Point reactor kinetics Neutronics thermal Hydraulics Reactivity Feedback

1. STABILITY ANALYSIS OF A NUCLEAR
REACTOR
Seminar Report under the guidance of
Prof. Suneet Singh
Alok Kumar
17i170013
May 5, 2018

2. Contents
1 Introduction 1
1.1 Need of Nuclear Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Need of stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Objective of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Thermal hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Neutronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Reactivity Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7.1 Coupling of Thermal hydraulics and Neutronics . . . . . . . . . . . 11
1.7.2 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7.3 Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Results 15
3 Conclusion 19
1

3. Abstract
Stability analysis of any system is extensively studied by researchers for a
long time.The nuclear reactor can be used as a system. The linear stabil-
ity in this system is only for small perturbations. mainly, there are three
dynamic processes neutronics, reactivity feedback and thermal hydraulics
in a nuclear reactor, which can be utilized in stability analysis. A math-
ematical model is developed which explains the working of these dynamic
process.This models contains a system of non linear differential equations.
This system is formed by coupling all the differential equation obtained by
three processes. This system is having some non linear terms, in order to
reduce the size of the system, the parameters are non dimensionalised. New
system can be analyzed for the purpose. The model is in reduced order,
all the differential equations representing the system behavior are carried
out in a reduced order. These coupled ordinary differential equations of of
nuclear reactor may reveal many complex phenomena which are analyzed
in this study. The associated coupled system gives the Jacobian matrix,
whose eigenvalues characterize the system behavior. Mainly the stability
analysis has been done in which bifurcation analysis is also introduced.
Bifurcation is introduced for the dynamics of the limit points. Behavior of
the system at different type of eigenvalues of the Jacobin matrix is also an-
alyzed. The regions of the stability depending upon bifurcations occurring
at the points also has analyzed. The limit cycles with describe global or lo-
cal stability as these trajectory will move on. The characteristic of dynamic
instabilities in a system are either self-sustained or diverging oscillations
in the system variables. These oscillations are caused by delay in system
parameters and hence strong feedbacks in system may lead to periodic
oscillations. The characteristics of these oscillations are mathematically
represented. For large perturbations the system behavior is analyzed by
bifurcation analysis. For these relatively large perturbations, the nonlinear
stability analysis of the system predicts the instability in the linearly stable
region. The local regions in the domain of the space of the study have also
determined by the bifurcation analysis. In this study we only did till linear
stability but elaborate about bifurcation analysis.

4. Chapter 1
Introduction
1.1 Need of Nuclear Energy
In this era, when energy demand per capita are increasing and the natural sources makes
the lines of anxiety on the forehead of humans. World is approaching towards the carbon
free energy efficient machinery and the technologies. But the needs are not ready to meet
the satisfaction due to increasing demand of per capita energy globally. Even with the
efficient renewable energy technologies, we can control the per capita energy need, it will
take much time and money to drive the whole country like India on solar, wind, hydro
or geothermal energy. So, from the strategic point of view to make country, with a huge
population, energy rich we should produce more nuclear energy with increasing factor of
renewable energy to reduce our coal energy share.
Most preferably, nuclear power is proving to be cheaper than the conventional sources
of power. the world is running out of the available fossil fuels, coal oil and natural gases.
The impact on the environment of the nuclear power plant is no more hazardous. The
size of nuclear power plant is also smaller that that of the conventional power plants.
The nuclear power plant does not produce any polluted gas during it’s performance. For
defense purposes, since nuclear power is much dense as compare to other conventional
sources so it can fuel the underseas fleet for a long time as compare to other fuels also
nuclear power plant does not consume oxygen like other conventional plants. So it is
useful in many ways as it is economically.
1.2 Need of stability analysis
Each dynamic system requires stability for a constant functioning. There are many per-
turbations occurring due to noise or applied perturbations, study in which we explain,
what the system is going to response against these perturbations is called stability anal-
ysis. In nuclear reactor also there are many parameters which continuously changes with
time, so it is necessary to know that with this change how the system is going to response.
In stability analysis some parameters are such that they give drastic change in the over-
1

