diff --git a/docs/source/bibliography/references.bib b/docs/source/bibliography/references.bib index 5a08d9d7d..79eb74285 100644 --- a/docs/source/bibliography/references.bib +++ b/docs/source/bibliography/references.bib @@ -4,7 +4,7 @@ @article{McNabb1963 year = 1963, journal = {Trans. Metall. Soc. AIME}, volume = 227, - pages = 618, + pages = 618 } @article{Longhurst1985, title = {{The soret effect and its implications for fusion reactors}}, @@ -33,7 +33,7 @@ @article{Delaporte-Mathurin2021 number = 3, pages = {036038}, issn = {0029-5515}, - url = {https://iopscience.iop.org/article/10.1088/1741-4326/abd95f}, + url = {https://iopscience.iop.org/article/10.1088/1741-4326/abd95f} } @phdthesis{Delaporte-Mathurin2022, title = {{Hydrogen transport in tokamaks : Estimation of the ITER divertor tritium inventory and influence of helium exposure}}, @@ -55,15 +55,57 @@ @article{Schmid2016 issn = {0031-8949}, url = {https://iopscience.iop.org/article/10.1088/0031-8949/T167/1/014025} } - @article{Guterl2019, - title={Effects of surface processes on hydrogen outgassing from metal in desorption experiments}, - author={Guterl, Jerome and Smirnov, RD and Snyder, P}, - journal={Nuclear Fusion}, - volume={59}, - number={9}, - pages={096042}, - year={2019}, - publisher={IOP Publishing}, - url = {https://iopscience.iop.org/article/10.1088/1741-4326/ab280a/meta} + title = {Effects of surface processes on hydrogen outgassing from metal in desorption experiments}, + author = {Guterl, Jerome and Smirnov, RD and Snyder, P}, + year = 2019, + journal = {Nuclear Fusion}, + publisher = {IOP Publishing}, + volume = 59, + number = 9, + pages = {096042}, + url = {https://iopscience.iop.org/article/10.1088/1741-4326/ab280a/meta} +} +@article{Pick1985, + title = {A model for atomic hydrogen-metal interactions—application to recycling, recombination and permeation}, + author = {Pick, MA and Sonnenberg, K}, + year = 1985, + journal = {Journal of Nuclear Materials}, + publisher = {Elsevier}, + volume = 131, + number = {2-3}, + pages = {208--220}, + url = {https://www.sciencedirect.com/science/article/abs/pii/0022311585904593} +} +@article{Hodille2017, + title = {Simulations of atomic deuterium exposure in self-damaged tungsten}, + author = {Hodille, EA and Zalo{\v{z}}nik, A and Markelj, S and Schwarz-Selinger, T and Becquart, CS and Bisson, R{\'e}gis and Grisolia, Christian}, + year = 2017, + journal = {Nuclear Fusion}, + publisher = {IOP Publishing}, + volume = 57, + number = 5, + pages = {056002}, + url = {https://iopscience.iop.org/article/10.1088/1741-4326/aa5aa5/meta} +} +@article{Schmid2021, + title = {On the use of recombination rate coefficients in hydrogen transport calculations}, + author = {Schmid, K and Zibrov, M}, + year = 2021, + journal = {Nuclear Fusion}, + publisher = {IOP Publishing}, + volume = 61, + number = 8, + pages = {086008}, + url = {https://iopscience.iop.org/article/10.1088/1741-4326/ac07b2/meta} +} +@article{Hamamoto2020, + title = {Comprehensive modeling of hydrogen transport and accumulation in titanium and zirconium}, + author = {Hamamoto, Yoshiki and Uchikoshi, Takeru and Tanabe, Katsuaki}, + year = 2020, + journal = {Nuclear Materials and Energy}, + publisher = {Elsevier}, + volume = 23, + pages = 100751, + url = {https://www.sciencedirect.com/science/article/pii/S2352179120300272} } \ No newline at end of file diff --git a/docs/source/theory.rst b/docs/source/theory.rst index 5ed6d4de7..55b39909c 100644 --- a/docs/source/theory.rst +++ b/docs/source/theory.rst @@ -275,38 +275,70 @@ where :math:`h` is the heat transfer coefficient and :math:`T_{\mathrm{ext}}` is Kinetic surface model ^^^^^^^^^^^^^^^^^^^^^ -Modelling hydrogen retention or outgassing might require considering the kinetics of surface processes :cite:`Guterl2019`. -A representative example is the hydrogen uptake from a gas phase, when the energy of incident atoms/molecules is not high enough to -to overcome the surface barrier for implantation. The general approach to accound for surface kinetics consists in -introducing hydrogen surface species :math:`c_\mathrm{s}`. Evolution of hydrogen surface concentration is governed by the atomic flux -balance at the surface, as sketched in diagram below: +Modelling hydrogen retention or outgassing might require considering the kinetics of surface processes. +A representative example is the hydrogen uptake from a gas phase, when