Theory

Overview

In this section the theoretical model is summarized, a detailed description is presented in the initial FLUNED paper (De Pietri et al., 2023).

The concentration of a radioactive species \(N_\alpha\) is treated as a passive scalar that does not alter the underlying flow or transport processes. In addition, only radioisotopes generated by single-step reactions and with constant parent concentration are considered.

With these assumptions, the equations implemented in the FLUNED solver are reported here.

Governing Conservation Equation

(1)\[\frac{\partial N_\alpha}{\partial t} + \nabla \!\cdot\! \bigl(\mathbf{U}\,N_\alpha\bigr) \;=\; \nabla \!\cdot\!\bigl(D_\alpha\,\nabla N_\alpha\bigr) + S_\alpha(N,\phi)\]

  • \(\mathbf{U}\) - coolant velocity vector

  • \(D_\alpha\) - molecular diffusivity of species \(\alpha\)

  • \(S_\alpha\) - volumetric source term (neutron interactions + decay)

Reynolds Decomposition

The Reynolds decomposition for turbulent flows is considered:

(2)\[N_\alpha(\mathbf{x},t) \;=\; \langle N_\alpha \rangle \;+\; N_\alpha'\]

By substituting (2) in (1) gives:

(3)\[\frac{\partial\langle N_\alpha\rangle}{\partial t} + \nabla\!\cdot\!\bigl(\langle\mathbf{U}\rangle\,\langle N_\alpha\rangle\bigr) + \nabla\!\cdot\!\langle \mathbf{u}' N_\alpha' \rangle \;=\; \nabla\!\cdot\!\bigl(D_\alpha\,\nabla\langle N_\alpha\rangle\bigr) + \langle S_\alpha(N,\phi)\rangle\]

Turbulent Closure - Gradient Diffusion Hypothesis

(4)\[\langle \mathbf{u}' N_\alpha' \rangle \;=\; -\,D_t\,\nabla\langle N_\alpha\rangle \;=\; -\,\frac{\nu_t}{\mathrm{Sc}_t}\;\nabla\langle N_\alpha\rangle\]

  • \(D_t\) - turbulent diffusivity

  • \(\nu_t\) - turbulent viscosity

  • \(\mathrm{Sc}_t \approx 0.7\) - turbulent Schmidt number

  • Valid only under isotropic-turbulence assumptions.

General Source Term

(5)\[\sum_{i\neq\alpha} (\langle\phi\sigma\rangle_{i\rightarrow\alpha} + \lambda_{i\rightarrow\alpha})\,N_i(t) \;-\; \sum_{i\neq\alpha} (\langle\phi\sigma\rangle_{\alpha\rightarrow i} + \lambda_{\alpha\rightarrow i})\,N_\alpha(t)\]

  • First (positive) sum - production of \(\alpha\) from parents \(i\).

  • Second (negative) sum - loss of \(\alpha\) via decay or transmutation.

  • \(\lambda_{x\rightarrow y}\) - instantaneous decay constant.

  • \(\langle\phi\sigma\rangle\) - neutron-flux-weighted reaction rate.

Simplified Source Term (Single-Step Transmutation + Decay)

Under the study’s assumptions (constant parent isotopes, single-step generation + decay):

(6)\[\langle S_\alpha(N,\phi)\rangle \;=\; \mathrm{RR}_\alpha(\phi)\;-\;\lambda_\alpha N_\alpha\]

  • \(\mathrm{RR}_\alpha\) - neutron-induced reaction rate producing isotope \(\alpha\).

  • \(\lambda_\alpha\) - decay constant of \(\alpha\).

Laminar-Flow Limit

For laminar regimes, (3) reduces to (1) with

  • \(D_t = 0 \quad (\nu_t = 0)\)

  • all variables in their instantaneous form (no ensemble averaging).