CHAPTER 5 Reactor Dynamics. Table of Contents

Transcription

1 1 CHAPTER 5 Reactor Dynamics prepared by Eleodor Nichita, UOIT and Benjamin Rouben, 1 & 1 Consulting, Adjunct Proessor, McMaster & UOIT Summary: This chapter addresses the time-dependent behaviour o nuclear reactors. This chapter is concerned with short- and medium-time phenomena. Long-time phenomena are studied in the context o uel and uel cycles and are presented in Chapters 6 and 7. The chapter starts with an introduction to delayed neutrons because they play an important role in reactor dynamics. Subsequent sections present the time-dependent neutron-balance equation, starting with point inetics and progressing to detailed space-energy-time methods. Eects o Xe and Sm poisoning are studied in Section 7, and eedbac eects are presented in Section 8. Section 9 is identiies and presents the speciic eatures o CANDU reactors as they relate to inetics and dynamics. Table o Contents 1 Introduction Overview Learning Outcomes... 3 Delayed Neutrons Production o Prompt and Delayed Neutrons: Precursors and Emitters Prompt, Delayed, and Total Neutron Yields Delayed-Neutron Groups Simple Point-Kinetics Equation (Homogeneous Reactor) Neutron-Balance Equation without Delayed Neutrons Average Neutron-Generation Time, Lietime, and Reactivity Point-Kinetics Equation without Delayed Neutrons Neutron-Balance Equation with Delayed Neutrons Solutions o the Point-Kinetics Equations Stationary Solution: Source Multiplication Formula Kinetics with One Group o Delayed Neutrons Kinetics with Multiple Groups o Delayed Neutrons Inhour Equation, Asymptotic Behaviour, and Reactor Period Approximate Solution o the Point-Kinetics Equations: The Prompt Jump Approximation1 5 Space-Time Kinetics using Flux Factorization Time-, Energy-, and Space-Dependent Multigroup Diusion Equation Flux Factorization Eective Generation Time, Eective Delayed-Neutron Fraction, and Dynamic Reactivity7

2 The Essential CANDU 5.4 Improved Quasistatic Model Quasistatic Approximation Adiabatic Approximation Point-Kinetics Approximation (Rigorous Derivation) Perturbation Theory Essential Results rom Perturbation Theory Device Reactivity Worth Fission-Product Poisoning Eects o Poisons on Reactivity Xenon Eects Reactivity Coeicients and Feedbac CANDU-Speciic Features Photo-Neutrons: Additional Delayed-Neutron Groups Values o Kinetics Parameters in CANDU Reactors: Comparison with LWR and Fast Reactors CANDU Reactivity Eects CANDU Reactivity Devices Summary o Relationship to Other Chapters Problems Reerences and Further Reading Acnowledgements List o Figures Figure 1 Graphical representation o the Inhour equation Figure 135 Xe production and destruction mechanisms Figure 3 Simpliied 135 Xe production and destruction mechanisms Figure Xe reactivity worth ater shutdown Figure 5 CANDU uel-temperature eect... 4 Figure 6 CANDU coolant-temperature eect... 4 Figure 7 CANDU moderator-temperature eect Figure 8 CANDU coolant-density eect List o Tables Table 1 Delayed-neutron data or thermal ission in 35 U ([Rose1991])... 6 Table CANDU reactivity device worth... 4

3 Reactor Dynamics 3 1 Introduction 1.1 Overview The previous chapter was devoted to predicting the neutron lux in a nuclear reactor under special steady-state conditions in which all parameters, including the neutron lux, are constant over time. During steady-state operation, the rate o neutron production must equal the rate o neutron loss. To ensure this equality, the eective multiplication actor, e, was introduced as a divisor o the neutron production rate. This chapter addresses the time-dependent behaviour o nuclear reactors. In the general time-dependent case, the neutron production rate is not necessarily equal to the neutron loss rate, and consequently an overall increase or decrease in the neutron population will occur over time. The study o the time-dependence o the neutron lux or postulated changes in the macroscopic cross sections is usually reerred to as reactor inetics, or reactor inetics without eedbac. I the macroscopic cross sections are allowed to depend in turn on the neutron lux level, the resulting analysis is called reactor dynamics or reactor inetics with eedbac. Time-dependent phenomena are also classiied by the time scale over which they occur: Short-time phenomena are phenomena in which signiicant changes in reactor properties occur over times shorter than a ew seconds. Most accidents all into this category. Medium-time phenomena are phenomena in which signiicant changes in reactor properties occur over the course o several hours to a ew days. Xe poisoning is an example o a medium-time phenomenon. Long-time phenomena are phenomena in which signiicant changes in reactor properties occur over months or even years. An example o a long-time phenomenon is the change in uel composition as a result o burn-up. This chapter is concerned with short- and medium-time phenomena. Long-time phenomena are studied in the context o uel and uel cycles and are presented in Chapters 6 and 7. The chapter starts with an introduction to delayed neutrons because they play an important role in reactor dynamics. Subsequent sections present the time-dependent neutron-balance equation, starting with point inetics and progressing to detailed space-energy-time methods. Eects o Xe and Sm poisoning are studied in Section 7, and eedbac eects are presented in Section 8. Section 9 identiies and presents the speciic eatures o CANDU reactors as they relate to inetics and dynamics. 1. Learning Outcomes The goal o this chapter is or the student to understand: The production o prompt and delayed neutrons through ission. The simple derivation o the point-inetics equations. The signiicance o inetics parameters such as generation time, lietime, reactivity, and eective delayed-neutron raction. Features o point-inetics equations and how they relate to reactor behaviour (e.g., reactor period). Approximations involved in dierent inetics models based on lux actorization.

