 4a: Steady-State BWR Experimental Test Problem

This experimental case is the IFA-432 rod irradiation performed at the Halden reactor in

Norway, a BWR [22]. The enrichment of this rod is slightly higher (10%) than those typically

found in LWRs but otherwise it is similar to a standard fuel rod. It is also shorter in length, but

the diameter and materials are appropriate. The geometry of this case is provided in Table 27 and

is shown in Figure 14.

Table 27: IFA-432 Fuel Rod Geometry

Geometry Value











Cladding type Zr-2



Fill gas pressure 101.3 kPa

Fuel rod pitch 14.22



# of time steps 50



50

Figure 14: IFA-432 Fuel Pin Image

The irradiation history of this rod has been simplified in order to bring it to a manageable

number of time steps as provided in Table 28 and shown in Figure 15. The original power versus

time history had much variation which has been smoothed out in order to preserve the overall

irradiation effect.

51

Table 28: IFA-432 Power History

Time Step

-

Time

(days)

Power

(kW/m)

Time Step

-

Time

(days)

Power

(kW/m)

1 0.00 0.00 26 355.06 28.97

2 0.10 3.28 27 377.67 28.85

3 0.20 6.56 28 392.18 29.86

4 0.30 9.84 29 413.05 30.71

5 0.40 13.12 30 437.73 32.11

6 0.50 16.40 31 475.03 31.08

7 2.23 23.43 32 500.15 27.63

8 28.97 30.82 33 524.51 25.23

9 53.70 26.14 34 536.26 26.77

10 63.14 26.53 35 563.16 31.02

11 97.94 32.45 36 590.49 29.80

12 125.17 33.07 37 617.39 24.57

13 132.06 33.15 38 622.89 21.73

14 152.43 29.21 39 655.36 24.08

15 157.86 27.38 40 673.66 25.43

16 163.56 30.54 41 680.78 25.47

17 181.05 31.05 42 693.35 27.04

18 196.11 27.96 43 705.61 26.87

19 206.87 28.56 44 715.82 26.92

20 231.83 29.63 45 729.09 27.60

21 257.14 29.96 46 754.66 27.68

22 275.96 30.20 47 772.92 27.58

23 300.91 33.29 48 783.91 28.66

24 317.30 31.35 49 791.76 25.28

25 328.50 30.72 50 809.80 0.00

52

0 100 200 300 400 500 600 700 8000

5

10

15

20

25

30

35

40

IFA-432 Irradiation History

Time (days)

Pow

er

(kW

/m)

Figure 15: IFA-432 Power History Plot

The axial power profile is an assumed shape since the original rod is fairly short axially.

These values are given in Table 29 and shown in Figure 16.

53

Table 29: IFA-432 Axial Power Profile

Location

(mm)

Relative Power

-

0.00 0.70

34.87 0.82

69.74 0.91

104.61 1.00

139.48 1.09

174.34 1.13

209.21 1.13

244.08 1.13

278.95 1.13

313.82 1.13

348.69 1.13

383.56 1.13

418.43 1.09

453.29 1.00

488.16 0.91

523.03 0.82

557.90 0.70

54

0 100 200 300 400 500

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

IFA-432 Axial Power Profile

Axial Position (mm)

Rela

tive P

ow

er

Figure 16: IFA-432 Axial Power Profile Plot


values are specific to this BWR experimental case configuration.





Cladding Roughness 0.2 μm 0.8 μm 0.1 μm Normal



Fuel Pellet Roughness 1.5 μm 2.5 μm 0.167 μm Normal

Rod Fill Pressure 0.08 MPa 0.12 MPa 0.0667 MPa Normal

55

2.2.8 – Case 4b: Transient BWR Experimental Test Problem

This case is modeled after the FK-1 pulse test performed at the Fukushima Daini BWR

[23]. The pulse is one in a series of short-term RIA tests which were performed after an initial

irradiation of the fuel. The initial burnup is equal to 45.5 GWd/MTU. The geometry and

dimensions of the FK-1 fuel rod are provided in Table 31.

Table 31: FK-1 Fuel Rod Geometry

Geometry Value











Cladding type Zr-2







56

The duration of the test is 1.000 seconds. The pulse’s power history is provided in Table

32 and is shown in Figure 17.

Table 32: FK-1 Transient Power History

Time (s)

Power (kW/m)

0.000 0.0

0.195 0.0

0.196 200.0

0.197 400.0

0.198 1000.0

0.199 1500.0

0.200 2500.0

0.201 10000.0

0.202 21000.0

0.203 45000.0

0.204 77500.0

0.205 95000.0

0.206 86000.0

0.207 60000.0

0.208 35000.0

0.209 20000.0

0.210 10000.0

0.211 5000.0

0.212 1500.0

0.213 1000.0

0.214 500.0

0.215 200.0

0.216 100.0

0.217 0.0

1.000 0.0

57

0.19 0.195 0.2 0.205 0.21 0.215 0.220

2

4

6

8

10

x 104 FK-1 Transient Pulse

Time (s)

Pow

er

(kW

/m)

Figure 17: FK-1 Transient Power History Plot

The coolant temperature history can also be applied to the pulse test. The values are

provided in Table 33. The transient time steps are shown in Table 34.

58

Table 33: FK-1 Transient Coolant Temperature History

Time

(s)

Coolant T

(K)

0.00 305

0.21 305

0.25 393

0.35 436

0.50 510

0.70 452

0.85 452

1.00 383

Table 34: FK-1 Transient Time Step Sizes


0.001 s 0.00 - 0.15 s

0.00001 s 0.15 - 0.25 s

0.001 s 0.25 - 1.00 s

The axial power profile is assumed to be flat, due to the small axial dimension (106.0

mm). The manufacturing uncertainties are the same as in Case 4a.

59

2.2.9 – Case 5a: Steady-State PWR Experimental Test Problem

The IFA-429 experiment is also included in the IFPE Database and it has been slightly

modified in order to make usable input values [24]. The main parameters for this case are shown

in Table 35 and Figure 18, which document the geometry of the fuel rod. The irradiation history

of this rod is provided in Table 36 and is shown in Figure 19. The axial power is given in Table

37 and shown in Figure 20.

Table 35: IFA-429 Fuel Rod Geometry

Geometry Value


Cladding ID 9.50 mm








Cladding type Zr-4





Coolant pressure 15 MPa


# of time steps 46



60

Figure 18: IFA-429 Fuel Rod Image

61

Table 36: IFA-429 Power History

Time Step -

Time (days)

Power (kW/m)

Time Step -

Time (days)

Power (kW/m)

1 0.0 0.00 24 886.6 24.25

2 0.1 5.39 25 958.4 19.65

3 0.2 10.77 26 1011.6 20.31

4 0.3 16.16 27 1021.9 19.86

5 0.4 21.54 28 1043.5 22.54

6 29.6 26.93 29 1079.8 22.13

7 81.6 22.82 30 1115.3 21.41

8 102.2 24.31 31 1178.7 24.09

9 130.7 20.27 32 1248.3 21.35

10 186.6 24.35 33 1298.9 21.35

11 221.6 24.77 34 1340.0 17.55

12 302.8 20.82 35 1377.8 19.39

13 392.5 22.30 36 1409.9 19.90

14 422.7 24.42 37 1444.6 17.78

15 463.3 24.47 38 1482.9 12.43

16 498.2 25.55 39 1514.7 11.90

17 543.2 21.53 40 1577.1 15.14

18 644.5 17.98 41 1608.9 21.74

19 708.0 26.08 42 1641.6 18.43

20 745.2 23.03 43 1688.2 21.62

21 775.7 24.06 44 1724.6 19.93

22 800.5 24.11 45 1763.6 17.91

23 829.2 22.47 46 1784.4 0.00

62

0 200 400 600 800 1000 1200 1400 16000

5

10

15

20

25

30

IFA-429 Irradiation History

Time (days)

Pow

er

(kW

/m)

Figure 19: IFA-429 Power History Plot

Table 37: IFA-429 Axial Power Profile

Location

(mm)

Relative Power

-

0.0 0.93

24.3 0.96

48.7 1.00

73.0 1.03

97.4 1.03

121.7 1.03

146.0 1.03

170.4 1.03

194.7 1.00

219.1 0.96

243.4 0.93

63

0 50 100 150 200

0.85

0.9

0.95

1

1.05

IFA-429 Axial Power Profile

Axial Position (mm)

Rela

tive P

ow

er

Figure 20: IFA-429 Axial Power Profile Plot


values are specific to this PWR experimental case configuration.

64

Table 38: Case 5 Manufacturing Tolerances





Gap Thickness 0.08 mm 0.12 mm 0.00667 mm Normal

U235 Enrichment 12.95 w/o 13.05 w/o 0.0167 w/o Normal

Case 5b - the transient PWR experimental test problem, Case 6a - the steady-state VVER

experimental test problem, and Case 6b - the transient VVER experimental test problem are not

ready at this moment and will be provided in Version 2.0 of the Specification on Phase II.

