ANALYTICAL MODELLING AND STUDY OF THESTABILITY CHARACTERISTICS OF THE ADVANCED

HEAVY WATER REACTOR

by

A. K. Nayak, P. K. Vijayan and D. SahaReactor Engineering Division

2000

Please be aware that all of the Missing Pages in this document wereoriginally blank pages

BARC/2000/E/011

GOVERNMENT OF INDIAATOMIC ENERGY COMMISSION

ANALYTICAL MODELLING AND STUDY OF THE

STABILITY CHARACTERISTICS OF THE ADVANCED

HEAVY WATER REACTOR

byA.K. Nayak, P.K. Vijayan and D. Saha

Reactor Engineering Division

BHABHA ATOMIC RESEARCH CENTREMUMBAI, INDIA

2000

BARC/2000/E/011

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Analytical modelling and study of the stability characteristics ofthe Advanced Heavy Water Reactor

104 p., 51 figs., 1 tab., 1 ill.

A.K. Nayak; P.K. Vijayan; D. Saha

Reactor Engineering Division, Bhabha Atomic Research Centre,Mumbai

Bhabha Atomic Research Centre, Mumbai - 400 085

Reactor Engineering Division,BARC, Mumbai

Department of Atomic Energy

Government

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BARC/2000/E/011

30 Date of submission: March 2000

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Bhabha Atomic Research Centre, Mumbai

42 Form of distribution: Hard copy

50 Language of text: English

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52 No. of references: 74 refs.

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60 Abstract: An analytical model has been developed to study the thermohydraulic and neutronic-coupleddensity-wave instability in the Indian Advanced Heavy Water Reactor (AHWR) which is a naturalcirculation pressure tube type Boiling Water Reactor. The model considers a point kinetics model for theneutron dynamics and a lumped parameter model for the fuel thermal dynamics alongwith the conservationequations of mass, momentum and energy and equation of state for the coolant. In addition, to study theeffect of neutron interactions between different parts of the core, the model considers a coupled multipointkinetics equation in place of simple point kinetics equation. Linear stability theory was applied to revealthe instability of in-phase and out-of-phase modes in the boiling channels of the AHWR. The resultsindicate that the design configuration considered may experience both Ledinegg and Type I and Type IIdensity-wave instabilities depending on the operating condition. Some methods of suppressing theseinstabilities were found out. In addition, it was found that the stability behavior of the reactor is greatlyinfluenced by the void reactivity coefficient, fuel time constant, radial power distribution and channel inletorificing. The delayed neutrons were found to have strong influence on the Type I and Type II instabilities.Decay ratio maps were predicted considering various operating parameters of the reactor, which are usefulfor its design.

70 Keywords/Descriptors : PHWR TYPE REACTORS; BWR TYPE REACTORS; REACTOR

STABILITY; DELAYED NEUTRONS; OSCILLATION MODES; TWO-PHASE FLOW; REACTOR

COOLING SYSTEMS; POWER DISTRIBUTION; SIMULATION; REACTOR SAFETY; URANIUM

OXIDES; SPECIFICATIONS; REACTIVITY; PLUTONIUM OXIDES; THORIUM OXIDES

71 INIS Subject Category: S21

99 Supplementary elements:

•in-

CONTENTS

ABSTRACT vi

1.0 INTRODUCTION 11.1 Instabilities in BWRs 21.1.1 Thermohydraulic instabilities in boiling two-phase

flow systems 21.1.1.1 Experimental investigations in boiling natural circulation loops 31.1.1.2 Experimental investigations in boiling forced circulation loops 41.1.1 ..3 Theoretical investigations in boiling systems 51.1.1.3.1 Investigations based on linear analysis 51.1.1.3.2 Investigations based on non-linear analysis 61.1.1.4 Investigations relevant to power reactors 61.1.2 Coupled neutronic-thermohydraulic oscillations 71:1.2.1 Computer codes to simulate coupled neutronic-thermohydraulic

instabilities 71.1.2.2 Fundamental studies on coupled neutronic-thermohydraulic

instabilities 82.0 MODELLING OF THE STABILITY BEHAVIOUR OF TWO-PHASE

NATURAL CIRCULATION 102.1 Steady State Solution 112.2 Linear Stability Analysis 132.3 Neutron Kinetics 182.4 Fuel Heat Transfer Model 192.5 Neutron Kinetics Model for Out-of-phase Oscillations 212.5.1 Coupled Multipoint Kinetics Model 212.5.2 Modal Kinetics Model for Analysing the

Out-of-phase Instabilities 242.6 Mode of Oscillation 252.7 Determination of threshold of stability, frequency of oscillation and

the Decay Ratio 253.0 POWER DISTRIBUTION IN THE AHWR CORE 264.0 MODEL VALIDATION 265.0 LEDINEGG TYPE INSTABILITY BEHAVIOUR OF THE AHWR 275.1 Influence of geometric and operating parameters

on Ledinegg type instability 285.2 Influence of axial power distribution on Ledinegg

type instability 285.3 GENERAL CONCLUSIONS 296.0 DENSITY WAVE INSTABILITY BEHAVIOUR OF THE AHWR 296.1 Influence of geometric and operating parameters

on density-wave instability 29

IV

7.0 COUPLED NEUTRONIC-THERMOHYDRAULICSTABILITY CHARACTERISTICS OF THE AHWR 31

7.1 Paramaters for Coupled Neutronic-Thermohydraulic Instability 32

7.1.1 Influence of void reactivity feed back 327.1.2 Influence of the fuel thermal time constant 337.2 Out-of-phase instability behaviour considering the coupled multipoint

kinetics model and modal point kinetics model 347.3 Influence of delayed neutrons on the stability 357.4 Effect of Inlet Orificing 357.5 Effect of Radial Power Distribution 367.6 Decay Ratio Maps for the AHWR 368.0 CONCLUSIONS 37

Nomenclature 39

References 39

TABLE 1: Details of inlet orificing indifferent channels of AHWR 44

Appendix 1 : Models employed to estimate two-phasefriction factor multiplier 45

Abstract

It is indispensable for the economical improvement in India to solve insufficientelectric power supply. The nuclear power option is one of the several proven technologiesfor large scale power production. A significant contribution of nuclear power in the totalenergy generation can also significantly reduce the green house gases. Realising this fact,India has been continuously pursuing to increase its share of electricity production fromnuclear. At present two BWRs and nine CANDU plants are in operation. Reprocessingtechnology of spent fuels and fabrication of MOX fuels have been well established.Under such circumstances, it is planned to develop the Advanced Heavy water Reactor(AHWR) for the purpose of utilising the vast thorium resource available in India. In theAHWR, neutron is moderated by heavy water, core is cooled by boiling light water innatural circulation and the fuel is a combination of plutonium-thorium and thorium-uranium oxides. With natural circulation being the designed heat removal mode, thereactor may experience various types of thermohydraulic instabilities which may furtherget coupled with the neutronics to induce power oscillations. Hence, it is essential topredict the stable and unstable zones of the reactor operation during the design stage sothat instability if found, methods of suppressing or procedures to avoid them can beworked out. For this purpose, theoretical models were developed by solving theconservation equations of mass, momentum and energy applicable to homogeneousequilibrium flow based on linear stability theory. Comprehensive models for the neutronkinetics (which include a point kinetics model for in-phase mode oscillation and acoupled multi-point kinetics model or modal point kinetics model for out-of-phase modeoscillation) and thermal dynamics of the fuel are incorporated into the above model toinvestigate the coupled neutronic-thermohydraulic instabilities. The characteristicequation was derived considering the interactions of the parallel multiple channels withthe downcomers when operating under natural circulation condition, and the feed backeffects due to the void reactivity and the Doppler reactivity. The stability behaviour of thesystem and the oscillation modes (i.e. in-phase and out-of-phase) were determined fromthe roots of the characteristic equation. The analytical model was first applied for theexperimental results of density-wave instabilities carried out under different geometricand operating conditions, which are a twin parallel boiling channel system under forcedcirculation conditions at atmospheric pressure (Tokyo Institute of Technology) and theATR simulation facility (Heat Transfer Loop, HTL) under natural circulation conditions.It is verified from the results that the analytical method proposed in this work cansimulate well the stable flow thresholds of density-wave instability. Next, the stabilityanalysis method is applied for the AHWR concept with the loop geometry as discussed inthe feasibility report.

The analytical results indicate that the Ledinegg type instability is of concern duringthe start-up of the AHWR but the density-wave instability may occur even at 7 MPapressure which is the rated power condition according to the reactor configuration

VI

considered for this study. It is also found that orificing at the inlet of channels which is anefficient method to suppress the density-wave instability in forced circulation system, isnot always effective to stabilise the natural circulation system.

Finally, the measures to prevent the Ledinegg type and density-wave instabilities arediscussed for the AHWR design under consideration. The effect of fuel thermal dynamicsand void reactivity coefficient on the stability of in-phase and out-of-phase modeoscillations among the boiling channels of the AHWR were investigated. The analyticalresults indicate that the void reactivity coefficient and fuel time constant have differentinfluences on the Type I and Type II instabilities in the AHWR core. The Type Iinstability is governed by the dominance of pressure drop due to gravity, whereas theType II instabilities occur due to the dominance of pressure drop due to friction in thetwo-phase region. It is also found that the coupled multi-point kinetics model and themodal point kinetics model predict the same threshold power of out-of-phase instability ifthe coupling coefficient in the former model is half the subcriticality used in the lattermodel. The stability behaviour of the reactor is found to be influenced by the radial powerdistribution and the channel inlet orificing. Further, the delayed neutrons were found todecrease the Type I instability but increase the Type II instability. Finally it wasrecommended to operate the reactor at channel inlet subcooling of less than 10 K so as tohave sufficient stability margin.

VII

1. INTRODUCTION

In order to utilise thorium in the country's nuclear power programme, it is importantto develop the Advanced Heavy Water Reactor (AHWR) which would generate most of itselectricity from U233, bred in-situ from thorium [1]. The major features of this reactor arethat it is a heavy water moderated, boiling light water cooled 750 MWth pressure tube typereactor with the objective to burn thorium in the core with a combination of (Th-U233)O2and (Pu-Th)O2 as the fuel. The reactor is based on well proven water reactor technologiesand would incorporate a number of passive safety features in its design. One of them is toadopt natural circulation core cooling during start-up, power-raising, rated power conditionand accidental conditions. This concept eliminates the recirculation pumps which arenormally present in conventional forced circulation BWRs. Fig. 1 shows the Primary HeatTransport System (PHTS) of the reactor as described in the feasibility report. As shown inthe figure, the PHTS contains many pressure tubes (about 408) which are housed inthe calandria, and are connected between the steam drums and header with equal numberof tail pipes and inlet feeder pipes respectively. One fourth of these channels (pressuretubes) are connected to one each of the four steam drums. From each steam drum, fourdowncomer pipes are connected to the common octagonal inlet header. Each channel hasinner diameter of 120 mm and contains 52 fuel rods and 8 water rods. The outer diameterof the fuel rod is 11.2 mm and that of the water rods is 6 mm. The heated length of thechannel is 3.5 m. The inner diameter of the inlet feeders is about 97 mm and that of the tailpipes is about 122 mm. The height of the outlet feeders above the core is about 30 m whichis large enough to promote required natural circulation flow rate. The steam drums havelength of 10 m and inner diameter of 3 m and are set horizontally. The steam separation willbe achieved by the gravity without using separators which may reduce natural circulationflow rate because of large pressure drop. Each downcomer connecting the header with thesteam drum is having a length of about 27 m and inner diameter of 290 mm. This is atypical configuration of the reactor PHT system considered in the present study. Undernormal operating condition the steam drum pressure is maintained at 70 bar. Boiling takesplace in the core and the two-phase mixture flowing out of the core is separated into steamand water in the steam drum. The separated steam flows into the turbine and an equal massrate of feed water enters the steam drum. The coolant circulates naturally in the primarycoolant system.

With the pressure tube type concept, it is possible to go for on line fuelling. However,natural circulation being the designed heat removal mode, it requires very tall riser pipes (ortail pipes) than a conventional forced circulation BWR of same power rating to achieve thedesired thermalhydraulic characteristics. This in turn may cause large two-phase pressuredrop in the riser portion as compared to that occurring in the single-phase portioncomprising of downcomers, inlet feeder pipes and single-phase portion of heated channels.Dominance of two-phase pressure drop may initiate thermo-hydraulic instabilities in thereactor. Also, it is not possible to throttle sufficiently at the inlet of each channel which maysuppress the instabilities at the cost of reduced flow, since natural circulation flow isdependent on the resistance of the channel. With several parallel boiling channels havingdifferent power and resistances connected between the header and the steam drum with verylong feeder pipes, the reactor may experience various types of instabilities during various

operating conditions from atmospheric to rated pressure and the power raising process.Occurrence of the thermo-hydraulic instabilities may further induce power oscillationsthrough the void reactivity coupling. Instabilities of any form is undesirable from the viewpoints of reactor operation, control and safety. It is required to predict the stable andunstable regions of the reactor operations during the design stage so that if instability isfound, methods of suppressing or procedures to avoid them can be worked out.

1.1 Instabilities in BWRs

Reviewing previous investigations, it is found that several studies have already beencarried out in the field of instabilities which may occur in boiling systems. Earlierinvestigations have shown that the most important instabilities in a BWR are purethermohydraulic and coupled neutronic-thermohydraulic instabilities [2]. The differencebetween the two instabilities is that in the latter case there is a power feed back in additionto the flow feed back. In general, conventional BWRs experience the coupled neutronic-thermohydraulic instabilities because the reactivity instabilities dominate the reactorresponse due to their additional feed back. However, in the case of a natural circulationBWR such as the AHWR, it may be required to raise the power during start-up under thecoolant condition of low pressure and low temperature without the addition of fissionenergy from physics design limitations. In addition, the coupled neutronic-thermohydraulicinstability behaviour also depends on the pure thermalhydraulic oscillatory behaviour ofthe system [3]. Hence, it is important to investigate the pure theonal hydraulic instabilitiesin boiling two-phase flow systems in addition to the coupled neutronic-thermohydraulicinstabilities.

