Abstract
In this study, based differential equations methods are used to solve equations because these methods are dependent on boundary value data more than other mathematical equations. We have calculated neutron flux, criticality and geometrical eigenvalue by using multi-group method and solving the neutron diffusion equation for finite and infinite cylindrical and spherical reactors in this study. For the calculation of the total neutron flux cross sections, we need the neutron diffusion equation. Thus, we have established the relationship between neuron flow and cross-section of neuron depending on neutron energy. Critical calculations have been made by comparing the results with MNCP (montecarlo n-partical) simulation methods. For necessary computer calculations, the programme, Wolfram-Matematica-7 has been used.
Highlights
Bessel differential equations are used for the calculations of neutron flux ( φ ) and criticality coefficient (K) and cylindrical geometric structure is taken into account as the reactor geometry
D is used for diffusion coefficient and L stands for diffusion length and is used for the calculation of diffusion length with the help of L2 = D where, a is absorbtion macroscopic cross section [13] [14] [15] [16]
Bessel differential equations are second order ordinary differential equations and they offer solutions in the cylindrical, spherical and polar coordinates and required physical parameters in the reactor can be obtained through the use of Bessel differential equations [8]
Summary
Bessel differential equations are used for the calculations of neutron flux ( φ ) and criticality coefficient (K) and cylindrical geometric structure is taken into account as the reactor geometry. We have calculated neutron flux, criticality and geometrical eigenvalue by using multi-group method and solving the neutron diffusion equation for finite and infinite cylindrical and spherical reactors in this study. In case of critical reactor, considering that the number of neutrons will not be changed by the time, the Boltzmann diffusion equation turns to: D.∇2Φ − ∑a Φ + S =0. The first term −D.∇2Φ stands for leakage neutrons, the second term ∑a Φ is for the absorption neutron and the last term S is for the neutron resources in the reactor core. The expression =S k∞ ⋅ ∑a ⋅Φ gives the number of absorption in reactor core and the expression ∑a Φ is in a unit time and volume
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