5. [?]
all system performance by varying little. This process is called bifurcation and study of
bifurcation is called bifurcation analysis. So we will also study bifurcation analysis in a
little.
1.2.1 Bifurcation Analysis
A bifurcation manifests itself with a change of the number of attractors in a nonlinear
dynamical system.In a bifurcation point at least one eigenvalue of the Jacobian is having
real part zero.Stationary bifurcation occurs when the single eigenvalue cross the stability
boundary transversally. Since as the parameters varies, the qualitative structure of the
flow can change.In other words the fixed points can created or destroyed or their stability
cam change.These changes which are qualitative are called bifurcations in dynamics and
the parameters values at which they occur are called the bifurcation points. As some
parameter values varies, the occurring of these bifurcations provides the models of tran-
sitions and instabilities to the system.
Here we will discuss some of these bifurcations:
• When two fixed point move toward each other, collide and mutually annihilate,
saddle point bifurcation occurs there.
• when there is the existence of a fixed point which can’t be destroyed for all values
of the parameters, transient bifurcation occurs.
• In a Hopf bifurcation the conjugate complex pair crosses the stability boundary. It is
the most important bifurcation in our study so we will discuss it in detail. It occurs
for two or more than two dimensional systems.According to Poincare-Andronov-
Hopf bifurcation theorem
”A system having autonomous ordinary differential equation is considered as
˙y = F(y, p) (1.1)
2

6. where y is an n dimensional variable and p is system parameter vector.The equilib-
rium solutions of the above system can be written as
F(y, p) = 0 (1.2)
”
In fact, this solution is a matrix of the same order as the order of the system of equa-
tions.Stability of fixed points of the system can be analyzed by nature of the eigenvalues
of this Jacobian matrix. These eigenvalues varies as system parameters varies. For a Hopf
bifurcation to occur following criterion should hold
• Eigenvalues of the Jacobian matrix should be purely imaginary.
• The transversal condition should followed by the eigenvalues i.e, the eigenvalues
should cross the the imaginary axis with non-zero speed at parameters equal to the
critical value.
T = Re[
dλ
dav
]av=av0,λ=+−iω=0 (1.3)
The conclusion of the Hopf theorem is the co-existence of of some specific type periodic
solutions. These periodic solutions are called limit cycle.
A limit cycle is an isolated closed trajectory. Isolated means that neighboring trajectories
are not closed; they spiral either toward or away from the limit cycle. If all neighboring
trajectories approach the limit cycle, we say the limit cycle is stable or attracting. Oth-
erwise the limit cycle is unstable, or in exceptional cases, half-stable. Stable limit cycles
model systems that exhibit self-sustained oscillations. In other words, these systems oscil-
late even in the absence of external periodic forcing. Hopf bifurcation can also be divided
into two part on the basis of limit cycle.
• Supercritical Hopf Bifurcation occurs when a stable spiral converts into an unsta-
ble spiral surrounded by a neighboring limit cycle.Stable limit cycle emerge from
this bifurcation and unstable focus gains the stability . For a supercritical Hopf
bifurcation to occur following criterion should hold
– The limit cycle should grow proportional to the square root of the difference
between the parameter and critical value of parameter(p − pc)
1
2 .
– The frequency of the limit cycle is given by Imλ at p = pc, where Iλ is imagi-
nary part of the eigenvalue.
• The sub critical Hopf bifurcation occurs where the limit cycle which is unstable
shrinks to zero amplitude and enveloped the origin. It occurs at p=0.
3

