stability analysis of boiling water nuclear reactors using parallel computing paradigms

7/29/2019 Stability Analysis of Boiling Water Nuclear Reactors Using Parallel Computing Paradigms

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Stability Analysis of Boiling Water NuclearReactors Using Parallel Computing

Paradigms

A Report submitted forevaluation of work done during PBI, 7th semester

by

Manu Rakesh(2008065)

under the guidance of

Dr. G. Dutta

Indian Institute of Information Technology, Design and

Manufacturing, Jabalpur

14 November, 2011


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Contents

List of Figures vii

Acknowledgement viii

1 Introduction 1

1.1 Motivation and Objective of Project Work . . . . . . . . . . . . . . . . 11.2 Outline of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 TH Model: A Brief Overview 3

2.1 Non-dimensionalised version of TH model . . . . . . . . . . . . . . . . 6

3 Analysis of out-of-phase instabilities 8

3.1 Model description and boundary conditions for out-of-phase instabilityanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.1 Boundary conditions to model out-of-phase instability . . . . . . 83.1.2 Solution technique to model out-of-phase instability . . . . . . . 9

4 Numerical Treatments 12

4.1 Generic Methods for Dense Matrix Inversion . . . . . . . . . . . . . . . 124.1.1 Sherman-Morrison Formula . . . . . . . . . . . . . . . . . . . . 124.1.2 LU Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 134.1.3 Cholesky Decomposition . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Generic Methods for Sparse Matrix Inversion . . . . . . . . . . . . . . . 144.2.1 Frontal Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.2 Biconjugate Gradient Stabilised Method . . . . . . . . . . . . . 15

4.3 Sparse Matrix Storage Schemes . . . . . . . . . . . . . . . . . . . . . . 15

4.4 Justification of Parallelization . . . . . . . . . . . . . . . . . . . . . . . 164.5 Finalized Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6 Matrix Inversion based SS Solution . . . . . . . . . . . . . . . . . . . . 17

4.6.1 The Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6.2 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6.3 Reducing The Condition Number . . . . . . . . . . . . . . . . . 20

5 Nuclear Coupled TH Model Integration 21

5.1 Neutronic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Fuel Heat Conduction Model . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3 Coupling Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3.1 Steady State Coupling Algorithm . . . . . . . . . . . . . . . . . 25

i


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CONTENTS ii

5.3.2 Transient Coupling Algorithm . . . . . . . . . . . . . . . . . . . 255.3.3 Algorithm to satisfy steady state TH boundary conditions . . . 27

6 Results 29

6.1 In-phase Oscillations with neutronic feedback . . . . . . . . . . . . . . 296.1.1 Time Savings Achieved . . . . . . . . . . . . . . . . . . . . . . . 33

7 Further Scope of Work 35

Bibliography 36


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Nomenclature

English symbols

A Area (m2)a Acoustic speed (m/s)Ci Precursor concentration of i

th delayed group (atoms/cm3)

cpf Specific heat of fuel rod (kJ/(kg.K))D Diffusion coefficient (cm)e Total specific internal energy (J/kg)ef Total specific flow energy (J/kg)flo Single-phase liquid friction factor (dimensionless)G Number of energy groups (dimensionless)g Acceleration due to gravity (m/s2)H Vertical height (m)h Specific enthalpy (J/kg)hf Specific enthalpy of saturated liquid (kJ/kg)

hfg Specific enthalpy difference between saturated vapour and liquid (J/kg)hG Gap conductance (W/m

2 K)h Heat transfer coefficient of the coolant (W/m

2 K)J Ratio of time step sizes for thermal-hydraulic and shape function

solvers [ J Natural number ] (dimensionless)K Isentropic bulk modulus (N/m2)k Thermal conductivity (W/(m.K))ki Inlet orifice coefficient (dimensionless)ke Exit orifice coefficient (dimensionless)keff Multiplication factor (dimensionless)khr

Orifice coefficient between heater and riser (dimensionless)L Length of channel; L = LH + LR (m)m Number of delayed neutron families (dimensionless)m Mass flow rate (kg/s)N Amplitude function (dimensionless)

Npch Phase change number

QTmThfg

vfgvf

(dimensionless)

Nsub Subcooling number

hfhinhfg

vfgvf

(dimensionless)

PH Heated perimeter (m)Pw Wetted perimeter (m)p Pressure (Pa)

p Pressure drop (N/m2

)Q Total power (W)

iii


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CONTENTS iv

q

Heat flux (W/m2)q

w Heat flux at wall (W/m2)

q

Volumetric heat generation rate (W/m3)Re Reynolds number (dimensionless)

r Radius (m)r At any pont r in reactor core

S Heater and riser area ratio; S =AHAR

(dimensionless)

s Specific entropy (J/(kg.K))T Temperature (K)t Time (s)u Fluid velocity in axial direction (m/s)us Characteristic fluid velocity (m/s)V Volume of reactor (cm

3)v Neutron velocity (cm/s)

vf Specific volume of saturated liquid (m3

/kg)vfg Specific volume difference between saturated vapour and liquid (m3/kg)

W Unknown variable as mass flow rate (kg/s)x Vapour quality (dimensionless)z Distance in axial direction (m)

Greek letters

Azimuthal angle (degree or radian)

Neutron flux (cm2/s) Void fraction (dimensionless) ( r , t ) Void fraction at 3-D space r in BWR and time t (dimensionless)r ( z , t ) Void fraction ofrth channel at axial location z and time t (dimensionless)t Total macroscopic cross section (cm

1)R Macroscopic removal cross section (cm

1)sgg Macroscopic scattering cross section from g

to g (cm1)f Macroscopic fission cross section (cm

1)a Macroscopic absorption cross section (cm

1) Total delayed fraction (dimensionless)

s Fraction of delayed fission neutrons appear in the sth

group (dimensionless) Kinematic viscosity (m2/s)(g

) Average number of neutrons released per fission in group g

(dimensionless)Pg Fraction of prompt fission neutrons emitted into group g (dimensionless)Dsg Fraction of delayed neutrons emitted into group g from decay of precursors

in family s (dimensionless)s Decay constant of delayed neutron precursor family s (s

1) Density (kg/m3)

Reactivitykeffkeff

(dimensionless)

Shear stress (N/m2)

Two-phase multiplier (dimensionless) Difference operator (Dimensionless)


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CONTENTS v

Time step used for amplitude function calculation (s) t Time step used for shape function calculation (s) t Time step used for thermal-hydraulic calculation (s) Matrix