the energy of incident atoms/molecules is not high enough to +overcome the surface barrier for implantation. The general approach to account for surface kinetics :cite:`Pick1985, Hodille2017, Guterl2019, Schmid2021` consists in +introducing hydrogen surface species :math:`c_\mathrm{s}`. -.. math:: - :label: eq_surf_conc - - \dfrac{d c_\mathrm{s}}{d t} = J_\mathrm{bs} - J_\mathrm{sb} + J_\mathrm{vs} - -where :math:`J_\mathrm{bs}` is the flux of hydrogen atoms from the bulk onto the surface, :math:`J_\mathrm{sb}` is the flux of hydrogen atoms from the surface -into the bulk, and :math:`J_\mathrm{sv}` is the net flux of hydrogen atoms from the vacuum onto the surface. +Evolution of hydrogen surface concentration is determined by the atomic flux balance at the surface, as sketched in the simplified energy diagram below. .. figure:: images/potential_diagram.png :align: center :width: 800 - :alt: Potential energy diagram for hydrogen near a surface of an endothermic metal. Energy levels are measured from the :math:`\mathrm{H}_2` state + :alt: Idealised potential energy diagram for hydrogen near a surface of an endothermic metal. Energy levels are measured from the :math:`\mathrm{H}_2` state Potential energy diagram for hydrogen near a surface of an endothermic metal. Energy levels are measured from the :math:`\mathrm{H}_2` state - -The connection condition between surface and bulk domains represents the Robin boundary condition for the diffusion problem. - -The Robin boundary condition can be used to account for kinetic processes occurring on a surface . The general approach consists in considering -a temporal evolution of hydrogen surface species (:math:`c_\mathrm{s}`): +The governing equation for surface species is: .. math:: :label: eq_surf_conc \dfrac{d c_\mathrm{s}}{d t} = J_\mathrm{bs} - J_\mathrm{sb} + J_\mathrm{vs} +where :math:`J_\mathrm{bs}` is the flux of hydrogen atoms from the subsurface (bulk region just beneath the surface) onto the surface, +:math:`J_\mathrm{sb}` is the flux of hydrogen atoms from the surface into the subsurface, and :math:`J_\mathrm{vs}` is the net flux of hydrogen +atoms from the vacuum onto the surface. The current model does not account for possible surface diffusion and, therefore, is limited to +one-dimensional hydrogen transport simulations. + +The connection condition between surface and bulk domains represents the Robin boundary condition for the hydrogen transport problem. + +.. math:: + :label: eq_subsurf_conc + + -D \nabla c_\mathrm{m} \cdot \mathbf{n} = \lambda_{\mathrm{IS}} \dfrac{\partial c_{\mathrm{m}}}{\partial t} + J_{\mathrm{bs}} - J_{\mathrm{sb}} + +where :math:`\lambda_\mathrm{IS}` is the distance between two interstitial sites in the bulk. + +.. note:: + + At steady state and :math:`x=0`, :eq:`eq_subsurf_conc` is reduced to :math:`D\frac{\partial c_\mathrm{m}}{\partial x}=J_\mathrm{bs}-J_\mathrm{sb}` + representing eq. (12) in the original work of M.A. Pick & K. Sonnenberg :cite:`Pick1985`. + +The fluxes for subsurface-to-surface and surface-to-subsurface transitions are defined as follows: + +.. math:: + :label: eq_Jbs + + J_\mathrm{bs} = k_\mathrm{bs} \lambda_\mathrm{abs} c_\mathrm{m} \left(1-\dfrac{c_\mathrm{s}}{n_\mathrm{surf}}\right) + +.. math:: + :label: eq_Jsb + + J_\mathrm{sb} = k_\mathrm{sb} c_\mathrm{s} \left(1-\dfrac{c_\mathrm{m}}{n_\mathrm{IS}}\right) + +where :math:`n_\mathrm{surf}` is the surface concentration of adsorption sites, :math:`n_\mathrm{IS}` is the bulk concentration of interstitial sites, +:math:`\lambda_\mathrm{abs}=n_\mathrm{surf}/n_\mathrm{IS}` is the characteristic distance between surface and subsurface sites, :math:`k_\mathrm{bs}` +and :math:`k_\mathrm{sb}` are the rate constants for subsurface-to-surface and surface-to-subsurface transitions, respectively. +Usually, these rate constants are expressed in the Arrhenius form: :math:`k_i=k_{i,0}\exp(-E_i / kT)`. Both these processes are assumed to take place +if there are available sites on the surface (in the subsurface). Possible surface/subsurface saturation is accounted for with terms in brackets. + +.. note:: + + In eq. :eq:`eq_Jsb`, the last term in brackets is usually omitted, since :math:`c_\mathrm{m} \ll n_\mathrm{IS}` is assumed. + However, this term is included in some works (e.g. :cite:`Hamamoto2020`) to better reproduce the experimental results. + + ------------ References ------------