hngd 1.0.1

Implementation of the Hydride Nucleation-Growth-Dissolution (HNGD) model. Simulates hydrogen behaviour in a 1D geometry using an explicit finite difference scheme (Euler method).

HNGD-Model

A package with an implementation of the Hydride Nucleation-Growth-Dissolution (HNGD) model, that simulates hydrogen behaviour in Zirconium alloys in a 1D geometry using an explicit finite difference scheme (Euler method).

Installation

It is highly recommended that you install this package in an isolated virtual environment. If you are using Anaconda, open Anaconda Prompt and create the environment HNGD-env with the latest Python version:

  conda create --name HNGD-env python

Then, activate this environment:

  conda activate HNGD-env

And install the package as:

  pip install hngd

If you are not using Anaconda, you can use a similar procedure with the venv module.

For developers

If you want an editable version of the code (i. e. a version for developers), clone this repository to the desired location, move to that directory (cd C:\path\to\install\HNGD-Model) and install the package as:

  pip install -e .

How to use

Input parameters

In order to do a simulation, there are five obligatory inputs needed:

• Experimental data
• Simulation data
• Thermal history
• Initial hydrogen concentration
• Model parameters

The following sections explains each of these input parameters in detail.

1. Experimental data

Data from the experiment. Consists on a tuple with the following information:

experimental_data = (slab length [m] -> float, 
                     flux at the left of the sample [wt.ppm m-2 s-1] -> float, 
                     flux at the left of the sample [wt.ppm m-2 s-1] -> float, 
                     liner solubility factor [/] -> float, 
                     liner width [m] -> float)

A liner is a thin, protective layer that is sometimes applied to the inner surface of the cladding. In the model, it is implemented through a local difference in solubility: liners are usually made of pure Zr and don't have much oxygen, causing the solubility in the liner to be lower than in the rest of the cladding. This difference in solubility is given by the liner solubility factor, and typical values range between 0.95-0.98.

For example:

experimental_data = (0.025, 0, 0, 1, 0) 

Corresponds to a slab with a length of 2.5 cm, with no hydrogen flux coming to the sample, and no liner.

2. Simulation data

Data from the simulation. Consists on a tuple with the following information:

simulation_data = (name -> str, 
                   number of nodes -> int, 
                   dt_div -> float, 
                   test_t_set [s] -> float, 
                   output_criterion -> float,
                   model -> str)

The name is used to name output files. The number of nodes corresponds to the 1-D spatial discretization. The time step obtained to ensure a stable solution for the forward Euler method can be divided by dt_div to produce a more accurate solution. Because time steps can be quite small, the solution is not saved at every time iteration. An output will be saved if test_t_set seconds have passed, or if the average relative difference between the current and last temperature or hydrogen concentration profile is greater than the output_criterion (for example, a value of 0.01 correspond to a 1% change between the current and last temperature/concentration profile). The model represent which model is used for the simulation: the original HNGD model or the modified mHNGD model.

For example:

simulation_data = ('test', 70, 2, 80, 0.01, 'mHNGD')

Corresponds to a 70-node discretization, where the stable time step is divided by 2, and the output will be saved every 80 seconds, or if the temperature or hydrogen profiles changed more than 1% with respect to the last time step, and the simulation is run using the mHNGD model.

3. Thermal history

Thermal treatment applied to the sample. Consists on a numpy array with the following shape:

temperature_data = np.array([[np.nan,  pos_1  ,  pos_2  , ... ,   pos_n ],
                             [time_1, temp_1_1, temp_2_1, ... , temp_n_1],
                             [time_2, temp_1_2, temp_2_2, ... , temp_n_2],
                             [time_3, temp_1_3, temp_2_3, ... , temp_n_3],
                             [  ... ,    ...  ,    ...  , ... ,    ...  ],
                             [time_k, temp_1_k, temp_2_k, ... , temp_n_k]])

The first row corresponds to the positions at which the temperature is defined [m]. The first column corresponds the different time stamps at which the temperature is defined [minutes]. The temperature at each position and time stamp is given in [°C]. Temperatures at intermediate positions and time stamps are obtained by linear interpolation.

For example:

temperature_data = np.array([[np.nan,   0, 0.025/2, 0.025],
                             [     0,  20,      50,    20],
                             [    38, 400,     200,   400],
                             [    48, 400,     400,   400],
                             [    68, 400,     400,   400],
                             [   108,  20,      80,    20]])

Corresponds to an initial temperature profile given by 20°C at the edges of the sample, and 50°C at the middle of the sample. Then, after 38 minutes the temperature increased to 400°C at the edges of the sample, and 200°C at the center. Ten minutes later, the temperature profile is uniform, at 400°C, and it is maintained for 20 minutes. Then, at 108 minutes the temperature is 20°C at the edges and 80°C at the middle.