65

Chapter 3 : Code Descriptions

Fuel performance codes are used to simulate nuclear processes which involve the fuel

inside of the core. The fuel may be analyzed under various situations such as typical irradiation or

its behavior can be simulated during abnormal events such as RIAs. These codes often contain

detailed information on materials and parameters within the fuel pins and are able to model the

properties as they change over the course of the simulation. The code user has many options to

specify various correlations within the code’s calculation. The nuclear fuel performance codes

used in this analysis are the Pacific Northwest National Lab’s (PNL) FRAPCON and

FRAPTRAN. These are both single fuel pin codes capable of calculating various fuel and

mechanical properties of the user-specified fuel geometry. The fuel pins are modeled and the

target parameter, the fuel centerline temperature, is recorded from the generated output files.

3.1 – FRAPCON

Steady-state reactor operation is analyzed with FRAPCON. This code calculates cladding

and fuel temperatures, various fuel properties and dimensions, and more, based on a variety of

inputs that the user specifies [19]. Inputs include the time steps for the program, properties of the

pellet such as its dimensions and density, the nodalization of the geometry, and the fuel

properties, the mole fractions of the chemicals present in the simulation, fuel enrichment, rod

dimensions, and other geometries. There are also various correlations that can be activated

depending on what is most accurate for the simulation. These correlations include more detailed

material properties and various models that track things such as cladding degradation. Also, many

66

included subroutines in FRAPCON can give very detailed results for certain parameters, mostly

physical properties of the fuel and cladding. During calculations, FRAPCON takes into account

many phenomena such as heat conduction, deformation, mechanical interaction of cladding and

fuel, fission gas release, internal gas pressure, oxidation and heat transfer through the rod to the

coolant. A sample of a PB-2 steady-state FRAPCON input file is shown in Appendix A.1.

3.2 – FRAPTRAN

The transient code is FRAPTRAN, which also models a single fuel rod but is designed

for more rapid changes in boundary conditions such as a pulse in power. These scenarios cause

much more fluctuation in the calculated parameters. The time steps are much smaller than those

used for the steady-state calculations, often on the order of 0.001 seconds or smaller. This code

can calculate data for time-dependent rod power scenarios [20, 21]. It provides radial and axial

temperature distributions as well as time dependence for the fuel rods analyzed. It is also useful

for determining physical parameters of the fuel such as diameter and fuel-cladding gap thickness,

the internal pressure of the fuel rod gases, the heat transfer coefficient at the cladding surface,

strains and stresses, and more details. The types of abnormal operational scenarios for which this

code is useful are in reactivity-initiated accidents (RIAs) such as rod ejection/drops and loss-of-

coolant accidents (LOCAs) [9]. The codes are capable of predicting deformation history of the

fuel rods as a function of time-dependent rod parameters given as inputs. An example PB-2

transient FRAPTRAN input file is shown in Appendix A.2. The output utilized from

FRAPTRAN is the optional plot file, which is designed to be used in a spreadsheet program for

easy plotting of various parameters at the specified time steps. Within this file is a line which

contains the fuel centerline temperature, and this value is read and written to a file for use within

the uncertainty analysis code.

67

3.3 – DAKOTA

Uncertainty analysis codes are coupled with these fuel performance codes in order to

perform a study on the effects of uncertainty on the fuel centerline temperature. These codes

compile and analyze the results in order to provide a statistical summary of the cases. Sandia

National Laboratory’s DAKOTA is used in this study, which is focused on confidence intervals

that are automatically generated within the code [28]. It is capable of being coupled with other

modeling codes such as FRAPCON and FRAPTRAN and relying on data produced by them. Its

main use in this study is to vary the input parameters on which the uncertainty study is performed

and to record the resulting target parameter values. Within DAKOTA, the parameters are defined

using their distribution types (such as normal or uniform) along with the values for mean and

standard deviation of each varied parameter. There is also the option to specify upper and lower

bounds which is utilized in order to prevent values outside of the provided uncertainty ranges.

The parameters can be sampled with Monte Carlo methods where the user supplies a seed. The

output of DAKOTA is customizable and provides statistical information relevant to this study.

The 95% confidence intervals as well as correlation matrices are provided for each of the

predicted target parameters. The DAKOTA input file used for the PB-2 transient case is shown in

Appendix A.3. An example of the DAKOTA output file from the PB-2 transient case is in

Appendix A.4.

3.4 – Other Scripts

In order to complete the code coupling between DAKOTA and the fuel performance

codes, there have to be several scripts which relay information from output files and store them

into another file. There is also a batch script which tells the computer which files to run and

68

which programs with which to run them. For this study, MATLAB creates the input files for

either FRAPCON or FRAPTRAN using the parameters file which is directly generated by

DAKOTA. After the fuel performance calculation is complete, Python searches for the target

parameter within the output file and stores it into a results file which is read by DAKOTA and

used to create a tabulated list of each case and its results. The batch script allows for slower codes

by pausing for several seconds in between steps in order to allow calculations to finish and

prevent overwriting the wrong files. Within the DAKOTA input file the driver batch script, as

well as the generated parameters file, are specified. These files are the ones to be read and written

by the other scripts during the coupling. The batch script and the Python script is modeled off of

C. Tracy’s scripts [29] and are adapted in order to work for both steady-state and transient cases.

In the transient cases, the fuel centerline temperature is requested at four different time steps, so

the Python script must store an array of these four values and pass them all back to DAKOTA

which will tabulate all four temperatures along with the values of each parameter in the current

case. An example of the PB-2 transient batch script is shown in Appendix A.5. The several

MATLAB files used in the PB-2 transient case are shown in Appendix A.6. The Python script for

the PB-2 transient case is provided in Appendix A.7. The cases were run using the command

prompt (cmd.exe) on a PC. A flowchart showing how the different scripts and codes are coupled

is shown in Figure 21.

69

Figure 21: Code Flowchart

70

Chapter 4 : Steady-State Case Results

Cases 1a and 2a of the UAM Phase II Benchmark, in Exercise 1, are the steady-state

cases which are long-term irradiations of the fuel pins. Case 1a corresponds to the BWR

simulation (PB-2) and Case 2a corresponds to the PWR simulation (TMI-1). These cases were

run using DAKOTA as the main driver and each were set to run for 100 cases. There was a single

target parameter, the fuel centerline temperature, which was recorded at each trial. The input data

and uncertainties were all taken directly from the Phase II report [30] and only manufacturing

tolerances and boundary conditions were varied in this study. The DAKOTA output provides the

95% confidence intervals on the mean value of the fuel centerline temperature as well as on the

standard deviation of this value. There are also partial correlation matrices for each case which

show the correlation of each of the input parameter on the target parameter. The maximum fuel

centerline temperatures were also recorded at each axial node for these two cases. This provides

data which show where along the fuel rod the input uncertainty propagates the most on the output

parameter.

4.1 – Peach Bottom Unit 2 Steady-State Results

The input parameters were varied according to the data found in Table 5 and Table 10. A

summary of the case is shown in Table 39.

71

Table 39: PB-2 Steady-State Case Summary

Reactor: PB-2 (BWR)

Case: Steady-State

Samples: 100

Random Seed (DAKOTA): 17

Parameters Varied: 10

Target Parameter: Maximum Centerline Temperature

Sample Mean: 1164.2 K

Sample Standard Deviation: 21.37 K

The 95% confidence intervals on the calculated mean and standard deviation from the

100 cases run are shown below in Table 40. These values are determined to have 95% confidence

due to the number of sample cases that were run.

Table 40: PB-2 Steady-State Confidence Intervals

Lower Bound Upper Bound

Mean 1160.0 K 1168.5 K

Standard Deviation 18.77 K 24.83 K

The maximum and minimum values recorded during the simulation are shown in Table

41 as a comparison to the mean value and to show the range of the maximum fuel centerline

temperatures recorded during the analysis. These values are important in showing how much of

an effect the uncertainties of the inputs can have on the output. This shows a range of nearly 100

Kelvin, or a change of -4.9% from the sample mean.

Table 41: PB-2 Steady-State Maximum and Minimum Values

Value % from Sample Mean

Minimum Value 1106.7 K -4.94%

Maximum Value 1204.6 K 3.47%

The partial correlation matrix, shown in Table 42, shows how variations of each

individual input parameter affect the target parameter. A value of larger magnitude (either

72

positive or negative) shows a stronger correlation than those that are closer to zero. In Table 42

the parameters with the most effect are the cladding-pellet gap thickness, the fuel density, and the

core power. These parameters are shown to strongly influence the maximum fuel centerline

temperature as calculated using FRAPCON.

Table 42: PB-2 Steady-State Partial Correlation Matrix

Input Parameter Partial Correlation to Output

Cladding Pellet Gap Thickness 0.964848

Cladding Thickness 0.073798

Fuel Density -0.967663

Cladding Roughness 0.049350

Fuel Pellet Roughness 0.240961

Fill Gas Pressure 0.143138

Coolant Flow Rate -0.098619

Coolant Inlet Temperature 0.530132

Core Pressure 0.087216

Core Power 0.996615

The resulting confidence intervals on the target parameter for the Peach Bottom steady-

state case can be further refined with more cases, but at a cost of calculation time. For the

purposes of this study, the 95% confidence on the 95% intervals is sufficient. Appendix A.8

shows the tabulated results from this case and what the variations for each parameter were. The

first six columns were the actual units of each parameter and the boundary conditions are

percentage changes (or change in Kelvin for the inlet temperature column) and the resulting target

parameter is shown in the final column.

