On an analytical evaluation of the flux and dominant eigenvalue problem for the steady state multi-group multi-layer neutron diffusion equation
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C. Ceolin
Abstract
In this work the authors solved the steady state neutron diffusion equation for a multi-layer slab assuming the multi-group energy model. The method to solve the equation system is based on an expansion in Taylor Series resulting in an analytical expression. The results obtained can be used as initial condition for neutron space kinetics problems. The neutron scalar flux was expanded in a power series, and the coefficients were found by using the ordinary differential equation and the boundary and interface conditions. The effective multiplication factor k was evaluated using the power method. We divided the domain into several slabs to guarantee the convergence with a low truncation order. We present the formalism together with some numerical simulations.
Kurzfassung
In dieser Arbeit lösen die Autoren die stationäre Neutronendiffusionsgleichung für mehrschichtige Bereiche und das Multi-Energiegruppenmodell. Das Verfahren, welches das Gleichungssystem löst, basiert auf einer Taylor-Reihenentwicklung und führt zu einem analytischen Ausdruck. Die erhaltenen Ergebnisse können als Anfangsbedingung für raumabhängige Neutronenkinetikprobleme verwendet werden. Der skalare Neutronenfluss wurde in eine Potenzreihe entwickelt und die Koeffizienten durch Verwendung der gewöhnlichen Differentialgleichung, der Randbedingungen und der inneren Anschlußbedingunen ermittelt. Der effektive Multiplikationsfaktor k wurde anhand der Potenzmethode bestimmt. Der Bereich wurde in mehrere Schichten unterteilt, um Konvergenz mit einem niedrigen Reihenabbruch zu garantieren. Der Formalismus wird zusammen mit einigen numerischen Simulationen vorgestellt.
References
1 Lemos, R. M.; Vilhena, M. T.; Silva, F. C.; Wortmann, S.: Analytical Solution for Two-group Diffusion Equations in a Multilayered Slab Using Laplace Transform Technique. Progress in Nuclear Energy50 (2008) 747–75610.1016/j.pnucene.2008.01.006Search in Google Scholar
2 Ganapol, B.: One-group steady state diffusion in a heterogeneous slab. Mathematics and Computers in Simulation80 (2010) 2142–215810.1016/j.matcom.2010.04.008Search in Google Scholar
3 Boyce, W. E.; DiPrima, R. C.: Elementary Differential Equations and Boundary Value Problems. John Wiley #38; Sons, Inc., USA (2001)Search in Google Scholar
4 Corliss, G.; Chang, Y. F.: Solving Ordinary Differential Equations Using Taylor Series, ACM Transactions on Mathematical Software8 (1982) 114–14410.1145/355993.355995Search in Google Scholar
5 Barrio, R.; Rodriguez, M.; Abad, A.; Blesa, F.: Breaking the limits: The Taylor series method. Applied Mathematics and Computation217 (2011) 7940–795410.1016/j.amc.2011.02.080Search in Google Scholar
6 Nahla, A. A.; Zayed, E. M. E.: Solution of the nonlinear point nuclear reactor kinetics equations. Progress in Nuclear Energy52 (2010) 743–74610.1016/j.pnucene.2010.06.001Search in Google Scholar
7 Nahla, A. A.: Taylor's series method for solving the nonlinear point kinetics equations. Nuclear Engineering and Design241 (2011) 1592–159510.1016/j.nucengdes.2011.02.016Search in Google Scholar
8 Mitchell, B.: Taylor series method for the solution of the point kinetics equations. Annals of Nuclear Energy4 (1977) 169–17610.1016/0306-4549(77)90062-7Search in Google Scholar
9 Duderstadt, J.; Hamilton, J.: Nuclear Reactor Analysis, John Wiley #38; Sons, USA (1976)Search in Google Scholar
10 Pollard, J. P.: AUS Diffusion Module POW checkout. 1-and-2- dimensional kinetics calculations, AAEC/387 (1977)10.1111/j.1365-2044.1977.tb11653.xSearch in Google Scholar
11 Quintero-Leyva, B.: The multi-group integro-differential equations of the neutron diffusion kinetics. Solutions with the progressive polynomial approximation in multi-slab geometry. Annals of Nuclear Energy37 (2010) 766–77010.1016/j.anucene.2010.02.005Search in Google Scholar
© 2014, Carl Hanser Verlag, München
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Articles in the same Issue
- Contents/Inhalt
- Contents
- Summaries/Kurzfassungen
- Summaries
- Technical Contributions/Fachbeiträge
- Decomposition analysis of the sodium void reactivity of the Korean sodium-cooled fast reactor
- Flow accelerated corrosion study in feeder pipes
- Analysis of the flow instability among channels of the OTSG in the naval craft NPP
- Experimental investigation of the MSFR molten salt reactor concept
- Investigation of neutronic behavior in a CANDU reactor with different (Am, Th, 235U)O2 fuel matrixes
- Atomistic nano-scale 3D simulations about effects of Cr percentage on the molecular dynamics parameters of Fe-9–12% Cr alloys at fusion reactor temperature conditions
- On an analytical evaluation of the flux and dominant eigenvalue problem for the steady state multi-group multi-layer neutron diffusion equation
- A finite volume approach to the problem of heat transfer in axisymmetric annulus geometry with internal heating element using local analytical solution techniques
- CO2 doped γ-irradiated hydroxyapatite for EPR dosimetry
- An investigation of the effect of the upper beryllium reflector on the moderator temperature coefficient of reactivity of miniature neutron source reactors