Vector

Superscript

r rth radial channelT Transpose Per unit length (m1)n nth time step

Subscript

c Heat conduction related quantityH Heaterg Energy group gk kth axial nodein Inletic Innerer cladnc Number of radial nodes for clad (dimensionless)nf Number of radial nodes for fuel pellet (dimensionless)

oc Outer cladR Risers Delayed neutron precursor family sT Totalw Wall Coolant2 Two-Phase1 Single phase

Acronyms

BC Boundary conditionBW R Boiling water reactorCF D Computational fluid dynamicsCR Control rodCRDA Control rod drop accidentDW O Density wave oscillationDN S Direct Numeric SimulationsE V E T Equal-velocity and equal-temperatureE V U T Equal-velocity and unequal-temperatureF A Fuel assembly


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CONTENTS vi

F C Forced circulationF CN Fuel compositionF HC Fuel heat conductionHZP Hot zero power (condition)

IC Initial conditionIQS Improved quasistaticLES Large Eddy SimulationsLOCA Loss of coolant accidentLV Laguna Verde (BWR)LW R Light water reactorME MacroelementMECA Method of characteristics analysisMSB Marginal stability boundaryN C Natural circulationN P P Nuclear power plantRP Relative powerT H Thermal-hydraulic


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List of Figures

2.1 Schematic diagram of a FC system . . . . . . . . . . . . . . . . . . . . 42.2 Schematic diagram of parallel channels lying between two plena . . . . 4

4.1 General Structure of Matrix to be solved for Single Channel SteadyState Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.1 Schematic Diagram of typical BWR loop. . . . . . . . . . . . . . . . . . 225.2 Schematic diagram of a fuel rod containing fuel pellet, gap and clad. . . 245.3 Three level time step structure. . . . . . . . . . . . . . . . . . . . . . . 255.4 Solution technique for nuclear coupled TH system during steady state. 265.5 Solution technique for nuclear coupled TH system during transient anal-

ysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.1 Variation of Power with Node Location for various channels duringSteady State Calculations. nsub = 0.603547; npch = 1.839268. . . . . . . 29

6.2 Variation of inlet mass flow rate with time for various channels during

core-wide mode of oscillations. nsub = 0.603547; npch = 1.839268. . . . 306.3 Time evolution of inlet and exit mass flow rates in phase plane diagram

at marginal stability boundary in presence of neutronic feedback effectsduring core-wide mode of oscillations. nsub = 0.603547; npch = 1.839268. 30

6.4 Variation of Relative Power with time during core-wide mode of oscil-lations. nsub = 0.603547; npch = 1.839268. . . . . . . . . . . . . . . . . 31

6.5 Temporal variation of relative power and core average void fractionduring core-wide mode of oscillations. nsub = 0.603547; npch = 1.839268. 31

6.6 Temporal variation of relative power and core average Fuel Temperatureduring core-wide mode of oscillations. nsub = 0.603547; npch = 1.839268. 32

6.7 Temporal variation of relative power and core average inlet and exitmass flow rates per channel during core-wide mode of oscillations. nsub= 0.603547; npch = 1.839268. . . . . . . . . . . . . . . . . . . . . . . . . 32

6.8 Instataneous 3-D power distribution at a particular axial fuel planeduring core-wide mode of oscillations when the BWR is operating atrated mass flow rate. nsub = 0.603547; npch = 1.839268. Power isdepicted on a scale of 0 to 12 105 Watts. . . . . . . . . . . . . . . . . 33

6.9 Instataneous 3-D power distribution at various axial fuel planes duringcore-wide mode of oscillations when the BWR is operating at rated massflow rate. nsub = 0.603547; npch = 1.839268; time t = 14.10 secs. Power

is depicted on a scale of 0 to 12 105

Watts. . . . . . . . . . . . . . . . 34

vii


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Acknowledgements

I would like to take this opportunity to express my sincere gratitude towards my guide,Dr. Goutam Dutta for his invaluable guidance, interest, encouragement throughoutthe course of the project work. Without his support and patience in guiding me, therealization of this project would not have been possible. The freedom he lended to me

while working on the project enabled me to bring the most out all circumstances.I would also like to thank my fellow batchmate Sachin Kumar. Discussions with

him were always fruitful.It would be wrong not to thank the staff members of the Computer Center. The

help provided by them with the computational resources at hand time and time again,helped making the completion of the project much easier.

Last but not the least, I would like to thank my mother for being the source ofinspiration to bring the best out of my potential.

Date: Manu Rakesh


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Chapter 1

Introduction

1.1 Motivation and Objective of Project Work

Two-phase flow boiling systems are inherently susceptible to TH instabilities, whichmay cause flow oscillations of constant or diverging amplitude. Sustained flow os-cillations may lead to mechanical forced vibration on components or system controlproblems in water cooled reactors where coolant water is used as moderator. Theseoscillations could induce boiling crisis, disturb control systems, or cause mechanicaldamage. It can lead to tube failure due to wall temperature increase and thermalfatigue of the tubes due to continuous cycling of wall temperature. This can cause thebreak down of fuel elements in nuclear reactors, leading to more serious accidents suchas release of radioactive materials.

Among the different types of flow instabilities, the density wave oscillations (DWOs)

are of primary concern in water-cooled nuclear reactors where limited boiling is allowedin the core. Therefore, the investigation is required to be focused on finding out thelimiting values of parameters such as, average channel pressure, the power density,maximum void fraction, etc., within which one must operate the reactor to precludesuch instabilities.

The designers job is to predict the threshold of flow instability so that one candesign around it and compensate for that. It is also required to find out the possibleremedies in avoiding such instabilities in the operational regime.

Now-a-days, designers rely heavily on computational models to aid them in thedesign process for Nuclear Reactors. Such computational models must be fairly accu-

rate and at the same time robust enough to parametrically evaluate the stability andresponse characteristics for a number of cases. But systems that follow the physics ofthe problem at micro-level accuracy, like LES and DNS face the constraints of takinga very long time to generate results for a set of input parameters such as a pair ofSub-cooling Number and Phase change number. To accurately distinguish the stablezone from the unstable zone in a phase-plane diagram using MSBs, literally hundredsof simlulations must be carried out.

