On the determination of boiling water reactor characteristics by noise analysis

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ON THE DETERMINATION OF

BOILING WATER REACTOR CHARACTERISTICS

BY NOISE ANALYSIS

ERIK KLEISS

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ON THE DETERMINATION OF

BOILING WATER REACTOR CHARACTERISTICS

BY NOISE ANALYSIS

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ON THE DETERMINATION OF

BOILING WATER REACTOR CHARACTERISTICS

BY NOISE ANALYSIS

PROEFSCHRIFT ter verkrijging van

de graad van doctor in de

technische wetenschappen aan

de Technische Hogeschool Delft,

op gezag van de Rector Magnificus

prof. ir. B.P.Th. Veltman,

te verdedigen op

donderdag 16 juni 1983

te 14.00 uur door

ERIK BERNARDUS J O H A N N E S KLEISS

natuurkundig ingenieur

geboren te Schiedam

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Dit proefschrift is goedgekeurd door

de promotor: Prof.dr.ir. H. V A N DAM

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Aan mijn ouders

Aan alle vrienden

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CONTENTS

SAMENVATTING 9 SUMMARY 11 CHAPTER 1. INTRODUCTION AND OUTLINE 13

CHAPTER 2. ANALYSES OF NEUTRON DETECTOR RESPONSE

TO BUBBLES IN A WATER MODERATED REACTOR 17

2.1 Introduction 17 2.2 Qualitative description of the detector response. 17

2.3 Computation of the detector response via

perturbation theory 18 2.4 Experimental setup and experiments 20

2.5 Results and discussion 22

2.6 Conclusions 28 Appendices 29 CHAPTER 3. A TWIN SELF-POWERED NEUTRON DETECTOR FOR STEAM

VELOCITY DETERMINATION IN A BOILING WATER REACTOR 31

3.1 Introduction 31 3.2 Construction 32 3.3 Experiments 33 3.4 Conclusions 36 CHAPTER 4. INCORE POWER FEEDBACK EFFECTS DEDUCED FROM

NEUTRON NOISE MEASUREMENTS 39

4.1 Introduction 39 4.2 Theoretical background 39

4.3 Application 43 4.4 Discussion 45

Appendix 46 CHAPTER 5. THE DETERMINATION OF THE REACTOR TRANSFER FUNCTION

FROM THE SPACE DEPENDENCE OF THE NEUTRON NOISE 49

5.1 Introduction 49 5.2 Experimental conditions 50

5.3 Estimation of the noise source distribution 50 5.4 The estimation procedure for the RTF 53

5.5 Results and discussion 56 5.6 Effects of pressure noise 61 5.7 Concluding discussion 65

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CHAPTER 6. AUTOREGRESSIVE MODELLING OF REACTOR NOISE SIGNALS 67

6.1 Introduction 67 6.2 Analysis of incore detector signals 68

6.3 Analysis of steam flow/vessel pressure/power relations 71

6.4 AR analysis of pressure controller 77 APPENDIX 1. THE CONTROL ROD EXPERIMENT 81

A l . l Introduction 81 A1.2 The input signal 82 A1.3 Execution of the experiments 84

A1.4 Estimation of the reactivity transfer function 91 A1.5 Calculation of the rod step reactivity effect 94 A1.6 Estimation of the void reactivity coefficient 96

A1.7 Pressure controller parameters 97 APPENDIX 2. THE AUTOREGRESSIVE MODELLING OF NOISE SIGNALS 101

A2.1 Introduction 101 A2.2 Time series modelling 104

A2.3 Practical aspects of MAR modelling 110

A2.4 Examples of AR modelling 117 APPENDIX 3. A MODEL FOR THE DYNAMICS OF A

BOILING WATER REACTOR 127

A3.1 Introduction 127 A3.2 Basic model equations 129

A3.3 Evaluation of model parameters 132

A3.4 Results 140 NAWOORD 147

Chapters 2 and 4 have been reprinted from Annals of Nuclear Energy with kind permission of Pergamon Press.

Chapter 3 has been reprinted from Nuclear Technology with kind permission of the American Nuclear Society.

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SAMENVATTING.

In dit proefschrift wordt de analyse van ruissignalen van kokend-water reacto-ren (BWR's) behandeld. Omdat de belangrijkste ruisbron het kookproces in de reactorkern is en de belangrijkste variabele de neutronenflux, wordt de invloed van stoombellen op de neutronenflux gedetailleerd onderzocht.

In Hoofdstuk 2 wordt een experiment beschreven dat is uitgevoerd om in een kleine, onderkritieke reactor de responsie te meten van het signaal van een neutronen detector op de passage van een enkele luchtbel. Op deze wijze wordt het elementaire proces in een BWR gesimuleerd. Een theoretisch model voor de beschrijving van de responsie is getest en de resultaten stemmen goed overeen met de metingen. Daarnaast wordt een kwalitatieve verklaring van de metingen gegeven.

De overige hoofdstukken hebben betrekking op ruismetingen die verricht zijn in de kerncentrale te Dodewaard. In Hoofdstuk 3 wordt de constructie besproken van een tweeling self-powered neutronen detector, die ontwikkeld is om de snel-heid te meten van de stoom in de splijtstofbundels. De detector-karakteristie-ken zijn bepaald en de detector blijkt goed geschikt te zijn voor deze toepas-sing.

In Hoofdstuk 4 wordt dieper ingegaan op het gedrag van de neutronenruis in het laagfrequente gebied. Hier worden afwijkingen van puntkinetica gevonden, die verklaard kunnen worden met een uitbreiding van de theorie van Hoofdstuk 2 naar vermogenscondities. Als een nuttig practisch gevolg blijkt het mogelijk te zijn om, uit het plaatsafhankelijk gedrag van de ruis, de overdrachtsfunctie tussen reactiviteit en reactorvermogen te bepalen.

In Hoofdstuk 5 wordt hier dieper op ingegaan. De uit de ruis bepaalde over-drachtsfuncties kloppen goed met die, welke gebaseerd zijn op onafhankelijke methoden: uit de meting van de responsie op een stapvormige beweging van een regelstaaf en uit modelberekeningen.

Hoofdstuk ó behandelt het gebruik van het autoregressief model voor de analyse van de samenhang van een aantal signalen. Voor wat betreft neutronenflux, druk en stoomdebiet zijn de belangrijkste ruisbronnen bepaald. De kookruis is

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langrijk, maar daarnaast treden ook onafhankelijke variaties in het stoomdebiet op (akoestische resonanties) en blijkt ruis in het regelsysteem van belang te zijn. Bovendien worden de overdrachtsfuncties tussen een aantal variabelen be-paald, waaruit schattingen verkregen kunnen worden voor enkele fysische groot-heden die van belang zijn bij het reactorbedrijf.

In drie appendices worden metingen en theoretische onderwerpen behandeld die nodig zijn in hoofdstukken 4-6, maar die niet direct over de analyse van reac-torruis gaan. In Appendix 1 worden de details besproken van de regelstaaf-stap experimenten, die werden gebruikt ter controle van de reactor overdrachtsfunc-tie die in Hoofdstuk 5 uit de ruis bepaald werd. Appendix 2 behandelt de ach-tergronden van de toepassing van het multivariate autoregressieve model bij ruisanalyse. Hierbij zijn enkele problemen die bij de practische toepassing kunnen onstaan opgelost en de methode wordt gedemonstreerd aan de hand van de analyse van enkele eenvoudige electrische netwerken. Tenslotte wordt in Appen-dix 3 een model afgeleid voor de dynamica van een kokend-water reactor en wor-den enkele resultaten voor de Dodewaard reactor gepresenteerd.

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SUMMARY.

This thesis deals with the analysis of the noise signals in boiling water reac-tors. As the main noise source is the boiling process in the core and the most important variable the neutron flux, the effect of the steam bubbles on the neutron flux is studied in detail.

Chapter 2 deals with an experiment, performed in a small subcritical reactor, to measure the response of a neutron detector to the passage of a single air bubble. In this way the elementary process in a BWR is simulated. A mathemat-ical model for the description of the response was tested and the results agree very well with the experiment. Also some attention is paid to the qualitative physical explanation.

The remaining chapters discuss noise measurements in the Dodewaard boiling water reactor in The Netherlands. Chapter 3 deals with the construction of a twin self-powered neutron detector, developed to perform steam velocity meas-urements in the core. Detector characteristics are measured and it appears that it is well suited for its purpose.

In chapter 4, the study concentrates on the low-frequency part of the neutron noise characteristics. Here deviations from a point-kinetics behaviour of the core are observed. An explanation can be obtained by an extension of the theo-ry discussed in Chapter 2 to at-power conditions. As a useful practical result, it appears possible to determine the reactor transfer function between reactivity input and reactor power output, from the space-dependence of the neutron noise.

