Neutron Point Kinetics with Ramp Reactivity

Analytical and numerical solutions of the NPKE with a linear reactivity ramp

Overview of the methodology

The present section contains the main equations used in the present work. Please, see the full manuscript for a more detailed description of them.

Neutron Point Kinetics Equations

The Neutron Point Kinetic Equations with a single group of delayed neutron precursors are given as:

$$ \begin{matrix}\dfrac{dn\left(t\right)}{dt}&=&\frac{\rho\left(t\right)-\beta}{\Lambda}n\left(t\right)+\lambda C\left(t\right)+q,\\\dfrac{dC\left(t\right)}{dt}&=&\frac{\beta}{\Lambda}n\left(t\right)-\lambda C\left(t\right).\\\end{matrix}\ . $$

Here \(\Lambda\) is the prompt generation time, \(\beta\) is the total delayed neutron fraction and \(\lambda\). In the present case, the ramp reactivity, \(\rho(t)\) has the following linear form:

$$ \rho(t) = a t + b, \quad a>0. $$

Modified Integration Method (MIM)

The proposed solutions were developed using a more efficient, formal and advanced Modified Integration Method (MIM), originally developed by Smets (1957). Such procedure consists of assuming that analytical solutions have the following form:

\[n\left(t\right)=\int_{\Omega}{\bar{n}\left(s\right)e^{st}ds},\ \ C_i\left(t\right)=\int_{\Omega}_i\left(s\right)e^{st}ds}\]

Integral Representation of \(n(t)\)

\[\begin{aligned} n(t)=\;& K_{1}\,e^{-\lambda t} \int_0^\infty e^{-y^{2}/2+\mathcal{E}_{3}(t)\,y}\, y^{\lambda\beta/a}\,dy \\[4pt] &+K_{2}\,e^{-\lambda t} \int_0^\infty e^{-y^{2}/2-\mathcal{E}_{3}(t)\,y}\,y^{\lambda\beta/a}\,dy \\[4pt] &+B_{p,1} \int_0^\infty \exp\!\left[ -\frac{1}{a}\left(\frac{\Lambda}{2}s^{2}+(\beta-b-at)s\right) \right] \,(s+\lambda)^{\lambda\beta/a}\,ds. \end{aligned}\]

For completeness, we recall that the auxiliary function \(\mathcal{E}_3(t)\) arises directly from the analytical reduction of the point kinetics equation with a linear reactivity. It is defined as

\[\mathcal{E}_3(t) = \frac{1}{\sqrt{2a\Lambda}} \left[ at + b - \beta + \frac{\Lambda\lambda}{2} \right],\]

which is obtained after completing the square in the exponent of the transformed equation for $n(t)$. The function $B_{p,1}$ appearing in the third integral term of \(n(t)\) follows analogously from the same reduction procedure and depends linearly on the parameters \(a\), \(b\), \(\beta\), \(\Lambda\), \(\lambda\) and \(q\), as detailed in the associated manuscript.

Delayed Neutron Precursor Concentration

Once \(n(t)\) is known, the concentration of delayed neutron precursors is obtained from

\[C(t) = C(0)\,e^{-\lambda t} + \frac{\beta}{\Lambda}\,e^{-\lambda t} \int_0^t e^{\lambda\tau}\,n(\tau)\,d\tau.\]

The numerical implementation of this integral is described in the Codes section.

RK4 Reference System

For validation purposes, a high-precision RK4 solver is applied directly to the NPKE system:

\[\frac{dn}{dt}=\frac{\rho(t)-\beta}{\Lambda}\,n(t)+\lambda\,C(t)+q, \qquad \frac{dC}{dt}=\frac{\beta}{\Lambda}\,n(t)-\lambda\,C(t),\]

with the same ramp reactivity \(\rho(t)=at+b\).

Further details and numerical parameters are provided in the Validation page.