4 4 The Essential CANDU First-order perturbation theory. Fission-product poisoning. Reactivity coeicients and eedbac. CANDU-speciic eatures (long generation time, photo-neutrons, CANDU start-up, etc.). Numerical methods or reactor inetics. Delayed Neutrons.1 Production o Prompt and Delayed Neutrons: Precursors and Emitters Binary ission o a target nucleus AX 1 Z X X A Z X X X occurs through the ormation o a compound nucleus which subsequently decays very rapidly (promptly) into two (hence the name binary ) ission products A m and B m, accompanied by the emission o (promp gamma photons and (promp neutrons: 1 AX AX 1 n Z X X Z X X, (1) The exact species o ission products A m and B m, as well as the exact number o prompt neutrons emitted,, and the number and energy o emitted gamma photons depend on the pm mode m according to which the compound nucleus decays. Several hundred decay modes are possible, each characterized by its probability o occurrence p m. On average, p prompt neutrons are emitted per ission. The average number o prompt neutrons can be expressed as: p m pm m () p. (3) Obviously, although the number o prompt neutrons emitted in each decay mode, positive integer (1,, 3 ), the average number o neutrons emitted per ission, p, is a ractional number. the incident neutron. pm as well as p pm, is a depend on the target nucleus species and on the energy o The initial ission products A m and B m can be stable or can urther decay in several possible modes, as shown below or A m (a similar scheme exists or B m ): A A m1 Am 1HE m A m3 1LE (mode 1) (mode ) ' (mode 3) (as A n (delayed neutron) m3. Fission products A m that decay according to mode 3, by emitting a low-energy beta particle, are

5 Reactor Dynamics 5 called precursors, and the intermediate nuclides A' m3 are called emitters. Emitters are daughters o precursors that are born in a highly excited state. Because their excitation energy is higher than the separation energy or one neutron, emitters can de-excite by promptly emitting a neutron. The delay in the appearance o the neutron is not caused by its emission, but rather by the delay in the beta decay o the precursor. I a high-energy beta particle rather than a lowenergy one is emitted, the excitation energy o the daughter nuclide is not high enough or it to emit a neutron, and hence decay mode does not result in emission o delayed neutrons.. Prompt, Delayed, and Total Neutron Yields d Although one cannot predict in advance which ission products will act as precursors, one can predict how many precursors on average will be produced per ission. This number is also equal to the number o delayed neutrons ultimately emitted and is called the delayed-neutron yield,. For incident neutron energies below 4 MeV, the delayed-neutron yield is essentially independent o the incident-neutron energy. I delayed neutrons are to be represented explicitly, the ission reaction can be written generically as: n X A B n n. (4) p p d d The total neutron yield is deined as the sum o the prompt and delayed neutron yields:. (5) d The delayed-neutron raction is deined as the ratio between the delayed-neutron yield and the total neutron yield: d p. (6) For neutron energies typical o those ound in a nuclear reactor, most o the energy dependence o the delayed-neutron raction is due to the energy dependence o the prompt-neutron yield and not to that o the delayed-neutron yield. This is the case because the latter is essentially independent o energy or incident neutrons with energies below 4 MeV..3 Delayed-Neutron Groups Precursors can be grouped according to their hal-lives. Such groups are called precursor groups or delayed-neutron groups. It is customary to use six delayed-neutron groups, but ewer or more groups can be used. For analysis o timerames o the order o 5 seconds, six delayedneutron groups generally provide suicient accuracy; or longer timerames, a greater number o groups might be needed. Partial delayed-neutron yields are deined or each precursor group. The partial delayed-neutron group yield represents the average number o precursors belonging to group that are produced per ission. Correspondingly, partial delayed-neutron ractions can be deined as: Obviously, d d. (7)

6 6 The Essential CANDU. (8) 1 Values o delayed-group constants or 35 U are shown in Table 1, which uses data rom [Rose1991]. Table 1 Delayed-neutron data or thermal ission in 35 U ([Rose1991]) Group Decay Constant, (s -1 ) Delayed Yield, d (n/iss.) Delayed Fraction, Total Simple Point-Kinetics Equation (Homogeneous Reactor) This section presents the derivation o the point-inetics equations starting rom the timedependent one-energy-group diusion equation or the simple case o a homogeneous reactor and assuming all ission neutrons to be prompt. 3.1 Neutron-Balance Equation without Delayed Neutrons The time-dependent one-energy-group diusion equation or a homogeneous reactor without delayed neutrons can be written as: n( r, ( r, D ( r, a( r, t, (9) where n represents the neutron density, is the macroscopic production cross section, a is the macroscopic neutron-absorption cross section, is the neutron lux, and D is the diusion coeicient. Equation (9) expresses the act that the rate o change in neutron density at any given point is the dierence between the ission source, expressed by the term ( r,, and the two sins: the absorption rate, expressed by the term a( r,, and the leaage rate, expressed by the term D ( r,. I the source is exactly equal to the sum o the sins, the reactor is critical, the time dependence is eliminated, and the static balance equation results: s( r) D s( r) as( r), (1) which is more customarily written as: D s( r ) as( r ) s( r ). (11)

7 Reactor Dynamics 7 To maintain the static orm o the diusion equation even when the ission source does not exactly equal the sum o the sins, the practice in reactor statics is to divide the ission source artiicially by the eective multiplication constant, e, which results in the static balance equation or a non-critical reactor: Using the expression or the geometric bucling: Eq. (1) becomes: 1 D s ( r ) as( r ) s( r ) e. (1) B g e D a, (13) s( r ) Bgs ( r). (14) Note that the geometrical bucling is determined solely by the reactor shape and size and is independent o the production or absorption macroscopic cross sections. It ollows that changes in the macroscopic cross sections do not inluence bucling; they inluence only the eective multiplication constant, which can be calculated as: e a DBg. (15) Because the value o geometrical bucling is independent o the macroscopic cross section, the shape o the static lux is independent o whether or not the reactor is critical. To progress to the derivation o the point-inetics equation, the assumption is made that the shape o the time-dependent lux does not change with time and is equal to the shape o the static lux. In mathematical orm: ( r, T( s( r), (16) where T( is a unction depending only on time. It ollows that the time-dependent lux ( r, satisies Eq. (14), and hence: ( r, Bg( r,. (17) Substituting this expression o the leaage term into the time-dependent neutron-balance equation (9) the ollowing is obtained: n( r, ( r, DBg ( r, a( r, t. (18) The one-group lux is the product o the neutron density and the average neutron speed, with the latter assumed to be independent o time: ( r, n( r, v. (19)

