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One of the technological challenges for hydrogen materials science is the currently active search for structural materials with important applications (including the ITER project and gas-separation plants). One had to estimate the parameters of diffusion and sorption to numerically model the different scenarios and experimental conditions of the material usage (including extreme ones). The article presents boundary value problems of hydrogen permeability and thermal desorption with dynamical boundary conditions. A numerical method is developed for TDS spectrum simulation, where only integration of a nonlinear system of low order ordinary differential equations is required. The main final output of the article is a noise-resistant algorithm for solving the inverse problem of parametric identification for the aggregated experiment where desorption and diffusion are dynamically interrelated (without the artificial division of studies into the diffusion limited regime (DLR) and the surface limited regime (SLR)).

Studies on the interaction of hydrogen isotopes with structural materials are mainly necessitated by problems in the energy industry, metal protection from hydrogen corrosion, and the design of chemical reactors [

One had to estimate the parameters of diffusion and sorption to numerically model the different scenarios and experimental conditions of the material usage (including extreme ones) and identify the limiting factors. Some particular problems of the hydrogen materials science related to the topic of this study were presented and investigated in [

Experimental practices usually employ various modifications of the penetration method and TDS. The results of measurements depend both on the unit design features and on the procedure of preparing samples for hydrogen permeability testing. A successive use of various methods often causes, for example, impurities to appear on the sample surface, which significantly affects the reproducibility of the results. These data are the input for the inverse problems of parametric identification, which are sensitive to the level of different errors. It is therefore advisable to aggregate experiments to improve the accuracy and informative value of the measurements. We suggest the following set-up of the “cascade” experiment.

A membrane heated to a fixed temperature served as the partition in the vacuum chamber. Degassing was performed in advance. A sufficiently high pressure of hydrogen gas was built up in steps at the inlet side. The penetrating flux was determined by mass-spectrometry in the vacuum maintained at the outlet side. This is the penetration method. Its advantage is a reliable determination of the diffusion coefficient by the Daines–Berrer method (based on the so-called lag time). It allows distinguishing between the bulk and the surface processes in the model, keeping in mind that surface parameters are significantly more difficult to estimate. When the steady state level of the penetrating flux is registered, we increase the inlet pressure and wait until a new steady state value is established. Using (at least) three pressure jumps at the inlet side we record the steady state flux values at the outlet side, thus determining “the degree of rectilinearity” of the isotherm. Then the pumping for vacuum building is stopped and the experiment proceeds as the “communicating vessels” method. When pressure values become nearly equal (the sample is almost uniformly saturated with hydrogen) it is possible to turn off the heating, create the vacuum at both sides of the membrane, and begin slowly reheating the sample (TDS-experiment). In addition, there is no depressurization of the diffusion cell and the sample surface remains uncontaminated by additional impurities. We will clarify the details of the aggregated experiment stages as we describe the method of solving the inverse problem of parametric identification. An important consideration is the uniqueness of the parameter estimates of the investigated model. Mathematicians are often reproached for “fascination with uniqueness theorems”. But after all, in justifying the choice of, for example, structural materials for the ITER project, the results obtained on thin laboratory samples are extrapolated to “walls”. Uniqueness allows for a correct recomputation.

Papers [

The main result of the paper is the method of parametric identification from experimental data. The difficulties of inverse problems solving in mathematical physics are well known. There is extensive mathematical literature and a number of specialized journals (inverse problems, ill-posed problems, etc.). In experimental practice, the inverse problem of multiparameter estimation is reduced to the one-factor-at-a-time method for DLR and SLR. In real life, however, a material is used in the presence of a dynamic “surface-bulk” interplay. Thoroughly elaborated techniques are available for estimation of the diffusion coefficient. The determination of desorption and dissolution parameters is far more complex (unless the temperature is artificially lowered to “turn off” processes in the bulk). The paper presents an algorithm allowing the estimation of desorption and dissolution coefficients where diffusion and surface processes interact intensively.

Let us briefly describe the experiment. A sample of a structural material preheated to a fixed temperature acts as a vacuum chamber barrier. The sample degassing is performed in advance. At the initial time moment, pressure is built up at the inlet side by puffing of a portion of molecular hydrogen. The declining pressure in the input chamber and increasing pressure in the output chamber are measured.