1.1.1 Thermohydraulic instabilities in boiling two-phase flow systems

State-of-the-art reviews on thermal hydraulic instabilities which may occur in boilingtwo-phase flow systems are provided in [4-7], These reviews suggest that several types ofinstabilities may appear in a boiling two-phase system; some of which can arise from thesteady state characteristics of the system such as

(a) flow excursion (Ledinegg type) instability and(b) relaxation instability (flow pattern transition, bumping, geysering,

chugging, etc.), and

the rest can be due to the dynamic nature of the system such as

(a) density-wave oscillations,(b) pressure-drop oscillations,(c) acoustic oscillations and(d) thermal oscillations.

However, the most commonly encountered instability in a BWR is the density-waveoscillation which may occur from the start-up to the rated operating conditions and may

induce power oscillations through the void reactivity coupling. The reason for occurrence ofthis type of instability as described by Yadigaroglu and Bergles [8] is as follows:

Consider an oscillatory subcooled inlet flow entering a heated boiling channel asdepicted in Fig. 2. The inlet flow fluctuations will create propagating enthalpy perturbationsin the single-phase region. The boiling boundary will respond according to the amplitudeand phase of the enthalpy perturbations at the point where the flow reaches saturation.Changes in the flow and in the length of single-phase region will combine to create anoscillatory single-phase pressure drop perturbation Spi. Enthalpy perturbations in the two-phase region will appear as quality and void fraction perturbations and will travel with theflow along the heated channel. The combined effects of the flow and void fractionperturbations and the variations of two-phase length will create a two-phase pressure dropperturbation 8p2. Since the total pressure drop across the boiling channel is imposed by theexternal characteristic of the channel, the two-phase pressure drop perturbations will createfeed back pressure drop perturbations of opposite sign in the single-phase region, which caneither reinforce or attenuate the imposed oscillations by creating a feedback flowperturbations. With correct timings, the perturbations can acquire appropriate phases andbecome self sustained. Under these conditions the system would be at the threshold ofstability since it would have the capability of oscillating without externally imposedperturbations. Recently, Rizwan-Uddin [9] identified that the axial variations in the two-phase mixture velocity is the origin of the density-wave instability in two-phase flowsystems. It can be said that the mechanism behind the occurrence of this type of instabilityis now well understood in addition to many other instabilities which are cited earlier.However, experimental and analytical investigations are still being carried out to predict theonset of instability accurately because the threshold of stable flow is found to be varied bythe differences in boiling loop geometry and their operating conditions, and to understandthe characteristics of oscillations under unstable operation. These are desirable for thedesign and safe operation of current and future BWRs.

1.1.1.1 Experimental Investigations in boiling natural circulation loops

Experimental investigations in two-phase natural circulation loops having single boilingheated channel have been carried out in [10-15]. They observed density-wave instability intheir experiments, which was found to increase with increase in channel exit restriction,inlet subcooling, and decrease in pressure, channel inlet restriction and downcomer level.Such behaviour was also observed in parallel heated channels of a two-phase naturalcirculation loop by Mathisen [16]. Chexal and Bergles [11] observed seven flow regimeswhen their loop was heated from cold condition, out of which three were steady and fourwere unstable. Lee and Ishii [12] found that the non-equilibrium between the phases such asflashing created flow instability in the loop. Kyung and Lee [13] investigated the flowcharacteristics in an open two-phase natural circulation loop using Freon-l 13 as the testfluid. They observed three different modes of oscillation with increase in heat flux such as

(a) periodic oscillation characterised by flow oscillations with an incubation period.

(b) continuous circulation which is maintained with the churn/wispy-annular flow pattern.This was found to be a stable operation mode in which the flow was found to increasewith heat flux first and then decrease with increase in heat flux, and

(c) periodic circulation characterised by flow oscillations with continuous boiling inside theheater section (i.e. there is no incubation period) and void fraction fluctuates from 0.6 to1.0 repeatedly. In this mode, mean circulation rate was found to decrease with increasein heat flux although the mean void fraction kept on increasing.

Jiang et. al. [14] observed three different kinds of flow instability such as geysering,flashing and density-wave oscillations during start-up of the natural circulation loop. Wu et.al. [15] observed that the flow oscialltory behaviour was dependent on the heating powerand inlet subcooling. Depending on the operating conditions, the oscillations can beperiodic or chaotic.

Fukuda and Kobori [17] observed two modes of oscillations in a natural circulationloop with parallel heated channels. One was the U-tube oscillation characterised by channelflows oscillating with 180° phase difference, and the other was the in-phase modeoscillations in which the channel flow oscillated alongwith the whole loop without anyphase lag among them. Aritomi et. al. [18] observed three kinds of instabilities during thepower raising process of a two-phase natural circulation loop with twin boiling channels,such as geysering, in-phase natural circulation oscillation and out-of-phase density-waveoscillations.

1.1.1.2 Experimental investigations in boiling forced circulation loops

Similar experiments to study flow instabilities in forced convection loops havingboiling heated channels have also been carried out by many investigators [19-24]. Aritomiet. al. [19] found that the flow oscillations in the parallel boiling channels were independentof the magnitude and nature of disturbance at the inlet of the channels. They also found thatwhen the flow conditions differed between the channels but where the individualcharacteristic frequencies and inertia masses were approximately equal, the instabilitybehaviour almost agreed with that of identical channels under the average operatingconditions of dissimilar channels. The system was found to be more stable with increase indifference of flow conditions between the channels. Takitani and Takemura [20] obsevedthat the phase difference between the inlet and outlet of flow rates in the boiling channelwas 180 . They also found that the threshold of instability was unaffected by the presenceor absence of super heated steam. Nakanishi et. al. [21] observed both fundamental andhigher mode of oscillations in their experiment. Wang et. al. [22] observed density-waveoscillations even up to 100 bar in their experiment. Xiao et. al. [23] observed pressure dropand thermal oscillations in addition to the density-wave oscillations which occurred both atlow and high flow quality in their experiments. Their experiments also revealed that thedensity-wave oscillations can appear at pressures up to 192 bar and disappear above 207bar. Fukuda et. al. [24] conducted experiments in parallel multiple channels (up to sevenchannels) using an apparatus with short or long riser pipes. They observed Type I density-wave instability in both the cases. The flow was found to be more stable with short riser

pipes than those with long riser pipes. They classified the oscillation modes as following: amode in which (n-x) channels oscillate in-phase and x channels oscillate out-of-phase, andanother mode with all channels having phase difference of 360/n degree with each other.

1.1.1.3 Theoretical investigations in boiling systems

Several studies based on linear and non-linear analyses have been carried out in the pastto investigate flow instabilities in two-phase flow systems. These analyses solve the energyand continuity equations to obtain the enthalpy distribution in the heated region. Themomentum equation is then integrated around the loop with this enthalpy distribution. Inthe linear analytical technique, the governing equations are linearised by perturbing aroundthe steady state and solved analytically to obtain the characteristic equation and the stabilityof the system is investigated from the roots of the characteristic equation. In the non-linearanalyses, the governing equations are solved numerically by using finite difference method.The linear stability method takes less CPU time to evaluate the threshold of stability.Hence, it is useful when analysing the mechanism of instability by plotting stability maps.But the accuracy of this method is limited within the threshold of instability. Beyond thethreshold, the non-linear oscillations appear which can not be predicted by the linearstability technique. For this situation, the non-linear analysis is usually employed. However,the problem with the non-linear analysis is the numerical diffusion which may appear in thecalculations and then it is difficult to interpret the physical instability of the system.

1.1.1.3.1 Investigations based on linear analysis

Analytical investigations based on linear stability analysis on flow instability in anatural circulation loop have been carried out in [25-28 •]. They assumed homogeneous two-phase flow in their analyses and validated their predictions with the test data. Saha andZuber [29] modified Ishii and Zuber's [25] model by taking in to account the thermal non-equilibrium effect between the phases. They found that the thermal non-equilibrium effectpredicts a more stable system at low subcooling when compared with the thermalequilibrium model.

Fukuda and Kobori [17] classified the two types of instabilities named as Ledinegg anddensity-wave oscillations in to eight different types depending on the dominance of pressuredrop components (viz., gravity, inertia, friction and acceleration) in the heated channel andnon-heated riser sections. Roughly, two main types of instability can be distinguished fromtheir study; one is the low frequency Type I instability caused by the dominance ofgravitational pressure loss term and the other is the high frequency Type II instabilitygoverned by the dominance of frictional pressure loss.

Furutera [26] found that the threshold of instability depends on the two-phase frictionfactor multiplier and the heat capacity in the subcooled boiling region considered in theanalytical model. Lee and Lee [27] predicted the threshold of instability for Ledinegg anddensity-wave oscillations and showed that the density-wave instability analysis is sufficientfor the stability analysis of two-phase flow system. Wang et. al. [28] predicted the threshold

of denisty-wave instability for a two-phase natural circulation loop and found that theinstabilities can appear at low as well as at high power levels. The stability of density-waveoscillations was found to decrease with decrease in diameter of riser pipes and systempressure.

Similar analyses have also been carried out by numerous investigators for forcedconvection boiling system [30-33]. Sumida and Kawai [30] derived expressions for thehydraulic stability of boiling channels in terms of flow impedance defined by the dynamicresponse of pressure drop to the inlet flow based on which they classified the density-waveinstability in to three main categories. Fukuda and Hasegawa [31] derived the mode ofoscillations of parallel boiling channels from the characteristic equation. Taleyarkhan et. al.[32] considered subcooled boiling, heat flux distribution in the channel, heater walldynamics, slip flow and transient mixing of flow paths in their analysis. They found thatsubcooled boiling destabilises the system and the system stability improves with ventilationbetween the channels. Clausse et. al. [33] found that for system having identical channels,the system can oscillate with all channels in-phase with the external loop while the channelsthemselves oscillating out-of-phase maintain a constant flow in the loop. If the channels aredifferent, complicated modes of oscillation involving coupling between the channels andthe loop can be formed.

1.1.1.3.2 Investigations based on non-linear analysis

Non-linear analyses based on numerical technique have been carried out by Gurugenciet. al. [34], Chatoorgoon [35] and Nigamutlin et. al. [36]. Gurugenci et. al. developed anumerical code to generate limit cycles of pressure drop and density-wave oscillations in aboiling upflow system in a channel. Chatoorgoon developed a computer code named as'SPORTS' which solves the conservation equations numerically with minimumapproximations, avoiding the use of property derivatives and matrix inversions. It alsopermits large and small time steps. This code can predict the limit cycle oscillations innatural circulation and forced circulation systems. Nigamutlin et. al. found that the heattransfer coefficient does not affect the threshold of stability while the thermal wall inertiahas a strong stabilising effect. Non-linear analyses also revealed presence of chaoticoscillations in two-phase flow systems by [37-39].

1.1.1.4 Investigations relevant to power reactors

Apart from all these fundamental studies cited above, there are investigations on flowinstabilities in power reactors. Krishnan and Gulshani [40] investigated the flowinstabilities in CANDU type PHWRs analytically and experimentally under two-phasenatural circulation conditions. They observed an intermittent flow oscillations that occurredat low channel exit qualities caused by periodic formulation and expulsion of slug bubblesin the boiling channel, and, an oscillatory flow instability that occurs at high channel exitqualities characterised by large amplitude sinusoidal type flow oscillations in the channel.Mochizuki [41] studied the instability behaviour of the ATR in a simulated experimentalfacility under natural circulation and forced circulation conditions. Also, he carried outnumerical investigations to clarify the various types of flow instabilities in the reactor.

Moreover, all these studies clarified that the density-wave oscillation is the mostcommon type instability which can occur in boiling natural circulation loops and in forcedconvection boiling parallel heated channels. Occurrence of this instability depends on thesystem pressure, channel inlet and outlet resistances, heater power and inlet subcoolingconditions. The oscillatory characteristics can become more complex with increase, indissimilarities among the boiling channels.

1.1.2 Coupled neutronic-thermohydraulic oscillations

As said before, in a BWR the neutronics can be coupled with the thermohydraulicoscillations to initiate reactivity oscillations in the core. This is because, in BWRs thepower generation is directly related to the neutron flux, which is a function of the reactivityfeed back which in turn depends strongly on the void fraction in the core. Thus when thevoid fraction oscillates due to the oscillations of the flow, the power oscillates according tothe reactivity feed back. The neutronic feed back depends on (1) the neutron dynamicswhich determine the power generated in the fuel, (2) the fuel dynamics, which determinethe power extracted from the fuel to the coolant and (3) the channel thermal hydraulics,which characterise the void fraction response to changes in heat flux and that include theinlet flow feed back via the downcomer, and (4) the reactivity feedback dynamics that relatethe void fraction to a reactivity value which affects the neutron dynamics. Fig. 3 shows atypical block diagram describing the interactions among the neutronics, fuel dynamics andthermal hydraulics. If we increase the external reactivity, the reactor power will increase(via GR). This causes an increase in fuel temperature (via GF) and the void fraction (via Ga).However, before the power generated by the neutronics can feed back through themoderator density, it has to change the fuel temperature to alter the heat flux from the fuelto the coolant, which depends on the fuel time constant. This may destabilise the thermalhydraulic instability by adding the phase delay but may stabilise it by filtering theoscillations having large frequencies [42]. Moreover, the effects of the thermal fuel timeconstant on the thermal hydraulic instability depends on the resonance frequency of itsoccurrence. It is also shown by March-Leuba and Rey [42] that the void reactivity can havelarge filtering effect (i.e. gain reduction) for frequencies higher than the inverse of thedensity-wave time constant. This is because if the power oscillation frequency is higherthan the density-wave characteristic time delay, the wave front will not have time to leavethe top of the channel before the next wave front is created. In this manner, the positive andnegative parts of the wave cancel each other and there is a significant decrease in overalldensity reactivity feed back. On the other hand, if the power oscillation is of low frequency,the cancelling effects is not produced and the gain is not reduced. Hence, the effect of thevoid reactivity feed back on the density-wave instability depends on the characteristic ofoscillation of the thermal hydraulics in the system.

