diff --git a/Figures/Chapter3/monoblocks/interface_condition/iter case/thermal_prop.pdf b/Figures/Chapter3/monoblocks/interface_condition/iter case/thermal_prop.pdf index a16f73c..fbf8ed0 100644 Binary files a/Figures/Chapter3/monoblocks/interface_condition/iter case/thermal_prop.pdf and b/Figures/Chapter3/monoblocks/interface_condition/iter case/thermal_prop.pdf differ diff --git a/Figures/Chapter3/monoblocks/parametric_study/thermal_prop.pdf b/Figures/Chapter3/monoblocks/parametric_study/thermal_prop.pdf deleted file mode 100644 index a16f73c..0000000 Binary files a/Figures/Chapter3/monoblocks/parametric_study/thermal_prop.pdf and /dev/null differ diff --git a/bibfile.bib b/bibfile.bib index 8a6401b..2311496 100644 --- a/bibfile.bib +++ b/bibfile.bib @@ -10566,3 +10566,22 @@ @article{mcnabb_new_1963 year = {1963}, pages = {618--627}, } + +@article{brown_endfb-viii0_2018, + series = {Special {Issue} on {Nuclear} {Reaction} {Data}}, + title = {{ENDF}/{B}-{VIII}.0: {The} 8th {Major} {Release} of the {Nuclear} {Reaction} {Data} {Library} with {CIELO}-project {Cross} {Sections}, {New} {Standards} and {Thermal} {Scattering} {Data}}, + volume = {148}, + issn = {0090-3752}, + shorttitle = {{ENDF}/{B}-{VIII}.0}, + url = {https://www.sciencedirect.com/science/article/pii/S0090375218300206}, + doi = {10.1016/j.nds.2018.02.001}, + abstract = {We describe the new ENDF/B-VIII.0 evaluated nuclear reaction data library. ENDF/B-VIII.0 fully incorporates the new IAEA standards, includes improved thermal neutron scattering data and uses new evaluated data from the CIELO project for neutron reactions on 1H, 16O, 56Fe, 235U, 238U and 239Pu described in companion papers in the present issue of Nuclear Data Sheets. The evaluations benefit from recent experimental data obtained in the U.S. and Europe, and improvements in theory and simulation. Notable advances include updated evaluated data for light nuclei, structural materials, actinides, fission energy release, prompt fission neutron and γ-ray spectra, thermal neutron scattering data, and charged-particle reactions. Integral validation testing is shown for a wide range of criticality, reaction rate, and neutron transmission benchmarks. In general, integral validation performance of the library is improved relative to the previous ENDF/B-VII.1 library.}, + language = {en}, + urldate = {2022-07-14}, + journal = {Nuclear Data Sheets}, + author = {Brown, D. A. and Chadwick, M. B. and Capote, R. and Kahler, A. C. and Trkov, A. and Herman, M. W. and Sonzogni, A. A. and Danon, Y. and Carlson, A. D. and Dunn, M. and Smith, D. L. and Hale, G. M. and Arbanas, G. and Arcilla, R. and Bates, C. R. and Beck, B. and Becker, B. and Brown, F. and Casperson, R. J. and Conlin, J. and Cullen, D. E. and Descalle, M. -A. and Firestone, R. and Gaines, T. and Guber, K. H. and Hawari, A. I. and Holmes, J. and Johnson, T. D. and Kawano, T. and Kiedrowski, B. C. and Koning, A. J. and Kopecky, S. and Leal, L. and Lestone, J. P. and Lubitz, C. and Márquez Damián, J. I. and Mattoon, C. M. and McCutchan, E. A. and Mughabghab, S. and Navratil, P. and Neudecker, D. and Nobre, G. P. A. and Noguere, G. and Paris, M. and Pigni, M. T. and Plompen, A. J. and Pritychenko, B. and Pronyaev, V. G. and Roubtsov, D. and Rochman, D. and Romano, P. and Schillebeeckx, P. and Simakov, S. and Sin, M. and Sirakov, I. and Sleaford, B. and Sobes, V. and Soukhovitskii, E. S. and Stetcu, I. and Talou, P. and Thompson, I. and van der Marck, S. and Welser-Sherrill, L. and Wiarda, D. and White, M. and Wormald, J. L. and Wright, R. Q. and Zerkle, M. and Žerovnik, G. and Zhu, Y.}, + month = feb, + year = {2018}, + pages = {1--142}, + file = {ScienceDirect Full Text PDF:D\:\\Logiciels\\data_zotero\\storage\\2I9PI5MF\\Brown et al. - 2018 - ENDFB-VIII.0 The 8th Major Release of the Nuclea.pdf:application/pdf;ScienceDirect Snapshot:D\:\\Logiciels\\data_zotero\\storage\\JFGW9MM5\\S0090375218300206.html:text/html}, +} diff --git a/chapters/appendix.tex b/chapters/appendix.tex deleted file mode 100644 index b8d84b8..0000000 --- a/chapters/appendix.tex +++ /dev/null @@ -1,4 +0,0 @@ -\setchapterstyle{lines} -\labpage{Molten Salt} -\chapter{Molten Salt} -% \blinddocument diff --git a/chapters/breeding_blankets.tex b/chapters/breeding_blankets.tex deleted file mode 100644 index af11cf9..0000000 --- a/chapters/breeding_blankets.tex +++ /dev/null @@ -1,5 +0,0 @@ -\setchapterstyle{lines} -\labpage{Breeding blankets} -\chapter{Breeding blankets} -\section{Intro} -\section{Summary} \ No newline at end of file diff --git a/chapters/chapter1/intro.tex b/chapters/chapter1/intro.tex index e64b7a5..2260fa4 100644 --- a/chapters/chapter1/intro.tex +++ b/chapters/chapter1/intro.tex @@ -199,7 +199,7 @@ \subsection{Triple product} So far, no fusion device has been able to even reach \textit{break-even} ($Q = 1$) (see \reffig{triple product vs T}). The record of $Q = 0.68$ by the European \gls{tokamak} \gls{jet} and was performed in 1997 \sidecite{mailloux_overview_2022}. The objective of the \gls{iter} \gls{tokamak}, currently under construction in France, is to demonstrate an amplification factor of $Q=10$ over \SI{400}{s} \sidecite{casper_development_2013}. -Note that \acrshort{iter} will not produce any electricity as this will be the role of a future fusion reactor: DEMO \sidecite{federici_overview_2014}. +Note that \acrshort{iter} will not produce any electricity as this will be the role of a future fusion reactor: \gls{demo} \sidecite{federici_overview_2014}. Other designs aim at demonstrating \gls{plasma} gain (i.e.