7. 1.3 Objective of the work
4

8. ( Copied from world Nuclear Association Site)
[?]
The objective of the work carried out is to understand the basic functioning of
the nuclear reactor. How different parameters varies and what is the impact of these
parameters. Here in the figure shown in the left side there is a chamber, in this chamber
the vertical line type structure is the fuel rods and the the fluid around it is the coolant.
Our study is all about the stability of this chamber. In chapter mathematical model all
the governing equation have been from this chamber. Also analysis of the effect of the
coefficient of void and fuel temperature reactivity on the reactor has been studied. On
the parameter curve the Hopf curve is actually stability boundary. These boundaries have
been considered as actual boundaries in the study. On the stability boundary the analysis
of sub and supercritical Hopf bifurcation has been done.Here in this study we will find
the mechanism of neutron flow and the heat flow and the nature of this flow. For this we
will develop some models for neutrons. All the three sections of the chapter mathematical
modeling have modeled according to the behavior of the neutrons, thermal interaction be-
tween coolant (fluid) and fuel rod, the reactivity feedback respectively. this mathematical
formulation is due to the physics in the nuclear reactor. For the neutronics part we take
the fact that for a constant operation of nuclear reactor the rate of generation of neutrons
during the reactions should balance with the loss due to leakage and absorption.
If the neutron yield per fission is greater than one than the reactor is called supercritical.
In other words this yield is also called multiplication factor which is equivalent to the
ratio of number of the neutrons in one generation and number of the neutron in preceding
generations. On the other hands if the multiplication factor is less than one the reactor is
5

9. called sub critical.finally if the multiplication factor is exactly one, generation of neutron
is constant i.e, the reaction proceeds at the constant rate and the reaction is said to be
critical.
The reactor is controllable due to this controllable multiplication factor. In a rector the
neutrons can disappear in two ways, one is the absorption and the other is leakage. The
reactor will operate at the critical situation if the rate of leakage plus the rate of absorp-
tion is equal to the rate of generation of the neutrons. Reaction, leakage, absorption and
the generation all the processes depend on the size and shape of the reactors.
Most of the neutron released in fission are emitted essentially at the instant time of the
fission, this fraction is about 99 percent. Rest of the one percent neutrons are emitted
after some time. The instant emitted neutron are called prompt and other are called
delayed neutrons. Though the fraction of delayed neutrons is very less as compare to
the prompt neutrons, they have great impact on the overall performance of the system.
Those Nuclei, which are having the delayed neutron during the fission process are said to
be the delayed neutron precursors. It is believed to be 20 such precursors.
Similarly for thermal hydraulics the major assumption is that of the lumped parame-
ters which implies that the temperature profile is uniform or the core is isothermally
distributed throughout.
1.4 Thermal hydraulics
This part of the modeling describes the heat energy transfer between the rod and coolant.
Here we are using the lumped parameters so the the temperature is uniformly distributed
throughout the fluid and the fuel rods. This temperature we will consider as the saturation
temperature of the state. The heat which is generated in the fuel rod by nuclear fission is
conduction through the fuel rod and convected to surrounding coolant in the flow channel.
Mathematical modeling of any system for the stability analysis describes the
response of the system against any perturbation. Here in this part of thermal hydraulics
first we will write all the governing equations for the heat transfer between the rod and
the coolant. Here we assume the transfer to be at a constant rate even if at the transient
condition. The steady state condition implies that the the temperature is same as the
saturation temperature of the coolant and it is throughout the same even on the bound-
aries.Thus the change in the temperature is calculated with reference to the saturation
i.e, steady state temperature similarly the change in the volume of voids is also taken in
account with reference to the steady state void volume. As the temperature is increasing
the volume of voids is also increasing so the difference is positive same as of the temper-
atures.
Thus the heat transfer per unit time from the rod to the fuel is :
Cf
Tf (t)
dt
= P(t) − U(Tf (t) − Tsat) (1.4)
hfg
vg
dv(t)
dt
= U(Tf (t) − Tsat) − P0 (1.5)
6