Consider hydrogen transfer through the sample (

Initial data are determined by the fact that the sample had been preliminarily degassed:

Nonlinear boundary conditions are derived from the material flux balance:

Hereinafter,

Let us clarify the experimental conditions. The volumes

It now remains to find the magnitudes of

Within the experimental unit the membrane is situated in a tube (which is heated to a predetermined temperature) between the inlet and the outlet chambers. The tube diameter is large enough to consider the equality of pressures as the criterion of thermodynamical quasi-equilibrium between the gas in the tube and in the chambers. The membrane temperature should be taken for the formula for the kinetic constant

It is clear from physical considerations that a quasi-stationary state is quickly established when the membrane is thin and the material has a sufficiently high hydrogen permeability coefficient: the diffusant concentration distribution is practically linear with respect to the thickness. In this sense, the results of numerical modelling based on the “general” model (the presented boundary-value problem) confirm its adequacy. Since near-to-surface concentrations cannot be measured, the Richardson approximation is usually used in practice to analyze the penetrating flux:

Differentiate (

The change of variables in (

Let us formulate step-by-step the numerical algorithm of modelling the pressures

(1) We fix

(2) The system of equations

(3) We numerically integrate the ODE system (

Computational experiments show that the model curves almost coincide (at

Observe the difference from the quasi-equilibrium model (the Richardson approximation), where the only parameter for approximation of the experimental pressure is the complex

The proposed model is adapted to the experimental conditions and the data range for alloys based on V group metals with high hydrogen permeability, in particular, data for vanadium alloys which are presented in [

Next we express the solutions of the standard boundary value problems with Dirichlet boundary conditions corresponding to the jumps of the inlet pressure.

From the computational point of view it is convenient to introduce the dimensionless time

The function

The establishment of fluxes

The result of connecting the stages into a single “experimental” curve of the penetrating flux

During the real experiment, the gas pressure inside the outlet volume

Only the operation of integration is needed to calculate the lag time (see below):

When the steady state permeability value is established during the penetration experiment, continuous pumping at the outlet and maintenance of constant pressure at the inlet are stopped. The aggregated experiment moves to the stage of “communicating vessels”: inlet pressure declines and outlet pressure grows (

We numerically solve the initial boundary problem of hydrogen permeability:

The membrane temperature is taken in the dependences of

Dynamics of pressures

For completeness, we briefly describe the method of estimating the diffusion coefficient proposed by Daines-Berrer. The curve of the flux

With a new zero time reference, integrating the expression

In experimental practices it is common to draw and analyze the isotherm, i.e., the curve of the steady state penetrating flux

Let us analyze the steady state flux balance equation:

A graphic illustration (using a minimal required set of the calculated model

Extrapolation of the isotherm of

Note that the obtained initial approximations are in good agreement with the original “forgotten” parameters as shown in Table

Original value | Approximate value | |
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Table

Thus, the first stage of the aggregated experiment is perceived as a preliminary estimation of the

Pressure values inside the chambers tend to equalize with time. The uniform equilibrium saturation

The thermal desorption flux of hydrogen from the sample is measured by mass spectrometer. What additional information can be obtained from the TDS experiment? The dependence

The heating

The surface is a significant potential energy barrier (see [

Model (

We shall hereinafter use a contracted notation for simplicity

We can take into account different channels of diffusion, but the information content of the TDS experiment is limited. Therefore, the coefficient

Seeking a write-up of the TDS peaks set, it is handier to use the following model:

Let us take a brief look at the equilibrium ratios at the accepted detail level of modeling. We assume that pressure and temperature are constant. Formally, equilibrium is characterized by all derivatives being equal to zero. Keeping in mind the extensively used Sieverts’ law, we shall observe proportionality to the

(1) On the surface the following ratio is applied:

(2) In the “surface–bulk” equilibrium we have (

(3) For definiteness, we take into account only one type of traps (

The “saturation-degassing” experiment provides information about a general average concentration

The curves

When experimentally estimating the hydrogen permeability of structural materials, Richardson approximation is often used for the penetration flux density:

Hence, we can estimate the

Identification of TDS spectra is required not only to reveal the causes of different thermal desorption peaks, but also to enable numerical extrapolation and generalization of the results received for laboratory samples (

The comparison of simulated and experimental TDS spectra with a focus on parametric identification requires only the surface concentration (

The accepted TDS degassing model is

For more comfortable modeling we turn to dimensionless variables using substitution rules:

We specify the factor

Hereafter, let us be guided by a maximum limit of surface concentration at around

In the meantime, this a priori restriction (arising from the assumption that stationary “balls” are ordered geometrically on a plane) is highly questionable. For a dynamics model, it appears that the concentration threshold

The functional differential equation of thermal desorption (

Let us run some transformations using the theory of Jacobi theta functions to explicitly extract the integrable singularity. We consider the presentation (

To be specific in the paper, we use data for nickel and steel (12Cr18Ni10Ti) [^{2}/s, ^{2}/s, ^{3},

The main role in the degassing dynamics belongs to quadratic desorption. We therefore approximate the integral term in (

Functions

Saturation degree.

