Analytic solutions and numerical method for a coupled thermo-neutronic problem

TL;DR

This work presents a simplified 1D model that couples neutron diffusion with enthalpy transport in a nuclear core, through a Doppler-like dependence of the diffusion coefficient on enthalpy. It develops two complementary solution strategies: (i) a numerical Crank–Nicolson method that reduces the coupled problem to a single scalar equation for the parameter $oldlambda$, ensuring a unique solution with proven convergence, and (ii) an analytical/semi-analytical framework using incomplete elliptic integrals to express the coupled density and enthalpy for various interpolations of the cross-section function $ extSigma(h)$. The results show that both approaches yield highly accurate solutions and that the benchmark quantity $k$ can vary considerably with the interpolation of $ extSigma$, even when neutron flux profiles are similar. These methods provide a rigorous, efficient means to analyze highly coupled thermo-neutronic systems and enable sensitivity studies via analytic expressions for $I_{oldlambda}$ and related quantities.

Abstract

We consider in this contribution a simplified idealized one-dimensional model in a nuclear core reactor coupling the diffusion equation on the neutron flux withthe enthalpy equation for the water which collects the heat produced by this idealized nuclear core. These equations are coupled through the dependency of thecoefficients of the diffusion equation in terms of the enthalpy. We propose a numerical method treating globally the coupled problem for finding its unique solution.Simultaneously, we use incomplete elliptic integrals to represent analytically the density of neutrons and the enthalpy in the fluid. Both methods lead to the samesolution with high accuracy. However, another quantity, generally used as a benchmark for comparing results, depends considerably on the approximation used forthe coefficients of the diffusion equation.

Figures (8)

• Figure 1: Homographic transformation for the computation of the integral $\, \int_0^{\,1} \!\! {{{\rm d} h}\over{\sqrt{\psi_\lambda (h)}}}$; case $\, p < 0 < 1 < g$.
• Figure 2: Homographic transformation for the computation of the integral $\, \int_0^{\,1} \!\! {{{\rm d} h}\over{\sqrt{\psi_\lambda (h)}}}$; case $\, 0 < 1 < p < g$.
• Figure 3: Homographic transformation for the computation of the integral $\, \int_0^{\,1} \!\! {{{\rm d} h}\over{\sqrt{\psi_\lambda (h)}}}$; case $\, p < g < 0 < 1$.
• Figure 4: Contribution $\, \psi^0 \,$ to the function $\, \psi_\lambda$. The case $\, \alpha>0 \,$ is on the left and the case $\, \alpha < 0 \,$ is splitted into two figures: two complex roots in the middle and three real roots on the right.
• Figure 5: Contribution $\, \psi^1 \,$ to the function $\, \psi_\lambda$. The case $\, \beta>0 \,$ is splitted into two sub-cases: two complex roots on the left and three real roots in the middle; the case $\, \beta<0 \,$ is presented on the right.

Theorems & Definitions (38)

• Lemma 2.1
• Theorem 3.1
• proof
• Proposition 3.1
• proof
• Theorem 3.2
• Proposition 3.2
• proof
• Lemma 3.1
• proof