1.1.2.1 Computer codes to simulate coupled neutronic-thermohydraulic instabilities

A lot of research on this area has been carried out in the past to model and understandthe instability behaviour in commercial BWRs. State-of-the-art review on this instabilityhas been provided by March-Leuba and Rey [42]. This review suggests that there are many

incidents of coupled neutronic-thermohydraulic instabilities in commercial BWRs in thepast. However, the topic generated added interest after the oscillations observed in the twooperating reactors, i.e., La Salle 2, USA [ 43] and in the Coarso, Italy [44]. The oscillationsin the La Salle 2 reactor was an in-phase mode oscillation in which the neutron flux, flowrate and void fraction almost oscillated in-phase with each other at all points in the corewhich eventually resulted in reactor scramming. The oscillations in the Coarso plant was anout-of-phase instability in which one half of the core oscillated 180° out-of-phase with theother half. The out-of-phase instability is difficult to detect with average power detectorsand may not therefore lead to an automatic scram of reactor if instability occurs. Hence, theout-of-phase instability is more dangerous compared to the in-phase mode instability.

Many investigators have attempted to model both the above types of instabilities bynumerical time domain and frequency domain analytical codes. Some of the well knownbenchmarked codes used to simulate the La Salle 2 and Coarso instabilities on time domainare TOSDYN-2 [45], STANDY [46], RETRAN [47] and TRACG [48]. TOSDYN-2includes a 3D neutron kinetics model coupled to a five equation thermal hydraulics modeland considers multiple parallel channels as well as the whole loop. STANDY incorporates3D neutron kinetics and parallel channel flow but does not consider the whole loopbehaviour. RETRAN considers ID and point kinetics neutron dynamics equation,homogeneous two-phase flow and single radial thermal hydraulic region. TRACG solvesfull 3D as well as ID point kinetics but takes a lot of CPU time to solve the instabilityproblem. These codes successfully simulated the instability behaviour in these plants andthe causes of instabilities were also found by them. In addition to these, there are manybenchmarked frequency domain analysis codes such as NUFREQ [49], LAPUR [50],FABLE [51], etc. NUFREQ is a family of codes such as NUFREQ-N, NUFREQ-NP andNUFREQ-NPW which calculate the reactor transfer function for the in-phase oscillationmode. The main differences among them are the ability to model the pressure as anindependent variable. LAPUR considers point kinetics model both for in-phase and out-of-phase mode of oscillations. The thermohydraulics part is modelled as up to seven flowchannels whose inlet flows are coupled dynamically between the two plenums of a BWR tosatisfy the pressure drop boundary conditions as imposed by the recirculation loopdynamics. FABLE can model up to 24 radial thermal hydraulic regions that are coupled topoint kinetics to estimate the reactor transfer function for the in-phase mode oscillation.

1.1.2.2 Fundamental studies on coupled neutronic-thermohydraulic instabilities

Fundamental studies in the field of BWR instabilities have also been carried out bynumerous investigators. Kleiss and Dam [52] derived the characteristic equation of anatural circulation BWR using linear stability analysis. Belblidia et. al. [53] derived ananalytical model to study the coupled neutronic-density wave oscillations using nodalapproach based on Avery's coupled core kinetics theory [54]. They evaluated the couplingcoefficient using the steady state flux distribution in the reactor core. This method wasuseful to consider the spatial kinetics effect which is otherwise not considered in the pointkinetics model. Their main findings was that the stability margin was greatly affected by thecoupling coefficient among the different regions of the core.

March-Leuba and Blakeman [55] provided one of the most succinct explanations of thephysical processes involved for the occurrence of out-of-phase instability as a competitionbetween the stabilising subcritical neutronics and the destabilising gain of the thermalhydraulic feed back. They pointed out that under certain circumstances, the destabilisingeffect of void feed back can excite the first harmonic, even if the fundamental (in-phase)mode is stable. The stability margin was related to the subcriticality of the first harmonic(i.e. the separation of the fundamental and first harmonic eigen values).

Hashimoto [56] showed by modal expansion that in the low flow and high power regionof a BWR, out-of-phase oscillation can occur for certain values of subcriticality and voidreactivity feed back. Rao et. al. [57] found out the influence of the fuel time constant andvoid reactivity coefficient on the instability of in-phase mode considering a single channelof the La Salle 2 reactor under its transient conditions. Uehiro et. al. [58] applied the aboveanalytical model to study the in-phase and out-of-phase instabilities in the parallel boilingchannels of the La Salle 2 reactor. They found out that the interaction between the channelsdue to neutron diffusion influence the out-of-phase oscillation.

Munoz-Cobo et. al. [59] developed a non-linear phenomenological model based onmodal expansion of neutron kinetics equation to study the in-phase and out-of-phaseoscillations in a BWR. They found out that the in-phase mode oscillations appear onlywhen the azimuthal mode has not enough thermal hydraulic feed back to overcome theeigen value separation but the out-of-phase oscillations can occur due to different thermalhydraulic properties of the two reactor lobes if modal reactivities have appropriate feedback gains.

Moreover, these studies reveal that the nuclear-coupled thermohydraulic instability is avery complex problem which depends on the behaviour of channel thermal hydraulicoscillations, radial and axial power distribution in the core, reactivity feed back effects dueto the void and fuel temperature, and the fuel time constant. These instabilities may occur inforced circulation BWRs under natural circulation conditions after a pump trip transientwhen the core exit quality is high due to low flow and high power. This may cause a largevoid reactivity coefficient and two-phase pressure drop to initiate instabilities. Suchinstabilities can also occur in a natural circulation BWR (for example, the Dodewaardreactor) as predicted recently by Van Bragt and Van der Hagen [60]. They considered asingle channel of the above reactor and found out that the stability behaviour of the reactoris strongly dependent on the state of thermohydraulic system [3]. They also observed thatthe Type-I and Type-II thermohydraulic oscillations which occur at low and high powersrespectively are influenced by the neutron and fuel dynamics in different manner.

From these studies, it may be anticipated that the AHWR may also experience boththermohydraulic and coupled neutronic-thermohydraulic instabilities under certain conditions.Experiments carried out in the simulation facility of the ATR [41] which is also a pressuretube type forced circulation BWR, showed thermohydraulic instabilities for certain operatingconditions when operated under natural circulation mode. It was also observed that both TypeI and Type II thermohydraulic instabilities can occur in the ATR up to a pressure of 7 MPa.With increase iti subcooling, the instability boundaries for both the above types were found to

be enhanced. Since the AHWR will be started from atmospheric condition by naturalcirculation, instabilities that occur at low pressures such as the Ledinegg type may appear inaddition to the density-wave instability observed in the ATR simulation facility. Hence, fromthe view point of the design of the AHWR, it is required to evaluate the stable and unstableregions from atmospheric to operating pressure conditions considering the effects of both thegeometry and operating conditions. Looking at the geometry of the AHWR, it can be observedthat the PHT System contains many parallel channels which are connected between the headerand the steam drum by equal number of very long inlet feeder pipes and tail pipes respectively.The stability analysis of such a system having multiple channels with different power,resistances and operating under natural circulation conditions is much more complex anddifferent from that of a forced circulation vessel type BWR, because the total flow in the PHTSystem of a natural circulation BWR is dependent on loop geometry, inlet subcooling and theheat applied, and the flow in the parallel channels depends on the pressure drop in eachchannel connected between the inlet header and the steam drum. Most of the previous analyseshave not considered the stability behaviour of such a natural circulation system. Further tothis, since each channel in the AHWR is connected to the steam drum and the header withvery long inlet and outlet pipes respectively, the period of density-wave oscillation is expectedto be much larger than a conventional vessel type BWR. The effects of neutron and fueldynamics on these low frequency density-wave oscillations occurring due to the interactions ofthe multiple long parallel paths having different power, resistances and with the external loop(i.e. downcomers) under natural circulation condition have not been studied earlier. Hence, itis required to develop analytical models to study the thermohydraulic and coupled neutronic-thermohydraulic instabilities in the AHWR for the in-phase and out-of-phase mode ofoscillations which may occur in the multiple boiling channels of the reactor from atmosphericto rated pressure conditions.

2.0 MODELLING OF THE STABILITY BEHAVIOUR OFTWO-PHASE NATURAL CIRCULATION

The mathematical model to be presented in this section will be applied to investigatethe stability behaviour of the Indian AHWR. Fig. 4 shows a simplified loop of AHWRconsidered for analysis. As shown in the figure, all the steam drums and downcomers arelumped in to an equivalent steam drum and downcomer respectively. All the channels in thecore are connected between the inlet header and steam drum by same number of inletfeeders and outlet feeders respectively. In the analysis it is assumed that

in the single phase region ;

(a) fluid is incompressible and

(b) Boussinesq approximation is valid for variation of density with temperature, and,

in the two-phase region;

(a) flow is homogeneous,(b) 5p/dt in energy conservation is neglected,(c) subcooled boiling is neglected,

10

(d) two-phases are in thermo-dynamic equilibrium,(e) no carry-over and carry-under in the steam drum (i.e. complete separation),(f) complete instantaneous mixing of feed water in the steam drum,(g) heat losses in the loop pipings are neglected and(hj inlet subcooling is constant for a particular power and feed water temperature

conditions.

With these assumptions the conservation equations of mass, momentum and energy for one-dimensional two-phase flow are given by

Mass

dw

' ck

Momentum

~A~a

Energy

f A2

c

= 0,

—(w2v) + ~v+2DAT •

i heated region,

f-0. (2,

dt ' &~\ 0 adiabaticregion. ^

State i . • '

P = f(P,h). (4)

The pressure drop due to bends, restrictions, spacers, etc. was estimated as

tpk =Kw2 IlpA2. (5)

2.1 Steady State Solution

The governing equations for the steady state conditions are obtained by dropping thetime derivatives. The Eqs. (1) to (3) can be written as

f-v.

I I

ch f qhA heated region, •

da ~ \ 0 adiabatic region.

Eqs. (6) to (8) can be solved analytically and expressions for pressure drop for single-phaseand two-phase regions can be obtained as

1-6 region

8Pm + > 2 1 2 D A l Pin adiabatic region,2 2 p v heatedregion,

2-6, region

a l(wahvfg I Ahk) + \gl{vf +(qhA(x-Lxp)v/i./whJ-K)}]

-~T=z'S + (Jw v/2DA ) heated region,(g I v) + (fw2v / 2 DA2) adiabatic region,

(10)

where Lspis the length of single-phase region in the heated channel and is obtained by a

heat balance as

The pressure drop for each channel with associated inlet and outlet feeders (A/?cA) can beestimated from equations (9) and (10). Since all the parallel channels are connectedbetween inlet header and; steam drum, (A/?^), for each parallel path from inlet header andsteam drum is same, i.e.

PH ~ PSD = &PH-SD = (APc*)l = (APc*>2 = (APrt)« • (12)

Similarly, the pressure drop between the steam drum and inlet header is estimated as

l A 2 , (13)

where AZ is the elevation difference between the steam drum water level and inlet headerand wt is the total loop flow rate and is equal to

r-l

12

The total PHT flow rate (w,) and flow rate in each channel (wdl)l connected between inletheader and steam drum are estimated by solving equations (12) to (14) together iterativelywith the condition that A/?W_VD + A/?S7)_H is equal to zero.

2.2 Linear Stability Analysis

The set of conservation equations (1) to (3) are linearised by superimposing of smallperturbations of w , h , p , v and qhover the steady state values as follows

w = wa + w ; h = ha + hi; p = pxx + p ; v = vxx +v';qh= qhxx + qh where

w" = w(x)ee";h =h(x)sex';p = p(x)se";v' = v(x)ees';qh =qh eexl (15)

In equation (15) e is a small quantity and w,h,p,v and qh are the averages of perturbedflow rate, enthalpy, pressure, specific volume and heat added per unit volume of coolantrespectively and s is the stability parameter. Substituting them in the conservationequations, the perturbed equations for the single-phase region and two-phase region of theloop can be written as follows.

1 -<(> region

Perturbed mass conservation equation is written as

(16)

Perturbed energy conservation equation can be written as

3i PinAs . \-{w'qhssA!wl) + qhAlwss heated region,

3c w^ [ 0 adiabatic region,

Perturbed momentum conservation equation is written as

^ + w'[(s I A) + ( > „ / PinDA2)) - (g/3pinh I Cp) = 0, (18)

2-<{> region

Perturbed mass conservation is written as

(19)

13

Perturbed energy conservation equation can be written as

+ As {. =\-[(w/wl)qhtSSA)]

1Wu 0

s heated region,

adiabatic region,(20)

Perturbed momentum conservation equation is written as

dp

A2)(v )(ch'

(21)

Solution of the perturbed equations for various segments of the loop can be obtained byintegration as follows.

In the single-phase region,

(a) heated region,

w = win = const., (22)

(23)

where hin is the perturbed inlet enthalpy at the inlet of core and x is the residence time of thefluid in the section.

- A / 7 = {si A)P,nDA2 m"sp

L W S V r _ r v i

w.

(b) adiabatic region,

w -wi» = const.,

(24)

(25)

(26)

14

-Ap = (si A)

In the two-phase region,

(a) heated region,

w =wsp

As vfg

ti =[Per>Lc

where

P =~ hv(r2 + Aspf I w j - I w; + q'hA I wx

(27)

(28)

(29)

(30)

r,=[-C1+(C12-4C2)0 5]/2,

r2=[-C1-(C1z-4C2n/2,

(31)

(32)

(33)

(34)

(35)

and tim is the perturbed enthalpy at the inlet of boiling region of the channel which can beestimated from Eq. (23).

C, =

w L +As v

* ' "2 h,,

+ i

15

(b) adiabatic region,

= *>m + K , /v.)(vA

(36)

(37)

(38)

-Ap =-

(39)

The perturbed pressure drop across the bends, orifices and spacers in the single-phaseregion is given by

Apk sp=(K,p/ p42)w ( Vwm . (40)

Similarly, the perturbed pressure drop due to bends, spacers and other restrictions are givenby

2w, (41)

For constant inlet subcooling, i.e. for the case hjn = const., the general expressions forperturbed flow rate, enthalpy and pressure drop in single-phase and two-phase regions canbe expressed as a function of single-phase perturbed flow rate in the steam drum, i.e.

in the single-phase region,

w =f(w'SD) = const.,

Ap =h(w'SD),

in the two-phase region,

(42)

(43)

(44)

(45)

(46)

16

Ap=hl(w'SD). (47)

For the geometry of AHWR consisting of multiple parallel channels coupled with thedowncomer, the perturbed equations can be written as follows.

For each parallel channel connected between header and steam drum,

where G(wm), is the sum of perturbed pressure drops in single-phase and two-phase regionsof ilh parallel channel.