10. Pt = Nt
P0
N0
(1.6)
whereCf is the heat capacity of the fuel element in the reactor core(Jk−1
), P(t) is power of
the rector(W), U is overall heat transfer coefficient, Tf (t) is the mean fuel temperature(K),
Tsat is mean coolant temperature(K),N(t) is the neutron concentration in the reactor core,
N0 is the steady state neutron density, hfg is the latent heat of vaporisation (JKg−1
) and
vg is the specific volume (m3
Kg−1
) of the coolant vapor.
1.5 Neutronics
In nuclear rectors, the energy is produced due to nuclear fission so nuclear fission is the
basic phenomenon of the nuclear reactors. In this fission reaction initially the fuel nu-
clei absorbs the neutrons and then emits those after the reaction. Average number of
neutrons emitted from fission varies to fuel material type. This emitted electron number
varies from 1.5 to 3.0. Also,these emitted neutron get slow down by different mechanisms
as scattering. When a neutron scatter to a nucleus it can be absorbed or can scatter at
an arbitrary angle. Neutrons emitted from fission slow down (for thermal reactors) and
causes other fissions. Which results in a chain reaction.As we discussed in chapter 1 that
If k(Multiplication factor) is less than 1.0, implies that number of neutrons is getting
lesser, so the reaction will also get slower and slower and reactor will shut down itself
automatically after some time. If k is greater than 1.0 it means number of neutrons is
increasing, so the power generated in the reactor will increase exponentially. For keeping
the the rate of reaction constant in the reactor we have to keep the value of multiplication
factor one. while K=1 reactor will work at steady state conditions. This is the criticality
condition which is the basis of the design parameters of a nuclear reactor. Neutron bal-
ance is the basis of the neutronics equations for the nuclear rectors. Which is equivalent
to the fact that number of neutrons generated in a control volume in addition with the
neutron come in from outside the volume should equal to the number of neutron absorbed
in the volume plus leaked out. So we will take the neutron concentration into account.
The fission reaction of the fuel material produced the source term for the equations. The
energy levels of neutrons decides the the fission cross section of the fissile material. An-
gular neutron flux also have importance. This energy, time, space and angle dependent
neutron balance equation is called neutron transport equation. Being a multi variate
equation it’s very hard to solve analytically so we shall take some assumption. Since the
neutrons have different energies so we will group them in different energy levels. Also, It’s
an reasonable approximation to reduce the energy dependence. To eliminate the angular
dependency of the neutron transport equation, the diffusion approximation is the most
appropriate assumption. As a result of these approximations neutron diffusion equation is
obtained which is the adequate equation for most of the neutronics calculations of nuclear
7

11. reactors. For a constant power level operation of a reactor rate of neutron production
via fission reactions should be exactly balanced by neutron loss via absorption or loss via
leakage. first we will discuss about some parameters which we will use later in actual
modeling in neutronics for a reaction to occur and the rate of reaction are dependent on
certain things which described below.
2.2.1 Neutron Diffusion Equation
In order to derive the neutron diffusion equation we will first define some
definitions
2.2.2 Neutron flux Density:
φ = n.v (1.7)
where n is neutron density cm−3
and V is neutron velocity (cm/s) both
depend on (r, E, t)and it’s a scalar quantity. In other words, Neutron flux
density is distance traveled by all neutron per unit volume, we will accept
here one group approximation so there is no dependence of energy.
2.2.3 Microscopic Cross Section
Microscopic Cross section is effective area of nuclide for interaction with
neutrons, number of reactions of neutron beam with atoms in a material
will be proportional to this microscopic cross section and atomic density
of material.
Σ = σ.n (1.8)
,
where σ is microscopic cross section (cm2
) and n is atomic density. σ de-
pends on (E),n on (r,t).
2.2.4 Macroscopic Cross Section
Σ is σ.n and is the probability for a reaction to occur per distance traveled
by a neutron.
2.2.5 Reaction Rate of a neutron
R = φ.x (1.9)
where,φ is Neutron flux density(cm−3
s−1
), Σ →is macroscopic cross section
8

