Solution of a new model of fractional telegraph point reactor kinetics using differential transformation method

Abstract

A new model of fractional telegraph point reactor kinetics FTPRK is introduced to approximate the time dependent Boltzmann transport equation considering new terms that contain time derivative of the reactivity and fractional integral of the neutron density. Caputo fractional derivatives and fractional Leibniz rule are used for such derivation. Cattaneoequation is applied to overcome the flaw of infinite neutron velocity and to describe the anomalous transport. Effect of the new term on the neutron behaviour is discussed. The new model is applied to both TRIGA reactor and to commercial pressured water reactor of a Three Mile Island type reactor, TMI-type PWR. Results for step, ramp and sinusoidal excess reactivities with thermal hydraulic feedback are presented and discussed for different values of anomalous sub-diffusion exponent, the fractional order, 0 < µ ≤ 1. To maintain the reactor safe at start-up after insertion of step reactivity and based on the concept of prompt jump approximation, the FTPRK model is simplified and solved analytically by Mittag–Liffler function. Physical interpretations of the fractional order µ and relaxation time τ and their effects on the behaviour of the neutron population are discussed. Also, the effect of a small perturbation in the geometric buckling on the neutron behaviour is discussed for finite reactor core. The new model is solved numerically using the fractional order multi-step differential transform method MDTM. The MDTM constitutes an easy algorithm based on Taylor's formula and Caputo fractional derivative. Two theorems with their proofs are introduced to solve the fractional system. Two major disadvantages of the method about the choice of the fractional order values and the step size length are addressed. We present a procedure which enables us to solve the system with appropriate values of fraction orders.