Analytical solution of the Telegraph Point Reactor Kinetics model during the cold start-up of a nuclear reactor

We solve the new model of the Point Reactor Kinetics (PRK) equations developed based on the Telegraph approximation of the neutron transport equation analytically for a linear insertion of reactivity typically introduced by lifting the control rods discontinuously and manually during the cold start-up of a subcritical nuclear reactor. The Telegraph model introduces a new parameter called the Relaxation Time (τ) and we study its impact on the analytical solution for this case of reactivity insertion and for several speeds of lifting the control rods. We find that as the Relaxation time increases, the solution response is relaxed behind that of the diffusion model which was solved earlier in the literature for the same case. On the other hand, as the Relaxation time tends to zero, we find that the response of the neutron density tends to that of the diffusion case yielding a verification for the new proposed solution. Moreover, when reducing the Control Rod lifting speed, the effect of the relaxation time and hence the Telegraph approximation is reduced and it approaches that of the diffusion. We discuss Both mathematical and physical reasons for the cause of these behaviors.