12. (s−1
). Both depends on (r, E, t) or (r, t) for one group assumption.
2.2.6 Reaction Rate Density
Reaction rate density is the total number of the reactions occurred per unit
volume per second or the reaction rate is the number of interactions per
unit volume per unit time.
Macroscopic cross section is the sum of the partial cross section for fission
and scattering etc. This means the reaction rate can also be subdivided
into these separate contributions.
Using Fick’s Law,the neutron current density
J = −D
dφ
dx
(1.10)
where J is Neutron current density (cm−2
s−1
), D is Diffusion Coefficient
(cm) φis neutron flux density (cm−2
s−1
), Neutrons diffusion occurs from
high to low density.
One -Dimensional Neutron Diffusion Equation:
dN
dt
= −
d
dx
(−D
dφ
dx
) − Σaφ + vΣf φ (1.11)
The first term dn
dt is the time derivative of neutron number density. If this
dn
dt = 0, we have a stationary neutron flux otherwise this is increasing or
decreasing with time.
− d
dx(−Ddφ
dx)
is flow of neutron in X direction. Here note that we have two minus signs
because leakage causes as lost term in balance equation.
-Σaφ-is absorption rate, this is also a loss term so with minus sign.
vΣf φ is a fission neutron source term. this is equal to the product fission
rate density and average number of neutron released in fission event. For
U-235 this average number is around 2.5.
For a three dimensional Flow the Diffusion equation is as follows:
dn
dt
= .(D φ) − Σaφ + vΣf φ (1.12)
and,
dn
dt
= D 2
φ − Σaφ + vΣf φ (1.13)
9

13. Eq(2.10) is the diffusion equation for a homogeneous reactor.
where represents the divergence in all three direction. Which implies
the global change in the neutron density. In the design process of a reac-
tor core it would very difficult to find fuel composition that will make the
reactor core exactly critical and the neutron flux density.
In a reactor core, the change in the neutron density is directly depen-
dent on neutron density and the density of delayed neutron.The coefficient
of change in neutron density is the ratio of the the difference between re-
activity or fractional excess multiplication factor(ρ) and delayed neutron
fraction and the neutron generation time(t). Similarly the delayed neu-
tron change coefficient is the ratio of the delayed neutron fraction (β) and
neutron generation time(t) and the decay constant is with the precursor
density of delayed neutron.
dN(t)
dt
= (
ρ(t) − β
δ
)N(t) + λ1C(t) (1.14)
dC(t)
dt
=
β
δ
N(t) − λ1C(t) (1.15)
P(t) = N(t)
P0
N0
(1.16)
Again, we can describe in terms of the reactivity or fractional excess multi-
plication factor and steady state fuel temperature(Tf(t)−T0
) and difference
between the void volume and steady state void volume(Vt − Vo) with fuel
temperature coefficient of reactivity(K−1
) and void coefficient of reactivity.
Since at the steady state, there is no loss, so we non dimensionalise the
parameters with reference to the steady state values.
1.6 Reactivity Feedback
When power changes in a nuclear reactor are large enough to influence the
reactivity value , the transient behavior of the rector is termed as nuclear
10

14. reactor dynamics.The influence of power on the reactivity has to quan-
tified in order to properly describe the dynamics behavior of the nuclear
rector. Reactivity feedback depends on the reactivity introduced by the
control rods, reactivity due to coolant void fraction and due to reactiv-
ity contributed by the fuel and moderator temperature feedback. In fact
Reactivity is the cumulative sum of all these factors. In this model the
assumption is that the reactivity feedback is only due to the fuel temper-
ature and the void formation. The reason behind ignoring other factors is
their lessor impact. Thus the reactivity feedback can be represented as
Reactivity shows two behavior in different reactors fist is that reactiv-
ity increases with temperature, in this case with increase in temperature
the reactivity increases which implies that the temperature increases in
this case the reactor in sort of unstable state and the second behavior is
that the reactivity decreases with increase in temperature, in this case the
change in the reactivity is in the negative direction which implies that as
the temperature increases the power reduces followed by decrease of the
temperature, thus this configuration of the nuclear reactor is stable.
ρ(t) = αf (Tf (t) − Tf0) + αv(
(Vt − V0)
v0
) (1.17)
which simply denotes the linear relation between the reactivity and the
change fuel temperature from steady state and the change in void volume
from steady state and the coefficient of proportionality are fuel tempera-
ture coefficient of reactivity and void volume coefficient of reactivity re-
spectively.
1.7 Stability Analysis
1.7.1 Coupling of Thermal hydraulics and Neutronics
This section of the modeling is called Coupled Neutronics thermal hy-
draulics. In this portion we will combine all the equations of thermal
hydraulics and neutronics. Thus our system of equations is appeared to be
as follows:
dN(t)
dt = (
αf (Tf (t)−Tf0)+αv(
(Vt−V0)
v0
)−β
δ )N(t) + λ1C(t)
11