Chapter 5 goes deeper into the practical elaboration of this method. The resulting transfer functions exhibit a good agreement with ones obtained by independent means: control rod step experiments and model calculations.

In Chapter 6 the relations between several variables are studied with the use of autoregressive modelling techniques. The main noise sources in the reactor are identified in as far neutron flux, pressure and steam flow are concerned. Boiling noise is important, but also independent steam flow variations (acous-tic waves) and control system noise play a substantial role. Furthermore the

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transfer functions between several variables were obtained, yielding estimates for a number of parameters involved in reactor operation.

Three appendices are present which are not directly involved in the analysis of the Dodewaard noise, but which discuss measurements and theories necessary in the Chapters 4 to 6. Appendix 1 treats the control rod step experiments, used for an experimental validation of the noise-based reactor transfer functions obtained in Chapter 5. Appendix 2 deals with the application of the multivari-ate autoregressive modelling technique to the study of noise signals. Some newly discovered topics are discussed and the method is demonstrated by the analysis of some simple networks.

Finally, in Appendix 3 a model is derived for the dynamics of the Dodewaard boiling water reactor and some results are presented.

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CHAPTER 1. INTRODUCTION AND OUTLINE.

Noise is a very general phenomenon in physical systems. It is the occurence of random fluctuations with time in the characteristic variables or output signals of the system under consideration. The causes for these fluctuations are many-fold; they range from the intrinsic probabilistic character of the elementary processes underlying the macroscopic system behaviour, to very clearly identif-iable external perturbations. Often the presence of noise is considered a nui-sance which limits the precision of measurements performed on the system, or which deteriorates system performance. However, i f the noise is accepted as a phenomenon in its own right, useful results can be obtained from its analysis. This will be discussed in the following.

A general approach is to consider the observed noise as being caused by one or more noise sources, acting as inputs to the system. The characteristics of the output noise are determined by both noise source and system properties, e.g. spectral densities of the noise sources and transfer functions of the linear(ized) system. If the characteristics of the noise sources are known by some method, the system properties can be obtained. If on the other hand the system characteristics are available, the properties of the noise sources can be studied. Both alternatives have found their application to nuclear reac-tors.

For the case that reactor system properties are known sufficiently accurate, the noise sources can be investigated and the presence of 'abnormal' noise be detected. Excessive mechanical vibration of reactor components (with danger of wear or fatigue), insufficient coolant flow and associated unexpected coolant boiling (danger of fuel damage), loose parts and leakages in components are examples of topics that received much attention (1,2). It will be clear that this type of noise analysis can be of use in the safe operation of nuclear power plants; due to the generally large sensitivity of these techniques, it is expected that failures may be detected in an early stage before substantial damage to the plant occurs.

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input to the system and used to measure system properties. Noise analysis is sometimes able to yield parameters that are not or only with difficulty obtain-able by other methods. Also, it may be impossible (technically or due to unde-sired interference with plant operation) to apply external input signals to the reactor system for the measurement of its response functions; the intrinsic noise is then a welcome alternative. The first applications of reactor noise analysis were based on this approach. The probabilistic nature of the fission process and neutron multiplication was already clear in the early years of reactor operation (3) and was used to measure e.g. criticality and neutron generation time. The success of these applications, often called zero-power reactor noise analysis, lies in the fact that the basic noise sources are very clear and the mathematical tools for its description were partly available from statistical physics (see e.g. 4). With the development of power reactors other noise generating processes, with less well-defined properties such as coolant density and temperature fluctuations due to turbulence and boiling, became important. Nevertheless, the noise sources can often be modelled sufficiently close or be measured to be used to obtain system properties. An early example is the stability monitoring of the EBWR (experimental boiling water reactor (5)), still a topic of interest(2). Another example is the measurement of coo-lant flow, also discussed in this thesis.

This type of noise analysis may contribute to a safe and economic operation of reactors, too. In the first place, measurements of physical parameters under operational conditions can be used as a check on the data and calculational methods applied in the design phase. This may lead to a reduction in conserva-tive margins thus improving economic performance. Furthermore, monitoring of the system during fuel cycle may reveal slow deterioration of instrumentation, controller performance, dependence of stability on power and burnup conditions, etc.

This thesis focusses primarily on the second approach: the use of the intrin-sic noise to measure system characteristics of boiling water reactors (BWR's). In these reactors, the boiling and steam transport processes in the core act as the dominating noise source. From reactor physical point of view, the interac-tion of this boiling noise with the neutron flux field in and around the reac-tor is one of the most interesting topics.

One description of the neutron flux response is the local/global concept of Wach and Kosaly (6). A steam bubble in the moderator gives rise to a neutron flux variation in its immediate vicinity (the local component, mainly due to decreased moderation). At the same time the neutron balance is influenced, so a reactivity effect occurs that affects the neutron flux in the core as-a-whole: the global component. This approach proved to be very fruitful in obtaining a qualitative understanding of the noise signals of incore neutron detectors used for the measurement of the steam velocity in the fuel bundles.

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More quantitative results came available after the introduction of a method, based on perturbation theory, for the description of the detector responses (7-9). Chapter 2 of this thesis deals with an experiment that was carried out to test the applicability of this theory in practice. The response of a neu-tron detector to air bubbles injected in a small subcritical reactor was meas-ured, thus simulating the BWR incore conditions in the laboratory.

The remaining chapters deal with the noise analysis of an actual BWR, the Dodewaard reactor; this is a small (54MWe), natural circulation boiling water reactor in The Netherlands. Chapter 3 deals with the development of a special neutron detector to be used for the accurate measurement of the steam velocity in the core of this reactor. As our research was not aimed at the study on two-phase flow and the validation of the design and operation programs, results obtained with this detector are not discussed. Instead, a study on the neutron flux behaviour is performed.

Apart from the explanation of the local noise component, the perturbation theo-ry is in principle suited for a description of the global component, up to then generally neglected. For this purpose, the method had to be extended to include the effects of power variations on neutron cross sections. This is done in Chapter 4. In this way an explanation is found for the observed space-dependent effects in the low-frequency region of the noise. A direct result is the quantitative determination of the at-power reactor transfer func-tion (RTF), outlined in Chapter 4 and discussed in more detail in Chapter 5. Due to the novelty of this rather indirect method, a validation must be obtained by comparison with independent methods. For this purpose experiments were performed with control rod movements to measure the RTF directly; these are discussed in Appendix 1. Also a theoretical model for the dynamic beha-viour of the Dodewaard reactor was developed and is presented in Appendix 3, In the course of the studies of Chapter 4 and 5, the necessity arose to intro-duce sophisticated methods for the analysis of the mutual interaction of many neutronic and process noise signals. The autoregressive modelling technique, not long before introduced into reactor noise work, seemed to offer good possi-bilities. Before a succesful application was possible, some pitfalls and theoretical problems had to be removed. These points are discussed in Appendix 2.

Apart from the applications of this method in Chapter 5, it appeared a powerful tool to obtain information on other reactor characteristics. Chapter 6 discusses the use of the developed methods to identify the most important noise sources in the reactor and to measure transfer functions between several vari-ables and determine the associated physical parameters.

The three appendices form a rather substantial part of this thesis. As the treated topics do not directly discuss the analysis of reactor noise, they are added as appendices to avoid a disturbance of the main line of the chapters. However, the presented material forms a significant part of the research

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formed.

This thesis is not meant as a documentation of the Dodewaard reactor and all its noise characteristics. The appreciation by the reader of the topics dis-cussed will depend on his involvement in the trade. Although many details and conclusions may be specific for the Dodewaard reactor, it is believed that the developed techniques are fairly well applicable to other reactors.

CHAPTER 1. References.

1. SMORN-II. 2nd. Specialists meeting on reactor noise. M.M.R.Williams(ed). Progress in Nucl.Energy 1,73-804,(1977). 2. SMORN-IIL 3rd. Specialists meeting on reactor noise.

M.M.R.Williams(ed). Progress in Nucl.Energy 9,(1982).

3. Courant, E.D. and P.R.Wallace. Phys.Rev.72, 1038-1048,(1949). 4. Williams,M.M.R. Random Processes in Nuclear Reactors,

Pergamon Press, (1974).

