NUREG/IA-0215 

 NSI RRC KI 90-12/01-03-07

 International 

 Agreement Report

 

Spatial Effects and Uncertainty Analysis for 

Rod Ejection Accidents in a PWR 

Prepared by: 

A. Avvakumov, V. Malofeev, V. Sidorov 

Nuclear Safety Institute 

Russian Research Centre 

“Kurchatov Institute” 

1, Kurchatov Sq., Moscow 

123182 Russia 

H. H. Scott, NRC Project Manager 

Office of Nuclear Regulatory Research 

U.S. Nuclear Regulatory Commission 

Washington, DC 20555-0001 

September 2007 

 

Prepared for 

U.S. Nuclear Regulatory Commission, Institute for Radioprotection and Nuclear Safety (France), 

and Ministry of Science and Technologies of the Russian Federation 

Published by 

U.S. Nuclear Regulatory Commission
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 NUREG/IA-0215 

 NSI RRC KI 90-12/01-03-07

 International 

 Agreement Report

 

Spatial Effects and Uncertainty Analysis for 

Rod Ejection Accidents in a PWR 

Prepared by: 

A. Avvakumov, V. Malofeev, V. Sidorov 

Nuclear Safety Institute 

Russian Research Centre 

“Kurchatov Institute” 

1, Kurchatov Sq., Moscow 

123182 Russia 

H. H. Scott, NRC Project Manager 

Office of Nuclear Regulatory Research 

U.S. Nuclear Regulatory Commission 

Washington, DC 20555-0001 

September 2007 

 

Prepared for 

U.S. Nuclear Regulatory Commission, Institute for Radioprotection and Nuclear Safety (France), 

and Ministry of Science and Technologies of the Russian Federation 

Published by 

U.S. Nuclear Regulatory Commission

iii

ABSTRACT 

A rod ejection accident is a design-basis event for a pressurized water reactor. It is well known that spatial 

effects play a very important role in this accident. In this study four cases using a model of Three Mile 

Island Unit 1 are considered: ejection of the central or peripheral control rod at the end of cycle or the 

beginning of cycle. Calculations were carried out using the BARS neutronic code coupled with the 

RELAP5/MOD3.2 thermal hydraulic code. This coupled code allows three-dimensional pin-by-pin 

neutronics and assembly-by-assembly thermal-hydraulics simulation of a transient. The results showed 

that the major parameters of the accident (the peak power and core energy deposition) were a function of 

spatial effects. Analysis of the dependence of the peak local fuel enthalpy on spatial effects was 

performed. Uncertainty analysis was carried out for the central control rod ejection accident at hot zero 

power conditions. The analysis of uncertainties was performed for the following parameters: local fuel 

enthalpy, maximum core power, and power pulse width. Calculated results showed that the uncertainty in 

key safety parameters would be determined to a great extent by the uncertainty in the control rod worth. 

The effect of initial core power on the above parameters was analyzed using a rod ejection accident 

starting from 33% of rated power. 

v

FOREWORD

Regulatory agencies in most countries require licensees to routinely analyze postulated reactivity-initiated 

accidents, and fuel damage criteria are established to assess the consequences of those accidents. At the 

time that high burnup was shown to affect such damage criteria, representatives from the Kurchatov 

Institute in Russia informed several research organizations that the Kurchatov Institute had performed 

related tests with high-burnup fuel, but the test results had not been evaluated. Subsequently, cooperative 

arrangements were made between the Kurchatov Institute, the French Institute for Radiological Protection 

and Nuclear Safety (IRSN, formerly IPSN), and the U.S. Nuclear Regulatory Commission to evaluate the 

test results. Those results have now been evaluated, supplemented, and published in two NUREG/IA 

reports (0156 and 0213). 

Although the test reactor data were the main interest of the collaboration, there was also interest in 

examining best-estimate methods available in Russia for plant safety analyses as a means of comparing 

code capabilities. This was especially interesting because the Russian neutron kinetics code, BARS, had 

a completely different developmental history than the codes being used in France and the U.S. 

Examination of the Russian code was accomplished by analyzing several postulated reactivity transients. 

The present report describes all of the BARS calculations for rod-ejection accidents in a PWR with UO2

fuel. Three other related reports are in preparation, discussing computations for boron-dilution accidents 

with UO2 fuel, rod-ejection accidents with mixed-oxide (MOX) fuel, and other related benchmarks. 

 Brian W. Sheron, Director 

 Office of Nuclear Regulatory Research 

 U.S. Nuclear Regulatory Commission 

vii

CONTENTS 

Page

Abstract .......................................................................................................... iii

Foreword ......................................................................................................... v

List of Figures .................................................................................................. viii

List of Tables .................................................................................................... ix

Executive Summary ............................................................................................ xi

Acknowledgements ............................................................................................ xiii

Abbreviations .................................................................................................... xiv

1 Introduction ................................................................................................. 1

 1.1 Background .......................................................................................... 1

 1.2 Objectives ........................................................................................... 2

 1.3 Outline of Report ................................................................................... 3

2 Spatial Effects for Rod Ejection Accidents ............................................................. 5

 2.1 RELAP-BARS Transient Calculations .......................................................... 5

 2.2 Nordheim-Fuchs Model ............................................................................ 25

 2.3 Results of Steady-State Calculations ............................................................ 31

3 Uncertainty Analysis ...................................................................................... 33

 3.1 Methodology ......................................................................................... 33

 3.2 Analysis for Central Rod Ejection ................................................................ 33

 3.3 Effect of Initial Core Power ........................................................................ 40

4 Conclusions ................................................................................................. 47

5 References ................................................................................................... 51

Appendices 

A Calculational Results for a REA with Increased Rod Worth ........................................ A-1
viii

LIST OF FIGURES 

Figure Page

2.1 One-Eight Core Layout (EOC Cases) ........................................................... 6 

2.2 One-Eight Core Layout (BOC Cases) ........................................................... 7 

2.3 Position of the Hottest Pins in Fuel Assembly during REAs ................................ 10 

2.4 Assembly-Average Power Distributions (Case EOC Centre) ............................... 11 

2.5 Assembly-Average Power Distributions (Case EOC Periphery) ........................... 12 

2.6 Assembly-Average Power Distributions (Case BOC Centre) .............................. 13 

2.7 Assembly-Average Power Distributions (Case BOC Periphery) ............................ 14 

2.8 Average Axial Power Distribution (Cases EOC and BOC Centre) ......................... 16 

2.9 Power, Reactivity and Peaking Factors vs. Time (Case EOC Centre) ..................... 17 

2.10 Power, Reactivity and Peaking Factors vs. Time (Case EOC Periphery) ................. 18 

2.11 Power, Reactivity and Peaking Factors vs. Time (Case BOC Centre) ..................... 19 

2.12 Power, Reactivity and Peaking Factors vs. Time (Case BOC Periphery) ................. 20 

2.13 Fuel Temperature and Enthalpy in Hot Pins vs. Time (Case EOC Centre) ............... 21 

2.14 Fuel Temperature and Enthalpy in Hot Pins vs. Time (Case EOC Periphery) ............ 22 

2.15 Fuel Temperature and Enthalpy in Hot Pins vs. Time (Case BOC Centre) ............... 23 

2.16 Fuel Temperature and Enthalpy in Hot Pins vs. Time (Case BOC Periphery) ............ 24 

2.17 Feedback Parameter vs. Power Peaking Factor ............................................... 28 

3.1 Core Power and Reactivity vs. Time ............................................................ 34 

3.2 Fuel Pellet Enthalpy Increment vs. Time ...................................................... 36 

3.3 Reactor Power vs. Time .......................................................................... 41 

3.4 Reactivity vs. Time ............................................................................... 42 

3.5 Assembly Averaged Power at Three States of the Core ..................................... 43 

3.6 Fuel Enthalpy in the Hottest Pellet vs. Time ................................................... 44 

 

A.1 Power and Reactivity vs. Time .................................................................. A-3 

A.2 Fuel Temperature and Enthalpy in Hot Pins vs. Time ........................................ A-4 
ix

LIST OF TABLES 

Table Page 

2.1 Steady-State Calculation Results for Rod Worths ............................................ 8 

2.2 Core Parameters in REAs ......................................................................... 9 

2.3 Some Neutronic Parameters of REAs .......................................................... 29 

2.4 Evaluated Neutronic Parameters of REAs ..................................................... 30 

2.5 Steady-State Calculational Results for EOC HZP Case ..................................... 32 

2.6 Steady-State Calculational Results for BOC HZP Case ...................................... 32 

3.1 Sensitivity of Fuel Pellet Enthalpy ............................................................. 38 

3.2 Sensitivity of Maximum Core Power and Power Pulse Width .............................. 39 

3.3 REA Parameters in Comparison with the HZP Case ......................................... 45 

 

A.1 Calculational Results for 1.47β EOC REA .................................................... A-2 

xi

EXECUTIVE SUMMARY 

This study was undertaken to analyze spatial effects in the course of rod ejection accidents. If the rod 

worth is sufficient to reach prompt critical i.e. greater than β, then its withdrawal initiates a fast power 

excursion which is terminated due to the negative reactivity feedback. This event is of interest from the 

point of view of prediction of the maximum increase in fuel enthalpy during the accident. Although, as 

numerous steady-state calculations show, the maximum value of control rod worth scarcely exceeds 1β in 

a PWR, such an accident is to be considered in a NPP safety analysis with conservatively increased value 

of the ejected rod worth. 

A key parameter of interest in a NPP safety analysis is the peak local fuel enthalpy, which establishes the 

acceptance criterion for unacceptable fuel damage in reactivity initiated accidents in a LWR. It is wellestablished that spatial effects play an important role in the REA consequences, in particular, the core 

peak power and energy deposition which can be approximately related to the fuel enthalpy rise under an 

adiabatic assumption (no heat transfer from the fuel to the coolant). Then using simplified analytical 

approximations it is easy to establish relationships between the major parameters of interest and spatial 

effects. An influence of spatial effects on the REA parameters is revealed in two opposite trends. On the 

one hand, the higher the power peaking factor, the higher the local fuel enthalpy. On the other hand, a 

higher value of the power peaking factor tends to slow down the total energy deposition in the core and, 

as a result, to mitigate the fuel enthalpy increase. This is due to the fact that the fuel and moderator 

feedback is stronger in the hottest regions of the reactor core and reduces the peak power and reactivity. 

In this study an analysis of REAs was carried out using the RELAP–BARS code, which allows a 3-D pinby-pin neutronic and assembly-by-assembly thermal-hydraulic simulation of a LWR. Four REAs with 

ejection of the central or peripheral control rod at the EOC or BOC initial conditions were analyzed using 

a PWR calculational model based on TMI-1. To provide identical initial conditions the worth of an 

ejected control rod was artificially fitted to 1.21β for all cases. The calculational results showed that the 

peak powers were different to a great extent: from 4.4 to 37.5 GW. In the cases with ejection of the 

peripheral rod, the peak power was lower compared with ejection of the central rod by 2.5 times at EOC 

and by 2.9 times at BOC. In turn, practically the same, but inverse relationship was found for the power 

peaking factors. The maximum fuel enthalpy increase was about 26 cal/g at BOC and 17-19 cal/g at EOC. 

From the point of view of the assembly-by-assembly representation, the central rod ejection accidents 

with rather small power peaking factors resulted in slightly higher values of the maximum fuel enthalpy 

increase. On the contrary, the pin-by-pin representation showed that the peak local fuel enthalpy was 

higher for the peripheral REAs with the highest power peaking factors. In the EOC cases, a higher value 

of the peak enthalpy was due to the fact that the hottest fuel pin was found in an assembly diagonally 

adjacent to the hot assembly. This fact indicates that a problem of proper definition of the hottest pin 

location is essential. In the considered case the incorrect definition of the hottest pin location using an 

assembly-by-assembly approach, can lead to 15%-underestimation in the peak local enthalpy rise. 

The point kinetics approximation within the Nordheim-Fuchs model allowed to derive simple relationship 

between the peak power, total energy deposition and the power peaking factor. It was found that in the 

framework of this model the peak power and total energy deposition are inversely proportional to the 

power peaking factor. Thus, if the peaking factor for fuel enthalpy increase is approximately the same as 

that for power, then it can be expected that the maximum assembly-average fuel enthalpy increase is 

practically independent on spatial effects. 
xii

Another important problem of a REA analysis deals with uncertainty in prediction of the rod worth by 

different calculational methods. As was found during this study, the BARS results for the rod worths were 

always higher compared with those calculated by the PARCS code. The most difference occurs for the 

peripheral rod nearest to the reflector region. At the EOC initial conditions this difference reaches 35%. 

