This document discusses mathematical models related to fusion reactors and their energy gain. It begins by explaining how particles gain energy from background plasmas through MHD waves and are scattered, affecting the energy gain factor. It then discusses characteristics of Gen-4 nuclear reactors including different reactivity control mechanisms. Reactor kinetics models involving delayed and prompt neutrons are also covered. Finally, the document discusses industrial applications of Gen-4 reactors using accelerator models and transient analysis for security and operating conditions.

1. Journal of Energy Technologies and Policy www.iiste.org
ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online)
Vol.1, No.4, 2011
1
Mathematics of Fusion Reactors and Energy Gain Factor
Model
Syed Bahauddin Alam1*,
Palash Karmokar†,Asfa Khan†, Md. Nagib Mahafuzz†, Farha Sharmin††, Tahnia Farheen, Md.
Abdullah-Al-Mamun, Md. Rashiduzzaman Bulbul, Hasan Imtiaz Chowdhury, Md. Abdul Matin*
Department of EEE, *Bangladesh University of Engineering and Technology (BUET), Dhaka
†University of Asia Pacific (UAP), Dhaka
††Development Research Network (D.Net)
baha_ece@yahoo.com
Abstract
Particles are accelerated by gaining the energy from back ground plasmas through MHD waves. Particles
are scattered by the MHD waves which are raised by instability of background plasmas. Probability that a
particle entering to the downstream will eventually return to the upstream energy gain factor when a
particle crosses and re-crosses the shock front. In this paper, Characteristic features of gen-4 nuclear
reactors, its reactivity control mechanism, characteristic features of fission reactors, reactor kinetics and
accelerator models have been discussed. As well as transient analysis of reactor via simulation and its
industrial viability has been discussed in this paper.
Keywords: Fusion Reactors, Energy Gain Factor Model, Neutron Life Time.
1. Introduction
Reactivity control and its safety is treated by assimilation of neutrons in the nuclear reactor. For controlling
reactivity, Gen-4reactors are robust enough. In these reactors different mechanisms are used for controlling
the mechanism of reactor core’s activity. In Gen-4 nuclear reactors heavy particle scattering may be done
because of smoothing of the reactor process. Heavy Particle scattering from an Electron and by this
mechanism reactivity and atom speed can be controlled. For PARR-1 Nuclear Reactor Computer-Aided
Testing and simulation has evolved. In the design of thermal reactor Resonance Escape Probability is one of
the important factors. Difference of the power-law index Equation which shows difference between Vietri &
Peacock
This equation becomes 0 when variance of the energy gain factor is 0. In a thermal reactor, most of the
neutrons are immersed after they have retarded to thermal energies. Thermal reactors are typical to diverse
escape probability. All of the fission neutrons must eventually be absorbed some wherein the reactor and
there having no ef ux of neutrons from an infinite nucleus. . In most reactor designs, various restraints ensue
in this heat departure the reactor chamber at a comparatively low temperature, so that trivial or none of it can
be retrieved as wattage. In a fusion power reactor a plasma must be exerted at a high temperature in order

