Grain Boundary Venting

This model represents venting of gas from grain-boundary to the free volume (gap). This is modeled either via a fractional coverage-driven sigmoid representing the interconnection of bubbles or via correlations linked to the fuel’s open porosity. The venting strength is expressed through a venting probability, applied as a sink term to the grain-boundary gas inventories.

The implementation follows Simulation::GrainBoundaryVenting().

Activation and options

The model is enabled by:

  • iGrainBoundaryVenting (input option)

Options available: - 0: No venting, the model returns immediately. - 1: Sigmoid-based model, it represents bubble interconnection. - 2: Open porosity-based model (Claisse and Van Uffelen correlation). - 3: Open porosity-based model (under development).

Inputs

  • iGrainBoundaryVenting (input option)

  • Intergranular fractional coverage for option 1

  • Intergranular fractional intactness for option 1

  • Fabrication porosity for options 2, 3

  • <Gas> at grain boundary for each gas system

  • Time step (physics variable)

Outputs

  • Intergranular venting probability, P_vent

  • Open porosity for options 2, 3

  • Updated grain-boundary inventories (after applying the venting sink)

Key variables

The model uses: - Intergranular fractional coverage: initial fractional coverage at the grain boundary. - Intergranular fractional intactness: measure of boundary integrity (increment and final value). - Intergranular vented fraction: sigmoid-based fraction of vented boundary. - Open porosity: fraction of interconnected porosity reaching the exterior. - Intergranular venting probability: effective venting probability used in the sink term. - <Gas> at grain boundary for each gas system (Xe, Kr, He, radioactive gases, …).

Model Option 1: Sigmoid-based vented fraction

For iGrainBoundaryVenting = 1, a sigmoid function is used to compute the vented fraction. First, an internal sigmoid variable is defined as:

\[s = F^{n}\,\exp\!\left(-\Delta I\right)\]

where:

  • \(F^{n}\) is Intergranular fractional coverage (initial value),

  • \(\Delta I\) is the increment of Intergranular fractional intactness.

Then the vented fraction is computed as:

\[f_{\mathrm{vent}} = \left(1 + a\,\exp\!\left(-b\,(s-c)\right)\right)^{-1/a}\]

with constants (as in the code):

  • \(a = 0.1\) (screw_parameter),

  • \(b = 10.0\) (span_parameter),

  • \(c = 0.43\) (cent_parameter).

The effective venting probability is defined by mixing the intact and non-intact fractions:

\[P_{\mathrm{vent}} = (1 - I^{n+1}) + I^{n+1}\,f_{\mathrm{vent}}\]

where \(I^{n+1}\) is the final value of Intergranular fractional intactness.

Reference: Pizzocri et al., D6.4 (2020), H2020 Project INSPYRE.

Model Option 2: Open porosity-based (Claisse & Van Uffelen)

For iGrainBoundaryVenting = 2, the probability depends on the open porosity $p_{mathrm{open}}$:

\[P_{\mathrm{vent}} = 1.54 \sqrt{p_{\mathrm{open}}}\]

The open porosity is calculated from Fabrication porosity ($p_{mathrm{fab}}$) as:

  • If $p_{mathrm{fab}} < 0.050$: $p_{mathrm{open}} = p_{mathrm{fab}} / 20$

  • If $0.050 leq p_{mathrm{fab}} leq 0.058$: $p_{mathrm{open}} = 3.10 , p_{mathrm{fab}} - 0.1525$

  • If $p_{mathrm{fab}} > 0.058$: $p_{mathrm{open}} = p_{mathrm{fab}} / 2.1 - 3.2 cdot 10^{-4}$

Reference: Claisse and Van Uffelen, JNM, 466 (2015).

Application to grain-boundary gas inventories

For each gas system, the inventory is updated by a sink term:

\[C_{\mathrm{gb}}^{n+1} = C_{\mathrm{gb}}^{n} - P_{\mathrm{vent}}\,C_{\mathrm{gb}}^{n}\,\Delta t\]

Implemented as: solver.Integrator( <Gas> at grain boundary, -P_vent, <Gas> at grain boundary increment )