The uncertainty analysis for the PWR central rod ejection accident starting from the HZP conditions 

carried out with the RALAP-BARS code showed that the uncertainties in the key parameters of the 

accident would be determined to a great extent by the uncertainty in ejected rod worth. For a rod worth of 

1.2β with the uncertainty of 15% (corresponding to two standard deviations), the uncertainty in local fuel 

enthalpy was estimated as 110%, the uncertainty in the maximum core power – as 216%, and the 

uncertainty in power pulse width – as 76%. The results demonstrated non-adiabatic nature of the transient 

and showed that the sensitivity of fuel enthalpy to the most of neutronic and thermal-hydraulic quantities 

strongly depends on time. Qualitatively, the RELAP-BARS results are agreed with the PARCS ones. 

The comparative study of the accident starting from 33% of rated power showed strong dependence of a 

number of the REA parameters on initial core power. Under the same rod worth of 1.2β, the peak power 

in the transient from 33% of rated power was by 3.5 times greater than that in the transient from HZP. 

Unlike the HZP case, a change in the hottest fuel pin location takes place due to power redistribution 

during the transient from the non-zero power. This phenomenon can lead to some difficulties to predict 

the hottest pin using assembly-by-assembly approach together with the pin reconstruction procedure to 

calculate fuel pellet enthalpy. In comparison with the HZP case, the REA from 33% of rated power leads 

to the increase in the maximum fuel pellet enthalpy up to 67 cal/g. 

In general, the conclusion was that the influence of spatial effects may be of interest not only in an 

analysis of reactivity initiated events, but also in interpretation of in-pile burst tests aimed to understand 

fuel rod behavior under severe accidents. The present study has also looked at the effect of the calculation 

approach (pin-by-pin against assembly-by-assembly) on the major parameter of a REA – the peak local 

enthalpy increase. 
xiii

ACKNOWLEDGEMENTS 

This work was performed at Russian Research Centre “Kurchatov Institute” as part of a project supported 

by the U.S. Nuclear Regulatory Commission (NRC), Institute for Radioprotection and Nuclear Safety 

(IRSN, France) and Ministry of Science and Technologies of the Russian Federation. The authors 

gratefully acknowledge Dr. David Diamond of Brookhaven National Laboratory (BNL, USA) for his 

work and leadership during the entire period of this project. 
xiv

ABBREVIATIONS 

BNL Brookhaven National Laboratory (USA) 

BOC Beginning of Cycle 

EOC End of Cycle 

FA Fuel Assembly 

HZP Hot Zero Power 

IRSN Institute for Radioprotection and Nuclear Safety (France) 

LWR Light Water Reactor 

NRC Nuclear Regulatory Commission (USA) 

NPP Nuclear Power Plant 

NSI Nuclear Safety Institute 

PWR Pressurized Water Reactor 

REA Rod Ejection Accident 

RRC KI Russian Research Centre “Kurchatov Institute” 

TMI-1 Three Mile Island Unit 1 

1-D One-Dimensional 

2-D Two-Dimensional 

3-D Three-Dimensional 
1

1 INTRODUCTION 

1.1 Background 

The rod ejection accident (REA) is a design-basis reactivity initiated event for a pressurized water reactor 

(PWR). The rod ejection scenario assumes a mechanical failure of the control rod drive mechanism and, 

as a result, ejection of a control rod to a fully withdrawn position due to the reactor coolant system 

pressure. If the ejected rod worth is sufficient, a rapid power excursion occurs, which is terminated by the 

negative fuel temperature (Doppler) feedback within a few hundredths of a second. 

From the point of view of a nuclear power plant safety analysis, peak local fuel enthalpy is a key 

parameter used as the acceptance criterion for fuel behavior in the course of reactivity initiated events in 

LWRs. It is clear that local fuel enthalpy depends on a spatial distribution of the energy deposition in the 

core during the accident; therefore, to determine a peak value of this parameter accurately, it is necessary 

to consider 3 D pin-by-pin neutronic model. In best-estimate nodal diffusion methods such a problem is 

usually split into two steps: first – a calculation of assembly-average power distribution, and second – 

peak power estimate within selected fuel assemblies by a pin-by-pin reconstruction method with further 

estimate of the peak local fuel enthalpy. This procedure has evident drawbacks compared with the direct 

pin-by-pin methods, especially when spatial effects are very complicated during the event. 

Analysis of a REA, as a rule, involves the following parameters of interest: the core peak power and the 

energy deposition, which can be approximately related to the fuel enthalpy rise under the assumption of 

an adiabatic process (i.e. no heat transfer to the coolant). A simple analytical approximation, based on the 

Nordheim-Fuchs model, allows to establish dependence of the peak power, pulse width and energy 

deposition on several parameters: the inserted reactivity (rod worth), delayed neutron fraction, prompt 

neutron lifetime, and a parameter which characterizes the Doppler feedback. As it was found the 

magnitude of the last parameter depended strongly on spatial effects during a transient. 

It is well-established that spatial effects play an important role in the REA consequences of interest. 

These effects are revealed in a dual manner. On the one hand, the higher the power peaking factor, the 

higher the local energy deposition and, consequently, the local fuel temperature (or enthalpy) increase. On 

the other hand, it is well-known that the Doppler feedback is stronger in the hottest fuel regions and 

reduces the peak power and reactivity. Besides, local heating of the coolant also decreases the peak power 

and reactivity. 

The REA analyses were performed using the RELAP–BARS code (Ref. 1), which allows a 3-D pin-bypin neutronics and assembly-by-assembly thermal-hydraulics simulation of a light water reactor (LWR). 

Calculations were carried out for four PWR REAs with ejection of the central or peripheral control rod at 

the end of cycle (EOC) or the beginning of cycle (BOC). To provide maximum non-uniformity of the 

power distribution in axial direction, all control rods were either fully inserted in the core or fully 

withdrawn during a transient. 
2

As it was mentioned the most important safety parameter of reactivity initiated accident is maximum local 

fuel pellet enthalpy. 3-D best-estimate neutronics methods are available to calculate local fuel pellet 

enthalpy; but unlike 1-D or 2-D very conservative methodologies, these methods do not guarantee 

conservative estimation in key safety parameters during such an accident. Therefore, it is important to 

determine the uncertainty in fuel enthalpy calculated by a best-estimate code. 

Recently a qualitative approach to an uncertainty analysis for the rod ejection accident (REA) was 

developed in Brookhaven National Laboratory (BNL, USA). For the REA, the fact that the physics of the 

transient is relatively well-known allowed the authors to define a simplified methodology to estimate the 

uncertainty in fuel enthalpy. The approach is based on using point kinetics in determining the quantities, 

which determine the uncertainty in fuel enthalpy instead of a very complicated consideration of 

uncertainties in cross sections. 

The approach was applied to the uncertainty analysis of a PWR REA at hot zero power conditions (HZP). 

The analysis took into account the point kinetics parameters, which were obtained from 3-D calculations 

and engineering judgement as to the uncertainty in those parameters. The results showed that the 

uncertainty in local fuel enthalpy would be determined primarily by the uncertainty in ejected rod worth 

and delayed neutron fraction. For an uncertainty in the former of 8% (one standard deviation) and the 

latter of 5%, the uncertainty in fuel enthalpy was 51% for control rod worth of 1.2β (β − delayed neutron 

precursor fraction). However, the authors considered only a few quantities of interest and their analysis 

was based on a conservative adiabatic assumption for fuel temperature calculation. 

1.2 Objectives 

The objective of this study was to analyze spatial effects in the course of rod ejection accidents with 

withdrawal of the central or off-central control rod. A PWR calculational model has 1/8 radial symmetry 

and includes fuel assemblies with different properties surrounded by the radial reflector. In an axial 

direction a uniform core nodalization scheme (axial layers with different properties) is used with the top 

and bottom reflectors. Although the BARS code allows a continuous axial distribution of power (due to 

use of the Fourier series expansion), hereinafter an axial power distribution is given in terms of used 

nodalization scheme (for the simplicity in a code intercomparison). 

In this study the following parameter is used to characterize spatial effects in the REAs: the total power 

peaking factor, Fq, defined as a ratio between the maximum fuel pellet power and an average pellet power 

over the core. It should be noted that “fuel pellet” term in this study is related to a calculational axial 

node, which is generally larger compared with the actual height of a fuel pellet. This definition of the 

power peaking factor is valid for the pin-by-pin approach. In an assembly-by-assembly representation the 

following definition of the power peaking factor, Fq

′

, is used: it is a ratio between the maximum power in 

an axial node of the hottest assembly to an average node power (provided that uniform axial nodalization 

is used). 
3

Another objective of this study is to analyze the uncertainty in peak fuel enthalpy, core power, and power 

pulse width for a REA in the TMI-1 PWR at HZP conditions. Sensitivity of these parameters to a variety 

of neutronic and thermal-hydraulic quantities of the core was studied using the pin-by-pin neutronic 

model together with more realistic thermal-hydraulic model that are implemented in the RELAP-BARS 

code. The effect of initial core power on key parameters of the accident was analyzed using calculational 

results for the TMI-1 REA starting from 33% of rated power. 

1.3 Outline of Report 

Section 2 focuses on the analysis of the RELAP–BARS calculational results for considered transients. 

This section also contains descriptions of the calculational models. The Nordheim-Fuchs model is used to 

understand the role of spatial effects on the major parameters of a REA. Evaluations of the feedback 

parameter as a function of the power peaking factor are derived. This section provides also a comparison 

of the rod worth steady-state calculations performed by different codes and possible uncertainties in the 

REA parameters due to uncertainty in the rod worth prediction. 

Section 3 focuses on the uncertainty analysis for the central rod ejection accident. The methodology of the 

uncertainty analysis for the PWR REA is described. In Section 3 the uncertainty in the peak fuel enthalpy, 

core power, and power pulse width is assessed to a number of neutronic and thermal-hydraulic quantities. 

The effect of the initial core power on parameters of the accident is analyzed by a comparative study of 

the TMI-1 REA starting from 33% of rated power. 

Appendix A contains the calculational results obtained by RELAP–BARS during modeling of a REA 

with ejection of the central rod with increased worth at EOC conditions. 

5

2 SPATIAL EFFECTS FOR ROD EJECTION ACCIDENTS 

2.1 RELAP-BARS Transient Calculations 

The reactor model was based on Three Mile Island Unit 1 (TMI-1) used as an international benchmark 

exercise (Ref. 2). The reference design for the PWR is derived from the reactor geometry and operational 

data of the TMI-1 NPP. Two different cores were used in this study. First core is assumed to be at the end 

of cycle (EOC), with a boron concentration in the coolant of 5 ppm, and equilibrium Xe and Sm 

concentration. The average burnup in the core is 40.7 GWd/t (the maximum burnup is about 58 GWd/t). 

Detailed specifications for this core are given in Ref. 2. Another core is assumed to be at the beginning of 

cycle (BOC) with the average core exposure of 18.2 GWd/t and boron concentration of 1700 ppm (Ref.3). 

The hot zero power (HZP) conditions are defined as follows: the reactor power is 2772 W (10-6 of rated 

power), fuel/coolant temperature is 551 K and core inlet pressure is 15.4 MPa. The reactor has one-eight 

core symmetry, as Figures 2.1 and 2.2 show. Initial steady-states are the same for both cores: control rod 

banks 1 to 4 are completely withdrawn, banks 5 to 7 are completely inserted. Figures 2.1 and 2.2 show 

only inserted control rod banks. For the simplicity it was assumed that the delayed neutron fraction was 

521.10 pcm at EOC and 632.34 pcm at BOC. 

For the calculations, a control rod cluster of Bank 7 is ejected from the core centre (Assembly H8) or 

periphery (Assembly N12) during 0.1 s to initiate a transient. The calculations were continued for 2.5 

seconds. This duration was chosen from the previous analysis of the rod ejection accidents where it was 

found that the fuel enthalpy increase has been arrested due to the reactor trip by that time (Ref. 4). In this 

study no reactor scram was assumed. Thus, four cases were studied: two types of REA (the central or 

peripheral rod ejection) at two initial conditions (EOC or BOC). 

Steady-state calculations show that an actual worth of the ejected rod is lower compared with the delayed 

neutron fraction, β. Table 2.1 presents the calculational results obtained by the BARS code at EOC 

(Ref. 5) and BOC. It is obvious, that these magnitudes are insufficient to reach prompt critical. Since a 

power excursion below prompt critical is out of interest, the control rod worth in each case of this study 

was fixed at 1.21β. This worth is possible only when there is a large distortion of flux distribution over 

the core: for instance, if the control rod at position M11 is out of the core throughout the transient with 

ejection of the peripheral control rod at position N12. Thus, to provide 1.21β worth of the ejected rod, the 

initial core conditions have to be changed. In cases of the central rod ejection there were changes in 

neutronic parameters of the ejected rod. When the peripheral rod was ejected, the control rod in Assembly 

M11 was assumed to be stuck out of the core and the ejected rod was partly inserted to the core to provide 

required 1.21β worth. 