An animation of this sample thermal treatment would be:

4. Initial hydrogen concentration

The hydrogen profile at the start of the experiment. Consists on a numpy array with the following shape:

initial_hydrogen_data = np.array([[pos_1, pos_2, pos_3, ..., pos_n],,
                                  [hyd_1, hyd_2, hyd_3, ..., hyd_n]])

The first row corresponds to the positions at which the hydrogen concentration is defined [m]. The second row corresponds to the hydrogen concentration at each of these positions [wt.ppm]. Concentration at intermediate positions are obtained by linear interpolation.

For example:

initial_hydrogen_data = np.array([[  0, 0.025],
                                  [205,   205]])

Corresponds to a uniform initial hydrogen profile of 205 wt.ppm.

5. Model parameters

The HNGD model has a large number of parameters regarding Hydrogen solubility and diffusion, formation energy of hydrides, kinetics of nucleation, growth and dissolution, etc. Default model parameters can be imported and used as:

import hngd.default

model_parameters = hngd.default.model_parameters

Then, model_parameters is a Python dictionary with the following keys and values:

• 'TSSd0': Pre-exponential term for TSSd [wt pmm] = 102000
• 'Q_TSSd': Activation energy for TSSd [J mol-1] = 35458.7
• 'TSSp0': Pre-exponential term for TSSp [wt ppm] = 30853.3
• 'Q_TSSp': Activation energy for TSSp [J mol-1] = 25249.6
• 'D0': Pre-exponential term of the diffusion coefficient [m2 s-1] = 1.08e-6
• 'Ed': Activation energy of the diffusion coefficient [eV at-1] = 0.46
• 'Q_star': Heat of transport of hydrogen in zirconium [J mol-1] = 25500
• 'V_star': Partial molar volume of hydrogen in the zirconium matrix [m3 mol-1] = 1.7e-6
• 'Eg': Activation energy for diffusion driven growth [eV at-1] = 0.9
• 'avrami_parameter': Avrami parameter for platelets [/] = 2.5
• 'Eth0': 0th coefficient of formation energy polynomial [eV at-1] = 0.5655
• 'Eth1': 1st coefficient of formation energy polynomial [eV at-1 K-1] = 4.0e-4
• 'Eth2': 2nd coefficient of formation energy polynomial [eV at-1 K-2] = 2.0e-7
• 'Eth3': 3rd coefficient of formation energy polynomial [eV at-1 K-3] = 3.0e-10
• 'K_G_mob0': Pre-exponential factor for diffusion-controlled growth [s-1] = 5.53e5
• 'K_G_th0': Pre-exponential factor for reaction-controlled growth [s-1] = 1.6e-5
• 'K_N_0': Pre-exponential factor for nucleation kinetics [s-1] = 2.75e-5
• 'K_D_0': Pre-exponential factor for dissolution kinetics [s-1] = 1110.13
• 'tau': tau for mHNGD [s] = 1e4
• 'g': g for mHNGD [wt.ppm] = 120
• 'delta': delta for mHNGD [/] = 1.05

Default model parameters can be modified. For example, to change the default diffusion coefficient given by Zhang et al. to the one given by Kearns, the corresponding parameters can be changed as:

model_parameters['D0'] = 7.90e-7
model_parameters['Ed'] = 0.47

Running a simulation

In order to run a simulation, the hngd.run module is imported. Then, set up the input parameters described in the previous section, and use the simulation function as:

import hngd.run
import hngd.default
import numpy as np

experiment_data = (0.025, 0, 0, 1, 0)

simulation_data = ('test', 70, 2, 80, 0.01, 'mHNGD')

temperature_data = np.array([[np.nan,   0, experiment_data[0]],
                             [     0,  20,                 20],
                             [    38, 400,                400],
                             [    48, 400,                400],
                             [    86,  20,                20]])


initial_hydrogen_data = np.array([[  0, experiment_data[0]],
                                  [205,               205]])

model_parameters = hngd.default.model_parameters

hngd.run.simulation(temperature_data, initial_hydrogen_data, experiment_data, 
                    simulation_data, model_parameters)

After the simulation is completed, the results will be saved in a txt file which contains the time, temperature, fraction of precipitation due to growth, fraction of precipitation due to nucleation, hydrogen concentration in solid solution and hydrogen concentration in hydrides.