5. Thie,J.A. Nucleonics 17,102-109, (October 1959).

6. Wach,D. and G.Kosaly. Atomkerenergie 23,244-250,(1974). 7. Dam,H.van. Atomkernenergie 25,70-71,(1975).

8. Dam.H.van. Atomkernenergie 27,8-14,(1976).

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17 Annals of Nuclear Energy. V o l . 6. pp. 385 to 398

Pergamon Press L t d 1979. Printed in G r e a i Britain

A N A L Y S I S O F N E U T R O N D E T E C T O R R E S P O N S E T O B U B B L E S I N A W A T E R M O D E R A T E D R E A C T O R

E . K L E I S S a n d H . V A N D A M

Interuniversify Reactor Institute, M e k e l w e g 15, Delft, T h e Netherlands

(Received 8 January 1979, in revised form 26 February 1979)

Abstract—The influence of air bubbles o n the signal of a neutron detector has been investigated in a water moderated subcritical assembly using an average response technique. Q u a l i t a t i v e and quantita-tive models are developed to explain the measured detector response. T h e quantitaquantita-tive one, based on perturbation theory, is i n g o o d agreement w i t h experimental results.

1. I N T R O D U C T I O N T h e f o r m a t i o n a n d t r a n s p o r t o f s t e a m b u b b l e s is a v e r y i m p o r t a n t s o u r c e o f the n o i s e i n b o i l i n g w a t e r r e a c t o r s . S t u d y o f t h i s n o i s e c a n , i n p r i n c i p l e , r e v e a l m u c h i n f o r m a t i o n t h a t is i m p o r t a n t for p l a n t o p e r -a t i o n , e.g. s t e -a m v e l o c i t i e s , v o i d f r -a c t i o n -a n d t h e r m -a l h y d r a u l i c i n s t a b i l i t i e s . I n r e c e n t years m u c h w o r k h a s b e e n c a r r i e d o u t i n t h i s field, b o t h e x p e r i m e n t a l l y a n d t h e o r e t i c a l l y . O n the e x p e r i m e n t a l side, v o i d v e l o c i t y a n d f r a c t i o n are o b t a i n e d a n d o t h e r t w o - p h a s e flow c h a r a c t e r i s t i c s w e r e s t u d i e d u s i n g the n o i s e s i g n a l s o f b o t h i n - a n d e x - c o r e n e u t r o n d e t e c t o r s ( A s h r a f A t t a et al, 1 9 7 8 ; v o n C e e l e n et al, 1 9 7 6 ; C r o w e et al, 1 9 7 7 ; K o s a l y et al, 1977a).

O n the t h e o r e t i c a l side, m o d e l s are d e v e l o p e d to c o m p u t e the r e s p o n s e o f a n i n - c o r e n e u t r o n d e t e c t o r to s t e a m b u b b l e s . T h e i d e a b e h i n d t h i s is t h a t o n c e t h e r e s p o n s e o f a d e t e c t o r t o s t e a m b u b b l e s is k n o w n c o r r e c t l y , i m p o r t a n t i n f o r m a t i o n a b o u t the b u b b l e flow c a n be d e r i v e d f r o m m e a s u r e m e n t s o f i n - c o r e n e u t r o n d e t e c t o r s i g n a l s . W a c h a n d K o s a l y (1974) p r o p o s e d the ' l o c a l - g l o b a l ' c o n c e p t , f o u n d e d o n o b s e r v e d i n t e r f e r e n c e effects i n the p o w e r s p e c t r a l d e n s i t i e s o f i n c o r e d e t e c t o r s . H e r e the g l o b a l c o m -p o n e n t o f the n o i s e w a s a s s u m e d t o be the r e a c t i v i t y n o i s e ( p a r t l y c a u s e d b y the s t e a m b u b b l e s ) , w h i l e the l o c a l effect w a s i n t e r p r e t e d as the flux c h a n g e i n the v i c i n i t y o f the d e t e c t o r . F o l l o w i n g this, m o d e l s w e r e c o n s t r u c t e d t o c o m p u t e the l o c a l p a r t o f the r e s p o n s e ( F u g e , 1 9 7 5 ; K o s a l y a n d M e s k o , 1 9 7 6 ; K o s a l y et al, 1 9 7 3 ; K o s a l y , 1975). A t h e o r e t i c a l b a s e w a s suggested b y v a n D a m ( 1 9 7 5 , 1 9 7 6 ) , w h o a p p l i e d t i m e d e p e n d e n t p e r t u r b a t i o n t h e o r y t o c o m p u t e t h e r e s p o n s e o f a n e u -t r o n d e -t e c -t o r -t o b u b b l e s . I n a -t w o - g r o u p m o d e l -the c o n n e c t i o n w i t h the l o c a l g l o b a l c o n c e p t w a s a p p a r -ent v i a t h e t w o r e l a x a t i o n l e n g t h s that a p p l y to the n e u t r o n t r a n s p o r t p r o c e s s ( v a n D a m , 1975, 1976; B e h r i n g e r et al, 1976).

T h i s p a p e r d e a l s w i t h the results o f e x p e r i m e n t s that w e r e c a r r i e d o u t i n o r d e r t o c h e c k the v a l i d i t y o f the p e r t u r b a t i o n m o d e l , a n d tries to get a better i n s i g h t i n t o the processes t h a t p l a y a r o l e i n the o r i g i n o f the r e s p o n s e . I n a s u b c r i t i c a l a s s e m b l y a B W R was s i m u l a t e d b y i n j e c t i n g a i r b u b b l e s i n t o the w a t e r m o d e r a t o r . A n e u t r o n s o u r c e a n d a n e u t r o n d e t e c t o r were p r e s e n t i n the s y s t e m a n d the i n f l u e n c e o f the b u b b l e s o n the s i g n a l o f the n e u t r o n d e t e c t o r was m e a s u r e d u s i n g a n a v e r a g e - r e s p o n s e t e c h n i q u e . T h e e x p e r i m e n t is d e s c r i b e d i n m o r e d e t a i l i n S e c t i o n 4. I n the next t w o S e c t i o n s a q u a l i t a t i v e m o d e l a n d a q u a n t i t a t i v e p e r t u r b a t i o n m o d e l for the d e s c r i p t i o n o f the d e t e c t o r r e s p o n s e a r e p r e s e n t e d . I n S e c t i o n 5 the e x p e r i m e n t a l results a r e d i s p l a y e d a n d c o m p a r e d w i t h t h e o r y . F i n a l l y , i n the a p p e n d i c e s s o m e c o m -m e n t s o n the c o -m p u t a t i o n s are g i v e n . 2. Q U A L I T A T I V E D E S C R I P T I O N O F T H E D E T E C T O R R E S P O N S E F r o m the r e a c t o r p h y s i c a l p o i n t o f v i e w , the s t e a m b u b b l e s i n the m o d e r a t o r c a n be t r e a t e d as l o c a l c h a n g e s i n the m a c r o s c o p i c c r o s s s e c t i o n s . T h e s e c h a n g e s affect the b e h a v i o u r o f the n e u t r o n p o p u l a -t i o n . I n -t h i s s e c -t i o n we w i l l c o n s i d e r -the c h a n g e s i n d e t e c t i o n p r o b a b i l i t y for n e u t r o n s , d u e t o c r o s s s e c t i o n c h a n g e s i n t h e r e a c t o r , i n a q u a l i t a t i v e w a y , s t a r t i n g f r o m the e l e m e n t a r y i n t e r a c t i o n s b e t w e e n n e u t r o n s a n d the m o d e r a t o r m a t e r i a l : c a p t u r e a n d s c a t t e r i n g . It is a p p r o p r i a t e t o s p l i t the effects o f s c a t t e r i n g i n t o t w o c o m p o n e n t s : e n e r g y - c h a n g e

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18 E . KLEISS a n d H . V A N D A M ( " m o d e r a t i o n " ) a n d d i r e c t i o n c h a n g e a n d f o l l o w i n g m o v e m e n t o f the n e u t r o n s (here i n d i c a t e d as " d i f f u s i o n " ) . W e a s s u m e n o c h a n g e s i n the fission c r o s s s e c t i o n b e c a u s e the b u b b l e s are f o r m e d i n the m o d e r a t o r m a t e r i a l a n d s e c o n d o r d e r effects l i k e s p e c t r a l c h a n g e s are n e g l e c t e d . 2.1. Capture I n t r o d u c t i o n o f a b u b b l e i n t o the m o d e r a t o r i m -plies r e m o v a l o f n e u t r o n c a p t u r i n g m a t e r i a l . T h e p r o b a b i l i t y o f the n e u t r o n b e i n g d e t e c t e d w i l l i n c r e a s e b e c a u s e it h a s a greater c h a n c e to r e a c h the d e t e c t o r . T h e r e s p o n s e w i l l o f c o u r s e be p r o p o r t i o n a l to the n u m b e r o f n e u t r o n s that u n d e r g o the i n f l u e n c e o f the b u b b l e . T h e r e s p o n s e w i l l a l s o be p r o p o r t i o n a l to the c h a n c e o f a n e u t r o n r e a c h i n g the d e t e c t o r . (In the next s e c t i o n this p r o b a b i l i t y w i l l be d e s c r i b e d by the a d j o i n t flux.) 2.2. Moderation T h e b u b b l e r e m o v e s m o d e r a t i n g m a t e r i a l so that n e u t r o n s w i l l have a s m a l l e r c h a n c e t o b e c o m e ther-m a l i z e d . If the n e u t r o n d e t e c t o r is o n e t h a t detects m a i n l y t h e r m a l n e u t r o n s (as m o s t d e t e c t o r s d o ) the r e s p o n s e w i l l be negative. If h o w e v e r the d e t e c t o r is m a i n l y s e n s i t i v e to fast n e u t r o n s the r e s p o n s e w i l l be p o s i t i v e . T h e r e s p o n s e w i l l be p r o p o r t i o n a l to the n e u t r o n flux a n d to the difference i n d e t e c t i o n p r o b -a b i l i t y for f-ast -a n d t h e r m -a l n e u t r o n s (-as d e s c r i b e d b y the difference o f the fast a n d t h e r m a l a d j o i n t flux i n the next s e c t i o n ) .