Thus arose the need for the present project, which strives to introduce a layer ofparallelization in the model so as to reduce the time taken during computation. Witha model capable of utilizing the power of the new generation of computers equipped

with multiple processors, it can be expected that ultimately, the design process ofNuclear Reactors will benefit significantly.

1


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CHAPTER 1. INTRODUCTION 2

1.2 Outline of the report

The report has been organized as follows:Chapter 2 is devoted to the introduction of the problem and TH model devel-

opment, with minor improvements done in the model proposed by Dutta and Doshi[1].Chapter 3 details the method used by Dutta, G. for simulating out-of-phase Oscil-

lations using the TH model developed.Chapter 4 deals with the mathematical model development for matrix inversion.

Firstly, it contains literature review of relevant techiniques and research work in thefield of matrix inversion and parallelization of the same. It also includes a study ofvarious storage schemes adopted for matrices.

Further, it includes a case study of an attempt made to perform the steady statecalculations using Matrix Inverson Method.

In Chapter 5, the model is integrated with a well validated 3D Nuclear Dynamics

model. This is done to simulate the heat generation in the reactor core closely. Here,the model parameters are made to confirm with the Laguna Verde Nuclear PowerPlant, Veracruze, Mexico.

Chapter 6, recounts the results of the simulations that were carried out over thecourse of the project work. It also attempts to explain the observations.


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Chapter 2

TH Model: A Brief Overview

Carrying forward the thermohydraulic model designed by Dutta, G. and Doshi, J. B.[1], the project aims to parallelise the computation involved in solving the model for

parallel channels assuming two-phase homogeneous medium.Presented below is a brief recapitulation of the said model.

t[U] + A(U)

z[U] = [D(U)] (2.1)

where U is a vector composed of three variables, A is a matrix which depends on thevector U and D is a vector containing allowances for mass, momentum and energytransfer across the system boundaries and between phases. For EVET model, theA matrix and D vector corresponding to the unknown variables in the vector U areshown below.

U =

Whp

(2.2)

A =

2WA

W2

2A

h

p

A W2

2A

p

h

a2

A

Wa2

A

p

h

Wa2

2A

p

h

a2

AWa2

A

h

p

Wa2

2A

h

p

D =

A

F + g dH

dz

+W2

A2

dAdZ

a2 Q + WFA p h

a2Q + WFA

h

p

where a is the acoustic speed.

a =

p

h

+1

h

p

1/2

(2.3)

This model requires the use of the following relationships for heat and momentumtransfer across the boundaries and phases.

Q =

q

wPHA

=

hPHA

(Tw T) (2.4)

3


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CHAPTER 2. TH MODEL: A BRIEF OVERVIEW 4

Reactor

core

(more details

in next figure)

Steam

Water

seperator

Steam

exit

Forced circulation

flow direction

Natural circulation

flow direction

Exit

Cold water Inlet

Makeup

water

Direction

controller

Pump

Heat exchanger

Flow adjuster

Unidirectionalvalve

Reservoir

Steam

water exitSaturated

Pressure

relief valveFilter

Figure 2.1: Schematic diagram of a FC system

Heati

ngsection

Heati

ngsection

Inlet throttle

valve

Largebypassline

Inlet plenum

Outlet plenum

(a) (b)

Exit throttle

valve

Flow in

Flow out

Figure 2.2: Schematic diagram of parallel channels lying between two plena


14/46


F =

wPw

A

=

4floDh

+K

l

W|W|

22a2(2.5)

where flo is the friction factor, is the appropriate two-phase multiplier and K/l isthe distributed loss coefficient. The choice of appropriate empirical equations for h

, and K/l is dependent on the particular problem under consideration. Equation ofstate, = (p, h) is used and it is to be noted that all the thermodynamic propertiesof our model are subjected to changes as the pressure drops along the axial directionof the channels.

On analysis of the eigenvalues of the matrix A (u, u +a, ua, all of which are real)it is found that the system is infact a group of hyperbolic equations. The PDEs arethen transformed into ODEs to ultimately obtain the compatibility equations along thecorresponding characteristics using substantial derivatives (i.e. d/dt /t + V ).

The equations can be used in a Lagrangian frame of reference as well as in aEulerian frame of reference. A Eulerian frame of reference is chosen where a finite

difference solution procedure can used, which is considerably faster than situations inwhich the Lagrangian frame is used, such as the method of characteristics analysis(MECA) method, which is accepted as a benchmark solution.

The compatibility equations are represented below in a compact form below.

B

t[U] + B

z[U] = [C] (2.6)

where the columns of B1 are the eigenvectors of A, is a diagonal matrix of eigen-values of A and C = B D. For EVET model, B, and C are as follows

B =

1

W

hp

Aa

W

hp

0 1 1

1 W

p

hAa W

p

h

=

Wa

+ aWa

Wa a

u + a u

u a

C =

D1 W h

p D2 +

Aa W h

p

D3

D3

+ D2

D1 W

h

p

D2 Aa

+ W

h

p

D3

C1C2

C3

The equation represented above in (1.6) is discretized into the following left and rightdifference equations:

Bnk

Un+1k Unk

t+ n

kBn

k

Un+1k Un+1k1

zk1= Cnk (2.7)

Bnk

Un+1k Unkt

+ nk

Bnk

Un+1k+1 Un+1kzk1

= Cnk (2.8)


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The use of the above quoted equations depends upon the flow characteristics at thepoint of interest, i.e. at a specific mesh point. Spatial derivatives are always ap-proximated by backward differences when the characteristic is positive and forwarddifferences when the characteristic is negative. For the present case of subsonic flow

(u < a), the spatial derivatives for C+

(i.e., 11 = u + a) and C0

(i.e., 22 = u) char-acteristic equations, are approximated by backward difference equations, whereas theC (i.e., 33 = ua) characteristic equations, are approximated by forward differenceequations. A matrix is now defined such that its diagonal elements are equal to 1if slope of characteristics is positive and 0 if slope of corresponding characteristic isnegative. Thus, in the case of subsonic flow,

=

1 0 00 1 0

0 0 0

Premultiplying the Eqs. (1.7) and (1.8) with and I respectively, and then addingthe both we get the following set of difference equations for the kth mesh point:

Mnk,k1

Un+1k + Mnk,k

Un+1k + Mnk,k+1

Un+1k+1 = Nnk (2.9)

2.1 Non-dimensionalised version of TH model

After arriving at the previous form of representing the model, we have now made anon-dimensionalised version of the same, presented below.