There are additional optional parameters for the simulation function:

• node: Change for which node of the spatial discretization the results are saved. Defaults to 1 (node at the left of the sample).
• save_results: If you want to save the results in a txt file. Defaults to True.
• save_last_concentration_profile: If you want to save the concentration profile of hydrogen in solid solution and hydrogen in hydrides for the last time step. Useful to compare to some experiments in the literature. Defaults to False.
• unit_time: Units of time in the output file, can be 's' (seconds), 'm' (minutes), 'h' (hours) or 'd' (days). Defaults to 's'.
• unit_temperature: Units of temperature in the output file, can be 'C' (celsius) or 'K' (kelvin). Defaults to 'K'.

Running an animation

In order to run an animation, the hngd.run module is imported. Then, set up the input parameters described in the previous section, and use the animation function as:

import numpy as np
import hngd.default_model_parameters
import hngd.run

experiment_data = (0.025, 0, 0, 1, 0)

simulation_data = ('test', 70, 2, 40, 0.01, 'mHNGD')

temperature_data = np.array([[np.nan,   0, experiment_data[0]],
                             [     0,  20,                 20],
                             [    38, 400,                400],
                             [    48, 400,                400],
                             [    86,  20,                20]])

initial_hydrogen_data = np.array([[  0, experiment_data[0]],
                                  [205,               205]])

model_parameters = hngd.default_model_parameters.model_parameters

animation = hngd.run.animation(temperature_data, initial_hydrogen_data, 
                               experiment_data, simulation_data, model_parameters)

The animation shows four plots: the evolution of the hydrogen profile (top, left), the evolution of the temperature profile (top, right), the evolution of hydrogen in solid solution as a function of temperature at a single node (bottom, left), and the evolution of temperature as a function of time at a single node (bottom, right).

There are additional optional parameters for the animation function:

• node: Which node is used to animate the temporal evolution. Defaults to the node at the middle of the sample (i. e., simulation_data[1]//2).
• save_animation: If you want to save the animation as a gif. Defaults to False. Note: When save_animation=True, the animation will not be displayed (to fix!).
• interval: Delay between frames in milliseconds. Defaults to 5.
• repeat: Whether the animation repeats when the sequence of frames is completed. Defaults to False. Note: Currently not working (to fix!).

Others

There are several other possibilities of the code, such as running a time-temperature-transformation experiment (TTT diagram) using hngd.run.TTT, or using hngd.run.debugger if you are a developer and want to modify the source code and need a debugger. Some of these applications can be found in the Examples folder of this repository.

Note for Developers

This code uses the open source, just-in-time (JIT) compiler from the numba library, which translates Python functions to optimized machine code at runtime using the industry-standard LLVM compiler library.

If you are making changes to the code and want to use the debugger to do some testing, you will have to disable numba. This can be done by setting:

 DISABLE_JIT: True

in the .numba_config.yaml file which is located at the hngd folder.

References

Some references for the HNGD model:

[1] Lacroix E., Modeling zirconium hydride precipitation and dissolution in zirconium alloys, PhD Thesis, The Pennsylvania State University (2019).

[2] Passelaigue F., Hydride Nucleation-Growth-Dissolution model: Implementation in BISON, MSc. Thesis, The Pennsylvania State University (2020).

[3] Passelaigue F., Lacroix E., Pastore G. and Motta A. T., Implementation and validation of the Hydride Nucleation-Growth-Dissolution (HNGD) model in BISON, Journal of Nuclear Materials 544, pp. 152683 (2021). https://doi.org/10.1016/j.jnucmat.2020.152683.

[4] Passelaigue F., Simon, P. A. and Motta, A. T., Predicting the hydride rim by improving the solubility limits in the Hydride Nucleation-Growth-Dissolution (HNGD) model, Journal of Nuclear Materials 558, pp. 153363 (2022). https://doi.org/10.1016/j.jnucmat.2021.153363.

[5] Bin Seo S., Duchnowski E. M., Motta A. T., Kammenzind B. F. and Brown N. R.,Sensitivity analysis for characterizing the impact of HNGD model on the prediction of hydrogen redistribution in Zircaloy cladding using BISON code, Nuclear Engineering and Design 393, pp. 111813 (2022). https://doi.org/10.1016/j.nucengdes.2022.111813.

[6] Lee C. and Lee, Y., Simulation of hydrogen diffusion along the axial direction in zirconium cladding tube during dry storage, Journal of Nuclear Materials 579, pp. 154352 (2023). https://doi.org/10.1016/j.jnucmat.2023.154352