STABILITY ANALYSIS OF ANALYSE DE STABILITE DE LA SPATIAL POWER DISTRIBUTION REPARTITION SPATIALE DE PUISSANCE IN RBMK-1000 REACTOR DANS LE REACTEUR RBMK-1000
by P Gulshani and A.R. Dastur par P. Gulshani et A.R. Dastur Atomic Energy of Canada Limited L'Energie Atomique du Canada Limitee CANDU Operations Operations CANDU and et B. Chexal B. Chexal Nuclear Safety Analysis Center Nuclear Safety Analysis Center Electric Power Research Institute Electric Power Research Institute 3412 Hillview Avenue 3412, Hillview Avenue Palo Alto. California, 94304 Palo Alto, California, 94304
December 1987 Decembre 1987
Atomic Energy L'Cnergie Atomique ot Canada Limited du Canada. Limttee CANDU Operations Operations CANDU
Slwridan Park RtMtrch Community Miuisuuga. Ontario L5K 1B2 Canada T«l (416) 823-9040 AECL-9424 STABILITY ANALYSIS OF ANALYSE DE STABILITY DE LA SPATIAL POWER DISTRIBUTION REPARTITION SPATIALE DE PUISSANCE IN RBMK-1000 REACTOR DANS LE REACTEUR RBMK-1000
by P Gulshani, A.R. Dastur and B. Chexa! par P. Gulshani. A.R Dastur et B. Chexal
Abstract R6sum6
A simple model describing the linearized dynamics o? a Un modele simple decrivant la dynamique Imearisee d'un coupled neutronic and thermohydraulic reactor system was reacteur neutronique et thermohydraulique couple, a ete used to analyze the stability of the radial and azimuthal utilise pour etudier la stablilite be la distribution de puis- power distribution in the boiling and spatially uncontrolled sance radiale et azimutale dans le reacteur RBMK-1000 RBMK-1000 reactor (similar to CHERNOBYL-4) A para- (semblable au reacteur Tchernobyl 4) bouillant et non metric survey of stability of the power distribution was contr6le au point de vue spatial, line etude parametrique carried out over a range of power levels, flowrates and de stabilite de la distribution de puissance a ete menee subcriticality of the power spatial modes The significance sur une serie de niveaux de puissance, de debits et de of the various feedback mechanisms such as the moder- sous-criticite des modes spatiaux de la puissance. L'lmpor- ator temperature, void, xenon and doppler reactivity are tance des divers mecanismes de retroaction comme la examined A map of instability trends in power spatial dis- temperature du moderateur, la reactivite due au coefficient tribution with the mode subcriticality and the reactor power de vide ainsi que la reactivite xenon et Doppler y sont and flowrate was developed. It is predicted that all lower etudies. Une courbe d'instabllite de la distribution spatiale modes of reactor power spatial distribution of interest are de puissance avec la sous-criticite et la puissance et debit unstable at all conditions in power and flowrate particular- du reacteur a ete elaboree. On prevoit que tous les modes ly at low power levels and flowrates. inferieurs interessants de repartition spatiale de la puis- sance du reacteur sont instables, dans toutes les condi- tions de puissance et de debit, et prmcipalement a de faibles debits et niveaux de puissance.
December 1987 Decembre 1987 ACKNOWLEDGMENT The authors would like to thank Dr. D.B. Primeau of the Atomic Energy of Canada Limited for his valuable suggestions and guidance in the course of this work. Thanks are also due to the reviewers from the Electric Power Research Institute and Atomic Energy of Canada Limited for their useful comments and constructive criticisms. This work was partially funded under a contract from Electric Power Research Institute, Palo Alto, California. CONTENTS
Section page 1 INTRODUCTION AND SUMMARY 1-1 2 FINDINGS AND CONCLUSIONS 2-1 3 DESCRIPTION OF RBMK-1000 REACTOR 3-1 4 ANALYSIS 4-1 The Model 4-1 Method of Solution - Linearization of Equations 4-6 5 RESULTS 5-1 Density Response and Power Coefficient Trends with 5-1 •. and Power at Fixed Flowrate and with Flowrate at Fixed Power Power Instability Trend with Xenon Reactivity Feedback Only 5-8 Power Instability Trend without Xenon Reactivity and with 5-14 without Moderator Temperature Feedbacks and Delayed Neutrons Power Instability Trends with All Major Reactivity 5-19 Feedbacks Included Instability Trend with Power and Subcriticality at 5-19 Fixed Flowrate Instability Trend with Flowrate and Subcriticality at 5-24 Fixed Power Trends in Density Response and Power Coefficient with 5-24 Power and Flowrate Instability Trend with Power and Flowrate for First Azimuthal Mode 5-30
6. REFERENCES 6-1 ILLUSTRATIONS Figure Page 3-1 Schematics of RBMK-1000 Reactor 3-2 3-2 Density Coefficient of Reactivity Versus Coolant Density Used 3-5 in Analysis of RBMK-1000 Reactor Linear Stability 3-3 Fuel Temperature Reactivity Coefficient Versus Fuel Temperature 3-6 Used in Analysis of RBMK-1000 Reactor Linear Stability 4-1 Reactor Model Components 4-2 4-2 Spatial Forms of Various Neutron Flux Mode 4-3 4-3 Feedback Loop Schematics 4-7 5-1 Possible Instability Modes in Neutron Flux Spatial Distribution 5-2 5-2 Change in Coolant Density Versus Power in RBMK-1000 Reactor at 5-3 100% Flowrate (Using Cosine Shape Flux Square Weighted Density) 5-3 Power Coefficient of Reactivity Versus Power at 100% Flowrate 5-5 in the RBMK-1000 Reactor with Void and Fuel and Moderator Temperature Reactivity Feedbacks Included (Using Cosine Shape Flux Square Weighted Density) 5-4 Power Coefficient of Reactivity Versus Power at 100% Flowrate 5-6 in RBMK-1000 Reactor with Only Void and Fuel Temperature Reactivity Feedbacks Included (Using Cosine Shape Flux Square Weighted Density) 5-5 Power Coefficient of Reactivity Versus Power at 100% Flowrate 5-7 in (Using Cosine Shape Flux Square Weighted Density) 5-6 Change in Coolant Density Versus Flowrate in RBMK-1000 Reactor 5-9 at 100% Power (Using Cosine Shape Flux Square Weighted Density) 5-7 Power Coefficient of Reactivity Versus Flowrate in RBMK-1000 5-10 Reactor at 100% Power (Using Cosine Shape Flux Square Weighted Density)
5-8 Amplification Factor Versus Subcriticality of Neutron Flux 5-11 Spatial Mode Excited by Xenon Reactivity Feedback Only and with No Delayed Neutrons at Various Power Levels in RBMK-1000 Reactor 5-9 Angular Frequency Versus Subcriticality of Neutron Flux Spatial 5-12 Mode Excited by Xenon Reactivity Feedback Only and with no Delayed Neutrons at Various Power Levels in>RBMK-1000 Reactor ILLUSTRATIONS (Cont'd.) Figure Page 5-10 Amplification Factor and Angular Frequency of Xenon-Feedback 5-13 Induced Oscillations Versus Subcriticality of Neutron Flux Spatial Mode in RBMK-1000 Reactor With All Other Feedbacks Neglected and with no Delayed Neutrons and at 100% Power and Flowrate 5-11A Amplification Factor Versus Power at Various Subcriticality of 5-15 Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Flowrate without Xenon Reactivity Feedback and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) 5-1 IB Amplification Factor Versus Power at Various Subcriticality of 5-16 Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Flowrate without Xenon Reactivity Feedback and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) (Figure 5-10A Continued) 5-12 Amplification Factor Versus Power at Various Subcriticality of 5-17 Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Flowrate with Delayed Neutron Kinetics and Including Major, Except Xenon, Reactivity Feedbacks (Using Cosine Shape Flux Square Weighted Density)
5-13 Amplification Factor Versus Power at Various Subcriticality of 5-18 Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Flowrate without Moderator Temperature and Xenon Reactivity Feedbacks and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) 5-1AA Amplification Factor Versus Power at Various Subcriticality of 5-20 Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Flowrate with Major Reactivity Feedbacks Included and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) 5-1AB Amplification Factor Versus Power at Various Subcriticality of 5-21 Neutron Flux Spatial Mode in RMBK-1000 Reactor at 100% Flowrate with Major Reactivity Feedbacks Included and with no Delayed Neutrons (Using Cosine-Shape-Flux Square Weighted Density) (Figure 5-13A Continued) 5-15 Angular Frequency Versus Power at Various Subcriticality of 5-22 Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Flowrate with Major Reactivity Feedbacks Included and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) ILLUSTRATIONS (Cont'd.)
