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Nuclear Reactor

Dynamics and Stability

Henryk Anglart

i

Nuclear Reactor Dynamics and

Stability

2011 Henryk Anglart All rights reserved

3

Preface

his textbook is intended to be an introduction to nuclear reactor kinetics, dynamics and stability for students of energy engineering and applied sciences as well as for professionals working in the nuclear field. The basic aspects of transient behavior of nuclear reactors are presented with focus on how to solve practical problems.

The textbook is organized into four chapters and each chapter is divided into several sections. Parts in the book of special interest are designed with icons, as indicated in the table to the left. “Note Corner” contains additional information, not directly related to the topics covered by the book. All examples are marked with a pen icon. Special icons are also used to mark sections with computer programs and suggested more reading.

Chapter one contains introductory information, such as classification of dynamic systems and description of basic approaches to analyze such systems. Chapter two and three are devoted to nuclear reactor kinetics and dynamics using the point-reactor approximation. Chapters four is dealing with dynamics and stability of two-phase flows in heated channels.

T I C O N K E Y

Note Corner

Examples

Computer Program

More Reading

C H A P T E R 1 - N U C L E A R R E A C T O R K I N E T I C S

4

Table of Contents

Preface i

1 Introduction ................................................................................................ 7

1.1 Purpose of the Book ....................................................................................... 7

1.2 Schematics of Basic Nuclear Systems ........................................................... 8

1.3 Historical Review ......................................................................................... 11 1.3.1 Reactor Transient Tests ...................................................................................... 11 1.3.2 Reactor Instability ............................................................................................... 11

1.4 Future Development .................................................................................... 12

1.5 Rudiments and Fundamental Approaches ................................................ 13 1.5.1 Types of Dynamical System ............................................................................... 13 1.5.2 Basic Terms Used in Dynamics .......................................................................... 14 1.5.3 Linear Stability Analysis .................................................................................... 16 1.5.4 Bifurcations......................................................................................................... 18 1.5.5 Time Domain Approach ..................................................................................... 19 1.5.6 Laplace Transform Approach ............................................................................. 24 1.5.7 Frequency Domain Approach ............................................................................. 26

2 Nuclear Reactor Kinetics ......................................................................... 33

2.1 Reactor Kinetics Models.............................................................................. 33 2.1.1 Delayed Neutrons ............................................................................................... 33 2.1.2 Derivation of Point Kinetics Equations .............................................................. 34 2.1.3 Equations for Six-Group Point Kinetics Model .................................................. 39 2.1.4 Equations for One-Group Point Kinetics Model ................................................. 40 2.1.5 Average Neutron Generation Time and Lifetime ............................................... 41

2.2 Normalized Point Kinetics Equations ........................................................ 41 2.2.1 Equilibrium Point of a Nuclear Reactor .............................................................. 42 2.2.2 Normalized Equations of Point Kinetics Model ................................................. 43

2.3 Solutions with Constant Reactivity ............................................................ 45 2.3.1 Solutions of Six-Group Point Kinetics Equations ............................................... 45 2.3.2 Solutions of One-Group Point Kinetics Equations ............................................. 48

2.4 Point Kinetics Model with Time-Dependent Reactivity ........................... 53 2.4.1 Small-Perturbation Approximation ..................................................................... 53 2.4.2 Numerical Solutions of Point Kinetics Equations ............................................... 55

2.5 Approximate Point Kinetics Models .......................................................... 59 2.5.1 The Prompt Jump Approximation ...................................................................... 59 2.5.2 The Prompt Kinetics Approximation .................................................................. 61 2.5.3 The Constant Delayed Neutron Source Approximation ..................................... 61


5

3 Nuclear Reactor Dynamics ...................................................................... 67

3.1 Reactivity Feedbacks .................................................................................. 67 3.1.1 Influence of Fuel and Moderator Temperature on Reactor Operations ...............67 3.1.2 Doppler Effect .....................................................................................................69 3.1.3 Reactivity and Reactivity Coefficients ................................................................70 3.1.4 Fuel Coefficient of Reactivity..............................................................................72 3.1.5 Moderator Coefficient of Reactivity ....................................................................72

3.2 Point Dynamics Model of PWR ................................................................. 73 3.2.1 Derivation of Point Dynamics Equations ............................................................73 3.2.2 Nuclear Reactor at Equilibrium ...........................................................................75 3.2.3 Normalized Point Reactor Dynamics Model .......................................................76 3.2.4 Reactor Dynamics in Presence of Small Perturbations ........................................79 3.2.5 Frequency Domain Analysis of Point Dynamics Model .....................................81

3.3 Point Dynamics Model of BWR ................................................................. 84 3.3.1 Formulation of BWR Dynamics Model ...............................................................84 3.3.2 Nuclear Reactor at Equilibrium ...........................................................................85

3.4 Nonlinear Effects ......................................................................................... 89

3.5 Nuclear Reactor Stability ........................................................................... 91 3.5.1 Open Loop Transfer Functions ............................................................................92 3.5.2 Closed Loop Transfer Functions..........................................................................92 3.5.3 Stability of Zero-Power Reactor ..........................................................................93 3.5.4 Stability of Closed Loop Systems ........................................................................94

4 Dynamics of Boiling Systems .................................................................. 99

4.1 Analysis of Two-Phase Flow Transients.................................................... 99

4.2 Flow Instabilities in Heated Channels ..................................................... 109 4.2.1 Classification of Instabilities .............................................................................109 4.2.2 Density Wave Oscillations ................................................................................111 4.2.3 Methods of Analysis of Two-Phase Flow Instabilities ......................................113 4.2.4 Frequency Domain Methodology ......................................................................113 4.2.5 Time Domain Approach ....................................................................................117

4.3 Instabilities in Heated Loops .................................................................... 119 4.3.1 Parallel Channel Instability ................................................................................119 4.3.2 Multiple Parallel Channels ................................................................................121 4.3.3 Boiling Loop Stability .......................................................................................121 4.3.4 Heated Wall Dynamics ......................................................................................121

APPENDIX A – LAPLACE TRANSFORMATION 125

APPENDIX B – NYQUIST STABILITY CRITERION 131

APPENDIX C – SELECTED STEAM-WATER DATA 135

APPENDIX D – BOILING CHANNEL STABILITY MODEL 137

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1 Introduction

he dynamics and stability of engineering equipment influence their economical, safe and reliable operation and for that reasons they are major areas of concern. Dynamics of many systems is their inherent property, and the systems are just designed to behave like that. There are however systems,

which are designed for stationary operations, but still, they will behave in a transient, dynamical manner. To such systems belong nuclear reactors, which preferably should operate at steady-state conditions during their whole lifetime. However, nuclear reactors are subject to various disturbances, which cause their transient behavior. For that reason it is necessary to investigate and understand their dynamical behavior.

Stability is another important property of engineering systems. Stability can be expressed using the everyday experience, which suggests that stable systems are inherently able to return to the initial stationary condition when subject to any external perturbation. On the contrary, unstable systems will transit to some other, completely new state, which significantly deviates from the initial one. Obviously, for technical equipment which is typically designed and optimized to operate at given desirable conditions, this type of behavior should be avoided. This is in particular true for nuclear reactors. If their operational conditions significantly deviate from the nominal ones, and the conditions undergo periodic oscillations, some important safety limits can be compromised. Eventually, such unsteady, unstable behavior – if not properly mitigated – could lead to reactor failures.

1.1 Purpose of the Book This book is intended to give an introduction to analysis and modeling of the transient behavior of nuclear reactors. Any transient behavior can be characterized by some measure of the time scale, which usually is a period of time during which a given transient parameter significantly changes (for example it changes e-folds), or undergoes a repeatable oscillation. In this book the focus will be on such transients, which have time scales much smaller than the reactor fuel cycle, however, much longer than the time scales of the reactor noise. It means that changes of reactor operational conditions due to fuel depletion or even due to poisoning will not be included in the present book. Nor transients resulting from the turbulent fluctuations of inlet velocity and temperature of coolant will be included.

The focus of this book is to elucidate the phenomena that govern the transient behavior of nuclear reactors which are subject to external perturbations. The reactor power response is investigated following sudden changes of such parameters as coolant temperature and/or mass flow rate at the inlet to the reactor core, or sudden

Chapter

1

T

C H A P T E R 1 - I N T R O D U C T I O N

8

change of reactivity due to movement of control rods. For that purpose simplified, practical computational models are developed.

To make the development of computational models as simple as possible, the Scilab (www.scilab.org) computational environment is used. The Scilab program is freely available and can be installed on various platforms. The package contains basic tools required for model development such as: time domain solvers, available through command ode(), frequency domain and transfer function tools applicable for linear systems defined by command syslin(), and integrated plotting functions. The programs developed in this book are freely available and can be downloaded from www.reactor.sci.kth.se/downloads .

1.2 Schematics of Basic Nuclear Systems Several different nuclear power systems have been developed in the past decades and their detailed description can be found elsewhere[1-1]. For the purpose of this book only two most wide-spread systems will be described and referenced: nuclear power plants with Pressurized Water Reactors (PWR) and with Boiling Water Reactors (BWR).

A schematic of a PWR plant is shown in FIGURE 1-1. It contains two circulation loops: the primary loop that carries coolant through the reactor core and the steam generators, which are usually operating in a once-through manner. The primary system is kept under high pressure (typically 15.5 MPa) to avoid boiling of coolant in the reactor core. The secondary loop is used to produce steam in steam generators. The steam flows through steam lines to turbines, where it expands and performes the shaft work. After condensation in the condenser, it returns as feedwater to steam generators.

FIGURE 1-1. Schematic of a nuclear power plant with Pressrized Water Reactor.

A schematic of BWR plant is shown in FIGURE 1-2. This type of plant contains the recirculation loop which is forcing coolant through the reactor core, and the main loop, which is circulating steam from the steam dome in the reactor pressure vessel to the turbines. After expansion in the turbine and performing the shaft work, steam is condensed in the condenser and then returnes as feedwater back to the reactor pressure vessel. Since steam is generated in the reactor pressure vessel, there is no need for steam generators. Thanks to this feature the plant design is significantly simplified.


9

However, the presence of the vapor phase in the reactor core influences the dynamic behavior of the system.

FIGURE 1-2. Schematic of a nuclear power plant with Boiling Water Reactor.

To be operated, nuclear power plants require sophisticated control systems. A simplified schematics of such systems used in PWRs and BWRs are shown in FIGURE 1-4 and FIGURE 1-4, respectively. Typically the system contains several dedicated controllers, such as the reactor power controller, the turbine controller, the reactor pressure controller and the water level controller. Each of the controllers collects plant signals as schematically shown in the figures.

FIGURE 1-3. Schematic of PWR control system. Sygnals: 1 – neutron flux, 2 – pressure in steam line, 3 – control rod position, 4 – steam pressure in pressurizer, 5 – valve position, 6 – heating power, 7 – water level in pressurizer, 8 – hot leg temperature, 9 – cold leg temperature, 10 – valve position, 11 – base signal, 12 – correction signal, 13 – feedwater inlet mass flux, 14 – water level in steam generator, 15 – steam mass flow rate, 16 – base signal, 17 – valve position.

Reactor power controller

Pressurizer pressure controller

Pressurizer level controller

Steam-generator level controller

Reactor

Pressurizer

Steam generator

Preheater

Steam

Feedwater

Hot leg

Cold leg

2

1 3

4

5

6

7

8

9 10

13 14 15 16

17

11 12

1-17 Signals


10

FIGURE 1-4. Schematic of BWR control system.

The operation point of a nuclear reactor is often described in terms of the reactor power and the core flow. A typical BWR power/core-flow map is shown in FIGURE 1-5. The map shows paths that are followed during the reactor start-up and shutdown. It also contains exclusion areas which should be avoided during the reactor normal operation. One such area indicated in the figure is the stability exclusion area. It is characterized by relatively high reactor power and low core flow rate. In this region the core is susceptible to instable behavior and thus the region should be avoided.

FIGURE 1-5. BWR power/core-flow map.


11

1.3 Historical Review 1.3.1 Reactor Transient Tests

Prediction of transients in Nuclear Power Plants (NPP) is important from the safety point of view. Such predictions give important information about plant behavior during transients, and in particular, they can reveal potential risks for failures that could lead to core damages.

Various transient tests have been performed in NPPs to learn about their behavior. The purpose of such tests is twofold: (a) they give a direct insight into plant performance, since major parameters of interests are measured and recorded; (b) they provide a valuable database for validation of computational tools.

In April 1977 such transient test was performed in Peach Bottom 2 power plant with General-Electric-designed BWR/4 reactor. Three turbine trip tests were performed at different power levels. The tests were concerned with transients following a sudden closure of the turbine stop valve. As a result of the valve closure, pressure waves were generated in the steam lines and propagated with relatively little attenuation into the reactor core. The induced core pressure oscillations resulted in significant changes of void fraction in the core, which, in turn, caused oscillations of reactor power. The data obtained in the Peach Bottom-2 tests were subsequently used in an International Benchmark program, where several computational codes were used by various organizations to predict the measurements[1-5].

A corresponding test for PWR was performed at Three-Mile-Island-1, with B&W-designed reactor. The analyzed transient was a main steam line break, which may occur as a consequence of the rupture of one steam line upstream of the main steam isolation valves. This event is characterized by significant space-time effects in the core caused by asymmetric cooling and an assumed stuck-out control rod after the reactor trip. The measured data were used for International Benchmark of various computational codes[1-6].

1.3.2 Reactor Instability

Interest into nuclear reactor instability started growing after first recorded incidents with Boiling Water Reactors (BWR). In 1978 the TVO-I reactor, which is ASEA-ATOM’s constructed BWR, underwent self-sustained oscillations leading to the reactor scram[1-3]. The oscillations had so-called out-of-phase mode, in which the mean reactor power remained approximately constant, whereas one half radially increased the power, and the other one decreased. Similar type of oscillations was observed in the Caorso plant in 1984 and it was the first event of this type that was widely described in the literature. However, only after the LaSalle event of March 1988[1-4] the BWR dynamics and stability issues attracted the attention of authorities and wider public. Shortly after this event (in 1990), an international workshop on BWR stability held in Long Island, USA, gathered more than 100 participants, to share their experience with modeling and analyzing the BWR dynamics and instabilities.

Until now several dozen instability events occurred worldwide: most of them taking place in the United States, Europe and Japan. It has been observed that BWRs are susceptible to the following types of instabilities:


12

• Control system instability caused by out-of-tune controllers. This is a malfunction of reactor control hardware and can be relatively easy removed by adjusting the controller gains.

• Channel thermal-hydraulic instability, in which a boiling channel can oscillate even without neutronic feedback, if the local pressure drops become out-of-phase with the inlet flow perturbations due to the density-wave effects.

• Coupled neutronic-thermal-hydraulic instability, also called reactivity instability, in which the density wave with void fraction variation is affecting the reactivity and power feedback.

While the first and the second type of instability can be easily removed or occurs only locally in blocked channels, the reactivity instabilities are major concern in BWRs.

Several dedicated stability test have been performed in nuclear power plants. In the period from 1989 to 1997 such measurements were performed in Forsmark 1 and 2 reactors. The measured data were subject to an International Benchmark program which started in 1999[1-7].

On October 26 1989 instability incident occurred in Ringhals-1 reactor, ASEA-ATOM-designed BWR. The incident occurred during reactor start-up, following the planned summer outage. At power 75% and 3720 kg/s core flow the power started to oscillate in a limit cycle at 0.5 Hz frequency. The amplitude of power oscillations reached 16%. After few minutes the oscillations were stopped by partial scram initiated by the operator. During measurements performed 6 hours prior to the incident, the local power range monitor signals were recorded as being out of phase. The largest noted phase difference was 130 degrees between the two halves of the core. The measurements revealed a local decay ratio of above 0.9. Following the incident, series of stability measurements were performed at Ringhals-1 reactor. In total 41 state points were measured during 4 consecutive fuel cycles (from 14th to 17th). The measured data were subsequently used to conduct an International Benchmark program[1-8].

1.4 Future Development The stability and dynamics of current nuclear reactors is quite well understood and widely analyzed, providing basis for safe and economic operation of existing nuclear power plants. However, the prediction tools require further development to assure high accuracy and reliability. In particular, the coupling between neutronics and thermal-hydraulics requires full three-dimensional treatment. Also, thermal-hydraulic modules will need to resolve temperature and density distributions on scales which are smaller then a single nuclear fuel pin.

The future reactor design development will focuse on new systems that are currently under consideration, namely Generation III+ and IV reactors. The characteristic feature of the former is the increasing presence of passive systems. Such systems, while desirable from the safety point of view, may introduce undesirable, difficult to model, instability features. The latter (re)introduce new type of coolants, such as supercritical water, liquid metals (such as lead-bismuth and sodium) and various gases (e.g. Helium). Some of the coolants (in particular supercritical water) undergo significant property


13

variation, similar to those during evaporation, which may lead to unstable behavior of the reactor cores. To analyze dynamics and stability of such system, a new generation of simulation tools will be required, combining Computational Fluid Dynamics (CFD) codes with fully three-dimensional treatment of neutronics.

1.5 Rudiments and Fundamental Approaches The kinetics, dynamics and instabilities of nuclear reactors are investigated using three fundamental approaches: the time domain approach, the Laplace transform approach and the frequency domain approach. Each of these approaches serves different purposes, as described in the following sections. The choice of the approach depends to a large extend on the type of the dynamical system under consideration. Even though a thorough classification is not available, some basic features of dynamical systems can be identified, as shortly described below.

1.5.1 Types of Dynamical System

The dynamical systems can be quite loosely classified into several types:

Type 0: Static Systems

This type is given only for reference. As the name suggests, such systems are stationary and do not change with time. Even though there are no systems in the nature that are strictly static, this assumption is very often made to make the analysis simple.

Type 1: Solvable Deterministic Systems

Such systems are characterized by existence of solutions in analytical, closed forms. The behavior of such systems can be strictly predicted at any time instant for given initial conditions. Such systems typically result from linearization and simplifications of certain existing real systems (such as a pendulum), serving as their mathematical models.

Type 2: Non-solvable Deterministic Systems

Systems of Type 2 are usually described by non-linear differential equations that have no analytical solutions in closed forms. The solution can be obtained by the numerical integration only. The response of such systems can be predicted with required accuracy, provided that a proper numerical approach is applied. An example of such system is the van der Pol oscillator.

Type 3: Deterministic Chaotic Systems

Such systems are characterized by solutions that are very sensitive to the initial conditions and which undergo sudden qualitative changes. Such systems are difficult to model and the results of predictions are usually evaluated in terms of mean values and variances. Many systems in the nature belong to this category, including the solar system. Complex technical systems, such as nuclear reactors can exhibit chaotic behavior as well. The best know example of the chaotic system is the Lorenz model of natural convection in the atmosphere.

The Lorenz equations have the following form,


14

(1.1)

xyzdt

dz

xzyxdt

dy

yxdt

dx

+−=

−−=

+−=

3

8

28

1010

Solution of the equations is shown in the xyz coordinates in FIGURE 1-6.

FIGURE 1-6. Solution of Lorenz equations in state coordinates xyz.

Type 4: Stochastic Systems

Stochastic systems have solutions that can be evaluated in terms of mean values and variances only. Typically the equations describing the stochastic systems contain coeeficients that can get random values.

1.5.2 Basic Terms Used in Dynamics

There are several terms that are used to describe the dynamical systems (and in particular nonlinear dynamical systems) that are introduced and explained in this section. The first important term is the dynamical system itself: in general it is a self-contained entity that exhibits some temporal behavior. Such system described with equations will constitute a model of the dynamical system.

State of a dynamical system is a variable or a list of variables (that is a vector) which are needed to uniquely determine the system current state. For the Lorenz system given by Eq. (1.1) the state is described by three variables: x, y and z. All possible states of a system are called a phase space. That is for the Lorenz system the phase space will be a three-dimensional space xyz.

Dynamics or equations of motion for a dynamical system describe a relation between the present and the future state of the system. For continuous dynamical systems, equations of motion are differential equations which can be either linear or nonlinear in nature, determining a linear or a nonlinear dynamical system, respectively.

Z

X

Y


15

Not always the future state of the dynamical system is uniquely determined by the present state. There are two important examples of such a situation, which apply to nuclear reactors:

• when the equations of motion explicitely depend on time (such systems are called nonautonomuous systems). This can occur in a nuclear reactor with insertion of a time-dependent reactivity,

• when the system is stochastic.

A solution of the the equations of motion starting from a given state point (that state point determines the initial conditions) is called an orbit or a trajectory. For continuous systems described with differential equations it is a curve in the phase space.

In some cases the orbits have special properties: if they start from a given point in the space phase, they come arbitrary close and arbitrary often to this or another point in the space phase. Sets of such points are called the non-wandering sets. There are four types of such sets:

1. Fixed points, which correspond to stationary solutions of the equations of motion.

2. Limit cycles, which correspond to periodic solutions.

3. Quasiperiodic orbits, which correspond to periodic solutions with at least two incommensurable frequencies (i.e., the ratio of the frequencies is an irrational number).

4. Chaotic orbits, which correspond to bound non-periodic solutions.

The first three types of orbits may occur in linear systems, whereas the fourth appears only in nonlinear systems.

A non-wandering set may be either stable or unstable. Changing a certain parameter (this is a so-called controlling parameter) of the system can change the stability of the non-wandering set. This is accompanied by the change of the number of non-wandering sets due to bifurcation. Change of stability and bifurcation always coincide.

There are two types of stability: a weaker (Lyapunov) and a stronger (asymptotic) one. The Lyapunov stability (also called marginal stability) occurs when every orbit starting in a neighborhood of a non-wandering set remains in its neighborhood. The asymptotic stability possesses all properties of the Lyapunov stability and in addition all orbits in the neighborhood approach the non-wandering set asymptotically. The stability of non-wandering sets is discussed in more detail in Section 1.5.3.

Non-wandering sets with asymptotic stability are called attractors. The basin of attraction is the set of all initial states approaching the attractor in the long time limit.


16

1.5.3 Linear Stability Analysis

Very often the system under consideration is described with a set of non-linear differential equations. To enable the analysis of the system with some of the well established approaches (such as the Laplace transformation approach), the equations have to be linearized.

Consider a system of non-linear ordinary differential equations,

(1.2) );( uxFx

=dt

d, 0)0( =x ,

where x is the vector of unknown functions, u is the vector of forcing functions, and F represents the right-hand-sides in the system of equations. The fixed point of the system is represented by vector x0, which satisfies the equation,

(1.3) 0);( 00 =uxF .

The forcing functions u0 are either equal to zero or to given constant values. The behavior of a dynamical system around any point (and in particular around the non-wandering sets) can be investigated using the linear stability analysis, also known as the first method of Lyapunov. For that purpose the function F is Taylor-expanded around the point x0 as follows,

(1.4) ( ) ( ) ( )2

00

0

xxx

FxFxxF

x

δδδ O+∂

∂+=+ .

Here

(1.5)

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=∂

∂

N

NNN

N

N

x

F

x

F

x

F

x

F

x

F

x

F

x

F

x

F

x

F

21

2

2

2

1

2

1

2

1

1

1

x

F

is the Jacobian matrix.

Taking x which is close to x0, that is x = x0 + δx and using Eq. (1.4) yields,

( )x

x

FuxF

xxxx

x

δδδ

0

);( 00

0

∂

∂+≅=

+=

dt

d

dt

d

dt

d

Thus, the system behavior around the non-wandering set, and in particular, the stability of the non-wandering set can be evaluated from the linearized differential equation,


17

(1.6) xAx

δδ

=dt

d, with ( ) 00 =xδ ,

where

(1.7)

0xx

FA

∂

∂= .

Assuming that the solutions of Eq. (1.6) have a form stet ex =)(δ , the equation yields,

(1.8) eeA s=⋅ .

This equation has a non-trivial solution for such values of s, which satisfy the following characteristic polynomial,

(1.9) 0=− IA s .

Here I is the unit matrix with dimensions N×N. Assuming that Eq. (1.9) has N single-valued roots, for fixed points, the solution of Eq. (1.6) has the following general form,

(1.10) ( ) ∑=

=N

k

ts

kkkeAt

1

exδ ,

where Ak are constant coefficients, ek are eigenvectors and sk are eigenvalues of the Jacobian matrix evaluated at the fixed point. If the coefficients of the Jacobian matrix are real, eigenvalues are real or complex conjugate numbers.

The solution given by Eq. (1.10) is valid only when xδ is small, it can be thus termed as the small-perturbation approximation. This is a very general method and usually gives some interesting conclusions concerning the behavior of dynamical systems. In particular, when real parts of all eigenvalues of the Jacobian matrix are negative, the fixed point is asymptoticallystable, as can be deduced from Eq. (1.10). Once analyzing the stability of any system, it is thus important to determine its fixed points (or in general, non-wandering sets) and the real parts of the eigenvalues of the Jacobian matrix.

Analytical calculation of the roots of the characteristic function is tedious for polynomials with the order greater than 2 and impossible for polynomials with the order grater than 5. In such cases the Routh and Hurwitz theorem can be used to evaluate the stability. This theorem states that the real parts of all roots of a polynomial are negative if and only if:

(1.11) 0...,0,0 21 >∆>∆>∆ N ,

where,


18

(1.12)

kkkkkkk

k

bbbbbbb

bbbbb

bbb

b

L

MOMMMMMM

L

L

L

625242322212

12345

123

1

01

01

000001

−−−−−−

=∆ ,

and the characteristic equation is as follows,

(1.13) 01

1

1 =++++=− −−

NN

NNbsbsbss LIA .

1.5.4 Bifurcations

A bifurcation manifests itself with a change of the number of attractors in a nonlinear dynamical system. In a bifurcation point, at least one eigenvalue of the Jacobian gets a zero real part. There are three generic types of so-called co-dimension-one bifurcations (here the term co-dimension refers to the number of control parameters for which fine tuning is necessary to get such a bifurcation). Two of them, which are relevant for continuous dynamical systems, are shortly described below.

Stationary bifurcation occurs, when a single real eigenvalue crosses the boundary of stability (that is the imaginary axis). This case is depicted in FIGURE 1-7. In Hopf bifurcation, a conjugated complex pair croses the boundary of stability, as shown in FIGURE 1-8.

FIGURE 1-7. Occurrence of the stationary bifurcation in a continuous dynamical system when the real eigenvalue crosses the stability boundary (shown with an arrow).

FIGURE 1-8. Occurrence of the Hopf bifurcation in a continuous dynamical system when the conjugated complex pair eigenvalues cross the stability boundary (shown with two arrows).

ω=Im(s)

λ=Re(s)

Stability boundary

ω=Im(s)

λ=Re(s)

Stability boundary


19

Crossing the boundary of stability indicates the bifurcation point, but it does not indicate how many solutions bifurcate or disappear at that point. Further analysis is necessary to determine the character of solutions after bifurcation points. This analysis is the subject of the bifurcation theory and is not covered in this book.

EXAMPLE 1-1. Consider a dynamical system that is described by the following set of two differential equations:

yyxgdt

dy

xbayxfdt

dx

−==

⋅+⋅==

);,(

);,( 2

α

αα

Here x and y are the state variables and α is the control parameter. There are two fixed points as follows:

( ) ( )( ) ( )0,

0,

2

1

bayx

bayx

α

α

−−=

−= , or assuming a = b = 1, ( ) ( )( ) ( )0,

0,

2

1

α

α

−−=

−=

yx

yx

The Jacobian at both fixed points are as follows:

( )

−

−=

−=

10

02

10

02,

1

1

αxyxJ

, ( )

−

−−=

−=

10

02

10

02,

2

2

αxyxJ

The characteristic polynomials for the two fixed points are:

( )( ) ( ) αααα

−−−−+=+−−−=−−

−−22112

10

02 2 sssss

s and

( )( ) ( ) αααα

−+−++=++−=−−

−−−22112

10

02 2 sssss

s

If α = -1, the first fixed point has the characteristic polynomial s2 – s – 2 = 0 with roots s1 = -1 and s2 = 2. That indicates an existence of a saddle point. At the same time the second fixed point has the characteristic polynomial s2 + 3s + 2 = 0 with roots s1 = -2 and s2 = -1, which indicates an existence of an attractor at that point. Similar analysis indicates that for α = 0 there is a double fixed point at the origin with eigenvalues s1 = 0 and s2 = -1, and for α > 0 there are no real fixed points.

MORE READING: An excellent introduction to the bifurcation theory can be found in an article by John David Crawford: “Introduction to bifurcation theory, “ Review of Modern Physics, Vol. 63, No. 4, October 1991. The article can be found at http://prola.aps.org/thumbnail/RMP/v63/i4/p991_1?start=0. Another good site to visit can be found at http://www.egwald.ca/nonlineardynamics/bifurcations.php

1.5.5 Time Domain Approach

The time domain approach is based on the time integration of the model differential equations. Taking a simple case of the first-order system described by the following differential equation,

(1.14) )()( tuxtgdt

dx=⋅+ ,


20

with the initial condition,

(1.15) ( ) 00 xx = ,

the solution can be obtained in a closed form after integration as follows,

(1.16) ( ) ( ) ( ) ( )( )

∫ ∫+= ∫−∫− t dttgdttgdttgdtetueextx

tt

0

''''''''

0 ''00 .

