Stoichiometry Deviation

This model computes the time evolution of the fuel stoichiometry deviation (\(x\) in \(\mathrm{UO_{2+x}}\)) under oxidising conditions. The model is activated via the input option iStoichiometryDeviation and supports both semi-empirical oxidation kinetics (options 1–4) and a mechanistic Langmuir-based approach (options 5–6). The model also updates the fuel oxygen partial pressure and oxygen potential using the Blackburn thermochemical relation.

Activation

The model is executed only if iStoichiometryDeviation is enabled (non-zero). If the option is disabled, the function returns and no variables are updated.

Inputs

The model uses:

  • iStoichiometryDeviation (input option)

  • Grain radius (sciantix variable) to compute \(S/V\)

  • Temperature (history variable)

  • Steam pressure (history variable)

  • Gap oxygen partial pressure (sciantix variable)

  • Equilibrium stoichiometry deviation (sciantix variable, computed by the UO₂ Thermochemistry model)

  • Time step (physics variable)

Key definitions

Surface-to-volume ratio

For the default grain-scale case, the surface-to-volume ratio is computed as:

\[\frac{S}{V} = \frac{3}{a}\]

where \(a\) is the grain radius (Grain radius). In option iStoichiometryDeviation = 6 the value is overridden and a fixed \(S/V = 225~\mathrm{m^{-1}}\) is used.

Temperature cut-off

If \(T < 1000~\mathrm{K}\), the model does not evolve the deviation:

  • Stoichiometry deviation is kept constant

  • Fuel oxygen partial pressure is set to zero

Semi-empirical oxidation kinetics (options 1–4)

For iStoichiometryDeviation = 1..4, the model follows a Carter-and-Lay type oxidation law, driven by the difference between the equilibrium deviation \(x_{\mathrm{eq}}\) and the current deviation \(x\) and scaled by \((S/V)\sqrt{P_{\mathrm{H_2O}}}\).

In the implementation, this is written in linear source–decay form:

\[\frac{dx}{dt} = -k\,x + s\]

with:

\[k = \alpha(T)\,\sqrt{P_{\mathrm{H_2O}}}\,\frac{S}{V}, \qquad s = \alpha(T)\,\sqrt{P_{\mathrm{H_2O}}}\,x_{\mathrm{eq}}\,\frac{S}{V}\]

where \(\alpha(T) = A \exp(-Q/T)\) is the surface exchange coefficient. Different parameter sets (\(A, Q\)) are used depending on the selected option:

  • Option 1: Cox et al. / Carter and Lay

  • Option 2: Bittel et al. (re-analysed)

  • Option 3: Abrefah et al.

  • Option 4: Imamura and Une (includes a normalisation by 0.12 in the steam term)

Time integration is performed using the solver decay form:

\[x^{n+1} = \mathrm{Decay}\left(x^{n},\,k,\,s,\,\Delta t\right)\]

via solver.Decay(...).

Mechanistic Langmuir-based model (options 5–6)

For iStoichiometryDeviation = 5 and 6, the oxidation kinetics are based on a mechanistic Langmuir-type formulation (Massih, 2018). In SCIANTIX the evolution equation is rewritten in the form:

\[\frac{dx}{dt} = K\left(1 - \beta\,e^{\alpha x}\right)\]

where:

  • \(K\) depends on temperature, steam pressure, and surface-to-volume ratio

  • \(\beta\) depends on the gap oxygen partial pressure

  • \(\alpha\) is treated as a constant (in the implementation \(\alpha = 57/2\))

The new value is computed using Newton iteration:

\[x^{n+1} = \mathrm{NewtonLangmuirBasedModel}\left(x^{n},\,\{K,\beta,\alpha\},\,\Delta t\right)\]

via solver.NewtonLangmuirBasedModel(...).

Thermochemical update: fuel oxygen partial pressure

After updating Stoichiometry deviation, the model computes the fuel oxygen partial pressure using the Blackburn thermochemical relation:

\[\ln p = 2\ln\left(\frac{x(2+x)}{1-x}\right) + 108\,x^2 - \frac{32700}{T} + 9.92\]

and returns:

\[p_{\mathrm{O_2}} = e^{\ln p}\]

This is implemented by BlackburnThermochemicalModel(...) and stored in:

  • Fuel oxygen partial pressure

Fuel oxygen potential

Finally, the fuel oxygen potential is computed from the oxygen partial pressure:

  • if \(p_{\mathrm{O_2}} = 0\), then the oxygen potential is set to 0

  • otherwise:

\[\mu_{\mathrm{O_2}} = RT \ln\left(\frac{p_{\mathrm{O_2}}}{0.1013}\right)\]

In the implementation this is stored as:

  • Fuel oxygen potential

Outputs

The model updates:

  • Stoichiometry deviation

  • Fuel oxygen partial pressure

  • Fuel oxygen potential

References