8 8 The Essential CANDU The neutron-balance equation can consequently be written as: n( r, v n( r, DBg v n( r, av n( r, t. () Integrating the local balance equation over the entire reactor volume V, the integral-balance equation is obtained: d dt n r t dv n r t dv DB n r t dv n r t dv. (1) (, ) v (, ) g v (, ) av (, ) V V V V The volume integral o the neutron density is the total neutron population nˆ( t ), which can also be expressed as the product o the average neutron density n( and the reactor volume: The volume-integrated lux ˆ ( as the product o the average lux ( V n( r, dv nˆ ( n( V. () can be deined in a similar ashion and can also be expressed and the reactor volume: (, ) ˆ r t dv ( ( V V. (3) It should be easy to see that the volume-integrated lux and the total neutron population satisy a similar relationship to that satisied by the neutron density and the neutron lux: ˆ ( ( r, dv n( r, v dv nˆ ( v V V. (4) With the notations just introduced, the balance equation or the total neutron population can be written as: dnˆ( v ˆ( ) v ˆ( ) v ˆ n t DBg n t a n( dt. (5) Equation (5) is a irst-order linear dierential equation, and its solution gives a ull description o the time dependence o the neutron population and implicitly o the neutron lux in a homogeneous reactor without delayed neutrons. However, to highlight certain important quantities which describe the dynamic reactor behaviour, it is customary to process its right-hand side (RHS) as ollows: dnˆ( DB dt g a 3. Average Neutron-Generation Time, Lietime, and Reactivity v nˆ (. (6) In this sub-section, several quantities related to neutron generation and activity are deined. Reactivity

9 Reactor Dynamics 9 Reactivity is a measure o the relative imbalance between productions and losses. It is deined as the ratio o the dierence between the production rate and the loss rate to the production rate. a g a g ˆ ˆ ˆ a DBg production rate - loss rate production rate ˆ DB Average neutron-generation time DB loss rate production rate e. (7) The average neutron-generation time is the ratio between the total neutron population and the neutron production rate. neutronpopulation nˆ nˆ 1 production rate ˆ nˆv v. (8) The average generation time can be interpreted as the time it would tae to attain the current neutron population at the current neutron-generation rate. It can also be interpreted as the average age o neutrons in the reactor. Average neutron lietime The average neutron lietime is the ratio between the total neutron population and the neutron loss rate: neutron population nˆ nˆ 1 loss rate ˆ ˆv a DBg a DBg n a DBg v. (9) The average neutron lietime can be interpreted as the time it would tae to lose all neutrons in the reactor at the current loss rate. It can also be interpreted as the average lie expectancy o neutrons in the reactor. The ratio o the average neutron-generation time and the average neutron lietime equals the eective multiplication constant: 1 a DBg v 1 a DBg v e. (3) It ollows that, or a critical reactor, the neutron-generation time and the neutron lietime are equal. It also ollows that, or a supercritical reactor, the lietime is longer than the generation time and that, or a sub-critical reactor, the lietime is shorter than the generation time. 3.3 Point-Kinetics Equation without Delayed Neutrons With the newly introduced quantities, the RHS o the neutron-balance equation (6) can be written as either:

10 1 The Essential CANDU or DB g a DB v nˆ ( v nˆ ( nˆ ( g a (31) DB 1 v ˆ( ) g a v ˆ( ) e ˆ DBg a n t DBg a n t n( DBg a. (3) The neutron-balance equation can thereore be written either as: dnˆ( nˆ( dt (33) or as: dnˆ( dt e 1 nˆ(. (34) Equation (33), as well as Eq. (34), is reerred to as the point-inetics equation without delayed neutrons. The name point inetics is used because, in this simpliied ormalism, the shape o the neutron lux and the neutron density distribution are ignored. The reactor is thereore reduced to a point, in the same way that an object is reduced to a point mass in simple inematics. Both orms o the point-inetics equation are valid. However, because most transients are induced by changes in the absorption cross section rather than in the ission cross section, the orm expressed by Eq. (33) has the mild advantage that the generation time remains constant during a transient (whereas the lietime does no. Consequently, this text will express the neutron-balance equation using the generation time. However, the reader should be advised that other texts use the lietime. Results obtained in the two ormalisms can be shown to be equivalent. I the reactivity and generation time remain constant during a transient, the obvious solution to the point-inetics equation (33) is: nˆ ( nˆ () e t. (35) I the reactivity and generation time are not constant over time, that is, i the balance equation is written as: dnˆ( ( n ˆ( t ) dt (, (36) the solution becomes slightly more involved and usually proceeds either by using the Laplace transorm or by time discretization. Beore advancing to accounting or delayed neutrons, one last remar will be made regarding the relationship between the neutron population and reactor power. Because the reactor power is the product o total ission rate and energy liberated per ission, it can be expressed as:

11 Reactor Dynamics 11 P( E ˆ ( ) ˆ iss t E iss n( v. (37) It can thereore be seen that the power has the same time dependence as the total neutron population. 3.4 Neutron-Balance Equation with Delayed Neutrons In the previous section, it was assumed that all neutrons resulting rom ission were prompt. This section taes a closer loo and accounts or the act that some o the neutrons are in act delayed neutrons resulting rom emitter decay Case o one delayed-neutron group As explained in Section.1, delayed neutrons are emitted by emitters, which are daughter nuclides o precursors coming out o ission. Because the neutron-emission process occurs promptly ater the creation o an emitter, the rate o delayed-neutron emission equals the rate o emitter creation and equals the rate o precursor decay. It was explained in Section.3 that precursors can be grouped by their hal-lie (or decay constan into several (most commonly six) groups. However, as a irst approximation, it can be assumed that all precursors can be lumped into a single group with an average decay constant. I the total concentration o precursors is denoted by C( r,, the total number o precursors in the core, C ˆ ( t ), is simply the volume integral o the precursor concentration and equals the product o the average precursor concentration C( and the reactor volume: (, ) ˆ C r t dv C( C( V V. (38) It ollows that the delayed-neutron production rate S ( r,, which equals the precursor decay rate, is: d Sd ( r, C( r,. (39) The corresponding volume-integrated quantities satisy a similar relationship: Sˆ ( ) ˆ d t C(. (4) The core-integrated neutron-balance equation now must account explicitly or both the prompt neutron source, ˆ p ( t ), and the delayed-neutron source: dnˆ( ˆ v ˆ( ) ( ) v ˆ( ) v ˆ p n t Sd t DBg n t a n( dt ˆ v ˆ( ) ( ) v ˆ( ) v ˆ p n t C t DBg n t a n(. (41) O course, to be able to evaluate the delayed-neutron source, a balance equation or the precursors must be written as well. Precursors are produced rom ission and are lost as a result o decay. It ollows that the precursor-balance equation can be written as:

