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Numerical Solution of Point Kinetic Equations for Nuclear Reactor
Patrick Bruneel, Ryan Green, Alex Scholtes
Advisor: Muhammad Usman, Ph.D.
Abstract: In this work, we will solve nuclear reactor point-kinetic equations. There are many numerical techniques to solve
these models. We will solve this coupled system of differential equations using the simplest methods, such as the Taylor
Series, Euler, and Runge-Kutta methods using MatLab program. We will then compare the accuracy of each method.
Introduction
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Nuclear fission is a process where the nucleus splits into
smaller fragments (fission products), hence releasing
energy.
The point kinetic equation help us to solve mathematical
models for neutron population density and the
concentration of delayed neutron precursors
To solve the point kinetic equations, we used Taylor
Series, Euler, and Runge Kutta methods to compare the
accuracy and efficiency of the first ordered coupled
differential equations.
Euler method is one of the simplest methods for
approximating differential equations.
Taylor series is a more theoretical than practical method.
This means that it does not differ greatly Euler’s except
expands upon the base derivative of this method with
additional order derivatives to find a more accurate
answer.
Runge-Kutta method is known as a highly accurate
method for solving different ordered differential
equations.
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Euler
Taylor Series
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Runge Kutta
Analysis
The Runge-Kutta Method seemed to have failed
with a varying reactivity. It appears to be a
limitation within the MatLab coding.
When running the first two tables, the RungeKutta Method had little error in finding the final
value of neutron population density.
The Euler method was efficient for solving the first
table but had trouble computing answers using a
higher reactivity, rho, for tables two and three.
The Taylor Series Method was quite consistent. It
managed to find answers slightly better than the
Euler Method due to its expanded form that
settles on a number closer to the exact answer.
Conclusion
Using the three mentioned mathematical
methods, we found the benefits and drawbacks of
using one versus another.
The Runge-Kutta Method seemed to be the most
effective way to find the answers for the systems
of the four tables.
Euler was a good method to use for how long it
took compared to how accurate it was but could
use slightly more depth to its differentiation.
Lastly, the Taylor Series seemed to get the best of
both worlds by find answers better than the Euler
Method and running slightly faster than the
Runge-Kutta Method.
References
1. A First Course in Differential Equations. Dennis Z. Gill.
2. A Taylor series solution of the reactor point kinetics equations, David
McMahon and Adam Pierson
Department of Nuclear Safety Analysis,Sandia National Laboratories,
Albuquerque, NM 87185-1141