2.3. Diffusion

T h i s effect is s o m e w h a t m o r e c o m p l e x t h a n the p r e -v i o u s o n e s b e c a u s e it is d e t e r m i n e d b y t w o o p p o s i n g processes, d e p e n d i n g o n the t r a v e l l i n g d i r e c t i o n o f the n e u t r o n . N e u t r o n s t r a v e l l i n g t o w a r d s the d e t e c t o r w i l l h a v e a n i n c r e a s e d p r o b a b i l i t y t o be d e t e c t e d i f scatter-i n g m a t e r scatter-i a l scatter-is t a k e n a w a y , w h scatter-i l e n e u t r o n s m o v scatter-i n g a w a y f r o m the d e t e c t o r w i l l h a v e a s m a l l e r c h a n c e t o b e reflected t o w a r d s the d e t e c t o r . T h e t w o p r o -cesses act s i m u l t a n e o u s l y , so the net effect w i l l d e p e n d o n the difference i n the n u m b e r o f n e u t r o n s g o i n g i n e i t h e r d i r e c t i o n , w h i c h is the c o m p o n e n t o f the net n e u t r o n c u r r e n t v e c t o r i n the d i r e c t i o n o f the d e t e c t o r . I n the next s e c t i o n t h i s d i r e c t i o n a l d e p e n -d e n c e w i l l be -d e s c r i b e -d b y the i n n e r p r o -d u c t o f the g r a d i e n t s o f t h e flux a n d a d j o i n t flux. S u m m a r i z i n g : the t o t a l r e s p o n s e o f a n e u t r o n d e t e c t o r t o a p a s s i n g b u b b l e is c o m p o s e d o f three c o m p o n e n t s : — a n a l w a y s p o s i t i v e c o m p o n e n t d u e t o d e c r e a s e d a b s o r p t i o n ; — a n a l w a y s n e g a t i v e o n e d u e t o d e c r e a s e d m o d e r -a t i o n , i n the c-ase o f -a t h e r m -a l n e u t r o n d e t e c t o r a n d — a g e o m e t r y d e p e n d e n t c o m p o n e n t d u e to d i f f u s i o n processes. C l e a r l y , the t o t a l r e s p o n s e d e p e n d s o n the m u t u a l p r o p o r t i o n o f the o p p o s i t e p r o c e s s e s a n d t h u s o n the c o n s t r u c t i o n d e t a i l s o f the specific r e a c t o r a n d the p o s i t i o n s o f b u b b l e a n d d e t e c t o r t h e r e i n . F i n a l l y , it s h o u l d be r e m a r k e d t h a t i n the f o r e g o i n g q u a l i t a t i v e a n a l y s i s o n l y the i n f l u e n c e o f the b u b b l e o n n e u t r o n s g o i n g d i r e c t l y t o the d e t e c t o r is c o n -s i d e r e d . C h a n g e -s i n the p r o d u c t i o n r a t e o f n e u t r o n -s a n d i n i n f o r m a t i o n t r a n s f e r r e d b y f i s s i o n c h a i n s is n e g l e c t e d ; i n o t h e r w o r d s , r e a c t i v i t y effects are n o t c o n s i d e r e d . I n fact o n l y the l o c a l c o m p o n e n t o f the r e s p o n s e is t h u s e v a l u a t e d . F o r i n s t a n c e , w i t h a t h e r m a l n e u t r o n d e t e c t o r the m o d e r a t i o n c o m p o n e n t o f the r e s p o n s e w o u l d a l w a y s be n e g a t i v e o n the b a s i s o f the p r e v i o u s a n a l y s i s ; b u t i n a r e a c t o r w h i c h is o v e r m o d e r a t e d , a p o s i t i v e r e a c -t i v i -t y effec-t o c c u r s w h i c h w i l l g i v e rise -t o a p o s i -t i v e " g l o b a l " c o m p o n e n t i n the r e s p o n s e . I n the next s e c t i o n a m o r e s o p h i s t i c a t e d m o d e l is d e v e l o p e d i n w h i c h the r e a c t i v i t y effects are ( i m p l i -c i t l y ) -c o n s i d e r e d . 3. C O M P U T A T I O N O F T H E D E T E C T O R R E S P O N S E V I A P E R T U R B A T I O N T H E O R Y T h e b e h a v i o u r o f the n e u t r o n p o p u l a t i o n i n a r e a c -t o r is d e s c r i b e d b y -the B o l -t z m a n n e q u a -t i o n : — 0 = B ^ + S. (1) dt H e r e <j> is the s t a t e - v e c t o r o f w h i c h the c o m p o n e n t s are the e n e r g y a n d / o r d i r e c t i o n d e p e n d e n t ( a n g u l a r ) n e u t r o n flux, d e l a y e d n e u t r o n p r e c u r s o r d e n s i t i e s , etc.; S is the s o u r c e v e c t o r a n d B is the t r a n s p o r t o p e r a t o r .

N o w we w i l l c o n s i d e r c r o s s s e c t i o n c h a n g e s d u e to b u b b l e s i n the m o d e r a t o r , g i v i n g rise t o c h a n g e s i n the n e u t r o n flux. W e split the fluxes a n d t h e t r a n s -p o r t o -p e r a t o r i n a c o n s t a n t a n d a fluctuating -p a r t : B = B „ + SB <t> = <po + &<t> a n d r e w r i t e (1), n e g l e c t i n g s e c o n d - o r d e r t e r m s : i,54> = B0S<t> + 5B<t>0 (2.1) d t B0<t>0 = " S o (2-2) F o u r i e r t r a n s f o r m i n g (2.1) a n d e l i m i n a t i n g the p r e c u r

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Analysis o f neutron detector response sor d e n s i t i e s leads t o

B(w)ö<Hu>) = -<5ß(<y)4>0 (3)

w h e r e B{a>) = B0( o ) ) — koE, E b e i n g the u n i t y m a t r i x .

In these e q u a t i o n s t h e f r e q u e n c y - d e p e n d e n c e o f t h e v a r i a b l e s is s h o w n e x p l i c i t l y . N o w p e r t u r b a t i o n t h e o r y ( B e l l a n d G l a s s t o n e , 1970) is a p p l i e d t o c o m p u t e the r e s p o n s e o f a n e u t r o n detect o r i n detecthe f r e q u e n c y d o m a i n . F i r s detect , a n a d j o i n detect o p e r a -t o r B+(cy) is c o n s t r u c t e d , o p e r a t i n g o n a d j o i n t func-t i o n s &lfunc-t;j&gfunc-t;* w h i c h h a v e func-the f o l l o w i n g p r o p e r func-t y :

(<t> * (to), B(co)<50(«))) = (<54H<y), B + (<« W>+ (<o)), (4)

i n w h i c h the i n n e r p r o d u c t i n c l u d e s i n t e g r a t i o n o v e r space a n d energy v a r i a b l e s . F o r the c o n s t r u c t i o n o f a n o p e r a t o r B+ w h i c h satis-fies (4) w e refer t o v a n D a m (1977). A d j o i n t f u n c t i o n s a r e g e n e r a t e d b y the e q u a t i o n B+{œ)<t> + (œ) + Y.d = 0. (5) In this e q u a t i o n Zd i s t h e m a c r o s c o p i c d e t e c t i o n c r o s s - s e c t i o n v e c t o r o f the n e u t r o n d e t e c t o r u n d e r c o n s i d e r a t i o n , t h u s f o r m i n g a k i n d o f " a d j o i n t s o u r c e " . C o m b i n i n g (3)-(5), w e finally get ( X „ 5 0 ( a ; ) ) = ( ^+( a ; ) , 5 B ( (O) 0 o ) . T h e L H S o f this e q u a t i o n g i v e s t h e f l u c t u a t i o n i n t h e d e t e c t o r c o u n t rate w h i c h w e define as t h e d e t e c t o r r e s p o n s e : R(a>) = (<l>+(<o),5B(a>)<l>Q). (6) H e r e the f r e q u e n c y d e p e n d e n c e is s h o w n . T h e a d j o i n t B(w) c a n be a p p r o x i m a t e d as f r e q u e n c y - i n d e p e n d e n t