Mn

k,k1Un+1k + M

n

k,kUn+1k + M

n

k,k+1Un+1k+1 = Nnk (2.10)

where

Mn

k,k1=

kB

k

=

+kWref +khref

W

h

p+kpref

Aa W

h

p

0 0khref+0kk

pref0 0 0

Mnk,k =

I

k + k1

I 2

k

Bk

=

(k1 + +k) (k + +k)W

h

p

(k1 + +k)Aa W

h

p

0 (k1 + 0k)(k1+0k)

k

(k + 2k) (k + 2k)W

p

h

(k + 2k)

Aa

+ W

p

h

Wrefhref

pref

Mn

k,k+1= (I )

kB

k

=

0 0 0

0 0 0k Wref k href

W

p

hk pref

Aa

+ W

p

h


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Nnk =

(I )zk + zk1

Dk

+

(I )k + k1

Bk

Unk

where k = zk/t.

The terms shown below are obtained from the steady state solution, with thepurpose of bringing the order of the terms in the coefficient matrix to near similar

values, so that absolute disparities are avoided in the coefficient matrix.

Wref = |Winlet|t=0

href = |hinlet|t=0

pref = |pinlet|t=0

With the application of this new system, the vector containing the unknowns is alsochanged as follows

U =

W

h

p

=

W/Wrefh/href

p/pref


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Chapter 3

Analysis of out-of-phase

instabilities

The BWRs are subjected to in-phase (core-wide) and out-of-phase (regional) modes ofoscillations. Dominance of one mode of oscillation over the other depends on variousoperating and boundary conditions. To safeguard the BWRs from both these types ofinstabilities, one needs to be more careful and conservative from safety point of view.Every effort is required to be made to detect which one of the two occurs first andthen, to suppress it. Recirculation loop has been modeled with the assumption of nopressure drop across the loop due to friction, inertia and the presence of various ex-core components except the pressure drop due to hydrostatic head difference whichensures that the parallel channels lying in the reactor core are subjected to constantpressure drop boundary condotions. Now, the focus is shifted to the analysis of out-of-

phase mode of oscillations when limited boiling is allowed in FC systems. The presentchapter deals with the methodology and the boundary conditions to be maintained tosimulate the out-of-phase instabilities in parallel channels of the reactor core.

3.1 Model description and boundary conditions for

out-of-phase instability analysis

The numerical model, characteristics based implicit finite difference scheme developedin the previous chapter, is used for the present case also, but with a little modifica-

tions in the boundary conditions. In the literature, the out-of-phase DWOs has beengenerally described as a result of constant pressure drop boundary conditions for allparallel channels lying in the reactor core while maintaining the total core inlet massflow rate constant. This kind of instability is characterized by self sustained out-of-phase oscillations in which the inlet mass flow rate increases in half of the reactor core,while it decreases in the remaining half.

3.1.1 Boundary conditions to model out-of-phase instability

Consider a case of two identical parallel channels, each representing a half of the corewith 180o phase difference, which are subjected to (i) fixed and common pressure dropboundary conditions across the channels and (ii) constant total inlet mass flow rate.

8


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CHAPTER 3. ANALYSIS OF OUT-OF-PHASE INSTABILITIES 9

A question arises how an asymmetric inlet flow distribution (indicated by 180o phasedifference) can take place for the two similar channels of same geometrical shape andsize under identical heat and other input conditions. It is to be noticed that one cantspecify velocity and pressure together as inlet boundary conditions for problem under

consideration since the governing equations are of hyperbolic in nature and it is a caseof subsonic flow condition. It implies the system of PDEs, representing the TH flowfields, are overspecified in terms of boundary conditions and the problem, as a result,becomes ill-posed for out-of-phase mode of oscillations. The problem, with these hypo-thetical boundary conditions, is solved and it is observed that (i) it allows only smallvariations in inlet mass flow rates for individual channels, and (ii) no convergence isobtained since the sum of the inlet mass flow rates of all individual channels is notequal to the specified total inlet mass flow rate. These results confirm the previousstudies made by Munoz-Cobo et al.[24]. Therefore, with these observations, it can beconcluded that the two parallel channels with identical input conditions, essentially,are undergoing in-phase oscillations (rather than out-of-phase oscillations) with anoverspecified total inlet mass flow rate boundary condition when a constant and com-mon pressure drop boundary condition is strictly maintained for both the channels.It is also to be noted that as far as in-phase instabilities are concerned, there is norestriction on the incoming mass flow rate at the entrance of each channel and it hasbeen described in previous chapter. In fact, parallel channels with in-phase instabili-ties will lead to oscillations for all individual channels separately and eventually, willresult in global oscillations in the total mass flow rate entering into the reactor core,and therefore, the total inlet mass flow rate for the parallet channels cant be heldconstant during the transients. Similar observations are found in studies by Lee etal. [25].

3.1.2 Solution technique to model out-of-phase instability

Next, to overcome the above mentioned difficulty in developing the model for out-of-phase mode of instability for two parallel channels, the following algorithm is adopted:

The problem has been converted into a case where an approximate constantpressure drop boundary condition is imposed with constant total inlet mass flowrate.

At every time step, the following boundary conditions are maintained for the

first channel:

Pressure inlet is specified and constant, i.e.

pin(t = t + t)

r=1= pinlet (3.1a)

Enthalpy inlet is specified and constant, i.e.

hin(t = t + t)

r=1= hinlet (3.1b)

Pressure outlet is specified and constant, i.e.

pout(t = t + t)

r=1

= poutlet (3.1c)


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Now, the transient channel equations are solved and one can find out the inlet

mass flow rate of first channel i.e.,

min(t = t + t)r=1

.

Since the total inlet mass flow rate is constant and specified ( mT), one can

calculate the mass flow rate at the entrance of the second channel and then, thefollowing boundary conditions are imposed for it:

Mass inlet is specified, i.e.

min(t = t + t)

r=2= mT

min(t = t + t)

r=1(3.2a)


hin(t = t + t)

r=2= hinlet (3.2b)

Pressure outlet is specified and constant, i.e.pout(t = t + t)

r=2= poutlet (3.2c)


pressure of the second channel i.e.,pin(t = t + t)

r=2= pinlet.

It is to be noted that pressure drop for the second channel, at present time step,is not constant.