Due to different delayed neutron fractions at EOC and BOC, an assumption of 1.21β worth means that the 

absolute value of the rod worth at BOC is greater by 21% compared with the EOC cases. For this reason 

it can be expected in this study that consequences of BOC REAs are to be more severe in terms of fuel 

enthalpy. (To compare the results for EOC REA with the same absolute value of the rod worth as at BOC, 

i.e. 1.47β, an additional calculation was carried out for the central rod ejection. The calculational results 

are given in Appendix A.) 
6

8 9 10 11 12 13 14 15 

H 

52.86 

Bank 7 

30.19 56.25 30.85 49.53 

Bank 7 

28.11 53.86 

Bank 6 

55.78 

K 

 57.94 30.80 55.43 29.83 53.95 

Bank 5 

25.55 49.17 

L 

 57.57 

Bank 6 

30.22 54.40 27.86 23.30 47.30 

M 

 49.71 

Bank 5 

28.85 52.85 40.94 

N 

 48.75 

Bank 7 

23.86 41.45 

O 

 37.34 

 52.86 

Bank 7 

 - fuel burnup (GWd/t) 

 - No. of regulating bank 

Figure 2.1. One-Eight Core Layout (EOC Initial Conditions) 
7

8 9 10 11 12 13 14 15 

H 

30.69 

Bank 7 

0.16 

Gd + B 

29.50 0.18 

Gd + B 

24.53 

Bank 7 

0.16 

Gd + B 

36.51 

Bank 6 

48.20 

K 

 32.26 0.17 

Gd + B 

29.30 0.17 

Gd + B 

29.25 

Bank 5 

0.15 

Gd + Gd 

40.34 

L 

 31.69 

Bank 6 

0.18 

Gd + B 

30.12 0.17 

Gd + B 

0.14 39.62 

M 

 24.52 

Bank 5 

0.18 

B 

31.73 26.73 

N 

 24.89 

Bank 7 

0.17 32.22 

O 

 24.82 

 30.69 

Bank 7 

 - fuel burnup (GWd/t) 

 - No. of regulating bank 

 Gd 

B 

 - 4 fuel pins with Gd 

 - B4C burnable poison rods 

Figure 2.2. One-Eight Core Layout (BOC Initial Conditions) 
8

Table 2.1. Steady-State Calculation Results for Rod Worths 

Case Index Rod Position Delayed Neutron 

Fraction, β (pcm) Rod Worth (pcm / β) 

EOC Centre Assembly H8 521.10 349.3 / 0.670 

EOC Periphery Assembly N12 521.10 472.8 / 0.907 

BOC Centre Assembly H8 632.34 273.5 / 0.432 

BOC Periphery Assembly N12 632.34 548.5 / 0.867 

Because of the REA start-up conditions corresponded to HZP with homogeneous thermal-hydraulic 

characteristics over the core, the RELAP thermal-hydraulics input deck was the same in all cases. An 

assembly-average representation of fuel and coolant parameters with 24 axial nodes was used to treat the 

core thermal-hydraulics. Besides, in each calculation a separate heat structure was chosen to represent the 

hottest fuel pin in the reactor core. The coolant thermal-hydraulic parameters for this heat structure 

corresponded to those for an assembly where the hottest pin was located. 

The following features were investigated: core power and reactivity, fuel assembly powers, temperatures 

and enthalpies including their local values. Table 2.2 presents major core parameters of four REAs 

calculated by RELAP–BARS. The peak values of fuel temperature and enthalpy occur at the end of the 

transient (2.5 s). Total fuel enthalpy may be defined by adding the initial value of about 17 cal/g. Position 

of the hottest pins within a 15×15 PWR standard fuel assembly (FA) is given in Figure 2.3. Axial position 

of the hottest fuel pellet is given in terms of axial nodes (totally 24) from the bottom of the core. The core 

height is 357.12 cm, therefore, the node size is 14.88 cm. Thus, Node 22 is located between 312.5 cm and 

327.4 cm from the core bottom. In the BOC cases the hottest pellet is located between Nodes 19 and 20, 

i.e. between 267.8 and 297.6 cm. The power peaking factors (total, Fq, radial, Fr, and axial, Fz) are given 

at 0.15-0.2 s (when they reach their maximum values) and at the end of the transient (2.5 s). 

Figures 2.4 through 2.7 give assembly-average power distributions obtained in four REAs at time when 

the core power reaches its maximum and at the end of the transient (2.5 s). All distributions are 

normalized to an average value and, therefore, the maximum value shows the assembly-average radial 

peaking factor (at the end of the transient they coincide with those given in Table 2.2). 

A comparison between the assembly-average and pin-by-pin radial peaking factors shows about 10%-

difference in the cases with ejection of the central rod and more then 25%-difference in the cases with 

ejection of the peripheral rod. It is very important in regard to definition of the hottest fuel pin by indirect 

methods, such as a flux reconstruction, in assembly-by-assembly calculations. Since the peripheral fuel 

assemblies have, as a rule, higher intra-assembly power peaking factor, it is of interest to consider them 

from the point of view of intra-assembly power distribution. For instance, in Case EOC Periphery it was 

found that in the vicinity of the ejected rod (position N12) there were 5 hot fuel assemblies: N12 and two 

symmetrical N13 and M12. Their mean powers were within 5%-uncertainty. In spite of the fact that the 

last assembly had the peak power, the hottest fuel pin belonged to Assembly N13, diagonally adjacent to 

the core reflector. In turn, the intra-assembly peaking factor for this assembly was 1.26 whereas for the 

rest of the hot assemblies it was 1.11. As a result, 15%-underestimation in fuel enthalpy increase for the 

hottest fuel pin can occur only due to incorrect definition of its position. 
9

Table 2.2. Core Parameters in REAs 

Case Index → EOC Centre EOC Periphery BOC Centre BOC Periphery 

Ejected Rod Worth 

(pcm / β) 630.9 / 1.2107 631.0 / 1.2109 765.7 / 1.2109 765.7 / 1.2108 

Peak Power 

(GW / % Nominal) 11.05 / 398.6 4.39 / 158.2 37.49 / 1352.5 13.07 / 471.4 

Time of Peak Power (ms) 335.9 298.2 254.2 243.0 

Power Pulse Width (ms) 61.9 64.5 39.8 41.4 

Core-Average Fuel 

Temperature (K) 590.2 568.8 664.3 594.9 

Maximum FA 

Temperature (K) 812.0 785.6 898.9 898.0 

Maximum Fuel Pin 

Temperature (K) 835.3 851.6 935.9 996.1 

Maximum FA Enthalpy 

Increase (cal/g) 18.88 16.87 25.52 25.45 

Max. Pin Enthalpy 

Increase (cal/g) 20.65 21.90 28.37 33.04 

Position of the 

Hottest Assembly H9 M12 H9 N13 

Position of the 

Hottest Fuel Pellet 

H9 

Node 22 

N13 

Node 22 

H9 

Nodes 19, 20 

N13 

Nodes 19, 20 

Pin-by-Pin Total Fq 

(max / at 2.5 s) 8.04 / 6.51 20.1 / 16.2 4.11 / 3.09 12.9 / 9.52 

Assembly-Average Fq′ 

(max / at 2.5 s) 7.42 / 6.03 16.0 / 13.0 3.71 / 2.79 10.2 / 7.51 

Pin-by-Pin Radial Fr 

(max / at 2.5 s) 2.91 / 2.66 7.42 / 6.61 2.73 / 2.48 8.76 / 7.40 

Assembly-Average Fr′ 

(max / at 2.5 s) 2.68 / 2.46 5.93 / 5.32 2.46 / 2.24 6.91 / 5.83 

Average Axial Fz 

(max / at 2.5 s) 2.76 / 2.45 2.71 / 2.44 1.50 / 1.25 1.47 / 1.29 
10

 

 R R 

 R 2 R 

 2,4 

1 R R R R 

 3 

 3 C 3 

 3 

1 R R R R 

 2 

 R R 

 R R 

 

 R - control rod guide tube 

 C - instrumentation tube 

 1 - EOC Centre 

 2 - EOC Periphery 

 3 - BOC Centre 

 4 - BOC Periphery 

Figure 2.3. Position of the Hottest Pins in Fuel Assembly during REAs 
11

8 9 10 11 12 13 14 15 

H 

2.070 

1.926 

2.631 

2.463 

1.857 

1.756 

1.710 

1.646 

0.649 

0.641 

0.757 

0.774 

0.336 

0.352 

0.236 

0.249 

K 

 1.950 

1.830 

1.982 

1.884 

1.338 

1.296 

1.222 

1.218 

0.561 

0.576 

0.835 

0.876 

0.334 

0.353 

L 

 0.874 

0.841 

1.321 

1.301 

1.091 

1.103 

1.215 

1.257 

1.002 

1.054 

0.351 

0.372 

M 

 0.634 

0.638 

1.128 

1.159 

0.938 

0.976 

0.699 

0.736 

N 

 0.617 

0.642 

0.986 

1.038 

0.457 

0.483 

O 

 t = tmax 

t = 2.5 s 

→ 

→

 0.564 

0.597 

 

 2.631 

2.463 

 - FA with peak power and the hottest pin 

Figure 2.4. Assembly-Average Power Distributions (Case EOC Centre) 
12

 0.068 

0.096 

0.068 

0.094 

0.056 

0.075 

0.091 

0.119 

0.101 

0.131 

 

 0.083 

0.119 

0.132 

0.186 

0.197 

0.274 

0.174 

0.238 

0.082 

0.109 

0.234 

0.303 

0.300 

0.385 

0.226 

0.285 

0.162 

0.201 

 

0.102 

0.146 

0.182 

0.258 

0.182 

0.253 

0.249 

0.339 

0.126 

0.167 

0.198 

0.253 

0.167 

0.210 

0.389 

0.486 

0.324 

0.401 

0.378 

0.461 

0.243 

0.291 

 

 0.119 

0.165 

0.232 

0.314 

0.244 

0.320 

0.305 

0.386 

0.186 

0.228 

0.394 

0.477 

0.382 

0.461 

0.436 

0.522 

0.277 

0.325 

0.491 

0.571 

0.243 

0.280 

 0.142 

0.186 

0.326 

0.411 

0.367 

0.447 

0.527 

0.625 

0.466 

0.544 

0.525 

0.606 

0.305 

0.345 

0.622 

0.700 

0.551 

0.619 

0.436 

0.489 

 0.237 

0.287 

0.582 

0.685 

0.608 

0.698 

0.771 

0.864 

0.431 

0.471 

0.804 

0.867 

0.724 

0.783 

0.845 

0.921 

0.741 

0.808 

0.275 

0.298 

 0.590 

0.675 

0.870 

0.966 

0.874 

0.937 

1.215 

1.271 

1.012 

1.047 

1.035 

1.071 

0.522 

0.541 

0.798 

0.835 

0.334 

0.348 

H 

 0.544 

0.580 

1.418 

1.465 

1.450 

1.464 

1.772 

1.759 

0.853 

0.833 

1.181 

1.149 

0.570 

0.554 

0.408 

0.398 

K 

 1.454 

1.461 

2.152 

2.111 

2.150 

2.059 

2.610 

2.458 

1.404 

1.314 

2.202 

2.067 

0.863 

0.813 

L 

 1.544 

1.474 

3.700 

3.465 

3.583 

3.310 

4.108 

3.789 

3.229 

2.998 

1.069 

0.998 

M 

 4.042 

3.726 

5.792 

5.316 

4.221 

3.856 

2.787 

2.564 

N 

 5.506 

5.011 

5.734 

5.241 

2.257 

2.067 

O t = 

t = 

tmax 

2.5 s 

→ 

→

 3.343 

3.050 

 

 4 5 6 7 8 9 10 11 12 13 14 15 

 5.792 

5.316 

- FA with peak power 

 5.734 

5.241 

- FA with the hottest pin 

Figure 2.5. Assembly-Average Power Distributions (Case EOC Periphery) 
13

 8 9 10 11 12 13 14 15 

H 

1.847 

1.712 

2.409 

2.237 

1.969 

1.844 

1.694 

1.620 

0.713 

0.709 

0.736 

0.758 

0.330 

0.349 

0.169 

0.180 

K 

 1.979 

1.840 

1.942 

1.831 

1.431 

1.377 

1.180 

1.178 

0.617 

0.637 

0.857 

0.906 

0.258 

0.275 

L 

 0.983 

0.943 

1.301 

1.278 

1.072 

1.085 

1.141 

1.184 

1.046 

1.105 

0.280 

0.298 

M 

 0.717 

0.726 

1.135 

1.170 

0.954 

0.996 

0.630 

0.666 

N 

 0.744 

0.780 

1.138 

1.201 

0.406 

0.431 

O 

 t = tmax 

t = 2.5 s 

→ 

→

 0.572 

0.608 

 

 2.409 

2.237 

 - FA with peak power and the hottest pin 

Figure 2.6. Assembly-Average Power Distributions (Case BOC Centre) 
14

 0.053 

0.080 

0.051 

0.076 

0.039 

0.056 

0.068 

0.095 

0.078 

0.108 

 

 0.072 

0.110 

0.116 

0.175 

0.200 

0.300 

0.174 

0.257 

0.079 

0.112 

0.234 

0.323 

0.303 

0.415 

0.196 

0.264 

0.139 

0.183 

 