2. Journal of Energy Technologies and Policy www.iiste.org
ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online)
Vol.1, No.4, 2011
2
that nuclear fusion can pass off. A system’s energy is lost to its surroundings is defined as Confinement
times . In a plasma device, whether enough fusion will occur to sustain a reaction is determined by
confinement times. Thermal Utilization factor of Fusion reactor and Prompt neutron lifetime. For an infinite
thermal reactor time required for neutron to slow down to thermal energies is small compared to the time
neutron spends as a thermal neutron before it is finally absorbed. Reactor kinetics model for delayed
neutrons and no delayed neutrons are twisted with prompt neutron lifetime. Industrial applications of gen-4
nuclear reactor are basically wide enough. Accelerator kinetics and its models are used in the reactors for
industrial applications. Transient analysis of nuclear reactors basically provides security information and its
operating condition at different valve position, temperature etc. In this paper, Characteristic features of
gen-4 nuclear reactors, its reactivity control mechanism, characteristic features of fission reactors, reactor
kinetics and accelerator models have been discussed.
2. Characteristics Features of Gen-4 Nuclear Reactors (CTFNR): Reactivity Control
By immersion of neutrons in the reactor fuel, secure reactivity command is fundamentally acted. For
ascertaining reactivity, Gen-4 reactors are robust enough. In these reactors different mechanics are exploited
for operating the mechanics of reactor core’s process. By insuring circulation rate through the jet pumps
short term reactivity commutes are performed. When the water through the core is changed magnitude,
because of neutron temperance, reactivity is as well increased. Control cruciform control vanes are required
for longer term reactivity. By the commixture of Gadolinium Oxide (GdO2 ) and U O2 pellets, reactivity
control for counterbalancing fuel burn up is rendered. Gadolinium is transmuted into low neutron
absorption cross-sectional and by that way more neutrons are imbibed in the reactor fuel. An electrons
scatters heavy electrons. Since smoothening of the reactor operation, in Gen-4 reactors, heavy particle
dispersion is done. As alpha is heavy charged particles, pass through matter and they interact through the
Coulombic force, predominately on the electrons of the medium as of they occupy most of the matter’s bulk.
Towards heavy charged molecules with kinetic energy (MeV range), the more minuscule separation energy
of an electron to the nucleus is tri ing. Thus, a “free” electrons at rest is that, with which an incident alpha
particle interacts. To analyze this scattering reaction, identify particles X and y as the electron. For this
scattering process, there is no change in the rest masses of the reactants, i.e., Q = 0. Now
2
cose e M e
e
E Mm E
M m
θ=
+
(1)
The maximum electron recoil energy and the maximum kinetic energy loss by the incident heavy particle,
occurs for
2
cos 1θ = (2)
Thus, the maximum energy of the recoil electron is
max( ) 4 /e e ME M E M= (3)
This is energy sufficient to free most electrons from their atoms and create an ion-electron pair. Virtually
collisions transferless energy from the alpha particle, and, consequently, tenners of grands of ionization and

3. Journal of Energy Technologies and Policy www.iiste.org
ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online)
Vol.1, No.4, 2011
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innervations fundamental interaction are requisite for an alpha with respective MeV of kinetic energy to
retard and become part of the ambient medium. Data processor (PC) accomplishes reactivity reckonings
from the static positive reactor period info for the control rod and accomplishes online acquisition of distinct
signals exploitation of the well-known in - hour equation as given below,
6
1
i
l it
β
ρ = 1 / Τ +
+ λ
∑ (4)
where, ρ= scheme reactivity, T = static reactor flow, l= neutron interim time period, and iβ , iλ the
fraction and decay constant of the thi group of delayed neutrons, respectively Testing for PARR-1
Nuclear Reactor. Resonance Escape Probability is one of the crucial factors out the contrivance of nuclear
reactor. Thermal reactors are distinctive to diverse escape probability. All of the fission neutrons must
eventually be absorbed somewhere in the reactor. However, some neutrons might be absorbed as retarding
by nuclei having absorption resonances at energies over the thermal region. Most of the neutrons are
assimilated in a nuclear reactor subsequently decompressing to thermal energies. If P is the probability that,
a fission neutron is not immersed in any of these resonances, then P is the Resonance Escape Probability.
/F F M MsM
N V I V
P e
− ξ ∑= (5)
Fusion energy gain factor, Q is the ratio of fusion power density to to the externally supplied power for
heating unit volume of plasma in steady state. Plasma must be maintained at a high temperature in a fusion
power reactor in order that nuclear fusion can occur. Various constraints ensue in this heat imparting the
reactor chamber at a relatively low temperature in virtually reactor excogitations,
so that minuscule or none of it can be recuperated as wattage. In these reactors, wattage is brought forth from
the fraction of the fusion power comprised in neutrons. The neutrons are not moderated by the obtuse
plasma in inertial confinement fusion or the magnetic fields in magnetic confinement fusion but are
absorbed in a encompassing “blanket”. Imputable to versatile exothermic and endothermic reactions, the
blanket may have a power gain factor a few per centum higher or lower than 100%, but that will be
neglected in our scheme. A fraction of the electrical power is re-circulated to run the reactor arrangements.
It is defined as,Let,
Pfus=fusion power density
Pheat =Power supplied by external power sources
Pelect=Produced electric power
frecirc=fraction of power used to run fusion reactor
Here,
Pelect= ηelect(1 – fc)Pfus
Pheat=(1 – fc)Pfus ηelect ηheat frecirc
Thus Fusion energy gain factor is,

