High-burnup Structure Porosity

This model describes the evolution of the high-burnup structure (HBS) porosity and related pore statistics (pore density, pore volume, pore radius, and xenon inventory in pores). The model is activated through the input option iHighBurnupStructurePorosity and follows the implementation in Simulation::HighBurnupStructurePorosity().

Activation and option flag

The model is executed only if:

  • iHighBurnupStructurePorosity > 0

Option summary

  • 0: not considered (HBS porosity forced to zero).

  • 1: empirical porosity increase after a burnup threshold, capped at a maximum porosity.

Inputs

The model uses:

  • iHighBurnupStructurePorosity (input option)

  • Burnup (sciantix variable, used to trigger porosity increase and as integration increment)

  • Time step (physics variable, used for time integration of pore density and xenon inventory)

It also uses UO₂HBS matrix properties (via matrices["UO2HBS"]):

  • pore nucleation rate

  • pore resolution rate

  • pore trapping rate

State variables updated include:

  • HBS porosity

  • HBS pore density

  • HBS pore volume

  • HBS pore radius

  • Xe in HBS pores

  • Xe atoms per HBS pore

  • Xe in HBS pores - variance

  • Xe atoms per HBS pore - variance

Model formulation (case 1)

Porosity evolution

For option 1, the porosity evolution is controlled by:

  • a burnup threshold \(B_{th} = 50\) (MWd/kg),

  • a constant porosity increment rate \(k = 1.3\times 10^{-3}\),

  • an upper porosity cap \(\phi_{max} = 0.15\).

If the current porosity is below \(\phi_{max}\) and burnup is below \(B_{th}\), the porosity increment is zero. Otherwise the increment is set to \(k\). If porosity exceeds the cap, it is limited to \(\phi_{max}\).

In the code, porosity is integrated with burnup increment:

\[\phi^{n+1} = \phi^n + \dot{\phi}\,\Delta B\]

where \(\Delta B\) is the burnup increment in the current step (Burnup increment).

The porosity is finally limited to:

\[\phi^{n+1} \le \phi_{max} = 0.15\]

Pore density (nucleation and resolution)

If \(\phi > 0\), the pore number density is evolved with a decay-type equation (nucleation as source, resolution as decay):

\[N_p^{n+1} = \mathrm{Decay}\left(N_p^n,\; k_{res},\; k_{nuc},\; \Delta t\right)\]

where:

  • \(k_{res}\) is matrices["UO2HBS"].getPoreResolutionRate()

  • \(k_{nuc}\) is matrices["UO2HBS"].getPoreNucleationRate()

  • \(\Delta t\) is the current Time step

If \(\phi = 0\), then HBS pore density is set to 0.

Pore volume and radius

When \(N_p > 0\), pore volume is obtained directly from porosity:

\[V_p = \frac{\phi}{N_p}\]

The pore radius is then computed from volume as:

\[R_p = 0.620350491\,V_p^{1/3}\]

Pore interconnection by impingement

The pore number density is further updated to account for pore interconnection by impingement. A limiting factor is computed as:

\[L(\phi) = \frac{2 - \phi}{2(1-\phi)^3}\]

and the interconnection rate is defined as:

\[k_{int} = 4\,L(\phi)\]

Pore density is then updated with a binary interaction law driven by the pore volume increment:

  • solver.BinaryInteraction(Np, k_int, ΔV_p)

After this update, pore volume and radius are recomputed consistently from the updated pore density.

Xenon in HBS pores

The xenon inventory in HBS pores (average concentration, at/m³) is integrated as:

\[C_{Xe,p}^{n+1} = C_{Xe,p}^{n} + \left( 2k_{nuc} + N_p\,(k_{trap}-k_{res}) \right)\Delta t\]

implemented via solver.Integrator(...) with:

  • \(k_{nuc}\) = UO2HBS.getPoreNucleationRate()

  • \(k_{trap}\) = UO2HBS.getPoreTrappingRate()

  • \(k_{res}\) = UO2HBS.getPoreResolutionRate()

If \(N_p > 0\), atoms per pore is:

\[n_{Xe,p} = \frac{C_{Xe,p}}{N_p}\]

Variance of xenon in pores

A variance-like quantity is also evolved and converted into a per-pore variance:

\[\sigma_{Xe,p}^{2} = \frac{C_{Xe,p}^{var}}{N_p}\]

The source term used in the code includes trapping/resolution contributions and a nucleation contribution proportional to \((n_{Xe,p} - 2)^2\). The evolution is integrated with the time step using solver.Integrator(...).

References

  • Spino et al. (2006), data basis for the empirical porosity evolution.