12 1 The Essential CANDU dcˆ( t ) ˆ ( ) ˆ d t C ( t ) dt v ˆ( ) ˆ d n t C(. (4) The system o equations (41) and (4) completely describes the time dependence o the neutron and precursor populations. Just as in the case without delayed neutrons, they will be processed to highlight neutron-generation time and reactivity. Because reactivity is based on the total neutron yield rather than the prompt-neutron yield, the prompt-neutron source is expressed as the dierence between the total neutron source and the delayed-neutron source: dnˆ( v ˆ( ) v ˆ( ) ˆ( ) v ˆ( ) v ˆ n t d n t C t DBg n t a n( dt v ˆ( ) v ˆ( ) ˆ DBg a n t d n t C(. (43) The RHS is subsequently processed in a similar way to the no-delayed-neutron case: DB v nˆ ( v nˆ ( Cˆ ( g a d DBg a d v ˆ( ) ˆ n t C( nˆ( Cˆ (. (44) The neutron-balance equation can hence be written as: dnˆ( nˆ( Cˆ ( dt. (45) The RHS o the precursor-balance equation can be similarly processed: ˆ d v ˆ( ) ( ) v ˆ( ) ˆ ( ) ˆ( ) ˆ d n t C t n t C t n t C(, (46) leading to the ollowing orm o the precursor-balance equation: dcˆ( t ) nˆ( Cˆ ( dt. (47) Combining Eqs. (45) and (47), the system o point-inetics equations or the case o one delayed-neutron group is obtained: dnˆ( nˆ( Cˆ ( dt dcˆ( t ) nˆ( Cˆ ( dt. (48)

13 Reactor Dynamics Case o several delayed-neutron groups I the assumption that all precursors can be lumped into one single group is dropped and several precursor groups are considered, each with its own decay constant, then the delayed-neutron source is the sum o the delayed-neutron sources in all groups: Sˆ ( ) ˆ d t C ( 1, (49) where C ˆ ( represent the total population o precursors in group. The neutron-balance equation then becomes: dnˆ( ˆ v ˆ( ) ( ) v ˆ( ) v ˆ p n t Sd t DBg n t a n( dt ˆ ˆ ˆ ˆ p g a 1 Processing similar to the one-delayed-group case yields: v n( C ( DB v n( v n(. (5) ˆ( ) ˆ dn t dt nˆ( C ( 1. (51) Obviously, precursor-balance equations must now be written, one or each delayed group : dcˆ ( t ) v ˆ( ) ˆ d n t C ( ( 1... ) dt. (5) Processing the RHS o Eq. (5) as in the one-delayed-group case yields: ˆ v ˆ( ) ( ) d v ˆ( ) ˆ ( ) ˆ( ) ˆ d n t C t n t C t n t C (. (53) Finally, a system o +1 dierential equations is obtained: ˆ( ) ˆ dn t nˆ( C ( dt 1 dcˆ ( t ) nˆ( ˆ C ( ( 1... ) dt, (54) representing the point-inetics equations or the case with multiple delayed-neutron groups. 4 Solutions o the Point-Kinetics Equations Following the derivation o the point-inetics equations in the previous section, this section deals with solving the point-inetics equations or several particular cases. The irst case involves a steady-state (no time dependence) sub-critical nuclear reactor with an external neutron source constant over time. An external neutron source is a source which is independent o the neutron lux. The second case involves a single delayed-neutron group, or which an

14 14 The Essential CANDU analytical solution can be easily ound i the reactivity and generation time are constant. Finally, the general outline o the solution method or the case with several delayed-neutron groups is presented. 4.1 Stationary Solution: Source Multiplication Formula Possibly the simplest application o the point-inetics equations involves a steady-state subcritical rector with an external neutron source, that is, a source that is independent o the neutron lux. The strength o the external source is assumed constant over time. I the total strength o the source is Ŝ (n/s), the neutron-balance equation needs to be modiied to include this additional source o neutrons. The precursor-balance equations remain unchanged by the presence o the external neutron source. Because a steady-state solution is sought, the time derivatives on the let-hand side (LHS) o the point-inetics equations vanish. The steady-state point-inetics equations in the presence o an external source can thereore be written as: nˆ ˆ ˆ C S 1 nˆ ˆ C ( 1... ). (55) Equation (55) is a system o linear algebraic equations where the unnowns are the neutron and precursor populations. This can be easily seen by rearranging as ollows: nˆ ˆ ˆ C S 1 nˆ ˆ C ( 1... ). (56) The system can be easily solved by substitution, by ormally solving or the precursor populations in the precursor-balance equations: ˆ C ˆ ( 1... ) n and substituting the resulting expression into the neutron-balance equation to obtain: (57) nˆ nˆ Sˆ 1. (58) Noting that the sum o the partial delayed-neutron ractions equals the total delayed-neutron raction, as expressed by Eq. (58), the neutron-balance equation can be processed to yield: The neutron population is hence equal to: 1 nˆ nˆ ˆ ˆ ˆ ˆ ˆ ˆ n n n n n S 1. (59) ˆn Sˆ. (6)