\ $Q > 1$) sooner than \acrshort{iter} and at a smaller scale (see \reffig{comparison reactors}). This is the case of \acrshort{sparc} and \acrshort{arc} developed by Commonwealth Fusion Systems and MIT \sidecite{sorbom_arc_2015,creely_overview_2020} or \acrshort{step} designed by the \gls{ukaea} \sidecite{wilson_steppathway_2020}. @@ -241,7 +241,7 @@ \subsection{Plasma-facing materials} Methane is then deposited on locations hard to access in the reactor \gls{trapping} tritium even more. For these two safety reasons, \gls{cfc} was replaced with \Gls{W} or \gls{Be} (or both). The advantage of \gls{Be} is that it creates a stable oxide. -The formation of this oxide layer consumes oxygen and reduces the oxygen level in the plasma. +The formation of this oxide layer consumes oxygen and reduces the oxygen impurity level in the plasma. \Gls{W} has a very high melting point (\SI{3422}{\celsius}) and retains less tritium \sidecite{pajuste_tritium_2021}. However, \Gls{W} being a high-Z element, eroded \Gls{W} will make the \gls{plasma} radiate and cool it down. @@ -282,10 +282,10 @@ \subsection{Divertor}\labsec{divertor section} \Glspl{monoblock} are typically made of a \Gls{W} substrate with a cooling pipe running through. This cooling channel is necessary to keep the component's temperature below its operating limit and exhaust heat. -Several \gls{monoblock} designs are currently studied for DEMO with varying dimensions, different materials for the cooling pipe or the interlayer, etc. \sidecite{vizvary_european_2020, huang_tungsten_2016, hirai_use_2016, domptail_design_2020}. +Several \gls{monoblock} designs are currently studied for \gls{demo} with varying dimensions, different materials for the cooling pipe or the interlayer, etc. \sidecite{vizvary_european_2020, huang_tungsten_2016, hirai_use_2016, domptail_design_2020}. The main candidate is the ITER-like design, the type of \gls{monoblock} that will be used in \acrshort{iter} \cite{hirai_use_2016}. -This design has a tungsten substrate with a CuCrZr cooling pipe and a Cu interlayer for compliance. -In \acrshort{iter}, \glspl{monoblock} will be \SI{12}{mm}-thick whereas they will be thinner (\SI{4}{mm}) in DEMO \sidecite{you_european_2018}. +This design has a \gls{W} substrate with a CuCrZr cooling pipe and a Cu interlayer for compliance. +In \acrshort{iter}, \glspl{monoblock} will be \SI{12}{mm}-thick whereas they will be thinner (\SI{4}{mm}) in \gls{demo} \sidecite{you_european_2018}. \begin{figure*} \includegraphics[width=\linewidth]{Figures/Chapter1/monoblock_to_divertor.pdf} @@ -437,7 +437,7 @@ \subsection{Safety} \section{Helium and Hydrogen in metals} -This Section summarises the main processes at stake (see \reffig{helium and hydrogen in metals sketch}) when helium and hydrogen particles interact with metals and with each other. +This Section summarises the main processes at stake (see \reffig{helium and hydrogen in metals sketch}) when \gls{He} and \gls{H} particles interact with metals and with each other. \begin{figure*}[h!] \begin{subfigure}{0.9\linewidth} @@ -472,7 +472,7 @@ \subsubsection{Hydrogen} \begin{figure} \centering \includegraphics[width=\linewidth]{Figures/Chapter1/cross_section_he3_neutron_capture.pdf} - \caption{Cross section of the $^3\mathrm{He(n,T)H}$ \cite{shimwell_xsplot_2021}.} + \caption{Cross section of the $^3\mathrm{He(n,T)H}$ reaction \cite{shimwell_xsplot_2021}.} \labfig{helium3 neutron capture cross section} \end{figure} @@ -486,7 +486,7 @@ \subsubsection{Helium}\labsec{sources of helium} Helium can also be produced in materials indirectly. Since tritium decays into helium-3 (see \refeq{tritium decay}), regions with high tritium \gls{retention} are expected to act as a source of helium over time \cite{shimada_tritium_2017}. Moreover, interactions of neutrons (from the fusion reactions) with metallic elements (e.g.\ tungsten or iron) can produce helium via transmutation \sidecite{watanabe_status_2011}. -This \emph{\gls{transmutation gas}} production has been estimated using well-established neutronics simulations (Monte-Carlo simulations modelling the path of neutroncs in matter) \sidecite{gilbert_neutron-induced_2013, gilbert_integrated_2012}. +This \emph{\gls{transmutation gas}} production has been estimated using well-established neutronics simulations (Monte-Carlo simulations modelling the path of neutrons in matter) \sidecite{gilbert_neutron-induced_2013, gilbert_integrated_2012}. Depending on the position in the DEMO \gls{divertor}, cumulative helium production over the course of three \glspl{fpy} could reach more than \SI{400}{appm}. @@ -548,13 +548,13 @@ \subsubsection{Diffusion} \end{equation} where $\langle R^2(t) \rangle$ is the mean squared displacement of the species. -This modelling technique was used to estimate the diffusion coefficient of \gls{H} in \gls{W} \sidecite{wang_molecular_2020, zhou_molecular_2016,kato_super-saturated_2015,liu_hydrogen_2014} and for He in \gls{W} \sidecite{faney_numerical_2013,faney_spatially_2014,faney_spatially_2015,sefta_surface_2013,perez_mobility_2017}. +This modelling technique was used to estimate the diffusion coefficient of \gls{H} in \gls{W} \sidecite{wang_molecular_2020, zhou_molecular_2016,kato_super-saturated_2015,liu_hydrogen_2014} and for \gls{He} in \gls{W} \sidecite{faney_numerical_2013,faney_spatially_2014,faney_spatially_2015,sefta_surface_2013,perez_mobility_2017}. % Experiments Diffusivity of hydrogen has also been measured experimentally in \gls{W} \sidecite{holzner_solute_2020, frauenfelder_solution_1969, anderl_hydrogen_1990}, copper and copper alloys (CuCrZr) \sidecite{anderl_deuterium_1992} and other metals. Note that the diffusion coefficients measured experimentally are usually effective coefficients accounting for \gls{trapping} effects (detailed below). % EUROFER \sidecite{montupet-leblond_permeation_2021,esteban_hydrogen_2007,aiello_hydrogen_2002}, -Because He tends to cluster (as explained below), measuring its diffusivity experimentally is extremely complicated and therefore most estimations of He diffusion coefficients are numerical. +Because \gls{He} tends to cluster (as explained below), measuring its diffusivity experimentally is extremely complicated and therefore most estimations of \gls{He} diffusion coefficients are numerical. \reffig{diffusivity materials} is a collection of diffusivity values found in literature (measured experimentally or computed) for tungsten, copper and CuCrZr. % The diffusivities of copper and CuCrZr are comparable (see \reffig{diffusivity solubility copper} and \reffig{diffusivity solubility cucrzr}). @@ -594,7 +594,7 @@ \subsubsection{Trapping at defects} \begin{figure*} \centering \includegraphics[width=0.75\linewidth]{Figures/Chapter1/tds_helium_nicolas.pdf} - \caption{H TDS spectra of pre-damaged W. Reproduced from \cite{ialovega_hydrogen_2020}.