5-16 Angular Frequency and Amplification Factor Versus Power for 5-23 Flux Spatial Mode With 20 mk Subcriticality in RBMK-1000 Reactor at 100% Flowrate With Major Reactivity Feedbacks Included and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) 5-17A Amplification Factor Versus Flowrate at Various Subcriticality 5-25 of Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Power with Major Feedbacks Included and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) 5-17B Amplification Factor Versus Flowrate at Various Subcriticality 5-26 of Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Power with Major Feedbacks Included and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) (Figure 5-16A Continued) 5-17B Amplification Factor Versus Flowrate at Various Subcriticality 5-26 of Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Power with Major Feedbacks Included and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) (Figure 5-16A Continued) 5-18 Angular Frequency Versus Flowrate at Various Subcriticality of 5-27 Neutron Flux Spatial Mode in RBMK-1000 Reactor at 100% Power with Major Feedbacks Included and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) 5-19 Change in Coolant Density Versus Power in RBMK-1000 Reactor at 5-28 Various Flowrates and at Fuel Frequency (Using Cosine Shape Flux Square Weighted Density) 5-20 Power Coefficient of Reactivity Versus Power in RBMK-1000 5-29 Reactor at Various Flowrates and at Fuel Frequency (Using Cosine Shape Flux Square Weighted Density) 5-21 Amplification Factor Versus Power for First Azimuthal Mode of 5-31 Neutron Flux Distribution of 7 mk Subcriticality in RBMK-1000 Reactor at Various Flowrates and with Major Reactivity Feedbacks Included and with no Delayed Neutrons (Using Cosine Shape Flux Square Weighted Density) 5-22 Amplification Factor Versus Power for First Azimuthal Mode of 5-32 Neutron Flux Distribution of 7 mk Subcriticality in RBMK-1000 Reactor at Various Flowrates and with Delayed Neutron Kinetics and Major, Except Xenon, Reactivity Feedbacks Included (Using Cosine Shape Flux Square Weighted Density)
IV TABLES Table Page 3-1 Major Thermohydraulic Characteristics of RBMK-1000 Reactor 3-3 3-2 Major Neutronic Characteristics of RBMK-1000 Reactor 3-4 4-1 Effects and Time Constants of Reactor Processes 4-3 Section 1 INTRODUCTION AND SUMMARY
This report presents the results of a parametric survey of instability in the radial and azimuthal power spatial distribution in the spatially uncontrolled (i.e. with no absorber rods in the reactor core) Soviet RMBK-1000 reactor (similar to CHERNOBYL-A) using a linear model of the reactor and the coupled thermohydraulic system. In this model, only small departures from reactor criticality and the given steady state reactor thermohydraulic conditions were considered. The RBMK-1000 reactor is boiling light water cooled and graphite moderated and contains enriched fuel (similar to CHERNOBYL-A). The reactor has positive lattice void and moderator temperature coefficients of reactivity. Both of these feedbacks have destabilizing effects on the reactor power distribution. Furthermore, the reactor is neutronically large (i.e., the reactor cross-sectional area is very large compared to the neutron migration area) and is characterized by a small neutron leakage. This characteristic is reflected in the low subcriticality of higher harmonic modes of power spatial distribution. Thus, a local reactivity change could cause a spatial mode of the reactor power to diverge in the absence of control.
The main objective of the parametric survey was to gain an understanding of general stability characteristics of power spatial distribution for this class of reactor. Thus, the parametric survey was used to obtain a better understanding of the relative importance of the various reactivity feedback mechanisms such as the fuel temperature, moderator temperature, void and xenon reactivity. The survey was also used to obtain a map of instability trends in power spatial distribution with the mode subcriticality, the reactor power and flowrate, etc. To achieve these objectives, simple linear models of the reactor and the coupled thermohydraulics of the fuel channel were used. The linear model for the reactor kinetics was derived by linearizing the equations governing the neutron diffusion and iodine and xenon concentrations about a given critical reactor operating condition. (In this linearization, only first order deviations in the variables from the operating condition were retained in the equations.) Similarly, the equations governing the dynamics of the core flow and fuel and moderator temperatures and the equation describing the reactivity feedbacks were linearized about a given reactor thermohydraulic operating condition. The equations were then Laplace transformed, combined and solved for the Laplace parameter. This parameter is simply a complex frequency whose real part determines the exponential rate of change in the neutron flux and whose imaginary part is the frequency of oscillations (if any) in the flux.
The assumptions in the linear models were based on the following considerations. The interest in the study was confined to fuel, coolant, moderator and xenon response with frequencies less than a fraction of a
1 - 1 hertz. For the study of the core response to small disturbances, it was considered reasonable to assume constant header and steam drum pressures and thereby examine the behaviour of the header-to-drum part of the circuit rather than the entire circuit. In the equation for the reactivity feedbacks, the reactivity coefficients of the coolant density, fuel temperature, moderator temperature and xenon concentration computed in Reference 1 were used.
To compute the channel coolant density response and, hence, the void reactivity feedback, a linear model of the header-channel-steam drum system was used. In this model, a constant header-to-drum pressure drop was imposed. The channel power was represented by a point heat source located at the position of the coolant boiling boundary, i.e., the entire channel power was assumed to act at this location. This simplifying approximation overestimated the coolant density response. The model predicted that the change in the mass flowrate as a consequence of constant header-to-drum pressure drop had negligible effect on the density response compared to that of the power. The response was predicted to depend only on the ratio of power to flow rather than on power and flow separately.
In the linear model for the reactor, prompt neutron cycle time was ignored as its contribution in the frequency range of interest (i.e., at and below a few cycles per second) is negligibly small due to short prompt neutron lifetime. A reduced two-group representation was used for the usual six groups of delayed neutrons. This approximation made possible a more detailed study of the effects of the reactivity feedback mechanisms. With this approximation, the linear model equations were solved to obtain closed-form solutions for the (exponential or e-folding) growth factor and the frequency for each oscillatory or non-oscillatory spatial mode of the reactor power as a function of the mode subcriticality, the reactor power, flowrate, pressure and core inlet fluid temperature. In the parametric survey presented in this report the system pressure and core inlet fluid temperature were held fixed at their nominal full power condition values.
The models described above give the following predictions.
At a given flowrate, the predicted channel coolant density response is small at low (near 20% for 100% flowrate) and high (near 100%) power levels. At these two power levels, the channel void nearly collapses and the boiling boundary is near the channel entrance respectively. The density response exhibits a maximum at an intermediate low power. At lower flowrates, the maximum in the density response occurs at lower power levels. The density response behaves in this manner because it depends on the power-to-flow ratio rather than on the power or/and the flow separately. The maximum response is, however, a constant independent of the power and flowrate and is about -0.65 (gm/cm3) per Full Power Unit (FPU) (-8.35 lbm/ftVFPU) at a frequency equal to that of the fuel.
