Journal of Energy Technologies and Policy ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online) Vol.1, No.4, 2011

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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

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Journal of Energy Technologies and Policy ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online) Vol.1, No.4, 2011

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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

Ee =

2 M + me

Mm e E M cos θ e

(1)

The maximum electron recoil energy and the maximum kinetic energy loss by the incident heavy particle, occurs for

cos 2 θ = 1

(2)

Thus, the maximum energy of the recoil electron is

( E e ) m ax = 4 M e E M / 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

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Journal of Energy Technologies and Policy 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,

ρ =1/ Τ +

6

βi

∑ l + λ it

(4)

1

where,

ρ = scheme reactivity, T = static reactor flow, l= neutron interim time period, and βi , λi the

fraction and decay constant of the i t h 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.

P=e

− N FVF I / ξM

∑ sM VM

(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,

3

Q=

1 (1 ? f c ) Pfus η爁electη heat

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(6) recirc

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, τ E is as follows,

τ

E

=

3nkT Plo s s

(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,

dn ( t ) k eff − 1 = n (t ) dt l'

(8)

The solution of this first-order differential equation is,

n ( t ) = n (0) exp[

k eff − 1 l'

n ( t )] (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,

keff ≠ 1

. (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.

td =

2VT (∑

a

II F +

∑

a

M )

(11)

Considering an infinite homogenous thermal reactor whose thermal flux must be independent of the

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position. Time dependent diffusion equation for thermal neutron is,

st − where

sT

∑

aφΤ

dn , dt

=

(12)

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,

dn = k ξ (1 − β ) ∑ dt aφ T

where n =

∑ Aeω

T

,C=

+

6

∑λC i =1

i

i

(13)

∑Beω

T

The complete solution for n is

n = no

β β −ρ

λρt β−ρ

e

−

ρ β −ρ

ρ−β

e

lp

(14)

Finally it is,

T = lP / (ka −1)

(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,

CR =∈ λ N =∈ N

0.693 T0.5

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,

N m in =

1.442C R m in T 0.5 atoms ∈

If the atomic weight of the radionuclide is A, the minimum mass of radionuclides in the sample is

M

m in

=

1 .4 4 2 C R m in T 0 .5 ∈ Na

(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.

V = 0.5B2 R2

q 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 waveâ€™s cycle. The force on the electron produced by the E-field is

Fa (t ) = 0.5eR

dB (t ) 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

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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, pp. 113-118, Malaysia, 29 Nov, 2010. Syed Bahauddin Alam, Hussain Mohammed Dipu Kabir, Md. Nazmus Sakib, Celia Shahnaz, Shaikh Anowarul Fattah, EM Shielding, Dosimetry Control and Xe(135)-Sm(149) Poisoning Effect for Nuclear Waste Treatment in 2010 IEEE International Power and Energy Conference, PECON 2010, pp. 101-106, Malaysia, 29 Nov,2010. Syed Bahauddin Alam, Hussain Mohammed Dipu Kabir, Md. Rishad Ahmed, A B M Rafi Sazzad, Celia Shahnaz, Shaikh Anowarul Fattah, Nuclear Waste Transmutation by Decay Energetics, Compton Imaging, Bremsstrahlung and Nuclei Dynamics in 2010 IEEE International Power and Energy Conference, PECON 2010, pp. 107-112, Malaysia, 29 Nov, 2010. Syed Bahauddin Alam, Hussain Mohammed Dipu Kabir, A B M Rafi Sazzad, Khaled Redwan, Ishtiaque Aziz, Imranul Kabir Chowdhury, Md. Abdul Matin, Can Gen-4 Nuclear Power and Reactor Technology be Safe and Reliable Future Energy for Developing Countries? In 2010 IEEE International Power and Energy Conference, PECON 2010, pp. 95-100, Malaysia, 29 Nov, 2010. Syed Bahauddin Alam, Md. Nazmus Sakib, Md Sabbir Ahsan, A B M Rafi Sazzad, Imranul Kabir Chowdhury, Simulation of Bremsstrahlung Production and Emission Process in 2nd International Conference on Intelligent Systems, Modelling and Simulation, ISMS2011, Malaysia, Phnom Penh (Cambodia) , 25-27 Jan, 2011. Syed Bahauddin Alam, Md. Nazmus Sakib, Md Sabbir Ahsan, Khaled Redwan, Imranul kabir, Simulation of Beta Transmutation by Decay Energetics in 2nd International Conference on Intelligent Systems, Modelling and Simulation, ISMS2011, Malaysia, Phnom Penh (Cambodia) , 25-27 Jan, 2011. Syed Bahauddin Alam, Md. Nazmus Sakib, A B M Rafi Sazzad, Imranul Kabir in Simulation and Analysis 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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11.Mathematics of Fusion Reactors and Energy Gain Factor Model

Published on May 13, 2012

baha_ece@yahoo.com Journal of Energy Technologies and Policy Keywords: Fusion Reactors, Energy Gain Factor Model, Neutron Life Time. 1

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