15 Reactor Dynamics 15 Note that the reactivity is negative, and thereore the neutron population is positive. Equation (6) is called the source multiplication ormula. It shows that the neutron population can be obtained by multiplying the external source strength by the inverse o the reactivity, hence the name. The closer the reactor is to criticality, the larger the source multiplication and hence the neutron population. Substituting Eq. (6) into Eq. (57), the individual precursor concentrations become: ˆ ˆ S C ( 1... ). (61) The source multiplication ormula inds applications in describing the approach to critical during reactor start-up and in measuring reactivity-device worth. 4. Kinetics with One Group o Delayed Neutrons Another instance in which a simple analytical solution to the point-inetics equations can be developed is the case o a single delayed-neutron group. This sub-section develops and analyzes the properties o such a solution. The starting point is the system o dierential equations representing the neutron-balance and precursor-balance equations: dnˆ( nˆ( Cˆ ( dt dcˆ( t ) nˆ( Cˆ ( dt. (6) For the case where all inetics parameters are constant over time, this is a system o linear dierential equations with constant coeicients, which can be rewritten in matrix orm as: d nˆ ( nˆ ( dt Cˆ ( Cˆ (. (63) According to the general theory o systems o ordinary dierential equations, the irst step in solving Eq. (63) is to ind two undamental solutions o the type: n e t C. (64) The general solution can subsequently be expressed as a linear combination o the two undamental solutions: n ˆ( t ) n n t 1 1t a e a1 e Cˆ( t ) C C 1. (65) Coeicients a and a 1 are ound by applying the initial conditions. To ind the undamental solutions, expression (64) is substituted into Eq. (63) to obtain:

16 16 The Essential CANDU and subsequently: d n n t t e e dt C C, (66) n n t t e e C C. (67) Dividing both sides by the exponential term and rearranging the terms, the ollowing homogeneous linear system is obtained, called the characteristic system: n C. (68) This represents an eigenvalue-eigenvector problem, or which a solution is presented below. First, the system is rearranged so that the unnowns are each isolated on one side, and the system is rewritten as a regular system o two equations: n C n C. (69) Dividing the two equations side by side, an equation or the eigenvalues is obtained:. (7) This is a quadratic equation in, as can easily be seen ater rearranging it to:. (71) The two solutions to this quadratic equation are simply:

17 Reactor Dynamics 17,1 4. (7) Once the eigenvalues are nown, either the irst or the second o equations (69) can be used to ind the relationship between n and C. In doing so, care must be taen that the right eigenvalue (correct subscrip is used or the right n - C combination. Note that only the ratio o n and C can be determined. It ollows that either n or C can have an arbitrary value, which is usually chosen to be unity. For example, i the second equation (69) is used, and i n and n 1 are chosen to be unity, the two undamental solutions are: 1 e t The general solution is then: 1 e t 1 1. (73) 1 1 nˆ( a e a e C t t 1 1t ˆ( ) 1. (74) 4.3 Kinetics with Multiple Groups o Delayed Neutrons Having solved the inetics equations or one delayed-neutron group, it is now time to ocus on the solution o the general system, with several delayed-neutron groups. The starting point is the general set o point-inetics equations: ˆ( ) ˆ dn t nˆ( C ( dt 1 dcˆ ( t ) nˆ( ˆ C ( ( 1,..., ) dt. (75) As long as the coeicients are constant, this is simply a system o irst-order linear dierential equations, whose general solution is a linear combination o exponential undamental solutions o the type:

18 18 The Essential CANDU n C C 1 t e. (76) There are +1 such solutions, and the general solution can be expressed as: nˆ( l n ˆ 1 ( ) l C t C1 a e l l l Cˆ ( C 4.4 Inhour Equation, Asymptotic Behaviour, and Reactor Period t l. (77) By substituting the general orm o the undamental solution, Eq. (76), into the point-inetics equations and ollowing steps similar to those in the one-delayed-group case, the ollowing characteristic system can be obtained: n nc 1 C nc ( 1,..., ). (78) The components C can be expressed using the precursor equations in (78) as: C n ( 1,..., ). (79) Substituting this into the neutron-balance equation in (78) yields: n n n 1. (8) Note that the component n can be simpliied out o the above and that by rearranging terms, the ollowing expression or reactivity is obtained: 1. (81) Equation (81) is nown as the Inhour equation. Its +1 solutions determine the exponents o the +1 undamental solutions. To understand the nature o those solutions, it is useul to attempt a graphical solution o the Inhour equation by plotting its RHS as a unction o and observing its intersection points with a horizontal line at y. Such a plot is shown in Fig. 1 or the case o six delayed-neutron groups.

19 Reactor Dynamics Figure 1 Graphical representation o the Inhour equation For large (positive or negative) values o, the asymptotic behaviour o the RHS can be obtained as: 1. (8) The resulting oblique asymptote is represented by the blue line in Fig. 1, and the RHS plot (shown in bright green) approaches it at both and. Whenever equals minus the decay constant or one o the precursor groups, the RHS becomes ininite, and its plot has a vertical asymptote, shown as a (red) dashed line, at that value. For, the RHS vanishes, as can be seen rom Eq. (81), and hence the plot passes through the origin o the coordinate system. Three horizontal lines, corresponding to three reactivity values, are shown in violet. The two thin lines correspond to positive values, and the thic line corresponds to the negative value. Figure 1 shows that the solutions to the Inhour equation are distributed as ollows: - 1 solutions are located in the - 1 intervals separating the decay constants taen with negative signs, such that 1 1. All these solutions are negative. The largest solution, in an algebraic sense, is located to the right o 1 and is either negative or positive, depending on the sign o the reactivity. It will be reerred to as or. The smallest solution, in an algebraic sense, lies to the let o and will be reerred to as min or. It is (obviously) negative as well. Note that because the generation time is usually less than 1 ms, and oten less than.1 ms, the slope o the oblique asymptote is very small. Consequently, is very ar to the let o, and hence 1. The importance o this act will become clearer later, when the prompt-jump approximation will be discussed. Overall, the solutions are ordered as ollows: min

20 The Essential CANDU... min 1. (83) It is worth separating out the largest exponent in the general solution described by Eq. (77) by writing: nˆ( l n n ˆ 1 ( ) l C t C1 C t 1 a e a e l l1 l Cˆ ( C C Furthermore, it is worth actoring out the irst exponential term: t nˆ( l n n ˆ 1 ( ) l C t C t 1 C 1 l t e a al e l1 l Cˆ ( C C Note that because is the largest solution, all exponents l that or large values o t, all exponentials o the type l e solution can be approximated by a single exponential term: nˆ( n ˆ C1 ( t ) C1 a e Cˆ ( C This expression describes the asymptotic transient behaviour. The inverse o is called the asymptotic period: 1 T With this new notation, the asymptotic behaviour can be written as: t t l. (84). (85) are negative. It ollows nearly vanish, and hence the. (86). (87) nˆ( n ˆ 1 ( ) t C t C1 T a e Cˆ ( C. (88) Beore ending this sub-section, a ew more comments are warranted. In particular, it is worth considering the solution to the point-inetics equation (PKE) or three separate cases: negative