The behaviour of the lattice power coefficient of reactivity with power and flowrate is similar to that of the density response. This indicates that the void reactivity feedback mechanism is dominant. The coefficient is
1 - 2 positive at nearly all power levels and flowrates of interest. (With the reactivity coefficients obtained from lattice physics calculations at a fuel burnup of 1300Q MWd/teU, the power coefficient is predicted near zero at normal operating condition.) The maximum in the power coefficient is nearly a constant independent of the power and flowrate and, at the fuel characteristic frequency, is about +12 mk/FPU. The growth or amplification factor o is predicted positive for the entire range of power and flowrate studied and for lower mode subcriticality. Thus, all spatial modes of the reactor power distribution of practical interest are unstable with core boiling. The dominant instability mechanism is shown to be the void-reactivity feedback for o greater than 10"3 per second. The destabilizing effects of the moderator temperature and xenon feedback are significant for o less than 10"' per second. Delayed neutrons have a relatively small stabilizing effect at all growth rates particularly for growth rates less than about 10"a per second. For most conditions of interest, the power instability is of exponential, i.e., non-oscillatory, nature. At lower flowrates, the maximum in o occurs at lower power levels as that in the density response. The maximum value of o, however, remains nearly constant independent of power and flowrate for a given subcriticality and is about +0.19 per second, i.e., a reactor period of about 5.26 seconds, for the first azimuthal mode which is 7 mk subcritical. Thus, the reactor is most unstable at low power levels and flowrates. The above instability trends in the power spatial distribution have the following implications for the reactor operation. At high and, in particular, at full power conditions, the destabilizing effect of the moderator temperature reactivity feedback dominates. Since this feedback mechanism is slow (with about 16 minute time constant), the reactor power is readily controlled manually or otherwise. At lower reactor power and flowrate experienced during startup and shutdown, the destabilizing effect of the void-reactivity feedback becomes dominant. This feedback mechanism is fast (i.e., on the order of the fuel time constant) and would cause the reactor power to undergo rapid excursion independently of the power-to-flux ratio maintained during startup and shutdown. Thus, at lower power and flowrate, special care in the design of the control system would be required.
1 - 3 Section 2 FINDINGS AND CONCLUSIONS
The parametric survey of instabilities in the radial and a^.imuthal power spatial distributions gives the following results. i. The density response and, hence, the lattice power coefficient of reactivity relevant to the spatial modes exhibits a maximum at a low power level for a given flowrate. This power is lower, the lower the flowrate. These trends in the density response and the power coefficient with power and flowrate are determined mainly by the void reactivity feedback. The power coefficient is positive at nearly all power levels and flowrates with core boiling. With the reactivity coefficients obtained from lattice physics calculations at a fuel burnup of 13000 MWd/teU, the power coefficient is predicted close to zero at normal operating condition. The density response and the power coefficient depend (mainly) on the power-to-flow ratio rather the power or/and the flow separately.
ii. The radial and azimuthal modes of power spatial distribution with subcriticality less than 25 mk are unstable at all power levels and flowrates. The instability is mainly of exponential (non-oscillatory) nature. The dominant instability mechanism is the void reactivity feedback for instability growth rates greater than 10"3 per second. The destabilizing effects of the xenon and moderator temperature reactivity feedbacks contribute significantly for instability growth rates less than 10"3 per second. The stabilizing effect of delayed neutrons is relatively small at all growth rates particularly for growth rates below about 10" * per second. iii. For a given mode subcriticality and flowrate, the maximum instability in power occurs at a low power level at which the void reactivity feedback exhibits a maximum. At lower flowrates, this maximum occurs at lower power levels. (At 100% flowrate, the power at which the instability in the power distribution is maximum is about 30% nominal.) iv. For a given mode subcriticality, the maximum e-folding (i.e., exponential) growth rate is a constant independent of the power and flowrate. For the first azimuthal mode, this rate is about +0.19 per second, i.e., a reactor period of about 5.26 seconds.
2 - 1 The significance of these instability trends in the power spatial distribution for the reactor operation is as follows. At high and, particularly, at full power conditions, the destabilizing effect of the moderator temperature reactivity feedback dominates. This feedback mechanism is slow and, hence, the reactor power is readily controlled. At lower reactor power and flowrate experienced during startup and shutdown, the destabilizating effect of the void-reactivity feedback becomes dominant. This feedback mechanism is fast and would cause the reactor power to increase independently of the power-to-flow ratio maintained during startup and shutdown. Thus, at lower power and flowrate, special care in the design of the control system would be required. (The linear model given in this report could easily be extended to study the problem of the reactor control.) It is therefore concluded, based on the reactor properties deduced from the literature, that in the small signal approximation, several spatial modes of power distribution in the spatially uncontrolled RBMK-1000 reactor are unstable at all, and particularly at low, power levels and flowrates.
2-2 Section 3 DESCRIPTION OF RBMK-1000 REACTOR
Figure 3-1 shows a schematic of the two-loop RBMK-1000 reactor and Tables 3-1 and 3-2 and Figures 3-2 and 3-3 give its main physical characteristics. The RBMK-1000 is a graphite moderated, boiling light-water cooled reactor using (2%) enriched uranium oxide fuel. It is a pressure tube reactor with vertical zirconium-niobium pressure tubes containing eighteen-element fuel bundles. Each pressure tube (i.e., channel) penetrates a graphite block. The square graphite block and the pressure tube contact through a graphite sleeve. The sleeve consists of rings of graphite which are alternately in contact with the pressure tube and the graphite block along the length of the tube. The stacked square graphite blocks are housed in a thin-walled cylindrical vessel. To prevent oxidation of graphite and improve heat transfer from the graphite blocks to the coolant, the reactor space is filled with a mixture of helium and nitrogen.
The two-phase mixture from the channels is fed through individual risers to four steam drum separators from which the steam goes to the turbines. The coolant water from the steam drum is fed to the individual fuel channels through downcomers, main circulating pumps, collector header, group dispensing headers and feeders. The flow to the individual channels is controlled by control valves. Because of the relatively high neutron absorption cross-section of light water, the RBMK-1000 reactor has a high positive void coefficient of reactivity as indicated in Figure 3-2. The moderator temperature coefficient of reactivity is also positive as indicated in Table 3-2. Thus, channel coolant voiding causes the reactor power to increase locally unless checked by the negative fuel temperature coefficient of reactivity (Figure 3-3) and/or control rod movement. (The lattice power coefficient of reactivity is shown below to be positive with core voidage indicating that the negative fuel temperature coefficient is not large enough to compensate for the positive void coefficient.) (The reactivity coefficients used in the analysis were obtained from lattice physics calculations at a fuel burnup of 13000 MWd/teU. System reactivity coefficients may be different because of the distribution of fuel temperature, coolant density and most importantly fuel burnup. The void coefficient of 30 mk/(100% void) at the working point and at the operating condition before the accident at CHERNOBYL as quoted by the Soviets (2) is obtained at a burnup slightly higher than 13000 MWd/teU.) Furthermore, the RBMK-1000 reactor is neutronically large (i.e., the reactor diameter is very large compared to the neutron mean free path). This is reflected in the relatively small subcriticality of the higher harmonic modes of the flux spatial distribution as seen in Table 3-2. The reactor is also characterized by small neutron leakage. Consequently, different regions of the reactor tend to act neutronically independently in response to a given reactivity perturbation. A local reactivity disturbance could result in large local distortion in neutron flux. Positive void reactivity feedback then causes further increase in the local flux until the region becomes critical on its own.