} + \caption{H TDS spectrum of pre-damaged W. Reproduced from \cite{ialovega_hydrogen_2020}.} \labfig{TDS example ialovega} \end{figure*} @@ -721,12 +721,12 @@ \subsubsection{Advection in liquids} For hydrogen diffusing in liquid LiPb (typically in a WCLL breeding blanket with a characteristic length $L \approx \SI{1}{m}$), with $D \approx \SI{1e-9}{m^2.s^{-1}}$ and $u \approx \SI{1e-4}{m.s^{-1}}$ \sidecite{dark_influence_2021}, $\mathrm{Pe} \approx 10^{5}$, which means that \gls{advection} dominates the mass transport and cannot be neglected. \subsubsection{Clustering} -Single He atoms implanted into the material diffuse rapidly due to the high W-He repulsion. -This high repulsive W-He interaction is such that interstitial He atoms preferably rearrange into groups of atoms in order to minimise the number of repulsive interactions \sidecite{hamid_molecular_2019, hammond_large-scale_2018}. +Single \gls{He} atoms implanted into the material diffuse rapidly due to the high W-He repulsion. +This high repulsive W-He interaction is such that interstitial \gls{He} atoms preferably rearrange into groups of atoms in order to minimise the number of repulsive interactions \sidecite{hamid_molecular_2019, hammond_large-scale_2018}. This phenomenon, called \emph{clustering}, was highlighted by \gls{dft} studies \cite{becquart_density_2009,dunn_rate_2013} and MD simulations \cite{henriksson_molecular_2006}. -Small clusters are themselves mobile as long as all the He atoms within the cluster are occupying interstitial position in the solid \gls{lattice}. -The activation energy for interstitial He atoms and clusters in \gls{W} ranges from 0.15 to \SI{0.45}{eV} according to Perez et al.\ \sidecite{perez_mobility_2017}. -He clusters will eventually grow by interacting with either interstitial He atoms or other clusters. +Small clusters are themselves mobile as long as all the \gls{He} atoms within the cluster are occupying interstitial position in the solid \gls{lattice}. +The activation energy for interstitial \gls{He} atoms and clusters in \gls{W} ranges from 0.15 to \SI{0.45}{eV} according to Perez et al.\ \sidecite{perez_mobility_2017}. +He clusters will eventually grow by interacting with either interstitial \gls{He} atoms or other clusters. Clustering of \gls{H} atoms is less clear and Henriksson et al.\ showed that \gls{H} atoms do not form bonds with other \gls{H} atoms in \gls{bcc} \gls{W} \sidecite{henriksson_difference_2005}. @@ -736,7 +736,7 @@ \subsubsection{Bubble nucleation} This process is called \gls{trap mutation} or \emph{\gls{self-trapping}} and the trapped clusters act as nuclei for bubble formation. \Gls{trap mutation} has been modelled in \gls{W} using \gls{dft} \sidecite{boisse_modelling_2014} and Monte Carlo computations \sidecite{de_backer_modeling_2015}. -It has been shown that this phenomenon depends not only on the number of He atoms in the cluster but also on temperature, position of the cluster to the free surface or even the crystal orientation \sidecite{blondel_modeling_2017, hu_interactions_2014, hu_dynamics_2014}. +It has been shown that this phenomenon depends not only on the number of \gls{He} atoms in the cluster but also on temperature, position of the cluster to the free surface or even the crystal orientation \sidecite{blondel_modeling_2017, hu_interactions_2014, hu_dynamics_2014}. At this point, the trapped cluster occupies the newly created \gls{W} \gls{vacancy} position. It is considered immobile since it would require either diffusion of another \gls{vacancy} next to it, or recombination of the Frenkel pair in order to diffuse \sidecite{morishita_nucleation_2007}. @@ -754,26 +754,26 @@ \subsubsection{Bubble growth} Condon and Schober \sidecite{condon_hydrogen_1993} reviewed the key mechanisms of bubble growth in metals. Each of these mechanisms can become dominant over another depending on the implantation and the metal conditions. -Bubbles can continue to grow by absorbing interstitial He atoms or mobile He clusters (i.e.\ that haven't self trapped). -Considering that vacancies are mobile in the solid, the volume of a bubble could also increase if a \gls{vacancy} or a \gls{vacancy} cluster interacts with a He bubble. +Bubbles can continue to grow by absorbing interstitial \gls{He} atoms or mobile \gls{He} clusters (i.e.\ that haven't self trapped). +Considering that vacancies are mobile in the solid, the volume of a bubble could also increase if a \gls{vacancy} or a \gls{vacancy} cluster interacts with a \gls{He} bubble. The same is true for He-vacancies or H-vacancies clusters. -There is no experimental evidence of He clustering with \gls{self-interstitial} \gls{W} atoms \sidecite{faney_spatially_2014}. +There is no experimental evidence of \gls{He} clustering with \gls{self-interstitial} \gls{W} atoms \sidecite{faney_spatially_2014}. % This process is described by cluster dynamics equations in which interaction between the clusters is governed by pairs of association and dissociation rates. -During the growth of a He bubble by absorbing He atoms, if the pressure increases until reaching a critical value, \gls{dislocation loop} punching can occur. +During the growth of a \gls{He} bubble by absorbing \gls{He} atoms, if the pressure increases until reaching a critical value, \gls{dislocation loop} punching can occur. During the punching event, a whole facet of \gls{W} atoms is pushed, and the vacant \gls{lattice} sites are absorbed by the bubble allowing the bubble to expand and reducing the pressure in it \sidecite{sefta_surface_2013}. The produced \gls{self-interstitial} \gls{W} atoms will likely be attracted by \textit{image forces} at the surface and will contribute to the roughening of the surface and/or formation of surface structures. -\Glspl{dislocation loop} happen at very high pressure and if the number of vacancies in the \gls{lattice} is low compared to the amount of He atoms. -This is the case when a high He flux is applied and the He ions energy is low so that no displacement damaged is produced \cite{sefta_surface_2013}. -If vacancies were created via He ions implantation, they could interact with existing He bubbles which would have