The perturbed pressure drop in the single-phase region between the steam drum level andheader is given by

Ap - G (w. ) . (49)

It may be noted that (wsl)) is also the total perturbed flow rate in the single-phase region

of the system and is the sum of the perturbed flow rates at the inlet of all parallel channels,i.e.

n

( XV ^ = 7 (XV ^ /'SO^

By substituting Eq. (50) into Eq. (49) and adding Eq. (48) to Eq. (49), the following sets of

simultaneous equations can be easily obtained for the condition that ApSD_lf + Ap,,_SD = 0 ,

i.e.

AwJi +G\(w,n)\ +Gr(w,n)2 +GT(wm)3 + + Gr(wm)n = 0 , (51a)

I yU 1 -*- ( j / yU 1 - ^ IT I yU 1 -+- IT,, ( \\) I "T" —~ ~~ "~~ ~~ ~~ -f" C 7 1 W ) ^~ \J \ J 1 D I

+GT(wm)2 +G r (wJ , + +G.,(w,;,)n +Gn(win)n =0. (51n)

17

Dividing equation (51) by Gr throughout the following matrix form of equation (51) can bewritten

G,/Gr

G2IGT 1

1 1 + G3 / GT

K,),

= 0. (52)

Eq. (52) can also be written as

and the characteristic equation is Ftj = 0,

(53)

(54)

where Fi} = (55)

To determine, the coupled neutronic-thermalhydraulic instabilities in the AHWR it isrequired to derive the relationship for the perturbed heat generation rate ((#*),) in thechannel by considering the neutron and fuel dynamics.

2.3 Neutron Kinetics

The point kinetics approximation is adopted for the neutron field dynamics as given by

dn( 5 6 )

m=\

The precursor concentration for each group can be obtained as

dt I(57)

where n is the neutron density, k is the effective multiplication factor, ft is the delayedneutron fraction, Xm and Cm are the decay constant and precursor concentration of delayed

18

neutrons of group m respectively. Eqs. (56) and (57) are linearised by perturbing over thesteady state as discussed before and the perturbed equations can be easily solved aftereliminating the steady state conditions to obtain

(58)

where n is the perturbed neutron density and k is the perturbed reactivity which is relatedto the void reactivity coefficient and Doppler coefficient as

k'=Cayav+CDTLav. (59)

In Eq. (59) ym and ffjav are the perturbed void fraction and fuel temperature

respectively averaged over the heated channel length. They can be estimated from thecoolant density and the fuel heat transfer equations as discussed below.

2.4 Fuel Heat Transfer Model

Assuming only radial heat transfer, the fuel heat transfer equation can be written as

(60)

where mf is the mass of fuel rods, Cf is the specific heat capacity of fuel, Hf is an

effective heat transfer coefficient, Q(t) is the heat generation rate in the fuel rods, 7} „,,(/)

is the length average fuel temperature, af is the heat transfer area of fuel rods and Tsat is the

coolant saturation temperature.

Perturbing Eq. (60) over the steady state for Tl av(t) and Q(t) and cancelling the

steady state terms, we get

TfMV(mfC/s + H/a/) = Q\ (61)

where Q is the perturbed heat generation rate in the fuel rod.

Applying the heat balance equation for the heat transfer from fuel to coolant

Hfaf(T/av-Txa!) = qhAcLc. (62)

Perturbing Eq. (62) over the steady state and cancelling the steady state terms we get

TfjOV=qhAcLJHfar (63)

1 19

Substituting Eq. (63) into Eq. (61) and rearranging we get

mfCfsl Hfaf

(64)

Since the heat generation rate in fuel is proportional to the neutron density, Eqs. (58) and(59) can be substituted into Eq. (64) to yield

6

where Gf = ——CDACLC

(65)

(66).

and rf =mfCf

is the fuel time constant.

The density of two-phase mixture is given by

get

r = -

(67)

Perturbing Eq. (67) over the steady state and cancelling for steady state condition, we

(68)v.

The channel average perturbed void fraction can be obtained by integration as

(69)

which can be approximated after some algebraic simplification as

. 1 v/gwhere y, =——- hfg

-er r'L* —&'* -eriL")r2 V 1

1 vfg 1

(70)

(71)

(72)

20

—Qu KKA(73)

(74)

(75)

(76)

Substituting Eq. (70) into Eq. (65) an expression for the perturbed heat added/unitvolume of coolant (qh) to any channel /' for a perturbation of channel inlet flow rate (wm)in the ilh channel can be easily obtained as given by

(77)-Gf\\i

Eq. (77) can be substituted into Eqs. (23) and (29) to obtain the perturbed enthalpy inthe single-phase and two-phase regions of any heated channel respectively. These can befurther substituted into Eqs. (24) and (36) to yield the perturbed pressure drop in single-phase and two-phase regions of any heated channel. Finally the characteristic Eq. (55)changes accordingly by substituting the respective perturbed pressure drop components intoEq.(52).

2.5 Neutron Kinetics Model for Out-of-phase Oscillations

During an out-of-phase instability, neutron diffusion from channel to channel may bean important factor due to change in void fraction among the boiling channels which thepoint kinetics model does not take into account. This may result in a different qh for anychannel than that calculated using Eq. (77). To analyse this problem, a coupled multipointkinetics model based on Baldwin's theory of neutron kinetics of coupled cores [61] hasbeen employed which was applied to investigate the oscillatory behaviour of multipleparallel channels of the AHWR. Further to this, the modal point kinetics model developedrecently by Hashimoto et. al. [62] was also applied to predict the out-of-phase modeoscillations of the AHWR and the results of the two models were compared.

2.5.1 Coupled Multipoint Kinetics Model

The model considers the reactor core to contain 'N' number of subcores which aresubcritical, isolated by reflectors and influence each other only through leakage neutrons

21

number of which is proportional to the average neutron flux over each subcore. Eachsubcore may contain one channel or group of channels having the same power andresistances.

Taking the model as 'N' bare homogeneous reactors, the coupled multipoint kineticsequation for the i,h subcore is given by

dnt(t) ^ 0 ( 1 / 7 , ) !

yJ*i

Pm.iJ{ , 4 f t ) (79)where <xy is the coupling coefficient that determines the reactivity contributed by the

interaction of j , h subcore with the ilh subcore. In the analysis, af> is assumed to be

constant which can be estimated from the steady state condition of Eq. (78) as

(80)

For small excess reactivity

,... "'•"

Perturbing Eqs. (78) and (79) and solving them together as obtained before for the pointkinetics equation, the perturbed neutron density in the i,h subcore can be obtained as

",,,, n,..../©,, (82)

where T, = i - V-£E=_ , (83)

Eq. (82) can be further written as

where Fj ~\ I <9; (86)

Eq. (85) can be written in expanded form for N subcores as

n,

•+aXHnN

+ai2n2

+a2HnN

+aiNnN

(87a)

(87b)

(87c)

n.(87d)

(87e)

lt\ matrix form

- a B - a

- a , - a.

-a. -a, -a.. . 1/

Nx)

* >

(88)

(89)

where T = < , .[- a,, for i

(90)

(91)

23

kn

or.\ri=r[Xit X,, . . . X,\ (92)

kn.

(93)

or, Nx\(94)

Since heat generation rate Q is proportional to the neutron density n

S o (95)

The perturbed heat added/unit volume of coolant for any subcore can be obtained bysubstituting Eq. (95) into Eq. (64).

2.5.2 Modal Kinetics Model for Analysing the Out-of-phase Instabilities

The out-of-phase oscillations in BWRs is a spatially dependent phenomenon with aneutron thermallhydraulic coupled feedback effect. It is characterised by power oscillationswith one half of the core increasing and the other half decreasing with a phase difference of180°. This instability is explained recently as a phenomenon in which the neutron higherharmonics mode is excited by the thermalhydraulic feedback effect [55, 62-64]. Thermalhydraulic excitation of the first azimuthal mode resulting in out-of-phase oscillations wereobserved in operating BWRs by Bergdahl and Oguma [64]. The in-phase (fundamental)oscillation mode has a keff equal to one in the equilibrium state. The higher harmonic

modes are all subcritical which could result in out-of-phase oscillations depending on thesubcriticality of the harmonic mode and the void reactivity feedback. A derivation of themodal point kinetics equation governing higher harmonics oscillations of the neutrondensity and precursor concentration is given in reference [62]. Following this theory, theperturbed neutron density considering the modal point kinetics equation can be expressedas

n

<'+tn . l

where p t = — - —

(96)

(97)

24

In Eq. (97) Xo and Xc represent the eigenvalue of the fundamental mode and higherharmonics. Hence, the subcriticality, pc, is identical to the eigenvalue separation of thefundamental and higher harmonics mode. The subcriticality can be evaluated by relating tothe geometric buckling using a one group diffusion theory as [63]

(98)

where D is the diffusion coefficient, AB2n is the geometric buckling difference between

harmonics and fundamental mode, v number of neutrons/fission and ^ is the fission

cross section. Eq. (58) can be replaced by Eq. (96) to analyse the out-of-phase instabilitiesin the AHWR core.

2.6 Mode of Oscillation

In the AHWR as shown in Fig. 1, both in-phase and out-of-phase oscillations mayoccur. During an out-of-phase oscillations, the thermal hydraulic parameters oscillateamong the boiling channels connected between the header and the steam drum with certainphase difference and amplitude ratio without any oscillations taking place in the externalloop (i.e. the downcomers). On the otherhand, during an in-phase mode of oscillations, theflow oscillations in the channels take place without any time lag and also with the samefrequency as in the external loop

From Eq. (48) we can express the ratio of perturbed flow rate oscillation between channels/ and j as

J J (99)

The quantity M + jN can be expressed as Re75 where R is the ratio of amplitude

(R = yj(M2 + ;V2)and 0 is the phase difference (0 = tan'1 (N / Mj). The nature ofoscillation, i.e. in-phase or out-of-phase can be determined by substituting the roots of thecharacteristic Eq. (54) into Eq. (99).

2.7 Determination of threshold of stability, frequency of oscillationand the Decay Ratio

The stability of the system is investigated by determining the roots of the characteristicequation (Eq. (54)). If s is the root of the characteristic equation given by s-rj + jarwhere rj is the real part and m is the imaginary part of the root. Then the system isconsidered to be stable if all TJ < 0, and unstable if any rj > 0. At the neutral threshold pointat least one of the 7 = 0.

The time period of oscillation is related to the absolute value of the imaginary part asT = 2K I \w\ and the frequency of oscillation is / = 1 / T.

25

The decay ratio defined as the ratio of two successive amplitude of impulse can be

obtained as DR = e2""1'^. The DR indicates the stability margin which the system can haveat any operating conditions. If it is less than one, the system is stable. If it is more than one,the system is considered to be unstable. At DR equal to one, the system is at the thresholdof stability.

3.0 POWER DISTRIBUTION IN THE AHWR CORE

In the AHWR, the power distribution in the core varies radially and axially. Fig. 5shows a typical power distribution in the core, which is considered in the present analysis.Table 1 gives the required channel inlet loss coefficients for different channels so as toachieve proper flow distribution in the channels in order to have nearly uniform channelexit qualities. It can be observed from Table 1 that the channel inlet loss coefficients varyfrom 0 to 250 depending on the radial power factors. Fig. 6 shows the axial powerdistribution in the core for different channel types. The channels with similar axial andradial power profiles, inlet orificing and resistances are grouped into one channel type. Soeach channel type will have different thermal hydraulic characteristics from the other. It canbe observed that the axial power profiles are more or less uniform for all the channel types.The slight increase in heat flux at the top edge of the core is due to the presence of heavywater reflector which is normally not present in conventional BWRs.

4.0 MODEL VALIDATION

It is required to validate the theoretical model discussed earlier with the test data. Forthis purpose, experimental data obtained from a simple forced circulation loop having twinboiling channels and that of the Heat Transfer Loop (HTL) that simulates the ATR wereused. The former experimental facility operates under atmospheric condition and boilingtakes place in both the heated channels. The details of this facility are given inreference[19]. The threshold of stability for the twin boiling channel system has beenpredicted and comparison of the predictions with the test data is shown in Fig. 7. It can beobserved that the predictions are in good agreement with the test data. However, it is alsorequired to validate the present model with the test data at higher pressures obtained fromlarge facilities having geometry similar to that of the AHWR. In literature, it is found thatthe ATR is the only pressure tube type BWR for which experimental data under naturalcirculation and forced circulation conditions are available [41]. Figs. 8 and 9 show acomparison of the predictions with the test data for the threshold of instability of the HTLoperating under forced circulation conditions. The results indicate that the theoretical modelcan predict the stability behaviour of boiling forced circulation loops having parallelchannels successfully. It is also important to validate the predictions on the steady statebehaviour of natural circulation/ because accurate predictions of the thresholds forLedinegg type and density wave stability boundaries depend on steady state predictions.Figs. 10 and 11 show a comparison of the predictions with the test data for Steady statenatural circulation flow rate of the HTL. It can be observed from both these figures that thehomogeneous model predicts larger flow rate than that obtained using.the slip model [65].This is because, the homogeneous model predicts larger void fraction in the channels and

26

hence larger driving buoyancy force, thus giving larger flow rate. As observed in thefigures, the present model predicts the steady state behaviour of the system at differentpowers and downcomer levels reasonably accurately; Fig. 12 shows a comparison of thepredictions with the test data for the threshold of instability of natural circulation for HTL.The predictions were carried out at different pressures and subcoolings. The threshold ofstability at any subcooling has been predicted as a function of non-dimensional powercalled phase change number and subcooling number as defined in reference [41]. Asobserved from these figures, the current model predicts the threshold of Type-I and Type-IIinstabilities accurately. Many previous researchers [25-28] have validated the applicabilityof homogeneous model for the stability analysis. However, their analyses have notconsidered the interaction of multiple parallel channels operating under natural circulationconditions.

5.0 LEDINEGG TYPE INSTABILITY BEHAVIOUR OF THE AHWR

The occurrence of Ledinegg type instability can be ascertained by investigating thesteady state behaviour alone. The criterion for this type of instability is given by

^p<0, (100)

where tsp} is the internal pressure loss in the system and Apd is the driving head due to

buoyancy. bp} includes all losses in the inlet feeders, core, outlet feeders, steam drum and

downcomers except the pressure drop due to gravity in the steam drum (liquid level),downcomers, header and inlet feeders. Apd includes the gravity head from the steam drumup to the core inlet. Hence, to check the occurrence of Ledinegg instability the variation of&pf and Apd as a function of flow rate is required.

