Hydrogen localization under thermal gradients in hydride forming metals

TL;DR

The paper develops a stochastic, fluctuation-aware thermodynamic framework for hydrogen transport under thermal gradients in hydride-forming metals, focusing on the α–δ phase transition in Zircaloy-4. It derives coupled heat and mass transport relations and a Boltzmann-type localization model where hydrogen in solid solution follows $n_α = n_u e^{-(β_0-β)Q^*}$ and phase precipitation is governed by the random enthalpy of mixing via $n = [1-\mathcal{P}(x)]n_α + \mathcal{P}(x)n_δ$ with $\mathcal{P}(x)$ Gaussian. The method is applied to Zircaloy-4 cladding under axial/radial gradients, implemented in a standalone solver, and yields hydrogen localization in colder regions (Soret effect) with δ-phase enrichment near inter-pellet gaps, in qualitative agreement with experimental observations. The approach can be integrated into fuel-rod codes to better predict hydride evolution under service conditions. Key results show Case B (larger temperature gradients) producing stronger δ-hydride localization than Case A, while parameter sensitivities (e.g., $Q^*$, $\boldsymbol{\upsigma}_H$) largely modulate the extent rather than the qualitative trend.

Abstract

Migration of hydrogen and hydride formation under thermal gradient leads to hydrogen redistribution in certain metals. These metals include zirconium, titanium, hafnium and their alloys with tendency to form hydrides. A computational method for hydrogen localization in such metals is presented. The method utilizes the heat flux in a steady state to compute temperature distribution (as input), and hydrogen mass flux under temperature gradient to determine hydrogen distribution both in solid solution and in the hydride phase in a two-dimensional setting. Hydrogen precipitation to hydride is determined by a solid solubility relation with an exponential function of the enthalpy of mixing per a van 't Hoff relation. The enthalpy of mixing is treated here as a stochastic variable subject to thermodynamic fluctuations. Henceforth, the Einstein-Boltzmann fluctuation theory is adapted to calculate the spatial distribution of hydrogen in solid solution and in the hydride phase. Hydrogen concentration gets localized in the colder region of the body (Soret effect). We apply the model to the case of a zirconium alloy, Zircaloy-4, which is a material for fuel cladding utilized in pressurized water reactors. Cladding continuously picks up hydrogen due to Zr oxidation during reactor service, which we take into account. Our calculated results, hydrogen concentration profiles are comparable to experimental observations reported in the literature.

Figures (7)

• Figure 1: Fuel pellet and cladding tube geometry schemata where $T_\mathrm{ci}$ and $T_\mathrm{co}$ denote cladding inner and outer surface temperatures, respectively.
• Figure 1: The probability density function $\mathcal{P}(x)$ according to Eq. eqn:error-fun-sol.
• Figure 2: An schematic geometry of missing pellet piece (chip) defining the chip height ($z_\mathrm{MC}$) and its depth ($r_\mathrm{MC}$).
• Figure 3: Calculated total hydrogen uptake concentration in the cladding $n_g$ as a function of time; cf. \ref{['sec:h-uptake']}.
• Figure 4: Cladding tube inner surface temperature $T_{ci}(z)$ close to the inter-pellet gap at $z=0$ for case A (a), case B (b). Temperature distribution (kelvin) across cladding wall (0.57 mm) close to the inter-pellet gap for case A (c) and case B (d). The radial element 1 is at the cladding outer radius and element 80 is located at the cladding inner radius.