the effect of increasing the volume and thus decreasing the pressure (assuming no change in temperature and no other implantation mechanism). +\Glspl{dislocation loop} happen at very high pressure and if the number of vacancies in the \gls{lattice} is low compared to the amount of \gls{He} atoms. +This is the case when a high \gls{He} flux is applied and the \gls{He} ions energy is low so that no displacement damaged is produced \cite{sefta_surface_2013}. +If vacancies were created via \gls{He} ions implantation, they could interact with existing \gls{He} bubbles which would have the effect of increasing the volume and thus decreasing the pressure (assuming no change in temperature and no other implantation mechanism). -Coalescence of He bubbles has been observed in \gls{md} simulations \sidecite{hamid_molecular_2019, hammond_helium_2019, zhang_simulation_2019} and would tend to increase the bubble size decreasing the bubble density at the same time. -This may not have an impact on He concentration on the macroscopic scale but might influence bubble bursting. +Coalescence of \gls{He} bubbles has been observed in \gls{md} simulations \sidecite{hamid_molecular_2019, hammond_helium_2019, zhang_simulation_2019} and would tend to increase the bubble size decreasing the bubble density at the same time. +This may not have an impact on \gls{He} concentration on the macroscopic scale but might influence bubble bursting. The pressure inside the bubble and the bubble radius are two parameters of interest and are correlated. -Sefta \sidecite{sefta_surface_2013} proposed to use the Wolfer equation of state in order to determine the number of He atoms contained in a He bubble based on its pressure, the latter being calculated from its radius and its surface tension. +Sefta \sidecite{sefta_surface_2013} proposed to use the Wolfer equation of state in order to determine the number of \gls{He} atoms contained in a \gls{He} bubble based on its pressure, the latter being calculated from its radius and its surface tension. One must be aware that if radii and pressure of bubbles computation is quite straightforward using \gls{md} \sidecite{zhang_simulation_2019} or cluster dynamics \sidecite{faney_spatially_2015} simulations it will be more complex to estimate these metrics considering a continuum model that does not keep track of every type of clusters but only a few of them. The only information \textit{a priori} available in this case is indeed the local helium concentration and an equivalence could be found by either having a high density of small bubbles or a low density of big bubbles. An effort has been made by Ialovega to measure the pressure inside helium bubbles using \gls{eels} \sidecite{ialovega_surface_2021}. @@ -800,8 +800,8 @@ \subsubsection{Blistering} Helium \gls{blistering} has been observed in \gls{W} at low temperature ($< \SI{1000}{K}$) \sidecite{baldwin_formation_2010}. Hydrogen \gls{blistering} was also observed in \gls{W} \sidecite{haasz_effect_1999}. Causey et al.\ also reviewed a wide range of studies showing H exposure leads to \gls{blistering} \sidecite{causey_hydrogen_2002}. -\Gls{blistering} was found to lead to local hardening in W due to the production of dislocations \sidecite{chen_irradiation_2019}. -It can also form cracks depending on the alloying elements in W and the microstructure \sidecite{ueda_hydrogen_2005}. +\Gls{blistering} was found to lead to local hardening in \gls{W} due to the production of dislocations \sidecite{chen_irradiation_2019}. +It can also form cracks depending on the alloying elements in \gls{W} and the microstructure \sidecite{ueda_hydrogen_2005}. Hydrogen \gls{blistering} was observed under high energy irradiation (typically from a few \si{keV} to \si{MeV}). These energies are orders of magnitudes higher than the ones expected in the \acrshort{iter} \gls{divertor} (see \reffig{divertor exposure conditions}). @@ -819,21 +819,21 @@ \subsubsection{Bursting} \end{figure} When a bubble grows near the surface and is over-pressurised, \gls{bursting} can occur (see \reffig{bubble bursting zhou}). -As the bubble size increases via \gls{loop punching}, the W \gls{lattice} is deformed and the ligament thickness decreases. -The latter can rupture which would make all the He atoms contained in the bubble to be released to the vacuum. -This is why He \gls{bursting} is characterised by sharp drops in the He \gls{inventory} \sidecite{hammond_helium_2019}. +As the bubble size increases via \gls{loop punching}, the \gls{W} \gls{lattice} is deformed and the ligament thickness decreases. +The latter can rupture which would make all the \gls{He} atoms contained in the bubble to be released to the vacuum. +This is why \gls{He} \gls{bursting} is characterised by sharp drops in the \gls{He} \gls{inventory} \sidecite{hammond_helium_2019}. Sefta and co-workers observed that \gls{bursting} is more likely to happen at high temperatures. This phenomenon contributes to surface roughening and could be the beginning of the formation of \gls{fuzz} \sidecite{sefta_helium_2013}. -Indeed, a \gls{bursting} event could either form a crater on the W surface or an empty cavity due to self-healing. -In the last case, called a \textit{pinhole} \gls{bursting} event, the cavity can be repressurised with He atoms. +Indeed, a \gls{bursting} event could either form a crater on the \gls{W} surface or an empty cavity due to self-healing. +In the last case, called a \textit{pinhole} \gls{bursting} event, the cavity can be repressurised with \gls{He} atoms. Blondel et al.