Gas Diffusion

This model computes the intra-granular diffusion of fission gases using a spectral (Booth-type) formulation. Three solver configurations are available, allowing increasing physical detail: a single-equation model, a coupled solution–bubble model, and an extended formulation including high-burnup structure (HBS) effects.

The model also performs mass balance to determine the gas concentration at the grain boundary and accounts for optional immediate release.

Activation

The diffusion model is executed when a diffusion solver option is selected via the iDiffusionSolver input flag (values 1, 2 or 3). If the flag is not set or set to a disabling value, the routine returns without updating diffusion variables.

Inputs

The model uses:

  • iDiffusionSolver (input flag: 1, 2, or 3)

  • iGrainBoundaryBehaviour (input flag)

  • Time step (physics variable)

  • Gas production rate

  • Gas decay rate

  • Grain radius

  • Diffusivities (solution and bubble)

  • Resolution and trapping rates

  • Restructured volume fraction (for HBS case)

Solver options

The solver is selected through:

iDiffusionSolver = 1, 2, or 3

Case 1 — One equation (effective diffusion)

A single spectral diffusion equation is solved for the total intragranular gas:

\[\frac{\partial C}{\partial t} = D_{\mathrm{eff}} \nabla^2 C + S - \lambda C\]

where the effective diffusivity is:

\[D_{\mathrm{eff}} = \frac{R}{R+T} D_{\mathrm{fg}} + \frac{T}{R+T} D_{\mathrm{b}}\]

with:

  • \(R\) resolution rate

  • \(T\) trapping rate

  • \(D_{\mathrm{fg}}\) lattice diffusivity

  • \(D_{\mathrm{b}}\) bubble diffusivity

The solution is obtained via:

solver.SpectralDiffusion(...)

The total gas is partitioned into solution and bubbles according to the equilibrium fraction:

\[f_{\mathrm{eq}} = \frac{R}{R+T}\]

Case 2 — Two coupled equations

Separate spectral equations are solved for:

  • Gas in intragranular solution

  • Gas in intragranular bubbles

The solver:

solver.SpectralDiffusion2equations(...)

updates both populations simultaneously.

The total intragranular gas is:

\[C_{\mathrm{grain}} = C_{\mathrm{solution}} + C_{\mathrm{bubbles}}\]

Case 3 — Three coupled equations (Xe with HBS)

This configuration is available for xenon in UO₂ with high-burnup structure.

Three populations are solved:

  • Intragranular solution

  • Intragranular bubbles

  • HBS region

The solver:

solver.SpectralDiffusion3equations(...)

includes:

  • Separate diffusivities in UO₂ and HBS

  • Production rates in both regions

  • Resolution and trapping

  • Radioactive decay

  • A restructuring sweeping term

The sweeping term is defined as:

\[S_{\mathrm{sweep}} = \frac{1}{1 - f_r} \frac{\Delta f_r}{\Delta t}\]

where \(f_r\) is the restructured volume fraction.

Grain boundary mass balance

For each gas species (non-HBS systems), the grain-boundary concentration is computed by mass conservation:

\[C_{\mathrm{gb}} = C_{\mathrm{prod}} - C_{\mathrm{dec}} - C_{\mathrm{grain}} - C_{\mathrm{released}}\]

Negative values are set to zero.

Immediate release option

If:

iGrainBoundaryBehaviour = 0

all gas at the grain boundary is instantaneously released:

\[C_{\mathrm{released}} = C_{\mathrm{prod}} - C_{\mathrm{dec}} - C_{\mathrm{grain}}\]

and the grain-boundary concentration is reset to zero.

Numerical method

All diffusion equations are solved using a spectral (Booth-type) formulation with a finite number of modes:

\[C(r,t) = \sum_{k=1}^{N} a_k(t) \frac{\sin(k \pi r / a)}{r}\]

where:

  • \(N\) is the number of modes

  • \(a\) is the grain radius

The spectral formulation provides a meshless and computationally efficient solution compatible with online coupling to fuel performance codes.

Outputs

  • <Gas> in intragranular solution

  • <Gas> in intragranular bubbles

  • <Gas> in HBS (when Case 3 is active)

  • Grain-boundary concentration

  • Released inventory (when applicable)