0.100 

0.154 

0.203 

0.311 

0.180 

0.269 

0.230 

0.336 

0.137 

0.195 

0.190 

0.260 

0.180 

0.241 

0.355 

0.472 

0.321 

0.420 

0.417 

0.540 

0.230 

0.293 

 

 0.138 

0.208 

0.229 

0.333 

0.239 

0.335 

0.297 

0.400 

0.205 

0.268 

0.379 

0.485 

0.370 

0.472 

0.427 

0.540 

0.313 

0.390 

0.519 

0.640 

0.197 

0.240 

 0.160 

0.224 

0.326 

0.436 

0.402 

0.517 

0.535 

0.668 

0.506 

0.620 

0.517 

0.626 

0.333 

0.397 

0.586 

0.693 

0.519 

0.612 

0.366 

0.429 

 0.274 

0.351 

0.594 

0.735 

0.673 

0.810 

0.782 

0.914 

0.490 

0.555 

0.775 

0.865 

0.682 

0.765 

0.754 

0.853 

0.730 

0.825 

0.204 

0.230 

 0.637 

0.765 

0.861 

0.997 

0.938 

1.040 

1.216 

1.307 

1.073 

1.137 

0.976 

1.037 

0.552 

0.588 

0.772 

0.831 

0.240 

0.258 

H 

 0.570 

0.628 

1.398 

1.477 

1.574 

1.617 

1.736 

1.746 

0.911 

0.900 

1.112 

1.089 

0.543 

0.530 

0.284 

0.278 

K 

 1.551 

1.579 

2.132 

2.104 

2.241 

2.143 

2.433 

2.271 

1.505 

1.392 

2.227 

2.062 

0.661 

0.616 

L 

 1.709 

1.623 

3.530 

3.253 

3.453 

3.119 

3.810 

3.429 

3.328 

3.022 

0.847 

0.778 

M 

 4.258 

3.815 

5.795 

5.120 

4.360 

3.844 

2.528 

2.256 

N 

 6.293 

5.504 

6.669 

5.833 

2.052 

1.809 

O t = 

t = 

tmax 

2.5 s 

→ 

→

 3.408 

2.981 

 

 4 5 6 7 8 9 10 11 12 13 14 15 

 6.669 

5.833 

- FA with peak power and the hottest pin 

Figure 2.7. Assembly-Average Power Distributions (Case BOC Periphery) 
15

For this study with relatively simple scenario of the transient starting from HZP, it was not so difficult to 

define the hottest pins in advance. It is clear, that as can be seen in Figures 2.1 and 2.2, in the cases with 

ejection of the central rod the hottest pins have to be located in Assembly H9 (adjacent to Assembly H8) 

with relatively low fuel burnup. Analogously, in cases with ejection of the peripheral rod the hottest pins 

are to be located somewhere in Assemblies M12, N13 or even N12 with the highest powers. But in many 

other cases with rather complicated spatial power distribution history, the flux reconstruction procedure 

may encounter serious difficulties. Another problem is use of this procedure for the peripheral assemblies 

adjacent to the core reflector where the flux reconstruction method may lead to significant uncertainty in 

pin powers. 

Core-average axial power distributions are given in Figure 2.8 for two cases with ejection of the central 

rod at EOC and BOC at two time moments: 0.2 s and the end of the transient. It can be seen from Figures 

2.4 through 2.8 and the data presented in Table 2.2, that the power peaking factor decreases due to the 

core heat-up during the transient: by 20% at EOC and by 25% at BOC. 

Figures 2.9 through 2.16 presents the following parameters as a function of time of the transients: the core 

power, reactivity, and peaking factors for the first second, and the fuel temperature and fuel enthalpy 

increase for the hottest fuel pins. As can be seen in Figures 2.9-2.12, the power excursion starts at about 

0.2 s. The power reaches a peak value at 0.24-0.34 s, then, due to negative reactivity feedback the power 

excursion is terminated. 

The power pulse width is 62-65 ms at EOC and about 40 ms at BOC, thus, the power pulse (with the right 

boundary of 10% of a peak power) lies within 0.2-0.4 s at EOC and within 0.2-0.3 s at BOC. Reactivity 

reaches its maximum at 0.13-0.15 s due to withdrawal of the ejected rod and continues to be constant 

during about 0.10-0.15 s. Then reactivity begins to drop as a result of the negative reactivity feedback due 

to fuel heat-up and, later, due to coolant heat-up. At the time of the core peak power, reactivity is equal to 

approximately 1β. The behavior of the total power peaking factor, Fq, is rather similar: firstly it increases 

due to the rod ejection, then there is a “plateau” area with further decrease due to the core heat-up. 

Figures 2.13 through 2.16 indicate that about 80% of the fuel enthalpy increase is produced during first 

0.5-0.6 s after the rod ejection. It should be noted also that in the cases with ejection of the peripheral 

control rod, 50% of the core energy is produced only in several fuel assemblies: 17 at BOC and 23 at 

EOC, i.e. about 10-13% of the total number of fuel assemblies in the core (177). 

Now consider an influence of spatial effects on the REA parameters. As it can be seen in Table 2.2 and 

Figures 2.9-2.16, noticeable differences in the power peak, fuel temperature and enthalpy increase occur. 

In the cases with ejection of the peripheral rod the peak power is lower than that in the cases with ejection 

of the central rod: by 2.5 times at EOC and by 2.9 times at BOC. On the other hand, practically the same, 

but inverse relationship occurs for the power peaking factor. The total core energy deposition can be 

characterized by a core-average fuel temperature increase. This parameter shows a similar tendency: 

corresponding ratio for the rod ejection cases is 2.2 at EOC and 2.6 at BOC. Consequently, it is worth to 

conclude that there is an evident correlation between the peak power, reactor energy deposition and the 

power peaking factor, Fq : the higher the Fq value, the lower the peak power and energy deposition in the 

core. However, it is clear, that a REA with the highest magnitude of Fq is more severe from the point of 

view of the local fuel enthalpy increase. Thus, in fact, the effect of the power peaking factor is revealed in 

two opposite directions, namely, a higher value of the peaking factor tends to decrease the total energy 

deposition in the core and, on the other hand, to increase the local fuel enthalpy. 
16

0 40 80 120 160 200 240 280 320 360

Core Heitght (cm)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Relative Power

Central Rod Ejection

(EOC HZP)

 t = 0.2 s

 t = 2.5 s

0 40 80 120 160 200 240 280 320 360

Core Heitght (cm)

0.0

0.3

0.6

0.9

1.2

1.5

1.8

Relative Power

Central Rod Ejection

(BOC HZP)

 t = 0.2 s

 t = 2.5 s

Figure 2.8. Average Axial Power Distribution (Cases EOC and BOC Centre) 
17

0.0 0.2 0.4 0.6 0.8 1.0

Time (sec)

0

100

200

300

400

Relative Power (% of Nominal)

0.0

0.4

0.8

1.2

1.6

Reactivity ($)

Central Rod Ejection

 (EOC HZP)

0.0 0.2 0.4 0.6 0.8 1.0

Time (sec)

2

4

6

8

10

Power Peaking Factor

Central Rod Ejection

(EOC HZP)

Pin-by-Pin

Assembly

Figure 2.9. Power, Reactivity and Peaking Factors vs. Time (Case EOC Centre) 
18

0.0 0.2 0.4 0.6 0.8 1.0

Time (sec)

0

40

80

120

160

Relative Power (% of Nominal)

0.0

0.4

0.8

1.2

1.6

Reactivity ($)

Peripheral Rod Ejection

 (EOC HZP)

0.0 0.2 0.4 0.6 0.8 1.0

Time (sec)

8

12

16

20

24

Power Peaking Factor

Peripheral Rod Ejection

(EOC HZP)

Pin-by-Pin

Assembly

Figure 2.10. Power, Reactivity and Peaking Factors vs. Time (Case EOC Periphery) 
19

0.0 0.2 0.4 0.6 0.8 1.0

Time (sec)

0

400

800

1200

1600

Relative Power (% of Nominal)

0.0

0.4

0.8

1.2

1.6

Reactivity ($)

Central Rod Ejection

 (BOC HZP)

0.0 0.2 0.4 0.6 0.8 1.0

Time (sec)

1

2

3

4

5

Power Peaking Factor

Central Rod Ejection

(BOC HZP)

Pin-by-Pin

Assembly

Figure 2.11. Power, Reactivity and Peaking Factors vs. Time (Case BOC Centre) 
20

0.0 0.2 0.4 0.6 0.8 1.0

Time (sec)

0

100

200

300

400

500

Relative Power (% of Nominal)

0.0

0.3

0.6

0.9

1.2

1.5

Reactivity ($)

Peripheral Rod Ejection

 (BOC HZP)

0.0 0.2 0.4 0.6 0.8 1.0

Time (sec)

4

6

8

10

12

14

Power Peaking Factor

Peripheral Rod Ejection

(BOC HZP)

Pin-by-Pin

Assembly

Figure 2.12. Power, Reactivity and Peaking Factors vs. Time (Case BOC Periphery) 
21

0.0 0.5 1.0 1.5 2.0 2.5

Time (sec)

500

600

700

800

900

Fuel Temperature (K) in Hottest Pellet

Central Rod Ejection (EOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

0 0.5 1 1.5 2 2.5

Time (sec)

0

5

10

15

20

25

Enthalpy Increment (cal/g) in Hottest Fuel Pellet

Central Rod Ejection (EOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

Figure 2.13. Fuel Temperature and Enthalpy in Hot Pins vs. Time (Case EOC Centre) 
22

0.0 0.5 1.0 1.5 2.0 2.5

Time (sec)

500

600

700

800

900

Fuel Temperature (K) in Hottest Pellet

Peripheral Rod Ejection (EOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

0 0.5 1 1.5 2 2.5

Time (sec)

0

5

10

15

20

25

Enthalpy Increment (cal/g) in Hottest Fuel Pellet

Peripheral Rod Ejection (EOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

Figure 2.14. Fuel Temperature and Enthalpy in Hot Pins vs. Time (Case EOC Periphery) 
23

0.0 0.5 1.0 1.5 2.0 2.5

Time (sec)

500

600

700

800

900

1000

Fuel Temperature (K) in Hottest Pellet

Central Rod Ejection (BOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

0 0.5 1 1.5 2 2.5

Time (sec)

0

5

10

15

20

25

30

35

Enthalpy Increment (cal/g) in Hottest Fuel Pellet

Central Rod Ejection (BOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

Figure 2.15. Fuel Temperature and Enthalpy in Hot Pins vs. Time (Case BOC Centre) 
24

0.0 0.5 1.0 1.5 2.0 2.5

Time (sec)

500

600

700

800

900

1000

Fuel Temperature (K) in Hottest Pellet

Peripheral Rod Ejection (BOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

0 0.5 1 1.5 2 2.5

Time (sec)

0

5

10

15

20

25

30

35

Enthalpy Increment (cal/g) in Hottest Fuel Pellet

Peripheral Rod Ejection (BOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

Figure 2.16. Fuel Temperature and Enthalpy in Hot Pins vs. Time (Case BOC Periphery) 
25

As can be seen in Table 2.2 both cases with ejection of the peripheral rod show lower core-average fuel 

temperature compared with the cases with ejection of the central rod: 569 K at EOC and 595 K at BOC 

against 590 K and 664 K, respectively. Practically the same picture is observed for the peak assemblyaverage temperatures at EOC: 786 K against 812 K. But in the BOC cases they are very close: 899 K and 

898 K. Consequently, the similar situation was found for the maximum assembly-average enthalpy 

increase: 16.9 cal/g against 18.9 cal/g at EOC, and about 25.5 cal/g in both BOC cases. 

Therefore, from the point of view of assembly-by-assembly representation, both cases with ejection of the 

central control rod result in higher values of the assembly-average enthalpy increase compared with 

ejection of the peripheral rod. The next step is a definition of the hottest fuel pin. Consider the EOC cases. 

If this procedure is performed for the hottest fuel assembly, then we need the intra-assembly peaking 

factors for Assembly H9 (Case EOC Centre) and Assembly M12 (Case EOC Periphery). For both 

assemblies these factors were about 1.1. Thus, it can be expected the following values for the maximum 

pin enthalpy increase: 18.6 and 20.7 cal/g for ejection of the peripheral and central rod, respectively. As 

can be compared with the data presented in Table 2.2, such an approach leads to about 18% 

underestimation in the maximum pin enthalpy for Case EOC Periphery. 

In the BOC cases the assembly-average parameters (the maximum fuel temperature and enthalpy) were 

found as approximately the same (899 K and 25.5 cal/g, respectively). The intra-assembly peaking factor 

was about 1.1 for Assembly H9 (Case BOC Centre) and about 1.27 for Assembly N13 (Case BOC 

Periphery). Thus, the maximum fuel pin enthalpy increase is estimated as 32.3 and 28.1 cal/g for ejection 

of the peripheral and central rod, respectively. The last values are close to those obtained from the pin-bypin calculations (see Table 2.2). 