Now, for the next time step, the conditions are reversed, and therefore, the

following boundary conditions are maintained for the second channel: Pressure inlet is specified and constant, i.e.

pin(t = t + t)

r=2= pinlet (3.3a)


hin(t = t + t)

r=2= hinlet (3.3b)

Pressure outlet is specified and constant, i.e.

pout(t = t + t)

r=2 = poutlet (3.3c)


mass flow rate of the second channel i.e.,

min(t = t + t)r=2

.

Since the total inlet mass flow rate is constant and specified ( mT), one cancalculate the mass flow rate at the entrance of the first channel and then, thefollowing boundary conditions are imposed for it:

Mass inlet is specified, i.e.

min(t = t + 2t)

r=1

= mT

min(t = t + 2t)r=2

(3.4a)


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hin(t = t + 2t)

r=1= hinlet (3.4b)

Pressure outlet is specified and constant, i.e.pout(t = t + 2t)

r=1= poutlet (3.4c)


pressure of the first channel i.e.,pin(t = t + 2t)

r=1= pinlet.

It is to be noted that pressure drop for the first channel, at present time being,is not constant.

Next, at time step t = t + 3t, first and second channels again will be subjected

to boundary conditions defined by Eqs. (3.1a) to (3.1c) and Eqs. (3.2a) to (3.2c)respectively. In the similar way, one can proceed for new time steps.

The procedure outlined above guarantees that the pressure drop is kept approx-imately constant throughout the full core.


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Chapter 4

Numerical Treatments

After the current thermo-hydraulic model is obtained in a suitably discretized form,for n number of nodes, we are left with a system of 3(n 1) linear equations [two

variables are known at the inlet, whereas one variable is known at the outlet]. Thissystem is first solved under steady state conditions to get a steady state solution. Thissolution is then used to perform transient analysis of the same problem.

For the current problem, obtaining a solution for the system is one of the compu-tationally intensive tasks that must be performed. Thus, the need is to reduce thetime spent on computation here so that the model may arrive at results in a quickerfashion.

Under brute force applications of calculating solutions to this problem by matrixinversion techniques, the computational complexity becomes (2/3)n3. This measureis almost universally derided. Under other schemes of obtaining solutions, the best

matrix inversion techniques give a complexity of 3n2. But there have been othermethods wherein reduced orders of computational complexity have been achieved.Some general methods are discussed below which briefly explain the concept adoptedin the most generic methods employed for obtaining solutions. After this, we willcome to discuss the solution methdology to be adopted for obtaining the solution tothe actual system that will be obtained from the discretization, which is a sparsematrix. The techiniques and methodologies to be used for our case will be discussedfirstly for serial application and then for parallel application.

4.1 Generic Methods for Dense Matrix Inversion

4.1.1 Sherman-Morrison Formula

The Sherman-Morrison formula [as shown below] [2] that is highly regarded amongstmathematicians as an efficient way to reduce computational complexity is a way bywhich the inverse of a matrix is not calculated each time from scratch; by treatingchanges in the original matrix as perturbations, updates the previously calculatedinverse by perturbations. This formula is a special case of the Sherman-Morrison-Woodbury formula [5] that makes rank-1 updates on the previously inverted matrix.

(A + uvT)1 = A

1

A1uvTA1

1 + vTA1u (4.1)

12


22/46

CHAPTER 4. NUMERICAL TREATMENTS 13

For the changes encapsulated by the column vectors u & v, the changes can be ac-counted for to obtain the inverse of the new changed matrix, provided that the inverseto the original unperturbed matrix is known.So although this method does not give a quicker method for directly calculating the

inverse of the matrix, it does give reduce the computational efforts required to finda solution to a suitable system of equations. The criteria of suitability and theirshortcomings are discussed below.

While this method can obtain complexities of 2n2 or n2, this is severely limitedby the assumption that the changes must be expressable in the form of the dyadicproduct u v. In the present problem scenario, this is highly unlikely to happen: theproblem is based on physical principles, and as such all the terms in the coefficientmatrix may change considerably at each time step. Also, given that the formulais based on differences expressed in the column vectors u and v, which admittedlymight be composed of very small values for very small time steps, the formula, thoughmathematically correct, may fail after considering how computers deal with numbers: precision may be lost, data values may get rounded-off, thus beating the purpose ofconsidering very small time steps.

On the other hand, literature has been presented by Kentaro, Linjie and Takashi,where they claim that implementations of the Sherman-Morrison formula may be par-allelized, albiet partially [3]. This makes the Sherman-Morrison formula a peculiarlyintriguing option for use in our application in the future, if the need and conditionsbe.

Also to be kept in handy for future reference is the Matrix Determinant Lemmashown below

det(A + uvT) = (1 + vTA1u) det(A) (4.2)

4.1.2 LU Decomposition

LU decomposition (also called LU factorization), the mathematical representation ofwhich is presented below, is a matrix decomposition which writes a matrix as theproduct of a lower triangular matrix and an upper triangular matrix. This decompo-sition is used in numerical analysis to solve systems of linear equations or calculatethe determinant of a matrix.

a11 a12 a13a21 a22 a23

a31 a32 a33

=

l11 0 0l21 l22 0

l31 l32 l33

u11 u12 u130 u22 u23

0 0 u33

(4.3)

An invertible matrix admits an LU factorization if and only if all its leading principalminors are non-zero. The factorization is unique if we require that the diagonal ofL(or U) consist of ones. If the matrix is singular, then an LU factorization may stillexist. In fact, a square matrix of rank k has an LU factorization if the first k leadingprincipal minors are non-zero, although the converse is not true. [6]

LU decomposition is generally considered as a modified form of Gaussian Elimina-tion.

LU decomposition frequently arrives at underdetermined systems of equations,wherein irrelevant factors can be set to arbitrary values. As such, it is often nec-

essary to put some restrictions on the L and U matrices, like making either matrix alower/upper unit matrix.


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The LU decomposition method has been widely studied and many efficient algo-rithms have been developed for this purpose. The LU decomposition method has alsobeen improvised to act efficiently on sparse matrices, which in our case is the nature ofthe coeffient matrix. These algorithms attempt to find sparse factors L and U. Ideally,

the cost of computation is determined by the number of nonzero entries, rather thanby the size of the matrix.

L11L21

0L22

U11 U12

0 U22

=

U111 L111 + U

111 U12U

122 L

122 L12L

111 U

111 U12U

122 L

122

U122 L122 L12L

111 U

122 L

122

Calculating the inverse of a decomposition by blockwise recursive method.