In particular, taking constant function g(t) = ω0=const and the step-function on the right-hand side: u(t) = 0 for t<0 and u(t) = u0 for t≥0, yields:

(1.17) ( ) ( ) ttte

ux

uexe

utx 000

0

00

0

00

0

0 1ωωω

ωωω−−−

−+=+−= ,

If the forcing function has the form u(t) = A sin(ωt), the solution is as follows,

(1.18) ( ) ( )ttA

exA

txt ωωωω

ωωωω

ω ωcossin022

0

022

0

0 −+

+

+

+= −

.

The first term on the right-hand-side of Eq. (1.18) decays with time, whereas the second term represents the persistent response of the system to the sinusoidal forcing function. As can be seen, the response signal will have a changed amplitude and phase as compared to the forcing function, but will have the same frequency of oscillations. The amplification factor and the phase shift can be obtained by transforming the term as follows,

( )

( ) ( ) ( )

( ) ( )θωθωω

ω

ωθωθθωω

ωωθω

ωω

ω

ωω

ωω

ωω

ωωωωω

ωω

++

=++

=++

=

−

+=−

+

tA

ttA

ttA

ttA

ttA

sincos

cossinsincoscos

costansin

cossincossin

22

0

0

22

0

0

22

0

0

0

22

0

0022

0

Here,

(1.19)

−=

0

arctanω

ωθ

is the phase shift, and

(1.20) ( ) ( ) ( )

22

0

2

0

2

22

0

02

22

0

0

22

0

0 1tan1cos

ωω

ω

ω

ωω

ωθ

ωω

ω

θωω

ω

+

=++

=++

=+

A

AAA


21

is the amplitude of the response signal. Thus, the system response can be written as,

(1.21) ( ) ( )θωωωωω

ω ω ++

+

+

+= −

tA

exA

txt

sin22

0

022

0

0 .

For constant and positive ω0, the first term on the right-hand-side of Eq. (1.21) decays with time, whereas the second term represents the persistent response of the system.

The system amplification is usually given in decibels, using the amplification at zero frequency as a reference,

(1.22) 0

lg20=

=ωatamplitudesignalresponse

amplitudesignalresponseL .

Using the above definition, the amplification of the first-order system is,

(1.23) 21

1lg20

β+=L ,

where

(1.24)

0ω

ωβ = .

FIGURE 1-9 shows the phase shift and amplification characteristics of the first order system.

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0.01 0.1 1 10 100

beta

theta

[d

eg

]

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

0.01 0.1 1 10 100

beta

L [

dB

]

(a) (b)

FIGURE 1-9. Characteristics of the first order system: (a) phase shift, (b) amplification.

Similarly, for the second-order system describing the forced oscillations with viscous dissipation the governing equation is as follows,


22

(1.25) takxdt

dxc

dt

xdm ωsin

2

2

=++ .

This equation can be transformed to the following form,

(1.26) tAxdt

dx

dt

xdωωζ sin2 2

02

2

=++ ,

where,

(1.27) m

c

2=ζ ,

(1.28) m

k=0ω .

The general solution of the equation in case of small damping ( 0ωζ < ) and initial

conditions ( ) 00 xx = and ( ) 00 xx && = is as follows,

(1.29)

( )

( )( )

( )( )θω

ωζωω

ωθωθζω

ωθωζωω

ωω

ζω

ζ

ζ

−+−

+

++

+−

−

++=

−

−

tA

ttAe

txx

txetx

t

t

sin

4

sincossin1

cossin

4

sincos

22222

0

022222

0

000

&

The first two terms result from the perturbation introduced by the initial conditions and they will decay with time, whereas the last term represents the system response when time t goes to infinity. It can be seen that the system response signal will have the same frequency as the forcing function, but the amplitude will be amplified with factor,

(1.30)

( ) 22222

0 4

1

ωζωω +−,

and the phase will be shifted with

(1.31)

−

−=

22

0

2arctan

ωω

ζωθ .

Using the definition of the system amplification given by Eq. (1.22) and the frequency ratio defined in Eq. (1.24), the amplification and the phase shift are given as,


23

(1.32) ( ) 2221

1lg20

ββ dL

+−= ,

(1.33)

−

−=

21arctan

β

βθ

d,

where

(1.34)

0

2

ω

ζ=d ,

is a dimensionless damping factor. The amplitude and phase shift characteristics are shown in FIGURE 1-10.

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0.01 0.1 1 10 100

beta

the

ta [

de

g]

d=0.1

d=1

d=10

-100

-80

-60

-40

-20

0

20

40

0.01 0.1 1 10 100

beta

L [

dB

]

d=0.1

d=1

d=5

(a) (b)

FIGURE 1-10. Characteristics of the second order system: (a) phase shift, (b) amplification.

An important parameter used in the evaluation of BWR stability is the decay ratio, which is defined as the ratio of two consecutive amplitudes in a given signal, as shown in FIGURE 1-11. The decay ratio can be calculated from the analytical solution as a ratio of the system response at time t0 + T to the value at t0, where T is the period of oscillations.

(1.35) T

t

Tt

ee

eDR

λλ

λ−

−

+−

==0

0 )(

.

The period of oscillations is obtained as

(1.36) ω

π2=T ,

thus, the decay ratio is as follows,


24

(1.37) ω

πλ2−

= eDR .

FIGURE 1-11. Definition of the decay ratio.

1.5.6 Laplace Transform Approach

Linear systems described with a set of linear ordinary differential equations with constant coefficients can be investigated using the Laplace transform approach. The definition of the Laplace transformation and a table with selected functions and their images are given in APPENDIX A.

One of the most useful properties of the Laplace transformation is that it can be used to derive the transfer function for dynamic systems. If any system is subject to a forcing function u(t) and the response of the system is described with a function x(t), then the system transfer function G(s) is defined as,

(1.38) ( ) ( )( )su

sxsG

ˆ

ˆ= ,

where ( )sx and ( )su are images (Laplace transformations) of functions x(t) and u(t),

respectively. Fundamental properties of system transfer functions are given in APPENDIX A.

Transfer functions are very useful, since they a characterizing the system which they are derived for. In particular, transfer functions can be used to investigate the signal amplification, phase shift and system stability, once moving from the Laplace domain to the frequency domain, as described in the next section.

In some cases transfer functions can be used directly for evaluation of the system stability. This is the case when the transfer function can be expressed as a polynomial quotient,

(1.39) ( ) ( )( )sD

sNsG = ,

A2

A1 DR=A2/ A1

t0 t0+T t


25

of two polynomials N(s) and D(s). The roots of the nominator polynomial N(s) are called the roots of the transfer function, whereas the roots of the denominator polynomial D(s) are called the poles of the transfer function. Thus, the transfer function can be written as,

(1.40) ( ) ( )( ) ( )( )( ) ( )N

M

pspsps

zszszsCsG

−−−

−−−=

...

...

21

21 .

Transfer function G(s) describes a stable systems when all poles p1, p2, …, pN have negative real parts.

If all coefficients in polynomials that determine a transfer function are real, then the poles of the transfer function will be either real or conjugate complex numbers. Their possible locations for a stable system, using the s-plane, are shown in FIGURE 1-12.

FIGURE 1-12. Locations of poles of transfer function on s-plane.

The location of poles on the s-plane suggests that the system response will be described by the following function,

( )θωλλ ++= tAeeAtx ttsin)( 3

1 .

Since λ3<λ, the first term will decay with time faster than the second term. Thus, the second term will dominate the system response. If both λ and ω are known, the decay ratio for the system response signal can be found as,

( ) ( )[ ][ ]

ωπλλλ

λ

θω

θω /2

0

0

sin

sin0

0

eetAe

TtAeDR

T

t

Tt

==+

++≅

+

.

The method of Laplace transformation can be successfully applied to determine dynamic system stability providing that all poles of the transfer function can be determined. However, in many cases finding poles of the transfer function can be very difficult or even impossible. This can occur when the transfer function is no longer expressed in terms of a rational polynomial but instead it contains complex transcendental functions. In such cases the properties of the dynamic system can be investigated in the frequency domain, as described in the next section.

Re(s)

Im(s)

λ

ω

-ω

s1

s2

s3

λ3


26

1.5.7 Frequency Domain Approach

Similarly as the Laplace transformation approach, the frequency-domain method can be used for linear systems only. If the system under consideration is non-linear, it should be first linearized around a certain operational point.

A frequency response of any system is the system behavior subject to a sinusoidal forcing function. Such forcing function can be expressed as,

(1.41) ( ) tutu ωsin0= ,

where u0 is the amplitude of the input signal (forcing function) and ω is the signal frequency expressed in radians per second.

When a linear system is subject to a sinusoidal signal at input, the system response will also be a signal of the same shape and frequency. However, the output signal will have different amplitude and phase. Thus, the output signal can be written as,

(1.42) ( ) ( )θω += txtx sin0 .

Examples of input (forcing) and output signals for a linear system are shown in FIGURE 1-13.

Output signal Input signal

FIGURE 1-13. A linear system response to a sinusoidal forcing function at steady-state. The system introduces a lag to the signal and reduces its amplitude.

The signal amplification factor has been previously defined as,

(1.43)

0

0

u

xM = ,

and the phase shift is defined as,

(1.44) πθ 2T

Tx= , radians.

Tx

u0

x0

T


27

It can be shown that the frequency response of a linear system can be obtained by substituting s with jω in the transfer function, and that:

• the amplification factor is given as ( )ωjGM = ,

• the phase shift is given as ( )ωθ jGarg= .

In addition, it can be shown that the amplification at steady-state is obtained as,

(1.45) ( ) ( )0lim0

0 GjGM ==→

ωω

.

The system amplification can be expressed in decibels, as defined in Eq. (1.22). In terms of the transfer function G(s), the amplification is as follows,

(1.46) ( )( )0

lg20G

jGL

ω= , decibels (dB).

The amplification factor and the phase shift can be plotted as a function of frequency ω. These plots are so-called frequency characteristics or Bode characteristics.

The frequency approach can give an answer whether the system is stable or not without specifically finding the poles of the transfer function. For that purpose the Nyquist plot is used, in which the imaginary part of G(jω) is plotted against the real part of G(jω). The principles of the Nyquist plot and the proof of the Nyquist stability criterion is given in APPENDIX B.

For the first-order system described by the following differential equation,

(1.47) )(0 tuxdt

dx=⋅+ ω , 0)0( =x

the transfer function is obtained after Laplace transformation of the equation,

(1.48) )(ˆ)(ˆ)0()(ˆ0 susxxsxs =⋅++ ω ,

which gives,

(1.49)

0

1

)(ˆ

)(ˆ)(

ω+==

ssu

sxsG .

The real and the imaginary parts of G(jω) are readily obtained as,

( )( )( ) 22

0

22

0

0

00

0

0

1)(

ωω

ω

ωω

ω

ωωωω

ωω

ωωω

+−

+=

+−+

+−=

+= j

jj

j

jjG .

thus,


28

[ ]22

0

0)(Reωω

ωω

+=jG ,

[ ]22

0

)(Imωω

ωω

+−=jG .

The module and the argument of G(jω) are now found as,

(1.50) 22

0

2

22

0

2

22

0

022 1ImRe)(

ωωωω

ω

ωω

ωω

+=

++

+=+= GGjG

From Eq. (1.46) the system amplification is found now as,

(1.51) ( )( ) 2

0

0

1

1lg20

1

1

lg200

lg20β

ω

ωωω

+=

+==

j

G

jGL ,

where β = ω/ω0. Similarly, the phase shift is found as,

(1.52) ( ) ( )βωθ −=== arctanRe

Imarctanarg

G

GjG .

As can be seen, identical expressions have been obtained for L and θ as previously derived in the time domain analysis.

The Bode plots for the first order system have been already shown in FIGURE 1-9. The Nyquist plot can be obtained in an analytical form by expressing the imaginary part of G(jω) with its real part. This can be achieved as follows,

[ ] [ ]00

22

0

0

22

0

)(Re)(Imω

ωω

ω

ω

ωω

ω

ωω

ωω jGjG −=

+−=

+−=

Thus,

[ ][ ] 0

)(Re

)(Imω

ω

ωω

jG

jG−=

The imaginary part can be now expressed as,

0

22

0

2

0

0 1

Re

Im1

Re

Im

Re

Im

Re

Im

Imω

ωω

ω

−+

−−=

−+

−−=

G

G

G

G

G

G

G

G

G ,

or,


29

0

22 ReImRe

ω

GGG =+ .

This equation represents a half-circle on the G-plane, which can be readily seen from the following form of the equation,

2

0

2

2

0 2

1Im

2

1Re

=+

−

ωωGG .

The Nyquist plot for the first order system is shown in FIGURE 1-14.

FIGURE 1-14. Nyquist plot for the first-order system.

The Bode and Nyquist characteristics can be plotted with dedicated Scilab functions nyquist() and bode(), as shown in the Example below.

EXAMPLE 1-2. Perform the Nyquist and Bode plots for first-order system using Scilab functions. SOLUTION: Assume first order system with ω0 = 1, that is the transfer function is G(s) = 1/(s+1). To plot the Nyquist and Bode plots, the transfer function is first defined as: s = poly(0,’s’); G = 1/(s+1); h = syslin(‘c’,G); The plots are then generated as:

nyquist(h,0.001,100,’First order system’); bode(h,0.001,100,’First order system’); where frequency interval was chosen between 0.001 and 100 rad/sec. The obtained plots are shown in the figures below.

ImG

ReG ω=0 ω → ∞

1/2ω0

1/2ω0

1/ω0


30

First order system

0.001

0.025

0.052

0.088

0.155

0.439

0.849

3.943

100

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

Nyquist plot

Re(h(2i*pi*f))

Im(h

(2i*

pi*

f))

FIGURE 1-15. Nyquist plot of the first order system with ω0 = 1 obtained with the Scilab program.

-3

10

-2

10

-1

10

0

10

1

10

2

10

-60

-50

-40

-30

-20

-10

0

Magnitude

Hz

db

-3

10

-2

10

-1

10

0

10

1

10

2

10

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Phase

Hz

de

gre

es

First order system

FIGURE 1-16. Bode plot of the first order system with ω0 = 1 obtained with the Scilab program.

N O M E N C L A T U R E

A amplitude A matrix d dimensionless damping factor DR decay ratio e eigenvector F Jacobian matrix G transfer function I unit matrix

j imaginary unit, 1−=j

L system normalized amplification M amplification factor s Laplace transform parameter t time T period of oscillations


31

u input (forcing) signal x output (response) signal

Greek

β frequency ratio

θ phase shift ω frequency of the forcing function

0ω system eigen frequency

Subscript 0 steady-state conditions

R E F E R E N C E S

[1-1] Anglart, H., Applied Reactor Technology, Compendium, KTH, 2009.

[1-2] Gialdi, E. et al., Core stability in operating BWR: Operational experience, Proc. SMORN-IV, Dijon, France, Pergamon Press, Oxford, 1984.

[1-3] Valtonen, K., RAMONA-3B and TRAB assessment using oscillation data from TVO-I, Proc. Int. Workshop on BWR Stability, OECD-NEA CSNI Report 178, pp. 205-231, 1990.

[1-4] NRC Bulletin 88-07 Supplement 1, Power Oscillations in Boiling Water Reactors, 1988.

[1-5] NEA/NSC/DOC(2001)1, Boiling Water Reactor Turbine Trip (TT) Benchmark, Volume I: Final Specification, February 2001.

[1-6] NEA/NSC/DOC(99)8, Pressurized Water Reactor Main Steam Line Break (MSLB) Benchmark, April 1999.

[1-7] NEA/NSC/DOC(99)9, Forsmark 1 & 2 BWR Stability Benchmark, May 1999.

[1-8] NEA/NSC/DOC(96)22, Ringhals 1 Stability Benchmark – Final Report, 1996.

E X E R C I S E S

EXERCISE 1-1: A dynamical system is described by the following differential equation: )(2

0 tFxx =+ ω&& .

Find the system response if 0)( FconsttF == and at .0,0 === xxt &


0 tFxx =+ ω&& .

Find the system response if attF =)( and at .0,0 === xxt &


0 tFxx =+ ω&& .

Find the system response if ateFtF

−= 0)( and at .0,0 === xxt &

EXERCISE 1-4: Perform the Nyquist plot for the second order system 03 =++ xxx &&& .

EXERCISE 1-5: Represent the second order system 03 =++ xxx &&& as a set of two differential equations. Find the fixed point of the system and analyze its stability using the first method of Lyapunov.


32

33

2 Nuclear Reactor Kinetics

he power generated in a nuclear reactor depends on several parameters such as the mass of the fissile material, the macroscopic fission cross section and the level of the neutron flux. Nuclear reactor kinetics is dealing with transient neutron flux changes resulting from a departure from the critical state. Such

situations arise during startup and shut-down of a reactor, or due to accidental disturbances in the reactor steady-state operation. In this Chapter the point kinetics equations are derived and solved. Various methods of solutions are applied and compared.

2.1 Reactor Kinetics Models 2.1.1 Delayed Neutrons

The emission of delayed neutrons has significant consequences on the transient behavior of reactors. Even though the fraction of delayed neutrons is small, it may play a dominant role in the over-all reactor behavior.

Nearly all of the neutrons produced due to a fission process are emitted without a noticeable delay after the fission. These neutrons are termed as prompt neutrons. The prompt neutrons are emitted by the direct fission products, immediately after the fission process.

Nuclei of some of the fission products may beta-decay into daughter nuclei which then immediately emit a neutron. Out of roughly 500 possible fission products, there are about 40 that possess this special property and they are called precursors of delayed neutrons (in short: precursors). The process of emitting delayed neutrons is schematically shown in XFIGURE 2-1X.

It would be a tremendous task to track separately all 40 precursors in analyzing reactor kinetics phenomena. In fact, the yield fractions and decay constants of all precursors are not even known exactly. Therefore, it is customary to represent all precursors by six groups (or families) of precursors. The yield fractions and decay constants of such groups are obtained experimentally by exposing a sample of fissionable material to a very short neutron pulse and then measuring the time behavior of the source of the delayed neutrons. The results obtained for U-233, U-235 and Pu-239 are shown in XTABLE 2-1X.

Chapter

2

T


34

FIGURE 2-1. Schematic of emission of delayed neutrons.

TABLE 2-1. Characteristics of delayed fission neutrons in thermal fission.

Approximate half-life [s] Number of delayed neutrons per 100 fissions Energy (MeV) U-233 U-235 Pu-239

55 0.053 0.060 0.024 0.25 23 0.197 0.364 0.176 0.46 6.2 0.175 0.349 0.136 0.41 2.3 0.212 0.628 0.207 0.45 0.61 0.047 0.179 0.065 0.41 0.23 0.016 0.070 0.022 - Total delayed 0.70 1.650 0.630 Total fission neutrons 249 242 293 Fraction delayed 0.0026 0.0065 0.0020

As in steady-state analyses of nuclear reactors, the complete neutron balance equations for the time dependent neutron flux depend on the neutron location in the three-dimensional space, the neutron direction vector and the neutron energy. For reactor kinetics applications, however, it is neither feasible nor necessary to use the same level of complexity. The equations can be condensed into purely time-dependent ones, so-called point kinetics equations. The name point kinetics merely indicates that all space and angle dependencies in equations are neglected and only the time dependence is left.

2.1.2 Derivation of Point Kinetics Equations

In the non-critical system the production and loss of neutrons are not in balance, and in the absence of an independent neutron source, the system can not be in the steady-state condition. For the present purpose, the principle of neutron conservation can be expressed in a general form as follows.

(2.1) ( ) SCt

N

i

iii ++−−=∂

∂∑

=1

1v

1TMF λφφβ

φ,

(2.2) NiCt

CiiDi

i ...,,1, =−=∂

∂λφβ F .

prompt neutron

delayed neutron

emitter (A,Z)

beta-decay

precursor (A,Z-1) final

nucleus (A-1,Z)

fission product


35

Equation (2.1) is a time and space-dependent neutron balance equation, where φ is

the neutron flux, Ci is the concentration of precursors in i-th group, S is the external

neutron source, ∑=N

i

1

ββ is the total yield of the delayed neutrons and iλ is the

decay constant of i-th group of precursors. The time and space dependent concentrations of precursors are given by Eqs. (2.2). The equations are given in general forms, where operators F, FD, M and Ti represent the production of fission neutrons, the production of the precursors of delayed neutrons, the migration and loss of neutrons and the conversion from i-th group precursors to neutrons, respectively. These operators can take various forms, depending on the particular model equations that are employed. Typical choices include transport equations, diffusion approximation equation or multi-group diffusion approximation equations. As an example, for one-group diffusion approximation, the operators are as follows,

(2.3) ( ) ( )rrM aD Σ+∇⋅−∇= , fD Σ== νFF Ti = 1.

The neutron flux φ can be factored as follows,

(2.4) ( ) ( ) ( )tntEtE s ⋅= ,,,,,, ΩrΩr φφ ,

where sφ is the shape function and n(t) is the amplitude function. The amplitude

function is a function of time only, whereas the shape function is a function of the same independent variables as the neutron flux, that is: r - the neutron position vector, Ω - the neutron direction vector, E - the neutron energy and t - time.

The amplitude function will have a simple physical interpretation if it is defined as,

(2.5) ( ) ( ) ( ) dEddtEEtn ΩrΩrΩr ,,,,,v

1φψ∫= .

Such definition suggests that the amplitude function is a mean weighted number of

neutrons in a nuclear reactor at time t. Here ( )E,,Ωrψ is an arbitrary weighting

function. If ( ) 1,, =EΩrψ then n(t) is the mean number of all neutrons in the reactor

at time t. Using Eq. (2.4) in (2.5) yields,

( ) ( ) ( ) ( )

( ) ( ) ( ) dEddtEEtn

dEddtntEEtn

S

S

ΩrΩrΩr

ΩrΩrΩr

,,,,,v

1

,,,,,v

1

φψ

φψ

∫

∫

⋅

=⋅=

,

thus,

(2.6) ( ) ( ) 1,,,,,v

1=∫ dEddtEE S ΩrΩrΩr φψ .

Equation (2.6) is called the normalization condition of the shape function. Multiplying the neutron balance equation (2.1) by the weighting function yields,


36

( ) SCnnnt

N

i

iiiSSS ψψλφψφψβψφ ++−−=

∂

∂∑

=1

1v

1TMF

or,

( ) SCnnt

ndt

dn N

i

iiiSSSS ψψλφψφψβψφψφ ++−−=

∂

∂+ ∑

=1

1v

1

v

1TMF .

The last equation can be integrated over all neutron positions, direction angles and energies. Using a shortcut notation dEdddx Ωr= , the integration yields,

( ) ∫∑ ∫∫∫∫ ++−−==

SdxdxCdxndxndxdt

dn N

i

iiiSSS ψψλφψφψβψφ1

1v

1TMF .

Further, the equation is divided by a normalization factor F, that will be determined later,

( )

∫∑ ∫

∫∫∫

+

+

−−=

=

SdxF

dxCF

dxF

dxF

ndt

dndx

F

N

i

iii

SSS

ψψλ

φψφψβψφ

11

111

v

11

1

T

MF

.

Defining,

( ) ∫=Λ dxF

t Sψφv

11,

yields,

(2.7)

( )

∫∑ ∫

∫∫

Λ+

Λ

+

−−

Λ=

=

SdxF

dxCF

dxF

dxF

n

dt

dn

N

i

iii

SS

ψψλ

φψφψβ

11

111

1

T

MF

.

In the similar manner, Eq. (2.2) is multiplied by iTF

ψΛ

1 and integrated to yield,

(2.8) ∫∫∫ Λ−

Λ=

ΛdxC

Fdx

FdxC

Fdt

diiiSDiiii TFTT ψλφψβψ

111.

Introducing the following definitions,

(2.9) ( ) ∫Λ= dxC

FtC iii Tψ

1


37

(2.10) ( ) ∫Λ= Sdx

FtS ψ

1,

(2.11) ( ) ∫= dxF

t SDiii φψββ FT1

,

(2.12) ( ) ∫= dxF

t Sφψββ F1

,

Eqs. (2.7) and (2.8) become,

(2.13) ( ) SCdxF

n

dt

dn N

i

iiS ++

−−⋅

Λ= ∑∫

=1

1λβφψ MF ,

(2.14) NiCndt

Cdii

ii,...,1, =−

Λ= λ

β.

The normalization factor F can be chosen in such a way that the coefficients in Eqs. (2.13) and (2.14) will have a simple physical interpretation. If this factor is defined as,

(2.15) ∫= dxF SφψF ,

it can be interpreted as the mean weighted number of neutrons created in a reactor due to fission per unit time at time t. In addition,

(2.16) ( )( ) ( )

ρφψ

φψ

φψ

φψφψ ≡

−=

−=−

∫∫

∫∫

∫dx

dx

dx

dxdx

FS

S

SF

MF

F

MFMF

1.

As can be seen, the above term is equal to the reactivity, since it is a ratio of the net neutron production to the total neutron production.

Finally, the point kinetics equations are as follows,

(2.17) SCndt

dn N

i

ii ++Λ

−= ∑

=1

λβρ

,

(2.18) NiCndt

Cdii

ii,...,1, =−

Λ= λ

β.

Equations (2.17) and (2.18) have been derived from the general neutron balance equations without making any approximations. For this reason the equations are termed as the “exact point kinetics equations”, where “exact” means that the time, space, energy and direction dependent neutronics is taken into account and lumped into the point kinetics equation. In particular, Eqs. (2.17) and (2.18) are equivalent to initial full equations, if the effective kinetics parameters are calculated as given by Eqs. (2.9) through (2.12).


38

To calculate the effective kinetics parameters it is necessary to chose proper shape and weighting functions. From the mathematical point of view any well defined function could be used as the weighting function. However, in practical applications the adjoint neutron flux is chosen as the weighting function, that is,

(2.19) ( ) ( )xx*

0φψ = ,

where ( )x*

0φ is a solution of the adjoint neutron balance equation (see Note Corner

below).

In addition, the shape function is taken equal to the neutron distribution function in the critical reactor,

(2.20) ( ) ( )xtxS 0, φφ = ,

and the neutron creation operators will take the values for the critical reactor as well,

(2.21) 0FF = , 0DD FF = ,

where subscript 0 refers to the critical state of the reactor.

NOTE CORNER: Adjoint neutron flux has a simple physical interpretation. Using the one-group diffusion approximation it can be easily shown that the adjoint neutron flux at a given point r0 is proportional to the reactivity change caused by an introduction or a removal of one neutron per second at that point. Adjoint neutron flux is obtained from a solution of the adjoint neutron balance equation. In such equation all operators are replaced with their adjoint forms. The adjoint operator is defined by the scalar product of two functions ϕ and ψ , each

out of the respective functional space: ( ) ∫= dxϕψψϕ, , where x represents in a short form all

independent variables of functions ϕ and ψ , and the integration is performed over the whole region

where the functions are determined. For a given linear operator L, its adjoint operator L* is defined by the following equality: ( ) ( )ϕψψϕ ∗= LL ,, . If in addition L = L*, then the operator is self-adjoint. It can

be easily shown that if L is a constant, then L = L*. Also it can be shown that the Laplace operator is

self-adjoint; that is *∆=∆ . For the first order derivatives, the following will be valid, however: (d/dt)* = -d/dt.

If the assumptions (2.19) through (2.21) are adopted, the kinetics parameters are as follows:

(2.22)

( ) ( )

( ) ( )∫

∫=Λ

dxxx

dxxx

00

*

0

0

*

0v

1

φφ

φφ

F,

(2.23) ( ) ( )

( ) ( )∫∫

=dxxx

dxxx Di

ii

00

*

0

00

*

0

φφ

φφββ

F

FT,

(2.24) ∑=

=N

i

i

1

ββ .


39

As an example, the neutron balance equations with the one-group diffusion approximation can be written as follows,

( ) SCDt

N

i

iiaf ++Σ−∇+Σ−=∂

∂∑

=1

21v

1λφφφνβ

φ,

NiCt

Ciifi

i ...,,1, =−Σ=∂

∂λφνβ ,

and since in steady-state, the time derivatives and the external neutron sources are equal to zero, the equations yield,

000

2

0 =Σ−∇+Σ φφφν af D .

Since all operators in the above equation are self-adjoint, the adjoint balance equation is identical,

0*

0

*

0

2*

0 =Σ−∇+Σ φφφν af D .

Further, since the boundary conditions for both equations are identical (zero flux at the extrapolated boundary), both solutions are identical, with possibly a constant multiplier c as follows,

(2.25) 0

*

0 φφ c= .

For the critical reactor and the one-group diffusion approximation the operators are as follows,

(2.26) fD Σ== ν00 FF , Nii ,...,1,1 ==T ,

thus,

(2.27)

( ) ( )

( ) ( )

( )

( ) f2

0

2

0

00

*

0

0

*

0

v

1v

1

v

1

Σ⋅⋅=

Σ==Λ∫

∫

∫

∫νφν

φ

φφ

φφ

dxxc

dxxc

dxxx

dxxx

fF

(2.28) ( ) ( )

( ) ( ) i

f

f

i

Di

iidxc

dxc

dxxx

dxxxβ

νφ

νφβ

φφ

φφββ =

Σ

Σ==

∫∫

∫∫

2

0

2

0

00

*

0

00

*

0

F

FT.

The reactivity could be found from Eq. (2.16), however, this would require the knowledge of the neutron flux distribution. As an approximation, the perturbation theory can be applied.