The axial distribution of the confidence intervals on the mean and standard deviation of

the fuel centerline temperatures was also studied. This was performed by recording the maximum

temperature value at each of the 17 axial nodes in this case and performing a statistical analysis

on the results to determine the confidence intervals. Figure 22 below shows the mean values at

each axial node along with triangles representing the upper and lower bounds. Figure 23 shows

73

the standard deviation at each axial node as well as the 95% confidence intervals. The range of

the intervals for both mean and standard deviation increases towards the center of the fuel rod,

where temperatures are highest. This is the location that has the most uncertainty in its

calculations.

Figure 22: PB-2 Mean Fuel Centerline Temperatures at each Axial Node with 95% CIs

74

Figure 23: PB-2 Standard Deviation of Fuel Centerline Temperatures at each Axial Node with 95% CIs

4.2 – Three Mile Island Unit 1 Steady-State Results

The input parameters for the PWR steady-state case were varied according to the data

found in Table 5 and Table 17. A summary of this case is shown in Table 43.

75

Table 43: TMI-1 Steady-State Case Summary

Reactor: TMI-1 (PWR)

Case: Steady-State

Samples: 100




Sample Mean: 1437.0 K

Sample Standard Deviation: 37.60 K

The 95% confidence intervals on the population’s mean and standard deviation are shown

below in Table 44. These values are determined to have 95% confidence due to the number of

sample cases that were run.

Table 44: TMI-1 Steady-State Confidence Intervals


Mean 1429.5 K 1444.4 K


The maximum and minimum values of the target parameter calculated during the

simulation are given in Table 45 as a comparison to the mean value and to show the range of the

maximum fuel centerline temperatures recorded during this case’s analysis. The range of this case

is over 150 Kelvin for the limited sample size. The minimum value is -5.7% different from the

sample mean. This is slightly larger variation than was found in the BWR case.

Table 45: TMI-1 Steady-State Maximum and Minimum Values


Minimum Value 1354.9 K -5.71%


The parameters with the strongest effect on the target parameter, as shown in Table 46,

are the cladding-pellet gap thickness, the fuel density, the coolant inlet temperature, and the core

76

power. The variations in the other parameters are mostly strong, except for the fuel enrichment

which shows a weak influence on the calculated maximum fuel centerline temperature.

Table 46: TMI-1 Steady-State Partial Correlation Matrix


Cladding Pellet Gap Thickness 0.997441

Cladding Thickness 0.671459

Fuel Density -0.996046

Fuel Enrichment 0.029899

Coolant Flow Rate -0.486394

Coolant Inlet Temperature 0.967434

Core Pressure -0.339541

Core Power 0.998757

The Three Mile Island case has results similar to the Peach Bottom case. The main

differences are due to the different uncertainties which were applied from the manufacturing

tolerances, since the same boundary condition changes were made for each of these two cases.

Minor differences in the geometry of PWRs can also cause some of the observed differences in

the calculated values.

The axial profile of the mean fuel centerline temperature as well as the standard

deviations shows the same pattern as found in case 1a. The range of the confidence intervals is

maximized towards the axial center of the fuel rod, where temperatures are greatest. Figure 24

and Figure 25 show these effects at each of the 17 axial nodes in this TMI-1 steady-state case.

77

Figure 24: TMI-1 Mean Fuel Centerline Temperatures at each Axial Node with 95% CIs

78

Figure 25: TMI-1 Standard Deviation of Fuel Centerline Temperatures at each Axial Node with

95% CIs

79

Chapter 5 : Transient Case Results

The transient cases for Exercise II-1 of the UAM benchmark for both the BWR and PWR

are power pulses. The boundary conditions change rapidly over a short period of time due to the

introduced reactivity which causes a jump in power level at a slight delay due to the feedback of

the system. For these cases, the fuel centerline temperature is requested at four different times: at

the beginning of the scenario, at the beginning of the power pulse, at the time corresponding to

the peak of the pulse, and at the end of the scenario. The maximum fuel temperature is not

requested at a certain elevation, just the highest value at the four specified times. DAKOTA is

able to calculate the 95% confidence intervals for each of these four temperature values. The

partial correlation matrices also give insight into which input parameters have the largest impact

on the target parameters.

5.1 – Peach Bottom Unit 2 Transient Results

The definitions for the PB-2 transient case are provided in Chapter 2 and the parameters

that have uncertainty values applied will be the same as from the steady-state case. Now that

there are four measured values (owing to the four different time steps at which data are requested)

there will be four confidence intervals calculated. Table 47 below provides a summary of the case

as it was run in DAKOTA. There are four rows for the sample means and standard deviations

which correspond to the four times as given. The temperature starts fairly low (near the coolant

inlet temperature value) until the pulse occurs, around 0.196 seconds into the transient scenario.

80

Table 47: PB-2 Transient Case Summary

Reactor: PB-2 (BWR)

Case: Transient

Samples: 100




(at specified times)

Times Specified:

0.000 s

0.196 s 0.206 s

1.000 s

Sample Means:

Time 1: 304.9 K Time 2: 305.2 K

Time 3: 1090.8 K

Time 4: 1517.4 K

Sample Standard Deviations:

Time 1: 5.73 K Time 2: 5.73 K

Time 3: 21.10 K

Time 4: 31.77 K

The confidence intervals were also calculated for each of the fuel centerline temperatures.

These values are given in Table 48 and are specific to each time during the scenario. The

variation increases with time, as expected, and this is shown as the standard deviations increasing

in magnitude after the second specified time. Figure 26 shows the spacing between the lower and

upper standard deviation bounds (represented as triangles) increasing over the duration of the

scenario. This is due to the propagation of uncertainties throughout the time steps and cumulating

towards the end of the transient. There are more possible variations in the fuel centerline

temperatures because of the many initial values the input parameters could have been.

81

Table 48: PB-2 Transient Confidence Intervals


Time 1 (0.000 s) Mean 303.7 K 306.0 K


Time 2 (0.196 s) Mean 304.0 K 306.3 K


Time 3 (0.206 s) Mean 1086.6 K 1095.0 K


Time 4 (1.000 s) Mean 1511.1 K 1523.7 K


Figure 26: PB-2 Transient Standard Deviation Confidence Bound Plot

The minima and maxima of the fuel centerline temperatures at each time step are also

noted in order to determine the range of values observed during the simulation.

82

Table 49 shows these values grouped by time step. The percentage distance from the

sample mean increase as time steps increase, which is expected as it follows the same trend as the

standard deviations.

Table 49: PB-2 Transient Maximum and Minimum Values


Time 1 (0.000 s) Minimum Value 295.6 K -3.06%








It is important to note that the times at which the data are recorded remain the same for

each case. The shapes of the temperature versus time curves may vary between each case so these

times will not always correspond to the same location on these curves. The partial correlations are

also calculated by DAKOTA and the impact of each parameter on the four temperature

recordings is shown in Table 50 below. The input parameters with the strongest effect on the

temperature at each time are the coolant inlet temperature, the fuel density, and the fuel pellet

diameter. The impact of the fuel density and the fuel pellet diameter changes over time as they

have a stronger correlation at the last two time steps than at the initial two time steps.

83

Table 50: PB-2 Transient Partial Correlation Matrix


Time 1 Time 2 Time 3 Time 4

Cladding Pellet Gap Thickness 0.127053 -0.064659 -0.151142 -0.137891

Fuel Pellet Diameter 0.058286 -0.153242 -0.995304 -0.994876

Fuel Density -0.159086 -0.423037 -0.999398 -0.999343

Cladding Roughness -0.082599 -0.123127 0.022850 -0.036884

Fuel Pellet Roughness 0.022509 -0.123964 -0.001842 -0.021738

Fill Gas Pressure 0.027231 -0.090430 0.029198 0.073896

Coolant Flow Rate -0.059321 0.159477 -0.105244 -0.000633

Coolant Inlet Temperature 1.000000 1.000000 0.999836 0.999544

Core Pressure -0.140494 0.137556 -0.241585 -0.236111

Core Power 0.204141 0.958708 0.999993 0.999992

5.2 – Three Mile Island Unit 1 Transient Results

As with the BWR case, the specifications for this test case are provided in Chapter 2 and

are utilized to create the model for the transient test of a Three Mile Island fuel pin. This scenario

involves a rapid introduction of reactivity which causes an abrupt pulse in the power. The

duration of this scenario is 0.400 seconds, compared to the PB-2 transient which was 1.000

seconds in length. Table 51 shows the general characteristics of this case along with the

calculated means and standard deviations at each of the four time steps of interest. The beginning

of the pulse is at 0.065 seconds and the center of the pulse occurs at 0.082 seconds.

84

Table 51: TMI-1 Transient Case Summary

Reactor: TMI-1 (PWR)

Case: Transient

Samples: 100



Target Parameter: Maximum Centerline Temperature (at specified times)

Times Specified:

0.000 s

0.065 s

0.082 s 0.400 s

Sample Means:

Time 1: 553.3 K

Time 2: 556.8 K

Time 3: 1145.8 K Time 4: 1634.9 K

Sample Standard Deviations:

Time 1: 6.02 K

Time 2: 6.02 K Time 3: 16.58 K

Time 4: 28.08 K

Table 52 contains the lower and upper confidence bounds on the calculated temperatures

and their standard deviations. As in the BWR case, the variation increases with time. This

increase is shown in Figure 27 as the spacing between the triangles (which represent the upper

and lower confidence bounds on the standard deviations) increases during the last two time steps.