\ proposed to model \gls{bursting} as a stochastic function of depth in the material rather than a calculation of the bubble pressure. They have also shown that simulation parameters have an impact on the \gls{retention} \sidecite{blondel_continuum-scale_2018}. % These differences are mainly due to 2D effects as more bursting events occur but with smaller bubbles. They have shown that the size of the reaction network size (using cluster dynamics) does not seem to have an influence (between 250 and 200) as the first bursting events happen with clusters of size $\text{He}_{80}$. % Other simulation parameters (depth of the sample, pre-existing vacancies, bubble growth trajectory...) don't affect the simulations results as they converge for long time steps (100 s). -If \gls{bursting} is not included in continuum simulations, the volume fraction of He present in W could become very large and the dilute limit approximation could no longer be valid \sidecite{sefta_surface_2013}. +If \gls{bursting} is not included in continuum simulations, the volume fraction of \gls{He} present in \gls{W} could become very large and the dilute limit approximation could no longer be valid \sidecite{sefta_surface_2013}. The correct metric for estimating \gls{bursting} probabilities must therefore be chosen with care. \subsubsection{W tendrils or ``nano-fuzz''} @@ -845,21 +845,21 @@ \subsubsection{W tendrils or ``nano-fuzz''} \labfig{w fuzz wright} \end{figure} -In 2012, Wright et al.\ \sidecite{wright_tungsten_2012} observed the formation of nanostructures on the surfaces of the W divertor of the reactor Alcator C-mod. -These nanostructures are made of W \glspl{tendril} (see \reffig{w fuzz wright}). -These structures are called W \gls{fuzz}, nano-fuzz or even fuzzy W. -Because a small portion of the \gls{divertor} grew W \gls{fuzz}, no conclusion was made regarding its influence on the \gls{plasma} operation. -However, if these structures were to be removed during \gls{plasma} operation via erosion, W atoms could be fed into the \gls{plasma}, affecting the \gls{tokamak} performances. -Moreover, this phenomenon could increase the W dust formation in the reactor and lead to contamination and safety issues \sidecite{grisolia_tritium_2015}. +In 2012, Wright et al.\ \sidecite{wright_tungsten_2012} observed the formation of nanostructures on the surfaces of the \gls{W} divertor of the reactor Alcator C-mod. +These nanostructures are made of \gls{W} \glspl{tendril} (see \reffig{w fuzz wright}). +These structures are called \gls{W} \gls{fuzz}, nano-fuzz or even fuzzy W. +Because a small portion of the \gls{divertor} grew \gls{W} \gls{fuzz}, no conclusion was made regarding its influence on the \gls{plasma} operation. +However, if these structures were to be removed during \gls{plasma} operation via erosion, \gls{W} atoms could be fed into the \gls{plasma}, affecting the \gls{tokamak} performances. +Moreover, this phenomenon could increase the \gls{W} dust formation in the reactor and lead to contamination and safety issues \sidecite{grisolia_tritium_2015}. W \gls{fuzz} has been observed at high temperature (>1000K), high flux (>\SI{1e21}{He^+.m^{-2}.s^{-1}}) and long exposure (t>\SI{1e2}{s}) \sidecite{baldwin_formation_2010, nishijima_sputtering_2011}. The reason of the \gls{fuzz} formation is still unclear but could be due to bursting events and/or accumulation of self interstitial W atoms at the surface \sidecite{baldwin_effects_2009, baldwin_helium_2008, woller_dynamic_2015, hammond_helium_2017}. -Thermal properties of the media are also impacted by the formation of W \gls{fuzz} \sidecite{wirtz_influence_2016} which could have a severe impact during ELM-like events. +Thermal properties of the media are also impacted by the formation of \gls{W} \gls{fuzz} \sidecite{wirtz_influence_2016} which could have a severe impact during ELM-like events. After 1h of \gls{plasma} implantation, nanostructuring can be found deep in the bulk (up to several hundred of $\mu$m). According to Baldwin and Doerner \sidecite{baldwin_formation_2010}, heavy alloying helps reduce the formation of He-induced \gls{fuzz}. -Takamura et al.\ showed \gls{fuzz} could be grown under relevant \gls{tokamak} conditions (high-flux He \gls{plasma} irradiation and surface temperature greater than \SI{1250}{K}) \sidecite{takamura_formation_2006}. +Takamura et al.\ showed \gls{fuzz} could be grown under relevant \gls{tokamak} conditions (high-flux \gls{He} \gls{plasma} irradiation and surface temperature greater than \SI{1250}{K}) \sidecite{takamura_formation_2006}. Baldwin et al.\ showed the \gls{fuzz} thickness evolved as a square root of the \gls{fluence} \cite{baldwin_effects_2009}. \begin{figure} [h!] @@ -869,19 +869,19 @@ \subsubsection{W tendrils or ``nano-fuzz''} \labfig{w fuzz mccarthy} \end{figure} -McCarthy et al.\ studied the formation of W \gls{fuzz} (see \reffig{w fuzz mccarthy}) at helium fluences ranging from \SI{4e23}{m^{-2}} to \SI{e25}{m^{-2}}, temperatures ranging from \SI{1050}{K} to \SI{1150}{K} and He ion energies from \SI{80}{eV} to \SI{100}{eV}. -They identified different \gls{fuzz} growth regimes depending on the He \gls{fluence} due to the change in porosity of the fuzzy layer during the growth process. -The rate of growth was found to be dependent on the temperature and the state (ion or atom) of the incident He particles. +McCarthy et al.\ studied the formation of \gls{W} \gls{fuzz} (see \reffig{w fuzz mccarthy}) at helium fluences ranging from \SI{4e23}{m^{-2}} to \SI{e25}{m^{-2}}, temperatures ranging from \SI{1050}{K} to \SI{1150}{K} and \gls{He} ion energies from \SI{80}{eV} to \SI{100}{eV}. +They identified different \gls{fuzz} growth regimes depending on the \gls{He} \gls{fluence} due to the change in porosity of the fuzzy layer during the growth process. +The rate of growth was found to be dependent on the temperature and the state (ion or atom) of the incident \gls{He} particles. -Recent modelling work also showed temperature could significantly affect the bursting of He bubbles and therefore the growth of W \gls{fuzz} \sidecite{niu_effect_2021}. +Recent modelling work also showed temperature could significantly affect the bursting of \gls{He} bubbles and therefore the growth of \gls{W} \gls{fuzz} \sidecite{niu_effect_2021}. -De Temmerman et al.\ concluded that temperature was the most critical parameter controlling W \gls{fuzz} growth \sidecite{de_temmerman_nanostructuring_2012}. +De Temmerman et al.\ concluded that temperature was the most critical parameter controlling \gls{W} \gls{fuzz} growth \sidecite{de_temmerman_nanostructuring_2012}. \subsubsection{Cracks} -The role of He implantation in cracks formation is still unclear since cracks have also been observed during pure thermal shock on W PFC \sidecite{wirtz_influence_2016}. +The role of \gls{He} implantation in cracks formation is still unclear since cracks have also been observed during pure thermal shock on \gls{W} PFC \sidecite{wirtz_influence_2016}. Under some specific conditions, cracks can close due to thermal expansion which induce frictional loads on the structure. -The formation of W nano-fuzz could also bridge those cracks as observed by Lemahieu et al.