2.1.3 Equations for Six-Group Point Kinetics Model

Equations X(2.17) and X(2.18) are the fundamental equations of reactor kinetics for the point-reactor model. In the following the overbar indicating the effective values of the kinetics parameters will be dropped, unless it introduces confusion. It should be


40

remembered, however, that they represent some mean weighted quantities, expressed by Eqs. (2.9) through (2.12). The six-group point kinetics model is summarized in TABLE 2-2..

TABLE 2-2. Point reactor kinetics model.

Reactor kinetics equation SCn

dt

dn

i

ii ++Λ

−= ∑

=

6

1

λβρ

Delayed-neutron precursor equations: i = 1,…,6 6,...,1, =−

Λ= iCn

dt

dCii

ii λβ

The values of decay constants and yields of delayed-neutron precursors are given in TABLE 2-3X.

TABLE 2-3. Decay constants and yields of delayed-neutron precursors in thermal fission of uranium-235.

Decay constants and yields of delayed-neutron precursors in thermal fission of uranium-235

t1/2, [s] iλ , [s-1] iβ ii λβ

55.7 0.0124 0.000215 0.0173 22.7 0.0305 0.00142 0.0466 6.22 0.111 0.00127 0.0114 2.30 0.301 0.00257 0.0085 0.61 1.1 0.00075 0.0007 0.23 3.0 0.00027 0.0001

Total 0.0065 0.084

2.1.4 Equations for One-Group Point Kinetics Model

Some properties of the point kinetics model can be investigated using a one-group approximation. The total yield of the one group of delayed neutrons is obtained as a sum of yields in all groups. For uranium-235 this value is shown in TABLE 2-3X and is equal to β = 0.0065. The decay constant can be obtained from a proper averaging, e.g.,

(2.29)

∑∑

=

=

=⇒=6

1

6

1

i i

ii i

i

λ

β

βλ

λ

β

λ

β.

Using data from TABLE 2-3, the equivalent decay constant for one-group assumption

for uranium-235 is obtained as 08.0084.00065.0 ≅=λ s-1.

The equations in one group approximation of point kinetics equations are as follows,

(2.30) SCndt

dn++

Λ

−= λ

βρ,

(2.31) Cndt

dCλ

β−

Λ= .


41

Here C represents the concentration of precursors of all groups of delayed neutrons.

In the next section special cases of the point kinetics model will be considered and their solutions will be found.

2.1.5 Average Neutron Generation Time and Lifetime

The average neutron generation time Λ has been derived for the one-group diffusion approximation in Eq. (2.27) and can be written in various forms as follows,

(2.32)

af kk

l

k

l

Σ===

Σ=Λ

∞∞

∞

v

1

v

1

ν.

Here l is the average neutron lifetime and k is the effective multiplication factor. The name “generation time” has been chosen since Λ represents the average time between

two birth events in successive neutron generations. Firstly, fΣ1 is the mean free path

for fission, that is, it is the average distance a neutron travels from its birth to a fission

event. Then, ( )ff t∆=Σ v1 is the average time between the birth of a neutron and a

fission event it may cause. Since ν -neutrons is released per fission, the averaged time between the birth of a neutron and the birth of a new generation is as follows,

(2.33) Λ=Σ

=∆

f

ft

νν v

1.

In a similar manner, the average traveling distance of a neutron between the birth and

the death (absorption or leakage) is ( )21 ga DB+Σ and the average neutron lifetime

can be obtained as,

(2.34) 222 1

1

v

11

v

1

gaga BLDBl

+Σ=

+Σ= ,

since the one-group diffusion length is given as aDL Σ= .

Eq. (2.34) yields,

(2.35) Λ⋅=Σ

=+Σ

=∞

∞

∞

kk

k

BL

k

kl

aga v1v

122

.

Equation X(2.35) indicates that the lifetime and the generation time are equal for a critical reactor. For subcritical reactor (k < 1) the neutron lifetime is shorter than the generation time and as a consequence, the neutron population will decrease. For supercritical reactor the lifetime will be longer and the neutron population will increase, whereas for a critical reactor the population remains constant.

2.2 Normalized Point Kinetics Equations The point-reactor model equations constitute a system of ordinary linear differential equations with variable coefficients, since the reactivity ρ and other kinetics


42

parameters are – in general - functions of time. For an arbitrary function ( )tρ it is thus

– in general - not possible to obtain a rigorous analytical solution of the system of equations. In such case it is necessary to employ approximate numerical solutions.

In the following the equilibrium of an operation reactor is investigated as well as normalized equations of the point kinetics model are derived. Such equations are very convenient since all initial conditions are suitably reduced to zero values.

It is convenient to represent the point reactor equations in a normalized excess form, using some well defined reference condition as a reference. A reasonable choice is to find the equilibrium (steady-state) parameters of the reactor and use them to derive the normalized excess equations for point reactor kinetics. For that purpose it is necessary to find the equilibrium point (that is the fixed point) of the model.

2.2.1 Equilibrium Point of a Nuclear Reactor

Equilibrium points of equations in XTABLE 2-2 corresponding to the reactor steady-state condition can be obtained by equating to zero the time derivatives. The following system of equations is obtained,

(2.36) 06

1

=++

Λ

−∑

=

SCni

iiλβρ

,

(2.37) 0=−Λ

iii Cn λ

β.

Substituting Eq. X(2.37) to (2.36) yields,

(2.38) 06

1

=+Λ

+

Λ

−∑

=

Snni

iββρ.

Since

(2.39) ∑=

=6

1i

iββ .

Equation (2.38) becomes,

(2.40) 0=+Λ

⇒+Λ

+

Λ

−SnSnn

ρββρ.

Equation (2.40) yields the equilibrium neutron density as,

(2.41) ρ

Λ−=

Sne .

Since ne cannot be negative, because this is the averaged weighted number of neutrons in the reactor, the reactivity must be negative if there are neutron sources in the reactor. It means that the equilibrium in the reactor with external neutron sources is


43

possible when the reactor is subcritical. In such situation the steady-state power of the nuclear reactor is determined by the neutron source efficiency, the reactivity and the mean neutron generation time.

When S = 0 (reactor has no external neutron sources), Eq. (2.41) yields:

(2.42) 0=enρ .

This equation is satisfied in three different cases:

a) 0,0 ≠= enρ , which defines the critical state of reactor

b) 0,0 =≠ enρ , reactor shut down condition

c) 0,0 == enρ , non-physical (critical reactor cannot have zero power)

The first equilibrium (or fixed) point can be chosen as the reference case for the normalization of the point kinetics equations.

2.2.2 Normalized Equations of Point Kinetics Model

The power of an operating reactor can change in a very wide range, up to 1010 times. Due to that it is useful to express the reactor power in terms of the steady-state power, which must be larger than zero for a critical reactor.

Assuming that the mean average neutron density is ne, Eq. X(2.37) yields,

(2.43) Λ

=i

eiie

nC

λ

β.

Introducing normalized excess variables,

(2.44)

e

e

n

nnx

−= ,

(2.45)

ie

ieii

C

CCy

−= ,

one gets

(2.46) ee nxnn += ,

(2.47) ieiiei CyCC += .

Substituting Xthe normalized variables into Xthe six-group point kinetics equations yields,

(2.48) xn

Syx

dt

dx

ei

iiΛ

++Λ

+Λ

+Λ

−= ∑=

ρρβ

β 6

1

1,


44

(2.49) 6,...,1=−= iyxdt

dyiii

i λλ ,

with the following initial conditions:

(2.50) 0)0( =x ,

(2.51) 6,...,10)0( == iyi .

Equation (2.48) and X(2.49) together with the initial conditions given by Eqs. (2.50) and X(2.51) constitute the normalized point kinetics equations. They describe the deviation of the reactor power from the initial, steady-state value. The model can be easily implemented using the Scilab environment. A script containing the point kinetics model is shown below.

COMPUTER PROGRAM: Point Reactor Kinetics Model

function [dy]= PKModel(t,y,yield,dconst,LAMBDA,ne)

// Point kinetics model

// dy - returned right-hand-side values

// t - time

// y - vector of unknown functions

// yield – yields of the precursors

// dconst – decay constants of precursors

// LAMBDA – generation time

// ne – neutron density at equilibrium

//

// Set parameters

//

NPre = length(yield); // Get number of precursors

neq = 1 + NPre; // Total number of equations

// Time functions

rho = InReactivity(t); // Reactivity as a function of time

S = InSource(t); // Neutron source as a function of time

//

// Calculate Right-Hand-Sides of differential equations

//

dy = zeros(neq,1); // Initialize RHS

dy(1) = (-sum(yield)*y(1) + yield*y(2:$) + rho + rho*y(1))/LAMBDA + S/ne;

for i = 1:NPre

dy(i+1) = dconst(i)*(y(1) - y(i+1));

end

endfunction

The point kinetics model PKModel() can be solved with the Scilab build-in solver for the ordinary differential equations. For that purpose a dedicated solver script PKODESol() has been implemented in Scilab. The model requires a specification of two additional functions: InReactivity(t) and InSource(t) which describe the time variation of the reactivity and the external sources, respectively. The usage of the point kinetics model implemented in Scilab is elucidated in the example below.


45

EXAMPLE 2-1. A critical reactor operated during a long period of time with constant power. At time t = 0 the reactivity was suddenly made 0.0022 positive. Assuming s

310−=Λ and one group of delayed neutrons with 108.0 −= sλ and 3105.6 −⋅=β , predict the relative power change due to the reactivity insertion.

Make a plot power versus time in time interval from 0 to 5 s. SOLUTION: Time dependent reactivity is given with the following function:

function [rho] = InReactivity(t)

if t < 0

rho = 0.;

else

rho = 0.0022;

end

endfunction

The vector of time instances at which the solution is found is given as: ->t=linspace(0,5,100);

Finally, the model is invoked as follows: ->PKODESol(t,6.5e-3,0.08,1.e-3,1);

The calculated time variation of the relative power as a function of time is shown in FIGURE 2-2.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.01.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Time, s

Po

we

r

FIGURE 2-2. Relative reactor power change after sudden insertion of 0.0022 reactivity at time t = 0.

2.3 Solutions with Constant Reactivity If the reactivity is constant, the model contains a set of linear ordinary differential equations with constant coefficient and can be solved analytically. This condition is valid in the case of the step change of reactivity from 0 to some finite (positive or negative) value ρ0. Two types of solutions will be obtained and compared: using the six-group and the one-group models.

2.3.1 Solutions of Six-Group Point Kinetics Equations

Point-kinetics equations can be easily solved for a step-change of the reactivity, when the reactor operated at steady state before the step change. For t > 0 the reactivity is then constant and Eqs. (2.48) with (2.49) are linear differential equations with constant


46

coefficients, which can be solved using, e.g., the Laplace transformation approach (see APPENDIX A),

(2.52) )(ˆ)(ˆ1

)(ˆ1

)(ˆ)(ˆ 06

1

0 sxn

sS

ssysxsxs

ei

iiΛ

++Λ

+Λ

+Λ

−= ∑=

ρρβ

β,

(2.53) )(ˆ)(ˆ)(ˆ sysxsys iiii λλ −= .

Here )(ˆ sx , )(ˆ syi and )(ˆ sS are Laplace transforms of functions x(t), xi(t) and S(t),

respectively, e.g.,

(2.54) ∫∞

−==0

)()()(ˆ dttxetxLsxst

.

The system of algebraic equations formed by Eqs. (2.52) and (2.53) has to be solved to

find )(ˆ sx and )(ˆ syi . From Eq. (2.53) one gets,

(2.55) )(ˆ)(ˆ sxs

syi

ii λ

λ

+= .

Combining Eq. (2.55) with (2.52) yields,

(2.56)

ei i

ii

n

sS

ssx

ss

)(ˆ1)(ˆ

1 06

1

0 +Λ

=

Λ−

+Λ−

Λ+ ∑

=

ρρ

λ

λββ,

or

(2.57)

ei i

i

n

sS

ssx

ss

)(ˆ1)(ˆ

11 00

6

1

+Λ

=

Λ−

+Λ+ ∑

=

ρρ

λ

β.

Finally,

(2.58) )(

)(ˆ1

)(

1)(ˆ 0

sM

sS

nssMsx

e

+Λ

=ρ

,

where

(2.59) Λ

−

+Λ+= ∑

=

06

1

11)(

ρ

λ

β

i i

i

sssM .

To find x(t) it is necessary to apply the inverse Laplace transformation to Eq. (2.58) and for that purpose the zero values of M(s) function have to be found. It can be shown that all roots of function M(s) are real and their location can be conveniently presented using graphical techniques.


47

Equation (2.59) can be written as:

(2.60) 0

6

1

)(1

10)( ρλ

β=≡

+Λ+Λ⇒= ∑

=

sFs

ssMi i

i.

As can be seen, finding roots of M(s) is equivalent to finding crossing points of

function F(s) with 0ρ . XFIGURE 2-3X demonstrates this approach.

FIGURE 2-3. A graphical demonstration of the roots of M(s) function.

Equation X(2.60) is called the inhour equation (which comes from inverse hour, when it was used as a unit of reactivity that corresponded to e-fold neutron density change during one hour) and is a 7th order algebraic equation with 7 roots. As shown in XFIGURE 2-3X, the first root is positive for positive reactivity. The root will change sign and become negative when reactivity is negative. All remaining roots are always negative.

When zero values sk of function M(s) are known, it is possible to find x(t) in a general form using the following formula, which is known from the theory of the Laplace transformation,

(2.61)

′+

Λ=

Λ= ∑

=

−6

0

010

)(

1

)0(

1

)(

1)(

k

ts

kk

kesMsMssM

Ltxρρ

,

or shorter,

(2.62) ∑=

+=7

1

)(k

ts

kkeAAtx ,

where

(2.63) )0(

10

MA

Λ=

ρ, 7,...,1,

)(

10 =′Λ

= ksMs

Akk

k

ρ.

( )sρ

s s1 s2

0ρ

s3 s4 s5 s6 s7


48

When 00 >ρ then s1 > 0, sk < 0, k = 2, …, 7. In such case the last terms in Eq. (2.62)

approach zero when time increases to infinity. In most cases the terms can be neglected after relatively short period of time following the step change of reactivity and only two first terms are significant. That is,

(2.64) ts

eAAtx 1

1)( += , s1 > 0.

When 00 <ρ then sk < 0, k = 1, …, 7 and the last terms in Eq. (2.62) decrease

relatively fast compared to the second term, which also decreases with time. It means that after short period of time Eq. (2.64) is approximately valid, however, now s1 < 0.

The reactor period, Tp, or e-folding time, is defined as the time required for the neutron density to change by a factor e, that is,

(2.65) pTt

entn/

0)( = .

Since roots sk have a unit of reciprocal time, Eq. (2.62) shows that for positive s1 (and thus positive reactivity) 1/s1 is the reactor period after the laps of sufficient time to permit the contribution of other terms to damp out. Consequently Tp = 1/s1 is called the stable reactor period. The quantities 1/s2, 1/s3, …, 1/s7 are sometimes referred to as transient reactor periods but they are negative and have no physical significance.

If reactivity 0ρ is negative, all roots will be negative and s1 will determine the slowest

rate of change of the neutron density and thus will ultimately yield a stable (negative) reactor period equal to 1/s1.

2.3.2 Solutions of One-Group Point Kinetics Equations

The general solution found in the previous section will be, for the sake of simplicity, analyzed for a case when only one group of delayed neutron is assumed. Equations (2.52) and X(2.53) become,

(2.66) )(ˆ)(ˆ1)(ˆ

)(ˆ)(ˆ 00 sxn

sS

s

sysxsxs

e Λ++

Λ+

Λ+

Λ−=

ρρββ,

(2.67) ( ) ( ) ( )sysxsys ˆˆˆ λλ −= .

The averaged decay constant λ in Eqs. (2.66) and (2.67) is defined as,

(2.68)

∑=

=6

1i i

i

λ

β

βλ ,

and

(2.69) ∑=

=6

1i

iββ .

Finding ( )sy from Eq. (2.67) and substituting into Eq. (2.66) yields,


49

(2.70)

( ) ( )( )( )

+

Λ−−

+=

+

Λ

Λ−

+

Λ−

Λ+

+

=

+Λ−

Λ

−+

+Λ

=

ee

e

n

S

sssss

s

n

S

sss

s

ss

n

sS

ssx

00

21

00

002

0

0 )(ˆ1

)(ˆ

ρλρ

λρλ

ρβ

λ

λ

λβρβ

ρ

.

It has been assumed in Eq. (2.70) that the source term is constant and equal to S0, whereas s1 and s2 are roots of the polynomial in the denominator and are as follows,

(2.71) 2

4 0

2

00

2,1

Λ+

+

Λ−

Λ±

+

Λ−

Λ−

=

λρλ

ρβλ

ρβ

s .

The inverse Laplace transformation of Eq. (2.70) yields the required solution in the time domain. It should be noted that Eq. (2.70) can be written as,

(2.72) ( )

( )( ) )(

)()(ˆ 00

21 ssM

sLA

n

S

sssss

ssx

e

=

+

Λ−−

+=

ρλ.

Using the formula given in APPENDIX A (see (A.6)), the inverse Laplace transform is found as,

(2.73) ( )( )

′+

′+=

− tsts

esMs

sLe

sMs

sL

M

LA

ssM

sLAL 21

)(

)(

)(

)(

0

0

)(

)(

22

2

11

11.

Finally,

(2.74) ( ) ( )

−

++

−

++

+

Λ= tsts

e

esss

se

sss

s

ssn

Stx 21

122

2

211

1

21

00)(λλλρ

,

and the neutron flux is obtained as,

(2.75)

( ) ( )

−

++

−

++

+

Λ+

=+=

tsts

e

e

ee

esss

se

sss

s

ssn

Sn

ntxntn

21

122

2

211

1

21

001

)()(

λλλρ .

EXAMPLE 2-2. The reactivity in a steady-state thermal reactor with no external neutron sources, in which the neutron generation time is 10-3 s, is suddenly made 0.0022 positive. Assuming one group of delayed neutrons, determine the subsequent change of neutron flux with time: 108.0 −= sλ , 3105.6 −⋅=β .

SOLUTION: The roots s1 and s2 are found as s1 = 0.03982… and s2 = -4.4198, and the equation for the neutron flux becomes,


50

( )tt

e eentn4198.403982.0 4844.04844.1)( −−= .

EXAMPLE 2-3. A critical reactor operated during a long period of time and the mean weighted number of neutrons was equal to ne = 106. Suddenly a source of neutrons with mean weighted yield equal to S0 = 106 s-1 was introduced into the reactor. Find the number of neutrons as a function of time, n(t). In calculations assume one group of delayed neutrons with 11.0 −= sλ , 3104.6 −⋅=β and the

mean generation time s310−=Λ .

SOLUTION: The point-kinetics equations with one group delayed neutrons are as follows,

xn

Syx

dt

dx

e Λ++

Λ+

Λ+

Λ−= 00 ρρββ ,

yxdt

dyλλ −= , with initial conditions, x(0) = y(0) = 0. Since the reactor was initially critical, the

reactivity 00 =ρ . The source term is the following function of time,

≥

<=

0

00)(

0 tS

ttS

Substituting the reactivity and the source term into the point-kinetics equations, and performing the Laplace transform yields,

sn

Ssysxsxs

e

1)(ˆ)(ˆ)(ˆ 0 ⋅+

Λ+

Λ−=

ββ ,

)(ˆ)(ˆ)(ˆ sysxsys λλ −= .

Combining the above two equations yields,

Λ++

+⋅=

βλ

λ

ss

s

n

Ssx

e 2

0)(ˆ.

To facilitate the inverse Laplace transform, the right-hand-side of the above equation will be expressed in terms of basic functions as follows,

( )

Λ++

++

+

Λ++

Λ+

=

Λ++

⋅+

Λ++⋅+

Λ++⋅⋅

⇒

Λ++

++=

Λ++

+

βλ

βλ

βλ

βλ

βλ

βλ

βλ

βλ

λ

ss

scabasb

ss

scsbssa

s

c

s

b

s

a

ss

s

2

2

2

2

22

.

The constants a, b and c are found from the following equations,

0=+ ca , 1=+

Λ+ ba

βλ , λ

βλ =

Λ+b .


51

And the constants are readily obtained as,

Λ+

=β

λ

λb

,

2

Λ+Λ

=β

λ

βa

, 2

Λ+Λ

−=β

λ

βc

.

Finally, the expression for ( )sx becomes,

Λ++

Λ+Λ

−

Λ+

+

Λ+Λ

=β

λβλ

ββ

λ

λ

βλ

β

sssn

Ssx

e

111)(ˆ

222

0.

Applying the inverse Laplace transformation yields,

⋅

Λ+Λ

−

Λ+

+

Λ+Λ

=

Λ+− t

e

etn

Stx

βλ

βλ

β

βλ

λ

βλ

β22

0)(,

and the number of neutrons is obtained as,

−

Λ+Λ

+

Λ+

+=

Λ+− t

e etSntn

βλ

βλ

β

βλ

λ1)(

20

.

XFIGURE 2-4 shows the variation of the excess normalized number of neutrons with time.

0 1 2 3 4 5 6 7 8 9 100.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

t, s

x

FIGURE 2-4. Excess normalized number of neutrons in a critical reactor as a function of time after

sudden insertion of a constant neutron source S = 106 s-1.

As can be seen the number of neutrons increases very fast during the first 0.5s of the transient and then almost linearly, for time t larger than 1s. This is because the exponential term in the solution is very small and the term linearly proportional to time prevails.


52

EXAMPLE 2-4: An under-critical reactor operated during a long period of time with the reactivity equal to βρ 90 −= . A source of neutrons with a mean weighted

yield equal to S0 = 106 s-1 was present in the reactor. Suddenly the source of neutrons was removed from the reactor. Find the number of neutrons as a function of time, n(t). In calculations assume one group of delayed neutrons with

11.0 −= sλ , 3105.6 −⋅=β and the mean generation time s310−=Λ .

SOLUTION: The equilibrium number of neutrons before the removal of the source can be calculated as,

4

3

36

0

0 10709.1105.69

1010⋅≈

⋅⋅

⋅=

Λ−=

−

−

ρ

Sne

The solution of the point-kinetics model with one group of delayed neutrons have been found as,

( ) ( )

−

++

−

++

+

Λ+=+= tsts

e

eee esss

se

sss

s

ssn

Snntxntn 21

122

2

211

1

21

001)()(λλλρ ,

where the roots are found as,

01.65,09.02

4

21

0

2

00

2,1 −=−=⇒Λ

+

+

Λ−

Λ±

+

Λ−

Λ−

= sss

λρλ

ρβλ

ρβ

.

Finally, taking S0 = 0, the solution becomes,

( )tteetn

⋅−⋅− ⋅+⋅⋅= 01.6509.04 9.01.010709.1)( .

XFIGURE 2-5X shows the number of neutrons as a function of time.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time, s

Neoutr

ons

FIGURE 2-5. Number of neutrons in a reactor as a function of time after sudden removal of a constant

neutron source S0 = 106 s-1.

As can be seen the number of neutrons rapidly drops from initial 17090 just below 2000 and it continues to drop, however with much smaller paste. FIGURE 2-6XX shows the same function for a longer time interval.


53

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time, s

Ne

utr

on

s

FIGURE 2-6. Number of neutrons in a reactor as a function of time after sudden removal of a constant

neutron source S0 = 106 s-1 in a time interval from 0 to 50 s.

The rapid drop of the number neutrons during the first 0.05 s (prompt jump) is now hardly visible. What can be seen is the slow drop of the number of neutrons from less than 2000 to 0 during approximately 50 s.

2.4 Point Kinetics Model with Time-Dependent

Reactivity In the previous Section the point kinetics equations were considered to be linear equations with constant coefficients due to a special value of the reactivity: it has been assumed that the reactivity is a step-function of time. In general case, when the reactivity is an arbitrary function of time (and even of power, but this will be considered in the next Chapter), the term )()( tntρ in Eq. (2.30) will cause that the the

methods described in the previous Section can not be applied (in particular, the Laplace transformation method will not work). To solve this type of equations other methods should be used, such as the small-perturbation approximation method, or the equations should be solved numerically. The former approach will be valid for small perturbations only, but it will in general lead to an analytical solution that can give a valuable insight into the behavior of the system under consideration. The latter method will provide a solution which is valid for any perturbation (both small and large), but the interpretation of the results is somewhat more complicated and requires a generation of plots and diagrams which then can be analyzed.

2.4.1 Small-Perturbation Approximation

The exact point kinetics equations can be turned into a linear system of ordinary differential equations with constant coefficients using a general approach presented in the Introduction. In this section the approximate equations will be derived using the perturbation method.

To get rid of the terms with the time-dependent coefficients, it is necessary to consider the perturbed equations by assuming that all parameters deviate from the equilibrium values by small amounts. That is, the following applies,


54

(2.76)

)()(

)()(

)()(

0

0

0

tt

tyyty

txxtx

iii

δρρρ

δ

δ

+=

+=

+=

.

Substituting Eq. (2.76) into (2.48) yields,

(2.77)

[ ] [ ] [ ]

[ ] [ ][ ])()()(

)(1

)()(

000

6

1

000

txxt

n

St

tyytxxdt

txxd

e

i

iii

δδρρδρρ

δβδβδ

+Λ

+++

Λ

+

++Λ

++Λ

−=+

∑=

.

The terms in Eq. (2.77) can be rearranged as follows,

(2.78)

43421small

e

i

ii

i

ii

txt

txxt

xn

St

tyytxxdt

txd

dt

dx

)()(

)()()(

)(11

)()(

000

00

6

1

6

1

000

δδρ

δρδρρδρρ

δββδββδ

Λ+

Λ+

Λ+

Λ++

Λ+

Λ

+Λ

+Λ

+Λ

−Λ

−=+ ∑∑==

.

The last term is a product of two small quantities which is negligibly small and will be neglected. Since the terms with “0” index satisfy the initial equation, the above equation can be written as,

(2.79) )()()(

)(1

)()( 0

0

6

1

txxtt

tytxdt

txd

i

ii δρδρδρ

δβδβδ

Λ+

Λ+

Λ+

Λ+

Λ−= ∑

=

.

In a similar manner, the equation for the precursors of delayed neutrons is transformed as,

(2.80) [ ] [ ] [ ] 6,...,1)()(

)(00

0 =+−+=+

ityytxxdt

tyydiiii

ii δλδλδ

,

or,

(2.81) 6,...,1)()()(

000 =−−+=+ ityytxx

dt

tyd

dt

dyiiiiii

ii δλλδλλδ

.

Again, “0” values satisfy the initial equation, which leads to,

(2.82) 6,...,1)()()(

=−= itytxdt

tydiii

i δλδλδ

.

Equations (2.79) and (2.82) are point-kinetics equations perturbed around an

equilibrium operational point defined by values 00 and ρx . It should be noted that the

equations constitute a system of linear ordinary differential equations with constant coefficients.


55

Assuming that x and yi correspond to small perturbations designed earlier by δx and δyi,

and that thanks to a proper normalization both 00 and ρx are equal to zero, the

equations can be written in the standard form as,

(2.83) Λ

+Λ

+Λ

−= ∑=

)()(

1)(

)( 6

1

ttytx

dt

tdx

i

ii

ρβ

β,

(2.84) 6,...,1)()()(

=−= itytxdt

tdyiii

i λλ ,

with initial conditions,

(2.85) 0)0( =x ,

(2.86) 6,...1,0)0( == iyi .

Applying the Laplace transformation, the system of equations formed by Eqs. (2.83) through (2.86) becomes,

(2.87) Λ

+Λ

+Λ

−= ∑=

)(ˆ)(ˆ

1)(ˆ)(ˆ

6

1

ssysxsxs

i

ii

ρβ

β,

(2.88) 6,...,1)(ˆ)(ˆ)(ˆ =−= isysxsys iiii λλ .

Equations (2.87))) and (2.88) give the following transfer function of the reactor:

(2.89)

++Λ

==

∑=

6

1

1

)(ˆ

)(ˆ)(

i i

i

ss

s

sxsG

λ

βρ,

which is often referred to as the zero-power reactor transfer function. More detailed analysis of this transfer function will be performed in the next Chapter.

2.4.2 Numerical Solutions of Point Kinetics Equations

The most general solution of the point kinetics equations can be obtained using the numerical integration. In this section two cases will be considered: the first case, when the reactivity is a linear function of time, and the second case, when the reactivity is a sine function of time.

The linear (ramp) and sinusoidal changes of reactivity have important practical applications. The ramp change of reactivity takes place when control rods are moved with constant velocity under normal reactor operation. The reactivity can be given by the following function,

(2.90) ( ) tt aρρ = ,


56

where aρ is the rate of increase of reactivity. Typical changes of n(t) for this type of

reactivity increase are shown in FIGURE 2-7,

FIGURE 2-7. Time-variation of n(t) for the ramp change of reactivity.

The curves in FIGURE 2-7 suggest the following form of the solution,

(2.91) ( ) 2ln atBtn +≈ ,

or

(2.92) ( )2

atAetn ≈ ,

where A, B and a are proper constants. The above qualitative analysis indicates that

when the reactivity linearly increases, the power increases as 2

te , whereas, as it was

shown earlier, with step-change of reactivity, the power change is proportional to te .

