The point reactor kinetics model is a reduced-order model of the physics of a nuclear reactor. It describes the dynamic behavior of the neutron density that of a nuclear reactor that includes the effects of 6 groups of delayed neutron precursors. It can help with the intuitive understanding of reactor dynamics. The feedback reactivity of the point reactor can be explained by doppler effects. Point kinetics separate the flux behavior into two factors, the first depending on space and time, and the second, only depending on time. The immediate neutron flux is generated from the fission event, and the delayed neutron power comes from the radioactive decay of the fission products. An 8th order Adams-Bashforth-Moulton method can be used to solve the point kinetics equation with adiabatic doppler effects. It also compensates for the response of a ramp reactivity. The assumptions that allow us to reduce the order of this model with Fick’s Law are: the angular flux is anisotropic, the neutron source term is isotropic, and there is a signal neutron energy. The reactor power is proportional to the neutron flux[^7]
$$P_{th}(t)\approx\bar{n}$$
Where \(\bar{n}\) is the neutron flux. [^7] In other words, the reactor power is proportional to the ratio between the neutron flux and the nominal average neutron density, multiplied by the nominal reactor power.[^7]
$$P_{th}(t)=\frac{\bar{n{}}}{N_{0,N}}$$
The point kinetic equations are
$$\frac{dn}{dt}=\frac{\rho(t)-\beta}{\Lambda}n(t)+\sum^m_{i=1}\lambda_iC_i(t)$$
$$\frac{dC_i}{dt}=\frac{\beta_i}{\Lambda}n(t)-\lambda_iC_i(t)$$
With \(\beta_i\) is the delayed fraction of neutrons for the i-th group, \(n(t)\) is the number of delayed neutrons. \(\lambda_i\) being the decay constant of the i-th group of delayed neutrons, and \(C_i\) is the precursor concentration of the i-th group of delayed neutrons. \(\Lambda\) is the mean neutron generation time. \(\rho\) is the reactivity. Typical initial conditions for this problem are
$$\begin{matrix}
n(0)=n_0\
c_i(0)=\frac{\beta_i}{\lambda_i\Lambda}
\end{matrix}$$
Any numerical integration scheme that cannot accurately approximate the point kinetics model cannot be used for space-time reactor kinetics. The higher-order methods have better prediction accuracies. You can ignore the reactivity effects if you are modeling transients below minute timescales.[^6] The point reactor kinetics equations are derived from the diffusion or transport equations that describe a neutron population.[^6] These equations are identical when represented in terms of reactor power rather than neutrons.[^6]
Nuclear Reactor Reactivity – change in multiplication factor of the system
Power Series Solution – method for solving the point reactor kinetics.
Piecewise Constant Approximation – another method for solving point reactor kinetics
New Analytical Method – reactor kinetics with fuel temperature feedback
Convergent Accelerated Taylor Series – used nonlinear reactivity for accurate solutions to the point kinetics equations
Enhanced Piecewise Constant Approximation
ITS2 Method – obtains exact solution to point kinetic reactor equations
ADM8 – 8th order Adams-Bashworth-Moulton used as integration methods
Fick’s Law
Doppler Effect of Thermal Reactors
Ramp Reactivity Insertions
Point Kinetics Decomposition Method – method for solving point kinetics
RELAP5– nuclear reactor safety code
RELAP-7 – state-of-the-art reactor safety program
Point Kinetics Reactivity – reactivity model
Delayed Neutron Precursor – uses 6 groups for simulations
Single flow channel reactor core model
Exponential Matrix Technique – used to solve the point-kinetic equations
Point Reactor Kinetics with Startup Source – identical equations + added source term
Fission Products – modeled as 6 groups.
TRACE – trace started with a point kinetics model
Thermal Hydraulic Reactor Model – tests reactor kinetics with an entire reactor simulation
TRIGA MK 2 Neutronics Model – uses 6-precursor point kinetics
Nuclear Submarine Model – uses the point kinetics equations for the reactor
Sources
- “On the numerical solution of the point reactor kinetics equations,” Nuclear Engineering and Technology, vol. 52, no. 6, pp. 1340–1346, Jun. 2020, doi: 10.1016/j.net.2019.11.034.[^1]
- E. Boeke, “Runge-Kutta Methods to Explore Numerical Solutions of Reactor Point Kinetic Equations”.[^2]
- _20043611[^3]
- H. Zhang, L. Zou, D. Andrs, H. Zhao, and R. Martineau, “POINT KINETICS CALCULATIONS WITH FULLY COUPLED THERMAL FLUIDS REACTIVITY FEEDBACK,” 2013.[^4}
- M. S. El-Genk, T. Schriener, A. Hahn, and R. Altamimi, “A Physics-based, Dynamic Model of a Pressurized Water Reactor Plant with Programmable Logic Controllers for Cybersecurity Applications”.[^5]
- Matthew Johnson, Scott Lucas, and Pavel Tsvetkov, “Modeling of Reactor Kinetics and Dynamics,” INL/EXT-10-19953, 989898, Sep. 2010. doi: 10.2172/989898.[^6]
- W. Chen, T. Liang, and V. Dinavahi, “Comprehensive Real-Time Hardware-In-the-Loop Transient Emulation of MVDC Power Distribution System on Nuclear Submarine,” IEEE Open J. Ind. Electron. Soc., vol. 1, pp. 326–339, 2020, doi: 10.1109/OJIES.2020.3036731.[^7]
Backlinks:
Delayed Fission Neutrons
Lumped Parameter Model
Nuclear Power Generation
Nuclear Reactor
Numerical Integration Methods
Ordinary Differential Equations
Pressurized Water Reactors
Reactor Dynamics
Reactor Kinetics
Reduced Order Models
Systems That Runge-Kutta Integrators Handle Poorly