(e.g. the p l a t e a u r e g i o n o f the r e a c t i v i t y transfer func-t i o n ) func-t h e n B(a&gfunc-t;), B* a n d 0 + (a>) a r e r e a l a n d f r e q u e n c y -i n d e p e n d e n t a n d (6) c a n t h e n be -inverse F o u r -i e r t r a n s f o r m e d i n t o R(t) = ( f , M W o ) (7) w h i c h gives t h e d e t e c t o r r e s p o n s e i n t i m e d o m a i n . In fact, a " p r o m p t r e s p o n s e " a p p r o x i m a t i o n is a p p l i e d h e r e : the s y s t e m relaxes i n f i n i t e l y fast t o a s t a t i o n a r y flux c o r r e s p o n d i n g t o the p e r t u r b e d s y s t e m a n d d e l a y e d effects d o n o t p l a y a r o l e . I n this fre-q u e n c y r e g i o n the space d e p e n d e n c e o f the d e t e c t o r r e s p o n s e t o c r o s s s e c t i o n changes, d e s c r i b e d b y t h e a d j o i n t flux a n d the s t a t i o n a r y flux, is exact a n d n o a p p r o x i m a t e m e t h o d s (e.g. p o i n t m o d e l o r q u a s i - s t a t i c m o d e l ( K o s a l y et ai, 1977b)) a r e needed. T h e a b o v e d e v e l o p e d m e t h o d is n o w a p p l i e d t o a t w o - g r o u p d i f f u s i o n a p p r o x i m a t i o n w i t h o n e g r o u p o f d e l a y e d n e u t r o n s . T h e d i f f u s i o n e q u a t i o n s a r e : d 0, = ( V D V -

L

t

, - X,„ +

v , I/ :( l - /?))t.,0, d( d d It - ß)vt<t> ( V 02V + V¡ÁC + S , [ ' | , + ß^2^-f2<t>l - AC. S p l i t t i n g the fluctuating q u a n t i t i e s i n t o a c o n s t a n t a n d a fluctuating part, F o u r i e r t r a n s f o r m i n g a n d e l i m i n -a t i n g t h e d e l -a y e d n e u t r o n terms, w e c -a n w r i t e for B(to):

ia>\ i ßicü \ „ / ßioj

, \ D2v2\ - ( ï .0 2 + - , i

flux <^ + (oi) is f r e q u e n c y d e p e n d e n t because t h e a d j o i n t R e m e m b e r i n g that i n m u l t i g r o u p d i f f u s i o n t h e o r y the

o p e r a t o r B+, w h i c h is r e l a t e d t o B(co), is f r e q u e n c y a d j o i n t o p e r a t o r is the t r a n s p o s e o f B(a>) a n d e l i m i n

-d e p e n -d e n t . a t i n g t h e f r e q u e n c y - -d e p e n -d e n c e ( p l a t e a u r e g i o n ) w e If w e restrict o u r s e l v e s t o a f r e q u e n c y r e g i o n w h e r e g e t : ' V D j i i , V - ( I „ , + X.Jv, + v . I ^ ^ d - ß), v2ZSl2 Vtntfl(i - $ V ß2i >2V - Za¡v2 T h e c o n s t a n t flux i n t h e r e a c t o r i s t h e s o l u t i o n o f x i + r * ' ? ! »

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-20

E . KLEISS and H . V A N D A M

T h e b u b b l e s c a u s e f l u c t u a t i o n s i n D , , D2, £ „ , , E0J a n d ZS l J. T h e f l u c t u a t i n g p a r t o f the t r a n s p o r t o p e r a t o r

n o w r e a d s

SB = [nDiV> * ~ VlSL" " VlSZ> » ' 0

w h e r e <5D, etc. have a v a l u e at the p o s i t i o n o f the b u b b l e a n d are z e r o elsewhere. U s i n g G r e e n ' s d i v e r g e n c e t h e o r e m , we c a n w r i t e for d e t e c t o r r e s p o n s e : R(w) = - JdK{5D,(co)V^, • V<j>; + 6D2(co)\<l>2-\<t>;

+ SZ.^Atf -

0 2+) i . (8) the result g i v e n b y v a n D a m (1976), o r R(t) = - jdV{6D,(l)\<l>i • v</>,+ + ...} (9) u s i n g the p r o m p t r e s p o n s e a p p r o x i m a t i o n . B e c a u s e o f the f r e q u e n c y i n d e p e n d e n c e we d e v e l -o p e d a useful t -o -o l f-or the c -o m p u t a t i -o n -o f the res p o n res e . T h e flux a n d a d j o i n t flux c a n n o w be c a l c u -l a t e d o n c e for e v e r y d e t e c t o r p o s i t i o n ( a n d every s o u r c e p o s i t i o n for s u b c r i t i c a l systems) u s i n g a n y k n o w n t e c h n i q u e . T h e m e t h o d o f c o m p u t a t i o n for this w o r k is d i s c u s s e d i n the a p p e n d i c e s . If the p r o m p t r e s p o n s e a p p r o x i m a t i o n is n o t a p p l i c a b l e the a d j o i n t flux s h o u l d be c o m p u t e d for every f r e q u e n c y o f i n t e r

-est s e p a r a t e l y , o r , i n the t i m e d o m a i n , a s p a c e - t i m e d e p e n d e n t r e a c t o r c o d e s h o u l d b e u s e d ( V a l k o a n d M e s k o , 1977). A n e x a m p l e o f the c o m p u t e d fluxes a n d a d j o i n t fluxes is g i v e n i n F i g . 3. H e r e the a x i a l d i s t r i -b u t i o n is p l o t t e d for g e o m e t r y 4 B (see S e c t i o n 4) for s o u r c e s t r e n g t h o f 1 n/s a n d a d e t e c t o r efficiency o f 1 c p s / u n i t flux. 4. E X P E R I M E N T A L S E T U P A N D E X P E R I M E N T S E x p e r i m e n t s w e r e c a r r i e d o u t i n the l i g h t w a t e r m o d e r a t e d , n a t u r a l u r a n i u m fuelled s u b c r i t i c a l a s s e m -b l y L I S A at D e l f t . It c o n s i s t s o f 253 fuel p i n s c o n t a i n i n g h o l l o w u r a n i u m c y l i n d e r s . T h e s e p i n s are p l a c e d i n a h e x a g o n a l g r i d s t r u c t u r e w i t h a p i t c h o f 45 m m , t h u s f o r m -i n g a c o r e w -i t h a h e -i g h t o f 82 c m a n d a n e q u -i v a l e n t d i a m e t e r o f 8 0 c m . T h e kc„ o f the s y s t e m is a b o u t 0.82 a n d the m o d e r a t o r - u r a n i u m v o l u m e r a t i o is 2.06. S o m e e x p e r i m e n t s were c a r r i e d o u t i n a m o d i f i e d c o r e i n w h i c h the fuel p i n s were r e - a s s e m b l e d i n a n o t h e r p a t t e r n (see F i g . 1). T h e M/F v o l u m e r a t i o i n this c o r e w a s 3.96, the c o r e h a v i n g a d i a . o f 92 c m a n d

© © © ©

o f © © (Of ©

© © © ©

aluminium cladding

F i g . 1. Lattice cell configuration for the different geometries. Left figure: N o r m a l fuel r o d pattern. Detector in rod 1, neutron source in r o d 3: configuration 1. Detector in rod 2, neutron source in r o d 3: configuration 2. Detector in rod 3, neutron source in r o d 2: configuration 3. Detector in r o d 1, neutron source i n r o d 2: configuration 4. Right figure: M o d i f i e d fuel r o d pattern. Detector in r o d

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21

Analysis of neutron detector response

S A M u l t i s c o l e r B P ' t -NS« f J N D B C Û V a l v e B C : B u b b l e C h a n n e l B P : B u b b l e P r o b e N D : N e u t r o n D e t e c t o r N S : N e u t r o n S o u r c e S A : S u b c r l t l c a l A s s e m b l y S i g n a l c o n d i t i o n -i n q A v e r a g e r P u l s e g e n e r a t o r