3 - 1 FEEDERS CORE CROUP MAIN DISPENSING CIRCULATING HEADERS PUMP
FIGURE 3-1 SCHEMATICS OF RBMK-1000 REACTOR
3-2 Table 3-1 MAJOR THERMOHYDRAULIC CHARACTERISTICS OF RBMK-1000 REACTOR*
Total Thermal Power (MW) (Btu/hr) 3200 (842,7) Total Core Flowrate (Mt/hr) (lbm/hr) 42 (19051) Number of Fuel Channels 1661 Number of Fuel Elements per Bundle 18 Fuel Element Sheath Outer Diameter (mm) (in) 11.95 (0.47) Zr - 2.5% Nb Pressure Tube Inner Diameter (mm) (in) 80 (3.15) Channel Flow Area (cm2) (in2) 24.12 (3.74) Core Height (m) (ft) 7 (22.97) Core Diameter (m) (ft) 11.8 (38.71) Number of Graphite Moderator Blocks 2000 Channel Square Array Lattice Spacing (cm) (in) 25 (9.84) Number of Main Circulation Pumps (Total/normally operating) 8/6 Pump Head (MPa) 1.5 The following parameter values are at nominal full power conditions: Core Inlet Fluid Temperature (°C) (°F) 270 (518) Core Outlet Average Quality (%) 14 Core Outlet Pressure (MPa) (psia) 7.1 (217.6) Fuel Average Temperature CO (°F) 546 (1015) Moderator Operating Temperature (maximum) (°C) (°F) 650 (1202) Maximum Temperature Drop From Graphite Cell Boundary to 150 (302) Pressure Tube (°C) (°F) Effective Heat Transfer Coefficient From Fuel to Coolant 1.563 (0.077) (kW/mJ0C) (Btu/sec.ft'.°F) Effective Heat Transfer Coefficient From Moderator to 0.161 (0.0079) Coolant (kW/m2°C) (Btu/sec.ft2.°F) Fuel Time Constant (s) 5.9 Moderator Time Constant (hr) 8
* This data was obtained from a number of reports (such as those in References 2 and 3) in the literature.
3 - 3 Table 3-2 MAJOR CORE NEUTRONIC CHARACTERISTICS OF RBMK-1000 REACTOR*
Energy Transferred to Graphite (from slowing down of neutrons 5.5 and absorption of gamma radiation)(%) Lattice Coolant Density Coefficient of Reactivity (mk/gm/cm3) See Figure 3-2 Lattice Fuel Temperature Coefficient of Reactivity (nk/°C) See Figure 3-3 Lattice Moderator Temperature Coefficient of +0.07 Reactivity (mk/°C) Xenon Concentration Coefficient of Reactivity (mk/FPU) -28 Neutron Flux (n/cmJ/s) A.I x 10*13 Xenon Absorption Cross-Section (cmJ) 3.5 x 10"le Subcriticality of First Azimuthal Mode (ink) 6 to 7.5 Subcriticality of Second Azimuthal Mode (mk) 15 to 17.7 Subcriticality of First Axial Mode (mk) 25
This data was obtained from a number of reports (such as those in References 2 and 3) in the literature and from lattice calculations.
3 - A 0.1 0.2 0.3 0.4 0.5 0.6 07 08
COOLANT DENSITY (gm/cm3)
FIGURE 3-2 DENSITY COEFFICIENT OF REACTIVITY VERSUS COOLANT DENSITY USED IN ANALYSIS OF RBMK-1000 REACTOR LINEAR STABILITY FUEL TEMPEHATURE (°C)
300 400 500 600 700 BOO 900 1OO0 1100 i 1 -10
-11-
>
Ul ir -12- mtt Q. UJ SO U1(J -13- ui u.
-14-
-15-
FIGURE 3-3 FUEL TEMPERATURE REACTIVITY COEFFICIENT VERSUS FUEL TEMPERATURE USED IN ANALYSIS OF RBMK-1000 REACTOR LINEAR STABILITY Section A ANALYSIS
To understand the nature of and determine the possible modes of instability in the power spatial distribution in the RBMK-1000 reactor, a parametric survey of the instability was carried out. This section describes the models, assumptions, approximations and the method of solution of the governing conservation equations used.
THE MODEL In a critical reactor, there is a balance between the neutron production (through fission), absorption (in the fuel, moderator and the reactor materials) and leakage (out through the reactor boundaries). A change (6k) in reactivity (i.e. change in the neutron balance) changes thp reactor operating conditions: power, neutron flux (6
6k = af • 6Tf + Oj,, • 6Tm + av • 6pc + ax • 6X + 6kp (4-1) Each of the feedback mechanisms is characterized by its own time constant. Table 4-1 indicates the order of magnitude of these time constants. The physical significance of these time constants is as follows. An unstable reactor mode has a time constant determined by a number of these mechanisms of similar or shorter time constants. Thus, feedback mechanisms with much longer time constants than that of the mode may be disregarded. A neutron kinetic model with no spatial control system was used. The feedback effects on reactivity were computed from the rate equations for xenon and iodine concentrations and from models for the fuel and moderator
4 - 1 Skn 6k REACTOR KINETICS
MODERATOR FUEL ELEMENT
6T, 6Tm
XENON 6X REACTIVITY FEEDBACK CHANNEL
MODERATOR TEMPERATURE REACTIVITY FEEDBACK
VOID REACTIVITY FEEDBACK
FUEL TEMPERATURE REACTIVITY FEEDBACK
FIGURE 4-1 REACTOR MODEL COMPONENTS A A
(a) FUNDAMENTAL (oo) MODE
(b) FIRST AZIMUTHAL (c) SECOND AZIMUTHAL (01) MODE (02) MODE
A A A A[ -
(d) FIRST RADIAL (e) SECOND RADIAL (f) THIRD AZIMUTHAL (10) MODE (11) MODE (03) MODE
FIGURE 4-2 SPATIAL FORMS OF VARIOUS NEUTRON FLUX MODE
4-3 Table 4-1 EFFECTS AND TIME CONSTANTS OF REACTOR PROCESSESS
PHYSICAL EFFECT PROCESS STABILIZING DESTABILIZING TIME CONSTANT(s) FUEL TEMPERATURE X 5-12 REACTIVITY FEEDBACK VOID REACTIVITY X 1 - 5 FEEDBACK
MODERATOR TEMPERATURE X 60 - 10* REACTIVITY FEEDBACK XENON CONCENTRATION X lO' - 105 REACTIVITY FEEDBACK
DELAYED X 1 - 2000 NEUTRON
NEUTRON X 10-" - 10-' LEAKAGE
PROMPT X io-J NEUTRON
4-4 and channel thermohydraulics. In these models, the governing non-linear dynamical equations were linearized about a given steady state reactor condition in power, neutron flux, reactivity, fuel and moderator temperatures, coolant mass flowrate, inlet fluid temperature and channel coolant quality. In the linear model for the reactor, the following approximations were used. i. The neutronics were approximated by the one group (or speed) diffusion equation for bare (i.e., unreflected) reactor with homogeneous composition coupled with the equations for the delayed neutron kinetics. ii. The neutron diffusion equation was reduced to the equation for the kinetics of a point reactor for each spatial mode of power distribution by expanding the neutron flux, multiplication factor and delayed neutron precursor concentration in terms of the spatial eigen-functions or eigen-modes of the steady state of the reactor. The various spatial modes are then distinguished in the point reactor model by their different subcriticalities. iii. The point reactor model kinetic equation for each mode was then reduced to a steady state equation by ignoring the short prompt neutron cycle time (i.e., the short time constant). The effect of delayed neutron kinetics on power spatial mode instability was studied using a reduced two-group representation for the usual six groups of delayed neutrons. The reactivity feedbacks were derived using the following linear models. iv. Lumped heat capacity models were used for the fuel and moderator graphite block. Effective heat transfer coefficients from the fuel sheath and moderator to the coolant were computed from the knowledge of the steady-state operating temperatures shown in Table 3-1. The corresponding time constants were then calculated.