Figures 13 (a) and (b) show how the lower and upper threshold for Ledinegg typeinstability is identified. From these figures it is seen that the Ledinegg type instabilityoccurs when the power is more than 285 MWth and less than 460 MWth for operatingpressure of 0.1 MPa and subcooling of 30 K. When the power is in between the abovespecified range, the internal pressure loss curve intersects the driving buoyancy curve atthree points (i.e. three operating points at a given power level) which makes the systemunstable. Thus at 30 K subcooling, the system can have two threshold points of instability;Like this, the threshold points for instability can be predicted at different subcoolings.

Since the Ledinegg type instability depends on the variation of pressure drop with theflow rate, it is required to investigate the influence of two-phase friction factor multipliermodels incorporated in the analysis on the Ledinegg type instability. Previous analysis ofFurutera [26] has shown that the threshold of density wave instability is greatly affected bychanging the two-phase friction factor multiplier models in the analysis. Hence, an attempthas been made here to study the influence of different two-phase friction factor multipliercorrelations on the threshold of Ledinegg type instability. Fig. 14 shows a comparison ofthe stability maps obtained by different friction factor multiplier models given in Appendix-

27

1. For this purpose, the system pressure considered is 0.1 MPa. It can be observed that thestability maps are hardly influenced by changing the friction factor models. This may bebecause of the dominance of the gravitational loss in the two-phase region of the systemover that of the frictional loss due to the presence of very long riser pipes in the AHWR. Inthe following analysis, the friction factor multiplier model of Baroczy [66] has been usedfor the stability analysis.

5.1 Influence of geometric and operating parameters on Ledinegg type instability

Figure 15 shows the stability maps at different pressures. It can be observed that thestability of the Ledinegg instability increases with an increase in pressure. This may be dueto the fact that with an increase in pressure the void fraction decreases with qualitysignificantly in the two-phase region, which can reduce the S-shaped variation of theirreversible losses (i.e. dbp{ Idv) responsible for the occurrence of the Ledinegg type

instability. Similar to the Type I and Type II density wave oscillations, two types ofLedinegg instabilities are observed at any subcooling depending on the operating power.With increase in pressure, the threshold power for the lower and the upper instabilityboundary moves to much higher power. The interesting thing which can be observed fromthe figure is that this instability almost vanishes for the AHWR (i.e. shifts beyond theoperating envelope of power for the AHWR) when the operating pressure is more than 0.7MPa. So this type of instability is of concern for the AHWR at low pressures, i.e. during thestart-up conditions. Experiments carried out by Aritomi et. al.[18] and Jiang et. al. [14] forstart-up of natural circulation boiling loops showed instability which were identified bythem as geysering, flashing, natural circulation oscillations and density wave oscillations. Apressure tube type natural circulation BWR may experience Ledinegg type instability inaddition to all those instabilities observed by these investigators during start-up. One simpleway of avoiding this instability for the AHWR is by maintaining subcooling less than 10 Kas observed from the figure.

During a small break LOCA situation, the water level in the downcomer may fallbelow the normal operating condition. Figure 16 shows typical stability map at variousdowncomer levels at steam drum pressure of 0.1 MPa. It can be observed that at a particularsubcooling, both upper and lower threshold powers for instability decrease with reductionof downcomer level. This may be because with reduction of downcomer level, the drivinghead (Apd) for the natural circulation flow decreases. Hence, the internal pressure losscurve (Apf) can intersect the driving head curve at lower power causing reduction of

threshold power for Ledinegg type instability. Similar behaviour is also observed for theeffects of riser height on Ledinegg type instability as observed in Fig. 17.

5.2 Influence of axial power distribution on Ledinegg type instability

The effect of axial power distribution on Ledinegg type instability is shown in Fig. 18.The axial power profiles considered are shown in Fig. 19. The results indicate that the axialpower profile has marginal effects on Ledinegg type instability for a pressure tube type

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BWR for constant total power. This is because, the pressure drop is governed mainly by thelong tail pipes and not by the core. *

5.3 GENERAL CONCLUSIONS

Extensive investigations were carried out to study the Ledinegg type instabilitybehaviour of the AHWR. It was found that this instability occurs at realtively high powerlevels (much above 100 MWth) when the operating pressure is very low (< 0.7 MPa). Theinstability can be avoided by pressurising the system above 0.7 MPa before raising thepower beyond 100 MWth.

6.0. DENSITY WAVE INSTABILITY BEHAVIOUR OF THE AHWR

The density wave instabilities that may appear at low powers or low qualities and highpowers or high qualities (Type-I and Type-II respectively) have been predicted for theAHWR.

Figure 20 shows the in-phase and out-of-phase density wave instabilities which mayoccur in the AHWR considering only the hottest channels of the reactor (i.e. Channel Type-3 having RPF = 1.231 and Kjn =0.0, Table 1). The stability boundary has been plotted onnon-dimensional N^ - Nxuh plane (same as the non-dimensional parameters defined in

reference [41]). As discussed before, during the in-phase mode oscillations the flowoscillations among the channels along with the external loop (i.e downcomers) occurwithout any phase difference among them. During the out-of-phase mode oscillations, theflow bscillations among the channels occur with some phase difference without anyoscillations taking place in the external loop. From this figure, it is clear that the out-of-phase mode oscillations are more dominant as compared to the in-phase mode oscillationsin the reactor (since the former is having less stable area than the latter) because of the extrasingle-phase friction in the downcomers which stabilises the in-phase mode oscillations.Also, it can be observed that the difference in the Type II instability boundary between thetwo modes of oscillations is much larger than that for Type I instability boundary which arein agreement with the results of Van Bragt and Van der Hagen [3]. However, the differencein the Type I instability boundary between two modes of oscillations is also significantunlike that observed in the Dodewaard reactor.

6.1 Influence of geometric and operating parameters on density-wave instability

Figure 21 shows the stability maps for density wave instabilities for the AHWR atvarious pressures considering all the channels of the reactor. Also, the same results areshown in Fig. 22 in dimensional plane. From these figures, it is observed that the stablezone increases with an increase in pressure and a decrease in inlet subcooling which isconsistent with the test results of Mochizuki [41] for the ATR simulation facility. It can beobserved that for the density wave instability, neutrally stable conditions occur for theAHWR boiling channels at the rated full power conditions (i.e. 750 MWth and 7 MPapressure) for core inlet subcooling of about 17 K.

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The present analytical model considers all 12 channel types (as shown in Table 1) andeach channel type lumps all similar parallel channels in to an equivalent channel. It isinteresting to study the effects of interaction of similar channels and dissimilar channelsoperating under natural circulation on the threshold of stability for the AHWR because, it isexpected that the high power channels are more unstable compared to the low powerchannels. So it may so happen that the stable channels can make the unstable channelsstable or vice-versa.

Figure 23 shows a comparison of the threshold of instability for natural circulationbetween a twin channel system consisting of maximum power rated channels (channelType-3 of Table 1) and system consisting of all the channels of the AHWR. In the formercase, the two channels were having the same power and zero inlet orificing coefficient, and,both were connected between the steam drum and inlet header by same number of inlet andoutlet feeders. In the later case, all channel types consisting of all 408 channels of reactorhave been considered. It can be observed from the figure that the maximum rated powertwin channel system has lower stable zone compared to the case which considers all thechannels of the reactor. It is evident that the effects of channels with lower radial powerfactors and larger inlet loss coefficients are to stabilise the unstable channels thus enhancingthe stability of the whole system.

Figure 24 shows the effect of number of parallel channels on threshold of instability.All the channels considered are the maximum power rated channels (i.e. channel Type-3) ofthe AHWR. In the analysis the downcomer area was changed in the same ratio as thenumber of channels. It can be seen that increasing the number of channels does not affectthe stability behaviour of Type-II instability at all. Previous experiments of Mochizuki [41]also confirms that increasing the number of channels having1 similar power distribution doesnot change the threshold of Type-II instability when operated under natural circulation.However, the threshold of Type-I density wave instability slightly decreases with anincrease in number of channels. Since Type-I instability occurs at low power conditions, thenatural circulation flow rate in the channels is also low because of less driving force. Withincrease in number of channels, the flow rate in the system increases not proportionate withthe number of channels due to the variation of frictional pressure drop in the downcomerwith change in area. This may slightly affect the Type I stability behaviour of the system.

Figure 25 shows the effect of orificing of one channel on the stability boundary of atwin channel system when both channels are heated with same power. Both the channels areconnected between the steam drum and the header by the same number of outlet feeders.The orificing loss coefficients considered in the analysis are typical to that required for theAHWR (see Table 1). In the analysis one of the channels does not have any orificing at theinlet and the orificing loss coefficient in the other channel has been varied from zero to250. It can be observed that the threshold of Type-II instability increases when the inletorificing loss coefficient increases from zero to 30. Further increase in inlet orificing !loss,coefficient decreases the threshold of Type-II instability. In a natural circulation system, theflow rate in the channel depends on the heating power and the channel resistance. Withincrease in inlet throttling coefficient for same heating power decreases the channel flowrate which in turn causes an increase in channel exit quality. This reduces the threshold

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power for instability for that channel which may cause the other channel to be unstable. Soincrease in orificing at channel inlet does not always increase the stability of a naturalcirculation system with multiple channels. But, the stable zone of Type-I instabilityincreases with increase in orificing loss coefficient at channel inlet.

When both channels have similar axial and radial power distribution and same inletorificing loss coefficient, with increase in inlet orificing coefficient, the stable zones forboth Type-I and Type-II instabilities increase as observed in Fig. 26. So to enhance thestability, it is required that the channels must have different heating power depending ontheir resistances so as to have nearly same channel exit quality. This problem may be ofconcern when a natural circulation pressure tube type BWR is heated by external heatingwithout fission energy during start-up. It is required that the heaters should supply differentpowers to respective channels depending on their inlet orificing so as to promote thermalhydraulic stability.

The effect of channel outlet resistance on the density-wave instability is shown in Fig.27. It can be observed that the stability of the system considerably improves with decreaseof channel outlet resistance because of reduction of two-phase pressure drop. So to enhancethe stability of natural circulation BWR, restrictions in the two-phase regions should beminimum.

The effects of downcomer water level on the threshold of instability is shown in Fig.28, It can be observed that with reduction in downcomer level, the threshold powers forType-II instability decreases and that for Type I instability increases. Moreover, the stabilityof density-wave instability decreases with decrease in the downcomer water level. This isbecause of the reduction of natural circulation flow rate in the system which increases thetwo-phase frictional pressure drop due to decrease in two-phase mixture density (or increasein channel exit quality) for the same heating power.

Figure 29 shows the effect of riser height on threshold of instability. In a naturalcirculation system, smaller the riser height, lesser is the channel flow rate and larger ischannel exit quality for same heating power. This gives larger two-phase pressure drop dueto large channel exit quality. Larger the riser height, larger is the channel flow rate whichmay cause larger two-phase pressure drop due to larger riser length. So a reduction or anincrease in riser height on stability of natural circulation system is competitive. This fact isobserved from the present work which predicts that change in riser height does not changethe threshold of stability significantly.

7.0 COUPLED NEUTRONIC-THERMOHYDRAULICSTABILITY CHARACTERISTICS OF THE AHWR

Before studying the characteristics of nuclear-coupled thermohydraulic oscillations, itis required to study the pure thermohydraulic oscillations of both the in-phase and out-of-phase modes in the AHWR. This investigation will suggest the nature of dependency of theneutronics on the thermal hydraulic behaviour of the system. Fig. 30 shows a typical flow

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stability map considering the hottest channels of the AHWR (i.e. channel Type 3 havingRPF = 1.231 and Kjn = 0.0). The analysis considers two parallel boiling channels alongwiththe associated inlet and outlet feeders of the reactor. The frequency of oscillations at thethreshold powers for both modes of oscillation have been estimated from the root of thepharacteristic equation and are shown in the above figure. It can be observed that thefrequency of oscillation for Type-II instability is larger compared to that for Type-Iinstability. This is because the Type-I instability occurs at low power when the flowvelocity is smaller under natural circulation conditions than that for the Type-II instabilitywhich occurs at much higher power. Hence, the fluid takes longer time to pass through thetwo-phase region compared to the Type-II oscillation case. In general, it is also found thatthe frequency of oscillation is very much less for the AHWR channels compared to thatpredicted for the Dodewaard natural circulation BWR [3]. This is because the period ofoscillation in the AHWR channels is very large due to large two-phase region of the tailpipes whose length is several times of the chimney height of the Dodewaard BWR.

7,1 Parameters for Coupled Neutronic Thermohydraulic Instability

It will be interesting to study the influence of the neutronic coupling on the lowfrequency thermohydraulic instability studied above. For this purpose, the reactivityparameters were used from the predictions of a 3D neutronics code named as SERIES forthe AHWR. According to this, the Doppler reactivity coefficient is evaluated from thefollowing relationship

C o =-3x l (T 5 +19 .34x l0 - 9 7 \ (101)

Eq. (101) is valid for the AHWR fuel considered in the analysis in the temperature range of300~800 K. Similarly, the void reactivity coefficient (Ca) is estimated to be -0.005 for coreaverage void fraction of 0.4 at the rated full power operating condition. The fuel thermaltime constant is also estimated as 8 s at the rated full power operating condition. All theseparameters are likely to change with burn-up and operating conditions. Hence, they arevaried over a wide range to study their influence on the low frequency thermohydraulicoscillations observed in the boiling channels of the AHWR.