\ \sidecite{lemahieu_h/he_2016}. +The formation of \gls{W} nano-fuzz could also bridge those cracks as observed by Lemahieu et al.\ \sidecite{lemahieu_h/he_2016}. \subsubsection{Reduction of thermal performances} @@ -897,24 +897,24 @@ \subsubsection{Reduction of thermal performances} % NOTE: an interesting study would be to investigate thermal constriction due to the presence of inhomogeneities (He bubbles) in which thermal conductivity is low compared to the one of the W. \subsection{He/H interactions} -Lee et al.\ studied the influence of He implantation on D \gls{retention}. -They showed with Elastic Recoil Detection depth profiles (up to 40 nm) that D is trapped where He is trapped and proposed that He bubbles produce secondary defects around them which can trap D \sidecite{lee_hydrogen_2007}. +Lee et al.\ studied the influence of \gls{He} implantation on \gls{D} \gls{retention}. +They showed with Elastic Recoil Detection depth profiles (up to 40 nm) that \gls{D} is trapped where \gls{He} is trapped and proposed that \gls{He} bubbles produce secondary defects around them which can trap \gls{D} \sidecite{lee_hydrogen_2007}. These defects can be interstitial loops produced by \gls{loop punching} or even vacancies created by stress field induced by overpressurised bubbles. -Lee et al.\ also suggested that no evidence had been found on trapping of D by chemisorption on the inner surface of a He bubble nor by molecular interaction with the He cluster. -The privileged mechanism is therefore the trapping of D in the defects made by the stress field induced by the He bubble to the crystalline structure of the W. -Bergstrom et al.\ however showed hydrogen can be located at the surface of He bubbles \sidecite{bergstrom_molecular_2017}. +Lee et al.\ also suggested that no evidence had been found on trapping of \gls{D} by chemisorption on the inner surface of a \gls{He} bubble nor by molecular interaction with the \gls{He} cluster. +The privileged mechanism is therefore the trapping of \gls{D} in the defects made by the stress field induced by the \gls{He} bubble to the crystalline structure of the W. +Bergstrom et al.\ however showed hydrogen can be located at the surface of \gls{He} bubbles \sidecite{bergstrom_molecular_2017}. -It has been shown by Ueda et al.\ that He implantation (even in small amounts) greatly affects \gls{H} blistering in W \sidecite{ueda_simultaneous_2009}. -With only 0.1\% of He in the ion beam one can observe that \gls{H} blistering is completely suppressed for temperature greater than \SI{653}{K}. +It has been shown by Ueda et al.\ that \gls{He} implantation (even in small amounts) greatly affects \gls{H} blistering in \gls{W} \sidecite{ueda_simultaneous_2009}. +With only 0.1\% of \gls{He} in the ion beam one can observe that \gls{H} blistering is completely suppressed for temperature greater than \SI{653}{K}. At lower temperature, \gls{H} blistering occurs but is significantly reduced. -This phenomenon is due to the fact that \gls{H} migration to the bulk and accumulation at grain boundaries is avoided by He bubbles at the near surface which act as a diffusion plug for H. +This phenomenon is due to the fact that \gls{H} migration to the bulk and accumulation at grain boundaries is avoided by \gls{He} bubbles at the near surface which act as a diffusion plug for H. The same phenomenon has been observed by Miyamoto et al.\ \sidecite{miyamoto_microscopic_2011} which contributes to reducing \gls{H} \gls{retention}. -Markelj et al.\ \sidecite{markelj_hydrogen_2017} showed however that He implantation can increase D \gls{retention} in the He clustering zone. -This suggests that observed reduction or increase of D \gls{retention} in mixed H-He \gls{plasma} experiment depend on the implantation depth. +Markelj et al.\ \sidecite{markelj_hydrogen_2017} showed however that \gls{He} implantation can increase \gls{D} \gls{retention} in the \gls{He} clustering zone. +This suggests that observed reduction or increase of \gls{D} \gls{retention} in mixed H-He \gls{plasma} experiment depend on the implantation depth. -Ialovega et al.\ performed sequential \gls{H} implantation/desorption cycles on W samples pre-damaged with He \sidecite{ialovega_hydrogen_2020}. +Ialovega et al.\ performed sequential \gls{H} implantation/desorption cycles on \gls{W} samples pre-damaged with \gls{He} \sidecite{ialovega_hydrogen_2020}. He bubbles were found in the near surface region (see \reffig{he bubbles ialovega}) and significantly different \gls{tds} spectra were observed after several \gls{H} implantations/desorptions (see \reffig{TDS example ialovega}). -This works gives evidence of an interaction between He and \gls{H} in \gls{W}. +This works gives evidence of an interaction between \gls{He} and \gls{H} in \gls{W}. \section{Problem definition} @@ -944,7 +944,7 @@ \section{Problem definition} \begin{table*} [h] \centering - \begin{tabular}{L{2cm} L{0.5cm} L{0.5cm} L{0.6cm} L{1.7cm} l l l } + \begin{tabular}{L{2cm} C{0.5cm} C{0.5cm} C{0.6cm} C{1.7cm} c c l } & 1D & 2D & 3D & Multimaterial & Heat transfer & Open-source & Programming language \\ \hline \\ \acrshort{tmap7} \cite{longhurst_tmap7_2008} & $\checkmark$ & & & $\checkmark$ & & & Fortran\\ diff --git a/chapters/chapter2/verification_and_validation/comparison_with_tmap7.tex b/chapters/chapter2/verification_and_validation/comparison_with_tmap7.tex index d2ba873..a640ebd 100644 --- a/chapters/chapter2/verification_and_validation/comparison_with_tmap7.tex +++ b/chapters/chapter2/verification_and_validation/comparison_with_tmap7.tex @@ -50,7 +50,8 @@ \end{align} \labeq{code comparison BCs} \end{subequations} -with $\varphi_\mathrm{imp} = \SI{5e23}{m^{-2}.s^{-1}}$ the implanted particle flux, $R_p = \SI{1.25}{nm}$ the implantation depth, $\mathbf{n}$ the normal vector and $K_\mathrm{CuCrZr} = 2.9 \times 