v. A header-channel-steam drum model with constant channel inlet fluid temperature and header-to-drum pressure drop was used to compute the change in the channel coolant density and the axial coolant temperature distribution. To obtain simple results for the density change in the two-phase region, the two-phase flow was assumed homogeneous (i.e., no slip) and the entire channel power was assumed to be concentrated at the axial location of the fluid boiling boundary. This point heat source model tends to overestimate the change in the density. The changes in the subcooled liquid and the two-phase densities were then cosine-flux square weighted and averaged over the channel length. In this averaging, the motion of the boiling boundary was computed assuming axially uniform channel power. A - 5 METHOD OF SOLUTION The above linearized equations for the neutron kinetics and the reactivity feedbacks were then Laplace transformed in the time variable and expressed in terms of the complex frequency (i.e., Laplace parameter) s = o + iu. This transformation is equivalent to expressing the change in the neutron flux 6$ (for a given mode, i.e., for a given subcriticality) in the form: 6
= G = H (4-5) 6kp 1 + N.H
The poles of G determine the system stability because each pole snadds the exponential term exp (snt) to the change 6$ in the neutron flux. Thus, the flux spatial mode is unstable if any of the poles of G, or equivalently the zeros of (1 + N.H), has positive real part. Nyquist plot (i.e., the plot of the imaginary versus the real parts of (1 + N.H) as s varies along the entire length of the imaginary axis) may be used to determine the number of roots of (1 + N.H) with positive real parts. However, the approximations i. to v. given in Section 4.1 reduce the characteristic equation 1 + N.H = 0 to the fourth degree polynomial:
4-6 FIGURE 4-3 FEEDBACK LOOP SCHEMATICS
4-7 s* + as3 + bs1 + cs + d = 0 (4-6) The coefficients a to d are known functions of the given reactor conditions. Eq. (4-6) was solved for the four roots using well-known closed-form solutions (A), Only one of the roots was found to be physical.
GU.INM1 A - 8 S7/U/07 Section 5 RESULTS
This section presents the results of a parametric survey of the stability of power spatial distribution in the RBMK-1000 reactor using the model described in Section A, This parametric survey consists of plots of the flux exponential growth or amplification factor (i.e., e-folding growth rate) o and the oscillation frequency u against power, flowrate, and mode subcriticality.
In each of these plots, the sign of o indicates a stable (for negative o) or unstable (for positive o) spatial mode of neutron flux (compare eq. (4-2)). The magnitude of o gives the exponential rate of growth (for positive o) or decay (for negative o) of the mode amplitude. Zero value of u indicates a mode which is either growing or decaying purely exponentially, i.e., in a non-oscillatory manner. A non-zero value of ui indicates either divergent or convergent oscillation in the flux or power distribution. Zero value of o for non-zero u indicates an instability threshold condition, i.e., oscillatory mode with constant, amplitude. These instability modes are depicted in Figure 5-1.
Plots of the channel coolant density response (i.e., the change in the density in response to a power change) and the power coefficient of reactivity against power and flowrate are also given in this section.
COOLANT DENSITY RESPONSE AND POWER COEFFICIENT TRENDS WITH SUBCRITICALITY AND POWER AT FIXED FLOWRATE AND WITH FLOWRATE AT FIXED POWER Figure 5-2 shows variation in the cosine-shape flux square weighted and axially-averaged channel coolant density response with power at 100% flowrate at zero and at the fuel characteristic frequencies. At each frequency and at a given power, the density response in Figure 5-2 was obtained by changing the power by a small amount and computing the resulting change in the density. This density change was then expressed in terms of a full power unit, i.e., a 100% change in the given power. In Figure 5-2, the response at the fuel frequency is lower because the rate of change of flux is too rapid for the graphite to respond. For a given frequency, the density response in Figure 5-2 exhibits a maximum at a low power level (about 32% power at 100% flowrate). For powers below and above this level, the response is lower, particularly for powers below that at which the response is maximum. Below the power level (about 19%) at which core boiling ceases, the response approaches zero linearly with power. (In the analysis in this report, nominal values of core inlet fluid temperature and system pressure given in Table 3-1 were used. With these values, no boiling is predicted below 19% power. In the pre-accident condition in the reactor at CHERNOBYL, there was apparently core boiling at 17% power. This condition would be predicted at slightly higher core inlet fluid temperature and/or lower system pressure than nominal.)
S6035SR/ K _ 1 0UL5HANI J x B7/12/07 TIME
X
V < o
FIGURE 5-1 POSSIBLE INSTABILITY MODES IN NEUTRON FLUX SPATIAL DISTRIBUTION POWER (% OF NOMINAL)
10 20 30 40 50 80 90 100 0.0
-1.0-1 FIGURE 5-2 CHANGE IN COOLANT DENSITY VERSUS POWER IN RBMK-1000 REACTOR AT 100% FLOWRATE (USING COSINE-SHAPE-FLUX SQUARE WEIGHTED DENSITY) The change in the density consists of contributions arising from (i) the change in the subcooled liquid density, (ii) the change in the two-phase density and (iii) the change in the position of the boiling boundary. The contribution from (i) is generally small. The contribution from (ii) is maximum at a certain power for which the mass in the two-phase region is maximum and decreases for power levels above and below this power level. (Note that this power level is different from that for the onset of boiling for which the density is maximum because for the latter power level, the boiling length is zero and hence, the change in the axially average density is small.) The contribution from (iii) is small at high power because the non-boiling length in the channel is short and the local power (per unit length) is small. Thus, a change in the power results in a small change in the liquid enthalpy over this length, and, hence, a small amount of the liquid is brought to the saturation temperature. The resulting change in the position of the boiling boundary is small and, hence, so is the change in the axially averaged density. (Note that the change in the density is proportional to the change in the boiling boundary position.) As the power is reduced, the non-boiling length increases and so does the local power (per unit length) and, hence, the axially averaged density change. Near the channel exit, the power (per unit length) is small due to the cosine flux shape. Therefore, the contribution from (iii) also exhibits a maximum at an intermediate power level. Thus, the change in the flux square weighted density exhibits a maximum at an intermediate power level as indicated in Figure 5-2.
Figure 5-3 shows the power coefficient of reactivity (i.e., the net reactivity change arising from the fuel and moderator temperature and void reactivity feedbacks per unit fractional change in power) versus power at zero and at the fuel characteristic frequency at 100% flowrate. (The reactivity coefficients shown in Table 3-2 and Figures 3-2 and 3-3 were used). At each frequency and at a given power, the change in reactivity resulting from a small change in the given power was computed. This reactivity change was then expressed in terms of a full power unit, i.e., a 100% change in the given power. In Figure 5-3, the power coefficient is lower at the fuel frequency because the flux changes too rapidly for the graphite moderator to respond. At a given frequency, the variation in the power coefficient with power in Figure 5-3 is similar to that of the coolant density response. The power coefficient (at 100% flowrate) exhibits a maximum at about 30% power and approaches zero linearly below about 19% power at which the core boiling ceases. With core boiling, the power coefficient in Figure 5-3 is positive at all power levels. Figure 5-4 shows the variation in the power coefficient with power at zero and at the fuel characteristic frequency at 100% flowrate with the moderator temperature reactivity feedback neglected. Comparison of Figures 5-3 and 5-A shows that the power coefficient at the fuel frequency is not affected significantly by the moderator temperature feedback because the frequency is too high for the moderator to respond. Thus, at this frequency, the effects of the fuel temperature and void reactivity feedbacks dominate. (At about 19% power, at which the core void nearly collapses and, hence, the void
5 - A + 25-1
61
-5 20 30 40 50 60 70 80 90 100
POWER (% OF NOMINAL)
FIGURE 5-3 POWER COEFFICIENT OF REACTIVITY VERSUS POWER AT 100% FLOWRATE IN RBMK-1000 REACTOR WITH VOID AND FUEL AND MODERATOR TEMPERATURE REACTIVITY FEEDBACKS INCLUDED (USING COSINE-SHAPE-FLUX SQUARE WEIGHTED DENSITY) + 25-
g; si + 15- SE 51 + 10- ztr UJ UJ en 6°- + 5- a.