4. Journal of Energy Technologies and Policy www.iiste.org
ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online)
Vol.1, No.4, 2011
4
( )c fus elect heat recirc
1
爁1 ? f P
Q
η η
= (6)
The one conduct of energy expiration that is autonomous of the confinement intrigue and practically
inconceivable to obviate is Bremsstrahlung actinotherapy. Alike the fusion power density, the
Bremsstrahlung power density devolves on the square of the plasma compactness, but it does not alter as
apace with temperature. Fig. 5 represents the quality factor and Fig. 6 shows the particle confinement times.
2.6 Energy Confinement times
In which 0.5 of a system’s energy is lost to its surroundings is defined as Confinement times. In a plasma
device, whether enough fusion will occur to sustain a reaction is determined by confinement times.
A simple expression for the optimal confinement for the optimal confinement time is given. In a plasma
ignition, the fusion power density that goes into heating the plasma Pheat, must exceed the power density
lost to the environment, Ploss . The energy confinement times, Eτ is as follows,
3
E
l o s s
n k T
P
τ = (7)
Where we have used the fact that the average kinetic the average kinetic energy of the electrons and ions
with 3K T/2. Considering a core in which the neutron cycle takes
'
l seconds to complete. The alteration
∆n in the entire count of thermic neutrons in one cycle at time t is (keff - 1)n(t),
Where n (t) is the amount of neutrons at the setting out of the cycle. Thus,
1( )
( )
'
effkdn t
n t
dt l
−
= (8)
The solution of this first-order differential equation is,
'
1
( ) (0) exp[ ( )]
effk
n t n n t
l
−
=
(9)
where n(0) is the neutron population at t = 0. In this framework, the neutron population and
therefore the reactor power alters exponentially soon enough, 1effk ≠
(10)
.
For an infinite thermal reactor, time expected for neutron to retard to thermal energies is minuscule equated to
the time neutron drops as a thermal neutron before it is finally engulfed. The interim between emanation of
the prompt neutrons and immersions in nuclear reactor is called Prompt neutron lifetime, lfp. Mean diffusion
time is td.
2 ( )
d
V T a a
I I
t
F M
=
+∑ ∑ (11)
Considering an infinite homogenous thermal reactor whose thermal flux must be independent of the

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ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online)
Vol.1, No.4, 2011
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position. Time dependent diffusion equation for thermal neutron is,
,t a
d n
s
d tφ Τ
− =∑ (12)
where Ts is the source density of neutrons into the thermal energy region, and n is the density of thermal
neutrons.
The rate of change of neutron density is,
6
1
(1 ) i i
a T i
dn
k C
dt
ξ
φ
β λ
=
= − +∑ ∑
(13)
where
T
n Aeω
=∑ ,
T
C Beω
=∑
The complete solution for n is
p
t
l
n no e e
ρ βλρ
β ρβ ρ
β ρ β ρ
−
−
= −
− − (14)
Finally it is,
/( 1)PT l ka= − (15)
3. Industrial Applications of GEN-4 Nuclear Reactor via Accelerator Model (IANRAM)
Radioisotopes can retraces ideally from a tracer bullet. To discern the tracer bullet in a sample,few atoms are
mostly necessitated. Let, the actinotherapy emitted by the disintegration of a distinct radionuclide with a half
time of T 0.5 can betraced in a sample distribution with an efficiency of e. If a sample contains N atoms of the
radionuclide,the calculated countrate is,
0.5
0.693
CR N N
T
=∈ λ =∈
To observe the comportment of the radionuclide label, this count rate must be greater than some minimal
count rate CRmin which is above the base count rate. Then the minimum number of radioactive subatomic
particle in the sample required to observe the comportment of the radioisotope is,
min 0.5
min
1.442CR T
N =
∈
atoms
If the atomic weight of the radionuclide is A, the minimum mass of radionuclides in the sample is
m in 0.5
m in
1.442
a
C R T
M
N
=
∈
(16)

