Journal of Nuclear Engineering & Technology Review Article
Mathematical Derivation and Stability Analysis of a Coupled Neutron Kinetics–Thermal Feedback Model for Nuclear Reactor Safety
Abstract
Nuclear reactor safety assessment requires accurate characterization of the coupled interaction between neutron population dynamics and thermal feedback during steady-state operation and transient conditions. This study develops a mathematically derived coupled neutron kinetics–thermal feedback model for analysing reactor stability and transient behaviour. The formulation integrates point reactor kinetics with delayed-neutron precursor dynamics, fuel-temperature evolution, coolant- temperature variation, and temperature-dependent reactivity feedback. The reactor is represented as a nonlinear dynamical system in which neutron density determines volumetric heat generation, while temperature variations modify reactivity and consequently influence neutron multiplication. Equilibrium operating conditions are first derived analytically, followed by linearization around the nominal operating point using a Jacobian-state representation. Eigenvalue analysis is then employed to characterize local dynamic stability and identify the influence of reactivity feedback, thermal inertia, neutron generation time, and delayed-neutron fractions on system behaviour. A Lyapunov- based formulation is additionally established to provide a mathematical criterion for asymptotic stability. The derived framework enables systematic evaluation of reactor power excursions, temperature responses, damping characteristics, and recovery behaviour following small perturbations in reactivity or operating conditions. Parametric analysis demonstrates how stronger negative temperature coefficients improve suppression of power excursions, whereas reduced thermal capacity and weaker feedback can increase transient sensitivity. The proposed analytical framework provides a compact mathematical basis for reactor safety-oriented stability assessment and can support the preliminary analysis of advanced nuclear reactor control and protection strategies.
Keywords
References (11)
- Bureš L. Comparison of analytical models for frequency response and linear stability of molten salt reactors. Nuclear Science and Engineering. 2026 Feb 26;200(sup1):S503-26.
- Godinez FA, Quezada-Garcia S, Fernandez-Anaya G, Quezada-Tellez LA, Polo-Labarrios MA. Application of a Novel Variable-Order Fractional Neutron Point Kinetics Model for Nuclear Reactor Reactivity Transient Analysis. Fractals. 2026 Jul 30;34(01):2550098.
- Arul AJ, Obaidurrahman K, Prasad RR, Singh OP. Nuclear reactor kinetics. InPhysics of Nuclear Reactors 2021 Jan 1 (pp. 373-447). Academic Press.)
- Gupta HP, Trasi MS. Asymptotically stable solutions of point-reactor kinetics equations in the presence of Newtonian temperature feedback. Annals of Nuclear Energy. 1986 Jan 1;13(4):203-7.
- Aboanber AE, Hamada YM. Power series solution (PWS) of nuclear reactor dynamics with Newtonian temperature feedback. Annals of Nuclear Energy. 2003 Jul 1;30(10):1111-22.
- Nahla AA. An analytical solution for the point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback model. Progress in Nuclear Energy. 2009 Jan 1;51(1):124-8.
- Li H, Chen W, Zhang F, Luo L. Approximate solutions of point kinetics equations with one delayed neutron group and temperature feedback during delayed supercritical process. Annals of Nuclear Energy. 2007 Jun 1;34(6):521-6.
- Nahla AA, Zayed EM. Solution of the nonlinear point nuclear reactor kinetics equations. Progress in Nuclear Energy. 2010 Nov 1;52(8):743-6.
- Gyftopoulos EP. Theoretical and experimental criteria for nonlinear reactor stability. Nuclear Science and Engineering. 1966 Sep 1;26(1):26-33.
- Zhang T, Shen S, Hu L, Liu X. Multi-physics coupling rapid prediction method and autonomous control method for heat pipe nuclear reactors. Annals of Nuclear Energy. 2026 Feb 1;227:111927.
- Ryazanov VV. Stochastic Safety Limits and Scale-Dependent Power Fluctuations in Nuclear Reactors: A Critical Scaling Approach. arXiv preprint arXiv:2605.15283. 2026 May 14.