Analytical Solution of a Gas Release Problem Considering Permeation with Time-Dependent Boundary Conditions

TL;DR

This paper presents an analytical solution for hydrogen diffusion in cylindrical materials considering time-dependent boundary conditions, enabling efficient and precise determination of material properties like solubility and diffusivity.

Contribution

It introduces an analytical method to solve diffusion equations with complex boundary conditions in gas release experiments, improving efficiency over numerical methods.

Findings

Explicit hydrogen flux calculation from analytical solution

Exact satisfaction of time-dependent boundary conditions

Accurate modeling of specimen-container interaction

Abstract

In this paper the determination of material properties such as Sieverts' constant (solubility) and diffusivity (transport rate) via so-called gas release experiments is discussed. In order to simulate the time-dependent hydrogen fluxes and concentration profiles efficiently, we make use of an analytical method, namely we provide an analytical solution for the corresponding diffusion equations on a cylindrical specimen and a cylindrical container for three boundary conditions. These conditions occur in three phases -- loading phase, evacuation phase and gas release phase. In the loading phase the specimen is charged with hydrogen assuring a constant partial pressure of hydrogen. Then the gas will be quickly removed by a vacuum pump in the second phase, and finally in the third time interval, the hydrogen is released from the specimen to the gaseous phase, where the pressure increase will be measured by an equipment which is attached to the cylindrical container. The investigated diffusion equation in each phase is a simple homogeneous equation, but due to the complex time-dependent boundary conditions which include the Sieverts' constant and the pressure, we transform the homogeneous equations to the non-homogeneous ones with a zero Dirichlet boundary condition. Compared with the time consuming numerical methods our analytical approach has an advantage that the flux of desorbed hydrogen can be explicitly given and therefore can be evaluated efficiently. Our analytical solution also assures that the time-dependent boundary conditions are exactly satisfied and furthermore that the interaction between specimen and container is correctly taken into account.