Fig. 2. Schematic survey a kf!t o f 0.85. A 252C f n e u t r o n s o u r c e was p l a c e d i n

the c e n t r e o f the c o r e i n s i d e a fuel p i n . A 3H e n e u t r o n

d e t e c t o r (0.64 c m dia.) w a s p l a c e d i n s i d e a n o t h e r fuel p i n . T h i s d e t e c t o r w a s p a r t l y c o v e r e d w i t h c a d m i u m , t h u s l e a v i n g a s e n s i t i v e l e n g t h o f 2.5 c m . A i r b u b b l e s w e r e i n j e c t e d i n t o the r e a c t o r i n a b u b b l e - c h a n n e l p o s i t i o n e d b e t w e e n the fuel p i n s . T h e b u b b l e s w e r e n o t g e n e r a t e d c o n t i n u o u s l y , b u t v i a a v a l v e t r i g g e r e d b y a p u l s e g e n e r a t o r (pulse r e p e t i t i o n t i m e 10 s). A t e v e r y pulse, the v a l v e w a s o p e n e d a n d

the experimental setup.

a b u b b l e w a s g e n e r a t e d w h i c h t r a v e l l e d u p w a r d i n the s y s t e m . T h e s i g n a l o f the n e u t r o n d e t e c t o r was, after a m p l i f i c a t i o n , s u b s t r a c t i o n o f the m e a n l e v e l a n d a n t i a l i a s i n g f i l t e r i n g fed i n t o a 2 5 6 p o i n t d i g i t a l s i g -nal a v e r a g e r . T h i s a v e r a g e r w a s t r i g g e r e d b y the s a m e p u l s e t h a t o p e n e d the v a l v e . I n this w a y a n average-r e s p o n s e t e c h n i q u e foaverage-r m e a s u average-r i n g the d e t e c t o average-r average- re-s p o n re-s e w a re-s p e r f o r m e d .

A b u b b l e p r o b e w a s p o s i t i o n e d i n the b u b b l e c h a n -nel, w h i c h detected the p a s s i n g o f a b u b b l e t h r o u g h 0.02 4>,4> + (see text) 0.01 - ! I i ¡ i 0 detector source position position

F i g . 3. N e u t r o n flux and adjoint flux along the bubble channel in geometry 4 B . Fast flux. T h e r m a l flux. Fast adjoint flux. T h e r m a l adjoint flux.

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E . KLEISS and H V A N D A M 400

0 40 80

Fig. 4. E x a m p l e of the bubble risetime distribution. H o r i -zontal axis: rise-time in seconds; vertical axis: counts per

channel. the c h a n n e l . T h e s i g n a l o f t h i s p r o b e w a s i n p u t to a m u l t i s c a l e r w h i c h was a l s o t r i g g e r e d b y the v a l v e c o n t r o l p u l s e . In this w a y a b u b b l e r i s e t i m e d i s t r i b u t i o n c o u l d be o b t a i n e d ( a n d f r o m t h i s a b u b b l e v e l -o c i t y d i s t r i b u t i -o n ) s i m u l t a n e -o u s t-o the r e s p -o n s e m e a s u r e m e n t . T h i s b u b b l e v e l o c i t y a p p e a r e d t o be not very c o n s t a n t d u e to u n k n o w n c i r c u m s t a n c e s , it c o u l d v a r y u p t o 20",, r.m.s. d u r i n g o n e e x p e r i m e n t (see F i g . 4). T h e i n f l u e n c e o f this v a r y i n g v e l o c i t y w i l l be d i s c u s s e d i n S e c t i o n 5. A s c h e m a t i c s u r v e y o f the e x p e r i m e n t a l setup is g i v e n i n F i g . 2. M e a s u r e m e n t s o f the r e s p o n s e h a v e b e e n c a r r i e d o u t for s e v e r a l p o s i t i o n s o f the n e u t r o n d e t e c t o r a n d the n e u t r o n s o u r c e a r o u n d the b u b b l e c h a n n e l . T h i s c h a n n e l was a l w a y s p o s i t i o n e d at the c e n t r e o f the c o r e , the n e u t r o n s o u r c e a l w a y s at the m i d d l e o f the c o r e h e i g h t at t w o different d i s t a n c e s f r o m the b u b b l e c h a n n e l .

A l l different c o m b i n a t i o n s o f the d e t e c t o r a n d s o u r c e p o s i t i o n s are c o d e d b y a n u m b e r a n d a letter, the n u m b e r a p p l y i n g to the g e o m e t r y t o p v i e w as d i s p l a y e d i n F i g . 1, the letter to the h e i g h t o f the d e t e c t o r i n the c o r e : A: 41 c m ( b e i n g m i d - h e i g h t ) ; B : 31 c m ; C: 21 c m ; D: 3 6 c m a n d E: 3 3 . 5 c m . 5. R E S U L T S A N D D I S C U S S I O N I n t h i s s e c t i o n b o t h e x p e r i m e n t a l a n d t h e o r e t i c a l results a r e p r e s e n t e d a n d c o m p a r e d . T h e r e s u l t s are d i s p l a y e d i n F i g s 6 - 1 0 . T h e re-s p o n re-s e re-s are n o r m a l i z e d to the c o m p u t e d a n d m e a s u r e d a v e r a g e c o u n t rate, r e s p e c t i v e l y , t o be i n d e -p e n d e n t o f the exact v a l u e o f s o u r c e s t r e n g t h a n d d e t e c t o r efficiency. I n the d i s p l a y e d c u r v e s n o c o n f i -d e n c e i n t e r v a l s are g i v e n , b u t a g o o -d i m p r e s s i o n o f

the a c c u r a c y c a n be g o t f r o m the s p r e a d o f the m e a s u r i n g p o i n t s i n the figures.

T o get a g o o d u n d e r s t a n d i n g o f the c o m p u t e d re-s p o n re-s e re-s we c o m p a r e there-se w i t h the q u a l i t a t i v e m o d e l , t h i s b e i n g d o n e w i t h the a i d o f F i g . 5.

In t h i s figure the five c o m p o n e n t s o f the d e t e c t o r r e s p o n s e are d r a w n for f o u r g e o m e t r i e s : I B , 2 B , 3 B a n d 4 B . T h e five c o m p o n e n t s are the c o n t r i b u t i o n s o f fast a n d t h e r m a l a b s o r p t i o n , fast a n d t h e r m a l diffu-s i o n a n d m o d e r a t i o n , adiffu-s they a p p e a r i n (9). T h e c u r v e diffu-s i n F i g . 5 are n o t n o r m a l i z e d t o the a v e r a g e c o u n t rate, b u t the v a l u e s c o r r e s p o n d t o a u n i t s o u r c e a n d d e t e c t o r s t r e n g t h . T h e q u a l i t a t i v e m o d e l p r e d i c t s for the a b s o r p t i o n effect a p o s i t i v e c o n t r i b u t i o n a n d for the m o d e r a t i o n effect a n e g a t i v e o n e . T h i s is i n ac-c o r d a n ac-c e w i t h the ac-c o m p u t e d results. T h e i n t e r e s t i n g difference b e t w e e n the p i c t u r e s is the b e h a v i o u r o f the d i f f u s i o n c o m p o n e n t s , w h i c h s h o u l d be d e p e n d e n t o n the d i r e c t i o n o f n e u t r o n c u r r e n t r e l a t i v e t o detec-t o r . W h e n detec-the b u b b l e is far a w a y adetec-t detec-the detec-t o p o r b o detec-t detec-t o m o f the c o r e , the net n e u t r o n c u r r e n t is d i r e c t e d o u t w a r d s , a w a y f r o m the d e t e c t o r , a n d a n e g a t i v e (but s m a l l ) c o n t r i b u t i o n is e x p e c t e d . T h i s is i n effect the case.