Table 52: TMI-1 Transient Confidence Intervals


Time 1 (0.000 s) Mean 552.1 K 554.5 K


Time 2 (0.065 s) Mean 555.6 K 558.0 K


Time 3 (0.082 s) Mean 1142.5 K 1149.1 K


Time 4 (0.400 s) Mean 1629.0 K 1640.2 K


85

Figure 27: TMI-1 Transient Standard Deviation Confidence Bound Plot

The extreme values from each of the four time steps are shown in Table 53 and the

percentage difference values are smaller in this PWR case than they were for the BWR case. This

could be due to the fact that there are fewer parameters being varied in this transient scenario,

with seven in this case as opposed to ten input parameters with variations in the BWR case. Also,

the amount of uncertainties in manufacturing tolerances has an effect on how much variation

from the mean value the minimum and maximum values will show. As before, the variations

86

increase in the latter two time steps as the propagation of uncertainty builds up and causes more

possible permutations of fuel temperatures.

Table 53: TMI-1 Transient Maximum and Minimum Values










Once again, the times at which the calculated fuel centerline temperatures are recorded

does not change even though the shapes of the power and temperature versus time curves might.

Table 54 provides information on the correlation between the input and output parameters and

shows which have the strongest effects. In this test case, the most influential input parameters are

the coolant inlet temperature, the core power, and the fuel density. Once again the fuel density

does not affect the temperature much until later in the scenario. The core power has the same

effect, due to the fact that the power is very low at the beginning of this scenario and does not

factor in until the reactivity-induced pulse begins at the second measurement time step.

Table 54: TMI-1 Transient Partial Correlation Matrix


Time 1 Time 2 Time 3 Time 4

Cladding Pellet Gap Thickness -0.237252 -0.237852 -0.236979 -0.237323

Fuel Pellet Diameter 0.203857 0.189461 -0.092999 -0.106738

Fuel Density -0.013317 -0.071713 -0.746796 -0.762610

Coolant Flow Rate 0.034241 0.033064 0.027408 0.029963

Coolant Inlet Temperature 0.998765 0.998636 0.893690 0.738424

Core Pressure -0.001312 0.000454 -0.000979 -0.000366

Core Power -0.179583 0.123781 0.984364 0.985816

87

5.3 – DAKOTA Seed Study

Due to the relatively small sample size compared to the much larger number of possible

permutations of the input parameters, there is reason to perform an analysis on the random seed

value within DAKOTA. The seed used for the rest of the cases, 17, is compared with several

other values in Table 55 below. The impact on the mean and confidence intervals are shown.

Each seed was run with 100 test cases as previously done. The highest and lowest values from

this selection of seeds are highlighted on Table 55. The difference in these values is close to 10

Kelvin, which is fairly small compared to the mean values but it is still worth noting. If more

cases were run for each seed value, it is expected that these values would converge for each seed.

Table 55: Random Seed Variation Results

Seed Mean at 1.000 s Lower CI Bound at 1.000 s Upper CI Bound at 1.000 s

5 1514.9 K 1508.4 K 1521.5 K

7 1516.9 K 1510.3 K 1523.4 K

9 1515.1 K 1508.3 K 1521.8 K

11 1514.9 K 1508.2 K 1521.5 K

13 1514.2 K 1507.7 K 1520.8 K

15 1519.7 K 1513.5 K 1525.9 K

17 1517.4 K 1511.1 K 1523.7 K

19 1512.8 K 1506.5 K 1519.0 K

21 1519.0 K 1512.3 K 1525.5 K

23 1513.3 K 1506.7 K 1520.0 K

25 1514.6 K 1507.9 K 1521.3 K

27 1510.6 K 1504.3 K 1516.9 K

29 1517.3 K 1510.9 K 1523.7 K

88

Chapter 6 : Conclusions

The propagation of uncertainty from the input parameters to output or target parameters

can be calculated using random sampling to simulate possible permutations of input values. The

amount of variation in the target parameter is given by the confidence bounds which are

established with 95% confidence for each test case. The results from the steady-state cases show a

maximum deviation of 5.7% from the sample mean for the output parameter during one of the

simulated runs. The transient cases have a maximum deviation of 4.5% from the sample mean.

The range of the calculated confidence bounds for each case is smaller than the range of observed

values during the simulation, but it can be said that a set of input parameters chosen from their

possible values will produce an output value found within this range 95% of the time, with 95%

confidence. For both of these types of test cases, the most influential parameter on the output

values is the fuel density. The uncertainty analysis performed in this study provides information

necessary to design Exercise II-1 of the UAM Benchmark, which includes more cases similar to

those analyzed.

Future work for this study would be to complete the remaining cases within Exercise II-1

of the UAM Benchmark, which covers VVER geometries as well, and to perform further analysis

of the remaining uncertainty sources. Other plans include converting the MATLAB scripts to

Python, or another similar machine language, such that the user does not require a license to

perform the code. Other uncertainty analysis codes such as SUSA, a GRS (Germany) code,

should be explored in order to detect and learn other capabilities that may help in a future

analysis. Other sources of uncertainty such as code uncertainties can be analyzed by rewriting

certain portions of the fuel performance codes, but this step would require direct access to the

source code which is not always available. As more details are available on uncertainties within

89

the nuclear power plant, these cases can be modified accordingly to reflect the specific

knowledge of each parameter’s possible variations.

90

Appendix

A.1 – FRAPCON Input File - PB-2 Steady-State Case

**********************************************************

* FRAPCON-3, steady-state fuel rod analysis code

* CASE DESCRIPTION: Base Irradiation for PB-2

* UNIT FILE DESCRIPTION

*--------------------------------------------------------

* Output:

* 6 STANDARD PRINTER OUTPUT

* Scratch:

* 5 SCRATCH INPUT FILE FROM ECH01

* Input: FRAPCON-3 INPUT FILE (UNIT 55)

**********************************************************

* GOESINS:

FILE05='nullfile', STATUS='UNKNOWN', FORM='FORMATTED',

CARRIAGE CONTROL='NONE'

*

* GOESOUTS:

FILE06='base_test.out',

STATUS='UNKNOWN', CARRIAGE CONTROL='LIST'

FILE66='base_test.plot',

STATUS='UNKNOWN', FORM='FORMATTED',CARRIAGE

CONTROL='LIST'