21 Reactor Dynamics 1 reactivity, zero reactivity, and positive reactivity. Negative reactivity In the case o negative reactivity, all exponents in the general solution are negative. It ollows that over time, both the neutron population and the precursor concentrations will drop to zero. O course, ater a long time, the asymptotic behaviour applies, which has a negative exponent. Zero reactivity In the case o zero reactivity, can be written as: vanishes, and all other l are negative. The general solution nˆ( l n n ˆ 1 ( ) l C t C1 C1 a a l e l1 l Cˆ ( C C t l. (89) Ater a suiciently long time, all the exponential terms die out, and the neutron and precursor populations stabilize at a constant value. Note that these populations do not need to remain constant rom the beginning o the transient, but only to stabilize at a constant value. Positive reactivity In the case o positive reactivity, is positive, and all other l are negative. Hence, ater suicient time has elapsed, all but the irst exponential term vanish, and the asymptotic behaviour is described by a single exponential which increases indeinitely. 4.5 Approximate Solution o the Point-Kinetics Equations: The Prompt Jump Approximation It was mentioned in the preceding sub-section that the smallest (in an algebraic sense) solution o the Inhour equation is much smaller than the remaining solutions. This important property will mae it possible to introduce the prompt jump approximation, which is the topic o this sub-section. By inspecting the Inhour plot in Figure 1, and eeping in mind the expression o the oblique asymptote given by Eq. (8), it is easy to notice that the oblique asymptote intersects the x-axis at: as. (9) It is also easy to see that: as. (91) 1 Assuming a reactivity smaller than approximately hal the delayed-neutron raction (equal to.65 according to Table 1), and assuming a generation time o approximately.1 ms, the

22 The Essential CANDU resulting value o as is approximately -3.5 s -1, which is much smaller than even the largest decay constant in Table 1 taen with a negative sign. That value is only -3 s -1. This shows that the ollowing inequality holds true:... min as 1. (9) The general solution o the point-inetics equations expressed by Eq. (77) can be processed to separate out the term corresponding to 1 : nˆ( 1 l n n n ˆ 1 1 ( ) l C t 1t C t C l 1 C 1 t e a e a 1 a l e l 1 l Cˆ ( C C C. (93) According to Eq. (9), or l t l are positive. The 1 only negative exponent is t, which is also much larger in absolute value than all 1 other exponents. It ollows that ater a very short time, t, the irst term o the RHS o Eq. (93) becomes negligible, and the solution o the point-inetics equations can then be approximated by:, all exponents o the type nˆ( 1 l l n n n ˆ ( ) l l C t 1t C 1 1 C1 l t C 1 e a 1 a l e a l e l l 1 l l Cˆ ( C C C Concentrating on the neutron population, its expression is: 1 t l. (94) l lt nˆ( aln e l. (95) Substituting this into the neutron-balance equation o the point-inetics system, the ollowing is obtained: Noting that the ollowing inequality holds true: 1 1 l l t l l t ˆ lal n e al n e C l1 l1 1. (96) l l,... 1, (97) the LHS o Eq. (96) can be approximated to vanish, and hence the equation can be approximated by:

23 Reactor Dynamics 3 which is equivalent to: 1 l l t ˆ al n e C l1 1, (98) nˆ ( ˆ C ( 1. (99) By adding the precursor-balance equations, the ollowing approximate point-inetics equations are obtained: nˆ ( ˆ C ( 1 dcˆ ( t ) nˆ( ˆ C ( ( 1,..., ) dt. (1) This system o dierential equations and one algebraic equation is nown as the prompt jump approximation o the point-inetics equations. The name comes rom the act that whenever a step reactivity change occurs, the prompt jump approximation results in a step change, a prompt jump, in the neutron population. To demonstrate this behaviour, let the reactivity change rom 1 to at time t. The neutron-balance equation beore and ater t can be written as: 1 nˆ( ˆ C ( ( t t) 1 nˆ( ˆ C ( ( t t) 1. (11) The limit o the neutron population as t approaches t rom the let, symbolically denoted as nˆ( t ), is ound rom the irst equation (11) to be equal to: nˆ( t ˆ ) C 1 1. (1) Similarly, the limit o the neutron population as t approaches t rom the right, symbolically denoted as nˆ( t ), is ound rom the second equation (11) to be equal to: nˆ( t ˆ ) C 1. (13) Taing the ratio o the preceding two equations side by side, the ollowing is obtained: nˆ( t ) nˆ( t ) There is thereore a jump nˆ equal to: 1. (14)

24 4 The Essential CANDU 1 1 nˆ ( t ˆ ˆ ˆ ˆ ˆ ) n( t ) n( t ) n( t ) n( t ) n( t ). (15) O course, the actual neutron population does not display such a jump; it is continuous at t. Nonetheless, a very short time t ater t (at t t ), the approximate and exact neutron populations become almost equal. Note also that Eq. (15) is valid only i both reactivities 1 and are less than the eective delayed-neutron raction. 5 Space-Time Kinetics using Flux Factorization In the previous sections, the time-dependent behaviour o a reactor was studied using the simple point-inetics model, which disregards changes in the spatial distribution o the neutron density. This section will improve on that model by presenting the general outline o spacetime inetics using lux actorization. The approach ollows roughly that used in [Rozon1998] and [Ott1985]. A complete and thorough treatment o the topic o space-time inetics is beyond the scope o this text. This section should thereore be regarded merely as a roadmap. The interested reader is encouraged to study the more detailed treatments in [Rozon1998], [Ott1985], and [Stacey197]. 5.1 Time-, Energy-, and Space-Dependent Multigroup Diusion Equation The space-time description o reactor inetics starts with the time-, space-, and energydependent diusion equation. An equivalent treatment starting rom the transport equation is also possible, but using the transport equation instead o the diusion equation does not introduce undamentally dierent issues, and the mathematical treatment is somewhat more cumbersome. The time-, space- and energy-dependent neutron diusion equation in the multigroup approximation can be written as ollows: 1 v g t g ( r, D ( r, ( r, ( r, ( r, ( r, ( r, g g rg g sg ' g g ' g ' g pg p g ' g ' dg g ' 1 ( r, ( r, ( r, ( r, ( r, C ( r, The accompanying precursor-balance equations are written as:. (16) c ( r, p ( r, g '( r, g '( r, C ( r, t g '. (17) Equations (16) and (17) represent the space-time inetics equations in their diusion approximation. Their solution is the topic o this section. It is advantageous or the development o the space-time inetics ormalism to introduce a set o multidimensional vectors and operators, as ollows: Flux vector