7.1.1 Influence of void reactivity feed back

Figures 31 and 32 show the effect of C,, on the threshold of stability for the in-phaseand out-of-phase mode of oscillations between two subcores respectively considering thepoint kinetics model for the neutron dynamics. Each subcore contains channels of ChannelType-3 (i.e. hottest channels) and hence both have the same power and resistance. Theresults indicate that with an increase in negative Ca the threshold power for stabilitydecreases for Type-I and increases for Type-II instabilities for both in-phase and out-of-phase mode oscillations. The influence is more at higher degree of subcooling. Moreover,the stability of Type I and Type II instabilities increases with an increase in negative voidreactivity feed back. To explain this phenomenon, the decay ratio (DR) (defined as the ratioof two successive amplitude of impulse response) for both Type I and Type II instabilities

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at different void reactivity coefficients are estimated and plotted in Figs. 33 and 34respectively. The channel power considered for Type I instability is 600 kW and that forType II instability is 4500 kW. The channels can experience the above thermohydraulicinstabilities at these powers as seen before in Fig. 22. From these figures it is clear thatwith an increase in negative void reactivity coefficient at any power, the DR decreases. Athigher power (i.e. Type II instability), the effect of void reactivity feed back is more upto aC.a value of -0.2. With further increase in negative Ca, the decrease of DR is not

significant. On the other hand at low power (i.e. Type I instability), the DR continuouslydecreases with increase in void reactivity coefficient. The corresponding frequency ofoscillations for both the above powers are shown in Figs. 35 and 36 respectively. Fromthese figures, it is clear that addition of neutronic feedback on the thermal hydraulicoscillations increases the frequency of oscillations for Type II instabilities. The frequencyof Type I instability does not change significantly with the coupling of neutronics withthermal hydraulics. Hence for these low frequency of the Type-I and Type-II oscillations(<0.08 Hz) for the AHWR geometry considered, addition of neutronic feed back can havesignificant stabilising effect due to the less phase lag between the fuel heat generation rateand channel thermohydraulic oscillation. The destabilising effect due to the increase in gainof the void reactivity feed back loop with increase in negative void reactivity coefficient asobserved in vessel type BWRs [3, 58 ] is not significant for the AHWR.

7.1.2 Influence of the fuel thermal time constant

The effect of fuel thermal time constant on the stability of in-phase mode oscillationfor the above case is shown in Fig. 37. The fuel thermal time constant depends on the fuelproperties, operational conditions and fuel burn-up. Van der Hagen [67] has shown thatwith the use of lumped parameter model it could be as low as 2 s. Hence, the fuel thermaltime constant in the present analysis has been varied over a wide range from its nominalvalue (at rated full power operating condition) to study its influence on the stability. It canbe observed that for a given Nsub, with increase in xf the threshold power for Type-II

instability decreases and that for Type I instability increases. Moreover, the stability of thereactor decreases with increase in fuel thermal time constant. The DR and frequency ofoscillation for both Type II and Type I instabilities have been estimated at different fuelthermal time constants and plotted in Figs. 38 and 39 respectively. It can be observed fromthese figures that with an increase in fuel thermal time constant the DR increasescontinuously for Type I oscillation and the increase in DR is not significant at higher zf

for Type II oscillations. With increase in r y , the frequency of oscillation decreases

continuously for Type II oscillations due to increase in delay time, but it does not changesignificantly for Type I oscillations whose frequency is very low. Previous studies byMarch-Leuba and Rey [42] have shown that changes in fuel thermal time constant haveboth stabilising and destabilising effects. They found out that the stabilising effect is due tothe inherent filtering of the oscillations having frequency greater than 0.1 Hz and thedestabilising effect is due to the phase delay to the feed back. As observed in this case, forsuch low frequency of oscillation, the phase delay is more significant to destabilise the

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Type-II and Type I oscillation for an increase in fuel thermal time constant than thefiltering effect. Similar behaviour is also observed for the effect of fuel thermal timeconstant on the out-of-phase mode oscillations as shown in Fig. 40.

7.2 Out-of-phase instability behaviour considering the coupled multipointkinetics model and modal point kinetics model

The above analyses consider the point kinetics model for the neutron dynamics whichmay be good enough for revealing the instability behaviour of in-phase mode of oscillationwherein the power oscillation among the subcores takes place without any phase lagbetween them. However, during an out-of-phase mode oscillation neutron interactionsamongst channels due to variation of their void fraction may affect the stability. This effecthas been predicted by the coupled multipoint kinetics model and the results are shown inFig. 41. The coupling coefficient which determines the degree of coupling between subcoreto subcore has been estimated considering the reactivity requirement for xenon overrideand regulation using the computer code SERIES. This is found to be about 3.5 mkconsidering the reactor core to be divided into two equal subcores. However, this has beenvaried to study its effect on the out-of-phase instability in the present calculation. It can beobserved from the figure that with increase in coupling coefficient the threshold power forType II instability decreases and Type I instability increases at higher subcooling. At zerocoupling coefficient, the results of point kinetics model match with that of the coupledmultipoint kinetics model. Moreover, the stability of the reactor decreases with increase incoupling coefficient. To analyse this effect, the decay ratio has been predicted for differentcoupling coefficients at channel power of 2500 kW and inlet subcooling of 15 K, and theresults are shown in Fig. 42. It can be observed that with an increase in couplingcoefficient, the DR increases initially and then remains almost constant. Hence, the effectof coupling coefficient on the out-of-phase instability is significant at high subcooling andup to a coupling coefficient of 0.04. From this it is clear that for studying the out-of-phaseinstability, the point kinetics model is not adequate.

The out-of-phase instability behaviour for the above case has also been predictedusing the modal point kinetics model. The subcriticality (pc) for the first azimuthal modewas estimated from Eq. (98) and found to be about 9 mk. From the theory of Nishina andTokashiki [68], the coupling coefficient to be used in the coupled multipoint kinetics modelcorresponding to this subcriticality is 4.5 mk. In fact, this value is not much different fromthe coupling coefficient used previously (i.e. 3.5 mk). However, it is of interest to verify theabove theory by comparing the coupled neutronic thermohydraulic stability maps predictedby the coupled multipoint kinetics model and the modal point kinetics model for the out-of-phase instability of the reactor in order to establish the relationship between the twomodels. Fig. 43 shows a comparison of the predictions between the two models. It can beobserved that both the models give exactly the same threshold power for instability for theabove condition. Hence, the coupling coefficient is analogous to the subcriticality of thereactor during an out-of-phase mode oscillation. With increase in subcriticality, the stabilityof out-of-phase mode oscillation is found to decrease in the same way as that with thecoupling coefficient. Moreover, the coupled multipoint kinetics model derived here can beused to study the but-of-phase instability in the reactor in stead of solving complex multi

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dimensional neutron diffusion equations. It has also more advantage over the modal pointkinetics equations, because when analysing the out-of-phase instability behaviour ofmultiple channels or subcores having different power and resistances, the coupledmultipoint kinetics model can handle the problem by specifying different values ofcoupling coefficient for different channels or subcores, which is not possible by the modalpoint kinetics model.

7.3 Influence of delayed neutrons on the stability

It is of interest to investigate the contributions of the delayed neutrons on the coupledneutronic-thermohydraulic stability. Fig. 44 shows a comparison of the stability mapbetween the cases which consider the delayed neutrons and without the delayed neutronsfor in-phase mode of oscillations considering two typical hottest channels of the reactor. Itcan be observed from this figure that the delayed neutrons destabilise the Type II instabilityand stabilise the Type I instability significantly. To explain the phenomena, the decay ratiohas been predicted at different channel power and the behaviour is shown in Fig. 45. It canbe observed from this figure that if delayed neutrons are not considered, the decay ratioincreases with decrease in channel power for channel power less than about 1500 kWwhich corresponds to the region of Type I instabilities in the reactor. Similarly, the decayratio increases with increase in channel power if channel power is more than about 1500kW which corresponds to the region of Type II instability. If delayed neutrons are notconsidered in the analysis, two different modes (i.e. fundamental and higher mode) ofoscillations were observed in Type I oscillations. The fundamental mode of oscillation hasa characteristic frequency closer to the thermohydraulic oscillation frequency of thesystem, i.e. the time period of oscillation is related to the circulation time of the two-phasemixture. The behaviour of the fundamental mode of oscillation in Type I instability issimilar to that of the case which considers the delayed neutrons in the analysis. The decayratio of higher mode of oscillation observed in Type I instability without consideration ofdelayed neutrons decreases with increase in channel power. The effect of delayed neutronsis to increase the DR for Type II instabilities. However, for Type I instabilities the delayedneutrons cause a decrease of DR, because the DR of higher mode of oscillations is veryhigh which determines the stability if delayed neutrons are not considered in the analysis.The corresponding frequency of oscillation is shown in Fig. 46. It can be observed from thefigure that there is almost no difference between the frequency for the cases with delayedneutrons and without delayed neutrons for Type II instabilities. In Type I instabilities, thefrequency of higher mode of oscillation is significantly higher than the fundamental modeof oscillation if delayed neutrons are not considered. However, the frequency of thefundamental mode of oscillations without delayed neutrons, is almost closer to the casewith delayed neutrons considered in the analysis.

7.4 Effect of Inlet Orificing

Figures 47 and 48 show the effect of inlet orificing of the channels on the stability ofin-phase and out-of-phase modes respectively. In the analysis it is considered that both thesubcores contain same number of channels having the same RPF and inlet orificingcoefficient. The results indicate that with increase in orifice coefficient the threshold power

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for Type-II instability increases considerably and that for Type-I instability does not changesignificantly at low subcooling conditions. Also, the stability of both in-phase and out-of-phase mode oscillations increases with increase in orifice coefficient. Similar behaviourwas also observed earlier for the effect of inlet orificing on the thermohydraulic stability ofparallel boiling channels of the AHWR. Hence* the neutronic coupled instabilitycharacteristics follow the behaviour of the thermohydraulic instability of the system.

7.5 Effect of Radial Power Distribution

Figure 49 shows the effect of interaction between two subcores having different RPFand inlet orificing coefficients on the threshold of out-of-phase mode oscillation. In theanalysis, one of the subcores is always considered common and it contains channels ofchannel Type-3 (i.e. RPF = 1.231 and Kin=0.0) and the other subcore is considered to bevaried with same number of channels having different RPF and inlet orificing coefficient insuch a way that its outlet quality almost remains the same as that for the companionsubcore. It can be observed that by reducing the power in one subcore greatly stabilises theout-of-phase mode of oscillation occurring in the other subcore having higher powerchannels thereby enhancing the stability of the system.

7.6 Decay Ratio Maps for the AHWR

The contour lines of constant decay ratio for both in-phase and out-of-phase modeoscillations at different powers and inlet subcoolings are shown in Figs. 50 and 51respectively. Also, the constant feed water temperature lines alongwith the constantchannel exit quality lines are shown in the same figures. These maps are useful for designof the reactor because it gives an indication of the stability margin of the reactor in terms ofthe decay ratio under various operating conditions. This is because the phase changenumber and subcooling numbers can not fully represent the stability behaviour of a naturalcirculation BWR like the AHWR. The flow in this reactor is dependent on the power andthe core inlet subcooling which depends on the feed water temperature. Essentially, theonly independent parameters on which the natural circulation characteristics are dependentare the channel power and the feed water temperature. More research needs to be carriedout to model and reveal the natural circulation stability behaviour on suitable non-dimensional plane.

From these maps it is clear that it is possible to have decay ratio less than 0.4 forsubcooling less than 10 K, which implies that the reactor can have sufficient stabilitymargin for operating conditions at the above subcooling. Also, it is observed that the Type Iinstability occurs in the reactor at channel exit quality of less than 10 %. With increase inchannel exit quality at a particular subcooling, the DR increases. At a particular power, withincrease of subcooling at core inlet increases the DR. A decrease of feed water temperatureat a constant power, increases the subcooling, which has destabilising effect since the DRincreases. Comparing figures 50 and 51, it is evident that the out-of-phase instability is ofmore concern in the AHWR since the DR is higher at any power and inlet subcooling

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8.0 Conclusions

Analysis was carried out to study the thermohydraulic and the coupled neutronic-thermohydraulic instability behaviour for a particular PHT geometry of the AHWR which isa natural circulation pressure tube type BWR being designed in India. The analytical modelsolves by linearising the conservation equations of mass, momentum and energy andequation of state for the coolant together with a point kinetics model for the neutrondynamics and a lumped parameter model for the fuel dynamics. In addition, the modelconsiders a coupled multipoint kinetics model in place of simple point kinetics equation tostudy the effect of interactions between different parts of the core due to neutron diffusionduring an out-of-phase oscillation. The following insights were obtained from this study.

1) The Ledinegg type instability may occur in AHWR depending on the operatingconditions. An increase in system pressure suppresses this type of instability and itdisappears when system pressure is higher than 0.7 MPa. This type of instability can beavoided as long as the inlet subcooling is less than 10 K after the initiation of boiling inthe core during the start-up of the AHWR.

2) Reduction of water level in the downcomer during a small break LOCA phase, lowersthe threshold power for both upper arid lower boundaries of Ledinegg type instability.

3) Decreasing the height of riser (outlet feeders in the AHWR) lowers the threshold powerfor both upper and lower boundaries of Ledinegg type instability.

4) The effect of axial power distribution on Ledinegg type instability is negligibly small.

5) Both Type-I and Type-II density wave instabilities may appear in the AHWR dependingon the operating condition.

6) The density wave instability stabilises with an increase in pressure and a decrease in inletsubcooling.

7) In a parallel channel system, throttling at the inlet of the channel like orificing does notalways enhance the stability of density wave instability because the flow rate in thechannel decreases with increase in throttling coefficient when operated under naturalcirculation condition. As a result, the channel exit quality increases for the same powerthereby tending to destabilise.

8) Existence of power variation among the channels stabilises density wave instability ifchannel exit quality is same.

9) The density wave instability increases with a reduction of water level in downcomer.

10) The effect of riser height on the threshold of density wave instability is negligibly smallfor the range of riser height studied.

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11) The frequency of oscillation of Type-II thermohydraulic instability is larger than thatfor Type-I instability due to larger natural circulation flow rate in the former case whichtakes less time to pass through the two-phase region.

12) With an increase in negative void reactivity coefficient (Ca), the instabilities of Type Iand Type II decrease in both in-phase and out-of-phase mode oscillations.

13) With an increase in negative void reactivity coefficient, the frequency of Type IIinstability increases initially and then remains almost constant, but that for Type Iinstability remains almost constant.

14) An increase in fuel thermal time constant increases the instabilities of Type I and TypeII for both in-phase and out-of-phase mode oscillations.

15) With an increase in fuel thermal time constant, the frequency of oscillations for TypeII instability decreases continuously and that for Type I instability does not changesignificantly.