/**********************************************************

Base Irradiation

$frpcn

im=49, nr=17, ngasr=45, na=17

$end

$frpcon

thkgap=0.00015,

thkcld=0.00094,

den=95.1,

roughc=0.0000005,

roughf=0.000002,

fgpav=1207002.5,

dco=0.01430,

totl=3.6576,

cpl=0.178

dspg=0,

dspgw=0,

vs=0

hplt=0.01067,

rc=0,

hdish=0,

dishsd=0.0047

91

enrch=3.00,

imox=0,

comp=0,

ifba=0,

b10=0,

zrb2thick=0,

zrb2den=90

fotmtl=2,

gadoln=0,

ppmh2o=0,

ppmn2=0

deng=0,

rsntr=0,

tsint=1872.59

icm=4,

cldwks=0.5,

catexf=0.05,

chorg=10

idxgas=1,

nunits=0,

zr2vintage=1

iplant=-3,

pitch=0.01875,

icor=0,

crdt=0,

crdtr=0,

flux=18*22100000000000000

crephr=10,

sgapf=31,

slim=0.05,

qend=0.3,

ngasmod=2

jdlpr=1,

nopt=0,

nplot=1,

ntape=0,

nread=0,

nrestr=0

ProblemTime=

0.1, 0.2, 0.3, 0.4, 50

100, 150, 200, 250, 300

350, 400, 450, 500, 550

600, 650, 700, 750, 800

850, 900, 950, 1000, 1050

1100, 1150, 1200, 1250, 1300

1350, 1400, 1450, 1500, 1550

1600, 1650, 1700, 1750, 1800

1850, 1900, 1950, 2000, 2050

2100, 2150, 2200, 2230

qmpy=

3.28 6.56 9.84 13.12 15.70

92

15.70 15.70 15.70 15.70 17.32

17.32 17.32 17.32 17.32 17.32

17.32 17.32 17.32 16.55 16.55

16.55 16.55 16.55 16.55 16.55

16.55 16.55 16.55 13.39 13.39

13.39 13.39 13.39 13.39 13.39

13.39 13.39 13.39 15.16 15.16

15.16 15.16 15.16 15.16 15.16

15.16 15.16 15.16 15.16

p2= 7140000

tw= 550

go= 3460

nsp=0

iq=0,

fa=1

x(1)=

0.0000, 0.2286, 0.4572, 0.6858, 0.9144

1.1430, 1.3716, 1.6002, 1.8288, 2.0574

2.2860, 2.5146, 2.7432, 2.9718, 3.2004

3.4290, 3.6576

qf(1)=

0.70, 0.82, 0.91, 1.00, 1.09

1.13, 1.13, 1.13, 1.13, 1.13

1.13, 1.13, 1.09, 1.00, 0.91

0.82, 0.70

jn=17

jst=1

$end

93

A.2 – FRAPTRAN Input File - PB-2 Transient Case

*********************************************************************

* FrapTran, Transient fuel rod analysis code *

* *

* CASE DESCRIPTION: Assessment - PB-2 Pulse

*

FILE05='nullfile', STATUS='scratch', FORM='FORMATTED', *

CARRIAGE CONTROL='LIST' *

FILE15='sth2xt', STATUS='old', FORM='UNFORMATTED' *

* *

FILE06='PB2_pulse.out', STATUS='UNKNOWN', CARRIAGE CONTROL='LIST' *

FILE66='PB2_pulse.plot', STATUS='UNKNOWN', FORM='FORMATTED', *

CARRIAGE CONTROL='LIST' *

FILE22='restart.PB2' , STATUS='old', FORM='FORMATTED' *

/********************************************************************

PB-2 Pulse

$begin

ProblemStartTime=0.0,

ProblemEndTime=1.0,

$end

$iodata

unitin=1,

unitout=1,

inp=1,

dtpoa(1)=0.1,

dtplta= 0.196, 0.000, 0.010, 0.196, 0.794, 0.206

trest=1.92672e8,

$end

$solution

dtmaxa(1)= 0.001, 0.0, 0.0005, 0.180, 0.001, 0.210, 0.001, 1.0,

naxn=17,

nfmesh=25,

ncmesh=5,

$end

$design

gapthk= 1.5e-4 ,

FuelPelDiam= 0.01212,

frden= 0.951 ,

roughc= 0.5 ,

roughf= 2.0 ,

gappr0= 0.69e6 ,

RodLength= 3.6576 ,

RodDiameter= 0.01430,

pelh= 0.01067,

rshd= 0.0 ,

dishd= 0.0 ,

dishv0= 0.0 ,

coldw= 0.0 ,

cldwdc= 0.04 ,

gfrac(1)= 1.0 ,

tgas0= 295.0 ,

pitch= 0.01875,

pdrato= 1.31 ,

94

$end

$power

RodAvePower =

000.0 0.000

000.0 0.195

200.0 0.196

400.0 0.197

1000.0 0.198

1500.0 0.199

2500.0 0.200

10000.0 0.201

21000.0 0.202

45000.0 0.203

77500.0 0.204

95000.0 0.205

86000.0 0.206

60000.0 0.207

35000.0 0.208

20000.0 0.209

10000.0 0.210

5000.0 0.211

1500.0 0.212

1000.0 0.213

500.0 0.214

200.0 0.215

100.0 0.216

000.0 0.217

000.0 1.000

AxPowProfile = 0.70, 0.0000, 0.82, 0.2286, 0.91, 0.4572,

1.00, 0.6858, 1.09, 0.9144, 1.13, 1.1430,

1.13, 1.3716, 1.13, 1.6002, 1.13, 1.8288,

1.13, 2.0574, 1.13, 2.2840, 1.13, 2.5146,

1.09, 2.7432, 1.00, 2.9718, 0.91, 3.2004,

0.82, 3.4290, 0.70, 3.6576

$end

$model

internal='on',

metal='on',

cathca=1,

deformation='on',

noball=1,

nthermex=1

$end

$boundary

heat='on',

press= 2,

pbh2(1)=

7.14e6 0.0

7.14e6 1.0

zone=1,

htclev= 3.6576,

htco=2,

htca(1,1)= 2.0e6, 0.0, 2.0e6, 1.0,

tem= 8,

95

tblka(1,1)=

305 0.00

305 0.21

393 0.25

436 0.35

510 0.50

452 0.70

452 0.85

383 1.00

$end

96

A.3 – DAKOTA Input File - PB-2 Transient Case

# DAKOTA INPUT FILE - dakota_pb2_pulse.in

strategy,

single_method

graphics

tabular_graphics_data

tabular_graphics_file 'temps_pb.out'

method,

sampling

samples = 100 seed = 17

sample_type random

response_levels = 100.0 100.0 100.0 100.0

model,

single

variables,

normal_uncertain = 6

means = 0.000150 0.01212 0.951 0.5 2.0 0.69e6

std_deviations = 4.33e-6 6.67e-6 0.003 0.1 0.167 0.0233e6

lower_bounds = 0.000137 0.01210 0.942 0.2 1.5 0.62e6

upper_bounds = 0.000163 0.01214 0.960 0.8 2.5 0.76e6

descriptors 'thkgap' 'FuelPelDiam' 'den' 'roughc' 'roughf' 'fgpav'

uniform_uncertain = 4

lower_bounds = -2.0 -10.0 -3.0 -5.0

upper_bounds = 2.0 10.0 3.0 5.0

descriptors = 'flow' 'temp' 'pres' 'powr'

interface,

fork

analysis_driver = 'fraptran_wrapper_pb_pulse.bat'

parameters_file = "params_pb.in"

results_file = "results.out"

responses,

response_functions = 4

response_descriptors = 'T_time_1' 'T_time_2' 'T_time_3' 'T_time_4'

no_gradients

no_hessians

97

A.4 – DAKOTA Output File nond.out - PB-2 Transient Case (Abridged)

Running serial executable in serial mode.

DAKOTA version 5.2 released 11/30/2011.

Subversion revision built Dec 13 2011 12:16:57.

Constructing Single Method Strategy...

Writing new restart file dakota.rst

methodName = nond_sampling

gradientType = none

hessianType = none

>>>>> Running Single Method Strategy.

>>>>> Running nond_sampling iterator.

NonD random Samples = 100 Seed (user-specified) = 17

------------------------------

Begin Function Evaluation 1

------------------------------

Parameters for function evaluation 1:

1.4982165177e-04 thkgap

1.2127545770e-02 FuelPelDiam

9.5426363671e-01 den

4.6574674765e-01 roughc

2.1196093110e+00 roughf

7.0782842631e+05 fgpav

-5.6124401651e-01 flow

8.1617829902e+00 temp

3.1181568978e-01 pres

-1.1477705673e+00 powr

blocking fork: fraptran_wrapper_pb_pulse.bat params_pb.in results.out

1 file(s) copied.

1 file(s) copied.

Active response data for function evaluation 1:

Active set vector = { 1 1 1 1 }

3.1316000400e+02 T_time_1

3.1348001099e+02 T_time_2

1.0841999512e+03 T_time_3

1.5039000244e+03 T_time_4

------------------------------


------------------------------


1.5346074793e-04 thkgap


9.5140557646e-01 den

5.0754122120e-01 roughc

2.0684810324e+00 roughf

6.8184707362e+05 fgpav

1.2373619974e+00 flow

8.7426875765e+00 temp

2.5744292415e+00 pres

4.8632668727e+00 powr


1 file(s) copied.

1 file(s) copied.

98

time step too large for cladding annealing model (under flow)

effective coldworks and fluences set to zero









3.1373999000e+02 T_time_1

3.1407998657e+02 T_time_2

1.1294000244e+03 T_time_3

1.5721999512e+03 T_time_4

------------------------------


------------------------------


1.4297651785e-04 thkgap


9.5332944972e-01 den

4.5299332629e-01 roughc

2.1171137718e+00 roughf

6.7557071797e+05 fgpav

-3.9760788064e-01 flow

4.0422387375e+00 temp

2.1100293053e+00 pres

-4.7840993362e+00 powr


1 file(s) copied.

1 file(s) copied.



3.0904000900e+02 T_time_1

3.0935000610e+02 T_time_2

1.0569000244e+03 T_time_3

1.4640000000e+03 T_time_4

------------------------------


------------------------------


1.4011255989e-04 thkgap


9.5610142756e-01 den

3.8898968907e-01 roughc

2.1649453430e+00 roughf

6.5632335356e+05 fgpav

-3.5099509638e-01 flow

4.3544896133e+00 temp

3.4177221358e-02 pres

2.5601319293e+00 powr


1 file(s) copied.

1 file(s) copied.