I I 10 20 30 40 50 60 70 80 90 100
POWER (% OF NOMINAL)
FIGURE 5-4 POWER COEFFICIENT OF REACTIVITY VERSUS POWER IN RBMK-1000 REACTOR AT 100% FLOWRATE WITH MODERATOR TEMPERATURE FEEDBACK NEGLECTED (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) reactivity feedback is small, the power coefficient is lower with the positive moderator temperature reactivity feedback neglected.) At zero frequency, however, the power coefficient in absence of the moderator temperature reactivity feedback is significantly lower than that (Figure 5-3) with this effect included as expected. However, at both frequencies, the behaviour of the coefficient with power is mainly determined by that of the void reactivity feedback (i.e., by the channel coolant density response). This result is readily deduced from a comparison of Figures 5-3 and 5-4 with Figure 5-5 which shows the pow?r coefficient versus power at 100% flowrate with only the void reactivity feedback included. Figure 5-5 shows that the behaviour of the power coefficient with power with only the void reactivity feedback included is nearly the same as that in Figures 5-3 and 5-4 where the other reactivity feedbacks are also included, (Note that the power coefficient in Figure 5-4 is everywhere shifted to lower values compared to that in Figure 5-5 due to the negative fuel temperature reactivity coefficient.)
Figure 5-6 shows the variation in the channel coolant density response with flowrate at zero and at the fuel characteristic frequency at 100% power. Again the response at the fuel frequency is lower as expected. At a given frequency and at 100% power, the density response decreases monotonically with decreasing flow as the non-boiling length in the channel decreases and the quality in the channel increases, and hence, the two-phase density decreases.
Figure 5-7 shows the variation in the power coefficient of reactivity with flowrate at zero and the fuel frequency at 100% power. The coefficient is lower at the fuel frequency as expected. At a given frequency, the coefficient is lower at lower flowrate as is the density response. At the fuel frequency, the power coefficient in Figure 5-7 is negative below about B2% flowrate as the density feedback becomes small and the fuel temperature feedback becomes dominant.
POWER INSTABILITY TREND WITH XENON REACTIVITY FEEDBACK ONLY Figure 5-8 shows the variation in the amplification factor o versus subcriticality for a range of power levels with only xenon reactivity feedback accounted for and with delayed neutron kinetics neglected. For a given subcriticality, the mode is more unstable or less stable (i.e., o is more positive or less negative) at higher power levels. The reason is that, at higher power levels, neutron flux is higher and, therefore, a small increase in the neutron flux causes larger decrease in xenon concentration which, in turn, causes further larger increase in the neutron flux. Since a mode of a given subcriticality is more unstable at higher powers, the subcriticality at the instability threshold increases with increasing power as Figure 5-8 shows. Figure 5-9 shows the variation in the oscillation frequency u with subcriticality and power level corresponding to o in Figure 5-8. Figure 5-8 shows more clearly the correspondence between o and ui at 100% power.
5-7 + 15 -
+ 10- zee UJUJ u? S
UJ II > UJ °
-5
POWER (% OF NOMINAL)
FIGURE 5-5 POWER COEFFICIENT OF REACTIVITY VERSUS POWER AT 100% FLOWRATE IN RBMK-1000 REACTOR WITH ONLY VOID REACTIVITY FEEDBACK INCLUDED (USING COSINE-SHAPE-FLUX SOUARE WEIGHTED DENSITY) FLOWRATE (% OF NOMINAL)
10 20 30 40 50 60 70 80 90 100 -0.02. I I I I
-0.06-
3 a. :t -0.10
UJ E Q &*
V 8 s -014 •ty'Cy So 1 I"- -0.18- 111 a.
-0.22-
-0.26-1 FIGURE 5-6 CHANGE IN COOLANT DENSITY VERSUS FLOWRATE IN RBMK-1000 REACTOR AT 100% POWER (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) + 20
16-
I! • 12- K
8-
O-i
+ 4-
AT FUEL FREQUENCY
-4- I I I 1 I I 10 20 30 40 50 60 70 SO 90 100
FLOWHATE (% OF NOMINAL)
FIGURE 5-7 POWER COEFFICIENT OF REACTIVITY VERSUS FLOWRATE IN RBMK-1000 REACTOR AT 100% POWER (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) + 12 n
+ 10-
IT + 8-
O + 6-
0. <
m + 2- < i
m '
10 15 20 25 30 35
SUBCRICITALITY (mk)
FIGURE 5-8 AMPLIFICATION FACTOR VERSUS SUBCRITICALITY OF NEUTRON FLUX SPATIAI- MODE EXCITED BY XENON-REACTIVITY FEEDBACK ONLY AND WITH NO DELAYED NEUTRONS AT VARIOUS POWER LEVELS IN RBMK-1000 REACTOR 10-
z POWER (%) 1 6-1 100 a. o m o
o 2-1
I I 40 10 15 20 25 30 35 SUBCRITICALITY (mk)
FIGURE 5-9 ANGULAR FREQUENCY VERSUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE EXCITED BY XENON-REACTIVITY FEEDBACK ONLY AND V/ITH NO DELAYED NEUTRONS AT VARIOUS POWER LEVELS IN RBMK-1000 REACTOR Figures 5-8, 5-9 and 5-10 show that at a given power level, the mode is non-oscillatory (i.e., u = 0) and unstable at low subcriticality and oscillatory and less unstable and then stable at higher subcriticalities. POWER INSTABILITY TREND WITHOUT XENON REACTIVITY AND WITH AND WITHOUT MODERATOR TEMPERATURE FEEDBACK AND DELAYED NEUTRONS Figure 5-11A and 5-11B show the variation in o with power at various values of spatial mode subcriticality at 100% flowrate with fuel and moderator temperature and void reactivity feedbacks accounted for but xenon reactivity feedback neglected. Figure 5-11A shows unstable (i.e., positive values of o) modes and Figure 5-11B shows stable (i.e., negative values of o) modes. For the range of power and subcriticalities shown, only non-oscillatory or exponential instability (i.e., with zero frequency) is predicted. It is shown, in Section 5.A.I, that xenon reactivity feedback tends to induce oscillatory instability particularly in modes at higher subcriticality. For a given power, the power distribution is more unstable (i.e., o is more positive) or less stable (i.e., o is less negative) at lower subcriticalities as expected. For a given subcriticality, the mode is most unstable at a certain low power level. The mode is less unstable for power levels below and above this level. In particular, below this power level, the power distribution rapidly approaches stability as the power is reduced. This behaviour in o resembles that of the density response in Figure 5-2 and power coefficient of reactivicy in Figure 5-3. Figure 5-12 shows the variation in o with power at various values of spatial mode subcriticality at 100% flowrate with the delayed neutron kinetics and the major, except xenon, reactivity feedbacks included. A two-group representation for the six groups of delayed neutrons was used. Comparison of Figures 5-11A and 5-12 shows that the delayed neutrons have significant stabilizing effect (i.e., renders o less positive) only for growth rates (i.e., for o values) higher than 10'2 per second. Thus, the delayed neutron kinetics does not affect the general trend in mode instability and may be neglected. It may be included (as is done in Section 5.4.4) if accurate values of the growth rates (i.e., o values) in the range above 10'' per second is needed. Figure 5-13 shows the variation in o without the moderator temperature (and xenon) reactivity feedback and with no delayed neutrons. Comparison of Figures 5-11A and 5-13 shows that the void reactivity feedback dominates above about o = 10~3 per second (i.e., for higher growth rates) as expected. At 100% flowrate, the maximum instability in Figure 5-11A occurs at about 30% power as does the density response in Figure 5-2. The rapid approach of power to stability in Figure 5-11A and the reduction in the density change in Figure 5-2 occur at about 19% power level at which channel boiling ceases. Thus, in the small signal approximation, all modes of power spatial distribution below about 25 mk subcriticality in the RBMK-1000 reactor at