These algorithms use the freedom to exchange rows and columns to minimize fill-in(entries which change from an initial zero to a non-zero value during the execution of

an algorithm).

4.1.3 Cholesky Decomposition

Cholesky decomposition, along with LU decomposition, is one of the most popular tri-angular factorizations. Cholesky decomposition of a symmetric positive definite (SPD)matrix (or Hermitian positive definite, HPD) is quite similar to LU decomposition: Ais represented as A = LL (or, essentially the same, as A = UU), where L is a lowertriangular matrix with strictly positive diagonal entries, and L denotes the conjugatetranspose of L. This is the Cholesky decomposition. However there are differences

too. First, there is no pivoting (but factorization is stable). Second, instead of twomatrices (L and U) we have only one matrix multiplied by itself. Finally, Choleskydecomposition require two times less operation than LU decomposition of SPD/HPDmatrix of the same size.

4.2 Generic Methods for Sparse Matrix Inversion

4.2.1 Frontal Solvers

A frontal solver, due to Irons [8] is an approach to solving sparse linear systems which

is used extensively in finite element analysis. It is a variant of Gauss eliminationthat automatically avoids a large number of operations involving zero terms. [9][10]A frontal solver builds a LU or LDLT of a sparse matrix given as the assembly ofelement matrices by assembling the matrix and eliminating equations only on a subsetof elements at a time. This subset is called the front and it is essentially the transitionregion between the part of the system already finished and the part not touchedyet. The whole sparse matrix is never created explicitly. Only parts of the matrixare assembled as they enter the front. Processing the front involves dense matrixoperations, which use the CPU efficiently. In a typical implementation, only the frontis in memory, while the factors in the decomposition are written into files. This is

meant to reduce the chances of access collisions in the workspace memory. The elementmatrices are read from files or created as needed and discarded. A multifrontal solver of


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Duff and Reid [11] is an improvement of the frontal solver that uses several independentfronts at the same time. The fronts can be worked on by different processors, whichenables parallel computing.

4.2.2 Biconjugate Gradient Stabilised Method

The Biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is aniterative method by H.A. Van der Vorst[13] which can be applied to unsymmetricsparse systems that are too large to be handled by direct methods such as the Choleskydecomposition, LU decomposition or Gaussian Elimination. It is a variant of thebiconjugate gradient method (BiCG) and has faster and smoother convergence thanits parent algorithm, the original BiCG as well as other variants such as the conjugategradient squared method.

Out of the methods introduced above, the BiCGSTAB has been shown to achievebetter rates of convergence than a number of methods[14] that also exist in scien-tific papers. But it is to be noted that implementation of the algorithms involvedin BiCGSTAB is complex and requires extensive knowlegde of the Krylov SubspaceMathematics. As such, we will strive to implement a frontal solver that can efficientlymake use of multiple processors and the fact that the stiffness matrix at hand is asparse matrix. From here onwards, this chapter will discuss the various data struc-tures, algorithms and processing methods that will be used to make efficient use ofcomputational resources.

4.3 Sparse Matrix Storage Schemes

We now introduce the Yale sparse matrix format that will help us to exploit theadvantages of large scale sparse matrices.

We consider a large-scale engineering system of sparse, linear equations, which canbe represented in the matrix notation as

A x = b (4.4)

Now, we consider the stiffness matrix to be a N R NC (= 5 5) unsymmetricalsparse matrix as shown in eq (2.4):

A

=

1 2 3 4 51 d1 a1 a22 b1 d23 d3 a34 b2 d45 d5

(4.5)

where NR and NC represent Number of Rows and Number of columns of matrixA respectively.


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The sparse matrix in eq (2.5) is represented in compact row storage as

IA

12345

NR + 1 = 6

=

1468

1011

= starting locations of the

first non-zero term for each row

JA

1234

56789

NCOEF1 = 10

=

1341

234245

= column number associated

with non-zero terms of each row

AN(1, 2, 3, 4, . . . , N C O E F 1)T = d1, a1, a2, . . . , d5T (4.6)

where NCOEF1 represents the number of non-zero terms in the matrix and can becalculated as

NCOEF1 = IA(NR + 1) 1 = 11 1 = 10 (4.7)

Further simplifications are possible in the representations given above for IA, JAand NCOEF1 if the stiffness matrix presents itself as a symmetric matrix. But thatis a luxury that we cannot afford to assume.

Here, it is only fair to note that this format only saves memory for N N Z g andthe Eq. (3.1) can be rewritten as

1

vg

t[g (r, t)] = . [Dgg (r, t)] Rgg (r, t) +

g1

g=1sggg (r, t) +

(1 )Pg

Gg=1

(g)fgg (r, t) +ms=1

DsgsCs (r, t) where g=1,2,...,G (5.3)


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CHAPTER 5. NUCLEAR COUPLED TH MODEL INTEGRATION 23

To solve Eq. (3.3), zero flux boundary condition is used at the free surface of thereactor core. The above equations can be written in more compact form using matrixnotation as follows

V1

t [] =M + F

P[] +

ms=1

Ds sCs (r, t) (5.4)

Cst

=FDs

[] sCs where (1 s m) (5.5)

where = [1 (r, t) , 2 (r, t) , . . . , G (r, t)]

T (5.6)

V =

v1 0 . . . 00 v2 . . . 0...

..

.

..

.

..

.0 0 . . . vG

(5.7)

FDs =Ds 1f1,

Ds 2f2, . . . ,

Ds GfG

(5.8)

Ds

=Ds1,

Ds2, . . . ,

DsG

(5.9)

M =

(. D1+ R1) 0 . . . 0s12 (. D2+ R2) . . . 0s13 s23 . . . 0

......

......

s1G s2G . . . (. DG + RG)

(5.10)

FP =

p11f1 p12f2 . . .

p1GfG

p21f1 p22f2 . . .

p2GfG

......

......

pG1f1 pG2f2 . . .

pGGfG

(5.11)

The neutronic model used is based on flux factorization approach and IQS method-ology.

The flux vector (r, t) is factorized as

= (r, t) N(t) (5.12)

where(r, t) = [1(r, t), 2(r, t), . . . , G(r, t)]

T (5.13)

is called the shape function vector and N(t) is the amplitude function.

The purpose of factoring the total flux (r, t) = (r, t) N(t) is to achieve theeconomy of the detailed time integration of the rapidly varying part N(t) of the flux,while treating the slowly varying part (r, t) in larger time intervals. Splitting of totalflux into two functions naturally implies that subsequent changes are to be made in

the governing equations which will actually be discretized in time-domain.