Thus, the pin-by-pin calculations show that the peak fuel pin enthalpy increase is higher in both cases 

with ejection of the peripheral rod: 21.9 cal/g at EOC and 33.0 cal/g at BOC against 20.7 cal/g and 28.4 

cal/g, respectively. 

To investigate the influence of the power peaking factor on major REA parameters it is of interest to 

consider a simple approximation as described in the next section. This approach based on a NordheimFuchs model (Ref. 6), which is well-known and widely used for analytical estimations of different 

neutronic parameters in pulsed reactors. 

2.2 Nordheim-Fuchs Model 

This model is valid when the rod worth is greater than the value of the delayed neutron fraction, β. Next 

assumption is that the reactivity change due to feedback is proportional to the energy deposition. We will 

consider also an adiabatic approximation when there is no heat transfer to the coolant. Then using the 

point kinetics equations, the reactivity is: 

ρ(t) = ρo − γ Q(t) (1) 

where ρo is the reactivity worth of the ejected rod, γ is the feedback parameter, and Q(t) is the energy 

deposition. As it can be shown (Ref. 6), the peak power, Pmax, and the total energy deposition, Qtot, are: 

Pmax = (ρo − β)

2 ⁄ (2Λγ) (2) 
26

and 

Qtot = 2 (ρo − β) ⁄ γ (3) 

where Λ is the prompt neutron lifetime. According to Eq. (1) the feedback parameter is: 

γ = −δρ ⁄ δQ (4) 

where δQ is the energy deposition increase and δρ is the corresponding change in the reactivity. The 

adiabatic assumption means that the only feedback is the Doppler effect, thus, δQ and δρ can then be 

expressed in terms of the change in fuel temperature, ΔT: 

δQ = ∫

v 

Cp ΔT dV (5) 

and 

δρ = < φ+, αD ΔTψ> (6) 

where Cp is the fuel heat capacity, V is the fuel volume in the core, φ+

 is the adjoint flux solution for the 

initial steady state, αD is the fuel temperature (Doppler) reactivity coefficient, and ψ is the neutron flux 

with the following normalization condition: 

< φ+,ψ> = 1 (7) 

where the inner product represents integral over energy and the core volume. For further analysis it is 

worth to simplify both Eq. (5) and Eq. (6) if the fuel heat capacity and the Doppler coefficient are 

assumed to be constant. Such an approximation yields: 

δQ = Cp V ΔTavr (8) 

and 

δρ = αD < φ+, ΔTψ> (9) 

where ΔTavr is the average change in fuel temperature over the core. In this case Eq. (4) can be expressed 

as: 

γ = γo < φ+, ΔTψ> ⁄ ΔTavr (10) 

where 

γo = −αD ⁄ (Cp V) . (11) 

It can be seen that γ = γo under the assumption of constant ΔT, i.e. ΔT = ΔTavr. In other words it means 

that γo represents the feedback parameter in case when fuel temperature changes by the same value in 

each fuel pellet over the core. This is possible when power distribution is uniform in both radial and axial 

directions. It is obvious, that γo is the lower limit of the feedback parameter (αD is always negative) and, 

thus, provides the maximum values of the peak power, Pmax, and the total energy deposition, Qtot, as seen 

from Eq. (2) and Eq. (3). To estimate the upper limit of γ, it is necessary to keep in mind that generally 

ΔT and weight functions φ+

 and ψ in Eq. (10) have quite different spatial distributions. Nevertheless, 

taking into account Eq. (7) the upper limit of the feedback parameter, γmax, can be estimated as follows: 
27

γmax = γo ΔTmax ⁄ ΔTavr (12) 

where ΔTmax is the maximum change in fuel temperature over the core. ΔTmax ⁄ ΔTavr is the peaking factor 

for the change in fuel temperature. Under the adiabatic assumption, its value practically coincides with 

the power peaking factor, Fq. Then, the upper limit of γ is: 

γmax = γo Fq . (13) 

Thus, both limits are proportional to the power peaking factor (taking into account that Fq is equal to 1 for 

the lower limit). Therefore, one can expect also that γ may be approximated as a linear function of Fq. In 

the present study, based on the RELAP-BARS calculations this evaluation was done. 

Figure 2.17 shows the relative feedback parameter, γ/γo, as a function of the pin-by-pin and assemblyaverage power peaking factors, Fq and Fq′, respectively. As shown in the figure, both dependencies, f(Fq) 

and f(Fq′), are approximately linear functions. More precise approximations for γ were found as follows: 

γ ≈ γo ( 0.38 Fq + 0.24 ) (14) 

and 

γ′ ≈ γo (0.50 Fq′ − 0.09 ) . (15) 

For the present analysis we can estimate the values of the feedback parameter, γ, during considered 

transients. At HZP conditions (T = 551 K) the value of the fuel heat capacity, Cp, is about 3.0 MJ/(m3

K). 

The fuel volume in the TMI-1 core is 9.2 m3

. The fuel temperature (Doppler) coefficient is about 

2.8 pcm/K. Therefore, according to Eq. (11): 

γo ≈ 1.0×10-6 MJ-1 . (16) 

This result was also confirmed by the RELAP–BARS calculation of Case EOC Centre under an 

assumption of the uniform heat-up over the core volume (i.e. when ΔT = ΔTavr). Besides aforementioned 

peak power and total energy deposition, the power pulse width at the pulse half, Δt1/2, can be estimated 

(within the Nordheim-Fuchs model) as follows: 

Δt1/2 = 3.53Λ/ (ρo − β) . (17) 
28

Assembly-average Pin-by-pin

0 4 8 12 16 20

Power Peaking Factor

0

2

4

6

8

Feedback Parameter (relative units)

Figure 2.17. Feedback Parameter vs. Power Peaking Factor 
29

The Nordheim-Fuchs model does not take into account the effects of delayed neutrons. For this reason an 

applicability of this approach is restricted by the power pulse area. For the present study it means that this 

model is valid only to 0.4 s at EOC and to 0.3 s at BOC. Instead of the total energy deposition, Qtot, as 

defined by Eq. (3), we will use the energy deposition at the power peak time, Qo, which equals to Qtot / 2. 

Table 2.3 presents some of the neutronic parameters of interest obtained from the RELAP–BARS 

calculations of the transients. The prompt neutron lifetime, Λ, feedback parameter, γ, and peaking factors, 

Fq and Fq′, were averaged over those limits. The core energy deposition, Qo, was defined as integral of the 

total core power up to the time of the power peak. 

Table 2.3. Some Neutronic Parameters of REAs 

Parameter EOC Centre EOC Periphery BOC Centre BOC Periphery 

Core Energy 

Deposition, Qo (GJ) 0.366 0.149 0.818 0.292 

Prompt Neutron 

Lifetime, Λ (s) 1.86×10-5 1.87×10-5 1.46×10-5 1.49×10-5

Feedback 

Parameter, γ (MJ-1) 3.25×10-6 7.48×10-6 1.69×10-6 4.72×10-6

Pin-by-Pin Total 

Peaking Factor, Fq

7.70 18.9 3.84 12.0 

Assembly-average 

Peaking Factor, Fq′ 6.95 15.2 3.47 9.48 

Using the data presented in Table 2.3 and the ejected rod worth (see Table 2.2) it is easy to evaluate the 

following parameters: Pmax, Qo, γ and Δt1/2 by Eq. (2), (3), and (15)–(17). Table 2.4 contains these 

evaluated parameters. 

A comparison between the data presented in Table 2.4 with the calculated results (see Tables 2.2 and 2.3) 

shows that the peak power, Pmax, and core energy deposition, Qo, are underestimated by 3–10%. 

Practically the same uncertainties are found for the power pulse width, Δt1/2. Both feedback parameters 

are estimated within 5% uncertainty. (It should be noted, that the approximations defined by Eq. (14) and 

Eq.(15) may differ for REAs with other neutronic parameters.) 
30

Table 2.4. Evaluated Neutronic Parameters of REAs 

Parameter EOC Centre EOC Periphery BOC Centre BOC Periphery 

Peak Power 

(GW / % Nominal) 9.97 / 360 4.32 / 156 36.0 / 1300 12.6 / 456 

Core Energy 

Deposition, Qo (GJ) 0.338 0.147 0.789 0.282 

Feedback 

Parameter, γ (MJ-1) 3.17×10-6 7.42×10-6 1.70×10-6 4.80×10-6

Feedback 

Parameter, γ′ (MJ-1) 3.38×10-6 7.51×10-6 1.65×10-6 4.65×10-6

Power Pulse 

Width, Δt1/2 (ms) 59.8 60.1 38.6 39.5 

Thus, these results confirm an assumption about strict correlation between the peak power, core energy 

deposition and as a result, fuel temperature (or enthalpy) increase, and the power peaking factor. 

Presented simplified estimations allow to conclude the following fact: the feedback parameter, γ, is 

approximately proportional to the power peaking factor. In its turn, both the peak power and energy 

deposition are inversely proportional to γ and, as a result, to the power peaking factor. 

Therefore, it may be concluded that if the adiabatic assumption were valid during a REA, it would be 

expected that the peak fuel enthalpy increase practically does not depend on spatial effects. In other 

words, in the transients with quite different power distributions in the radial and axial directions, but with 

the same neutronic parameters, ρo and β, the results for the peak fuel enthalpy increase are to be obtained 

as very similar. 

Of course, a REA may be considered as an adiabatic process in terms of energy deposition only within a 

narrow time interval limited by the power pulse area when a very fast rise and decrease of power takes 

place. When power decreases to about 10% of its peak value, effects of delayed neutron become distinct. 

After the power pulse, the energy deposition process is rather mild. During the present study with 1.21β

REAs it was found the following. Just after the termination of the power pulse (at 0.4 s at EOC or 0.3 s at 

BOC) about 3-4% of fuel energy release is transmitted to the coolant; this value becomes about 20% at 

1 s. Besides, the prompt fraction of energy deposition (within the limits of the power pulse when the 

adiabatic assumption is valid) gives only about 30-40% of the total energy deposition at the end of the 

transient. On this evidence, it is clear that the above reasoning should be considered as very approximate. 

It should be also taken into account that the feedback parameter, γ, changes by several times during the 

transient as calculated by RELAP-BARS. As it can be seen in Figures 2.13 and 2.14 for the EOC cases, 

the difference between the assembly-average fuel enthalpy increase reaches 10% already at 0.4 s. From 

the other hand, as shown in Figures 2.15 and 2.16, for the BOC cases this difference is very small. 

Nevertheless, above mentioned simplified approach may be very useful to estimate some important 

parameters of a REA. 
31

2.3 Results of Steady-State Calculations 

Recently an intercomparison of PWR REA calculations performed by different codes has been carried out 

(Ref. 5). In the framework of that study a number of steady-state calculations were done using the 

following neutronic codes: BARS (Ref. 1), PARCS (Ref. 7) and CRONOS2 (Ref. 8), developed in the 

Russian Federation, the United States, and France, respectively. PARCS and CRONOS2 are assembly-byassembly nodal diffusion codes. During that study they used the same two-group cross-sections generated 

with the CASMO-3 code. BARS used four-group lambda-matrices generated with the TRIFON code. The 

reactor model was the same as defined in Section 2 for Case EOC Centre (see also Figure 2.1). 

Among those steady-state calculations there were calculations of each control rod in Bank 7 at HZP 

conditions. Table 2.5 presents the results of the steady-state calculations of the rod worths. Last column 

indicates average deviations of the BARS data from the PARCS and CRONOS2 results. Comparison of 

the rod worths shows that the differences between the PARCS and CRONOS2 results do not exceed 3%. 

The BARS result is slightly higher for Rod H8 (by 1%); the deviation for Rod H12 is 8%. The most 

deviation was found for Rod N12 (about 35%). 

The same steady-state calculations but at BOC HZP conditions (see Figure 2.2) have been performed by 

the PARCS code (Ref. 9). Table 2.6 presents those results in comparison with the BARS data. As can be 

seen in Table 4.2, the most deviations occur for Rod H8 (18%) and Rod N12 (15%). The deviation for 

Rod H12 (9%) is practically the same as was found for EOC HZP case. 

From a comparison of the data presented in Tables 2.5 and 2.6, the following observations can be 

obtained. The BARS results indicate that the heaviest rod (Rod N12) has a similar worth in terms of 

delayed neutron fraction, β, (about 0.9β) in both cases (at BOC its worth is 16% higher in absolute units). 

The PARCS results show that at BOC conditions the worth of Rod N12 is higher by 14% in absolute units 

and by 39% in terms of β. Nevertheless, 0.91β worth seems to be the highest one. For this reason to reach 

a prompt critical (i.e. a rod worth of more than 1β) in reality it is necessary to change the initial control 

rod arrangement in the core. If Rod M11 of Bank 5 is assumed to be stuck-out at the upper position (i.e. 

out of the core), the worth of Rod N12 can be significantly increased. As the BARS steady-state 

calculations show, the worth of Rod N12 is as follows: 

• 907 pcm or 1.74 β at EOC HZP conditions; 

• 954 pcm or 1.51 β at BOC HZP conditions. 