15. dC(t)
dt = β
δ N(t) − λ1C(t)
Cf
Tf (t)
dt = P(t) − U(Tf (t) − Tsat)
hfg
vg
dv(t)
dt = U(Tf (t) − Tsat) − P0
When we write a system of equation, it is convenient to combine the varios
parameters into a non dimensional numbers. This process is called non
dimensionalising. By non-dimensionalisation we can reduce The Form of
the system as:
where
x1 =
N − N0
N0
, x2 =
C − C0
C0
, x3 =
Tf − Tf0
Tf0 − Tsat
, x4 =
V − V0
V0
, τ =
βt
δ
(1.18)
Also in order to simplify this system of equation we take some dimension-
less parameters as:
af =
αf (Tf0 − Tsat)
β
, P =
δP0β
Cf (Tfo − Tsat)
, av =
αv
β
, q =
ρvgP0
hfgβV0
(1.19)
thus the resultant system is ,
dx1
dt = −x1(t) + x2(t) + af x3(t) + avx4(t) + af x1(t)x3(t) + avx1(t)x4(t)
dx2(t)
dt = b(x1(t) − x2(t))
dx3(t)
dt = P(x1(t) − x3(t))
dx4(t)
dt = qx3(t)
1.7.2 Linear stability
In this system first equation is having product of two variables ,which we
can ignore because of their higher order.Now we will find the Jacobian
formed by this system of equation.
12

16. 1.7.3 Jacobian Matrix
The Jacobian matrix for this system of equation is
|A|=
−1 1 af av
b −b 0 0
p 0 −p 0
0 0 q 0
now, put |A − λI| = 0, where I is the identity matrix and ”||” repre-
sents the identity matrix. Characteristic equation of this Jacobian matrix
is:
λ4
+λ3
(P +b+1)+λ2
(bP +P −Paf )−λ(Pqav +Pbaf )−Pbqav = 0 (1.20)
The system of equations has only one equilibrium solution that is
[x1, x2, x3, x4]T
=[0, 0, 0, 0]T
. So we will check the stability by varying the
the void coefficient of reactivity(αv) i.e, by varying the av. Non linearity
occurs due the void coefficient of reactivity(αv) because it depends on N(t),
Tf and the void formation in the core. In order to calculate the eigenvalues,
by putting av = 0, we got λ = 0 let λ = iω, ω > 0
we have ,
ω4
− ω2
(P + bP − Paf ) − Pbqav = 0 (1.21)
ω3
(1 + b + P) + ω(Pqav + Pbaf ) (1.22)
From these equations we can write the expression for af and av
af = −
ω2
(ω2
+ b2
− P + b)
P(b2 + ω2
(1.23)
av = −
ω2
(ω2
P + ω2
+ b2
P + bP)
Pq(b2 + ω2)
(1.24)
For the given Jacobian matrix, If all the eigenvalues have negative
real part, then the system is stable. From instability point of view if any
of the eigenvalue of the system is positive, the system will be unstable.
The system is said to be unstable if real part of at least one eigenvalue
is positive. Moreover, the systems having real part of the eigenvalues are
oscillating.
The Hopf bifurcation occurs when the pair of conjugate eigenvalues
crosses the imaginary axis transversally and the point where the system
13

17. loses its stability via Hopf bifurcation is called the Hopf point. For a Hopf
Bifurcation, the Transversal condition can be defined mathematically as
T = Re[
dλ
dav
]av=av0,λ=+−iω=0 (1.25)
dλ
dav
=
Pq(λ + b)
4λ3 + 3λ2(P + b + 1) + 2λ(bP + P − Paf )
(1.26)
The numbers of free parameters is dependent on the co-dimension for a bi-
furcation to occur.The eigenvalues of Jacobian matrix of the model varies
as one of parameters of the system changes. Saddle node and Hopf bi-
furcation are co-dimension one bifurcation detected and which could be
examines by eigenvalues. So av and af are those parameters which give
bifurcation. Using transversal condition and the criterion given in the in-
troduction part one can determine that which type of bifurcation occurring
due to variation in these parameters. The stability of a state indicates to-
wards stability of all the possible states. So the perturbation applies of
generated here is an average of these partial perturbations.
14