3.0935000600e+02 T_time_1

3.0967999268e+02 T_time_2

99

1.1084000244e+03 T_time_3

1.5423000488e+03 T_time_4

<<<<< Function evaluation summary: 100 total (100 new, 0 duplicate)

Statistics based on 100 samples:

Moment-based statistics for each response function:

Mean Std Dev Skewness Kurtosis

T_time_1 3.0485869995e+02 5.7343388808e+00 6.6311776943e-02 -1.1557148017e+00

T_time_2 3.0518499939e+02 5.7314046703e+00 6.6653010408e-02 -1.1558526273e+00

T_time_3 1.0907670020e+03 2.1103436210e+01 -8.1328171969e-02 -8.1737888441e-01

T_time_4 1.5173660059e+03 3.1772217580e+01 -1.1190799521e-01 -8.6211637073e-01

95% confidence intervals for each response function:

LowerCI_Mean UpperCI_Mean LowerCI_StdDev UpperCI_StdDev

T_time_1 3.0372088271e+02 3.0599651719e+02 5.0347887912e+00 6.6614395833e+00

T_time_2 3.0404776436e+02 3.0632223442e+02 5.0322125343e+00 6.6580309835e+00

T_time_3 1.0865796224e+03 1.0949543815e+03 1.8528961454e+01 2.4515339647e+01

T_time_4 1.5110617086e+03 1.5236703031e+03 2.7896224529e+01 3.6908998968e+01

Probability Density Function (PDF) histograms for each response function:

PDF for T_time_1:

Bin Lower Bin Upper Density Value

--------- --------- -------------

1.0000000000e+02 3.1469000200e+02 4.6578787586e-03

PDF for T_time_2:


--------- --------- -------------

1.0000000000e+02 3.1501998901e+02 4.6507304023e-03

PDF for T_time_3:


--------- --------- -------------

1.0000000000e+02 1.1311999512e+03 9.6974403351e-04

PDF for T_time_4:


--------- --------- -------------

1.0000000000e+02 1.5750999756e+03 6.7792015223e-04

Partial Correlation Matrix between input and output:

T_time_1 T_time_2 T_time_3 T_time_4

thkgap 1.27053e-01 -6.46594e-02 -1.51142e-01 -1.37891e-01

FuelPelDiam 5.82859e-02 -1.53242e-01 -9.95304e-01 -9.94876e-01

den -1.59086e-01 -4.23037e-01 -9.99398e-01 -9.99343e-01

roughc -8.25989e-02 -1.23127e-01 2.28499e-02 -3.68839e-02

roughf 2.25092e-02 -1.23964e-01 -1.84174e-03 -2.17379e-02

fgpav 2.72307e-02 -9.04299e-02 2.91976e-02 7.38957e-02

flow -5.93210e-02 1.59477e-01 -1.05244e-01 -6.32813e-04

temp 1.00000e+00 1.00000e+00 9.99836e-01 9.99544e-01

pres -1.40494e-01 1.37556e-01 -2.41585e-01 -2.36111e-01

powr 2.04141e-01 9.58708e-01 9.99993e-01 9.99992e-01

Partial Rank Correlation Matrix between input and output:

T_time_1 T_time_2 T_time_3 T_time_4

thkgap 1.50529e-03 -1.64250e-01 -2.12541e-02 -4.59848e-02

FuelPelDiam 1.61813e-02 -1.96452e-02 -2.81600e-01 -3.17588e-01

den -1.30412e-02 -7.73539e-02 -6.51100e-01 -7.56530e-01

roughc -1.47575e-02 2.53362e-02 5.25677e-02 5.52623e-02

roughf -2.22847e-02 3.70859e-02 -1.83725e-02 1.02800e-01

fgpav -1.03599e-02 -8.80877e-02 1.45942e-02 5.13353e-02

flow 1.34308e-02 1.37806e-01 1.48050e-01 1.78099e-01

temp 1.00000e+00 9.99978e-01 8.42216e-01 7.82552e-01

pres 4.93528e-04 -4.33761e-02 -2.59259e-01 -1.93601e-01

powr 1.16839e-02 8.12258e-02 9.94245e-01 9.96986e-01

100

A.5 – Batch Script for the PB-2 Transient Case

@echo off

copy /Y %1 params_copy.in

matlab -minimize -nodesktop -nojvm -nosplash -r scriptDriver;quit; %1

fraptran_file.inp

PING 1.1.1.1 -n 1 -w 5000 >NUL

copy /Y fraptran_file.inp fraptran.inp

PING 1.1.1.1 -n 1 -w 5000 >NUL

FRAPTRAN1_4

C:\Python27\python fraptran_plot_to_results_out.py PB2_pulse.plot %2

101

A.6 – MATLAB Scripts - PB-2 Transient Case

A.6.1 – Driver Script (PB-2 Transient Case)

%Script Driver - DAKOTA with MATLAB

%FRAPTRAN Cases - Transient

%PB-2 Specific Driver Script

%All parameter variations

clear; close all; clc;

%% Specify the paths of files used by the script driver for PB-2's transient case

dataFile= 'params_copy.in'; %parameter data file

inputFileOrig= 'origInput.inp'; %template for the input file

inputFile= 'fraptran_file.inp'; %name of created input file

%% Specify the BC search values

%there is no flow rate specified in FRAPTRAN

param_1= ' RodAvePower ='; %power

param_2= ' pbh2('; %pressure

param_3= ' tblka(1,1)='; %inlet T

%% Read in from Original Input File to get BC original values

%get the parameters stored into matrices

[BC_A BC_B BC_C]= inputReader(param_1,param_2,param_3,inputFileOrig);

%BC_A is power, BC_B is pressure, BC_C is inlet temp

%They each have a 2nd column (as found in the input file for FRAPTRAN)

%they are column matrices of different lengths

%need to be manipulated by the factors in the following section

manu_params= [' gapthk= ,'; %manufacturing parameters

' FuelPelDiam= ,'; %as found in the fraptran file

' frden= ,';

' roughc= ,';

' roughf= ,';

' gappr0= ,'];

%% Define Parameters from params input file (as generated by DAKOTA)

%get vector of parameters as 'params'

params= params_reader(dataFile);

%% Create new matrix of variables that are changed

%apply the parameter % changes as needed

%Flow rate: 2.0%

%Inlet T: 10K

%Pressure: 3.0%

%Power: 5.0%

n= length(params); %the last 4 values in 'params' are the BC variations (from the

params.pb file)

f_flow= params(n-3); %percentage change for flow rate (not used in FRAPTRAN)

f_temp= params(n-2); %in units of Kelvin (not percentage!)

f_pres= params(n-1); %percentage change for pressure

f_powr= params(n); %percentage change for power

powr_means= BC_A; %original values of power

pres_means= BC_B;

temp_means= BC_C;

102

%these holds the changes for the BC parameters as cells

[powr_mat pres_mat temp_mat]= ...

BC_paramMaker(f_powr,f_pres,f_temp,BC_A,BC_B,BC_C);

%powr_mat, pres_mat, temp_mat are 2 column arrays

%% Update manufacturing tolerance strings

p_1= num2str(params(1),8); %manufacturing tolerance values as strings w/ 8 decimal places

p_2= num2str(params(2),8);





manu_params(1,16:16+length(p_1)-1)= p_1; %rewrite the strings for output

manu_params(2,16:16+length(p_2)-1)= p_2;





manu_params

%% Create input file

%The manufacturing parameters are stored in manu_params

%The BC values are in powr_mat, pres_mat, and temp_mat

inputFileMaker(inputFileOrig,inputFile,manu_params,powr_mat,pres_mat,temp_mat);

A.6.2 – Input Parameter Reader Script (PB-2 Transient Case)

function [power,pressure,inlet_T]= inputReader(p_1,p_2,p_3,fileName)

%stores the 3 BC original values from the origInput.inp file

%this is the fraptran-specific input file reader

%need to store times along with the power values in order to make input file creation

easier

%export the matrix of these values for use in the driver script

fin= fopen(fileName);

c= fgetl(fin);

BC_1= p_1;

BC_2= p_2; %#ok<NASGU>

check_2= ' press= ';

BC_3= p_3; %#ok<NASGU>

check_3= ' tem= ';

i= 0;

power= zeros(1,2);

pressure= zeros(1,1);

inlet_T= zeros(1,1);

while ~feof(fin)

if length(c)>= length(BC_1)

if strcmp(c(1:length(BC_1)),BC_1)

c= fgetl(fin);

while ~isempty(c)

i= i+1;

A= textscan(c, '%f %f');

103

power(i,1:2)= [A{1} A{2}]; %power matrix with times

c= fgetl(fin);

end

end

end

if length(c)>= length(check_2)

if strcmp(c(1:length(check_2)),check_2)

A_1= textscan(c,'%s %f %s'); %A_1 holds the # of press= #,

num_pressure_values= A_1{2};

c= fgetl(fin);

for i=1:num_pressure_values

c= fgetl(fin);

A= textscan(c,'%f %f');

pressure(i,1:2)= [A{1} A{2}]; %pressure matrix with locations

end

end

end

if length(c)>= length(check_3)

if strcmp(c(1:length(check_3)),check_3)

A_1= textscan(c,'%s %f %s'); %A_1 holds the # of tem= #,

num_temp_values= A_1{2};

c= fgetl(fin); %#ok<NASGU>

for i=1:num_temp_values

c= fgetl(fin);

A= textscan(c,'%f %f');

inlet_T(i,1:2)= [A{1} A{2}]; %inlet T matrix with times

end

end

end

c= fgetl(fin);

end

fclose(fin);

end

A.6.3 – DAKOTA Parameter File Reader Script (PB-2 Transient Case)

function [params]= params_reader(dataFile)

%Params_xyz.in reader from file 'dataFile' (params.in)

%exports vector of the parameters altered in the case

% dataFile= 'params_tmi.in';

fin= fopen(dataFile);

c= fgetl(fin);

str_b= 'variables'; %beginning string to parameter section

str_e= 'functions'; %ending string after parameter section

i= 0; %parameter index

params= zeros(1,1); %parameter vector

k= 0; %stop point for parameter detection

j= 0; %start point for parameter detection

while ~feof(fin)

104

if strcmp(c(length(c)-length(str_b)+1:length(c)),str_b)

c= fgetl(fin);

j= 1;

while k~=1;

if strcmp(c(length(c)-length(str_e)+1:length(c)),str_e)

k= 1;

end

if k~=1 && j==1

i= i+1;