W h e n the b u b b l e is c l o s e r t o the d e t e c t o r , e.g. o n the l i n e d e t e c t o r - s o u r c e , the r e s p o n s e is p o s i t i v e i f the b u b b l e is i n b e t w e e n d e t e c t o r a n d s o u r c e a n d n e g a t i v e if the b u b b l e is at the o t h e r side o f e i t h e r o f t h e m . T h u s , a p o s i t i v e c o n t r i b u t i o n is e x p e c t e d i n I B a n d 4 B , a n d a n e g a t i v e o n e i n 2 B a n d 3 B . T h i s , t o o , is i n a c c o r d a n c e w i t h F i g . 5 e x c e p t for a s m a l l p o s i t i v e p e a k i n the fast d i f f u s i o n c o n t r i b u t i o n i n 3 B . T h i s c a n be e x p l a i n e d b y a c l o s e r l o o k at the geo-m e t r y : b e c a u s e the b u b b l e is v e r y n e a r t o the d e t e c t o r , a n d the s o u r c e a n d d e t e c t o r are at 10 c m different h e i g h t , there is a s m a l l r e g i o n w h e r e the n e u t r o n c u r -rent has a p o s i t i v e c o m p o n e n t i n the d i r e c t i o n o f the d e t e c t o r . In this s i t u a t i o n a s m a l l p o s i t i v e c o n t r i b u -t i o n s h o u l d arise, as i-t does. In 2 B -the s a m e effec-t o c c u r s . A s a c o n c l u d i n g r e m a r k it c a n be s t a t e d t h a t the c o m p u t e d r e s p o n s e s c a n be u n d e r s t o o d f r o m the v i e w p o i n t o f the e l e m e n t a r y processes. I n F i g s 6 - 1 0 the m e a s u r e d a n d c o m p u t e d r e s p o n s e s are s h o w n for c o m p a r i s o n . H o w e v e r , d i r e c t c o m p a r i -s o n i-s n o t p o -s -s i b l e b e c a u -s e the m e a -s u r e d r e -s p o n -s e -s are i n f l u e n c e d b y b u b b l e v e l o c i t y v a r i a t i o n s (see S e c -t i o n 4). T h e s e v e l o c i -t y v a r i a -t i o n s h a v e a s m o o -t h i n g effect o n the r e s p o n s e , so that d e t a i l is l o s t . I n p r i n -c i p l e t h i s p r o b l e m -c a n be s o l v e d b y " d e -c o n v o l u t i n g " the m e a s u r e d r e s p o n s e w i t h the m e a s u r e d b u b b l e v e l o c i t y d i s t r i b u t i o n , b u t b e c a u s e o f the s t a t i s t i c a l u n c e r t a i n t i e s i n b o t h t h i s w a s p r a c t i c a l l y i m p o s s i b l e . T h e r e -fore the c o m p u t e d r e s p o n s e w a s c o n v o l u t e d w i t h the

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Analysis of neu tron detector response 0 4 0 4 0 80 1 ___ ••• I—— -I ! 1 ' '. , 'r i V « , 1 i 1 Í 1 i 3 B i i i i i J i i i'' 0 4 0 80 i . / X ' l 4 B 4 0 8 0 4 0 80

F i g . 5. The five components of the response as function of the bubble height for geometries ill. IB. 3 B a n d 4B. Fast absorption. T h e r m a l absorption. M o d e r a t i o n . Fast diffu-sion. T h e r m a l diffudiffu-sion. H o r i z o n t a l axis: bubble position (cm): vertical axis: detector response

in 1 C T7 counts/s.

m e a s u r e d v e l o c i t y d i s t r i b u t i o n , t h u s s i m u l a t i n g the e x p e r i m e n t , the r e s u l t s are a l s o g i v e n i n F i g s 6 - 1 0 . T h e s e figures c o n s i s t of. f r o m left to r i g h t , the p l o t s : — t h e c o m p u t e d r e s p o n s e , for a b u b b l e o f 3.5 c m3,

— t h i s c o m p u t e d r e s p o n s e c o n v o l u t e d w i t h the v e l -o c i t y d i s t r i b u t i -o n a n d

— t h e m e a s u r e d r e s p o n s e .

A l l c u r v e s are p l o t t e d a g a i n s t the a x i a l p o s i t i o n o f the b u b b l e i n the c h a n n e l . F o r the left c u r v e t h i s is o b v i o u s : for the c e n t r e a n d r i g h t c u r v e s , w h i c h are m e a s u r e d i n t i m e d o m a i n , this c o u l d be d o n e u s i n g the a v e r a g e b u b b l e v e l o c i t y as o b t a i n e d f r o m the v e l -o c i t y d i s t r i b u t i -o n . I n t h i s w a y a n easier c -o m p a r i s -o n b e t w e e n e x p e r i m e n t s a n d t h e o r y is p o s s i b l e . A r a t h e r g o o d c o r r e s p o n d e n c e b e t w e e n the c a l c u l a t e d a n d the m e a s u r e d r e s p o n s e s exists, a p a r t f r o m differences i n the m a g n i t u d e o f the r e s p o n s e o f c o m p o n e n t s thereof.

I n a l l p l o t s a m a j o r n e g a t i v e p e a k exists d u e to the m o d e r a t i o n effect, a n d a s m a l l e r p o s i t i v e o n e i n the s i t u a t i o n s w h e r e s o u r c e a n d d e t e c t o r were n o t at the s a m e a x i a l p o s i t i o n . T h i s p o s i t i v e peak is p a r t l y

d u e t o the a b s o r p t i o n effect a n d p a r t l y t o the d i f f u s i o n effect (cases IA-B, 4 , 4 - C . 5A-B). In the cases 2 a n d 3 the d i f f u s i o n effect s h o u l d g i v e a n e g a t i v e c o n t r i b u -t i o n , w h i c h agrees w i -t h -the e x p e r i m e n -t .

In cases A o n l y a n e g a t i v e p e a k is v i s i b l e because a l l p e a k s c o i n c i d e a n d the largest ( m o d e r a t i o n ) d o m i n a t e s . L o w e r i n g the d e t e c t o r i n the c o r e leads t o a shift a n d b r o a d e n i n g o f the d i f f u s i o n p e a k ; the m o d e r a t i o n a n d d i f f u s i o n p e a k s shift to different h e i g h t s . T h i s c a n be e x p l a i n e d b y the fact that, d u e to the different r e l a x a t i o n l e n g t h s o f a d j o i n t a n d r e g u -lar flux, the m o d e r a t i o n a n d a b s o r p t i o n p e a k s w i l l be near the d e t e c t o r , w h i l e the d i f f u s i o n p e a k w i l l h a v e its m a x i m u m a b o u t w h e r e the b u b b l e passes the l i n e s o u r c e - d e t e c t o r .

A n i n t e r e s t i n g difference exists b e t w e e n g e o m e t r i e s 1 a n d 2, w h e r e the m o d e r a t i o n a n d a b s o r p t i o n effects s h o u l d be the s a m e b u t the d i f f u s i o n effect s h o u l d c h a n g e its s i g n . T h i s i n d e e d a p p e a r s t o be the case, e s p e c i a l l y i n c o n f i g u r a t i o n B.

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E . KLEISS and H . V A N D A M

figurations 4 a n d 5. H e r e the d e t e c t o r a n d s o u r c e are i n the s a m e p o s i t i o n s b u t the M/F r a t i o is i n c r e a s e d i n c o n f i g u r a t i o n 5. T h i s s h o u l d l e a d t o a r e d u c t i o n o f the m o d e r a t i o n effect b e c a u s e the difference b e t w e e n the fast a n d t h e r m a l a d j o i n t fluxes decreases.

T h e q u e s t i o n a r i s e s : W h i c h effects c a u s e the exist-i n g dexist-ifference exist-i n m a g n exist-i t u d e b e t w e e n the c o m p u t e d a n d the m e a s u r e d responses? S o m e m e a s u r e m e n t s agree w e l l , w h i l e o t h e r s are t o o large o r h a v e a n i n -c o r r e -c t r a t i o b e t w e e n the p o s i t i v e a n d n e g a t i v e p e a k s . T h e s e differences m a y be e x p l a i n e d b y the f o l l o w i n g c o n s i d e r a t i o n s . T h e v o l u m e o f the b u b b l e w a s subject t o s l i g h t v a r i -a t i o n s , d u r i n g -a single m e -a s u r e m e n t -as w e l l -as b e t w e e n successive m e a s u r e m e n t s . A l l c a l c u l a t i o n s w e r e d o n e u s i n g a b u b b l e v o l u m e o f 3.5 c m3.

In the c a l c u l a t i o n o f the c r o s s s e c t i o n set that w a s u s e d for the c o m p u t a t i o n s s o m e s i m p l i f y i n g a p p r o x i -m a t i o n s a r e -m a d e that affect the c r o s s s e c t i o n s . T h e s e h a v e b e e n c a l c u l a t e d b y the G G C - 4 c o d e w h i c h uses a h o m o g e n i z e d r e a c t o r m o d e l w i t h the s a m e n e u t r o n e n e r g y s p e c t r u m for a l l the m a t e r i a l s i n a l l r e a c t o r

z o n e s . S p e c t r a l differences i n the different m a t e r i a l s w o u l d g i v e a ( s l i g h t l y ) different c r o s s s e c t i o n set, e s p e c i a l l y i n the t h e r m a l r e g i o n .

T h e h e t e r o g e n e i t i e s i n the r e a c t o r a l s o give rise to d e v i a t i o n s o f the flux t h a t w a s c o m p u t e d for a h o m o -g e n e o u s s y s t e m . T h e s e h e t e r o -g e n e i t y effects h a v e b e e n a c c o u n t e d for i n a n a p p r o x i m a t e w a y a n d are c o m -m e n t e d o n i n A p p e n d i x 3.