FIGURE 2-8 shows a comparison of the relative power change in a reactor following the step and the ramp (linear) change of reactivity. In both cases the same reactivity equal to 0.0022 is introduced. For the step-insertion case, all the reactivity is inserted at time t = 0. For the ramp insertion case the reactivity is linearly increased from 0 to 0.0022 during time interval equal to ∆t = 6.6 s, and then it is kept constant. For a comparison, FIGURE 2-9 shows the power increase when the ramp insertion of the reactivity is performed 100-times faster, and the final reactivity 0.0022 is achieved after ∆t = 0.066 s. In this case the two solutions almost coincide, as it could be expected. It can be concluded from the figures that the step-change of reactivity is a good approximation for the case with the ramp insertion of reactivity, provided that the insertion rate is very fast. If the insertion rate is slow, the two solutions will differ and the step-change solution predicts higher power values during the first seconds of the transient.

ln n

t

23 aa ρρ > 12 aa ρρ > 1aρ


57

0 1 2 3 4 5 6 7 8 9 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Time, s

Re

lative

Po

we

r

Ramp insertion

Step insertion

FIGURE 2-8. Relative reactor power increase after ramp ( ( ) tt 31025.3 −⋅=ρ ) and step ( 0022.00 =ρ )

insertion of reactivity.

0 1 2 3 4 5 6 7 8 9 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Time, s

Re

lative

Po

we

r

Ramp insertion

Step insertion

FIGURE 2-9. Relative reactor power increase after ramp ( ( ) tt 325.0=ρ ) and step ( 0022.00 =ρ )

insertion of reactivity.

The sinusoidal change of reactivity is used for determination of the reactor transfer function using the so called pile oscillator. In this case the reactivity is the following function of time,

(2.93) ( ) tt ωρρ sin0= ,

where 0ρ and ω are the amplitude and the frequency of the reactivity oscillations,

respectively. Typical reactor power oscillations caused by harmonic reactivity oscillations are shown in FIGURE 2-10. The figure shows a comparison of solutions obtained with the exact one-group model and with the small-perturbation-approximation model. As can be seen the approximate solution can be considered correct only during the first period of time after the insertion of the reactivity (first 0.1 – 0.2 s), as it could be expected.


58

0 5 10 15 20 25 30 35 40 45 50-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time, s

Po

we

r

Exact solution

Small perturb. approx.

FIGURE 2-10. Power oscillation in a reactor subject to sine reactivity oscillation with 0022.00 =ρ and

14.3=ω rad/s. Comparison of the exact solution with the small-perturbation-approximation solution.

The small-perturbation approximation predicts power oscillations that have constant amplitude, constant mean value and constant frequency, equal to the frequency of oscillations of the pile oscillator. The solution obtained with the exact model indicates, however, that both the amplitude and the mean value of oscillations increase with time.

The influence of the reactivity amplitude on power oscillations is shown in FIGURE 2-11. It can be clearly seen, that the reactor power oscillations have a divergent character. The divergence rate increases with increasing amplitude of the reactivity oscillations.

0 5 10 15 20 25 30 35 40 45 50-2

0

2

4

6

8

10

12

Time, s

Po

we

r

Rho0 = 0.0022

Rho0 = 0.0065

FIGURE 2-11. Power oscillation in a reactor subject to sine reactivity oscillation with frequency

14.3=ω rad/s and two values of reactivity amplitude: 0022.00 =ρ and 0065.00 =ρ .

The figures indicate that the small perturbation approximation gives an oscillatory solution which agrees with the exact solution only for small time intervals right after the initiation of the perturbation. As a result the two solutions diverge from each other quite rapidly, already during the first cycle. Moreover, the small perturbation


59

approximation is not able to capture the long-term behavior of the reactor, since it predicts persistent, constant-amplitude power oscillations, whereas the exact solution is diverging with time.

2.5 Approximate Point Kinetics Models Even though a solution of the exact point kinetics equations can be obtained relatively easy using computers, in certain situations it is convenient and also enough to predict the kinetics only approximately. Three such approximations are discussed in the following sections.

2.5.1 The Prompt Jump Approximation

In the Prompt Jump Approximation (PJA) the rapid power change due to prompt neutrons is neglected, corresponding to taking 0=Λ in the point kinetics equations. In the full form, the equations are as follows,

(2.94) xn

Syx

dt

dx

ei

iiΛ

++Λ

+Λ

+Λ

−= ∑=

ρρβ

β 6

1

1,

(2.95) 6,...,1=−= iyxdt

dyiii

i λλ ,

Multiplying both sides of the first equation with Λ yields,

xn

Syx

dt

dx

ei

ii ρρββ +Λ+++−=Λ ∑=

6

1

Since 0=Λ , then

(2.96) ( )ρβ

ρβ

ρβ −+

−= ∑

=

6

1

1

i

ii ytx

The PJA consists of 6 ordinary differential equations with constant coefficients given by Eq. (2.95) and one algebraic equation describing the reactor excess power in terms of the concentration of the precursors of delayed neutrons.

Assuming one group of precursors, the equations are as follows,

(2.97) yxdt

dyλλ −= ,

(2.98) ( )ρβ

ρβ

ρβ −+

−= ytx

1.

The two equations can be combined into a single solvable ordinary differential equation with time-dependent coefficients,


60

(2.99) )()( taytadt

dy=− ,

ρβ

λρ

−=)(ta , 0)0( =y

Equation (2.99) has a closed-form solution as follows,

(2.100) ( ) ( ) ( )( )

∫ ∫=−∫

t dttadttadtetaety

t

0

''''''''0 .

Substituting the solution to Eq. (2.98) yields,

(2.101) ( ) ( ) ( )( )

ρβ

ρ

ρβ

β

−+∫

−= ∫

−∫t dttadtta

dtetaetxt

0

''''''''0 .

In particular, when

constta =−

=0

0)(ρβ

λρ,

the solution is as follows,

(2.102) ( )0

0

0

10

0

ρβ

ρ

ρβ

β ρβ

λρ

−+

−

−= −

t

etx .

The solutions given by Eq. (2.102) for various values of βρ0=r and for 108.0 −= sλ are shown in FIGURE 2-12.

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5

t [s]

x(t

) [-

]

FIGURE 2-12. Solutions obtained from PJA for different values of βρ0=r and using one group of

precursors with 108.0 −= sλ .

r=0.5

r=0.4

r=0.3

r=0.2

r=0.1

r=0.0


61

2.5.2 The Prompt Kinetics Approximation

The Prompt Kinetics Approximation (PKA) is somehow opposite to the PJA, since it neglects the influence of the delayed neutrons on the reactor kinetics, and considers only the prompt neutrons. The balance equation in the PKA is as follows,

(2.103) xn

S

dt

dx

e Λ++

Λ=

ρρ, ( ) 00 =x .

If there are no external neutron sources, the equation becomes,

(2.104) ( )1+Λ

= xdt

dx ρ.

This equation can be integrated as follows,

( )( )

∫∫ ′Λ

′=

+′

′ tx

tdt

x

xd

00 1

ρ,

and the following exact solution, valid for any reactivity function ρ(t’), is obtained,

(2.105) ( )( )

10

−=∫ ′

Λ

′ttd

t

etx

ρ

.

If, in particular, the reactivity is constant and equal to ρ0, the excess normalized reactor power will be given as,

( ) 10

−= Λt

etx

ρ

.

2.5.3 The Constant Delayed Neutron Source Approximation

The Constant Delayed neutron Source (CDS) is based on the assumption that the production of the delayed neutrons is constant and equal to the production at the beginning of the transient. In this approximation, the balance equations are as follows,

(2.106) xn

Syx

dt

dx

ei

iiΛ

++Λ

+Λ

+Λ

−= ∑=

ρρβ

β 6

1

1,

(2.107) ( ) 6,...,100 =−= iyx iii λλ .

Since x(0) = 0, then yi = 0 for i = 1,…,6. As a result, the first equation becomes,

(2.108) xn

Sx

dt

dx

e Λ++

Λ+

Λ−=

ρρβ.

The above equation can be solved using the initial condition x(0) = 0.

This equation can be solved analytically. For that purpose it is written in a more compact form as,


62

)()( tbxtadt

dx=+ , where:

Λ

−=

ρβ)(ta and

en

Stb +

Λ=

ρ)( .

For the initial condition x(0) = 0, the general solution is as follows,

(2.109) ( ) ( ) ( )( )

∫ ∫= ∫− t dttadttadtetbetx

t

0

''''''''0 .

In particular, assuming S = 0 and const== 0ρρ , the solution is given as,

(2.110)

−

−= Λ

−− t

etx0

1)(0

0

ρβ

ρβ

ρ.

The solutions are shown for various positive values of βρ0=r in FIGURE 2-13

and for negative values in FIGURE 2-14. As can be seen, the solutions represent with good accuracy the details of the prompt jump, whereas for increasing time, the solutions are converging to constant values. The latter is due to the neglect of the delayed neutrons.

The solution given with Eq. (2.110) demonstrates the influence of time-variation of delayed neutrons on reactor kinetics. If the concentration of precursors were not changing with time, insertion of any positive reactivity smaller than β would result with a limited increase of the neutron flux. However, if the inserted reactivity is larger than β, the neutron flux will exponentially increase with time to infinity. For ρ0>β the reactor becomes prompt critical, that is the prompt neutrons are enough to sustain the chain reaction.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

t [s]

x(t

) [-

]

FIGURE 2-13. Solutions obtained from CDS for different positive values of βρ0=r and s001.0=Λ .

r=0.5

r=0.4

r=0.3

r=0.2

r=0.1


63

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 1 2 3 4 5

t [s]

x(t

) [-

]

FIGURE 2-14. Solutions obtained from CDS for different negative values of βρ0=r and s001.0=Λ .


B buckling C concentration of delayed neutron precursors D diffusion coefficient E neutron energy F neutron production operator k effective multiplication factor l average neutron lifetime

L diffusion length, aDL Σ=

M migration and neutron removal operator n neutron density; amplitude function R reactor core radius s Laplace transform parameter S external neutron source t time T delayed neutrons to fission neutrons operator v mean neutron velocity x non-dimensional point kinetics variable z coordinate

β fractional yield of the delayed neutrons

φ neutron flux

λ decay constant ν total neutron yield per fission ρ reactivity

Λ average neutron generation time ψ weighting function

ω frequency

r=-0.1

r=-0.2

r=-0.3

r=-0.4

r=-0.5


64

aΣ macroscopic cross section for neutron absorption

fΣ macroscopic cross section for fission

Ω neutron direction angle

Subscript 0 critical or steady-state reactor conditions e equilibrium i pertinent to i-th group ∞ infinite medium

Superscripts * adjoint

R E F E R E N C E S

[2-1] Ott, K.O. and Neuhold, R.J., Introductory Nuclear Reactor Dynamics, ANS public., La Grange Park, Illinois, USA, 1985.

[2-2] Glasstone, S. and Sesonske, A., Nuclear Reactor Engineering, 3rd Ed., Van Nostrand Reinhold Company, 1981.

E X E R C I S E S

EXERCISE 2-1: Based on the point kinetics model implemented in Scilab, write your own program to solve the point kinetics equations with both 1 and 6 delayed-neutron precursors. Assume reactivity and external sources as given functions of time. Assume Λ ,

iλ and iβ to be known, user-specified

constants. Use two methods: (a) direct time integration of differential equations and (b) Laplace-transformed solution of small perturbation approximation equations.

EXERCISE 2-2: Apply models from EXERCISE 2-1 to solve EXAMPLE 2-1XX. Compare your solution with the solution given in XEXAMPLE 2-1X.

EXERCISE 2-3: Apply models from EXERCISE 2-1 to solve XEXAMPLE 2-1X but assume that the reactivity has been made 0.0022 negative. All other data are the same.

EXERCISE 2-4: Apply models from EXERCISE 2-1 to solve XEXAMPLE 2-1X. Compare two solutions: with one group and six groups of precursors. Use data from XTABLE 2-3X.

EXERCISE 2-5: Apply models from EXERCISE 2-1 to find the neutron density as a function of time n(t) in a reactor where the reactivity was step-changed from βρ 100 −= to βρ 5.91 −= . Weighted mean

source intensity is S = 105 s-1. Assume one group of delayed neutrons with 11.0 −= sλ , 3107 −⋅=β .

Assume s310−=Λ .

EXERCISE 2-6: Apply models from EXERCISE 2-1 to solve XEXAMPLE 2-2. Compare the numerical solution with the analytical solution obtained in XEXAMPLE 2-2X.

EXERCISE 2-7: Apply models from EXERCISE 2-1 to solve XEXAMPLE 2-3X. Compare the numerical solution with the analytical solution obtained in XEXAMPLE 2-3X.


65

EXERCISE 2-8: A constant neutron source was suddenly introduced into a critical reactor. Some time after the source introduction, it was observed that the neutron flux in the reactor increases linearly with

the rate 108 m-2 s-1 /s. Determine the mean weighted neutron source S , assuming one group of the delayed neutrons with the following parameters: λ = 0.1 s-1, β = 7.5 10-3, Λ = 10-4 s.

EXERCISE 2-9: During the reactor start-up the reactivity was step-changed from βρ 150 −= to

βρ 101 −= . The mean weighted neutron source was S = 106 s-1 and the generation time was equal to

Λ = 10-4 s. Assuming one-group of delayed neutrons with λ = 0.1 s-1 and β = 7.0 10-3, derive the

expression for the neutron density as a function of time.

EXERCISE 2-10: Determine the reactivity value 1ρ knowing that the ratio of the mean values of

neutron fluxes in a sub-critical reactor with βρ 100 −= and 1ρ is equal to 1.3.

EXERCISE 2-11: A reactor oscillator (that is a rod made of strong neutron absorber moving harmonically in the reactor core) was used to determine the transfer function of a reactor operating at low power. Determine the reactor power response to the sine change of the reactivity ( ) tt ωρρ sin0= .

Assume β = 7.5 10-3, λ = 0.1 s-1, Λ = 10-3 s, βρ 05.00 = , f = 0.01 Hz.

67

3 Nuclear Reactor Dynamics

uclear reactor dynamics is concerned with reactor power transients in which the reactivity feedback caused by the reactivity change due to the power change plays an important roll in over-all reactor behavior. In this Chapter the reactor dynamics equations will be derived and several example

solutions of reactor dynamics problems will be shown.

3.1 Reactivity Feedbacks When power changes in a nuclear reactor are large enough to influence the value of the reactivity, the transient behavior of the reactor is termed as the nuclear reactor dynamics. As can be expected, the influence of power on the reactivity has to be quantified in order to properly describe the dynamic behavior of the nuclear reactor.

3.1.1 Influence of Fuel and Moderator Temperature on Reactor Operations

During operation of a nuclear reactor the energy released due to nuclear fissions is transferred to the coolant. The resulting temperature distribution in the fuel and coolant (in BWRs even the void fraction distribution) is a subject of the thermal-hydraulic analysis of the nuclear reactor. The temperature distribution, which in a general case is a function of both the time and the location, is influencing the values of microscopic cross-sections for various nuclear reactions caused by neutrons. As a result the reactivity will depend on the temperature changes.

From a practical point of view it is important to know what the influence of temperature on reactivity is, and how it will influence the operation of the nuclear reactor. In general the following two cases can be considered:

• Reactivity increases with the temperature: in this case the increasing reactivity will cause the increase of the reactor power, which, in turn, will cause the increase of temperature, etc. It means that in this case the reactor will be inherently unstable.

• Reactivity decreases with temperature: in this case the decreasing reactivity will cause the decrease of the reactor power which will be followed by the decrease of the temperature, and so on. Clearly the reactor will be inherently stable in such a case.

The conclusion is that reactors should be constructed in such a way which assures the decreasing reactivity in function of increasing reactor power. This can be achieved by using proper materials and a proper nuclear reactor configuration.

Chapter

3

N

C H A P T E R 3 – R E A C T O R D Y N A M I C S A N D S T A B I L I T Y

68

The reactivity changes with temperature because the reactivity depends on macroscopic cross sections, which themselves involve the atomic number densities of materials in the core,

(3.1) ),(),(),( ttNt rrr σ=Σ .

The atomic density N(r,t) depends on the reactor power level because:

a) material densities depend on temperature T,

b) the concentration of certain nuclei is constantly changing due to neutron interactions (poisons and fuel burnup).

The microscopic cross section is given in Eq. (3.1) as an explicit function of the spatial location r and the time t. This must be done since the cross sections that appear in one-speed diffusion model are actually averaged over energy spectrum of neutrons that appear in the reactor core. Since this spectrum is itself dependent on the temperature distribution in the core and hence the reactor power level, this dependence must be taken into account in Eq. (3.1).

The reactivity variation with the temperature is the principal feedback mechanisms determining the inherent stability of a nuclear reactor with respect to short-term fluctuations in the power level. In principle, the reactivity feedback could be evaluated by solving heat transfer equations, both in fuel and coolant regions and predicting the temperature distribution in the reactor core for a given reactor power. However, such approach would lead to a very complex system of partial nonlinear differential equations. Typical simplification is based on the so-called “lumped-parameter” model in which the major parts of the reactor core are represented by a single averaged value of temperature, such as an average fuel temperature, moderator temperature and coolant temperature.

The subject of the reactor dynamics analysis is to accommodate the core average temperatures such as TF (fuel) and TM (moderator) in suitable models of reactivity feedback. To this end one can write the reactivity change as a sum of two contributions,

(3.2) )()()( Ptt fext δρδρρ += .

The reactivity in Eq. (3.2) is measured with respect to the equilibrium power level P0 (for which the reactivity is just equal to zero) and is a superposition of the “externally” controlled reactivity insertion (for example, such as by the control rod movement) and “internal” reactivity change due to inherent feedback mechanisms, indicated here as a function of the power level.

For steady-state power level P0, Eq. (3.2) becomes,

(3.3) )(0 00 Pfρρ += ,


69

which merely states that to sustain the criticality of the system one has to supply

external reactivity 0ρ to counteract the negative feedback reactivity )( 0Pfρ . The

incremental reactivities appearing in Eq. (3.2) are then defined as,

(3.4) 0)()( ρρδρ −= tt extext ,

(3.5) )()()( 0PPP fff ρρδρ −= .

The feedback mechanism in reactivity is schematically shown in FIGURE 3-1XX.

extδρ

fδρ

Neutron

kinetics

extδρ Incremental power

Σ +

-

Feedback

mechanisms

FIGURE 3-1. Reactivity feedback mechanisms.

As can be seen, the output signal from the system, which in this case is the incremental power, affects the input signal, which is the reactivity. For stability analysis, it is interesting to investigate the feedback signal characteristics (gain and phase shift), since this will determine the over-all stability of the closed-loop system.

3.1.2 Doppler Effect

With increasing material temperature, the nuclei will move with increasing speed. Since the nuclei move in a chaotic manner, their relative velocity against a mono-energetic neutron flux will no longer be constant and will have some distribution. This is equivalent to a situation in which neutrons would have a certain energy distribution when approaching stationary nuclei. This is the so-called Doppler effect, in analogy to a similar phenomenon known in acoustics and optics.

Due to the Doppler effect, the number of nuclear reactions caused by mono-energetic neutrons will depend on the temperature of the material. Without going into details it can be mentioned that with the increasing material temperature the microscopic absorption cross section will increase (e.g. of U-238) resulting in decreasing reactivity. Also the microscopic fission cross-section will decrease with the temperature leading to additional reduction of the reactivity.

One can estimate the influence of the Doppler effect on reactivity using the expression for the resonance escape probability as,

(3.6)

Σ⋅

⋅−=

s

F INp

ξexp ,

where for the metallic uranium and the uranium dioxide fuel at 300K temperature, the effective resonance integral I is given as,


70

(3.7) [ ] [ ]bM

AIb

M

AI UOU 5.8445.4;5.8195.2

2+=+= .

The temperature dependence of the integral I is described by the following correlation obtained from experimental data,

(3.8) ( )[ ]3001)300()( −+= TKITI β ,

where, for 238UO2:

(3.9)

×+×= −−

M

A33 107.4101.6β .

A and M in Eqs. (3.7) and (3.9) are area (in m2) and mass (in kg), respectively, of a fuel rod.

3.1.3 Reactivity and Reactivity Coefficients

When the effective multiplication factor k remains constant from one generation of neutrons to another, it is possible to determine the number of neutrons at any particular generation by knowing the number of neutrons at “zero” generation, N0, and the value of k. Thus, after n generations the number of neutrons is equal to

nkNN 0= .

In particular, if in the preceding generation there are 0N neutrons, then there are

kN0 neutrons in the present generation. The change of the number of neutrons

expressed as a fraction of the present number of neutrons is referred to as reactivity and is expressed as,

(3.10) k

k

kN

NkN 1

0

00 −=

−=ρ .

As can be seen, reactivity is a dimensionless number; however, since its value is often rather small, artificial units are defined. From the definition given by Eq. (3.10), the value of reactivity is in units of kk /∆ . Alternative units are % kk /∆ and pcm (percent milli-rho). The conversions between these units are as follows,

(3.11) pcm10%1001 5=∆=∆ kkkk ,

(3.12) pcm1001.0%1 3=∆=∆ kkkk ,

(3.13) kkkk ∆=∆= −− %1010pcm1 35.

EXAMPLE 3-1. Calculate the reactivity in a reactor core when k is equal to 1.002 and 0.998. SOLUTION: The reactivity is as follows: for k= 1.002, ρ = (1.002-1/1.002 = 0.001996 ∆k/k = 0.1996 % ∆k/k = 199.6 pcm. For keff = 0.998: ρ = (0.998-1/0.998 = 0.002 ∆k/k = 0.2 % ∆k/k = 200 pcm.


71

Other units often used in reactor analysis are dollars ($) and cents. By definition, 1$ is reactivity which is equivalent to the effective delayed neutron fraction, β , and, as can

be expected, one cent (1c) reactivity is equal to one-hundredth of a dollar.

As already mentioned, the dependence of the reactivity on temperature has an important influence on the reactor dynamics and stability. In particular, a reactor will be stable when the reactivity is a decreasing function of temperature. Needless to say that the dependence of the reactivity on various parameters should be known. However, it is not possible (nor necessary for most practical purposes) to estimate such functions with all details. Instead a linearized form of the function is applied to evaluate the reactivity change due to various parameter changes, that is,

(3.14)

...

...,...),,(

+++=

+∂

∂+

∂

∂+

∂

∂≅

M

M

TF

F

TC

C

T

M

M

F

F

C

C

MFC

TTT

TT

TT

TT

TTT

δαδαδα

δρ

δρ

δρ

δρ,

where M

T

F

T

C

T ααα ,, are the coolant, fuel and moderator temperature coefficient of

reactivity, respectively. These coefficients play important role in safe operation of nuclear reactors.

A single temperature coefficient of reactivity can be defined as a partial derivative of the core reactivity with respect to temperature,

(3.15) ∑∑ =∂

∂→

∂

∂≡

j

j

T

j j

TTT

)(αρρ

α .

Here j indicates that separate temperatures in the reactor (j = C for coolant, j = F for fuel, etc.) are taken into account.

The two dominant temperature effects in most reactors are the change in resonance absorption (Doppler effect) due to fuel temperature change and the change in the neutron energy spectrum due to changing moderator or coolant density (due to temperature, pressure or void fraction changes).

Noting that,

(3.16) T

k

kT

k

kTT

∂

∂≅

∂

∂=

∂

∂≡

112

ρα ,

one obtains,

(3.17) M

T

F

T

MF

TT

k

kT

k

kααα +=

∂

∂+

∂

∂=

11.

Now changes in fuel temperature TF do not affect the shape of the thermal neutron energy spectrum. It should be mentioned, however, that fuel and moderator temperature effects can not always be separated. In LWRs a change in moderator temperature will significantly change the moderator density, thereby influencing


72

slowing down and hence resonance absorption. In spite of such interference, it is customary to analyze both coefficients separately.

3.1.4 Fuel Coefficient of Reactivity

The fuel temperature coefficient has an important influence on reactor safety in case of a large positive reactivity insertion. A negative fuel temperature coefficient is generally considered more important than a negative moderator temperature coefficient. The reason is that the negative fuel coefficient starts adding negative reactivity immediately, whereas the moderator temperature cannot turn the power rise for several seconds.

This coefficient is also called the prompt reactivity coefficient or the fuel Doppler reactivity coefficient. Its value can be readily obtained from Eq. (3.6) as,

(3.18)

FFF

F

TdT

dI

Ip

dT

dp

pdT

dk

k

1ln

11===α .

Using Eq. (3.8) in X(3.18) yields,

(3.19)

F

F

TTKp 2)300(

1ln

βα

−= .

3.1.5 Moderator Coefficient of Reactivity

When the moderator is at the same time used as a coolant (this is the case in LWRs), the moderator coefficient of reactivity will in principle reflect the influence of the coolant density changes on the reactivity.

The dominant reactivity effect in water-moderated reactors arises from changes in moderator density, due to the thermal expansion of the coolant fluid or due to the void formation. The principal effect is usually the loss of moderation that accompanies a decrease in moderator density and causes a corresponding increase in resonance absorption. It can be calculated as follows,

(3.20)

M

M

sM

F

MMM

M

TdT

dN

pI

N

N

dT

d

pdT

dp

pdT

dk

k

=

−===

1lnexp

111

σξα .

Since dNM/dTM is negative and may be quite large, particularly if the coolant temperature is close to the saturation temperature, the reactivity coefficient can also be large.

Typical values of reactivity coefficients are given in TABLE 3-1X.

TABLE 3-1. Reactivity coefficients[3-1].

Type of coefficient BWR PWR HTGR LMFBR

Fuel Doppler (pcm/K) -4 to -1 -4 to -1 -7 -0.6 to -2.5

Coolant void (pcm/%void) -200 to -100 0 - -12 to +20


73

Moderator (pcm/K) -50 to -8 -50 to -8 +1.0 -

Expansion (pcm/K) ~0 ~0 ~0 -0.92

3.2 Point Dynamics Model of PWR A simple point dynamics model of PWR should take into account the influence of the fuel and the moderator temperature on the reactivity. Since the moderator - which is at the same time the reactor coolant - is not boiling, the moderator reactivity feedback will result from the density changes caused by temperature changes of single-phase coolant. Thus, the simplest point dynamics model of PWR should take into consideration the time variations of the fuel and coolant temperature.

3.2.1 Derivation of Point Dynamics Equations

The equations of point-reactor dynamics model consist of the point-reactor kinetics equations complemented with a system of equations to describe the non-nuclear (thermal-hydraulic) reactor parameters. In general, the equations are as follows,

(3.21) ∑=

+

Λ

−=

6

1i

iiCndt

dnλ

βρ,

(3.22) iiii Cn

dt

dCλ

β−

Λ= , i = 1, …, 6,

(3.23) ( )KfC uuut ,...,,)( 210 ρρρρ ++= ,

(3.24) ( )mKk

k wwuuunfdt

du,...,;,...,,, 121= .

The two first equations are the point kinetics equations, which have already been discussed in the previous Chapter. It should be noted that the source term S is neglected in the point reactor dynamics equations since it is usually small compared to other terms. The third equation expresses the dependence of the reactivity on various

parameters: 0ρ is the initial reactor reactivity with withdrawn control rods and in

“cold” condition, )(tCρ is the reactivity change due to the control system (control

rods, fluid absorbents, etc.) and fρ is the reactivity change due to non-nuclear

parameter (e.g. temperature) changes.

Since non-nuclear parameters uk, k=1, …, K do not change much in respect to the steady-state (equilibrium) values, Eq. (3.23) can be written as,

(3.25) k

K

k

kC ut ∑=

++=1

0 )( αρρρ .

There are various possible formulations of Eq. (3.24) leading to a variety of point dynamics models. In the simplest case of a point dynamics model suitable for PWRs,


74

the influence of the coolant and the fuel temperature is taken into account, and the corresponding equations are as follows,

(3.26) CCCC Tna

dt

dTγ−= ,

(3.27) FFFF Tna

dt

dTγ−= ,

(3.28) F

F

TC

C

TC TTt ααρρρ +++= )(0 .

The above model may be improved by a more accurate treatment of the heat transfer between coolant and fuel using the Newton’s equation of cooling. The heat balance for fuel can be written as,

(3.29) ( )CFF

FpFF TThna

dt

dTcm −−= ,

and the corresponding heat balance for coolant is as follows,

(3.30) ( ) ( )CinCexpCCCF

CpCC TTcWTTh

dt

dTcm −−−= ,

where mF and mC are the mass of fuel and coolant in the core, respectively, cpF and cpC are specific heat of fuel and coolant, h is the heat transfer coefficient between fuel and coolant, WC is the coolant mass flow rate [kg/s] and TCex and TCin are the coolant exit and inlet temperature, respectively. Assuming further that,

(3.31) ( )CinCexC TTT +=

2

1,

the model is closed, that is, the number of unknowns is equal to the number of equations. The complete system of equations for the point dynamics model is summarized in XTABLE 3-2.

TABLE 3-2. Point reactor dynamics model.

Reactor kinetics equation

∑=

+

Λ

−=

6

1i

iiCndt

dnλ

βρ

Delayed-neutron precursor equations: i = 1,…,6

ii

ii Cndt

dCλ

β−

Λ=

Fuel heat balance ( )

CFFF

pFF TThnadt

dTcm −−=


75

Coolant heat balance ( ) ( )CinCpCCCF

C

pCC TTcWTThdt

dTcm −−−= 2

Reactivity changes F

F

TC

C

TC TTt ααρρρ +++= )(0

The point dynamics model is schematically illustrated in FIGURE 3-2.