18. Chapter 2
Results
As we discussed in introduction part that the existence of Hopf bifurca-
tion depends on the co-dimension of the eigenvalues.Here all the graphs
are reprinted from papers and drawn for system parameters. For a co-
dimension two system if both the eigenvalues are having real part negative,
the system is strongly stable. For any global perturbation system does not
lose its stability. fig:Converging Limit Cycle Reprinted from [ []] In fig[]
If at least one eigenvalue has positive real part, the system goes to
instability because positive exponent. fig: Diverging Limit Cycle Reprinted
from [ []] In fig[] the curve shows an unstable limit cycle. For given system
parameters av and af ( []) unstable limit cycle occurs.
fig: Supercritical Hopf Bifurcation Caption[ []]reprinted In the op-
eration of the nuclear reactors,only the exact stable is considered as the
stable state, which is stable for small and large perturbation.
fig: Caption[ []]reprinted
When two parameters of the system Vary the generalized Hopf bi-
furcation occurs. This bifurcation is basically the transition point between
the supercritical and sub critical Hopf or vice versa.These points are either
initial or the final point of a limit cycle. As parameter ,void coefficient of re-
activity, is varied, one varied, one Hopf bifurcation is found on the stability
curve. Continue the analysis, the coefficient void and fuel temperature of
reactivity are varied simultaneously, generalized Hopf(GH). Once the Hopf
bifurcation point is detected, the bifurcation of limit cycle is checked by
changing void coefficient of reactivity. Once the Hopf bifurcation of limit
point bifurcation of limit cycle is seen,here the generalized Hopf point will
15

19. [?]
[?]
divide the supercritical and sub critical Hopf bifurcation. This lead to
plotting the global stability boundary.
This is the more described form of fig in which different part of
stability have been explained.The fig: Caption[ []]reprinted
As we have seen that the non dimensionalisation Reduces the steps
or complexity of the solution.
This graph is drawn between av and af . It can be observed that for
a positive value of av the steady state operation is unstable. Therefore for
a nuclear reactor to design it is always kept in mind that the void reactivity
coefficient should always be negative. Event slightly positive void reactivity
coefficient will give an instability resulting in accidents. however, with a
positive fuel temperature coefficient (af ) of reactivity, the equilibrium could
be stable provided that void coefficient of reactivity (av) is small enough
which we can clearly seen in the figure in zoomed portion of the region.
So for the security purpose fuel rods are designed to be having negative
void coefficient of reactivity af . For a small margins of the stability regions
for af where the range of av corresponding to stable equilibrium decreases
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20. [?]
with an increase in the value of af . Thus for a larger margin of variation in
av due to variation in operating conditions it is desirable to have a negative
fuel temperature coefficient of reactivity.
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21. [?]
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22. Chapter 3
Conclusion
From the analysis in this work, the given system of equation is solved sim-
ply by non dimensionalising, thus non dimensionalisation combines many
parameters into one and reduces the order of complexity also same process
can be done in a non linear system. For bifurcation analysis the parame-
ters which we are using here are the void coefficient of reactivity (av) and
fuel temperature coefficient of reactivity af , so in designing of reactor these
parameters are mostly considered for criticality gain.
We can conclude that with a decrease in the fuel temperature coefficient of
reactivity to further negative values the bifurcation becomes supercritical.
For a supercritical bifurcation the steady state operation is stable to any
perturbation caused by external or by internal variations in the linearly
stable region. This observation implies that reactors with high negative
values of the fuel temperature coefficient of reactivity are inherently safer
in operation as compared to those with small negative fuel temperature
coefficient where the bifurcation is sub critical in nature.
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