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A cyclotron comprises of two D-shaped realms cognised as dees. The oscillation degenerates with the
magnetic field in the dees continually contributing the charge back to the gap. As the charge is in the dee, the
force field in the gap is inverted, so the charge is once again accelerated across the gap. There is a magnetic
flux vertical to the plane of the page in each dee. All time the charge cuts through the gap it picks up speed.
These cause the half-circles in the dees to step-up in radius and finally the charge issues from the cyclotron at
high speed. The terminal kinetic energy is fundamentally independent of the potential drop in the gap, but the
kinetic energy is proportional to the square of the magnetic field, so increasing the magnetic field is the way
to increase the kinetic energy. An ion reaching the exit port at radius R with speed v has an energy given by,
K=qV , where v is the equivalent potential difference.
2 2
0.5
q
V B R
m
= (17)
If the magnetic field increases, there is a varying flux connecting the loop of electrons and so an induced
e:m:f: which accelerates the electrons. As the electrons get quicker they motive a larger magnetic field to
extend moving at an invariant radius, which is rendered by the modifying field; the issues are proportional, so
the field is always impregnable enough to keep the electrons in orbit. The magnetic field wont to cause the
electrons draw in a circle is also the one used to speed up them, though the magnet must be cautiously
contrived so that the field intensity at the orbit radius is equal to half the average field strength relating the
orbit. The E-field is, The field is changed by passing an alternating current through the primary coils and
particle acceleration occurs on the first quarter of the voltage sine wave’s cycle. The force on the electron
produced by the E-field is
( )
( ) 0.5a
dB t
F t eR
dt
= (18)
However, if the total energy is much greater than the rest energy then E = pc is an effective estimation. When
the magnetic field is at its strongest value, the subatomic particles have maximum energy. But the formula
used for the cyclotron will not work for betatron because the electron will be relativistic. As the centripetal
force is again rendered by the Lorentz force, Betatron is still used in industry and medicine as they are the
very compact accelerators for electrons.
5. Conclusion
In this paper, Characteristic features of gen-4 nuclear reactors, its reactivity control mechanism,
characteristic features of fission reactors, reactor kinetics and accelerator models have been discussed. As
well as transient analysis of reactor via simulation and its industrial viability has been discussed in this
paper.
References
Syed Bahauddin Alam et. al, Modeling of Physics of Beta-Decay using Decay Energetics, in American
Institute of Physics (AIP) Proceedings, 2012.
Syed Bahauddin Alam et. al Dosimetry Control and Electromagnetic Shielding Analysis, in American

7. Journal of Energy Technologies and Policy www.iiste.org
ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online)
Vol.1, No.4, 2011
7
Institute of Physics (AIP) Proceedings, 2012.
Syed Bahauddin Alam et. al Transient and Condition Analysis for Gen-4 Nukes for Developing Countries,
in American Institute of Physics (AIP) Proceedings, 2012.
Syed Bahauddin Alam et al., Methodological Analysis of Bremmstrahlung Emission , published in the
World Journal of Engineering and Pure and Applied Science, pp: 5-8,Volume: 1, Issue: 1 , Research |
Reviews | Publications, June 2011.
Syed Bahauddin Alam et al., Mathematical Analysis of Poisoning Effect, published in the World Journal of
Engineering and Pure and Applied Science, pp: 15-18, Volume:1, Issue: 1 Research | Reviews |
Publications , June 2011.
Syed Bahauddin Alam , Md. Nazmus Sakib, Md. Rishad Ahmed, Hussain Mohammed Dipu Kabir, Khaled
Redwan, Md. Abdul Matin, Characteristic and Transient Analysis of Gen-4 Nuclear Power via Reactor
Kinetics and Accelerator Model in 2010 IEEE International Power and Energy Conference, PECON 2010,
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Aziz, Imranul Kabir Chowdhury, Md. Abdul Matin, Can Gen-4 Nuclear Power and Reactor Technology be
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Conference, PECON 2010, pp. 95-100, Malaysia, 29 Nov, 2010.
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Conference on Intelligent Systems, Modelling and Simulation, ISMS2011, Malaysia, Phnom Penh
(Cambodia) , 25-27 Jan, 2011.
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of Beta Transmutation by Decay Energetics in 2nd International Conference on Intelligent Systems,
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of Advanced Nuclear Reactor and Kinetics Model in 2nd International Conference on Intelligent Systems,
Modelling and Simulation, ISMS2011, Malaysia, Phnom Penh (Cambodia) , 25-27 Jan, 2011.

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