FIGURE 3-2. Schematics of the point reactor dynamics model with reactivity, coolant inlet temperature and coolant mass flow rate as external forcing functions.

To solve the point dynamics equations it is necessary to specify the initial conditions. For that purpose, the reactor equilibrium condition will be determined.

3.2.2 Nuclear Reactor at Equilibrium

Equilibrium state of a nuclear reactor is described by Eqs. (3.21) through (3.24) in which the time derivatives are equal to zero and the reactivity is constant. In particular, using the model given in XTABLE 3-2, the equilibrium is described by the following system of algebraic equations,

(3.32) 06

1

=+

Λ

−∑

=iiiCn λ

βρ,

(3.33) 0=−Λ

iii Cn λ

β,

(3.34) ( ) 0=−− CFF TThna ,

(3.35) ( ) ( ) 02 =−−− CinCpCCCF TTcWTTh ,

(3.36) F

F

TC

C

T TT ααρρ ++= 0 .

Two cases have to be considered. If n = 0 then Ci = 0, TC = TF = TCi and this type of equilibrium corresponds to a shut-down reactor. When 0≠n then 0=ρ and the

following expressions are obtained (index e means the “equilibrium” value),

Tcin(t), Wc(t) TF

TC

Tcex

)(tCρ


76

(3.37) e

i

iei nC

Λ=

λ

β, ,

(3.38) ( ) 0,, =−− eCeFeF TThna ,

(3.39) ( ) ( ) 02 ,,,,, =−−− eCineCpCeCeCeF TTcWTTh ,

(3.40) eF

F

TeC

C

T TT ,,0 ααρρ ++= .

Solution of the system of equations yields,

(3.41) ( )

( )

++

++−=

hcWa

Tn

F

TF

T

C

T

pCeC

F

eCin

F

T

C

T

e

ααα

ααρ

,

,0

2

1, e

i

iie nC

Λ=

λ

β,

(3.42) eF

pCeC

eCineF nahcW

TT

++=

1

2

1

,

,, ,

(3.43)

pCeC

eFeCineC

cW

naTT

,

,,2

+= .

The above equations determine steady-state operation conditions of a nuclear reactor.

3.2.3 Normalized Point Reactor Dynamics Model

Introducing normalized variables,

(3.44)

e

e

n

nnx

−= ,

(3.45) 6,...,1, =−

= iC

CCy

ie

ieii ,

(3.46)

eF

eFF

FT

TTz

,

,−= ,

(3.47)

eC

eCC

CT

TTz

,

,−= ,

and substituting them into equations in XTABLE 3-2X the following normalized equations of point reactor dynamics are obtained,

(3.48) xyxdt

dx

i

iiΛ

+Λ

+Λ

+Λ

−= ∑=

ρρβ

β 6

1

1,


77

(3.49) 6,...,1=−= iyxdt

dyiii

i λλ ,

(3.50) C

eFpFF

eC

F

pFFeFpFF

eFF zTcm

hTz

cm

hx

Tcm

na

dt

dz

,

,

,

+−= ,

(3.51) ( )

wuTm

TWzw

m

Ww

Tm

TTW

uTm

TWz

cm

hWcz

Tcm

hT

dt

dz

eCC

eCineC

C

C

eC

eCC

eCineCeC

eCC

eCineC

C

pCC

eCpC

F

eCpCC

eFC

⋅+⋅−−

−++

−=

,

,,,

,

,,,

,

,,,

,

,

222

22

,

(3.52) FeF

F

TCeC

C

TC zTzTt ,,)( ααρρ ++= ,

with zero initial conditions.

Equation (3.51)) contains two “forcing functions” defined as,

(3.53)

eCin

eCinCin

T

TTu

,

,−= ,

(3.54)

eC

eCC

W

WWw

,

,−= .

The forcing functions are the dimensionless coolant inlet temperature, u(t), and the dimensionless coolant flow rate, w(t).

The derived model is represented by a set of non-linear ordinary differential equations with time-dependent coefficients and one additional algebraic equation describing the total reactivity. Such model can be solved using a numerical approach only. An example of the numerical implementation of the model using the the Scilab environment is presented below.

COMPUTER PROGRAM: Point Dynamics Model

function [dy]=

PDModel(t,y,yield,dconst,LAMBDA,ne,cpF,cpC,mF,mC,WCe,aF,h,alph

aF,alphaC,TCine,TCe,TFe)

// Point Dynamics model

// dy - returned RHS values

// t - time

// y - vector of unknown functions

// yield - yields of the precursors of delayed neutrons

// dconst - decay constants of the precursors of delayed neutrons

// LAMBDA - generation time

// ne - neutron density at equilibrium

// cpF - fuel specific heat

// cpC - coolant specific heat

// mF - mass of fuel

// mC - mass of coolant

// WCe - coolant flow at equilibrium

// aF - power-to-neutron-density factor

// h - fuel-coolant heat flux coefficient

// alphaF - fuel (Doppler) reactivity coefficient

// alphaC - moderator/coolant reactivity coefficient


78

// TCine - coolant inlet temperature at equilibrium

// TCe - coolant temperature at equilibrium

// TFe - fuel temperature at equilibrium

//

// Set parameters

//

NPre = length(yield);

neq = 3 + NPre; // Total number of equations

rho = InReactivity(t);

u = InCoolTemp(t);

w = InCoolFlow(t);

//

// Calculate Right-Hand-Sides of differential equations

//

dy = zeros(neq,1); // Initialize RHS

rho = rho + alphaC*TCe*y(3) + alphaF*TFe*y(2);

dy(1) = (-sum(yield)*y(1) + yield*y(4:$) + rho + rho*y(1))/LAMBDA;

dy(2) = (aF*ne*y(1)/TFe - h*y(2) + h*TCe*y(3)/TFe)/(mF*cpF);

dy(3) = (h*TFe*y(2)/TCe - (2*cpC*WCe+h)*y(3))/(mC*cpC) ...

+ 2*WCe*(TCine*u - (TCe-TCine)*w)/(mC*TCe) ...

- 2*WCe*w*(y(3) - TCine*u/TCe)/mC;

for i = 1:NPre

dy(i+3) = dconst(i)*(y(1) - y(i+3));

end

endfunction

The above Scilab function can be used with any ordinary differential equation solver to find the solution. For that purpose a dedicated solver script is provided: PDODESol(). In addition three forcing functions have to be supplied: InReactivity(t) – which gives the external reactivity as a function of time, InCoolTemp(t) – which gives the inlet coolant temperature as a function of time and InCoolFlow(t) – which provides the coolant mass flow rate as a function of time. The usage of the point dynamics model implemented in Scilab is elucidated in the example below.

EXAMPLE 3-2. Predict the neutron density change in a reactor following a positive step change

of reactivity with 0.0022 at time t = 1 s. Use the following reactor data: s001.0=Λ , λ = 0.1 s-1, β = 0.0075, cpF = 200 J kg-1 K-1, cpC = 4000 J kg-1 K-1, mF = 40000 kg, mC

= 7000 kg, WC,e = 8000 kg s-1, TCin,e = 550 K, aF = 7x106 J m3 s-1, ne = 200 m-3, F

Tα = -10-5 K-1, C

Tα = -5x10-5 K-1, h = 4x106 J K-1 s-1. Plot the calculated neutron

density as a function of time in the time interval from 0 to 5 s.

SOLUTION: The reactivity time change is given in the InReactivity() function as follows: function [rho] = InReactivity(t)

if t < 1

rho = 0.;

else

rho = 0.0022;

end

endfunction The other forcing functions have default zero value and do not need to be updated. The input parameters are specified at the Scilab prompt as follows: ->yield=7.5e-3; dconst=0.1; LAMBDA=1.e-3; ne=200; cpF=200; cpC=4000; mF=40000;

mC=7000; WCe=8000; aF=7e6; h=4e6; alphaF=-1e-5; alphaC=-5e-5; TCine=550; Two input parameters are still missing: the equilibrium fuel and coolant temperatures. These parameters can be found from Eqs. (3.42) and (3.43), and are specified in Scilab as follows: ->TFe=TCine+aF*ne*(1/(2*WCe*cpC)+1/h);


79

->TCe=TCine+aF*ne/(2*WCe*cpC);

The vector of time instances at which the solution will be found is defined as: ->t=linspace(0,5,100); Finally, the model solver is invoked as follows: ->PDODESol(t,yield,dconst,LAMBDA,ne,cpF,...

cpC,mF,mC,WCe,aF,h,alphaF,alphaC,TCine,TCe,TFe,’blue’); The calculated neutron density change with time is shown in FIGURE 3-3.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0200

210

220

230

240

250

260

270

280

Time, s

ne

, 1

/m^3

FIGURE 3-3. Neutron density change in a reactor following a sudden insertion of positive reactivity

0.0022 at time t = 1 s.

3.2.4 Reactor Dynamics in Presence of Small Perturbations

The point dynamics model can be written in a short form as,

(3.55) );( uxFx

=dt

d, 0)0( =x ,

where x is the vector of unknown functions, which for one-group approximation is as follows,

(3.56) [ ]TFC zzyx ,,,=x ,

u is the vector of the forcing functions,

(3.57) [ ]TC wu,,ρ=u

and F represents the right-hand-sides in the point dynamics model.

The fixed (or equilibrium) point of the reactor is represented by vectors x0 and u0, which satisfy the equation,

(3.58) 0);( 00 =uxF .


80

The present model has been formulated in terms of the excess normalized variables, using the equilibrium point as the reference. Due to that the fixed point of the model is given as,

(3.59) [ ]T00000 =x , and [ ]T0000 =u .

One of the important questions concerning the operation of nuclear reactors is their behavior in the vicinity of the fixed point, since typical power reactors stay in this condition during the most part of their lifetime. In particular, one should investigate whether the reactor is stable around the fixed point.

The behavior of a dynamical system around any point (and in particular around the fixed point) can be investigated using the first method of Lyapunov. For the present model the linearization can be easily achieved by dropping the product terms

Czwwux ⋅⋅ ,,ρ and the equations will be transformed into a set of linear equations

with constant coefficients,

(3.60) Λ

+Λ

+Λ

−= ∑=

ρβ

β 6

1

1

i

ii yxdt

dx,

(3.61) 6,...,1=−= iyxdt

dyiii

i λλ ,

(3.62) C

eFpFF

eC

F

pFFeFpFF

eFF zTcm

hTz

cm

hx

Tcm

na

dt

dz

,

,

,

+−= ,

(3.63) ( )

wTm

TTW

uTm

TWz

cm

hWcz

Tcm

hT

dt

dz

eCC

eCineCeC

eCC

eCineC

C

pCC

eCpC

F

eCpCC

eFC

,

,,,

,

,,,

,

,

2

22

−

−++

−=

,

(3.64) FeF

F

TCeC

C

TC zTzTt ,,)( ααρρ ++= .

The Jacobian matrix for the one-group approximation is as follows,

(3.65)

+−

−

−ΛΛ

−

=∂

∂

pCC

eCpC

eCpCC

eF

eFpFF

eC

pFFeFpFF

eF

cm

hWc

Tcm

hT

Tcm

hT

cm

h

Tcm

na

,

,

,

,

,

,

200

0

00

00

λλ

ββ

x

F.

The eigenvalues of the Jacobian are as follows,


81

(3.66) 01 =s ,

(3.67)

( ) eCFCpFpCeCFpFpCFpFCpC

FpFCpCC

eC

FpFCpC

WmhmccWmccmhcmhc

mcmcm

W

mc

h

mc

hs

,

22

,

,

2

82

2

1

22

−+⋅+⋅

×−−−−=,

(3.68)

( ) eCFCpFpCeCFpFpCFpFCpC

FpFCpCC

eC

FpFCpC

WmmhccWmccmhcmhc

mcmcm

W

mc

h

mc

hs

,

22

,

,

3

82

2

1

22

⋅−+⋅+⋅

×+−−−=,

(3.69) λβ

−Λ

−=4s .

The obtained expressions indicate that the Lyapunov exponents are always real and negatives for the second, the third and the fourth eigenvalue, whereas it is always zero for the first one.

3.2.5 Frequency Domain Analysis of Point Dynamics Model

The linearized point-dynamics model can also be investigated in the frequency domain. Equations (3.60) through (3.64) can be Laplace transformed and the following transfer functions can be derived,

(3.70) )(ˆ

)(ˆ)(

s

sxsH

Cρ= ,

(3.71) )(ˆ

)(ˆ)(

su

sxsL = ,

(3.72) )(ˆ

)(ˆ)(

sw

sxsM = ,

(3.73) )(ˆ

)(ˆ)(

su

szsN F= .

Transfer functions (3.70) through (3.73) describe the nuclear reactor response to various forcing functions.

Transfer function H(s) describes the nuclear reactor power response to a change of the external reactivity. The block diagram of a reactor undergoing this type of forcing function is shown in XFIGURE 3-4.


82

FIGURE 3-4. Block diagram of a reactor with a perturbation of the external reactivity.

The transfer functions GR(s), HF(s) and HC(s) describe the reactor response, the fuel feedback and the coolant feedback, respectively. The transfer functions can be derived from Eqs. (3.60) through (3.64) in the following way.

Performing the Laplace transformation of the point dynamics equations yields:

(3.74) Λ

+Λ

+Λ

−= ∑=

)(ˆ)(ˆ

1)(ˆ)(ˆ

6

1

ssysxsxs

i

ii

ρβ

β,

(3.75) 6,...,1)(ˆ)(ˆ)(ˆ =−= isysxsys iiii λλ ,

(3.76) )(ˆ)(ˆ)(ˆ)(ˆ,

,

,

szTcm

hTsz

cm

hsx

Tcm

naszs C

eFpFF

eC

F

pFFeFpFF

eFF +−= ,

(3.77) ( )

)(ˆ2

)(ˆ2

)(ˆ2

)(ˆ)(ˆ

,

,,,

,

,,

,

,

,

swTm

TTWsu

Tm

TW

szcm

hWcsz

Tcm

hTszs

eCC

eCineCeC

eCC

eCineC

C

pCC

eCpC

F

eCpCC

eF

C

−−

++

−=

,

(3.78) )(ˆ)(ˆ)(ˆ)(ˆ,, szTszTss FeF

F

TCeC

C

TC ααρρ ++= .

The transfer function GR(s) can be readily obtained from Eqs. (3.74) and (3.75) as follows,

)(ˆ sCρ )(ˆ sρ

)(ˆ sTρ

)(ˆ sFρ

)(ˆ sCLρ

)(ˆ sx

)(ˆ

)(ˆ)(

s

sxsGR

ρ=

)(ˆ

)(ˆ)(

sx

ssH F

F

ρ=

)(ˆ

)(ˆ)(

sx

ssH CL

C

ρ=


83

(3.79)

+Λ+Λ

=

∑=

6

1

11

1)(

i i

i

R

ss

sG

λ

β.

Laplace transforms )(ˆ szF and )(ˆ szC can be expressed in terms of )(ˆ sx solving Eqs.

(3.76) and (3.77) and noting that 0)(ˆ)(ˆ == swsu . Equation (3.77) yields,

(3.80) ( ) )(ˆ2

)(ˆ,,

,sz

hWcscmT

hTsz F

eCpCpCCeC

eF

C++

= .

Substituting Eq. (3.80) into (3.76) yields,

(3.81) )(ˆ

2

)(ˆ

,

2

,

sx

hWcscm

hhscmT

nasz

eCpCpCC

pFFeF

eFF

++−+

= .

Since )(ˆ)(ˆ, szTs FeF

F

TF αρ = , the following expression for HF(s) is obtained:

(3.82)

++−+

==

hWcscm

hhscm

na

sx

szTsH

eCpCpCC

pFF

eF

F

TFeF

F

T

F

,

2

,

2

)(ˆ

)(ˆ)(

αα.

In a similar manner the coolant feedback transfer function can be obtained as,

(3.83)

( ) )(2

)(

)(ˆ

2)(ˆ

)(ˆ)(

,

,

,

,

,

,

,,

sHhWcscm

h

sx

szT

cm

hWcs

Tcm

hT

T

T

sx

szTsH

F

eCpCpCC

F

T

C

T

FeF

F

T

pCC

eCpC

eCpCC

eF

eF

F

T

eC

C

TCeC

C

T

C

++

=+

+

==

α

α

α

α

αα

.

The block diagram shown in FIGURE 3-4X can be replaced with a simplified one shown in FIGURE 3-5.

FIGURE 3-5. Simplified block-diagram of a reactor subject to a perturbation of external reactivity.

)(ˆ sCρ )(ˆ sx

)(ˆ

)(ˆ)(

s

sxsH

Cρ=


84

The transfer function of the simplified block-diagram H(s) can be obtained as,

(3.84) [ ] )()()(1

)(

)(ˆ

)(ˆ)(

sGsHsH

sG

s

sxsH

RCF

R

C +−==

ρ.

3.3 Point Dynamics Model of BWR In BWR the major moderator feedback is caused by the change of the density due to boiling and the change of vapor content in the moderator. However, under low-void conditions, the feedback will also be caused by the density change due to temperature variations in the single-phase regions of the moderator/coolant.

3.3.1 Formulation of BWR Dynamics Model

A simple model of BWR dynamics, which contains the essential properties of BWR behavior has been formulated as follows[3-2],

(3.85) ( ) ( ) ( ) ( ) ( )

Λ++

Λ

−=

ttytx

t

dt

tdx ρλ

βρ,

(3.86) ( ) ( ) ( )tytx

dt

tdyλ

β−

Λ= ,

(3.87) ( ) ( ) ( )tzatxa

dt

tdz21 −= ,

(3.88) ( ) ( )t

dt

tdtρ

ρα = ,

(3.89) ( ) ( ) ( ) ( )tatatzk

dt

tdt

tαρρ

ρ43 −−⋅= .

The first two equations of the model describe the neutronic behavior of the reactor, where x(t) is the excess neutron density normalized to the steady-state neutron density, and y(t) is the excess delayed neutron precursor concentration, also normalized to the initial steady-state precursor concentration. The third equation represents the time behavior of the excess average fuel temperature, z, and the last two equations describe the time dependence of the excess void reactivity feedback. The total reactivity change is given as,

(3.90) ( ) ( ) ( )tzttF

Tαρρ α += .

As can be seen, the total reactivity feedback consists of the void and the fuel temperature reactivity feedback.

The model given by Eqs. (3.85) through (3.90) has been applied to an analysis of the stability test 7N at the Vermont Yankee reactor[3-4]. The recommended[3-2] constants in the model are given in the Table below.


85

TABLE 3-3. Parameters in the BWR dynamics model valid for Vermont Yankee test 7N.

Parameter Value Units

a1 25.04 K s-1

a2 0.23 s-1

a3 2.25 s-1

a4 6.82 s-2

k0 -3.70 x 10-3 K-1 s-2

F

Tα -2.52 x 10-5 K-1

β 0.0056 -

Λ 4.00 x 10-5 s

λ 0.08 s-1

The parameter k, which is proportional to the void reactivity coefficient and the fuel heat transfer coefficient, determines the gain of the feedback and, thus, defines the linear stability of the reactor point model. The value k0 given in the Table is the critical value above which the model becomes unstable.

3.3.2 Nuclear Reactor at Equilibrium

The full power (fixed) operating point is obtained by setting time derivatives in Eqs. (3.85) through (3.89) equal to zero. That is,

(3.91) ( ) ( )tytx λβ

+Λ

−=0 ,

(3.92) ( ) ( )tytx λβ

−Λ

=0 ,

(3.93) ( ) ( )tzatxa 210 −= ,

(3.94) ( )ttρ=0 ,

(3.95) ( ) ( ) ( )tatatzk t αρρ 430 −−⋅= .

In addition, the total reactivity must be equal to zero,

(3.96) ( ) ( )tztF

Tαρα +=0 .

The full power point corresponds then to the following state variable vector,


86

(3.97) [ ] [ ]TT

tzyx 0,0,0,0,0,,,,00 == ρραx .

The Jacobian matrix of the system is as follows,

(3.98)

−−

−

−Λ

Λ

−

=∂

∂

34

21

00

10000

000

000

000

aak

aa

λβ

λβ

x

F.

The eigenvalues of the matrix are as follows,

(3.99) 01 =λ , 22 a−=λ , 2

4 4

2

33

4,3

aaa −±−=λ ,

Λ

Λ+−=

λβλ5 .

Substituting the parameters from

TABLE 3-3, the following eigenvalues are obtained,

(3.100) 01 =λ , 23.02 −=λ , j357.2125.14,3 m−=λ , 08.1405 −=λ .

As can be seen only the first eigenvalue has a non-negative real part. Using the constants given in

TABLE 3-3 and applying an external reactivity perturbation:

(3.101) ( ) ( )ttex πρ 10cos0022.0= , for 0 < t < 0.1 s, otherwise ( ) 0=texρ

a limit cycle solution is obtained, as shown in FIGURE 3-6.

0 20 40 60 80 100 120 140 160 180 200-2

0

2

4

6

8

10

12

Time, s

No

rm.

Exce

ss V

ari

ab

le

Power

Fuel temperature


87

FIGURE 3-6. Reactor normalized excess power and fuel temperature as a function of time after step-insertion of reactivity; k = -3.7e-3.

0 2 4 6 8 10 12-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Fuel temperature

Po

we

r

FIGURE 3-7. Phase-space representation of limit cycle after step-insertion of reactivity; k = -3.7e-3.

To investigate the behavior of the model, the coefficient k is first changed to -3.5×10-3 and then to -3.9×10-3. The model response is shown in FIGURE 3-8 through FIGURE 3-11.

0 20 40 60 80 100 120 140 160 180 200-1

0

1

2

3

4

5

6

7

8

Time, s

No

rm.

Exce

ss V

ari

ab

le

Power

Fuel temperature

FIGURE 3-8. Development of power and fuel temperature as a function of time with k=-3.5×10-3.


88

0 1 2 3 4 5 6 7 8-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fuel temperature

Po

we

r

FIGURE 3-9. Phase-space representation of the model response with k=-3.5×10-3.

0 20 40 60 80 100 120 140 160 180 200-2

0

2

4

6

8

10

12

14

16

Time, s

No

rm.

Exce

ss V

ari

ab

le

Power

Fuel temperature

FIGURE 3-10. Development of power and fuel temperature as a function of time with k=-3.9×10-3.


89

-2 0 2 4 6 8 10 12 14 16-1.0

-0.5

0.0

0.5

1.0

1.5

Fuel temperature

Po

we

r

FIGURE 3-11. Phase-space representation of the model response with k=-3.9×10-3.

In the case of k = -3.5×10-3 the amplitude of oscillations is decreasing and the system is returning to the equilibrium point with zero excess power and fuel temperature. When k is changed to -3.9×10-3, the oscillation amplitude is growing and the system undergoes the limit cycle oscillations.

3.4 Nonlinear Effects For some special cases it is possible to compare the linear and non-linear solutions and evaluate the reactor stability using analytical methods in both cases. As an example, consider a simplified version of the point-dynamics model with one-loop feedback. The delayed neutrons will be taken into account indirectly by assuming

λβλβ ≈+Λ→Λ . It will be assumed that T(t) is an over-all average reactor

temperature and the rate of change of T(t) in the reactor is proportional to the deviation of the reactor power P(t) from its equilibrium, steady state value P0. Thus, the system can be described as follows,

(3.102) )()(

)()()( 0 tP

tTtP

t

dt

tdP

Λ

−+=

Λ

−=

βαρβρ,

(3.103) 0)()(

PtPdt

tdTM C −= ,

with initial conditions:

P(t=0) = P0, and T(t=0) = T0.


90

Here MC is the system total mass multiplied by the average heat capacity of fuel and

coolant and the system reactivity )(tρ has only two parts: constant term 0ρ and the

temperature feedback term )(tTα . Introducing non-dimensional variables

[ ] 00 /)()( TTtTt −=ϑ , 0/)()( PtPt =θ and 0/ tt=τ into Eqs. (3.102) and (3.103)

yields,

(3.104) 1)0(),()()(

=Λ′

+= θτθ

ταϑκ

τ

τθ

d

d,

(3.105) ( ) 0)0(,1)(1)(

=−= ϑτθµτ

τϑ

d

d.

Here new constants have been introduced to reflect the variable change.

Equations (3.104) and (3.105) can be solved analytically. Dividing both equations with each other yields,

(3.106) 1)(

)()(

)(

)(

−Λ′

+=

τθ

τθταϑκµ

τϑ

τθ

d

d,

or

(3.107) )()(

)()(

1)(τϑ

ταϑκµτθ

τθ

τθdd

Λ′

+=

−.

Both sides of Eq. (3.107) can be integrated as follows,

(3.108) 1)(ln)()(

)()()(

)(

1)(C

ddd +−=−=

−∫ ∫∫ τθτθ

τθ

τθτθτθ

τθ

τθ,

(3.109) 2

2)(2

1)()(

)(Cd +

+

Λ′=

Λ′

+∫ ταϑτκϑ

µτϑ

ταϑκµ .

Thus, the solution of Eq. (3.107) reads as follows,

(3.110) C+Λ′

+=−

µταϑτκϑτθτθ 2)(

2

1)()(ln)( .

The constant C can be found from the initial conditions:

(3.111) 1)0(2

1)0()0(ln)0( 2 =⇒+

Λ′

+=− CC

µαϑκϑθθ .

Thus the final solution of the nonlinear problem is as follows,

(3.112) 1)(ln)()(2

1)( 2 −−=

Λ′

+ τθτθ

µταϑτκϑ .


91

If now )(tϑ is plotted as a function of )(tθ for different values of κ , closed orbits,

looking like distorted circles, will be obtained, as shown in FIGURE 3-12X. A simple analysis shows that as κ increases, both )(tϑ and )(tθ remain finite for all values of

time t. Both these quantities oscillate but remain limited and do not diverge as ∞→t .

FIGURE 3-12. Limit cycle oscillations of reactor power and temperature.

Equations (3.104) and (3.105) can be linearized and solved using e.g. the Laplace transform approach. Thus, a direct comparison of the linear and nonlinear model is possible.

3.5 Nuclear Reactor Stability The stability of nuclear reactors can be analyzed by using two main approaches:

• the time-domain approach, when the reactor dynamics equations are solved in time domain and the stability property of the reactor is evaluated based on the calculated power oscillations,

• the frequency-domain approach, in which the reactor-dynamics equations are linearized around a fixed point and the stability properties of the system are investigated based on the location of poles of the transfer function.

The first approach is quite straightforward, providing that a proper numerical model for the reactor dynamics is available. The advantage of this approach is that all nonlinear effects are included, and the predictions are valid even far from the fixed (unperturbed steady-state) point. The disadvantage is usually the high computational cost.

rel. power

rel . temp. difference

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

k = 1.0e-3

k = 4.0e-3

k = 6.0e-3

( )tϑ

( )tθ


92

The second approach is valid only for small perturbations, thus it can be used for the prediction of the instability threshold, but not to predict the nonlinear behavior of the system far from the equilibrium point. The main advantage of the frequency domain approach is that it is fast running and computationally inexpensive. The frequency domain approach is outlined in the following sections.

3.5.1 Open Loop Transfer Functions

The Laplace transform representation is particularly useful in the development of the important property of a system called the transfer function (see Appendix A). In a general sense, the transfer function is a mathematical expression which describes the effect of a physical system on the signal transferred through it. For a system shown in XFIG. 2-4X, the transfer function G(s) is defined as a ratio of the Laplace transform of the output signal, Y(s) to the Laplace transform of the input signal, U(s); G(s) = Y(s)/U(s).

G(s)

U(s) Y(s)

FIGURE 3-13. Open loop system.

3.5.2 Closed Loop Transfer Functions

The system shown in XFIGURE 3-13 represents a so-called open-loop system, that is, a system without feedback. A system with feedback, also called as a closed loop system, is shown in FIGURE 3-14X.

G(s) U(s) Y(s)

Σ

H(s)

FIGURE 3-14. Closed loop system with feedback.

G(s) represents the system (forward) transfer function and H(s) is the feedback transfer function.

Performing a summation of signals at the input to the system yields,

(3.113) [ ] )()()()()( sYsGsYsHsU =⋅+ .

Thus, the transfer function for the system with feedback (closed loop transfer function) becomes,

(3.114) )()(1

)(

)(

)()(

sGsH

sG

sU

sYsGT

⋅−=≡ .


93

Both the forward and multiple feedback reactor transfer functions have been derived in previous sections.

3.5.3 Stability of Zero-Power Reactor

Reactor kinetics equations represent a typical example of an open-loop system. The input signal is then the reactivity and the output signal is the neutron flux. The transfer function, referred usually as a zero-power reactor transfer function, is defined as,

(3.115) )(ˆ

)(ˆ)(

s

sxsG

ρ= .

Using the linearized reactor point kinetics equations derived in the previous Chapter

(with dropped non-linear terms, which holds for small perturbations and βρ 1.0< ),

the transfer function of the zero power reactor is obtained as,

(3.116)

+Λ+

⋅Λ

==

∑=

6

1

11

11

)(ˆ

)(ˆ)(

i i

i

ss

s

sxsG

λ

βρ.

The gain ( )ωjG and the phase shift angle ( ))(arg ωjG of the transfer function are

shown in XFIGURE 3-15X and XFIGURE 3-16,X respectively.

10−2

10−1

100

101

102

−25

−20

−15

−10

−5

0

5

10

15

20

Frequency, 1/s

Gain

, decib

ele

s

Λ=10−3

s

Λ=10−4

s

Λ=10−6

s

FIGURE 3-15. Gain plot for various neutron generation time values.