16) The coupled multipoint kinetics model predicts that with an increase in couplingcoefficient the stability of out-of-phase mode oscillations decreases at high subcoolingand for larger value of coupling coefficient (> 0.04), the threshold power for stabilitydoes not change significantly. Also, for zero coupling coefficient, there is no differencein the stability boundary between the point kinetics model and coupled multipointkinetics model.

17) The effect of subcriticality in the modal point kinetics model on the out-of-phaseoscillations is similar to that of the coupling coefficient in the coupled multipointkinetics equation. Both the models predict the same threshold power for stability forout-of-phase oscillations if the coupling coefficient is half of the subcricality.

18) Delayed neutrons increase the Type II instabilities and decrease the Type I instabilities.When delayed neutrons are not considered in the analysis, higher mode of oscillationsare observed in the Type I instability.

19) Effect of inlet orificing of the channels is to stabilise both in-phase and out-of-phasemode of oscillations.

20) Radial power distribution among the subcores stabilises the out-of-phase mode ofoscillations.

21) The effects of increase in channel inlet subcooling, channel exit quality and decrease offeed water temperature are to increase the decay ratio for both in-phase and out-of-phasemode oscillations in the reactor. The reactor has better stability margin at lowersubcooling.

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Nomenclature

af heat transfer area (m2)A cross sectional area (m2)Ca void reactivity coefficient (Ak/k/Ay)CD Doppler coefficient (Ak/k/AT)Cm(t) delayed neutron precursor

concentration of group mCp specific heat (J/kg-K)D hydraulic diameter (m)f Darcy friction factorg acceleration due to gravity (m/s2)h enthalpy (J/kg)hfg latent heat of vapourisation (J/kg)Hf heat transfer coefficient (W/m2K)k(t) effective multiplication factorK loss coefficient1 prompt neutron life time (s)L length of section (m)nif mass of fuel rods (kg)n(t) neutron densityNpCh phase change number (vj//whfg)NSUb subcooling number (Ahsub/hfg)p pressure (N/m2)qh heat applied/unit volume of

coolant (W/m3)s stability parametert time (s)T temperature (K)v specific volume (m /kg)Vfg vg-Vf(m3/kg)

w mass flow rate (kg/s)2 axial distance (m)

Greek Symbols

a coupling coefficientP delayed neutron fractiona volumetric thermal expansion coefficient (K"1)y void fractionA. decay constant of delayed

neutron of group mvj/ power (W)T fluid residence time (s)if fuel time constant (s)A differencep density (kg/m3)Subscriptsav averagec corech channeld downcomerf liquidg vapour

H headerin inlet of sectionk loss due to restriction

sat saturationSD steam drumsp single phasess steady statess,av average steady statesub subcoolingt totaltp two-phase

References

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39

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42

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43

TABLE 1: Details of inlet orificing in different channels of AHWR

channel type

1

2

3

4

5

6

7

8

9

10

11

12

Radial PowerFactor

0.761

1.011

1.231

1.124

1.011

1.124

1.150

1.150

1.103

1.016

0.909

0.935

Inlet LossCoefficient

250

0

0

0

30

30

30

30

80

80

130

130

No. of Channelsin a Channel type

60

40

4

28

4

12

40

28

44

52

52

44

44

Appendix 1 : Models employed to estimate two-phase friction factor multiplier

(a) Baroczy [66]

(b) Homogeneous model (Owens [73])

(Al)

where <pfo is the two phase friction factor multiplier, pf is the density of saturated liquid,

plp is the density of two-phase mixture.

(c ) Homogeneous model (Cicchitti [74])

0.25

where x is the quality, pf and p.K are the dynamic viscosity of the saturated liquid and

vapour respectively.

(d) Lockhart-Martinelli [69]

£ 1 ( A 3 )

where Xlt =(G, IG &f\v f I v ̂ \Mf / ^ ) ° \ (A4)

C=20,where Gf and GK are the mass flux of saturated liquid and vapour respectively, vf and vK

are the specific volume of the saturated liquid and vapour respectively,

(d) Martinelli-Nelson [70]

where Xtl = (Gf IGg)M(v, Iv^f5{p., Ip.g)°', (A6)

C = 1.364 +18.23log(/?c. / p) for p > 6.9 A/Pa, (A7)C = 7.0 + 5.986 log(pc/p) for /?< 6.9 AZ/'a, (A8)

where pc is the critical pressure and p is the system pressure.

(e)Chisholm-Laird[71]

A2 = 1 + + —r , (A9)

45

where Xu ={Gf IGKtil\vf IvRt\Mf I^Ktn\ (A 10)

(f) Sekoguchi et. al. [72]

£ = 0.38 Re°;' [l + (Gx / G, )(vK / V / ) ]°" , (All)Rsh=GfD/fif, (A 12)

where D is the diameter of the pipe.

46

Steam

Steam drum Steam drum

Outletfeeder

Upper channelV J extension

Fuelchannel

Feed water

CalandriaHeader

Inlet feeder

Fig. 1 Schematic of AHWR Primary Heat Transport System

Ap=constant3

w.,

channel exit

two-phase region [2]

Z

boiling boundary

single-phase region f I

Fig.2 Typical boiling channel

• channel inlet

externalreactivity f *\_

6Dopplercoefficient

voidreactivity

G,

C.void fraction

Fig. 3 Typical block diagram showing the feed back mechanismamong the neulronics. fuel dynamics and thermal hydraulics

steam

core

PSD

feed water

Fig. 4 Schematic of the simplified loop of AHWR considered in analysis

AB

C

DEFG

HJKLMN0PQRST

UVwXY

1

881010

2

7788810

3

66

67

7108

4

9596

767

10rd

5

9544

56788

6

33444

5578

7

23999

94678

10

8

9223334567

10

9

92

223315678

10

1C

922

23

344678

10

11

99292

994678

10

CHANNELTYPE

1

2

3

4

5

67

89

1011

12

CHANNELGROUP

9

2

1

3

2

34

56

7

810

NO. OFCHANNELSIN A TYPE

60

40

4

28

4

124028

44

52

52

44

RPF

0.761

1.011

1.231

1.124

1.011

1.124

1.151.15

1.103

1.016

0.909

0.935

CHANNEL GROUP

EACH CHANNEL IN A GROUP HASINDENTICAL POWER DISTRIBUTION

CHANNEL TYPE

EACH CHANNEL IN A PARTICULARTYPE HAS IDENTICAL POWERDISTRIBUTION AS WELL AS ORIFICING

NOTE - NUMBERS INDICATE CHANNELGROUP NUMBER.

Fig. 5 Channel Power Distributionin the AHWR Core

2.0

1.6

x

COOsz<D

TO

a5TO

(0

76oo

1.2

0.8

-Type-1-Type-3Type-5

-Type-7Type-9

-Type-11

Type-2Type-4Type-6

Type-8Type-10^Type-12

0.4

0.00.76 1.52 2.27

Axial distance (m)

Fig. 6 Axial power distribution in different channel types

3.03

Ic(0x:O

3.0

2.5

2.0

1.5

1.0

• iChannel flow rate

• •V

• \

-

m

-

-

1

• | •= 0.0333 kg/sec

\

Stable

• i

i • i

• experimentprediction

-

unstable

-

I • I •

75 80 85 90

Inlet temperature (°C)

95 100

Fig. 7 Stability map comparison between predictions and experiment

5000

4500

4000

3500

3000<D

|

oi 2500cc

2000

1500

1000

500

Pressure - 7 MPa

Stable

Unstable

Prediction -1.54 kg/secPrediction - 2.84 kg/secPrediction - 3.2 kg/sec

T Experiment -1.54 kg/sec• Experiment - 2.84 kg/sec• Experiment - 3.2 kg/sec

10 15 35 40 4520 25 30

Subcooling (K)

Fig. 8 Comparison of threshold of instability between the prediction and the test data of ATRsimulation facility under forced circulation

4.0

3.5 -

3.0 -

2ifi2

1IDc

2.5 -

2.0 -

£ 1.5 -CO

O

1.0

0.5 -

0.0

I ' 1

Pressure - 7 MPa

Stable

1 . 1

1

DO

^ ^ ^

1 ' 1 '

D o "

O-C3 > ^

Unstable

Experiment - short riserExperiment - long riser

- Prediction - long riserPrediction - short riser

I . I .

0 2 3

Channel power (MW)

Fig. 9 Comparison of threshold of instability between prediction and test data of ATRsimulation facility under forced circulation

1.5

1.0

1I

5="55

O

0.5

0.0

Pressure = 6 MPa,Subcooling = 10 K

(a)

1.5

I1

Ijj 0.5O

0.0

/ ..'••

/ 8o

D Level - 10m (test)o Level - 5 m (test)— Level -10 m (with slip model)

Level - 1 0 m (without slip model)Level - 5 m (with slip model)Level - 5 m (without slip model)

100 200 300

Channel power (kW)

400

• Level - 7 m (test) o Level - 3 m (test)Level - 7 m (with slip model)Level - 7 m (without slip model)Level - 3 m (with slip model)

— Level - 3 m (without slip model)

(b)

100 200 300

Channel power (kW)

400

500

500

Fig. 10 Comparison of predicted natural circulation flow ratewith the test data of ATR simulation facility for various levelsin the downcomer

2.0

• 4 - *

2oIDccroxiO

1.0 h

0.5 h

0.0

1 ' 1

Pressure = 6 MPaChannel Power = 145 kW

-

o

A

• l . l

DOA

"*-—.

1 I • l ' l

- Level - 1 0 m (without slip model)

Level - 1 0 m (with slip model)

- Level - 5 m (without slip model)

- Level - 5 m (with slip model)

- Level - 1 m (without slip model)

- Level - 1 m (with slip model)

Level -10 m (test)

Level - 5 m (test)

Level - 1 m (test)

o • -6- :;

A " * " " £ -

, I , I . 1

0 10 20 4030

Subcooling (K)

Fig. 11 Prediction of steady state natural circulation flow rateat different downcomer level

50

1.6

1.2

o

0.8

0.4

Prediction - 7 MPaPrediction - 5 MPaPrediction - 1 MPa

• Experiment - 7 MPaO Experiment - 5 MPaA Experiment - 1 MPa

Unstable

0.0Unstable

I

0.00 0.03 0.06 0.09 0.12Nsub

Fig. 12 Comparison of threshold of instability between theprediction and the test data of ATR simulation facility

0.15

0.6

0.5

0.4TO

Q -

Q.

2 0.3T3

COV)(1)

)riving head-250 MW (stable)

MW (unstable)MW (neutrally stable)

(a)

0.2

0.1

0.0

1.0

0.8 -

S. 0.6sQ.2•a

I 0.4ina>

-if§

I

1000

"s \ I

V^Pressure = 0.1 MPaSubcooling = 30 K

2000 3000

Flow rate (kg/sec)

4000

0.2 -

0.01000 2000 3000

Flow rate (Kgis)

4000

1 1

-

4

, 1

Driving head400 MW (unstable)460 MW (neutrally stable)500 MW (stable)

i

i

\

\

\

i

\

\

i""«

1 i

(b)

VT":

XJ -

1

Fig. 13 Typical stable, unstable and neutrally stable behaviour

800

700

600

500

I 400Q_

300

200

100

0

Pressure = 0.1 Mpa

• BaroczyHomogeneousiHomogeneous2Lockhart-MartinelliMartinelli-NelsonChisholm-LairdSekoguchi

10

Stable

Stable

4020 30

Subcooling (K)

Fig. 14 Comparison of stability maps obtained by different friction factor models

50

1000

800

600

S. 400

200

0.1 MPa0.3 MPa0.5 MPa0.7 MPa

Stable

00 10 20 5030 40

Subcooling (K)

Fig. 15 Effect of Pressure on Ledinegg type instability

60

800

600 -

400 -

OQ.

200 -

Pressure = 0.1

38.9 m34 m

"~ . . . . . . OQ mC.XJ 111

—

-

• I •

1

MPa

Stable

i

• i i | . • i

/ ^y' . - ' * Unstable -

/ ' ' '

Stable

I . I .

0 10 4020 30Subcooling (K)

Fig.16 Effect of downcomer Level on Ledinegg type instability

50

900

750 -

600 -

450 -

IQ.

300 -

150 -

I

Pressure = 0.1

25.73 m20.0 m15.0m

-

i

i

MPa

Stable

I

l l i 1 i

y*

y^ **'"

S^ ' ' ' " • ' -

^y <'' . - • * "

/ > ' - : -

y^ ^' .-'' Unstable

^ • •

Stable

I . I .

0 10 4020 30

Subcooling (K)

Fig. 17 Effect of riser height on Ledinegg type instability

50

2 -

oo.

cc03x:O

1 -

0

1 ' 1

Pressure = 0.1 MPa

UniformActual flux profileChopped cosine

Stable

^ ^ ^

I . I

• i

Stable

I

I

Unstable

_ _ _

I

0 10 20 30

Subcooling (K)

Fig. 18 Effect of axial flux profile on Leddinegg instability

40 50

2.0

1.5 -

c 1-0-e-

0.5 -

0.0

1 ' 1 • 1

UniformMaximum power ratedchannel flux profile

Chopped cosine

y

yy

yy

yy

y

I . I . I

1

-

—

! . *

NN

N.N.

"V

1

0.00 0.76 2.28 3.041.52

Axial length (m)

Fig. 19 Axial flux distributions in the core considered in the stability analysis

0.7

0.6

rP = 7 MPa

out-of-phasein-phase

oQ.

0.5

0.4

0.3

0.2

0.1

0.0

Unstable

\\

Unstable

0.00 0.02 0.04 0.06 0.08 0.10

Nsub

Fig. 20 In-phase and out-of-phase thermohydraulicstability boundary in boiling channels of the AHWR

0.12 0.14

0.6

0.4 -

0.2 -

0.0

1

' Stable ***"•-..

i--"-"* __

• i

i

7MPa

5MPa

3MPa

^ -» •*"

1

1

X

s

^ "

\

1

Unstable

Unstable

i

—

-

1

0.01 0.02 0.03 0.04 0.05

Nsub

0.06

Fig. 21 Effect of pressure on the threshold of instability considering all channels

0)

oQ.

1,500.0

1,312.5

1,125.0

937.5

750.0

562.5

375.0

187.5

n n

" • •

—

—

Stable

!