The situation when a single control rod of any regulating bank is found as stuck-out at the upper or 

intermediate position, is possible as a result of any control system malfunction during a rector trip. Then 

if a reactor reaches a critical level at HZP, ejection of a control rod neighboring with a stuck rod may lead 

to a prompt critical. 

In conclusion it should be noted that 35% uncertainty in the rod worth as was found during the above 

mentioned intercomparison between the PARCS and BARS steady-state results (see Table 2.5), may lead 

to a significant uncertainty in consequences of a REA, especially if the ejected rod worth is above 1β. For 

instance, in a 1.2β REA a 35% uncertainty in the ejected rod worth may results in about 300% uncertainty 

in the core energy deposition and, consequently, in the peak fuel enthalpy increase, as it can be seen from 

Eq. (3) presented in the previous section. 
32

Table 2.5. Steady-State Calculational Results for EOC HZP Case 

Parameter BARS PARCS CRONOS2 Deviation 

Worth of Rod H8 (pcm/β) 349 / 0.670 347 / 0.666 345 / 0.662 3 / 0.006 

Worth of Rod H12 (pcm/β) 204 / 0.391 188 / 0.361 188 / 0.361 16 / 0.030 

Worth of Rod N12 (pcm/β) 473 / 0.907 344 / 0.660 353 / 0.677 125 / 0.240 

Table 2.6. Steady-State Calculational Results for BOC HZP Case 

Parameter BARS PARCS Deviation 

Worth of Rod H8 (pcm/β) 273 / 0.432 230 / 0.363 43 / 0.069 

Worth of Rod H12 (pcm/β) 164 / 0.259 150 / 0.237 14 / 0.022 

Worth of Rod N12 (pcm/β) 548 / 0.867 477 / 0.754 71 / 0.113 
33

3 UNCERTAINTY ANALYSIS 

3.1 Methodology 

The methodology of the uncertainty analysis is close to that developed in BNL. It is based on a sensitivity 

study to global quantities that are explicitly used in point kinetics equations or can be taken into account 

implicitly in point kinetics through thermal-hydraulic feedback. The approach does not require validity 

for the adiabatic assumption and is based on the non-adiabatic thermal-hydraulic model realized in the 

RELAP5/MOD3.2 code (Ref. 3). 

Assuming that a safety parameter (y) is a function of a number of above quantities (x) and the random 

error in each quantity is normally distributed, the square of the uncertainty in the parameter y can be 

written: 

(δy/y)2 = Σ(Sx)

2

(δx/x)2 (18) 

where δx/x is the uncertainty in the quantity x, Sx is the sensitivity of the parameter y to the quantity x, 

and the summation is over all quantities of interest. 

It was studied the uncertainty in the following safety parameters (y): 

• peak local fuel enthalpy, 

• maximum core power, 

• power pulse width. 

The sensitivities Sx to the quantities x were obtained from 3-D pin-by-pin calculations for different 

quantities x using the RELAP-BARS coupled code. The uncertainties in the neutronic and thermalhydraulic quantities δx/x were estimated by engineering judgement, using evidence from available 

references and validation results for the BARS code. 

3.2 Analysis for Central Rod Ejection 

The uncertainty analysis in safety parameters was carried out for the TMI-1 PWR with a high burnup 

core. The reactor of 2772 MW rated power, having one-eight symmetry, contains fuel assemblies with 

fuel burnup ranged from 23 up to 58 GWd/t (at the end of the cycle) as shown in Figure 2.1. The REA 

was defined for the central control rod at HZP with an ejection time of 100 ms (Ref. 5). The reference 

(without scram) transient duration was 2.5 s. Figure 3.1 shows the core power and the reactivity as a 

function of time for the reference case up to the end of the transient (2.5 s). 
34

0 0.5 1 1.5 2 2.5

Time (sec)

0

0.4

0.8

1.2

1.6

Reactivity ($)

0

1

2

3

4

Relative Reactor Power

Figure 3.1 Core Power and Reactivity vs. Time 
35

The following values of the key parameters were obtained as reference ones: 

• ejected rod worth − 1.21β; 

• peak power − 398.6% of rated power; 

• time of peak − 336 ms; 

• power pulse width − 61.9 ms; 

• maximum fuel pellet enthalpy − 37.6 cal/g; 

• maximum increase in fuel pellet enthalpy − 20.6 cal/g. 

In the reference transient the local fuel enthalpy reaches its maximum value at the end of the transient. To 

estimate a real time when the fuel enthalpy reaches its maximum, an additional transient with scram was 

calculated. In this transient it was supposed that reactor scram occurs with 0.45 s delay at 35% of rated 

power. Control rods movement during the scram was modeled with a speed of 155.8 cm/s (Ref. 2). 

Figure 3.2 shows the fuel pellet enthalpy increment as a function of time for both reference and additional 

transients. The peak of 17.4 cal/g in the fuel enthalpy increment occurs at the time of 0.785 s for the 

transient with scram. The reference transient overestimates the fuel enthalpy by less than 0.5% at that 

moment. 

The following neutronic and thermal-hydraulic quantities were taken into consideration during the 

uncertainty analysis: 

• reactivity worth of the ejected rod (ρ), 

• delayed neutron precursor fraction (β), 

• fuel temperature (Doppler) reactivity coefficient (αd), 

• moderator density reactivity coefficient (αm), 

• pellet heat capacity (Cp), 

• gap conductance (hg), 

• pellet conductivity (Kf), 

• clad-moderator heat transfer coefficient (hw), 

• fraction of energy deposited directly in the moderator (γ), 

• radial power peaking factor for the pellet (Fp). 

Sensitivity Sx was obtained as a result of corresponding calculation of the transient with perturbed 

quantities. Different variations of these neutronic and thermal-hydraulic quantities from their reference 

values were used. The reference value of the delayed neutron precursor fraction was perturbed by –10%; 

the value of the Doppler reactivity coefficient was changed by –9%; the value of the moderator density 

reactivity coefficient was increased by 2.1 times. All table data for pellet heat capacity, gap conductance, 

pellet conductivity, and clad-moderator heat transfer coefficient were increased by 10%. The energy 

deposited directly in the moderator was not taken into account in the reference transient. To estimate the 

sensitivity to this quantity, the values of 2% and 5% were considered for the fraction of energy deposited 

in the moderator. In the reference calculation the radial power distribution in the pellet was assumed as 

uniform one. To estimate the sensitivity to this distribution, the parabolic power distribution was 

considered with the peaking factor of 1.05. 
36

0 0.5 1 1.5 2 2.5

Time (s)

0

5

10

15

20

25

Fuel Enthalpy Increment (cal/g)

Legend:

without scram

with scram

Figure 3.2. Fuel Pellet Enthalpy Increment vs. Time 
37

The most important quantity is the reactivity worth of ejected control rod. The sensitivity of the maximum 

fuel enthalpy to the control rod worth strongly depends on both reference and perturbed values of the rod 

worth. To obtain the conservative estimation for this sensitivity it is necessary to use the perturbation 

determined by engineering judgement of the uncertainty in the rod worth instead of arbitrary small 

perturbation. Unfortunately, the BARS pin-by-pin neutronic model does not allow increasing the rod 

worth by more than 4% against its reference value (1.21β). To consider larger perturbation in the control 

rod worth the following formula can be applied to the sensitivity of the fuel enthalpy to the control rod 

worth: 

Sρ = Sρ1(1-β/ρ)/(1-β/ρ1) (19) 

where Sρ is the sensitivity for the required perturbed value of the rod worth (ρ) and Sρ1 is the sensitivity 

for the perturbed value ρ1 (ρ > ρ1). Sρ1 was obtained from RELAP-BARS calculation with the rod worth 

increased by 3.7% in comparison with the reference value (ρ0 = 1.21β). This formula was derived using a 

simple expression for the energy deposition obtained in a frame of the Nordheim-Fuchs approximation 

(Ref. 10). The formula was checked using another value for ρ1 obtained by increasing the reference value 

ρ0 by 2.4%. Both results are very close. 

The following values were obtained for the uncertainties in the neutronic and thermal-hydraulic 

quantities: 

• reactivity worth of the ejected rod − 15%; 

• delayed neutron precursor fraction − 5%; 

• fuel temperature reactivity coefficient − 15%; 

• moderator density reactivity coefficient − 5%; 

• pellet heat capacity − 8%; 

• gap conductance − 110%; 

• pellet conductivity − 25%; 

• clad-moderator heat transfer coefficient − 10%; 

• fraction of energy deposited in the moderator − 20%; 

• radial power peaking factor for the pellet − 5%. 

The uncertainty in the calculated rod worth was estimated as equivalent to two standard deviations based 

on the data presented in Ref. 10. The uncertainties in the delayed neutron precursor fraction, the Doppler 

coefficient, and pellet heat capacity were taken from Ref. 10. The maximum uncertainty in the Doppler 

coefficient, obtained from the BARS validation results (Ref. 12) does not exceed the uncertainty 

estimated in Ref. 10 using engineering judgement. The uncertainty in the moderator density reactivity 

coefficient was obtained from the BARS validation results (as a result of comparisons with precise Monte 

Carlo calculations) (Ref. 12). The uncertainty in gap conductance was estimated taking into account that 

the gap closure could take place. It was obtained by a calculation of the transient with the closed gap. The 

uncertainty in pellet conductivity was estimated using the data from handbook (Ref. 13), and the 

uncertainty in the clad-moderator heat transfer coefficient was taken from (Ref. 14). The uncertainties in 

the fraction of energy deposited directly in the moderator and in the radial power peaking factor for the 

pellet were estimated using engineering judgement. 
38

It should be noted that unlike the maximum core power and power pulse width, the maximum fuel pellet 

enthalpy is a local parameter. So, the uncertainty in the local power should be taken into account to 

estimate the uncertainty in the fuel enthalpy together with above-mentioned quantities. A perturbation in 

local power is not calculated by RELAP-BARS directly without perturbations other quantities. Therefore, 

the approach proposed in Ref. 10 was used to estimate contribution of the uncertainty in local power to 

the uncertainty in fuel pellet enthalpy. If one assumes that the increase in fuel pellet enthalpy is 

proportional to the local power form factor FL, then the contribution of the uncertainty in local power to 

the uncertainty in fuel enthalpy can be considered as the addition of (δFL

/ FL)

2

 to the formula (18). FL is 

defined as the fuel pellet power divided by the total core power. Based on the analysis carried out in Ref. 

10, the uncertainty of 8% was taken for FL. 

Table 3.1 presents the calculational results for the sensitivity of fuel pellet enthalpy. 

Table 3.1. Sensitivity of Fuel Pellet Enthalpy 

Quantity PARCS BARS (at = 0.785 s) BARS (at = 2.5 s) 

(x) Sx Sx (Sx)

2

(δx/x)2 Sx (Sx)

2

(δx/x)2

ρ 5.5 7.14 1.147 4.36 0.428 

β -4.0 -3.05 0.023 -1.34 0.004 

αd -1.0 -0.90 0.018 -0.77 0.013 

αm - -0.18 < 10-3 -0.39 < 10-3

Cp 0.9 0.04 < 10-3 0.15 < 10-3

hg - -0.09 0.010 -0.12 0.017 

Kf - -0.04 < 10-3 -0.20 0.003 

hw - -0.07 < 10-3 -0.08 < 10-3

γ - -0.03 < 10-3 -0.02 < 10-3

Fp - -0.20 < 10-3 -0.23 < 10-3

The RELAP-BARS results for the sensitivity of fuel enthalpy Sx and for contributions of each quantity x 

to the uncertainty in fuel enthalpy (Sx)

2

(δx/x)2 are presented at two time moments: the time of the 

maximum fuel enthalpy in the transient with scram (t=0.785 s) and for the end of the transient (t=2.5 s). 

For comparison, corresponding results for the sensitivity of fuel enthalpy obtained using the PARCS code 

(Ref. 10) are presented in Table 3.1 too. The results demonstrate that the sensitivity of fuel enthalpy to the 

most of quantities strongly depends on time because of non-adiabatic nature of the transient. Maximum 

contribution to the uncertainty in fuel enthalpy is due to the uncertainty in rod worth. 
39

Qualitatively, the RELAP-BARS results are agreed with the PARCS ones. However, the PARCS results 

give conservative estimation for the sensitivity to the most of the quantities in comparison with the 

RELAP-BARS results, because the adiabatic approximation was used in the PARCS calculations. Table 

3.1 shows larger value of the sensitivity to rod worth obtained using BARS at t = 0.785 s, because a 

perturbation of +15% in rod worth was considered using Eq. (19) instead of a very small perturbation 

used in PARCS. Note that the BARS gave the value of 5.14 for the sensitivity for a perturbation in rod 

worth of +3.7% and the value of 4.93 for a perturbation of +2.4%. 

The resulting uncertainty in the maximum fuel pellet enthalpy (the transient with scram) obtained using 

the BARS calculations is 110% for a rod worth of 1.2β. The uncertainty in fuel pellet enthalpy is 69% at 

the end of the reference transient. 