5-13 40-
+ 30- o K +20. -6 a« 5 u. I + 10 o in g o ui cc u u. 3 0, — - - 2 < m -10- 10 IS 20 25 30 40 SUBCRITiCAUTY (mk) FIGURE 5-10 AMPLIFICATION FACTOR AND ANGULAR FREQUENCY OF XENON-FEEDBACK INDUCED OSCILLATIONS VERSUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR WITH ALL OTHER FEEDBACKS NEGLECTED AND WITH NO DELAYED NEUTRONS AND AT 100% POWER AND FLOWRATE + 10-, < u SUBCRITICALITY a (mk| < 20 80 100 POWER (% OF NOMINAL) FIGURE 5-11A AMPLIFICATION FACTOR VERSUS POWER AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% FLOWRATE WITHOUT XENON REACTIVITY FEEDBACK AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) 5-15 - io-t- g o : - 10-5 - SUBCRITICALITY (mk) SUBCRITICAUITY (mk) I 1 20 40 60 80 100 POWER (<*b OF NOMINAL) FIGURE 5-11B AMPLIFICATION FACTOR VERSUS POWER AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% FLOWRATE WITHOUT XENON REACTIVITY FEEDBACK AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) (FIGURE 5-10A CONTINUED) 5-16 + 1 -1 + It)-1- SUBCRITICALITY (mk) < o u. 0. 5 • 10-6, 40 60 BO 100 POWER (% OF NOMINAL) FIGURE 5-12 AMPLIFICATION FACTOR VERSUS POWER AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% FLOWRATE WITH DELAYED NEUTRON KINETICS AND INCLUDING MAJOR, EXCEPT XENON, REACTIVITY FEEDBACKS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) 5-17 + 10-5 2 SUBCRITICAUTY (mk) .10-6 40 60 BO 100 POWER {% OF NOMINAL) FIGURE 5-13 AMPLIFICATION FACTOR VERSUS POWER AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% FLOWRATE WITHOUT MODERATOR TEMPERATURE AND XENON REACTIVITY FEEDBACKS AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE 5-1B 100% flowrate are unstable at all power levels between 19% and 100% power due to void (for growth rates greater than 10"3 per second) and moderator temperature (for growth rates below 10"3 per second) temperature reactivity feedbacks. POWER INSTABILITY TRENDS WITH ALL MAJOR REACTIVITY FEEDBACKS INCLUDED Instability Trend with Power and Subcriticality at Fixed Flowrate Figures 5-14A and 5-1AB show the variation in o with power at 100% flowrate for various mode subcriticalities with the fuel and moderator temperature, void and xenon reactivity feedbacks accounted for and with delayed neutrons neglected. Comparison of Figures 5-1AA and 5-11A where xenon reactivity feedback is neglected, shows that, at 100% flowrate at all power levels, the xenon reactivity feedback increases mode instability for growth rates below about 10"' per second. As shown in Section 5.3, the destabilizing effect of the void reactivity feedback dominates for growth rates greater than 10'3 per second and that of the moderator temperature reactivity feedback is significant below 10'3 per second. Figure 5-15 shows the variation in the oscillation frequency corresponding to the conditions in Figures 5-14. Figure 5-16 shows more clearly the correspondence between o and w for the flux spatial mode with 20 mk subcriticality. As mentioned in Section A.3, instability in the power spatial modes are non-oscillatory in the absence of xenon reactivity feedback. Figure 5-15 shows that xenon reactivity feedback tends to induce oscillatory instability particularly at higher mode subcriticality. For low values of mode subcriticality, the frequency is non-zero only at low power levels. At higher mode subcriticality, two branches of non-zero frequency curves appear one for a range of low and another for a range of high power levels with a branch of zero frequency line at intermediate power levels (Figure 5-16). At still higher mode subcriticality, the branches merge into a single non-zero frequency curve. Thus, at subcriticalities of interest (i.e., below 20 mk), the instability in Figure 5-1AA is non-oscillatory, i.e., is of exponential nature. In summary, Figure 5-1AA £>hows that, in the small signal approximation and at 100% flowrate, several modes of power spatial distribution in the RBMK-1000 reactor are unstable at all power levels between 19% and 100% power. This instability is caused mainly by the void reactivity feedback for growth rates greater than 10"3 per second and is maximum at about 30% power. The destabilizing effects of the moderator temperature and xenon reactivity feedbacks are significant for growth rates below 10'3 per second. The instability is generally non-oscillatory. The stabilizing effect of delayed neutrons is relatively small at all growth rates particularly for o values below 10"2 per second. 5 - 19 + 10 •> + 1 - ,10-1- +10-2- SUBCRITICALITY (mk) c o o + 10-3, 0. ,10-" + 10-5 • 10-6 20 40 60 100 POWER (% OF NOMINAL) FIGURE 5-14A AMPLIFICATION FACTOR VERSUS POWER AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% FLOWRATE WITH MAJOR REACTIVITY FEEDBACKS INCLUDED AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) - 10-6, Io < o -10-5. 30 SUBCRITICALITY (mk) 25 30 SUBCRITICALITY (mk) 20 40 60 60 100 POWER (% OF NOMINAL) FIGURE 5-14B AMPLIFICATION FACTOR VERSUS POWER AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% FLOWRATE WITH MAJOR REACTIVITY FEEDBACKS INCLUDED AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) (FIGURE 5-13A CONTINUED) 5-21 SUBCRITICALITY (mk) SUBCRITICAUTY (mk| 10 I 3- w z 5 DC. 2- 3 111 o UJ 1 - (9 < 0- I I 1 I 10 20 30 40 50 60 70 60 100 POWER (% OF NOMINAL) FIGURE 5-15 ANGULAR FREQUENCY VERSUS POWER AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% FLOWRATE WITH MAJOR REACTIVITY FEEDBACKS INCLUDED AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) - + 50 CO - +40 I 3- 3 S H ° I IE o O g HI 2- - + 20 o s H a. o u. u_ 1 - •+10 3 t 0- 10 20 30 40 50 60 70 80 90 100 POWER (% OF NOMINAL) FIGURE 5-16 ANGULAR FREQUENCY AND AMPLIFICATION FACTOR VERSUS POWER FOR NEUTRON FLUX SPATIAL MODE WITH 20 mk SUBCRITICALITY IN RBMK-1000 REACTOR AT 100% FLOWRATE WITH MAJOR REACTIVITY FEEDBACKS INCLUDED AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) Instability Trend with Flowrate and Subcriticality at Fixed Power Figures 5-17A and 5-17B show the variation in the amplification or the growth factor o with flowrate at various subcriticalities at 100% power with the major reactivity feedbacks included and with no delayed neutrons. At a given flowrate, the spatial distribution is more unstable or less stable at lower mode subcriticality as expected. For a given subcriticality, the mode is less unstable or more stable at lower flowrates. Figure 5-17A shows that, at 100% power, all modes of power spatial distribution with subcriticality below approximately 25 mk are unstable at flowrates between 19% and 100%. Figure 5-18 shows the frequencies of the stable and unstable modes in Figures 5-17. Instability in all modes with subcriticality between 2 and 15 mk is predicted to be non-oscillatory (i.e., with zero frequency) at all flowrates. At higher mode subcriticality, the instability is oscillatory with higher frequency at lower flowrates. At still higher mode subcriticality, the frequency begins to decrease with decreasing flowrates. Trends in Density Response and Power Coefficient with Power and Flowrate Figure 5-19 shows variation in the channel coolant density response with power at various flowrates. For a given flowrate, the density response exhibits a maximum at a certain low power level. Below and above this power level, the response is lower, particularly for power levels below that at which the maximum