25 Reactor Dynamics 5 r, t r, t Φ g (18) Precursor vector ξ (, ) r t dgc ( r, (19) Loss operator g g rg g sg ' g g ' M r, t D ( r, ( r, ( r, ( r, ( r, ( r, g ' g (11) Prompt production operator Fp ( r, pg ( r, p ( r, g '( r, g '( r, g ' (111) Precursor production operator or precursor group F (, ) d r t dg ( r, C ( r, (11) Inverse-speed operator where g ' g is the Kronecer delta symbol. 1 v 1 v g g ' g, (113) Using these deinitions, the time-dependent multigroup diusion equation can be written in compact orm as: v t 1 Φ r, t M( r, Φr, t Fp( r, Φr, t The precursor-balance equations can be written as: 1 ξ ( r,. (114) ( r, d ( r, r, t ( r, ( 1,..., ) t ξ F Φ ξ. (115) As a last deinition, or two arbitrary vectors Φ( r, and Ψ( r,, the inner product is deined as: Φ, Ψ g V core g ( r, g ( r, dv. (116) 5. Flux Factorization Expressing a unction as a product o several (simpler) unctions is nown as actorization. It is a well-nown act rom partial dierential equations that trying to express the solution as a product o single-variable unctions oten simpliies the mathematical treatment. It is thereore reasonable to attempt a similar approach or the space-time inetics problem. A irst step in this approach is to actorize the time-, energy-, and space-dependent solution into a unction

26 6 The Essential CANDU dependent only on time and a vector dependent on energy, space, and time. The unction dependent only on time is called an amplitude unction, and the vector dependent on space, energy, and time is called a shape unction. The sought-or lux can thereore be expressed as: Φr, t p( Ψ( r,. (117) Such a actorization is always possible, regardless o the deinition o the unction p(. In this case, the unction p( is deined as ollows: 1 p( w( r), v Φ( r,, (118) where w( r ) is an arbitrary weight vector dependent only on energy and position: w r wg r. (119) According to its deinition, p( can be interpreted as a generalized neutron population. Indeed, i the weight unction were chosen to be unity, p( would be exactly equal to the neutron population. From the deinition o the lux actorization, it ollows that the shape vector Ψ( r, satisies the ollowing normalization condition: 1 w( r), v Ψ( r, 1. (1) Substituting the actorized orm o the lux into the space-, energy-, and time-dependent diusion equation, the ollowing equations (representing respectively the neutron and precursor balance) result: dp( dt t r, t p( r, t p( ( r, r, t 1 1 v Ψ v Ψ M Ψ p 1 p( F ( r, Ψ r, t ξ ( r,. (11) ( r, p( d ( r, r, t ( r, ( 1,..., ) t ξ F Ψ ξ. (1) The precursor-balance equation can be solved ormally to give: t t ( tt ') ( r, ( r,) e e p( t ') d ( r, t ') ( r, t ') dt ' ξ ξ F Ψ. (13) By taing the inner product with the weight vector w( r ) on both sides o the neutron-balance equation and the precursor-balance equation, the ollowing is obtained: dp( w dt p( w p( w 1 1 r ; v Ψr, t p( wr ; v Ψr, t r ; M( r, Ψr, t d dt r ; F ( r, Ψr, t wr p 1 ; ξ ( r,. (14)

27 Reactor Dynamics 7 t w( r ); ξ ( r, p( w( r ); F ( r, Ψ r, t w( r ); ξ ( r, ( 1,..., ) d. (15) Equations (14) and (15) can be processed into more elegant orms ain to the point-inetics equations. To do this, some quantities must be deined irst which will prove to be generalizations o the same quantities deined or the point-inetics equations. 5.3 Eective Generation Time, Eective Delayed-Neutron Fraction, and Dynamic Reactivity The ollowing quantities and symbols are introduced: Total production operator Dynamic reactivity Eective generation time ( F( r, Fp ( r, Fd ( r,. (16) w( r ), F( r, Ψ( r, w( r ), M( r, Ψ( r, w( r ), F( r, Ψ( r, ( 1 w( r), v Ψ( r, w( r), F( r, Ψ( r, Eective delayed-neutron raction or delayed group w( r ), Fd ( r, Ψ( r, ( w( r ), F( r, Ψ( r, Total eective delayed-neutron raction. (17). (18). (19) ( ( 1. (13) Group (generalized) precursor population ˆ C ( t ) w ( r ), ξ ( r, t ). (131) With the newly introduced quantities, Eqs. (14) and (15) can be rewritten in the amiliar orm o the point-inetics equations: ( ( p ( p( ˆ C ( ( 1 ˆ ( C ( ) ( ) ˆ t p t C ( ( 1,..., ) t (. (13) O course, to be able to deine quantities such as the dynamic reactivity, the shape vector