1 1

Unstable

N.X

i

• i

3MPa7MPa5MPa

NN

Unstable

, i

—

-

—

—

0 5 10 15

Subcooling (K)

Fig. 22 Effect of pressure on threshold of instability considering all channels

20

0.6

0.4

t

0.2

0.0

Pressure = 7 MPa

Maximum power rated twin channelSystem considering all the channels

Unstable

Unstable

0.02 0.04 0.06

Nsub

0.08

Fig. 23 Comparison of threshod of instability between maximum power ratedtwin channel system and system considering all the channels of reactor

0.6

0.4 -

uQ.

z

0.2 -

0.0

1

Pressure

—

i

i

= 7MPa

Stable

i

I

NN

\

1

1

= 2= 5= 8

Unstable

\

Unstable

• i •

i

-

—

i

0.01 0.02 0.03 0.04 0.05Nsub

0.06

Fig. 24 Effect of number of parallel channels on the threshold of stability

0.6

0.4

oQ.

0.2

0.0

Pressure = 7 MPa ~—•——~ K< — 0, K 2 ~~ 0

K., = 0, K2 = 30

K1 = 0, K2 = 80

""•'•"" K.̂ — 0, K.2 — 130

--••• K., =0,K2 = 250

Unstable

Unstable

0.02 0.04

NSUb

0.06

Fig. 25 Effect of orificing of one channel on the stability behaviour of twinchannel system when both channels are heated with same power

1.2

0.8

uQ.

0.4

Pressure = 7 MPa

1^=80, K2=80

•K,=30, ^=30

K^O, K2=0

^=130,^=130

Unstable

Stable

0.0

Unstable

0.00 0.04 0.08 0.12

Nsub

Fig. 26 Effect of inlet orificing of both channels on stablity boundary

uQ.

0.8

0.7 -

0.6

0.5 -

0.4 -

0.3

0.2 -

0.1 -

0.0

1

Pressure

-

—

-

_

— ^

—

•

-

—

7

i

= 7MPa

*•-

\

\

Stable ^ x\

1

1

— — ̂

>

N•%

1

1

Kout " 2 5

out "•**

^out " 0 0

N

NV

Unstable

i

Unstable

NN

Ss

sy

1

NV

-

—

-

—

ss

s—

-

—

0.02 0.04 0.06 0.08

Nsub

Fig. 27 Effect of channel outlet resistance on threshold of instability

0.5

0.4

0.3

oQ.

0.2

0.1

0.0

I ' TPressure = 7 MPa

Level = 39 m

Unstable

Level = 32 m

Unstable

0.01 0.02 0.03 0.04 0.05

Nsub

Fig. 28 Effect of downcomer level on threshold of instability

0.06

0.5

0.4 -

0.3 -

I

0.2 -

0.1 *-

0.0

1 • 1 • I

Pressure = 7 MPa

Stable

. I . I . I

1 i

height = 15 mheight = 20 mheight = 26 m

N Unstable

Unstable

i

i

-

>

0.01 0.02 0.03 0.04 0.05

Nsub

Fig. 29 Effect of riser height on threshold of stability

0.06

uQ.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00.00

0.07337

0.0736

Stable

P = 7 MPa

0.066

0.0573

0.01680.0155

• out-of-phasein-phase

Unstable

kv0.056

\ 0.0501

0.0436

0.0371

.-- -* 0.02469

0,0173

0.0332

Unstable

0.02 0.04 0.06 0.08 0.10

Nsub

Fig. 30 In-phase and out-of-phase thermohydraulic

stability boundary in boiling channels of the AHWR

ua.

V.I

0.6

0.5

0.4

0.3

:

0.2

0.1

nn

' i

— • • • . , - * " * *

_

— ' C

a

c a—

T , •''

—

—

I

1

= -0.02

1Pressure = 7

1 %v*

>= - 0.005

= 0.0

I 1

8s

\% oi*\ ^

Stable

I

MPa

V\\

\ N

\*\

\\

\\

I\

\

*

\

• *

>

\

\

N \*\ ^

\

\

i

\\\

ss

•

1

1 1 1 .

—

_

Unstable-

• • • • : . . ' • ± * > ' ' —

Unstable

I . I .

0.00 0.02 0.04 0.06 0.08

Nsub

0.10 0.12 0.14

Fig. 31 Effect of void reactivity coefficient on stability of in-phase mode of oscillation

oQ.

0.5

0.4

0.3

0.2

0.1

0.0

1 1 ' I •

Pressure =

^ ^ > = 8s

•

\ ^

S N\

*

Stable

-

I . I .

i

7 MPa

\\

s\

% v\ N\

> \

•••

•

1 .

1 • 1 •

Ca = - 0.02

Ca = -0.005

C = 0.0a

* \ . Unstable

\ .

\

Unstable

I . I .

I

-

—

-

—

_

_

—

I

0.02 0.04 0.06 0.08 0.10

Nsub

0.12

Fig. 32 Effect of void reactivity coefficient on stability for out-of-phase mode of oscillation

2.0

1.6

1.2

a:Q

0.8

0.4

0.0

Pressure = 7 MPa

Channel Power = 600 kW

-0.8 -0.6 -0.4

C

-0.2 0.0

Fig. 33 Effect of void reactivity coefficient on the decay ratio for Type I instability

1.4

1.2 -

1.0 -

0.8 -

Q

0.6 -

0.4 -

0.2 =

0.0

I • 1

Pressure = 7 MPaChannel power = 4500 KWAT s u b=15K

l . I

1 i •

/—

I.

I.

I

i

-0.8 -0.6 -0.4

C

-0.2 0.0

Fig. 34 Effect of void reactivity coefficient on the decay ratio for Type II instability

NI

0.03

0.02

0.01

1 i

—

-

—

i

i i • i •

Pressure = 7 MPa

Channel Power = 600 (kW)

ATsub = 15K

tf = 8s

-

l . i .

-0.8 -0.6 -0.4

C

-0.2 0.0

Fig. 35 Effect of void reactivity coefficient on the frequency for Type I instability

0.12

0.10

N

0.08

0.06

0.04-0.8 -0.6

Pressure = 7 MPaChannel power = 4500 KWATsub = 15KT r = 8 S

I

-0.4

C

-0.2 0.0

Fig. 36 Effect of void reactivity coefficient on the frequency of oscillation for Type IIinstability

0.7

0.6 -

0.5 -

0.4 -

0.3 -

0.2 -

0.1 -

0.0

I

—

-

-

Tf =

I

^ .

2s

8s

16s

I

I

l

l 1

Pressure = 7

m •»

1

= - 0.005

*\

Stable

I

1

MPa

t

1 '

Unstable

Unstable

i

*

-

—

-

—

—

—

I

0.02 0.04 0.06 0.08 0.10

Fig. 37 Effect of fuel time constant on the stability for in-phase mode of oscillation

Q

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.00

Pressure = 7 MPa

Channel power = 4500 KW= 15K

C =-0.005a

205 10 15

xf(s)

Fig. 38(a) Effect of fuel time constant on the decay ratio of Type II instability

25

3.0

2.5

2.0

A rrCC 1-5Q

1.0

0.5

0.0

• I • I

Channel Power = 600 KW

Pressure = 7 MPa

Ca = - 0.005

ATsub=15K

^ ^ ^ ^

- •

-

. I . I

• i i i •

—

-

—

-

-

— — — " • " " " "

—

-

_

I . I .

0 5 10 15 20 25

xf(s)

Fig. 38(b) Effect of fuel time constant on the decay ratio Type I instabilityfor in-phase mode of oscillation

NI

LL

u.uou

0.075

0.070

0.065

n nan

. • • j i

Channel Power = 4500

Preesure = 7 MPa

ATsub = 1 5 K

Ca = - 0.005

-

l

I • I

kW

1 —

I . I

—

-

10xf (s)

15 20

Fig. 39 (a) Effect of fuel time constant on the frequency of oscillation forType II instability

25

0.05

0.04

0.03

N

cCD

0.02

I ' I

Channel Power = 600 kW

Pressure = 7 MPa

Ca = - 0.005

0.01

0.000 5 10 15 20

xf(s)

Fig. 39 (b) Effect of fuel time constant on the frequency of Type I oscillation

25

0.5

0.4

0.3

0.2

0.1

nn

1 i

^ * \

—

m

Pressure

—

—

I

I

= 7MPa

005

l

• I . I .

'

Stable

• i • i .

i • i •

x f =16s

xf = 8 s

—

Unstable

Unstable

I . I .

0.01 0.02 0.03 0.04 0.05

NSub

0.06 0.07 0.08

Fig. 40 Effect of fuel time constant on stability behaviour of out-of-phase mode of oscillation

pch

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

I

—

Pressure = 7

Ca = - 0.005

xf = 8 s

—

—

—

• *

• • I

MPa

Stable

i •

a 1 2 =

a 1 2 =

a 1 2 =

point

i

i • i

+0.0035

+0.007

+0.0

kinetics model

Unstable

Unstable

I I I !

i

—

—

—

—

-

-

—

I

0.01 0.02 0.03 0.04 0.05 0.06

Nsub

0.07

Fig. 41 Effect of coupling coefficient on stability behaviour of AHWR for out-of-phasemode of oscillation

0.08

1.15

1.10 -

1.05 -

Q

1.00

0.95 -

0.90

1 1

\

i

i • . i

Pressure = 7 MPaPower = 2500 kWATsub = 15K

Ca = - 0.005

I . I

I

-

-

—

0.00 0.02 0.080.04 0.06

a12

Fig. 42 Effect of coupling coefficient on decay ratio for out-of-phase mode oscillation

0.10

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

subcriticality = 0.007subcriticality = 0.014

—•—a1 2 = 0.0035

—O—a12 = 0.007

Pressure = 7 MPaCa = - 0.005

Unstable

Unstable

I I

0.01 0.02 0.03 0.04 0.05

NSub

0.06 0.07

Fig. 43 Comparison of stability maps between coupled multi point kinetics modeland modal point kinetics model for out-of-phase mode of oscillation

0.08

oQ.

z

0.9

0.8 -

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

1 • 1

with delayed neutronswithout delayed neutrons

Stable \

-

- ^

\ . i

I

Pressure = 7MPa -

C = - 0.0050t "

xf=8s

—

v Unstable"s. —

Unstable

• i i

0.00 0.03 0.06 0.09 0.12

Fig. 44 Effect of delayed neutrons on the stability of in-phase mode of oscillation

4.0

3.5 -

3.0 -

2.5 -

5 2.0

1.5 -

1.0 -

0.5 -

0.0

I • 1

— Fundamental mode with_ delayed neutrons

Fundamental mode withoutdelayed neutronsHigher mode withoutdelayed neutrons

-

-

"~ \ ^ i.—*—***r̂ ^ *

i . i

i i i

m

ATsub=10K

Ca=-0.005

xf=8 sec

^ - ;

i

0 1500 3000

Channel Power (kW)

Fig. 45 Effect of delayed neutrons on the Decay Ratio

4500 6000

1.0

0.8 -

0.6 -N

I<ucr 0.4 -

0.2 -

0.0

1 • • 1

Fundamental Mode withdelayed neutronsFundamental Mode withoutdelayed neutronsHigher Mode withoutdelayed neutrons

y'

1

ATsub=10K

Ca=-0.005

xf=8 sec.

— •

•

i •

—

—

—

-

-

1

0 1500 3000 4500

Channel Power (kW)

Fig. 46 Effect of delayed neutrons on frequency of oscillation

6000

•5a.

1.2

0.9

0.6

0.3

n n

—

—

Stable

m

i

**

\\

\«

4

1

1

\

•*

i # -

yf

*

\

\

\

1

Pressure = 7

Ca = - 0.005

xf = 8 s

s.\

\\

\\

\\

\\

\

\

1

1

MPa

\

\

1

—

V\

\\

\

-"

Unstable

I

K: =in

Kin =

Kin =

Kin =

\\

\

1

0

30

70

130

Unstable

>

•

I

—

—

• -

; ~

_ _

• • —

0.00 0.07 0.14 0.21 0.28Nsub

Fig. 47 Effect of inlet orificing on stability behaviour of AHWRfor in-phase mode of oscillation

0.35

oa.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

I

Pressure

- ^ ca = -o.

" ^ ' tf = 8s^ I

_ "V

N_ N

-

Stable \

-

• i t

i • • i

= 7MPa

005

s

\X • \ •

0 ^

1 . 1

1 ' 1 '

. . . . . . K jn = 0 -

K i n - 30

K: =70in .

Kin = 130

N. Unstable

\X

\

— - ^ ^ ^ "

Unstable

I . I .

0.00 0.04 0.08 0.12 0.16 0.20

Nsub

0:24

Fig. 48 Effect of inlet orificing on stability behaviour of AHWR for out-pf-phase modeof oscillation

1.2

0.9 -

« 0.6 -

0.3 -

0.0

1

Pressure = 7

- Ca =-0.005

x =8s

-

-

-

—

I ' l

MPa

*

Stable

l . i

i

RPF1 =

KJM = 0 (

nnr _— — — r \ r r « —

K ini = 0.(

RPF1 = :

Kim = 0-(

XX

XX

•>.. "** xX . X

"**x .^

- - • •

. 1

1

1.231

3; Kjn2 =

1.231;

3= K in2 ••

1.231

3: Kin2 =

XX

X

I

; RPF2 =

= 130.0

RPF2 =

= 80.0

; RPF2 =

= 30.0

1 1 '

= 0.909

1.103

= 1.124

Unstable

\ N

X

1

Unstable

i

0.00 0.03 0.06 0.09

Nsub

0.12 0.15 0.18

Fig. 49 Effect of radial power factor on stability behaviour of AHWR for out-of-phasemode of oscillation

2000

1750

0 255 10 15 20

Subcooling (K)

Fig. 50 Decay ratio map of AHWR for in-phase mode of oscillation

30

266.7

233.3

DR = 0.4DR = 0.6

0.035

2000

1750

1500

1250

* 1000a)oQ.

0)O

o

750

500

250

0-6 0 246 12 18

Subcooling (K)

Fig. 51 Decay ratio map of AHWR for out-of-phase mode of oscillation

30

267

233

200

167

133O

&TO

cr100

67

33

36

Published by: Dr. Vijai Kumar, Head Library & Information Services DivisionBhabha Atomic Research Centre, Mumbai - 400 085, India.