The RELAP-BARS calculational results for the sensitivity of the maximum core power and power pulse 

width are given in Table 3.2. 

Table 3.2. Sensitivity of Maximum Core Power and Power Pulse Width 

Quantity Maximum Core Power Pulse Width 

(x) Sx (Sx)

2

(δx/x)2 Sx (Sx)

2

(δx/x)2

ρ 13.87 4.328 -4.98 0.558 

β -10.77 0.290 3.09 0.024 

αd -0.97 0.021 < 0.01 < 10-3

αm -0.01 < 10-3 -0.02 < 10-3

Cp 1.05 0.007 0.01 < 10-3

hg < 0.01 < 10-3 < 0.01 < 10-3

Kf < 0.01 < 10-3 < 0.01 < 10-3

hw -0.01 < 10-3 -0.01 < 10-3

γ -0.03 < 10-3 -0.01 < 10-3

Fp < 0.01 < 10-3 < 0.01 < 10-3

The most contribution to the total uncertainty in the maximum core power and power pulse width is due 

to the uncertainty in rod worth as well as in case with fuel pellet enthalpy. Contributions of other 

quantities to the resulting uncertainties are very small. The resulting uncertainties were estimated as 216% 

in the maximum reactor power and 76% in power pulse width. 
40

3.3 Effect of Initial Core Power 

To analyze the effect of initial core power on key safety parameters, a calculation of the ejection of the 

central rod in the TMI-1 PWR starting from 33% of rated power was carried out using RELAP-BARS. To 

simplify a comparison of the results for HZP and non-zero power conditions, the same maximum value of 

1.2β for the reactivity was considered in REA from 33% of rated power. The same RELAP input deck 

was used in the RELAP-BARS calculation as in the HZP case. To reach a criticality for the initial steady 

state, an additional withdrawal of control rods at the core periphery was done in the BARS input deck. No 

other changes were done in the HZP input deck. 

Figures 3.3 and 3.4 show the reactor power and reactivity as a function of time for both HZP and 33% of 

rated power cases. In the last case the core power reaches its maximum value of approximately 14% of 

rated power at about 0.13 seconds. This peak value by more than 3.5 times exceeds corresponding value 

for the zero power case under the same maximum value of reactivity. 

The reasons for this very large difference are the following. 

• The decrease of 80% in the Doppler reactivity coefficient for the non-zero power case in comparison 

with the HZP one. This produces the factor of 1.8 in the resulting difference. 

• The increase of 3% in the “net” (without feedback) reactivity inserted by the control rod for the nonzero power case in comparison with the HZP one. Note that in the HZP case the “net” reactivity 

inserted by the control rod and the maximum reactivity are the same. This increase produces the 

factor of 1.55. 

• The “net” effect of initial power on peak power. This produces the factor of 1.15 in the resulting 

difference. 

• The increase of 10% in pellet heat capacity for the non-zero power case in comparison with the HZP 

one. This produces the factor of 1.1. 

Unlike the HZP case, in the non-zero power case the effect of feedback appears during the ejection of the 

control rod. This provides power pulse behavior as faster and sharper compared with the HZP case. 

Figure 3.5 shows assembly averaged radial power at three states of the core: at the initial steady state 

(0 s), at the time of the peak power (0.13 s), and at the end of the transient (2.5 s). Unlike the HZP case, 

significant deformations in radial power distribution take place after the rod ejection in the non-zero 

power case. Assembly powers differ up to 14.6% at the time of the peak power and at the end of the 

transient. 

Figure 3.6 shows fuel enthalpy in the hottest fuel pellet as a function of time for two assemblies: K10 and 

H9. Up to the time of 1.6 s the maximum value of enthalpy occurs in the pellet located in assembly K10, 

but at the end of the transient assembly H9 contains the pellet with the maximum enthalpy. Change in the 

hottest pin location takes place due to power redistribution during the transient. This phenomenon can 

lead to some difficulties in a prediction of the hottest pin using assembly-by-assembly approach together 

with the pin reconstruction procedure to calculate fuel pellet enthalpy. 

Table3.3 summarizes the calculation results of the effect of initial power on parameters of the accident. 
41

0 0.2 0.4 0.6 0.8 1

Time (sec)

0

3

6

9

12

15

Relative Reactor Power

from HZP

from 33% of rated power

Figure 3.3. Reactor Power vs. Time 
42

from HZP

from 33% of rated power

0 0.5 1 1.5 2 2.5

Time (sec)

-0.4

0

0.4

0.8

1.2

1.6

Reactivity ($)

Figure 3.4. Reactivity vs. Time 
43

 8 9 10 11 12 13 14 15 

H 

0.213 

1.936 

1.690 

1.169 

2.514 

2.227 

1.298 

1.826 

1.654 

1.458 

1.687 

1.580 

0.647 

0.641 

0.636 

0.974 

0.830 

0.878 

0.747 

0.589 

0.646 

0.391 

0.299 

0.333 

K 

 1.139 

1.936 

1.726 

1.613 

2.133 

1.949 

1.242 

1.383 

1.305 

1.247 

1.202 

1.202 

0.664 

0.561 

0.595 

1.141 

0.893 

0.984 

0.471 

0.361 

0.402 

L 

 1.278 

1.502 

1.398 

1.445 

1.495 

1.443 

1.157 

1.051 

1.071 

1.355 

1.112 

1.191 

1.189 

0.927 

1.021 

0.438 

0.334 

0.372 

M 

 0.704 

0.665 

0.669 

1.217 

1.040 

1.091 

1.021 

0.817 

0.883 

0.785 

0.608 

0.671 

N 

 0.664 

0.537 

0.579 

1.062 

0.826 

0.908 

0.499 

0.383 

0.424 

O 

 at t = 0 s Æ

at t = 0.13 s Æ

at t = 2.5 s Æ

 0.607 

0.464 

0.514 

 

Figure 3.5. Assembly Averaged Power at Three States of the Core 
44

0 0.5 1 1.5 2 2.5

Time (sec)

30

40

50

60

70

Fuel Enthalpy (cal/g)

Legend:

FA # K10

FA # H9

Figure 3.6. Fuel Enthalpy in the Hottest Pellet vs. Time 
45

Table 3.3. REA Parameters in Comparison with the HZP Case 

Parameter From 33% of rated power From HZP 

Maximum inserted reactivity (β) 1.22 1.21 

Peak power of the core (GW) 38.7 11.0 

Time of peak power (ms) 130 336 

Power pulse width (ms) 44 62 

Position of the hottest assemblies K10 and H9 H9 

Peak power of the fuel pin (MW) 2.82 0.835 

Maximum fuel pellet enthalpy (cal/g) 66.7 37.6 

Maximum fuel enthalpy increment (cal/g) 28.7 20.6 

Minimum of coolant outlet density (g/cc) 0.691 0.755 

Comparison between the HZP case and the REA from 33% of rated power shows that the difference in 

the maximum fuel enthalpy increment is about 40% and the maximum fuel pellet enthalpies differ by 

about 29 cal/g (37.6 versus 66.7 cal/g). 

47

4 CONCLUSIONS 

This study was undertaken to analyze spatial effects during rod ejection accidents. If the ejected rod worth 

is sufficient to reach prompt critical, i.e. greater than β, the result is a fast power excursion terminated by 

negative reactivity feedback from the increase in fuel temperature. This event is of interest from the point 

of view of prediction of the maximum increase in fuel enthalpy during the accident. Although, as 

numerous steady-state calculations show, the maximum value of control rod worth scarcely exceeds 1β in 

most PWRs, such an accident is frequently considered in NPP safety analyses with a conservatively 

increased value of the ejected rod worth. 

A key parameter of interest in an NPP safety analysis is the peak local fuel enthalpy, which establishes 

the acceptance criterion for unacceptable fuel damage in reactivity initiated accidents. It is wellestablished that spatial effects play an important role in the REA consequences, in particular, the core 

peak power and energy deposition, which is approximately the fuel enthalpy rise under an adiabatic 

assumption (no heat transfer from the fuel to the coolant). Using simplified analytical approximations it is 

possible to establish relationships between the major parameters of interest and spatial effects. To 

characterize them, the power peaking factors, Fq and Fq′ are used: the former is related to the pin-by-pin 

representation of the core power, and the latter to the assembly-by-assembly one. 

Localized spatial effects impact the REA parameters in two different ways. On the one hand, the higher 

the power peaking factor, the higher the local fuel enthalpy. On the other hand, a higher value of Fq tends 

to slow down the total energy deposition in the core and, as a result, to mitigate the fuel enthalpy increase. 

This is due to the fact that the fuel and moderator feedback is strongest in the hottest regions of the reactor 

core and reduces the peak power and reactivity. 

In this study an analysis of REAs was carried out using the RELAP5–BARS code (Ref. 1), which uses a 

3-D pin-by-pin neutronic and assembly-by-assembly thermal-hydraulic simulation of a LWR. Four REAs 

with ejection of the central or peripheral control rod and at EOC or BOC initial conditions were analyzed 

using a PWR model based on TMI-1. The same EOC PWR model was used for a previous study of the 

REA using different methods from U.S.A., France, and Russian Federation (Ref. 5). 

To provide identical initial conditions the worth of an ejected control rod was artificially fitted to 1.21β

for all cases. Duration of the transient was limited to 2.5 s. An assembly-average representation of fuel 

and coolant parameters with 24 nodes in an axial direction was used to treat the core thermal-hydraulics. 

A separate heat structure was chosen to represent the hottest fuel pin in the reactor core (the coolant 

parameters were the same as for the fuel assembly where the hottest pin was located). 

The results showed that peak power varied from 4.4 to 37.5 GW. In the cases with ejection of the 

peripheral rod, the peak power was lower compared with ejection of the central rod by a factor of 2.5 at 

EOC and 2.9 at BOC. An inverse relationship was found for the power peaking factors. The maximum 

assembly-average fuel temperatures were very close (within about 1 K) for the BOC cases. 

Corresponding values for the maximum fuel enthalpy increase were approximately 25.5 cal/g. For the 

EOC cases those values differed by 2 cal/g (18.9 and 16.9 cal/g for EOC center and EOC periphery, 

respectively). 
48

Thus, from the point of view of the assembly-by-assembly representation, the central rod ejection 

accidents with rather small power peaking factors resulted in slightly higher values of the maximum fuel 

enthalpy increase. On the contrary, the pin-by-pin representation showed that the peak local fuel enthalpy 

was higher for the peripheral REAs with the highest power peaking factors. In the EOC cases, a higher 

value of the peak enthalpy was due to the fact that the hottest fuel pin was found in Assembly N13 (see 

Figure 2.5) diagonally adjacent to the hot assembly M12. This fact indicates that defining the hottest pin 

location is problematic. In this case, the incorrect definition of the hottest pin location using an assemblyby-assembly approach, can lead to a 15% underestimation of the peak local enthalpy rise. Therefore, this 

should be taken into account in an uncertainty analysis of reactivity initiated accidents. 

The point kinetics approximation within the Nordheim-Fuchs model results in simple relationships for the 

peak power, total energy deposition and, with certain modifications, the power peaking factor. It was 

found that in the framework of this model the peak power and total energy deposition are inversely 

proportional to the power peaking factor. Thus, if it is assumed that the peaking factor for fuel enthalpy 

increase coincides with that for power, then it can be expected that the maximum assembly-average fuel 

enthalpy increase is practically independent of spatial effects. 

In spite of the fact that the Nordheim-Fuchs model is valid only within a narrow time interval limited by 

the power pulse area, this simplified approach (with the point kinetics parameters evaluated in advance) 

may be successfully used, in particular, for estimation of the major parameters of REAs. 

Another important problem in REA analysis is the uncertainty in prediction of the rod worth by different 

calculational methods. As was found during this study, the BARS results for the rod worths were always 

higher compared with those calculated by the PARCS code. The most significant difference occurs for the 

peripheral rod N12 (see Figures 2.1 and 2.2) nearest to the reflector region. At the EOC initial conditions 

this difference reaches 35%. The suspected reason for this discrepancy is the different treatment of the 

reflector boundary conditions in BARS vs nodal diffusion codes such as PARCS. More research would 

be needed with, for example, more precise calculational methods in order to resolve this difference. 

As the steady-state calculations of the rod worths show, a rod worth of 0.91β—below prompt critical--

seems to be the highest one. Thus, to provide the value of 1.2β for rod worth, it is necessary to artificially 

adjust the neutronics parameter of the ejected rod. Another, more realistic, approach is to assume that 

there is a withdrawn rod near the ejected control rod. For instance, in the case of ejection of the peripheral 

rod N12 (with “stuck” rod M11) the inserted reactivity value may reach 1.74β at EOC HZP conditions. 

In summary, the objective of this study was to understand the role of spatial effects on the outcome of 

different rod ejection accidents. The influence of spatial effects may be of interest not only in an analysis 

of reactivity initiated events, but also in interpretation of in-pile burst tests aimed to understand fuel rod 

behavior under severe accidents. The present study has also looked at the effect of the calculational 

approach (pin-by-pin against assembly-by-assembly) on the major outcome of an REA – the peak local 

enthalpy increase. 