10^{-14}\cdot \exp{(-1.92/(k_B\cdot T))}$ the recombination coefficient of the CuCrZr (in vacuum) expressed in \si{m^4.s^{-1}} \sidecite{anderl_deuterium_1999}. +% $\varphi_\mathrm{imp} = \SI{5e23}{m^{-2}.s^{-1}}$ the implanted particle flux, $R_p = \SI{1.25}{nm}$ the implantation depth, $\mathbf{n}$ the normal vector and +with $K_{r,\mathrm{CuCrZr}} = 2.9 \times 10^{-14}\cdot \exp{(-1.92/(k_B\cdot T))}$ the recombination coefficient of the CuCrZr (in vacuum) expressed in \si{m^4.s^{-1}} \sidecite{anderl_deuterium_1999}. The Dirichlet boundary condition on $\Gamma_\mathrm{top}$ for the hydrogen transport corresponds to a flux balance between the implanted flux and the flux that is retro-desorbed at the surface (see \refsec{triangle model}). The temperature profile in TMAP7 was fixed on the temperature profile produced by FESTIM (see \reffig{temperature}). diff --git a/chapters/chapter3/monoblocks.tex b/chapters/chapter3/monoblocks.tex index 25b96fc..8447efe 100644 --- a/chapters/chapter3/monoblocks.tex +++ b/chapters/chapter3/monoblocks.tex @@ -44,7 +44,7 @@ \section{Model description}\labsec{model description} \end{align} \labeq{bc thermal monoblock} \end{subequations} -where $\varphi_\mathrm{heat} = \SI{10}{MW.m^{-2}}$, $T_\mathrm{coolant} = \SI{323}{K}$ and $h = \SI{70000}{W.m^{-2}.K^{-1}}$. +where $T_\mathrm{coolant} = \SI{323}{K}$ and $h = \SI{70000}{W.m^{-2}.K^{-1}}$. \begin{subequations} \begin{align} @@ -54,14 +54,14 @@ \section{Model description}\labsec{model description} \end{align} \labeq{bc H transport monoblock} \end{subequations} -where $\varphi_\mathrm{imp} = \SI{1.6e22}{m^{-2}.s^{-1}}$, $R_p = \SI{1}{nm}$, and $K_\mathrm{r, \, CuCrZr} = 2.9 \times 10^{-14}\cdot \exp{(-1.92/(k_B\cdot T))}$ the recombination coefficient of the copper alloy (in vacuum) expressed in \si{m^4.s^{-1}} \sidecite{anderl_deuterium_1999}. +where $K_\mathrm{r, \, CuCrZr} = 2.9 \times 10^{-14}\cdot \exp{(-1.92/(k_B\cdot T))}$ the recombination coefficient of the copper alloy (in vacuum) expressed in \si{m^4.s^{-1}} \sidecite{anderl_deuterium_1999}. \begin{table*} \centering \begin{tabular}{p{1.7cm} R{3cm} R{3cm} R{1.8cm} R{1cm} R{1.8cm} R{1cm}} & \multicolumn{2}{c}{Thermal properties} & \multicolumn{4}{c}{Hydrogen transport properties}\\ \hline - Material & $\rho \cdot C_p \newline(\si{J.K^{-1}.m^{-3}})$ & $\lambda \newline(\si{W.m^{-1}.K^{-1}})$ & $D_0 \newline(\si{m^2.s^{-1}})$ & $E_\mathrm{diff} \newline(\si{eV})$ & $S_0 \newline(\si{m^{-3}.Pa^{-0.5}})$ & $E_\mathrm{S} \newline(\si{eV})$\\ + Material & $\rho \cdot C_p \newline(\si{J.K^{-1}.m^{-3}})$ & $\lambda \newline(\si{W.m^{-1}.K^{-1}})$ & $D_0 \newline(\si{m^2.s^{-1}})$ & $E_D \newline(\si{eV})$ & $S_0 \newline(\si{m^{-3}.Pa^{-0.5}})$ & $E_\mathrm{S} \newline(\si{eV})$\\ \hline \\ W \cite{frauenfelder_solution_1969,fernandez_hydrogen_2015}& % @@ -173,7 +173,7 @@ \subsection{Thermal behaviour}\labsec{monoblock thermal behaviour} \caption{Thermal behaviour of the monoblock.} \end{figure*} -The average surface temperature $T_\mathrm{surface}$ therefore increases linearly with the heat load and can be modelled by \refeq{thermal behaviour law} (see \reffig{surface temperature as a function of heat flux}). +The average surface temperature $T_\mathrm{surface}$ therefore increases linearly with the heat load and can be fitted by \refeq{thermal behaviour law} (see \reffig{surface temperature as a function of heat flux}). \begin{equation} T_\mathrm{surface} = 1.1 \times 10^{-4} \cdot \varphi_\mathrm{heat} + T_\mathrm{coolant} \labeq{thermal behaviour law} @@ -249,10 +249,10 @@ \subsection{Influence of cycling}\labsec{influence of cycling} Simulating these transient cycles would require stepsizes of $\approx \SI{10}{s}$ in order to capture the ramp-up and ramp-down phases. Simulating one cycle would therefore require more than 60 steps (excluding the resting phase). -On the other hand, FESTIM has an adaptive stepsize feature allowing the stepsize to increase (resp. decrease) when steps are solved in less (resp. more) than 5 Newton iterations. +On the other hand, FESTIM has an adaptive stepsize feature allowing the stepsize to increase (resp. decrease) when steps are solved in less (resp. more) than five Newton iterations. Therefore, if a continuous plasma exposure was simulated, the adaptive stepsize would allow the stepsize to increase up to thousands of seconds, reducing a lot the simulation time. -To verify the validity of this approximation, 1D simulations were run with plasma cycles or continuous exposure. +To verify the validity of the continuous exposure approximation, 1D simulations were run with plasma cycles or continuous exposure. For the cycled simulation, both the heat flux $\varphi_\mathrm{heat}$ and the particle flux $\varphi_\mathrm{imp}$ were varied from zero during the resting phases to their nominal values during the plateau phase (see \reffig{plasma cycle}). Two cases were run: diff --git a/chapters/chapter3/monoblocks/interface_conditions.tex b/chapters/chapter3/monoblocks/interface_conditions.tex index ffd97f6..c6d8d17 100644 --- a/chapters/chapter3/monoblocks/interface_conditions.tex +++ b/chapters/chapter3/monoblocks/interface_conditions.tex @@ -28,7 +28,7 @@ \includegraphics[width=\linewidth]{Figures/Chapter3/monoblocks/interface_condition/iter case/retention_chemical_pot.pdf} \caption{Retention (continuity of chemical potential).} \end{subfigure} - \caption{Concentration fields at $t=\SI{2e7}{s}$, $\varphi_\mathrm{heat} = \SI{7}{MW.m^{-2}}$.} + \caption{Influence of interface conditions on concentration fields at $t=\SI{2e7}{s}$, $\varphi_\mathrm{heat} = \SI{7}{MW.m^{-2}}$.} \labfig{concentration fields w/wo chemical potential} \end{figure} @@ -44,7 +44,7 @@ For the case at \SI{6}{MW.m^{-2}}, differences start to appear after \SI{3e6}{s} (\SI{7e5}{s} at \SI{7}{MW.m^{-2}}). After \SI{2e7}{s} of continuous exposure, the absolute difference at \SI{6}{MW.m^{-2}} was 25 \% and 55 \% at \SI{7}{MW.m^{-2}}. -This time of appearance of differences is identical to the time required for the hydrogen to migrate up to the W/Cu interface. +This time of appearance of differences corresponds to the time required for the hydrogen to migrate up to the W/Cu interface. This is explained by the high solubility ratio between W, Cu and CuCrZr leading to a higher concentration of mobile particles in CuCrZr (see \reffig{concentration fields w/wo chemical potential}) and therefore a higher trapping rate. Since the trap density in Cu is low, the global inventory is not affected by it. @@ -59,7 +59,7 @@ \includegraphics[width=\linewidth]{Figures/Chapter3/monoblocks/interface_condition/retention_concentration_short_exposure.pdf} \caption{continuity of mobile concentration.