28 8 The Essential CANDU Ψ( r, must be nown or approximated at each time t. Dierent shape representations Ψ( r, lead to dierent space-time inetics models. All lux-actorization models alternate between calculating the shape vector Ψ( r, and solving the point-inetics-lie equations (13) or the amplitude unction and the precursor populations. The detailed energy- and spacedependent lux shape at each time t can subsequently be reconstructed by multiplying the amplitude unction by the shape vector. 5.4 Improved Quasistatic Model The improved quasistatic (IQS) model uses an exact shape Ψ( r,. By substituting the ormal solution to the precursor equations (13) into the general neutron-balance equation (11), the ollowing equation or the shape vector is obtained: dp( 1 1 v Ψ r, t p( r, t p( ( r, r, t dt t v Ψ M Ψ p( F ( r, Ψ r, t p t t ( tt ') ξ () e e p( t ') Fd ( r, t ') Ψ( r, t ') dt ' 1. (133) The IQS model alternates between solving the point-inetics-lie equations (13) and the shape equation (133). The corresponding IQS numerical method uses two sizes o time interval. Because the amplitude unction varies much more rapidly with time than the shape vector, the time interval used to solve the point-inetics-lie equations is much smaller than that used to solve or the shape vector. Note that, other than the time discretization, the IQS model and method include no approximation. The actual shape o the weight vector w( r ) is irrelevant. 5.5 Quasistatic Approximation The quasistatic approximation is derived by neglecting the time derivative o the shape vector in the shape-vector equation (133). The resulting equation, which is solved at each time step, is: dp( 1 v Ψ r, t p( M( r, Ψ r, t dt t t ( tt ') p( Fp ( r, Ψ r, t ξ () e e p( t ') Fd ( r, t ') Ψ( r, t ') dt ' 1. (134) The resulting shape is used to calculate the point-inetics parameters, which are then used in the point-inetics-lie equations (13). As in the case o the IQS model, Equation (134) is solved in conjunction with the point-inetics-lie equations (13). Aside rom the slightly modiied shape equation, the quasistatic model diers rom the IQS model in the values o its pointinetics parameters, which are now calculated using an approximate shape vector. 5.6 Adiabatic Approximation The adiabatic approximation completely does away with any time derivative in the shape equation and instead solves the static eigenvalue problem at each time t:

29 Reactor Dynamics 9 1 M( r, Ψr, t F ( r, Ψ( r,. (135) The resulting shape is used to calculate the point-inetics parameters, which are then used in the point-inetics-lie equations (13). As in the case o the IQS and quasistatic models, Equation (135) is solved in conjunction with the point-inetics-lie equations (13). 5.7 Point-Kinetics Approximation (Rigorous Derivation) In the case o the point-inetics model, the shape vector is determined only once at the beginning o the transient (t=) by solving the static eigenvalue problem: 1 M( r,) Ψr F ( r,) Ψ( r ). (136) The resulting shape is used to calculate the point-inetics parameters, which are then used in the point-inetics-lie equations (13). Because the shape vector is not updated, only the point-inetics-lie equations (13) are solved at each time t. In act, they are now the true point-inetics equations because the shape vector remains constant over time. This discussion has shown that the point-inetics equations can also be derived or an inhomogeneous reactor, as long as the lux is actorized into a shape vector depending only on energy and position and an amplitude unction depending only on time. 6 Perturbation Theory It should be obvious by now that dierent approximations o the shape vector lead to dierent values or the inetics parameters. It is thereore o interest to determine whether certain choices o the weight vector might maintain the accuracy o the inetics parameters even when approximate shape vectors are used. In particular, it would be interesting to obtain accurate values o the dynamic reactivity, which is the determining parameter or any transient. The issue o determining the weight unction that leads to the smallest errors in reactivity when small errors exist in the shape vector is addressed by perturbation theory. This section will present without proo some important results o perturbation theory. The interested reader is encouraged to consult [Rozon1998], [Ott1985], and [Stacey197] or detailed proos and additional results. 6.1 Essential Results rom Perturbation Theory Perturbation theory analyzes the eect on reactivity o small changes in reactor cross sections with respect to an initial critical state, called the reerence state. These changes are called perturbations, and the resulting state is called the perturbed state. Perturbation theory also analyzes the eect o calculating the reactivity using approximate rather than exact lux shapes. First-order perturbation theory states that the weight vector that achieves the best irst-order approximation o the reactivity (e.g., or the point-inetics equations) when using an approximate (rather than an exac lux shape is the adjoint unction, which is deined as the solution to the adjoint static eigenvalue problem or the initial critical state at t=: M * ( r,) Ψ * r, F * ( r,) Ψ * r,. (137) The adjoint problem diers rom the usual direct problem in that all operators are replaced by

30 3 The Essential CANDU their adjoint counterparts. The adjoint A * o an operator A is the operator which, or any arbitrary vectors Φ( r, and Ψ( r,, satisies: * Φ, AΨ A Φ, Ψ. (138) The reactivity at time t can thereore be expressed as: * * ( r,), ( r, ( r, ( r,), ( r, ( r, ( Ψ F Ψ Ψ M Ψ * Ψ ( r,), F( r, Ψ( r,. (139) The remaining point-inetics parameters can be expressed similarly using the initial adjoint as the weight unction. An additional result rom perturbation theory states that when the adjoint unction is used as the weight unction, the reactivity resulting rom small perturbations applied to an initially critical reactor can be calculated as: * * Ψ ( r,), F( r, Ψ( r,) Ψ ( r,), M( r, Ψ( r,) ( * Ψ ( r,), F( r,) Ψ( r,), (14) where the symbols represent perturbations (changes) in the respective operators with respect to the initial critical state. Equation (14) oers a simpler way o calculating the reactivity than Eq. (139) because it does not require recalculation o the shape vector at each time t. Note that, within irst-order o approximation, the calculated reactivity is also equal to the static reactivity at time t, deined as: ( 1 1 (. (141) In act, perturbation theory can also be used to calculate the (static) reactivity when the initial unperturbed state is not critical. In that case, the change in reactivity is calculated as: 1 1 e 1 Ψ, FΨ Ψ, MΨ * * e * e e Ψ, FΨ. (14) In Eq. (14), the subscript or superscript denotes the unperturbed state. Finally, or oneenergy-group representations, the direct lux and the adjoint unction are equal. It ollows that in a one-group representation, the reactivity at time t can be expressed as: Ψ( r,), F( r, Ψ( r,) Ψ( r,), M( r, Ψ( r,) ( Ψ( r,), F( r,) Ψ( r,) Vcore ( r,) Vcore ( r,) dv a dv. (143)