The uncertainty analysis for the PWR central rod ejection accident starting from the HZP conditions 

carried out with the RELAP5-BARS code showed that the uncertainties in the key parameters of the 

accident would be determined to a large extent by the uncertainty in ejected rod worth. For a rod worth of 

1.2β with an uncertainty of 15% (corresponding to two standard deviations), the uncertainty in local fuel 

enthalpy was estimated as 110%, the uncertainty in the maximum core power as 216%, and the 

uncertainty in power pulse width as 76%. 
49

The results demonstrated the non-adiabatic nature of the transient and showed that the sensitivity of fuel 

enthalpy to most of the neutronic and thermal-hydraulic quantities strongly depends on time. 

Qualitatively, the RELAP5-BARS results are in agreement with the PARCS ones. 

The comparative study of the accident starting from 33% of rated power showed strong dependence of a 

number of the REA parameters on initial core power. Under the same rod worth of 1.2β, the peak power 

in the transient from 33% of rated power was 3.5 times greater than that in the transient from HZP. Unlike 

the HZP case, a change in the hottest fuel pin location takes place due to power redistribution during the 

transient from the non-zero power. This phenomenon can lead to some difficulties in predicting the 

hottest pin using the assembly-by-assembly approach with pin reconstruction to calculate fuel pellet 

enthalpy. In comparison with the HZP case, the REA from 33% of rated power leads to an increase in the 

maximum fuel pellet enthalpy to 66.7 cal/g. 

51

5 REFERENCES 

1. A. Avvakumov, V. Malofeev, and V. Sidorov, “Analysis of Pin-by-Pin Effects for LWR Rod Ejection 

Accident,” NUREG/IA-0175, 2000. 

2. K. N. Ivanov, et al., “Pressurized Water Reactor Main Steam Line Break (MSLB) Benchmark; 

Volume I: Final Specifications,” OECD Nuclear Energy Agency, NEA/NSC/DOC(99)8, April 1999. 

3. N. K. Todorova and K. N. Ivanov, “Project Report on Task 1 of BNL Sub-Contract “Core Model – 

PARCS/RELAP5,” The Pennsylvania State University, 2000 (available at www.nrc.gov in NRC’s 

public document system, ADAMS, as ML070660645). 

4. A. Avvakumov, et al., “3-D Pin-by-Pin Modeling of Rod Ejection RIA in VVER-1000,” Nuclear 

Safety Institute of Russian Research Centre “Kurchatov Institute”, Report No. 90-12/1-33-97, 1997 

(available at www.nrc.gov in NRC’s public document system, ADAMS, as ML070670258). 

5. D. J. Diamond, et al., “Intercomparison of Results for a PWR Rod Ejection Accident,” Nuclear 

Engineering and Design, 208, 2001. 

6. G. Bell and S. Glasstone, “Nuclear Reactor Theory,” Van Nostrand Reinhold Company, 1970. 

7. T. J. Downar et al., “PARCS: Purdue Advanced Reactor Core Simulator,” Proc. International 

Conference on the New Frontiers of Nuclear Technology: Reactor Physics, Safety and HighPerformance Computing (PHYSOR 2002, Seoul, Korea), American Nuclear Society, October 2002. 

8. J.-J. Lautard, S. Loubière, C. Fedon-Magnaud, “CRONOS2, A Modular Computational System for 

Neutronic Core Calculations,” IAEA Specialists Meeting on Advanced Calculational Methods for 

Power Reactors, Cadarache, France, IAEA-TECDOC-678, 1990 (available at www.nrc.gov in NRC’s 

public document system, ADAMS, as ML070670239). 

9. B. P. Bromley, Memo to D. J. Diamond, Brookhaven National Laboratory, May 2, 2001 (available at 

www.nrc.gov in NRC’s public document system, ADAMS, as ML062140704). 

10. D.J. Diamond, A.L. Aronson, and C.Y. Yang, “A Qualitative Approach to Uncertainty Analysis for 

the PWR Rod Ejection Accident,” Transactions of the American Nuclear Society, 83, 416, 2000. 

11. “RELAP5/MOD3.2 Code Manual,” NUREG/CR-5535, 1995. 

12. A. Avvakumov, V. Malofeev, “Validation of a Pin-by-Pin Neutron Kinetics Method for LWRs,” 

Proceeding of the International Twenty-Sixth Water Reactor Safety Information Meeting, Bethesda, 

Maryland, October 1998, NUREG/CP-0166, Volume 3, June 1999. 

13. P.L. Kirillov, Y.S. Yuriev, V.P Bobkov, “Handbook on Thermal-Hydraulic Calculations,” in Russ., 

Energoatomizdat, Moscow, 1990. 

14. I.I. Novikov, K.D. Voskresensky, “Applied Thermal-Dynamics and Heat Transfer,” in Russ., 

Gosatomizdat, Moscow, 1961. 

APPENDIX A 

CALCULATIONAL RESULTS FOR A REA WITH INCREASED 

ROD WORTH 

A-1

As it was above mentioned, the BOC delayed neutron fraction was approximately 21% higher compared 

with that at the EOC conditions (632 pcm against 521 pcm). This means that at the same relative rod 

worth expressed in terms of β, there are two different absolute values of the rod worth at BOC and EOC. 

For instance, for a 1.21β rod worth the absolute values are 631 pcm and 765 pcm at EOC and BOC, 

respectively. In this respect it would be interesting to compare the results for EOC REA with the same 

absolute value of the rod worth as at BOC, i.e. 765 pcm (1.47β). 

This section contains the calculational results of a REA with ejection of the central rod with increased 

worth obtained by the RELAP–BARS code. The absolute value of the rod worth corresponds to that 

presented in Table 2.2 for the BOC cases, i.e. 765.7 pcm or 1.47β in terms of the EOC delayed neutron 

fraction. The reactor model and the accident scenario were the same as described in Section 2. To provide 

required 1.47β, the initial arrangement of the control rod banks was changed. As was found it was 

impossible to obtain such a high reactivity only by changing neutronic parameters of the ejected rod. For 

this reason, four control rods of Bank 2 located at Assembly K9 diagonally adjacent to Assembly H8 were 

partly inserted before the REA (see Figure 2.1). Then they were also withdrawn together with the ejected 

rod H8 with the same speed. 

Table A.1 summarizes the calculational results of 1.47β EOC transient with ejection of the central control 

rod. Figures A.1 and A.2 show the following parameters as a function of time of the transient: the core 

power, reactivity, fuel temperature, and fuel enthalpy increase for the hottest fuel pin. 

The power pulse, as can be seen in Figure A.1, is very narrow, the pulse width is less than 30 ms. The 

power excursion produces rapidly the core heat-up, and due to the negative fuel and moderator 

temperature feedback, is terminated. As a result of the strong feedback, the reactivity drops up to 0.06β

(against 0.3-0.4β in 1.21β transients) at about 0.7s. After a while the reactivity becomes to rise as a result 

of the positive reactivity insertion due to the core cooldown. (This effect was not observed during the 

1.21β REA analysis, nevertheless, as was reported in Ref. 5, the Doppler and moderator reactivity effects 

were of the same order. 

It is obvious that sooner or later the negative moderator feedback component becomes to decrease due to 

forced circulation of the coolant with the core inlet temperature of 551 K.) 

As can be seen in Figure A.2 the fuel temperature/enthalpy reaches a peak value at about 0.5 s, then 

gradually decreases. The maximum fuel pellet temperature is 995 K, which corresponds to the enthalpy 

increase of 33 cal/g. The last value is higher by about 60% in comparison with Case EOC Centre, i.e. 

1.21β REA (see Table 2.2). This is in contrast to the simplified estimation: according to Eq. 3 it should be 

expected at least twice as high (or more precisely, about 46 cal/g). 
A-2

Table A.1. Calculational Results for 1.47β EOC REA 

Parameter Value 

Ejected Rod Worth (pcm / β) 765.7 / 1.4695 

Peak Power (GW / % Nominal) 48.83 / 1762 

Time of Peak Power (ms) 206.4 

Power Pulse Width (ms) 29.5 

Maximum Core-Average Fuel Temperature (K) 610.6 

Maximum Fuel Assembly Temperature (K) 960.5 

Maximum Fuel Pin Temperature (K) 995.1 

Maximum Assembly Enthalpy Increase (cal/g) 30.27 

Maximum Pin Enthalpy Increase (cal/g) 32.96 

Time of Maximum Pin Enthalpy Increase (ms) 507.0 

Position of the Hottest Assembly H9 

Position of the Hottest Fuel Pellet (Axial Layer) H9 (Node 22) 
A-3

0.0 0.5 1.0 1.5 2.0 2.5

Time (sec)

0

400

800

1200

1600

2000

Relative Power (% of Nominal)

0.0

0.4

0.8

1.2

1.6

2.0

Reactivity ($)

Central Rod Ejection

 (EOC HZP)

0.0 0.1 0.2 0.3 0.4 0.5

Time (sec)

0

400

800

1200

1600

2000

Relative Power (% of Nominal)

0.0

0.4

0.8

1.2

1.6

2.0

Reactivity ($)

Central Rod Ejection

 (EOC HZP)

Figure A.1. Power and Reactivity vs. Time 
A-4

0.0 0.5 1.0 1.5 2.0 2.5

Time (sec)

500

600

700

800

900

1000

1100

Fuel Temperature (K) in Hottest Pellet

Central Rod Ejection (EOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

0 0.5 1 1.5 2 2.5

Time (sec)

0

5

10

15

20

25

30

35

Enthalpy Increment (cal/g) in Hottest Fuel Pellet

Central Rod Ejection (EOC HZP)

Hottest Fuel Pellet

Average Pellet in Hot FA

Figure A.2. Fuel Temperature and Enthalpy in Hot Pins vs. Time 
BIBLIOGRAPHIC DATA SHEET

(See instructions on the reverse)

NRC FORM 335

(9-2004)

NRCMD 3.7

U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER

 (Assigned by NRC, Add Vol., Supp., Rev.,

 and Addendum Numbers, if any.)

NUREG/IA-0215

3. DATE REPORT PUBLISHED

MONTH

September

YEAR

2007

4. FIN OR GRANT NUMBER

Y6723

2. TITLE AND SUBTITLE

Spatial Effects and Uncertainty Analysis for Rod Ejection Accidents in a PWR

5. AUTHOR(S)

A. Avvakumov, V. Malofeev, V. Sidorov

6. TYPE OF REPORT

7. PERIOD COVERED (Inclusive Dates)

2001-2005

8. PERFORMING ORGANIZATION - NAME AND ADDRESS (If NRC, provide Division, Office or Region, U.S. Nuclear Regulatory Commission, and mailing address; if contractor,

Russian Research Center - Kurchatov Institute

9. SPONSORING ORGANIZATION - NAME AND ADDRESS (If NRC, type "Same as above"; if contractor, provide NRC Division, Office or Region, U.S. Nuclear Regulatory Commission, 

Division of Risk Assessment and Special Projects

provide name and mailing address.)

and mailing address.)

10. SUPPLEMENTARY NOTES

11. ABSTRACT (200 words or less)

A rod ejection accident is a design-basis event for a pressurized water reactor. It is well known that spatial effects play a very

important role in this accident. In this study four cases using a model of Three Mile Island Unit 1 are considered: ejection of the

central or peripheral control rod at the end of cycle or the beginning of cycle. Calculations were carried out using the BARS

neutronic code coupled with the RELAP5/MOD3.2 thermal hydraulic code. This coupled code allows three-dimensional

pin-by-pin neutronics and assembly-by-assembly thermal-hydraulics simulation of a transient. The results showed that the major

parameters of the accident (the peak power and core energy deposition) were a function of spatial effects. Analysis of the

dependence of the peak local fuel enthalpy on spatial effects was performed. Uncertainty analysis was carried out for the central

control rod ejection accident at hot zero power conditions. The analysis of uncertainties was performed for the following

parameters: local fuel enthalpy, maximum core power, and power pulse width. Calculated results showed that the uncertainty in

key safety parameters would be determined to a great extent by the uncertainty in the control rod worth. The effect of initial core

power on the above parameters was analyzed using a rod ejection accident starting from 33% of rated power.

12. KEY WORDS/DESCRIPTORS (List words or phrases that will assist researchers in locating the report.)

control rod ejection accident

pellet enthalpy

neutron kinetics

PARCS computer code

BARS computer code

spatial effects

uncertainty analysis

code assessment

14. SECURITY CLASSIFICATION

13. AVAILABILITY STATEMENT

unlimited

(This Page)

unclassified

(This Report)

unclassified

15. NUMBER OF PAGES

16. PRICE

NRC FORM 335 (9-2004) PRINTED ON RECYCLED PAPER

Moscow 123182

Russia

Office of Nuclear Regulatory Research

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Washington, DC 20555-001