A= textscan(c, '%f %s');

params(1,i)= A{1};

end

c= fgetl(fin);

end

end

if k==1 && j==1

break

end

end

fclose(fin);

end

A.6.4 – Parameter Updater Script (PB-2 Transient Case)

function[powr_mat,pres_mat,temp_mat]=

BC_paramMaker(f_1,f_2,f_3,powr_orig,pres_orig,temp_orig)

%powr, pres, temp is the order

%powr_orig is the matrix of the original power values used in FRAPTRAN

%pres_orig is the matrix of the original pressure values in FRAPTRAN

%temp_orig is the matrix of the original inlet T values in FRAPTRAN

%f_n are the variations (in % or Kelvin) specified as the uncertainty (divide % by 100)

powr_mat= zeros(1,2);

pres_mat= zeros(1,2);

temp_mat= zeros(1,2);

powr_mat= [powr_orig(:,1)+f_1*powr_orig(:,1)/100 powr_orig(:,2)];

pres_mat= [pres_orig(:,1)+f_2*pres_orig(:,1)/100 pres_orig(:,2)];

temp_mat= [temp_orig(:,1)+f_3 temp_orig(:,2)];

end

105

A.6.5 – Fuel Performance Code Input File Creator Script (PB-2 Transient Case)

function []= inputFileMaker(origFile,newFile,manu_params,powr_mat,pres_mat,temp_mat)

fin= fopen(origFile);

ftemp= fopen(newFile,'w');

text_check= '$design';

c= fgetl(fin);

%% Manufacturing Tolerance portion

%This while loop prints the first section of the input (up to the $design)

while ~feof(fin)

if length(c)>= length(text_check)

if strcmp(c(1:length(text_check)),text_check)

fprintf(ftemp, c);

fprintf(ftemp, '\n');

break;

else

c

ftemp

fprintf(ftemp, c);


end

else

fprintf(ftemp, c);


end

c= fgetl(fin);

end

%This loop prints out the new parameters into the $design section

%only the first 6 parameters, for now

param_check_1= ' RodLength=';

c= fgetl(fin);

[x y]= size(manu_params);

while ~feof(fin)

if length(c)>= length(param_check_1)

if strcmp(c(1:length(param_check_1)),param_check_1)

fprintf(ftemp, c);


break;

else

for i=1:x

fprintf(ftemp, manu_params(i,:));


end

for i=1:x

c= fgetl(fin);

end

break

end

else

fprintf(ftemp, c);


end

c= fgetl(fin);

end

%% BC portion

param_check_1= ' RodAvePower =';

param_check_2= ' pbh2(';

106

param_check_3= ' tblka(';

spacer= ' '; %an attempt to beautify the file

while ~feof(fin)



fprintf(ftemp, c);


break;

else

fprintf(ftemp, c);


end

else

fprintf(ftemp, c);


end

c= fgetl(fin);

end

%This loop prints out the new parameters into the $design section

%only the first 6 parameters, for now

% param_check_1= ' RodLength=';

%after reaching ' RodAvePower =' run a loop through until isempty(c)

%print the new values of power_mat along with the times side by side

%also need to include the times, which are stored as the 2nd column in A

c= fgetl(fin);

for i=1:length(powr_mat)

power_value= num2str(powr_mat(i,1));

power_time= num2str(powr_mat(i,2));

power_l= length(power_value);

power_value_spaced= [spacer(1:length(spacer)-power_l+1) power_value];

power_line= [' ' power_value_spaced ' ' power_time];

fprintf(ftemp, power_line);


c= fgetl(fin);

end

while ~feof(fin)



fprintf(ftemp, c);


break;

else

fprintf(ftemp, c);


end

else

fprintf(ftemp, c);


end

c= fgetl(fin);

end

for i=1:length(pres_mat)

pressure_value= num2str(pres_mat(i,1));

pres_time= num2str(pres_mat(i,2));

pressure_line= [' ' pressure_value ' ' pres_time];

fprintf(ftemp, pressure_line);



end

c= fgetl(fin);

107

while ~feof(fin)



fprintf(ftemp, c);


break;

else

fprintf(ftemp, c);


end

else

fprintf(ftemp, c);


end

c= fgetl(fin);

end

for i=1:length(temp_mat)

inlet_T_value= num2str(temp_mat(i,1));

temp_time= num2str(temp_mat(i,2));

inlet_T_line= [' ' inlet_T_value ' ' temp_time];

fprintf(ftemp, inlet_T_line);



end

c= fgetl(fin);

while ~feof(fin)

c= fgetl(fin);

fprintf(ftemp, c);


end

fprintf(ftemp, c); %print final line

fclose(ftemp);

fclose(fin);

end

108

A.7 – Python Script - PB-2 Transient Case

import sys #gets info from the batch script

fraptran_output_file = sys.argv[1] #imports the first argument provided in the batch file

dakota_output_file = sys.argv[2] #imports the second argument provided in the batch file

from array import *

max_t_values = array('f',[0,0,0,0]) #array which holds max temp values

i=0 #counter for indexing the array

max_tv = None #set an intial value for max_tv (for the error message)

in_fd = open(fraptran_output_file) #in_fd is the input file (the .plot file from

fraptran)

#Search for lines beginning with '139 ' which indicates centerline temperatures

for l in in_fd: #search through the lines in the file

if l[0:6] == '139 ': #check if there is a match to this string

tv_strings = l.rstrip().split() #split the line with space as the delimiter

tmp_values = map(float,tv_strings) #convert these strings to float #s

max_tv= max(tmp_values) #store the maximum temperature from the array in max_tv

max_t_values[i]= max_tv #store the max_tv value for this time step in the array

i+=1 #update the index for the array which holds the 4 values of max temperature

in_fd.close()

if max_tv is None: #prints an error message if the .plot file doesn't have any values

print "no values found for temperature in" % frapcon_output_file

#Write the maximum temperatures to results.out to be used in DAKOTA

out_fd = open(dakota_output_file,'w') #write to the results.out file

print >> out_fd, "%f" % max_t_values[0], max_t_values[1], max_t_values[2],

max_t_values[3]

#print the 4 values of max temperature across as columns in results.out

out_fd.close()

109

A.8 – DAKOTA Tabulated Results File (PB-2 Steady-State Case, Abridged)

Case # thkgap thkcld den roughc roughf fgpav

1 0.00015 0.000942 95.42636 4.66E-07 2.12E-06 707828.4

2 0.000153 0.000943 95.14056 5.08E-07 2.07E-06 681847.1

3 0.000143 0.000939 95.33294 4.53E-07 2.12E-06 675570.7

4 0.000145 0.00094 95.24615 5.00E-07 1.83E-06 718697.1

5 0.000148 0.00094 95.46341 5.17E-07 2.24E-06 700696

… … … … … … …

96 0.000146 0.000939 95.21038 4.27E-07 2.16E-06 667824.6

97 0.000146 0.00094 94.5245 4.12E-07 1.95E-06 702811.8

98 0.000154 0.00094 94.79317 4.62E-07 2.24E-06 706613.9

99 0.000138 0.00094 94.93615 7.15E-07 1.82E-06 712830.1

100 0.00014 0.000939 95.61014 3.89E-07 2.16E-06 656323.4

Case # flow temp pres powr Temperature (K)

1 -0.56124 8.161783 0.311816 -1.14777 1150.961

2 1.237362 8.742688 2.574429 4.863267 1204.6

3 -0.39761 4.042239 2.110029 -4.7841 1120.894

4 1.149834 -1.75006 2.98785 4.560995 1189.678

5 -0.44583 3.499298 -2.52038 -2.61898 1137.856

… … … … … …

96 -1.99394 2.886893 2.699306 4.864026 1193.05

97 -0.93218 -6.053 1.532091 -4.63069 1138.472

98 -0.1152 6.704671 1.218674 -2.91699 1156.011

99 -1.28318 0.371025 -1.15622 2.440223 1166.583

100 -0.351 4.35449 0.034177 2.560132 1156.633

110

A.9 – DAKOTA Tabulated Results File (PB-2 Transient Case, Abridged)

Case # thkgap FuelPelDiam den roughc roughf fgpav

1 0.00015 0.012128 9.54E-01 4.66E-01 2.119609 707828.4

2 0.000153 0.012135 9.51E-01 5.08E-01 2.068481 681847.1

3 0.000143 0.012115 9.53E-01 4.53E-01 2.117114 675570.7

4 0.000145 0.01212 9.52E-01 5.00E-01 1.828187 718697.1

5 0.000148 0.012121 9.54E-01 5.17E-01 2.239981 700696.0

… … … … … … …

96 0.000146 0.012117 9.52E-01 4.27E-01 2.156266 667824.6

97 0.000146 0.012118 9.45E-01 4.12E-01 1.945802 702811.8

98 0.000154 0.012121 9.48E-01 4.62E-01 2.236025 706613.9

99 0.000138 0.012118 9.49E-01 7.15E-01 1.817132 712830.1

100 0.00014 0.012114 9.56E-01 3.89E-01 2.164945 656323.4

Temperature (K)

Case # flow temp pres powr time_1 time_2 time_3 time_4

1 -0.56124 8.161783 0.311816 -1.14777 313.16 313.48 1084.2 1503.9

2 1.237362 8.742688 2.574429 4.863267 313.74 314.08 1129.4 1572.2

3 -0.39761 4.042239 2.110029 -4.7841 309.04 309.35 1056.9 1464

4 1.149834 -1.75006 2.98785 4.560995 303.25 303.59 1120.3 1562.9

5 -0.44583 3.499298 -2.52038 -2.61898 308.5 308.81 1070.5 1485

… … … … … … … … …

96 -1.99394 2.886893 2.699306 4.864026 307.89 308.23 1126.7 1570.6

97 -0.93218 -6.053 1.532091 -4.63069 298.95 299.26 1056.1 1467.1

98 -0.1152 6.704671 1.218674 -2.91699 311.7 312.02 1075.8 1491.6

99 -1.28318 0.371025 -1.15622 2.440223 305.37 305.7 1109.3 1545.4

100 -0.351 4.35449 0.034177 2.560132 309.35 309.68 1108.4 1542.3

References

1. “Uncertainty Analysis in Modeling UAM-2006 Workshop”, Summary Record,

NEA/NSC/DOC (2006)15.