33/46


Figure 5.2: Schematic diagram of a fuel rod containing fuel pellet, gap and clad.

5.2 Fuel Heat Conduction Model

The general heat conduction equation for the fuel pin and clad is given below:

t[c(Tc) hc(r, t)] = . [kc(Tc)Tc(r, t)] + q

(r, t) (5.14)

which can be reduced to the following form with the assumption that the axial tem-perature diffusion is neglected.

c(Tc) cpc(Tc)

t[Tc(r, t)] =

1

r

t

kc(Tc) r

Tc(r, t)

r

+ q(r, t) (5.15)

FHC equation is solved when initial condition is provided for the whole domain andboundary conditions are prescribed at the required places as below:

r[Tc(r, t)]

r=0

= (5.16)

kc(Tc)

r[Tc(r, t)]

r=rnf

= hG [Tnf1(t) Tnf(t)] (5.17)

kc(Tc)

r[Tc(r, t)]

r=rnf+1

= hG [Tnf1(t) Tnf(t)] (5.18)

kc(Tc)

r[Tc(r, t)]

r=roc=rnf+nc

= h [Tw(t) T(t)] (5.19)

where nf is the number of radial fuel nodes, nc is the number of clad nodes and asingle node is used for the gap between fuel and clad. To specify connective boundarycondition between the wall and the coolant, Dittus-Boelters heat transfer correlationis used for single-phase region and Chens correlation for the boiling region.


34/46


Figure 5.3: Three level time step structure.

5.3 Coupling Methodology

The coupled model is a combination of three modules and these are: TH, neutronic andFHC solvers. These modules are combined together to calculate the reactor dynamicsfor steady state and transient situations. The description of the steady state (Fig. 3.4)and transient (Fig. 3.5) coupling algorithms is as follows:

5.3.1 Steady State Coupling Algorithm

1. Steady state versions of all these three modules are used during the presentcalculation.

2. At first, the TH properties for all the channels are assumed. Using these, thespatial distribution of void fraction and temperature throughout the reactor coreare obtained.

3. Cross section look-up tables are used and neutronic properties are updated ac-cordingly depending on the void fraction and temperature of the locations, andcontrol rod (CR) positions.

4. Neutronic solver is then used and power generated in each mesh is calculated.5. Next, the temperatures of fuel elements are obtained using the FHC solver. Using

appropriate heat transfer coefficient, the convective heat flux passing to the THchannels is obtained.

6. TH solver is then used and it provides updated TH properties for all the channels.

7. Next, steps 2 to 6 are repeated until the convergence is achieved.

5.3.2 Transient Coupling Algorithm

1. It is to be noted that all TH and neutronic properties are known at previoustime step (time : t) throughout the reactor.


35/46


Figure 5.4: Solution technique for nuclear coupled TH system during steady state.

2. Transient initiator can be in the form of reactivity driven incidents like CRmotions or it can be through perturbations given in the TH channels or in theneutronic properties.

3. To start the iterations, the TH properties at new time step are assumed first.

Using these, the spatial distribution of void fraction and temperature throughoutthe reactor core for that time step are obtained.

4. Cross section look-up tables are used and neutronic properties are updated ac-cordingly throughout the reactor core.

5. Neutronic transient solver is used and shape function (r, t + t) and powerQ(r, t + t) are calculated where t is the time step used for shape and powercalculations.

6. Neutronic solver also calculates reactivities (t + i) and amplitude functionsN(t + i) at smaller time levels where i = it, i 1.0 (i O[10

3]).

7. Next, the amplitude functions with the help of linear interpolations are evaluatedat every fixed time intervals measured by t. Then, power is linearly interpolatedat all smaller and fixed time intervals t with respect to amplitude functionsof corresponding time levels. It helps in calculating the power Q(r, t + j(t))for all j (j = 1, 2,...,J). It is to be noted that t is the time step used forTH calculations. Three level time step structure used in the transient couplingcalculations is shown in Fig. 3.4

8. Next, FHC solver is used and temperatures of fuel elements are obtained for alltime levels t + j(t) where j = 1, 2,...,J.


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Figure 5.5: Solution technique for nuclear coupled TH system during transient analysis.

9. Next, the convective heat fluxes are calculated and then, the TH solver is usedto update TH properties for all channels and intermediate time levels t + j(t)where j = 1, 2,...,J.

10. TH properties calculated at time step t + J(t) t + t helps us in going backto step (4) and then the following steps are repeated until the convergence isachieved.

11. If the convergence is achieved, it allows us to go for next time step t + 2t andthen, all the previous steps are to be performed.

5.3.3 Algorithm to satisfy steady state TH boundary condi-

tions

The solution methodology adopted to satisfy the steady state boundary conditions forthe TH boundary channels is as follows:

1. At first, mr=1k=0 is assumed. It is to be noted that, prk=0 and h

rk=0 are already

specified for all radial channels, i.e., (r = 1, 2, . . . , R).

2. Steady state TH solver is then used and all unknowns are calculated. It helps in

calculating

pr=1

.


37/46


3. Now, mr=2k=0 is assumed and then

pr=2

is calculated using the TH solver. It

is observed that the

pr=1

=

pr=2

= p. Then, several iterations are

made to search mr=2k=0 for which

p

r=1 =

p

r=2 = p.

At the end of the convergence, mr=2k=0 is calculated which ensures common pressuredrop between the first and second channels.

4. Step 3 is followed for all other radial channels and then, we calculate the errorin total mass flow rate distribution, i.e., we observe mT

Rr=1 m

rk=0 = 0.

5. Next, a better guess of mr=1k=0 is made and steps 2 4 are performed repeatedly

until it is observed that

mT

Rr=1 m

rk=0

.

6. The procedure adopted ensures distribution of total mass flow rate among thechannels keeping the same pressure drop across them.


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Chapter 6

Results

6.1 In-phase Oscillations with neutronic feedback

During core-wide mode of oscillations, the entire reactor core is in phase, oscillatingthe mass flow rates for the channels and power for all fuel bundles together. Steadystate axial power profile for various fuel pins are shown in Fig. 6.1 when the BWRis working at a particular operation point denoted by rated mass flow rate, inletsubcooling (Nsub = 0.603547) and reactor power (Npch = 1.839268) at steady statecondition indicating a bottom peaked power shape for all channels. An estimate ofradial power distribution also can be obtained from the power profile for 10 differentfuel pins.