T h e effect o f the c a d m i u m s h i e l d a r o u n d the detec-t o r is n e g l e c detec-t e d . P e r t u r b a t i o n t h e o r y itself m i g h t b r e a k d o w n here, c o n s i d e r i n g the l a r g e b u b b l e d i m e n s i o n s . O n e w o u l d e x p e c t t o h a v e a s m a l l e r r e s p o n s e t h e n , a n d this t e n d -e n c y c a n b-e n o t i c -e d i n m a n y o f th-e p i c t u r -e s . C o n s i d e r i n g the i n f l u e n c e o f the b u b b l e v e l o c i t y d i s t r i b u t i o n o n the r e s p o n s e , u n c e r t a i n t i e s h e r e i n w i l l a l s o affect the c o r r e s p o n d e n c e b e t w e e n t h e o r y a n d e x p e r i m e n t . N o w we w i l l c o n s i d e r the e x p e r i m e n t i n c o n n e c t i o n w i t h the l o c a l / g l o b a l c o n c e p t a n d i n c o m p a r i s o n w i t h a s i m i l a r e x p e r i m e n t c a r r i e d o u t i n a c r i t i c a l f a c i l i t y ( F u g e ex ai, 1 9 7 7 ; V a l k o a n d M e s k o , 1977).

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Analysis of neutron detector response - I 1 1 1 r— . . . i ! i i i i

]

• 2 0 — i i — i — i

-f

: 80cm O O Fig. 7. I n the p e r t u r b a t i o n m o d e l the r e s p o n s e is d e t e r -m i n e d b y the a d j o i n t flux w h i c h c a n be c h a r a c t e r i z e d b y t w o r e l a x a t i o n l e n g t h s i n a t w o - g r o u p m o d e l . T h e s e r e l a x a t i o n l e n g t h s are, for a z e r o p o w e r r e a c t o r , 1, * L , (2 * N/ i G t a T ) . H e r e L is the t h e r m a l n e u t r o n d i f f u s i o n l e n g t h , T the F e r m i age a n d G(OJ) the reactivity transfer f u n c t i o n . 11 a n d 12 w i l l b e o f the o r d e r o f 1, r e s p e c t i v e l y , 8 0 c m i n the p l a t e a u r e g i o n for a c r i t i c a l l i g h t w a t e r r e a c t o r . T h e s e d i s t a n c e s c a n c l e a r l y be i d e n t i f i e d w i t h a s h o r t -r a n g e o -r " l o c a l " s e n s i t i v i t y a n d w i t h a l o n g - -r a n g e o -r " g l o b a l " o n e . I n this e x p e r i m e n t , h o w e v e r , the s e c o n d r e l a x a t i o n l e n g t h is a b o u t 6 c m , w h i c h c e r t a i n l y d o e s n o t deserve the n a m e " g l o b a l " . I n this sense the r e s p o n s e s c o m -p u t e d here are l o c a l r e s -p o n s e s w h i l e a g l o b a l effect d o e s n o t o c c u r . T h e g l o b a l effect has a l s o b e e n i d e n t i -fied as the r e a c t i v i t y c o m p o n e n t o f the n o i s e . I n t h i s e x p e r i m e n t the r e a c t i v i t y effect o f the b u b b l e is n e g l i -g i b l e ; it w o u l d -g i v e rise t o r e l a t i v e r e s p o n s e s o f 1 0 ~6 o n l y . S u m m a r i z i n g , i n the s t r o n g l y s u b c r i t i c a l s y s t e m n o g l o b a l p a r t o f the r e s p o n s e c o u l d be r e c o g n i z e d i n the e x p e r i m e n t s a n d i n the t h e o r e t i c a l t r e a t m e n t thereof. H o w e v e r , i n a c r i t i c a l s y s t e m the p e r t u r b a t i o n t h e o r y w o u l d give r e s p o n s e s t h a t c a n , p a r t l y , be c o n -s i d e r e d a-s g l o b a l . T h e r e s p o n s e s m e a s u r e d i n t h i s e x p e r i m e n t s h o w r e s e m b l a n c e w i t h a n e x p e r i m e n t d e s c r i b e d e a r l i e r ( F u g e el al, 1 9 7 7 ; V a l k o a n d M e s k o , 1977). T h e r e a n a v e r a g e - r e s p o n s e m e t h o d w a s a p p l i e d to a c r i t i c a l r e a c t o r a n d r e s p o n s e s w e r e o b t a i n e d t h a t s h o w e d s i m i l a r b e h a v i o u r , h a v i n g p o s i t i v e a n d n e g a t i v e c o m -p o n e n t s . T h e r e are, h o w e v e r , i m -p o r t a n t differences. O u r r e s p o n s e s are s t r i c t l y l o c a l , b u t h a v i n g a fine s t r u c t u r e w i t h n e g a t i v e m o d e r a t i o n a n d s o m e t i m e s p o s i t i v e d i f f u s i o n c o m p o n e n t s . T h e r e s p o n s e s i n the c r i t i c a l s y s t e m h a d a n e g a t i v e c o m p o n e n t w h i c h c o u l d be i d e n t i f i e d as g l o b a l , a n d a l o c a l c o m p o n e n t t h a t h a d a less p r o n o u n c e d s t r u c -ture. I n o u r o p i n i o n t h i s is m a i n l y d u e t o the fact that d i f f u s i o n effects p l a y a m i n o r r o l e b e c a u s e the flux g r a d i e n t s are m u c h s m a l l e r i n the c r i t i c a l s y s t e m .

(28)
(29)
(30)

28 E . K L E I S S and H . V A N D A M - 0 7 -

1

"58 II 80cm 0 t —i — i — i 1 — i — i — i — r ~ - i — i 1 — i — i — i r-80cm 0 80cm F i g . 10.

Figs 6-10. C o m p u t e d and measured responses as function of the b u b b l e height for configurations 1-5 for different detector heights. H o r i z o n t a l axis: bubble p o s i t i o n ; vertical axis: response i n % o f

average count rate.

T h e g l o b a l b e h a v i o u r o f the r e s p o n s e w a s c l e a r f r o m the r e s p o n s e o f a n o u t - o f - c o r e d e t e c t o r .

H o w e v e r , the g l o b a l c o m p o n e n t was s l i g h t l y differ-ent for differdiffer-ent d e t e c t o r s . T h i s s h o w s t h a t the g l o b a l c o m p o n e n t is n o t s t r i c t l y the r e a c t i v i t y effect o f the b u b b l e . T h i s s p a c e d e p e n d e n c e o f t h e g l o b a l c o m -p o n e n t c a n b e e a s i l y u n d e r s t o o d i n the -p e r t u r b a t i o n m o d e l : different d e t e c t o r s w i l l h a v e different a d j o i n t fluxes a n d t h u s different r e s p o n s e s . 6. C O N C L U S I O N S T h e a v e r a g e r e s p o n s e m e t h o d is a g o o d o n e for m e a s u r i n g the effect o f a i r b u b b l e s o n t h e s i g n a l o f a n e u t r o n d e t e c t o r i n a ( s u b c r i t i c a l ) r e a c t o r . T h e p e r t u r b a t i o n m o d e l is w e l l a p p l i c a b l e t o de-s c r i b e the a b o v e - m e n t i o n e d e x p e r i m e n t . S o m e differ-ences b e t w e e n m e a s u r e m e n t s a n d t h e o r y d o exist b u t they are n o t c o n s i d e r e d as e s s e n t i a l . In the f r e q u e n c y r e g i o n c o n s i d e r e d , the r e s p o n s e o f the s y s t e m o n the b u b b l e c a n be c o n s i d e r e d as p r o m p t . T h i s f a c i l i t a t e s the c o m p u t a t i o n o f the re-s p o n re-s e , for n o a c c o u n t h a re-s t o b e g i v e n t o d e l a y e d (or " m e m o r y " ) effects.

T h e m e a s u r e d a n d c o m p u t e d r e s p o n s e s c a n b e c o n s i d e r e d as the " l o c a l " r e s p o n s e i n the l o c a l / g l o b a l c o n -cept. R e a c t i v i t y ( o r l o n g r a n g e ) " g l o b a l " effects are n e g l i g i b l e i n t h e s t r o n g l y s u b c r i t i c a l s y s t e m u n d e r c o n s i d e r a t i o n . T h e r e s p o n s e s c a n b e u n d e r s t o o d f r o m the v i e w -p o i n t o f a q u a l i t a t i v e m o d e l . R E F E R E N C E S A d i r J., C l a r k S , F r o e h l i c h R . and T o d t C . J . (1967) R e p o r t GA 7157, San Diego. A s h r a f A t t a M , F r y D . N . , M o t t J . E . a n d K i n g W . T . (1978) Nucl. S r i . Engng 66, 264.

Behringer K . , K o s a l y G . and K o s t i c L j . (1976) Report EIR-bericht 303, W u r e n l i n g e n .

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