2. “Expert Group on Uncertainty Analysis in Modeling”, Mandate and Programme of Work, NEA/NSC/DOC(2006)17.

3. “Uncertainty Analysis in Modeling First Workshop (UAM-1)”, Summary Record,

NEA/NSC/DOC(2007)17.

4. Technology Relevance of the “Uncertainty Analysis In Modeling” Project for Nuclear Reactor Safety, NEA/NSC/DOC(2007)15.

5. J. Solis, K. Ivanov, B. Sarikaya, A. Olson, and K. Hunt, “BWR TT Benchmark. Volume

I: Final Specifications”, NEA/NSC/DOC(2001)1.

6. K. Ivanov, T. Beam, A. Baratta, A. Irani, and N. Trikouros, “PWR MSLB Benchmark.

Volume 1: Final Specifications”, NEA/NSC/DOC(99)8, April 1999.

7. “Core Design and Operating Data for Cycles 1 and 2 of Peach Bottom 2”, EPRI NP-563, Research Project 1020-1, June 1978.

8. J. Papin, B. Cazalis, J. M. Frizonnet, E. Federici, F. Lemoine (IRSN/DPAM), “Synthesis

of CABRI-RIA Tests Interpretation”, Forum Eurosafe, Paris, November 2003.

9. “Neutronics/Thermal-hydraulics Coupling in LWR Technology”, Vol. 1, CRISSUE-S-WP1: Data Requirements and Databases Needed for Transient Simulations and

Qualification, ISBN 92-64-02083-7, NEA No. 4452, OECD 2004.

10. “Neutronics/Thermal-hydraulics Coupling in LWR Technology”, Vol. 2, CRISSUE-S-WP2:State-of-the-art Report (REAC-SOAR), ISBN 92-64-02084-5, NEA No. 5436,

OECD 2004.

11. P. M. Chantoin (CEA), E. Sartori (OECD/NEA), J. A. Turnbull, “The Compilation of a Public Domain Database on Nulcear Fuel Performance for the purpose of Code

Development and Validation”, OECD/NEA, IFPE Database, June 1997.

12. G. M. O’Donnell, H. H. Scott, R. O. Meyer, “A New Comparative Analysis of LWR Fuel

Designs”, NUREG-1754, USNRC, 2001.

13. A. Avvakumov, V. Malofeev, V. Sidorov (KI), H. H. Scott (NRC), “Spatial Effects and

Uncertainty Analysis for Rod Ejection Accidents in a PWR, NUREG/IA-0215, USNRC,

2007.

14. K. J. Geelhood, W. G. Luscher, C. E. Beyer et al., “Predictive Bias and Sensitivity in

NRC Fuel Performance Codes", NUREG/CR-7001, PNNL-17644, PNNL, Richland,

WA, 2009.

15. L. Pagani, “On the Quantification of Safety Margins”, Dissertation, Massachusetts Institute of Technology, Cambridge, MA, 2004.

112

16. K. Ivanov, M. Avramova, S. Kamerow, I. Kodeli, E. Sartori, E. Ivanov, O. Cabellos,

“Benchmark for Uncertainty Analysis in Modeling (UAM) for Design, Operation, and Safety Analysis of LWRs, Volume I: Specification and Support Data for the Neutronics

Cases (Phase I)”, Version 2.0, March 2012.

17. G. A. Berna (GABC), C. E. Beyer (PNNL), K. L. Davis (INEEL), D. D. Lanning

(PNNL), “FRAPCON-3: A Computer Code for the Calculation of Steady-State, Thermal-Mechanical Behavior of Oxide Fuel Rods for High Burnup”, NUREG/CR-6534 Vol. 2,

PNNL-11513, 1997.

18. M. E. Cunningham, C. E. Beyer, P. G. Medvedev (PNNL), G. A. Berma (GABC), H. Scott (NRC), “FRAPTRAN” A Computer Code for the Transient Analysis of Oxide Fuel

Rods”, NUREG/CR-6739, Vol. 1, PNNL-13576, 2001.

19. D. D. Lanning, C. E. Beyer (PNNL), G. A. Berna (GABC), “FRAPCON-3: Integral Assessment”, NUREG/CR-6534, Volume 3, PNNL-11513, 1997.

20. M. E. Cunningham, C. E. Beyer, F. E. Panisko, P. G. Medvedev (PNNL), G. A. Berna

(GABC), H. H. Scott (NRC), “FRAPTRAN: Integral Assessment”, NUREG/CR-6739,

Vol. 2, PNNL-13576, 2001.

21. B. Collins, A. Yanov, T. Downar (UMich), M. Klein, W. Zwermann (GRS), M. Jessee,

M. Williamns (ORNL), “Application of PARCS to Uncertainty Analysis of TMI

Minicore”, Presented at the UAM-5 Meeting, Stockholm, April 2011.

22. D. D. Lanning, E. R. Bradley, “Final Irradiation and Postirradiation Data from the

NRC/PNL Instrumented Assembly IFA-432”, HPR-329/7.

23. T. Sugiyama, T. Nakamura, K. Kusagaya, H. Sasajima, F. Nagase, T. Fuketa, “Behavior of Irradiation BWR Fuel under Reactivity-Initiated-Accident Results of Tests FK-1, -2,

and -3”, JAERI-Research 2003-033, January 2004.

24. H. Devold, H. Wallin, “PIE Results from the High Burnup Rods CD/CH (IFA-429) and

Comparison with In-Pile Data”, HWR-409, October 1994.

25. U. Mertyurek, M. W. Francis, I. C. Gauld, “SCALE 5 Analysis of BWR Spent Nuclear

Fuel Isotopic Compositions for Safety Studies”, ORNL/TM-2010/286, December 2010.

26. S. S. Wilks, “Determination of Sample Sizes for Setting Tolerance Limits”, The Annals of Mathematical Statistics, 12, 91-96, 1941.

27. H. Glaeser, “GRS Method for Uncertainty and Sensitivity Evaluation of Code Results

and Applications”, Presented at NURETH-14, Toronto, Ontario, Canada, 2011.

28. B. Adams, K. Dalbey, M. Eldred, L. Swiler et al, “DAKOTA, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty

Quantification, and Sensitivity Analysis – Version 5.2 User’s Manual”, Sandia National

Laboratories, SAND2010-2183, 2011.

29. C. H. Tracy, “FRAPCON Fuel Centerline Temperature Sensitivity and Uncertainty Study

using DAKOTA”, M.Eng Paper, Department of Mechanical and Nuclear Engineering,

The Pennsylvania State University, 2012.

30. T. Blyth, M. Avramova, K. Ivanov, E. Royer, E. Sartori, O. Cabellos, “Benchmark for

Uncertainty Analysis in Modeling (UAM) for Design, Operation, and Safety Analysis of

113

LWRs, Volume II: Specification and Support Data for the Core Cases (Phase II)”,

Version 1.0, May 2012.

31. B. Ivanov, K. Ivanov, P. Groudev, M. Pavlova, and V. Hadjiev, “VVER-1000 Coolant

Transient Benchmark (V1000-CT). Phase 1 – Final Specification”.

NEA/NSC/DOC(2002)6.

32. L. Yegorova, K. Lioutov, N. Jouravkova (KI), et al., “Experimental Study of Narrow Pulse Effects on the Behavior of High Burnup Fuel Rods with Zr-1% Nb Cladding and

UO2 Fuel (VVER Type) under Reactivity-Initiated Accident Conditions: Program

Approach and Analysis of Results”, NUREG/IA-0213, Vol. 1, 2006.

33. L. Yegorova, K. Lioutov, N. Jouravkova (KI), et al., “Experimental Study of Narrow

Pulse Effects on the Behavior of High Burnup Fuel Rods with Zr-1% Nb Cladding and

UO2 Fuel (VVER Type) under Reactivity-Initiated Accident Conditions: Test Conditions and Results”, NUREG/IA-0213, Vol. 2, 2006.

34. Committee on the Safety of Nuclear Installations, “VVER-Specific Features Regarding

Core Degradation - Status Report”, NEA/CSNI/R(98)20, NEA/OECD September 1998.

Documents

FUEL PERFORMANCE CODE BENCHMARK FOR UNCERTAINTY ANALYSIS IN LIGHT WATER