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

0 10 20 30 40 50 60 70

Power(W)

Axial Nodes (Dimensionless)

Variation of Power with Node Location during Steady State

Channel 1Channel 2Channel 3Channel 4Channel 5Channel 6Channel 7Channel 8Channel 9

Channel 10

Figure 6.1: Variation of Power with Node Location for various channels during SteadyState Calculations. nsub = 0.603547; npch = 1.839268.

29


39/46

CHAPTER 6. RESULTS 30

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40 45 50

Massflow

rate(kg/s)

Time s

Variation of Inlet Mass Flow Rates with time

Ch1Ch2Ch6

Figure 6.2: Variation of inlet mass flow rate with time for various channels duringcore-wide mode of oscillations. nsub = 0.603547; npch = 1.839268.

9

10

11

12

13

14

15

16

17

18

19

6 8 10 12 14 16 18 20

ExitMassflowrate(kg/s)

Inlet Mass flow rate ( kg/s )

Channel 6

(a) State at 10.75 seconds

9

10

11

12

13

14

15

16

17

18

19

6 8 10 12 14 16 18 20

ExitMassflowrate(kg/s)

Inlet Mass flow rate ( kg/s )

Channel 6

(b) State at 48.63 seconds

Figure 6.3: Time evolution of inlet and exit mass flow rates in phase plane diagram atmarginal stability boundary in presence of neutronic feedback effects during core-widemode of oscillations. nsub = 0.603547; npch = 1.839268.

Figure 6.2 shows the temporal variation of inlet mass flow rates for three differentchannels at the same operating point. The Fig. 6.2 also illustrates that the frequencyof oscillations for all the channels is same and the inlet mass flow rate variation of eachof them is undergoing limit cycle oscillations after a few initial periods at the operatingcondition mentioned. The development of limit cycle oscillations for the channel 6 isconfirmed in a phase plane diagram shown in Fig. 6.3.


40/46


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25 30 35 40 45 50

Relativepower(dimensionless)

i

Sub-Cooling Number = 0.603547; Phase Change Number = 1.839268.

Relative power

Figure 6.4: Variation of Relative Power with time during core-wide mode of oscilla-tions. nsub = 0.603547; npch = 1.839268.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

Relativepower(dimensionless

)

Void(dimensionless)

i

Relative powerCore average void

Figure 6.5: Temporal variation of relative power and core average void fraction duringcore-wide mode of oscillations. nsub = 0.603547; npch = 1.839268.

Global average core power oscillation with time is shown in Fig. 6.4 and thecareful observation of the figure indicates that the average relative power of the coreshifts slightly to higher side from initial value 1.0. Overall influence of void-Dopplerfeedback effects on the average power of the reactor core is observed in Figs. 6.5 and

6.6. Figure 6.5 establishes the negative void feedback effects on reactor power, whileFig. 6.6 indicates the delayed effect of fuel temperature on core power.


41/46


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20650

651

652

653

654

655

656

657

658


Temperature(K)

Relative powerCore average temperature

Figure 6.6: Temporal variation of relative power and core average Fuel Temperatureduring core-wide mode of oscillations. nsub = 0.603547; npch = 1.839268.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 2010

11

12

13

14

15

16

17

18


Massflow

rate(kg/s)

Relative powerInlet mass flow rateExit mass flow rate

Figure 6.7: Temporal variation of relative power and core average inlet and exit massflow rates per channel during core-wide mode of oscillations. nsub = 0.603547; npch =1.839268.


42/46


(a) t = 32.0 secs (b) t = 33.1 secs

Figure 6.8: Instataneous 3-D power distribution at a particular axial fuel plane duringcore-wide mode of oscillations when the BWR is operating at rated mass flow rate.nsub = 0.603547; npch = 1.839268. Power is depicted on a scale of 0 to 12 105 Watts.

Figure 6.7 links the time variation of relative power and also the inlet and exitmass flow rates per channel and it shows the transportation delays associated with theprocesses involved.

3-D power distribution in a particular axial plane at two different time instances isshown in Fig. 6.8. Figures 6.8a and 6.8b also reveal that the total decrease or increaseof BWR power is taking place apparently at uniform rate throughout the reactor coreduring the present mode of oscillations.

3-D power distribution in various axial planes at the same time instance t =14.1 secs is shown in Fig. 6.9. Figures 6.9a, 6.9b, 6.9c, 6.9d and 6.9e reveal thatthe total decrease or increase of BWR power is consistent with the power states de-picted in the steady state results shown in Fig. 6.1, thus supporting our hypothesisthat An estimate of radial power distribution also can be obtained from the powerprofile for 10 different fuel pins.

6.1.1 Time Savings Achieved

Considering the fact that the problem size itself does not lend much weight to thecomputational workload, time savings achieved were significantly lesser than what a

large problem size would allow. Reasons for this were discussed in section 4.4.Still we were able to observe time savings upto the tune of approximately 12 to

13% on a consistent basis for 62 test cases with different values of nsub and npch. Itwas generally observably that the parallized code could simultaneously utilize upto 3processors on a Quad-Core desktop computer. The computational particulars relatedto the Core-wide oscillation study are listed below.

Neutronic Core Grid Size 28 28 62TH Core Grid Size 10 31Number of TH Variables 3 per node


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(a) Axial Plane 1 (b) Axial Plane 2

(c) Axial Plane 3 (d) Axial Plane 4

(e) Axial Plane 5

Figure 6.9: Instataneous 3-D power distribution at various axial fuel planes duringcore-wide mode of oscillations when the BWR is operating at rated mass flow rate.nsub = 0.603547; npch = 1.839268; time t = 14.10 secs. Power is depicted on a scale of0 to 12 105 Watts.


44/46

Chapter 7

Further Scope of Work

The new software developed has been made to confirm to the previous results obtainedafter the modifications were introduced. We can now include into the model the effects

of ex-core components such as the downcomer, the lower horizontal section and theupper horizontal section (steam delivery pipeline, turbine etc.) in both core wide andcore-regional modes of oscillations. This will give us a thorough understanding ofevents that can occur during the operation of a forced ciirculation nuclear reactor.

Various parametric studies can be done to study the effects of different factors suchas fuel burn-up level, etc.

With a sound foundation of a model supported by theory, one can now proceed invarious directions as per requirements and interests.

35


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