response occurs. The reason for this behaviour in the density response is given in Section 5.1. Except at its maximum value, the density response, at lower flowrates, is lower at power levels above and is higher at power levels below that at which the density is a maximum. The maximum density response is a constant at all power levels and flowrates and is about -0.65 gm/cm'/FPU. At lower flowrates, the maximum in the density response occurs at lower power because the density response depends on the power-to-flow ratio rather than on the power or/and flowrate separately. Thus, the maximum in the response shifts to give a constant power-to- flowrate ratio. Figure 5-20 shows the variation in the power coefficient of reactivity with power at various flowrates. For a given flowrate, the coefficient exhibits a maximum at a low power level close to that for the density response. Below and above this power level, the coefficient is small, particularly for power levels below that at which the coefficient is maximum. Except at its maximum value, the power coefficient, at lower flowrates, is smaller at power levels above and larger at power levels below the power at which the coefficient is maximum. The reason is that, at lower flowrates, the maximum in coefficient occurs at lower power. The density response in Figure 5-19 and, hence, the void reactivity feedback is responsible for this behaviour in the power coefficient. The maximum in the power coefficient is nearly a constant (as is that in the density) and is about +12 mk/FPU. The maximum value is slightly larger at lower flowrates because the power is lower and, hence, the stabilizing effect of the fuel temperature feedback is lower. The power coefficient is positive at nearly all power levels and flowrates. 5-24 10-1 •10-2. SUBCRITICALITY (mk) • 10-3. O o ,10-"- o o. < 10-5 _ -10-6 20 40 60 80 100 FLOWRATE (% OF NOMINAL) FIGURE 5-17A AMPLIFICATION FACTOR VERSUS FLOWRATE AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% POWER WITH MAJOR FEEDBACKS INCLUDED AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) 5-25 s I o a. i -10"5 " SUBCRITICALITY (mk) I I I 20 40 60 80 100 FLOWRATE ( FIGURE 5-17B AMPLIFICATION FACTOR VERSUS FLOWRATE AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% POWER WITH MAJOR FEEDBACKS INCLUDED AND WITH NO DEALYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) (FIGURE S-17A CONTINUED) 5-26 5 T SUBCRITICALITY (ink) a 3 111 2- a i V ro i o 0- 2-15 10 20 30 40 50 60 70 80 90 100 FLOWRATE (% OF NOMINAL) FIGURE 5-18 ANGULAR FREQUENCY VERSUS FLOWRATE AT VARIOUS SUBCRITICALITY OF NEUTRON FLUX SPATIAL MODE IN RBMK-1000 REACTOR AT 100% POWER WITH MAJOR FEEDBACKS INCLUDED AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) POWER (% OF NOMINAL) 00 40 50 60 90 100 -0 1 -0.2 100 el FLOWRATE (% OF NOMINAL) i — -0.3 I -0.4 yjd. on aU.J -0.5 -06 ^ p0WER ,N RBMK]M O SOUABE DESSIT?) ™EOUENCY (USING COS.1.E SHAPE FLUX • 20-1 20 40 60 30 100 FLOWRATE (% OF NOMINAL) UJ a. + 15- Mill V -5- I 10 20 30 40 50 60 70 BO 90 100 POWEH (% OF NOMINAL) FIGURE 5-20 POWER COEFFICIENT OF REACTIVITY VERSUS POWER IN RBMK-1000 REACTOR AT VARIOUS FLOWRATES AND AT FUEL FREQUENCY (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) Instability Trend with Power and Flowrate for First Azimuthal Mode Figure 5-21 shows the variation in the growth factor o with power at various flowrates for the first azimuthal spatial mode of power distribution with 7 mk subcriticality. The results in Figure 5-21 were obtained with the major reactivity feedbacks (i.e., fuel and moderator temperature, void and xenon reactivity) accounted for but delayed neutron kinetics ignored. For the range of conditions in power and flowrate in Figure 5-21, the mode is predicted to be unstable and non-oscillatory (i.e., with zero frequency). For a given flowrate, Figure 5-21 shows that the mode is most unstable at a certain low power level. Below this power level, the mode rapidly approaches stability. Above this power level, the mode is less unstable. Except at the maximum instability, the mode, at lower flowrate, is less unstable at power levels above and more unstable at power levels below that at which the mode is most unstable. The reason is that, at lower flowrates, the maximum in o occurs at lower power. The maximum value of o is nearly a constant independent of power and flowrate (as is that in the power coefficient in Figure 5-20) and is about +0.28 per second. This corresponds to a reactor period of about 3.57 seconds. As shown in previous sections, the behaviour of o is determined mainly by the destabilizing effect of the void reactivity feedback for o greater than 10"' per second and by the destabilizing effects of the moderator temperature and xenon reactivity feedback for o less than 10"J per second. Figure 5-22 shows the variation in the growth factor o with power at various flowrates for the first azimuthal mode with 7 mk subcriticality. The results in Figure 5-22 were obtained with the delayed neutron kinetics and the major, except xenon, reactivity feedbacks included. Comparison of Figures 5-21 and 5-22 shows that, for the conditions considered, the delayed neutron kinetics has a significant stabilizing effect (i.e., renders o less positive) for o values higher than 10"* per second. Thus, for the first azimuthal mode, the maximum value of o is + 0.19, i.e., a reactor period of 5.26 seconds, in presence of delayed neutrons (Figure 5-22) compared to a value of + 0.28 in absence of delayed neutrons (Figure 5-21). As shown in Section 5.4.1, the destabilizing effect of xenon reactivity feedback is important for values of o lower than about 10"* per second. Thus, in Figure 5-22 where the effect of xenon reactivity feedback is ignored, the mode is less unstable than in Figure 5-21 for o values lower than 10"* per second. 5-30 + 1 _ UI20 40 60 UBO 100 FLOWRATE (% OF NOMINAL) (X o g o a. < • 10-5 - <• 10-6 20 POWER (% OF NOMINAL) FIGURE 5-21 AMPLIFICATION FACTOR VERSUS POWER FOR FIRST AZIMUTHAL MODE OF NEUTRON FLUX DISTRIBUTION OF 7 mk SUBCRITICALITY IN RBMK-1000 REACTOR AT VARIOUS FLOWRATES AND WITH MAJOR REACTIVITY FEEDBACKS INCLUDED AND WITH NO DELAYED NEUTRONS (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) 5-31 FLOWHATE ("^ OF NOMINAL) : Ull20 40 60 80 l100 + 10-6 20 40 100 POWER (% OF NOMINAL) FIGURE 5-22 AMPLIFICATION FACTOR VERSUS POWER FOR FIRST AZIMUTHAL MODE OF NEUTRON FLUX DISTRIBUTION OF 7 mk SUBCRITICALITY IN RBMK-1000 REACTOR AT VARIOUS FLOWRATES AND WITH DELAYED NEUTRON KINETICS AND MAJOR, EXCEPT XENON, REACTIVITY FEEDBACKS INCLUDED (USING COSINE SHAPE FLUX SQUARE WEIGHTED DENSITY) 5-32 Section 6 REFERENCES 1. P.S. Chan, "Reactor Physics of RBMK-1000", Electric Power Research Institute. EPRI report, 1987 December. 2. USSR State Committee On The Utilization of Atomic Energy, "The Accident At The CHERNOBYL Nuclear Power Plant And Its Consequences", Information Compiled For The IAEA Experts' Meeting. Vienna. 1986 August 25-29. 3. National Nuclear Corporation Report, "The Russian Graphite Moderated Channel Tube Reactor", U.K., 1976; V.S. Romanenko, "Survey of Neutron Physics Investigations On The RBMK-1000 Reactor, Paper presented at Risley, 1977 February 7-8, under The SCUAE/UKAEA Agreement For Exchanges Of Information On Pressure Tube Reactors; W. Mitchell, III, "Design Features Of The Soviet RBMK-1000/CHERNOBYL-4 Reactor", Report Prepared For The Advisory Committee On Reactor Safeguards, 1986 May 4. A. M. Abramowitz and I.A. 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