The gain plot (often called the Bode diagram) indicates that an open-loop reactor system tends to be unstable as frequency becomes small, since the gain becomes infinite when frequency approaches zero. Thus, a reactor without feedback is intrinsically unstable.


94

10−2

10−1

100

101

102

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency, 1/s

Ph

ase

an

gle

, d

eg

ree

s

Λ=10−3

s

Λ=10−4

s

Λ=10−6

s

FIGURE 3-16. Phase plot for various neutron generation time values.

3.5.4 Stability of Closed Loop Systems

Reactor dynamics equations represent a typical example of a closed loop system. This is due to the presence of multiple feedbacks which exist in this case, as discussed in previous sections.

It is interesting to consider a special case of the open loop in which,

(3.117) )()()()( sUsHsGsU =⋅⋅ .

In this case the system will be self-excited if the feedback loop is closed. There will be no need for any external input signal since the feedback signal is in-phase with the external input. Equation (3.117) can be written as,

(3.118) [ ] 0)()()(1 =⋅− sUsGsH .

Since the input perturbation is arbitrary, the condition for the instability is,

(3.119) 0)()(1 =⋅− sGsH .

Equation (3.119) is called the characteristic equation of the system and is the same as the denominator of the closed loop transfer function, given by Eq. (3.114).

One can observe that the roots of the characteristic function will be poles of the closed loop transfer function. If the closed loop transfer function has the form,

(3.120) as

sF−

=1

)( ,

the original f(t) of the transfer function F(s) is,

(3.121) atesFLtf == − )()( 1.

Transfer function F(s) has a pole at s = a, which corresponds to the zero value of its denominator s – a. If now Re(a) > 0 the function f(t) will diverge with time which


95

indicates unstable system. In other words, roots of the characteristic equation may imply exponential divergence in the time domain. The limiting case when Re(a) = 0 is the marginal stability case in which a perturbation causes the system to oscillate sinusoidally, but not diverge with time. Naturally system is stable when Re(a) < 0, since then any perturbation will damp out with time.

One can formulate the following stability criterion for a closed loop system:

“The necessary and sufficient condition for the closed loop system to be stable to small perturbations is that all the roots of the characteristic equation have negative real parts”.

If one can factor the closed loop transfer function using partial fraction expansion, the roots can be easily determined. This is the case when the closed loop transfer function has a form of a rational polynomial. In the case of complicated transcendental algebraic equations direct evaluation of roots is not trivial, however, and an approach using the Nyquist Criterion (see Appendix B) has proven to be an efficient way of investigation of the system stability.


C concentration of delayed neutron precursors D diffusion coefficient E neutron energy G reactor forward transfer function h heat transfer coefficient H reactor feedback transfer function

j imaginary unit 1− k effective multiplication factor l average neutron lifetime

L diffusion length, aDL Σ=

n neutron density; amplitude function s Laplace transform parameter S external neutron source t time T temperature W mass flow rate x excess normalized reactor power y excess normalized concentration of precursors z excess normalized reactor temperature

Greek

M

Tα moderator reactivity coefficient F

Tα fuel (Doppler) reactivity coefficient

β fractional yield of the delayed neutrons

φ neutron flux

λ decay constant ν total neutron yield per fission ρ reactivity


96

Λ average neutron generation time

aΣ macroscopic cross section for neutron absorption

fΣ macroscopic cross section for fission

Subscript 0 critical or steady-state reactor conditions e equilibrium i pertinent to i-th group of delayed neutrons F fuel M moderator ∞ infinite medium

R E F E R E N C E S

[3-1] Duderstadt, J.J. and Hamilton, L.J., Nuclear Reactor Analysis, John Wiley & Sons, Inc., 1980.

[3-2] March-Leuba, J., Cacuci, D.G. and Perez, R.B., “Nonlinear Dynamics and Stability of Boiling Water Reactors: Part 1 – Qualitative Analysis,” Nucl. Sci. Eng., Vol. 93, pp. 111-123, 1986.

[3-3] Ott, K.O. and Neuhold, R.J., Introductory Nuclear Reactor Dynamics, ANS public., La Grange Park, Illinois, USA, 1985.

[3-4] Sandoz, S.A. and Chen, S.F., Trans. Am. Nucl. Soc., Vol. 45, p. 727, 1983.

E X E R C I S E S

EXERCISE 3-1: Derive transfer functions of a detailed and a simplified block-diagram of a nuclear reactor undergoing perturbation of the inlet temperature of coolant.

Hint: The detailed diagram in this case is as shown in figure below. As can be seen the perturbation of the inlet coolant temperature u(s) is transformed in the block with transfer function Ku(s) and next is summed with temperature feedback effects and enters the reactor with transfer function GR(s) as the total reactivity change )(ˆ sρ . All transfer functions shown in figure can be derived from (3.74)

through (3.78) assuming )(ˆ sCρ = w(s) = 0.

FIGURE 3-17. Figure for XEXERCISE 3-1X. Block diagram of a reactor with a perturbation of the inlet

temperature of coolant.

)(ˆ sx )(sGR

)(sH F

)(sH C

Ku(s) )(ˆ su


97

EXERCISE 3-2: Derive transfer functions of a detailed and a simplified block-diagram of a nuclear reactor undergoing perturbation of the inlet mass flow rate of coolant.

Hint: The detailed diagram in this case is as shown in figure below. As can be seen the perturbation of the inlet coolant flow w(s) is transformed in the block with transfer function Kw(s) and next is summed with temperature feedback effects and enters the reactor with transfer function GRw(s) as the total reactivity change )(ˆ sρ . All transfer functions shown in figure can be derived from (3.74)X

through (3.78)X assuming )(ˆ sCρ = u(s) = 0.

FIGURE 3-18. Figure for XEXERCISE 3-2X. Block diagram of a reactor with a perturbation of the inlet

mass flow rate of coolant.

EXERCISE 3-3: Extend the program from EXERCISE 2-1 for modeling of the reactor dynamics problems. Assume that the reactivity, the coolant mass flow rate and the inlet coolant temperature are given functions of time. All other parameters are user-defined constants.

EXERCISE 3-4: Using the transfer functions H(s), L(s) and M(s) derived in this Chapter and in XEXERCISE 3-1X and XEXERCISE 3-2X, evaluate the stability of the following reactor: s001.0=Λ , λ = 0.1 s-1, β = 0.0075, cpF = 200 J kg-1 K-1, cpC = 4000 J kg-1 K-1, mF = 40000 kg, mC = 7000 kg, WC,e = 8000

kg s-1, TCin,e = 550 K, aF = 7x106 J m3 s-1, ne = 200 m-3, F

Tα = -10-5 K-1, C

Tα = -5x10-5 K-1, h = 4x106 J K-1

s-1. Perform the Nyquist plot and use the Nyquist criterion to evaluate the reactor stability. Find roots of the characteristic equation and evaluate the reactor stability based on roots values. Compare the two methods.

EXERCISE 3-5: Using the time-domain model of the reactor point dynamics developed in XEXERCISE 3-3, determine the time response of the reactor caused by a step change of reactivity. Use the same reactor data as in XEXERCISE 3-4X. The reactivity change is described as )(05.0)( ttC βθρ = , where )(tθ is

the step function. Plot the reactor power as a function of time.

EXERCISE 3-6: Using the time-domain model of the reactor point dynamics developed in XEXERCISE 3-3X, determine the response of a reactor caused by a step change of the inlet coolant temperature. Use the same reactor data as in XEXERCISE 3-4X. The inlet temperature change is described as

)(55)( ttTCin θδ = K, where )(tθ is the step function. Plot the reactor power as a function of time.

EXERCISE 3-7: Using the time-domain model of the reactor point dynamics developed in XEXERCISE 3-3, determine the response of the reactor caused by a step change of the inlet coolant mass flow rate. Use the same reactor data as in XEXERCISE 3-4X. The inlet coolant mass flow rate change is described as

)(1.0)( , tWtW eCC θδ = kg s-1, where )(tθ is the step function. Plot the reactor power as a function of time.

)(ˆ sx )(sGR

)(sH F

)(sH C

Kw(s) )(ˆ sw


98

99

4 Dynamics of Boiling

Systems ynamics of boiling systems has important implications on over-all dynamics of nuclear power plants. For that reason it is necessary to investigate the dynamic characteristic of channels with phase change. The first section of this Chapter is dealing with analytical solutions of transient two-phase flows

in boiling channels subject to sudden perturbations of inlet or heating conditions. The channel response is analyzed in time domain. The second section is devoted to a study of boiling channel stability. The governing two-phase flow equations are linearized and the channel stability is investigated in the frequency domain. In the third and last section of this Chapter a similar stability analyses are performed for boiling loops.

4.1 Analysis of Two-Phase Flow Transients The analysis of transients in boiling loops and systems involves the modeling of several coupled phenomena such as: single-phase and two-phase flows in heated channels and volumes, boiling heat transfer, phase separation and mixing, transient heat conduction in solid structures, effects of void fraction on power generation (e.g. due to neutronic feedback in BWRs), and others. In this section the basic equations that describe all these processes will be given.

The simplest description of dynamics of a boiling channel is usually based on a one-dimensional drift-flux or homogeneous two-phase flow model. The following conservation equations are used in the one-dimensional drift-flux model.

Mass conservation equation for the liquid phase,

(4.1) ( )[ ] ( ) AAjz

At

lll Γ−=∂

∂+−

∂

∂ραρ 1 .

Mass conservation equation for the vapor phase,

(4.2) [ ] ( ) AAjz

At

vvv Γ=∂

∂+

∂

∂ραρ .

Energy conservation equation for the liquid phase,

(4.3) ( )( )[ ] ( )Hllllll PqAji

zApi

t′′=

∂

∂+−−

∂

∂ραρ 1 .

Chapter

4

D

C H A P T E R 4 - D Y N A M I C S O F B O I L I N G S Y S T E M S

100

Energy conservation equation for the vapor phase,

(4.4) ( )[ ] ( )Hvvvvvv PqAji

zApi

t′′=

∂

∂+−

∂

∂ραρ .

Momentum conservation equation for the two-phase mixture,

(4.5)

( ) 0sin2

4

1

22

,

,2

2

=+

−+

+∂

∂+

∂

∂+

∂

∂

∑ ϕρρ

δφξφ

ρ

gG

zzD

C

z

pAG

zAt

G

m

li

iiloi

h

lof

lo

M.

Here the Dirac delta function )( izz −δ has been used to indicate positions of local

pressure losses. Equation (4.5X) employs the static and dynamic mixture densities, defined as,

(4.6) vlk

k

km αρρααρρ +−==∑ )1( ,

(4.7) ( )

( )

122

12

1

1−−

−

−+=

= ∑

αραραρρ

lvk kk

k

M

xxx.

The closure relationship for void fraction in the drift-flux model is as follows,

(4.8)

vj

v

UjC

j

+=

0

α .

Here,

(4.9)

v

v

Gxj

ρ= ,

and

(4.10) ( )

l

l

xGj

ρ

−=

1 ,

are superficial velocities of vapor and liquid phase, respectively, Uvj is the drift velocity and C0 is the void concentration parameter. The channel cross section area A in Eqs. (4.1)X through (4.5X) can be, in general, a given function of the axial distance z.

The transient two-phase flow model described by Eqs. (4.1)X through (4.10)X is quite complex and includes five conservation equations. Solution of such a system is quite a demanding task, and for some purposes a simple model can be used, where only three conservation equations are formulated and solved.


101

For the homogeneous equilibrium model (HEM) the conservation equations are as follows.

Mass conservation equation for the mixture:

(4.11) 0=∂

∂+

∂

∂

z

G

t

mρ.

Energy conservation equation for the mixture:

(4.12) ( )[ ] ( )

HmMm Pqz

AGi

t

Api′′−=

∂

∂+

∂

−∂ ρ.

Momentum conservation equation for the mixture:

(4.13)

( ) 0sin2

4

1

2

1

2

,

,2

2

=+

−+

+∂

∂+

∂

∂+

∂

∂

∑=

ϕρρ

δφξφ

ρ

gG

zzD

C

z

pAG

zAt

G

m

l

n

i

iiloi

h

lof

lo

m ,

Closure relationships for mixture thermodynamic enthalpy, im, and mixture center-of-mass enthalpy, iM, are as follows:

(4.14) ( ) xixii gfm +−= 1 ,

(4.15) ( )

m

ggff

M

iii

ρ

αραρ +−=

1.

Mixture densities are given by Eqs. (4.6)X and (4.7X), whereas the remaining notation is conventional.

A similar three-equation model can be obtained by assuming slip between phases and using, e.g., the drift-flux model equations as the starting point. In such model the original mass and energy conservation equations are replaced by modified continuity equation and the void propagation equation.

Assuming the thermodynamic equilibrium between liquid and vapor phases

( flgv ρρρρ == , , etc., where subscript g is used to indicate the saturated vapor and

subscript f the saturated liquid phase) and combining mass conservation equations (4.1XX) and (4.2) yields the following volumetric continuity equation:

(4.16) z

p

dp

dj

dp

dj

t

p

dp

d

dp

dv

z

j g

g

gf

f

fg

g

f

f

fg∂

∂

+−

∂

∂

+

−−Γ=

∂

∂ ρ

ρ

ρ

ρ

ρ

ρ

αρ

ρ

α1.

Combining energy conservation equations (4.3X) and X(4.4) yields,


102

(4.17)

( )

∂

∂

+−

∂

∂

−−−

+′′

=Γ

z

p

dp

dix

dp

diG

t

p

dp

di

dp

di

i

Ai

Pq

fgfg

g

f

f

fg

fg

H

αραρ 111

.

In addition, from mass conservation equations and using Eq. X(4.8) one can derive the void propagation equation,

(4.18) z

C

z

p

dp

dj

t

p

dp

d

zC

t

Kg

g

gg

gg

K∂

∂−

∂

∂−

∂

∂−

Γ=

∂

∂+

∂

∂α

ρ

ρ

ρ

ρ

α

ρ

αα,

where,

(4.19) gjK UjCC += 0 .

EXAMPLE 4-1: Find the two-phase mixture density as a function of the axial distance and time for a boiling channel with a step change in inlet velocity. Assume uniformly heated channel with saturated inlet conditions, homogeneous flow model and constant system pressure.

With this assumptions Eq. X(4.16) becomes,

(4.20) Ω=Γ=∂

∂fgv

z

j .

From Eq. X(4.11) one gets,

(4.21) ( )0=

∂

∂+

∂

∂+

∂

∂=

∂

∂+

∂

∂=

∂

∂+

∂

∂

zj

z

j

tz

j

tz

G

t

m

m

mmmm ρρ

ρρρρ ,

or,

(4.22) Ω−=∂

∂−=

∂

∂+

∂

∂mm

mm

z

j

zj

tρρ

ρρ .

Equation X(4.20) can be integrated as,

(4.23) )(),( tfztzj +Ω= ,

where f(t) is a function of time only. Its value can be found substituting z = 0 in Eq. X(4.23),

(4.24) )()(),0( tjtftj in== ,

and Eq. X(4.23) becomes,

(4.25) )(),( tjztzj in+Ω= .

Assuming that the inlet velocity is given as,

(4.26)

>

≤=

0

0)(

2

1

tforj

tforjtjin

.


103

and taking into account the equation for a trajectory of a particle moving with velocity j,

(4.27) ztjjdt

dzin Ω+== )( ,

one gets,

(4.28) ( )

( )[ ]

>−Ω

≤Ω

−Ω

−+

Ω=−Ω

Ω−Ω

0if1

0if),(

0

2

0

2121

0

0

0

tej

tj

ejj

ej

ttztt

ttt

,

for t > 0, where t0 is the particle entrance time (z(t0) = 0).

Solving Eq. X(4.22) along particle trajectories (i.e., along the characteristic dz/dt = j) and combining with

Eq. X(4.27) yields,

(4.29)

Ω+

Ω>

Ω+

Ω+

Ω≤≤

−−+

Ω

=

Ω

2

2

2

1

12

1

2

1

1ln1

if

1

1ln1

0for

),(

j

zt

j

z

j

zt

ej

jj

j

j

j

z

tzf

t

f

m ρ

ρ

ρ

.

From Eq. X(4.25) it is obvious that the step change of inlet velocity is immediately transferred along the whole channel. That is at time t1 = 0 the superficial velocity at the channel exit is equal to j(L,0) = j1 + ΩL, whereas at any time t2 > 0, this velocity will be j(L,t2) = j2 + ΩL. This is obviously a non-physical behavior caused by the fact that pressure changes are neglected.

An instant change of the superficial velocity in the whole channel does not mean an instant change of the mixture density, void fraction and enthalpy. All these properties will change gradually as the particles will move along the channel.

Assuming that a particle enters the channel at time t0 = 0, it will arrive at a position z1 in the channel at

time

Ω+

Ω=

2

1

1 1ln1

j

zt . From Eq. X(4.29) results that the mixture density at that location will start

changing between time t = 0 and time t = t1 and then it will remain constant.

Having expression given by Eq. X(4.29) for the mixture density, one can derive similar time-and-location dependent expressions for the void fraction and the thermodynamic mixture enthalpy, using Eqs. X(4.17)

and X(4.14), respectively. FIGURE 4-1X shows the time evolution of the mixture density in a boiling channel with 70 bar pressure, pipe diameter equal to 10 mm, heat flux equal to 6·105 W m-2, velocity of the saturated water at the inlet equal to 2 m s-1. At time t = 0 the inlet velocity is reduced to 1.5 m s-1. The evolution of the mixture density is shown at different location from the inlet to the channel. As can be seen, at z = 0.5 m the density of the mixture drops from the initial value of 363 kg m-3 to a new value of 310 kg m-3 within 0.22 s, whereas at z = 3.5 m density drop is from 90 to 70 kg m-3 within 0.6 s.


104

-0.2 0.0 0.2 0.4 0.6 0.8 1.050

100

150

200

250

300

350

400

Time, s

Density, kg/m

^3

z = 0.5 m

z = 1.0 m

z = 1.5 m

z = 2.0 m

z = 2.5 m

z = 3.0 m

z = 3.5 m

FIGURE 4-1. Mixture density evolution with time at different locations in a boiling channel: pressure 70 bar, pipe diameter 10 mm, heat flux 6·105 W m-2, inlet velocity 2 m s-1, reduced to 1.5 m s-1 at t = 0.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

100

200

300

400

500

600

700

800

Distance, m

Density, kg/m

^3

t = 0.0 s

t = 0.1 s

t = 0.2 s

t = 0.5 s

t = 1.0 s

FIGURE 4-2. Mixture density along axial distance at different time instances in a boiling channel: pressure 70 bar, pipe diameter 10 mm, heat flux 6·105 W m-2, inlet velocity 2 m s-1, reduced to 1.5 m s-1

at t = 0.

The mass flux of the mixture can be calculated as ( ) ( ) ( )tztzjtzG m ,,, ρ= . Substituting functions

given by Eqs. X(4.25) and X(4.29X), the result is as shown in


105

-0.2 0.0 0.2 0.4 0.6 0.8 1.01100

1150

1200

1250

1300

1350

1400

1450

1500

Time, s

Ma

ss f

lux,

kg

/m^2

.s

z = 0.5 m

z = 1.0 m

z = 1.5 m

z = 2.0 m

z = 2.5 m

z = 3.0 m

z = 3.5 m

FIGURE 4-3. Mixture mass flux evolution with time at different locations in a boiling channel: pressure 70 bar, pipe diameter 10 mm, heat flux 6·105 W m-2, inlet velocity 2 m s-1, reduced to 1.5 m s-1 at t = 0.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.01100

1150

1200

1250

1300

1350

1400

1450

1500

Distance, m

Ma

ss f

lux,

kg

/m^2

.s

t = -0.0 s

t = +0.0 s

t = 0.2 s

t = 0.5 s

t = 1.0 s

FIGURE 4-4. Mixture mass flux along axial distance at different time instances in a boiling channel: pressure 70 bar, pipe diameter 10 mm, heat flux 6·105 W m-2, inlet velocity 2 m s-1, reduced to 1.5 m s-1

at t = 0.

It is interesting to note that the mass flux change consists of two parts: the first drop is caused by the sudden change of the inlet velocity. This drop takes place in the whole channel at the time of the inlet velocity change (t = 0). The second drop is caused by the change of the mixture density, which is transported in the boiling channel with the flow velocity.


106

EXAMPLE 4-2: Perform similar analysis as in XEXAMPLE 4-1X but using the drift-flux model. All other assumptions are the same.

The governing equations are obtained from Eqs. X(4.16) and X(4.18) and, after taking into account the assumptions, will be as follows:

(4.30) Ω=Γ=∂

∂fgv

z

j ,

and

(4.31)

−Ω=

∂

∂−

Γ=

∂

∂+

∂

∂αα

ρ

αα0C

v

v

z

C

zC

t fg

gK

g

K

.

The boundary/initial conditions are as follows,

(4.32) ( ) 0,0 =tα ,

(4.33)

>

≤=

0for

0for)(

2

1

tj

tjtjin

.

Integration of Eq. X(4.30X) yields,

(4.34) ztjtzj in ⋅Ω+= )(),( .

Assuming that,

(4.35) zCCCdt

dzinKK Ω+== 0,

,

EEEquation (4.31) can be re-written as,

(4.36)

−Ω=≡

∂

∂+

∂

∂α

ααα0C

v

v

Dt

D

zdt

dz

t fg

g ,

where D/Dt is the total derivative.

Equation X(4.35) can be integrated and the particle trajectories can be obtained. If a particle enters the channel at time t0 < 0, then Eq. X(4.35) becomes,

(4.37) zCUjCdt

dzgj Ω++= 010

,

or,

(4.38) ∫ ∫=

Ω++⇒=

Ω++

z t

tgjgj

dtzCUjC

dzdt

zCUjC

dz

0 0100100

.

The integration is performed from the channel inlet (z = 0, t = t0) to a certain position (z,t<0) and it yields,

(4.39) ( ) )(lnln1

00

10

010

0

0

010

0

ttCUjC

zCUjCttzCUjC

C gj

gj

z

gj −Ω=

+

Ω++⇒−=Ω++

Ω

,


107

(4.40) [ ]1);()(

0

10

000 −

Ω

+= −Ω ttCgj

eC

UjCttz .

Equation (4.40) describes the particle path from the inlet at time t = t0 < 0 until t = 0. Substituting t = 0 into Eq. (4.40X) one can obtain the axial position of the particle at the time of the step change of the inlet velocity:

(4.41) [ ]1);0( 00

0

10

00 −Ω

+== Ω− tCgj

eC

UjCtzz .

To find the particle trajectory for t > 0 one has to integrate Eq. (4.35X) from z = z0 and t = 0 to certain location (z,t>0). Eq. X(4.35X) now becomes,

(4.42) zCUjCdt

dzgj Ω++= 020

,

and the solution is,

(4.43) ( ) tCzCUjC

zCUjCtzCUjC

C gj

gj

z

z

gj Ω=

Ω++

Ω++⇒−=Ω++

Ω0

0020

020

020

0

ln0ln1

0

,

(4.44) ( ) tCtCgjeze

C

UjCttz

ΩΩ +−Ω

+= 00

0

0

20

0 1);( .

Substituting z0 given by Eq. X(4.41X) into (4.44X) yields,

(4.45) ( ) [ ]

( )

Ω

+−

Ω

−+

Ω

+

=−Ω

++−

Ω

+=

Ω−Ω

ΩΩ−Ω

0

2012

0

10

0

10

0

20

0

000

0000 11);(

C

UjCe

jje

C

UjC

eeC

UjCe

C

UjCttz

gjtCttCgj

tCtCgjtCgj

.

Paths described by Eq.(4.45 X) are valid for t0 < 0. The solutions for t0 > 0 can be readily obtained from Eq. X(4.44X) substituting z0 = 0 and t = t – t0. The result is as follows,

(4.46) ( )[ ]1);( 00

0

20

0 −Ω

+= −Ω ttCgj

eC

UjCttz .

The void fraction distribution can be obtained by integration of Eq. X(4.36X) and using Eqs. X(4.45) with X(4.46X). The resultant expression is as follows,

(4.47)

>

+

Ω+

−

≤<

Ω

−−

Ω

++

+

Ω−

=

Ω

)(for

1

11

)(0for1

1

),(

*

20

00

*

12

0

20

10

00 0

ztt

UjC

zCCv

v

ztt

ejj

C

UjCz

UjC

CCv

v

tz

gj

fg

g

tCgj

gj

fg

g

α

where,

(4.48)

+

Ω+

Ω=

gjUjC

zC

Czt

20

0

0

* 1ln1

)(.


108

The mixture density can be found from Eq. (4.6X) and using expression for void fraction given by Eq. X(4.47X). In the same manner one can obtain the mixture enthalpy from Eq. X(4.14X) or (4.15).

EXAMPLE 4-3: Solve same problem as in XEXAMPLE 4-1X assuming subcooled liquid at the inlet to the channel.

The energy equation in the single phase flow region is,

(4.49) A

Pq

z

iG

t

i Hlll

′′=

∂

∂+

∂

∂ρ .

Since ljG ρ= , the equation becomes,

(4.50)

l

Hlll

A

Pq

z

ij

t

i

Dt

Di

ρ

′′=

∂

∂+

∂

∂≡ .

Integrating Eq. (4.50) along characteristics dz/dt = jin(t) yields,

(4.51) tA

Pqiti

l

Hll

ρ

′′=− )0()( .

Here i(0) = in is the inlet enthalpy, which is assumed to be constant. The residence time of liquid particle in the subcooled region can be found from Eq.(4.51 X) as,

(4.52) ( ) const1 =−′′

= inf

H

l iiPq

At

ρφ

.

Integrating Eq. (4.30X) for z > λ(t), where λ(t) is the boiling boundary yields,

(4.53) [ ])()(),( tztjtzj in λ−Ω+= .

Again solving along characteristics dz/dt = j(z,t) yields,

(4.54) [ ] [ ]∫+

−Ω−−ΩΩ−+−+=

t

tt

ttttdjeetjjtjttz

01

01 )()(),( 2

)()(

012120

φ

φ ττλτφ

,

when t > 0.

For a step change of the inlet velocity, Eq. X(4.54X) becomes,

(4.55)

( )

( ) ( )

( )

>>

Ω−+

Ω

≤<≤+

Ω−+−+

Ω

≤>

Ω−+

Ω

−+

Ω

=

−−Ω

−−Ω

−Ω−Ω−

0andfor1

0andfor1

0andfor1

),(

01122

010111121

0112121

0

01

01

10

ttttjej

ttttttjtjjej

ttttjejj

ej

ttz

ttt

ttt

ttt

φφ

φφφ

φφ

φ

φ

φ

Solving Eq. X(4.22) along trajectories z(t,t0),

(4.56) ( )0),( 0

tt

fm ett−Ω−= ρρ ,

and eliminating t0 from Eqs.(4.55 X) and X(4.56) yields the solution for ),( tzmρ .


109

EXAMPLE 4-4: Consider uniformly heated boiling channel subject to a step change in the wall heat flux. Other assumptions are the same as in XEXAMPLE 4-1.

Equations X(4.20X) and X(4.22X) are still valid with,

(4.57

>Ω

≤Ω=Ω

0for

0for

2

1

t

t .

Integrating the equation of the characteristics given by XEq. (4.27)X yields,

(4.58) ( )[ ] [ ]

( )[ ]

>−Ω

≤−Ω

+−Ω

=−Ω

ΩΩΩ−Ω

0if1

0if1

),(

0

2

0

210

02

22012

tej

tej

eej

ttzttin

tintttin

,

for t > 0. Next, integrating Eq. (4.22X) along the trajectories z(t,t0) and eliminating t0, the density variation

),( tzmρ can be obtained as,

(4.59)

Ω+

Ω>

Ω+

Ω+

Ω≤≤

Ω

Ω−Ω+

Ω

Ω+

Ω

=

Ω

in

in

f

int

in

f

m

j

zt

j

z

j

zt

ej

z

tz2

22

2

2

2

12

2

11

1ln1

if

1

1ln1

0for

),(

2

ρ

ρ

ρ

.

4.2 Flow Instabilities in Heated Channels Various thermal-hydraulic instabilities have been observed in two-phase flows. As already mentioned, instabilities are undesirable since they may degrade system control and performance, erode thermal margins and lead to mechanical damages. Nuclear reactors have to be designed and operated in such a way that sustainable instabilities are avoided. This requires a good understanding of the phenomena that govern various modes of instabilities in nuclear power plants as well as knowledge of stability margins, which are valid for particular system and operating conditions.

4.2.1 Classification of Instabilities

Two-phase instabilities can be conveniently classified into static instabilities, which can be explained in terms of steady-state laws, and dynamic instabilities, which require a consideration of the transient conservation equations.

Static Instabilities:

1. Excursive (i.e., Ledinegg) instabilities are non-periodic flow transients. Instabilities of this type plagued early low-pressure fossil boilers, since flow excursions could lead to burn-out of the boiler tubes. Ledinegg instability may occur in heated channels with low system pressure and low inlet loss coefficient, where pressure drop may decrease with increasing flow. If the pump characteristics has less negative slope, excursive instability will occur.