Effect of defects

Effect of the surface

The greatest contribution to the integral is made by the value

The sought function

Qualitatively, thermal desorption spectra of metal structural materials have a typical form. Figure

TDS-spectra.

Let us focus on TDS spectra for tungsten because these spectra are qualitatively different from the previous ones. The aforementioned parameter values “for tungsten” are, of course, formal. Besides microimpurities, the parameters depend on how the sample surface was treated. This is especially true because in practice a plate is thin, the volume is small, and the effect of surface processes is more vivid. This is one of the reasons for such a high variation of the quantitative estimates of hydrogen permeability parameters. Another reason (but not the last) is the following. Different models are used for experimental data treatment (although coefficients formally have the same name). For the model data accepted here, a narrow splash followed by a lengthy movement to the second peak (less visible) was observed. In this model the sample has no high-capacity traps (standard diffusion equation), but more detailed consideration is given to the surface (see (

Figure

Dynamics of surface processes and diffusion.

Surface parameters

Diffusion afflux to the surface

Let us briefly present the numerical modeling

Set the parameters

The low order ODE system of type (

The graph

The presented algorithm of numerical modeling allows quick scanning of different scenarios and operating conditions of a material (including the heating law and extrapolation of the results with

Since the function

Note that the right-hand sides of the equations now depend only on one estimated dissolution parameter

Formally, we prolong the straight line segment until it crosses the coordinate axes. The crossing with the

The algorithm is laborious due to iterative use of the procedure of numerical solving of the initial problem for the ODE system. It requires some effort and familiarity with mathematical packages, but it is much easier and quicker to use the standard built-in operation than to perform iterative solution of the original nonlinear boundary value problem with three varying parameters.

The presented linearization method is illustrated in Figure

Estimation of desorption and dissolution parameters.

The initial value of

It is advisable to aggregate the thermal desorption (TDS) and penetration experiments (with and without vacuum pumping) to make the measurements substantially more informative for further estimation of the hydrogen permeability parameters and to improve the accuracy of parameter identification. This paper suggests a cascade experiment technique and the corresponding mathematical software.

The identification algorithm uses only integral operators thus ensuring the noise resistance of experimental data treatment. The penetration method with vacuum pumping is characterized by a significant measurement error, and data on the penetrating flux are required (and this, in turn, requires a more accurate determination of the vacuum system characteristics). The model of the dissolved hydrogen concentration jump at the inlet side is not very precise either. We are brought to a conclusion that the “communicating vessels" stage, where molecular hydrogen pressures are measured over a long time, is characterized by a much higher accuracy of measurements.

The first stage of the aggregated experiment is perceived as preliminary estimation of the diffusion

The final stage deals with the problem of identification of the spectra of hydrogen thermal desorption. This problem is of high relevance for the nuclear power industry. Qualitatively, the identification consists of revealing the causes of desorption peaks. It is usually assumed that TDS peaks appear due to hydrogen release from traps with different binding energies. Here, it was demonstrated using a diffusion model for a homogeneous material that if surface processes are taken into account, two-peak spectra can be obtained even for very thin experimental samples. The tendency to resort to the “theory of different traps” as the only explanation is understandable, but the volume of our samples was near zero for the trapping capacity to manifest itself.

The nonlinear boundary value problem (standard diffusion equation with dynamical boundary conditions) is reduced to the functional differential equation for the surface concentration, because nothing but desorption dynamics is required to plot a TDS spectrum. An effective numerical algorithm oriented to the use of mathematical packages (including freeware) is proposed. The main final output of this part of the paper is a geometrically transparent method for solving the inverse problem of surface parameter identification where desorption and diffusion in the bulk are dynamically interrelated.

The authors declare that there are no conflicts of interest regarding the publication of this article.

All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

The study was carried out under state order to the Karelian Research Centre of the Russian Academy of Sciences (Institute of Applied Mathematical Research KarRC RAS).