2. Flow regime relaxation instabilities are caused by the pressure-drop characteristics of the different flow regimes. For example, pressure drop in


110

slug flows is less then in bubbly flow with the same flow rates of gas and liquid. If a system is operating in the bubbly flow regime near the flow regime boundary, a small negative perturbation in liquid flow rate may cause a transition to slug flow. As a result, pressure drop in the channel will reduce. If the channel operates at a constant pressure drop condition (as in case of a large number of parallel channels), more liquid will enter the channel to satisfy the boundary conditions. This, in turn, may cause the system to return to bubbly flow regime.

3. Nucleation instabilities include bumping and geysering phenomena. These instabilities are characterized by a periodic relaxation of the metastable condition that builds up due to insufficient nucleation sites. In particular, if the liquid superheat builds up until the existing nucleation sites are activated, rapid boiling and expulsion of the resultant two-phase mixture may occur.

Dynamic Instabilities:

1. Density-wave oscillations can occur in both diabatic and adiabatic two-phase systems and in diabatic single-phase systems. Generally speaking, density wave oscillations are caused by the lag introduced into the thermal-hydraulic system by the finite speed of propagation of density perturbations. This type of instability is one of the most important and of practical concern in BWRs and will be discussed in more detail in the following part of this section.

2. Pressure-drop oscillations can occur in loops having a negative slope (similar to the situation described for the excursive instability) and containing a compressible volume (e.g. an accumulator). In such systems excursions may occur periodically.

3. Flow regime excited instabilities can occur when a particular flow regime, normally slug flow, induces a periodic disturbance in the system operating state. If this disturbance is at a frequency that is close to the natural frequency of the two-phase system, a resonance can occur.

4. Acoustic instabilities may occur in two-phase system having the proper combination of geometric characteristics and sonic speed. As in single-phase gas flows, organ-pipe-type standing waves can be set up when a pressure pulse propagates through two-phase mixtures flowing in a conduit. On reaching an area change or obstruction, the change in acoustic impedance causes a pressure pulse of opposite polarity to propagate in the opposite direction. If the excitation frequency and geometry of the conduit is such that an integral number of one-quarter wavelengths can fit within it, the standing waves may appear. Such acoustic-induced channel pressure drop oscillations of large amplitude have been observed for subcooled systems operating in the negative-slope region of the system pressure-drop versus flow curve.

5. Condensation-induced instabilities are known to lead to large water-hammer-type loads; however, their nature is not fully understood. A typical example is the so-called chugging phenomena that have been observed in the vent pipes of steam relief valves which are submerged in a liquid pool. When the steam first exits into the subcooled pool of liquid it is normally at a high enough


111

velocity to form a jet within the pool. However, as the steam flow rate drops off, the condensation rate in the pool may be large enough to completely collapse the steam jet, and cause a liquid slug to surge up into the discharge line. Subsequently, the steam can heat up the interface of the liquid slug to saturation, allowing the pressure of the discharging steam to increase such that it blows the slug back into the liquid pool. A cyclic process can occur with large inertial loads associated with the liquid slug motion being transmitted to the walls of the vessel containing the pool.

4.2.2 Density Wave Oscillations

To understand the mechanisms of density wave oscillations, consider an air/water flow channel connected to a tank filled with water, (see FIGURE 4-5X, depicting a system investigated by Svanholm and Friedly). Water is entering into the pipe through an inlet orifice and gas is constantly introduced into the pipe and mix with water at a certain distance downstream from the inlet orifice. To simplify the analysis, the following assumptions are made. The density of the air is negligible in relation to that of the water and the volume occupied by the water is negligible compared to that of the air. Finally, slip is also neglected.

FIGURE 4-5. Gas/liquid flow channel.

The airflow determines the velocity of the two-phase mixture and the density of the mix is proportional to that of the liquid. Assuming that the pipe has no losses except at the inlet and exit, the hydraulic head loss at the inlet orifice is,

(4.60) 2

2

lininin

Up

ρξ=∆ .

The pressure drop at the exit is,

(4.61) 2

2

exexexex

Up

ρξ=∆ .

Here Uin and Uex are mean inlet and exit velocities, respectively, ξin and ξex are the inlet

and outlet pressure loss coefficients, respectively, and exρ is the outlet (two-phase)

Gas inlet

Liquid

inp∆

exp∆

totp∆

Uex Uin


112

density. The constant pressure head, provided by the water tank, is always equal to the sum of these two pressure drops, i.e.

(4.62) ghppp lexintot ρ=∆+∆=∆ .

For example, opening the inlet orifice for a short time induces a density wave that propagates through the pipe and passes the exit orifice after a time τ. During the passage, the corresponding pressure drop at the exit increases temporarily. Since the total pressure drop always remains constant, the increased pressure drop over the exit will bring a corresponding pressure decrease at the inlet. This means that less water is sprayed into the channel creating a sudden decrease in density, which induces the wave that propagates through the channel. This dynamic process causes density wave propagation in the channel.

If the amplitude of the waves is converging the system is said to be stable and if the amplitude is diverging it is said to be unstable. These finite propagation times, τ, induce time-lag effects and phase-angle shifts between the channel pressure drop and the inlet flow, which may cause self-exciting oscillations.

In general, any increase in the frictional pressure drop in the liquid region has a stabilizing effect, since this pressure drop is in phase with the inlet flow and acts to damp the fluctuations. Inlet orificing can be used to stabilize an unstable flow. An increase in the two-phase region pressure drop has a destabilizing effect, since the pressure drop is out of phase with the inlet flow, due to the wave propagation time, τ. Thus, an exit flow restriction is a strongly destabilizing factor.

For a fixed channel geometry, an increase in inlet velocity has a stabilizing effect in terms of heat flux, because the extent of the two-phase flow region and the density change due to boiling are significantly reduced by the increase in inlet velocity. An increase in the system pressure has a stabilizing effect in terms of exit quality, since at higher pressure the density change due to phase change is less significant.

Density-wave instabilities can be further classified as follows:

1. Loop instabilities

2. Parallel-channel instabilities

3. Channel-to-channel instabilities

4. Neutronically-coupled instabilities

The most important modes of density-wave instabilities are loop and parallel-channel instabilities. The parallel-channel mode corresponds to a system of a big number of channels connected in parallel, in which a constant pressure condition governs flow through each of the channels. The principles of density-wave instability that are described in this section correspond just to this mode of instability. The loop instability is very similar; however, the boundary condition of zero pressure drop in the loop is imposed.


113

4.2.3 Methods of Analysis of Two-Phase Flow Instabilities

A rigorous stability analysis of boiling systems is difficult due to the nonlinear mathematical form of the underlying conservation equations and is only possible if some simplifying assumptions are made. In particular, if the threshold of instability is of interest, linearized models are often used. Such models are obtained by perturbing the governing equations around a given steady-state operating point. The linearized model is next Laplace-transformed and a frequency domain methodology is used to study the system stability.

While the linear stability analysis can be used to determine the instability threshold, this approach does not provide information concerning other characteristics of nonlinear systems, such as the magnitude and frequency of any limit cycle oscillations. For this purpose a non-linear stability analysis must be performed.

In general, the methods of linear analysis of two-phase flow instabilities consist of the following steps (so-called frequency-domain methodology):

1. Linearize governing equations around steady-state operating point 2. Obtain transfer functions 3. Examine properties of roots of characteristic equations

The methods of nonlinear analysis of the –phase instabilities are as follows:

1. Hopf’s bifurcation method

2. Method of Lyapunov

3. Harmonic quasi-linearization (describing function methods)

4. Fractals as measure of strange attractors (chaotic vibrations)

Nonlinear methods are beyond the scope of the present course and will not be discussed here (the interested reader may consult literature, e.g.,[4-1]). In what follows the basic principles behind the linear methods will be discussed.

4.2.4 Frequency Domain Methodology

Frequency domain methodology has been previously used to investigate the stability of a nuclear reactor. As a short remainder, a generic block diagram of a negative feedback-control system is shown in XFIGURE 4-6. This is a simple linear system with a single input and single output (SI/SO).

-

+G(s)

H(s)

In OutΣ

FIGURE 4-6. Block diagram of a feedback control system


114

The transfer function G(s) is called the forward loop transfer function, and H(s) the feedback loop transfer function. The output and input are related to each other as follows,

(4.63) [ ] )()()()()( sGsXsHsXsX outinout ⋅⋅−= ,

from which the output signal can be obtained as

(4.64) )()()(1

)()( sX

sHsG

sGsX inout ⋅

⋅+= .

The transfer function G/(1+GH) is called the closed loop transfer function and 1+GH=0 is the characteristic equation of the close loop. If any of the roots of 1+GH will appear in the right-halve of the complex plane, the closed system will be unstable.

Perturbation analysis

The first order perturbation of a time dependent function )(tf can be viewed as a first

order Taylor’s series expansion,

(4.65) )())((0

0 txx

fftxff δδ

∂

∂=−≡ ,

where 0f is the value of the unperturbed function. Consider a few examples:

Let axcxf 1)( = . The perturbed equation is,

00

1

01)(x

xa

f

fxxacxf

a δ=

δ⇔δ=δ −

Let dt

dxxf =)( : then,

(4.66) dt

xd

dt

dx )(δ=

δ .

Let dzzxGxf

xg

xg

∫=)(

)(

2

1

),()( : then, based on the Leibniz rule,

(4.67) 110220 ))(,())(,(),()(02

01

gxgxGgxgxGdzzxGxf

g

g

δ−δ+δ=δ ∫ .

Perturbations are quite similar to differential calculus. The only real difference is that perturbations are taken around steady state point and all variables not perturbed are denoted by a subscript zero to indicate the steady state value.


115

Derivation of perturbed equations

The time-dependent pressure drop in a boiling channel can be obtained from the integration of the momentum equation (e.g. Eq. X(4.5X) or X(4.13X)). The integration can be simplified by dividing the channel into two parts: a single-phase flow part, which is stretching from the inlet to the boiling boundary, located at distance λ from the inlet, and the two-phase flow part, stretching from the boiling boundary to the outlet from the channel, see XFIGURE 4-7X.

z = 0

λ=z

z = L

Two phase

Single phase

FIGURE 4-7. Single-phase and two-phase sections of a boiling channel.

Using the homogeneous equilibrium model, the pressure drop across each section is given as,

(4.68)

( )

∑∫∈

+

++∂

∂+

∂

∂

=−=∆

φρ

ξρρρ

λ

λ

φ

12

)(

2

41

)(

2

0

2,

2

1

Li l

iil

lh

lof

l

inH

zGdzg

G

D

C

z

G

t

G

ppp

,

for the single phase flow part and,

(4.69)

( )

∑∫∈

+

++

∂

∂+

∂

∂

=−=∆

φρ

ξφρρ

φρ

λ

λ

φ

22

)()(

2

4

)(

22

,

2,2

2

2

Li l

iiiilo

L

m

lh

lof

lo

M

exH

zGzdzg

G

D

CG

zt

G

ppp

H ,

for the two-phase flow part. Here λ is the position of the boiling boundary, φ1L is the

non-boiling length of the channel and φ2L is the boiling length of the channel.

Combining Eqs. X(4.68X) and X(4.69) with equations for mass and energy conservation, perturbing around a steady state point and then Laplace transforming yields the


116

following general expressions for the single-phase and two-phase pressure drop perturbations,

(4.70) ( )inHHHinHH

isqsjsp ˆ)(ˆ)(ˆ)( ,3,2,11 δδδδ φ Γ+′′′Γ+Γ=∆ ,

(4.71) ( )inHHHinHH

isqsjsp ˆ)(ˆ)(ˆ)( ,3,2,12 δδδδ φ Π+′′′Π+Π=∆ .

Here HHH ,3,2,1 and, ΓΓΓ are transfer functions of the perturbations of inlet velocity,

heat volumetric source and inlet enthalpy, respectively, on the perturbation of the

single-phase pressure drop. HHH ,3,2,1 and, ΠΠΠ are the corresponding transfer

functions for the perturbation of the two-phase pressure drop. Derivation of Eqs. X(4.70X) and X(4.71X), as well as expressions for transfer functions can be found in Appendix D.

Stability criterion

As an example, parallel channels with constant power and constant inlet enthalpy will

be considered. In such case, 0ˆˆ ==′′′inH iq δδ and since the total pressure drop is

constant,

(4.72) ( ) ( ) 021 =∆δ+∆δ φφ HHpp .

Combining Eqs. X(4.70X) and X(4.71X) with X(4.72) yields

(4.73) [ ] 0ˆ)()( ,1,1 =Π+Γ inHH jss δ .

In the case of parallel channel instabilities density-wave oscillations are manifested by self-sustained oscillations in the channel inlet flow rate caused by a feedback between the pressure drop perturbations in the single and two-phase portions of the channel. Using the technique of linear system control theory, the parallel channel stability can be considered as a feedback system. The appropriate block diagram is shown in XFIGURE 4-8.

1

,1

−Γ H H,1Π 1−

( )φδ

1Hp∆ ( ) φδ2Hp∆

totinj ,ˆδ

totinj ,ˆδ

extinj ,ˆδ

FIGURE 4-8. Block diagram for the parallel channel model.

The block diagram shown in this figure yields the following relationship between the

external perturbation extinj ,ˆδ and the system response, totinj ,

ˆδ ,


117

(4.74) ( )( )

extin

H

Htotin j

s

sj ,

,1

,1,

ˆ

1

1ˆ δδ

Γ

Π+

= .

It follows from this equation that the characteristic equation of the boiling channel is,

(4.75) ( )( )

01,1

,1=

Γ

Π+

s

s

H

H.

The necessary and sufficient condition for the boiling system under consideration to be stable to small perturbations is that all the roots of the characteristic equation, X(4.75)X have negative real parts.

In the case of complicated transcendental algebraic equations, such as those that occur in Eq. (4.75X), direct evaluation of roots is not trivial, and approach using the Nyquist Criterion (see Appendix B) has proven to be an efficient way of investigating system stability.

For boiling parallel channel,

(4.76) ( )( )s

ssG

H

H

,1

,11)(

Γ

Π+= ,

and its poles are equivalent to the combined poles of ( ) ( )[ ]ss HH ,1,1 Γ+Π and zeros of

( )sH,1Γ . From the shape of transfer functions ( )sH,1Π and ( )sH,1Γ is clear that all the

singularities of ( ) ( )[ ]ss HH ,1,1 Γ+Π are removable, whereas ( )sH,1Γ has no zeros

within C, thus, P = 0. Consequently, the parallel channel model derived herein is stable

if, and only if, the Nyquist plot of ( ) ( )ss HH ,1,1 ΓΠ does not encircle the point (-1,0).

4.2.5 Time Domain Approach

In addition to the frequency-domain approach, the time-domain approach is used in practical calculations of the nuclear reactor stability. In such calculations, the dynamic behavior of the system caused by a certain perturbation of input parameters is calculated as a function of time. XFIGURE 4-9X shows an example of the perturbation of an input parameter (this could be for instance the inlet subcooling) and an example of the output parameter (this could be for instance the reactor power).


118

FIGURE 4-9. Examples of a perturbation and the resulting answer.

The answer shown in XFIGURE 4-9X is typically obtained from an analysis performed with a transient code where both reactor kinetics and reactor thermal-hydraulics equations are solved simultaneously. If the answer has an oscillatory character then after a certain period of time the least damped eigenfrequency will dominate. The amplitude ratio, or the so-called decay ratio, A2/A1 is a practical measure how effective the damping is and how far from the instability the system is. For undamped system A2/A1 = 1. In such case the system is at the threshold of the instability. For decay ratio A2/A1 < 0.25 the system is well damped.

Stability map of a boiling channel

It is instructive to represent the dynamic behavior of a boiling channel on a so-called stability map. One such map proposed by Ishii and Zuber is shown in XFIGURE 4-10X. The map shows the region where the boiling channel is unstable in function of two non-dimensional parameters: the subcooling number and the phase-change number. The subcooling number is defined as follows,

(4.77) ( )( )

fgg

infgf

subi

iiN

ρ

ρρ −−= ,

and the phase-change number as,

(4.78) ( )

fggfin

HHgf

pchiAj

LPqN

ρρ

ρρ 0′′−

= .

Here LH is the heated length, PH is the heated perimeter, 0q ′′ is the heat flux, A is the

channel cross-section area, jin is the inlet velocity, iin is the inlet enthalpy and ifg is the latent heat.

Per

turb

atio

n

Answ

er

A1

A2

t

t


119

FIGURE 4-10. Stability map proposed by Ishii and Zuber.

It can be seen in XFIGURE 4-10X that stability boundary curve for higher inlet subcoolings is nearly parallel with the line of a constant exit quality xex, given by,

(4.79) ex

g

gf

pchsub xNNρ

ρρ −−= .

Ishii used this observation and derived a simple stability criterion for the high inlet subcoolings as follows,

(4.80) ( )

gf

g

ex

exin

exxρρ

ρ

ξϕ

ξϕξ

−

++

++≤

1

22.

The boiling channel is stable as long as Eq. X(4.80X) is satisfied. Here inξ and exξ are the

inlet and outlet loss coefficients, respectively and ϕ is the friction number given as,

(4.81)

h

Hf

D

LC

2=ϕ .

4.3 Instabilities in Heated Loops 4.3.1 Parallel Channel Instability

Parallel-channel system is often a part of a boiling loop. In particular, a core of a BWR is a multi-channel system composed of a number of parallel boiling channels (fuel assemblies) having common inlet and outlet plena. Clearly, all such channels must satisfy an equal-pressure-drop boundary condition. XFIGURE 4-11X shows a schematic of a loop containing parallel boiling channels.

Npch

Nsub

xex=0 xex=1

Two-phase flow

Subcooled liquid

Superheated vapor

Constant exit quality line

Unstable

Stable Stability boundary


120

FIGURE 4-11. A schematic of a loop containing parallel boiling channels.

The parallel channels may operate at different conditions. In LWR reactor core, the operating powers of channels vary depending on the radial location of the channel in the core. Inlet mass flow rate is typically regulated for different zones in the core. Due to that, usually one channel will be least stable in the core. If the number of channels is large and only one of them becomes unstable, the effect of oscillations in this single channel will be small and will not affect the operation of the other, stable channels. Therefore the analysis of the onset of the so-called parallel channel instabilities can be done by considering the most unstable channel subject to a constant pressure drop boundary conditions.

At steady-state conditions, pressure drop in each channel is given as,

(4.82)

l

k

N

i

kikilo

l

k

klk

l

k

kh

klofkk

GGrgLr

G

D

LCrp

loc

ρξφ

ρρ

ρ 2

22

1

,

2

,,

2

,2,4

2

,

,,,3

+++=∆− ∑

=

,

Here index k denotes specific channel, which may have individual geometry and local loss coefficients and Nloc is the number of local losses, including the inlet and the outlet pressure loss.

The constancy of the pressure drop for all channels implies that,

(4.83) pppp N ∆=∆==∆=∆ ...11 .

The mass conservation equation for parallel channels is as follows,

(4.84) WAGN

k kk =∑ =1,

Parallel channels

Lower Plenum

Upper Plenum Condenser

Downcomer

Pump


121

where N is the total number of parallel channels, W is the total mass flow rate through the core and Ak is the cross section area of channel k. Eqs. X(4.82X) through (4.84X) constitute a system of N+1 equations with N+1 unknowns (Gk and ∆p). Solution of the equation will give the individual mass flow rate in each channel.

4.3.2 Multiple Parallel Channels

In this situation the constant pressure drop condition is valid. Assuming a constant power of the heater and ignoring changes in the lower plenum temperature, the pressure drop perturbation of the channel can be obtained from Eqs. (4.70X) and (4.71X) as,

(4.85) ( ) ( ) ( ) [ ]

inH

inHH

fHHH

WsG

WssA

ppp

δ

δρ

δδδ φφ

)(

)()(1

,1,121 =Π+Γ=∆+∆=∆.

If ( ) 0=∆ Hpδ Eq. X(4.85) will always have a trivial, steady-state solution inWδ . It may

also have periodic nonzero solution provided that

(4.86) 0)()()( ,1,1 =Π+Γ= sssG HHH ,

for some 0≠= ωjs . In this case the channel is self-excited and will undergo self-

sustained periodic oscillations at the angular frequency ω.

4.3.3 Boiling Loop Stability

The analysis of transients in boiling loops and systems involves the modeling of several inter-related phenomena, such as: single-phase and two-phase flows in channels and plena, boiling heat transfer, phase separation and mixing, unsteady heat conduction and others. The equations for heat transfer and flow in different parts of the system can be combined and solved using proper boundary conditions and using various possible external perturbations. Additional possible models of different parts are shortly described below.

Mixing in large plenum: boiling loop contains additional elements which need modeling. Examples are lower and upper plenum, pump, condenser and pipelines. Modeling of lower and upper plena requires some assumptions about mixing. Using perfect mixing homogeneous flow model, lumped-parameter mass and energy conservation equations become, respectively,

(4.87) ∑∑==

−=Nj

j

jex

Ni

k

kin

pm

p WWdt

dV

1

,

1

,

,ρ,

(4.88) ( )

dt

dpViWiW

dt

idV p

Nj

j

jexjex

Ni

k

kinkin

pmpm

p +−= ∑∑== 1

,,

1

,,

,,ρ.

4.3.4 Heated Wall Dynamics

So far the thermal energy added to a boiling channel was considered in the form of a given heat flux q’’, at the channel wall. In reality the heater power rather than the wall heat flux should be used as a given controlled parameter. Any changes in the power


122

generated in the heater will be transmitted to the coolant with a delay depending on the heater geometry and material properties. Two distinct models can be considered:

One dimensional heat conduction model: in this case the lateral heat transfer (in fuel rods it will be heat transfer through pellet, gas gap and cylindrical cladding.) can be described by one-dimensional heat conduction equation,

(4.89)

p

hc

qTa

t

T

ρ

′′′+∇=

∂

∂ 2,

with wall heat flux given as,

(4.90)

walln

Tq

∂

∂−=′′ λ .

Here T(r,t) is the temperature of the heater, q’’’ is the volumetric heat generation rate and ah is the thermal diffusivity of the heater.

Lumped-parameter model: for simplicity, the distributed parameter model can be replaced with a lumped-parameter model as follows,

(4.91) HHw

HHpH PqAqdt

dTAc ′′−′′′=,ρ ,

with wall heat flux given as,

(4.92) ( )lw TThq −=′′ ,

where h is the effective heat transfer coefficient, Tl is the coolant bulk temperature, AH the heater cross-section area.

Pressure boundary conditions: the boundary condition for a boiling loop is obtained by summing-up pressure drops over the closed flow around the loop,

(4.93) 0=∆−∆ pumploop pp ,

where pumpp∆ is the pump head (for a natural circulation loop this part is equal to

zero) and loopp∆ is pressure drop in the rest of the loop.


A cross-section area cp specific heat C0 drift-flux distribution parameter Cf Fanning friction coefficient D diameter g gravity constant G mass flux


123

h heat transfer coefficient i specific enthalpy j superficial velocity L length p pressure P perimeter q’ linear heat q’’ heat flux q’’’ volumetric heat source Q heat r radius t time T temperature U velocity V volume W mass flow rate x quality z axial coordinate

Greek α void fraction λ heat conductivity ν molecular kinematic viscosity µ molecular dynamic viscosity

ρ density

Γ evaporation rate

Subscript f fluid; saturated liquid phase g gas phase; saturated vapor h hydraulic H heated k pertinent to phase k l liquid phase m mixture v vapor w wall

R E F E R E N C E S

[4-1] Lahey R.T. Jr. Ed. Boiling Heat Transfer, Modern Developments and Advances, ISBN 0-444-89499-3, 1992.

E X E R C I S E S

EXERCISE 4-1: In a BWR fuel assembly operating with the total power 1.5 MW and having constant inlet mass flow rate 2.4 kg/s the inlet coolant velocity was suddenly reduced with 10%. Plot the change of void fraction, mixture density, mixture enthalpy and mixture velocity at 3.3 m distance from the inlet to the assembly. Use homogeneous model and assumptions adopted in XEXAMPLE 4-1X. Use steam-water property at 70 bar pressure. The assembly has total length L = 3.65 m, cross-section area A = 0.0016 m2, heated perimeter PH = 0.7m, hydraulic diameter Dh = 0.01 m. Assume saturated conditions at the inlet and uniform distribution of heat flux along the assembly.


124

EXERCISE 4-2: Plot the stability map for a boiling channel with length LH = 3.6 m, hydraulic diameter D = 10 mm and pressure p = 70 bar. The inlet loss coefficient is equal to 4 and the outlet is equal to 0. Use the Blasius formula for the Fanning friction factor Cf assuming Re = 105. What is the maximum exit quality to keep the channel stable?

125

Appendix A - Laplace

Transformation

Many analysis techniques center on the use of transformed variables to facilitate mathematical treatment of the problem. In the analysis of continuous time dynamical systems, the use of the Laplace transform predominates.

The Laplace transform are used to convert time domain relationships to a set of equations expressed in terms of the Laplace operator 's'. Thereafter, the solution of the original problem is affected by simple algebraic manipulations in the 's' or Laplace domain rather than the time domain. The one-sided Laplace Transform of a time function f(t) is defined as:

( )badttfedttfetfLsF

b

a

st

ba

st <<=== ∫∫−

∞→→

∞− 0)(lim)()()(

00

. (A-1)

The transform associates a unique result or image function F(s) of the complex variable s = a+jb with every single-valued object or original function f(t) (t real) such that the improper integral (A-1) exists. F(s) is called the (one-sided) Laplace transform of f(t).

It is customary to design the image function with the capital letter, corresponding to the original function, and replace t argument with s. That is, U(s) is a Laplace transform of a function u(t). Another notation, used throughout this book is to use a ‘hat’ symbol

for the image function, that is ( )su is an image of u(t).

Basic properties of the Laplace transform are as follows:

The Laplace transform is linear:

)()()()()()( 212121 sFsFtfLtfLtftfL +=+=+ (A-2)

Laplace transforms of derivatives are given by the following expression:

)0()( fssFfLdt

dfL −=′=

(A-3)

In general:

Appendix

A

A P P E N D I X A – L A P L A C E T R A N S F O R M

126

)0(...)0()( 11 −− −−−==

nnnn

n

n

ffssFsfLdt

fdL (A-4)

Laplace transforms of selected functions are given below.

Step function

Consider a function f(t) = Ku(t), where u(t) is a step-function defined as,

≤

>=

00

01)(

tfor

tfortu

Applying the Laplace transformation yields,

( )s

K

s

Ke

s

KdteKdtetKutKuL

t

t

ststst =−−=−===

∞=

=

−∞

−∞

−

∫∫ 10)()(000

.

Impulse (Dirac delta) function

Strictly speaking Dirac delta function is not a function, but a so-called distribution and

symbolically described as ϕδ , , However, here the function form will be used, in

which the delta function is defined as follows,

∫∞

∞−

=+∞=≠= 1)(,)0(,00)( dtttfort δδδ

From the definition of the Laplace transformation one obtains,

1)( =tL δ .

Exponential function

atetf −=)( ,

( ) ( ) ( )asas

eas

dtedteeeLtastasstatat

+=−

+−=

+−===

∞

+−∞

+−∞

−−−

∫∫1

1011

000

.

To return to the time-domain from the Laplace domain, inverse Laplace transform is used. Again this is analogous to the application of ant-algorithms and as in the use of logarithms, tables of Laplace Transform pairs help to simplify the task, see Table A-1.

Table A-1. The Laplace transform pairs.

f(t) F(s)

x(t)+y(t) X(s)+Y(s)

A P P E N D I X A – N Y Q U I S T ’ S S T A B I L I T Y C R I T E R I O N

127

A f(t) A F(s)

f’ sF(s)-f(0)

fn )0(...)0()( 11 −− −−− nnn ffssFs

1 1/s

t 1/s2

ate

−

as +

1

tωsin , tωcos

22 ω

ω

+s,

22 ω+s

s

nt , atn

et 1

!+n

s

n,

( ) 1

!+

−n

as

n

Of particular interest for nuclear-reactor kinetics are the following properties of the Laplace transformation.

If L(s) and K(s) are polynomials with non-zero, single-valued roots and the degree of the polynomial L(s) is lower than the degree of polynomial K(s) = n, then:

∑=

−

′=

n

k

ts

k

k kesK

sL

sK

sLL

1

1

)(

)(

)(

)( (A-5)

( )( ) ∑

=

−

′+=

n

k

ts

kk

k kesKs

sL

K

L

ssK

sLL

1

1

)(

)(

0

0

)(

)( (A-6)

The following example illustrates the use of the Laplace transform in the system analysis, showing how it is used to solve linear ordinary differential equation (ODE).

Consider the first order process, described by the following differential equation:

)()()(

tUKtYdt

tdY⋅=+τ (A-7)

Here Y(t) is the output variable and U(t) is the “forcing “ input. The time-domain solution of this ODE will be found using the Laplace transform approach. Let define new variables, which are deviations from steady state solutions,

00 )()()()( UtUtuYtYty −=−= (A-8)

A P P E N D I X A – L A P L A C E T R A N S F O R M

128

Here Y0 is the steady-state value that Y(t) will attain given a steady input U0. Assuming that initially the system is at the steady-state condition, the following is valid:

0)0()0( == uy (A-9)

Substituting ((A-8) into (A-7) yields,

)()()(

tuKtydt

tdy⋅=+τ (A-10)

Application of the Laplace transform gives,

)()()0()( sKUsYyssY =+−τ

or, since y(0) = 0,

)()()( sKUsYssY =+τ (A-11)

Equation (A-11) can be transformed as follows,

s

K

sU

sYsG

τ+==

1)(

)()( (A-12)

G(s) is called the transfer function of the considered process and describes the relationship between input U(s) and the output Y(s). The time-domain solution of ((A-11) depends on the shape of the input function. Assuming a unit step change in u(t), i.e. u(t) = 1, t > 0, ((A-12) yields,

( ) ( )

+−⋅=

+⋅=⋅

+=

ssK

ssK

ss

KsY

ττ

τ

τ 1

11

1

11

1)( (A-13)

Using Table A-1, the time solution to ((A-13) is found as,

( ) ( ) 0

// 1)(1)( YeKtYeKtytt +−=⇒−= −− ττ (A-14)

Transfer functions play a central role in the analysis of dynamic systems behavior since they fully describe the relation between input-output pairs. In the previous example, the transfer function G(s) encapsulates the behavior of the system given by ((A-7) in the ratio given in ((A-12).

Transfer function possesses several interesting features:

• Transfer functions are independent of the form of the input.

• Transfer functions obey algebraic rules.

• Commutative property G1(s)G2(s) = G2(s)G1(s)

A P P E N D I X A – N Y Q U I S T ’ S S T A B I L I T Y C R I T E R I O N

129

• Associative property G1(s) + G2(s) = G2(s) + G1(s)

• Transfer functions are linear functions

Transfer functions can be visualized using block diagrams, as shown below:

Block diagram Transfer Function

G(s)

U(s) Y(s)

)()(

)(sG

sU

sY=

G1(s)

U(s) Y(s) G2(s)

)()()(

)(21 sGsG

sU

sY=

G1(s)

U(s) Y(s)

+

G2(s)

)()()(

)(21 sGsG

sU

sY+=

G(s)

U(s) Y(s)

Σ +

-

)(1

)(

)(

)(

sG

sG

sU

sY

+=

131

Appendix B – Nyquist

Stability Criterion

The Nyquist stability criterion is used to investigate stability of any (open or closed) system using its frequency characteristics. Almost always the criterion is used for closed systems, though. A useful, but not the most general formulation of the Nyquist criterion is as follows: if the mapping of the transfer function of an open system G(s)*H(s) on the plane GH encircles point (-1,0) then the system is unstable if it is closed. The criterion is illustrated in Figure B-1.

Figure B-1: Nyquist plots of a stable and unstable closed system. Curve A does not encircle point (-1,0) and the system is stable. Curve B encircles point (-1,0) and the system is unstable.

The Nyquist criterion can be easily proved, but first some introductory on complex variables and functions is required.

A complex variable s can be expressed as,

s = x + j y (B-1)

here x and y are real numbers and j is the unit imaginary number: 1−=j . Another

representation of the complex variable can be obtained in the polar coordinates:

Appendix

B

-1

ω=∝ ω=0

Re(GH)

Im(GH) Plane GH

A

B

A P P E N D I X B - N Y Q U I S T ’ S S T A B I L I T Y C R I T E R I O N

132

( ) ϕϕϕ jerjrs ⋅=+= sincos . (B-2)

Here syxr =+= 22 is the absolute value (norm, modulus) of the complex

number s and ( )xyarctan=ϕ is the argument of the complex number.

A complex function ( ) ( ) ( ) ϑjefyxjvyxusf =+= ,, associates one (single-valued

function) or more (multiple-valued function) values of complex dependent variable f(s) with each value of the complex independent variable s.

A single-valued function f(s) is called analytic (regular, holomorphic) at point s = a if and only if f(s) is differentiable throughout the neighborhood of s = a.

A singular point or singularity of the function f(s) is any point where f(s) is not analytic. The singularity can be a removable singularity if and only if f(s) is finite throughout a neighborhood of s = a, except possibly at s = a itself. The singularity can be a pole of order m if and only if (s-a)mf(s) (but not (s-a)m-1f(s)) is analytic at s = a.

The points s for which f(s) = 0 are called the zeros (roots) of f(s).

Function f(s) is meromorphic throughout a certain region D if and only if its only singularities throughout D are poles.

Nyquist criterion

Let f(s) be meromorphic throughout the bounded region inside and continuous on a

closed contour C on which ( ) 0≠sf . Let Z be the number of zeros and P be the

number of poles of f(s) inside C, respectively, where a zero or pole of order m is counted m times. Then

( )( )

PZdf

f

j C−=

′∫ ς

ς

ς

π2

1. (B-3)

For P = 0 one obtains the principle of the argument

π

ϑ

2

CN∆

= , (B-5)

where ϑC∆ is the variation of the argument ϑ of f(s) around the contour C. This

equation means that f(s) maps a moving point s describing the contour C once into a moving point f(s) which encircles the f plane origin N = 1, 2, … times if f(s) has, respectively, 0, 1, 2, … zeros inside the contour C in the s plane. The above criterion is used for locating zeros and poles of f(s) and is known as the Nyquist criterion.

As an example, consider a function f(s) that has two zeros (s = z1 and s = z2) and one pole (s = p1). The function can be given as

( ) ( )( )( )1

21

ps

zszssf

−

−−= (B-6)

A P P E N D I X B – N Y Q U I S T ’ S S T A B I L I T Y C R I T E R I O N

133

The location of the zeros and the pole are shown in Figure B-2.

Figure B-2: Location of zeros and poles on s plane.

Assume that point s is moving along the contour and encircles it. The argument of the number (s-z1) will increase by π2 . It means that with a single encirclement of the contour C by point s, the argument of function f(s) will increase by π2 for each zero which is inside of the contour. In a similar way it is obtained that the argument of function f(s) decreases with π2 for each pole located inside of the contour C. As a consequence, the mapping of f(s) on plane f will encircle the plane origin the corresponding number of times. In the considered case, f(s) will encircle the origin only once, since its argument will increase with (2 -1) multiplied by π2 .

The Nyquist criterion is very useful in the theory of stability. To evaluate the stability of any system, it is necessary to find the locations of zeros of the characteristic function of the system. If any of the zeros is located on the right-hand-side of the s plane, then the system is unstable. The contour on the s plane is shown in Figure B-3.

Re(s)

Im(s) Pole

Zeros

s

z1

Contour C

z2 p1

A P P E N D I X B - N Y Q U I S T ’ S S T A B I L I T Y C R I T E R I O N

134

Figure B-3: Contour in the s plane that covers the whole right-hand-side half-plane when ∞→R .

An example of the corresponding mapping of function f(s) on the f plane is shown in Figure B-4.

Figure B-4: Mapping of function f(s) for s encircling the right half-plane as shown in Fig. B-3.

R

CR

C+

C-

Re(s)

Im(s)

C+

C-

CR ω=0

ω=∝ Re(f)

Im(f)

135

Appendix C - Selected

Steam-Water Data

Steam-water properties at saturation for pressure p = 7 MPa (tsat = 285.83 °C)

Phase Density Viscosity Enthalpy Conductivity Spec. heat

[kg m-3] [Pa s] [J kg-1] [W m-1 K-1] [J kg-1 K-1]

Water 739.724 9.12488 10-5 1267660 0.571881 5402.48

Steam 36.525 1.89601 10-5 2772630 0.0629146 5356.59

Steam-water properties at saturation for pressure p = 15.5 MPa (tsat = 344.79 °C)

Phase Density Viscosity Enthalpy Conductivity Spec. heat

[kg m-3] [Pa s] [J kg-1] [W m-1 K-1] [J kg-1 K-1]

Water 594.38 6.83274 10-5 1629880 0.458470 8949.98

Steam 101.93 2.31084 10-5 2596120 0.121361 14000.6

Appendix

C

137

Appendix D –Boiling

Channel Stability Model

Expressions for pressure drop perturbations in a boiling channel are derived in this Appendix. It is assumed that the boiling channel has a uniform heat flux distribution and a uniform shape (and thus area cross section) along its length. All flow parameters are averaged across the flow area, thus the model is one-dimensional. The model presented in this Appendix follows the derivation presented in [D-1].

For a given system pressure, p, and the inlet velocity of the subcooled liquid,jin,0 , the steady state mass flux in the channel, G0, is given by,

0,0 inf jG ρ= . (D-1)

Here for simplicity it is assumed that the fluid density in the single phase (subcooled)

region is constant and equal to the saturation liquid velocity fρ . The inlet subcooling

is 0,subi∆ and the uniform axial heat flux is 0q ′′ . The steady state length of the single-

phase (subcooled) region, 0λ , can be evaluated from the energy equation for the

homogenous flow model:

( ) ( )t

p

A

Pq

z

iG

t

iHmMm

∂

∂+

′′=

∂

⋅∂+

∂

⋅∂ ρ.

Here mρ is the two-phase mixture density, PH is the heated perimeter, A is the

channel cross-section area, and the mixture enthalpy im and iM are defined as follows,

( ) xixii gfm +−= 1

( )

m

ggff

M

iii

ρ

αραρ +−=

1.

For steady-state both terms given below are equal to zero, thus,

0=∂

∂

t

p,

and

Appendix

D

A P P E N D I X D – B O I L I N G C H A N N E L S T A B I L I T Y

138

( )0=

∂

⋅∂

t

iMmρ ,

so the expression for 0λ becomes,

H

sub

Pq

iAG

0

0,0

0 ′′

∆=λ . (D-2)

In the two-phase region, the volumetric flux can be obtained as,

( ) 0,000 )( injzzj +−Ω= λ , (D-3)

where

fg

fgH

i

v

A

Pq00

′′=Ω . (D-4)

The homogenous density can be obtained from,

)()(

0

00

zj

Gzm =ρ . (D-5)

For transient analysis Equations (D-1)-(D-5) generalize to

)()( tjtG infρ= , (D-6)

( ) )()(),(),( tjtzztztj in+−Ω= λ , (D-7)

fg

fgH

i

v

A

Pztqzt

),(),(

′′=Ω , (D-8)

),(

),(),(

ztj

ztGztm =ρ . (D-9)

Using the homogeneous equilibrium model, the pressure drop across single-phase and two-phase section is given as,

( ) ∑∫∈

+

++∂

∂+

∂

∂=−=∆

φρ

ξρρρ

λλ

φ

12

)(

2

41)(

2

0

2,

2

1

Li l

i

il

lh

lof

l

inH

zGdzg

G

D

C

z

G

t

Gppp ,

for the single phase flow part and,

( ) ∑∫∈

+

++

∂

∂+

∂

∂=−=∆

φρ

ξφρρ

φρ

λλ

φ

22

)()(

2

4)(

2

2

,

2,2

2

2

Li l

i

iiilo

L

m

lh

lof

lo

M

exH

zGzdzg

G

D

CG

zt

Gppp

H

,


139

for the two-phase flow part. Here λ is the position of the boiling boundary, φ1L is the

non-boiling length of the channel and φ2L is the boiling length of the channel.

These equations can be linearized and Laplace transform to obtain,

( ) )(ˆ2

4)(ˆ4

2

0

0

0

01

1

sgD

CGsjG

DCsp f

hf

f

in

Li

i

h

ffHλδρ

ρδξ

λρλδ

φ

φ

++

++=∆ ∑

∈

(D-10)

( ) ( )

∑

∫

∈

Φ

+

+

++Ω

−

Ω+

++Ω

+

+Ω+=∆

φ

ρδδξ

λδρ

δρδ

δρδλ

2

0

),(ˆ2

)(),(ˆ

)(ˆ2

4

),(~

),(ˆ)(2

4)(

),(ˆ4

)(

2

00

0,000

0

2

000

0002

Li

imi

ii

lin

h

f

m

H

f

L

h

f

mH

zszj

zsjG

sgjGD

CG

dzzsGzsgzjD

Czj

zsjGD

Czsp

H

(D-11)

The next step is to relate the variables in Equation (D-10) and (D-11) to the channel external perturbations.

The volumetric flux perturbation can be obtained from Equation (D-7) as

( ) )()(ˆ),(ˆ),(ˆ 00 sjszszzsj inδλδδλδ +Ω−Ω−= . (D-12)

The boiling boundary perturbation, λδ ˆ , can be expressed in terms of ),(ˆ 0λδ si by

integrating the energy equation from 0λ to )(tλ and then perturbing it. This gives,

)(),(),(0

00 t

AG

Pqtiti H δλλλ

′′=− . (D-13)

Taking into account that )(),( 000 λλ iiti f == and Laplace-transforming yields

),(ˆ),(ˆ)(ˆ 00

0

0

0 λδλ

λδλδ sii

sii

vGs

subfg

fg

∆−=

Ω−= . (D-14)

The enthalpy perturbation can be obtained from the energy equation for the homogenous flow model, perturbing and Laplace-transforming this equation yields,

( )

−

′′

′′′′=+

0,00

0

0,

ˆ),(),(ˆ

),(ˆ

in

inH

in j

j

q

zsq

AG

qPzsi

j

s

dz

zsid δδδ

δ. (D-15)


140

The specific relationship between q ′′δ and q ′′′δ depends on the model used to

quantify the heated wall-dynamics and, in general, the following expression can be obtained,

),(ˆ)(ˆ)(),(ˆ)( 21 zsTsqsZzsqsZ WH δδδ =′′′+′′ . (D-16)

If a lumped parameter model is used to describe the heater dynamics, the following equation is obtained,

HHHW

HHpH PqAqdt

dTAc ′′−′′′=,ρ , (D-17)

where Hρ is the heater material density, Hpc , is the heater specific heat and AH is the

effective heater cross-section area.

Thus,

sAc

PsZ

HHpH

H

,

1 )(ρ

−= ,

and

sc

sZHpH ,

2

1)(

ρ−= .

In the single-phase region, WT , is given by the Newton’s law of cooling, as,

φ1h

qTT bW

′′=− . (D-18)

where Tb is the bulk fluid temperature.

Taking into account that

a

in

in

j

tjhth

=

0,

0,11

)()( φφ ,

where normally 8.0=a , yields,

( )0,

00,1

ˆˆˆˆ

in

inbW

j

jaqTThq

δδδδ φ ′′+−=′′ , (D-19)

where

pfb ciT ˆˆ δδ = .


141

Here, cpf is the fluid specific heat.

Equations (D-16) and (D-19) can be combined to obtain ),(ˆ zsq ′′δ in the single-phase

region as a function of )(ˆ sqH′′′δ , )(ˆ sjinδ and ),(ˆ zsiδ . Substituting the result into

Equation (D-15) and rearranging, we get,

( )00,

)(ˆ)(

)(ˆ)(),(ˆ)(

ˆ

q

sqs

j

sjszsis

dz

id H

in

in

′′

′′′+=+

δβ

δθδφ

δ (D-20)

with,

( ))(1)(

10,10

0,1

0, sZhAcG

hP

j

ss

pf

H

in φ

φφ−

+= , (D-21)

−

−

′′= 1

)(1)(

10,10

0

sZh

a

AG

qPs H

φ

θ , (D-22)

−

′′=

)(1

)()(

10,1

20,1

0

0

sZh

sZh

AG

qPs H

φ

φβ , (D-23)

Solving the differential Equation (D-20), 0=z to 0λ=z , with the boundary

condition )(ˆ)0,(ˆ sisi inδδ = gives

′′

′′′+−+= −−

00,

)()(

0

)(ˆ)(

ˆ)(1

)(

1ˆ),(ˆ 00

q

sqs

j

jse

siesi H

in

ins

in

s δβ

δθ

φδλδ λφλφ (D-24)

Now the expression for )(ˆ sλδ becomes,

inHin isqsjss ˆ)(ˆ)(ˆ)()(ˆ 321 δδδλδ Λ+′′′Λ+Λ= , (D-25)

where

[ ]0)(

0,

01 1

)(

)()(

λφ

φ

θλ s

insub

es

s

jis

−−∆

−=Λ , (D-26)

[ ]0)(

0

02 1

)(

)()(

λφ

φ

βλ s

sub

es

s

qis

−−′′∆

−=Λ , (D-27)

[ ]0)(03 )(

λφλ s

sub

ei

s−

∆−=Λ , (D-28)


142

In order to eliminate Ωδ from the expression for jδ in Equation (D-12) we can use an

equation that describes the heated wall dynamics combined with the Newton’s law of cooling.

( )msatW TTpCq

1

)( −=′′ , (D-29)

Here C(p) is a known function of pressure and m is a constant.

Perturbation of Eq. (D-29) yields,

( ) )(ˆ)(ˆˆ3

0

0, sqZsqq

mTTT HHsatWW

′′′=′′′′′

−= δδδ . (D-30)

Substituting this result into Equation (D-16) yields

( ) HHfg

fgHqZqi

AsZZ

vPsZs ˆˆ

)(

)()(ˆ

4

13

2 ′′′=′′′−

=Ω δδδ . (D-31)

Substituting Equations (D-25) and (D-31) into (D-12) we obtain an expression

without Ωδ ,

[ ] ( )[ ]

)(ˆ)(

)(ˆ)()()(ˆ)(1),(ˆ

30

204010

sis

sqssZzsjszsj

in

Hin

δ

δλδδ

ΛΩ

−′′′ΛΩ−−+ΛΩ−= (D-32)

To get an expression for ),(ˆ zsmρδ we can rewrite the mass equation as,

0=Ω+∂

∂+

∂

∂m

mm

zj

tρ

ρρ. (D-33)

and if one perturbs and Laplace-transforms this equation, and uses the continuity

equation, Ω=∂∂ zj , one gets,

( )( )

( )( )

( )( )

)(ˆˆ),(ˆ),(ˆ),(ˆ

0

0

0

0

0

szj

z

zj

zsjz

dz

dzs

zj

szs

dz

d mmmm Ω−−=

Ω++ δ

ρδρρδρδ (D-34)

Substituting Equations (D-3) and (D-5) into this equation gives

( )( ) ( )

Ω+−

Ω−

Ω+−Ω=

Ω+−

+Ω+

2

00,0

3

00,0

2

0

0

00,0

0 )(ˆ),(),(ˆ

1),(ˆ

inin

m

in

mjz

s

jz

zsjGzs

jz

szs

dz

d

λ

δ

λ

δρδ

λρδ (D-35)

In order to integrate this equation, a boundary condition must be established at

0λ=z . One can integrate the energy equation for the homogenous flow model from

0λ=z to λ=z ,


143

),(),(),(),( 00

0

λλρλλρρλ

λ

tjttjtdzt

mmm −+

∂

∂∫ . (D-36)

After perturbing and Laplace-transforming this equation one gets,

( )),(ˆ)(ˆ),(ˆ0

0,

0 λδδρ

λρδ sjsjj

s in

in

f

m −= (D-37)

and this can be rewritten, using Equation (D-13), as,

)(ˆ),(ˆ0,

0

0 sj

sin

f

m λδρ

λρδΩ

= . (D-38)

Now one can integrate Equation (D-35) from 0λ to z with this boundary condition.

This yields an equation on the form )()()()( zgzyzfzy =+′ and can be solved by

using the technique with integrating factor. Using equation (D-7) and (D-12) and

assuming that )(ˆ),(ˆ szs Ω=Ω δδ gives the result,

)(ˆ)(

1

)(1

)(ˆ)()(

1

)(ˆ)()(

),(ˆ

2

0

,0

0

1

0

,

2

0

0

0

0

1

0

,

2

0

0

0

00

1

0

0,

0

0

0

szj

jG

szj

j

sjzj

G

szj

j

szj

G

szj

jszs

oin

s

oin

in

s

oin

s

in

m

ΩΩ−

−

−Ω−

Ω

−

+Ω−

Ω

Ω−

=

−

Ω

−

Ω

−

Ω

δ

δ

λδρδ

(D-39)

Assuming that the only local losses are those at the inlet and the exit of the channel and substituting Equation (D-25) into (D-10) results in the following expression for the single-phase pressure drop perturbation, depending on perturbations in the inlet

velocity, injδ , inlet enthalpy, iniδ and in the heater internal heat generation Hq ′′′δ .

( )inHHHinHH

isqsjsp ˆ)(ˆ)(ˆ)( ,3,2,11 δδδδ φ Γ+′′′Γ+Γ=∆ , (D-40)

where

Λ

++++=Γ )(

2

44 1

0,

0,0

0,

00,1 s

j

g

D

Cj

DC

j

sG

inh

fin

in

h

f

in

H ξλλ

, (D-41)


144

)(2

4

0,

0,

0, sj

g

D

CjG i

inh

fin

Hi Λ

+=Γ , 3,2=i . (D-42)

With the same assumptions as for (D-40) and substituting equation (D-25), (D-32) and (D-39) into (D-11) we get expressions for the pressure drop perturbations in the two-phase region as,

( )inHHHinHH

isqsjsp ˆ)(ˆ)(ˆ)( ,3,2,12 δδδδ φ Π+′′′Π+Π=∆ . (D-43)

where,

( ))()()( 1210,1 ssFsFGH Λ−=Π , (D-44)

( ))()()( 3220,2 sFssFGH −Λ−=Π , (D-45)

)()( 320,3 ssFGH Λ−=Π , (D-46)

( )( )

( )( )+−Ω−

Ω+

Ω−

Ω−−+

Ω−= −Ω

12

2

4)( 0

2

0

2

0

0

00

0

2

1exs

h

Hfex ess

s

D

LC

s

ssF

τλτ

( )( )( )( )+−

Ω−Ω−

Ω+ −Ω

122

402

00

00,exs

h

fine

ssD

Cj τ

( )( )( ) +

−

Ω+−

Ω−+ −Ω−

111 0

00,

0 exex s

in

es

esj

g ττ

( )( )( )

−

Ω−

Ω++ −Ω exs

ex es

τξ 012

10

0 . (D-47)

( ) ( )( )

( )( )+−Ω−

Ω+

Ω−

Ω−Ω−+

Ω−

Ω= −Ω

12

2

4)( 0

2

0

2

0

0

000

0

2

02

exs

h

Hfex es

s

s

s

D

LC

s

ssF

τλτ

( )( )( )( )

( )( ) +Ω+−

Ω−

Ω+−

Ω−Ω−

Ω+

Ω−−−Ω0

0

0

0,

2

00

00, 00 122

4exexex ee

sj

ge

ss

s

D

Cjs

in

s

h

fin τττ

( )

( )( )( )

−

Ω−

Ω−−Ω−++ −Ω−Ω exex ss

ex

inh

fine

se

j

g

D

Cj ττξ 00 12

12

1

2

4

0

00

0,

0, (D-48)

( )( ) ( )

( )( )

+−Ω−

Ω−

ΩΩ−−−

Ω

Ω+= −Ω

12

)( 0

2

0

0,0

0,

00

2

0

0

03

exsin

exinH es

jj

s

sL

ssF

ττλ

( ) ( )( )( )

( )( )−

Ω−Ω−

−−−

Ω−−−+

−Ω

00

22

0,

0

0

0,2

02

1

2

4 0

ss

ejL

s

jL

D

C exs

in

H

in

H

h

f

τ

λλ


145

( )[ ] ( )[ ] +

−−−

ΩΩ−− −Ω− exex s

es

es

g ττ1

11

10

00

( )( )

( )( ) )(12

4

0

0,

00 sZe

s

jL ex

sin

Hex

−

Ω−−−+ −Ω ξλ τ . (D-49)

Ω=

0,

0

0

)(1

in

Hex

j

Ljτ . (D-50)

Considering local losses at other locations for the single-phase region yields,

Λ

++++=Γ ∑

∈

)(2

44 1

0,

0,0

0,

00,1

1

sj

g

D

jC

DC

j

sG

inh

inf

Li

i

h

f

in

H

φ

ξλλ

. (D-51)

For the two-phase region, only the exξ -terms in F1, F2 and F3 are changed. For F1 we

get:

( )( )

( )( )+−Ω−

Ω+

Ω−

Ω−−+

Ω−= −Ω

12

2

4)( 0

2

0

2

0

0

00

0

2

1exs

h

Hfex ess

s

D

LC

s

ssF

τλτ

( )( )( )( )+−

Ω−Ω−

Ω+ −Ω

122

402

00

00,exs

h

fine

ssD

Cj τ

( )( )( )+−

Ω+−

Ω−+ −Ω−

111 0

00,

0 exex s

in

es

esj

g ττ

( )( )( )∑

−

Ω−

Ω++ −Ω

i

s

iie

s

τξ 012

10

0 . (D-52)

F2 transforms into,

( ) ( )( )

( )( )12

2

4)( 0

2

0

2

0

0

000

0

2

0

2 −Ω−

Ω+

Ω−

Ω−Ω−+

Ω−

Ω= −Ω exs

h

Hfex es

s

s

s

D

LC

s

ssF

τλτ

( )( )( )( )

( )( ) +Ω+−

Ω−

Ω+−

Ω−Ω−

Ω+

Ω−−−Ω0

0

0

0,

2

00

00, 00 122

4exexex ee

sj

ge

ss

s

D

Cjs

in

s

h

fin τττ

( )

( )( )( )

−

Ω−

Ω−−Ω−++ −Ω−Ω∑ ii ss

i

i

inh

fine

se

j

g

D

Cj ττξ 00 12

12

1

2

4

0

00

0,

0,


146

( )

( )( )( )

−

Ω−

Ω−−Ω− −Ω−Ω∑ ii ss

i

i es

eττξ 00 1

21

2

1

0

00 . (D-53)

and F3,

( )( ) ( )

( )( )

+−Ω−

Ω−

ΩΩ−−−

Ω

Ω+= −Ω

12

)( 0

2

0

0,0

0,

00

2

0

0

03

exsin

exinH es

jj

s

sL

ssF

ττλ

( ) ( )( )( )

( )( )−

Ω−Ω−

−−−

Ω−−−+

−Ω

00

22

0,

0

0

0,2

02

1

2

4 0

ss

ejL

s

jL

D

C exs

in

H

in

H

h

f

τ

λλ

( )[ ] ( )[ ] +

−−−

ΩΩ−− −Ω− exex s

es

es

g ττ1

11

10

00

( )( )

( )( ) )(12

4

0

0,

00 sZe

s

jL i

sin

Hi

−

Ω−−−+ −Ω ξλ τ . (D-54)

where

Ω=

0,

0

0

)(1

in

ii

j

zjτ . (D-55)

Calculation of the individual heat transfer coefficients involves the following dimensionless numbers:

The Nusselt number, Nu λ

hDh ⋅=Nu . (D-56)

The Prandtl number, Pr λ

µ⋅= pc

Pr . (D-57)

The Reynolds number, Re µ

ρ⋅⋅= hDv

Re . (D-58)

The following equation is used:

( ) 33.08.0 PrRe023.0PrRe,Nu == φ . (D-59)

This yields:

hDh

λ33.08.0PrRe023.0

= . (D-60)

The friction-factor can be calculated with the Blasius correlation:


147

25.0Re

0791.0=fC . (D-61)

R E F E R E N C E S

[D-1] Lahey R.T. Jr. Ed. Boiling Heat Transfer, Modern Developments and Advances, ISBN 0-444-89499-3, 1992

151

INDEX

Adjoint neutron flux ............................... 38

Amplification factor ......................... 20, 26

Amplitude function ................................ 35

Asymptotic stability............................... 15

Attractor ................................................ 15

Basin of attraction ................................. 15

Bifurcation ............................................. 15

Bode characteristics ............................. 27

Channel thermal-hydraulic instability ... 12

Chaotic orbit .......................................... 15

Characteristic polynomial ..................... 17

Control system instability ..................... 12

Coupled neutronic-thermal-hydraulic

instability ................................. 12

Daughter nuclei ..................................... 33

Decay ratio ..................................... 23, 118

Density wave oscillations .................... 111

Doppler effect ........................................ 69

Dynamic instabilities ........................... 109

Dynamical systems ................................ 14

First method of Lyapunov ...................... 16

First-order system ................................. 19

Fixed point ............................................. 15

Frequency characteristics .................... 27

Frequency response .............................. 26

Fuel Doppler reactivity coefficient ........ 72

Fuel temperature coefficient ................. 72

Hopf bifurcation ..................................... 18

Jacobian matrix ..................................... 16

Limit cycle ............................................. 15

Linear stability analysis ........................ 16

Lyapunov stability ................................. 15

Marginal stability ................................... 15

Neutron generation time ....................... 41

Neutron lifetime ..................................... 41

Nonautonomuous systems .................... 15

Non-wandering sets ............................... 15

Normalized point dynamics equations .. 76

Normalized point kinetics equations .... 44

Nyquist plot............................................ 27

One group approximation ...................... 40

Orbit ................................................. 15

Percent milli rho - pcm .......................... 70

Phase shift ........................................20, 26

Phase space........................................... 14

Phase-change number ......................... 118

Point dynamics equations ..................... 73

Point kinetics equations ........................ 37

Poles of the transfer function ................ 25

Precursors of delayed neutrons ............ 33

Prompt criticality ................................... 62

Prompt neutrons .................................... 33

Prompt reactivity coefficient ................ 72

Quasiperiodic orbit ................................ 15

Reactivity ............................................... 37

Reactivity instability ............................. 12

Reactor period ....................................... 48

Routh and Hurwitz theorem................... 17

Second-order system ............................. 21

Shape function ....................................... 35

Six-group point kinetics model .............. 40

Stability map ........................................ 118

Stable reactor period ............................. 48

State of a dynamical system ................. 14

Static instabilities ............................... 109

Stationary bifurcation ............................ 18

Subcooling number .............................. 118

Trajectory .............................................. 15

Transfer function ................................... 24

Transient reactor periods ...................... 48

Zero-power reactor transfer function ... 55

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