} \end{subfigure} - \caption{Retention fields at $t=\SI{6.1e4}{s}$.} + \caption{Influence of interface conditions on retention fields at $t=\SI{6.1e4}{s}$.} \labfig{retention fields w/wo chemical pot short exposure} \end{figure} @@ -69,5 +69,4 @@ Moreover, outgassing flux through the cooling pipe greatly depends on the boundary condition imposed at the cooling surface. Therefore, in order to assess the impact of interface conditions on the outgassing flux through the cooling pipe, uncertainties must first be lifted regarding the recombination process occurring on surfaces in contact with water. -Since this work is motivated by the estimation of the divertor inventory, the concentration continuity assumption is therefore valid. -Moreover, only a few monoblocks are exposed to high heat fluxes and most of the divertor is at the coolant temperature (this will be explained further in \refch{Divertor inventory estimation}). +Since this work is motivated by the estimation of the divertor inventory, the concentration continuity assumption is therefore valid since only a few monoblocks are exposed to high heat fluxes and most of the divertor is at the coolant temperature (this will be explained further in \refch{Divertor inventory estimation}). diff --git a/chapters/chapter3/monoblocks/parametric_study.tex b/chapters/chapter3/monoblocks/parametric_study.tex index 422a72a..238d27c 100644 --- a/chapters/chapter3/monoblocks/parametric_study.tex +++ b/chapters/chapter3/monoblocks/parametric_study.tex @@ -1,5 +1,5 @@ Monoblocks in a fusion reactor will be exposed to a wide range of exposure conditions (heat and particle fluxes) and their behaviour in terms of hydrogen transport will change based on these conditions. -In ITER, these fluxes can reach $\approx$ \SI{10}{MW.m^{-2}} and $\approx$ \SI{e24}{H.m^{-2}.s^{-1}}. +In ITER, these fluxes can reach $\approx$ \SI{10}{MW.m^{-2}} and $\approx$ \SI{e24}{H.m^{-2}.s^{-1}} (see \reffig{divertor exposure conditions}). The distribution of these fluxes depend on many operation parameters. One way of simulating a whole divertor would be to simulate each and every monoblock for a given scenario along one Plasma-Facing Unit. diff --git a/chapters/chapter4/divertor.tex b/chapters/chapter4/divertor.tex index 89ac22e..a39972d 100644 --- a/chapters/chapter4/divertor.tex +++ b/chapters/chapter4/divertor.tex @@ -5,7 +5,7 @@ \chapter{Divertor inventory estimation}\label{Chapter4}\labch{Chapter4} % \section{Introduction} -This Chapter focusses on the estimation of the \gls{H} \gls{inventory} in the \glspl{divertor} of \gls{west} and \gls{iter}. +This Chapter focusses on the estimation of the \gls{H} \gls{inventory} in the \glspl{divertor} of \acrshort{west} and \gls{iter}. This estimation relies on the \gls{monoblock} behaviour law computed in \refch{Chapter3}. This behaviour law allows rapid evaluations of the \glspl{monoblock} \gls{H} \gls{inventory} for any exposure condition. Inputs are taken from \gls{soledge}-EIRENE \cite{bufferand_three-dimensional_2019} and \gls{solps} \cite{kaveeva_solps-iter_2020} plasma simulations. @@ -22,7 +22,7 @@ \subsection{Plasma simulations} \begin{figure}[h!] \centering \includegraphics[width=0.95\linewidth]{Figures/Chapter4/coordinates.pdf} - \caption{Geometry of \gls{west} and \gls{iter} \glspl{divertor}.} + \caption{Poloidal cross section of \gls{west} and \gls{iter} showing the \glspl{divertor} in red.} \labfig{reactors} \end{figure} This Section describes the parameters of the plasma simulations. @@ -74,8 +74,15 @@ \subsubsection{\gls{solps} runs} \begin{figure}[h!] \centering - \includegraphics[width=\linewidth]{Figures/Chapter4/example.pdf} - \caption{Method of \gls{divertor} \gls{H} \gls{inventory} estimation based on the surface concentration, the surface temperature and the behaviour law obtained in \refch{Chapter3}.} + \begin{overpic}[width=\linewidth]{Figures/Chapter4/example.pdf} + % \linethickness{2pt} + \thicklines + \put(55,36){\color{black}\vector(-1, 0){30}} + \put(25,75){\color{black}\vector(0, -1){30}} + \put(35,45){\color{black}\vector(1, 1){30}} + \end{overpic} + + \caption{Method of WEST \gls{divertor} \gls{H} \gls{inventory} estimation based on the surface concentration, the surface temperature and the behaviour law obtained in \refch{Chapter3}.} \labfig{behaviour law example} \end{figure} @@ -112,7 +119,7 @@ \subsection{Estimation of exposure conditions} \begin{equation} c_\mathrm{surface, \, i} = \frac{R_{p, \mathrm{i}} \ \varphi_\mathrm{imp, \,i}}{D(T_\mathrm{surface})} \end{equation} -where $R_{p, i}$ is the implantation depth in \si{m}, $\varphi_{\mathrm{imp}, \,i}$ is the implanted particles flux in \si{m^{-2}.s^{-1}} and $D$ is the \gls{H} diffusion coefficient in \si{m^{2}.s^{-1}}. +where $R_{p, i}$ is the implantation depth in \si{m}, $\varphi_{\mathrm{imp}, \,i}$ is the implanted particles flux in \si{m^{-2}.s^{-1}} and $D$ is the \gls{H} diffusion coefficient in \si{m^{2}.s^{-1}} (see \refsec{triangle model}). Finally, the implanted flux can be expressed as: \begin{equation} @@ -137,22 +144,24 @@ \subsection{Estimation of exposure conditions} All of these steps have been automated and packaged into a tool called divHretention. divHretention can directly interpret \gls{solps}/\gls{soledge} data and produce a distribution of \gls{monoblock} \gls{inventory} as in \reffig{behaviour law example}. +\marginnote{ The source-code of the tool is under version control and openly available via GitHub under a MIT licence \cite{delaporte-mathurin_irfmdivhretention_2021}. The divHretention python package is distributed via PyPi \cite{delaporte-mathurin_divhretention_nodate}. Moreover, all the results obtained in this Chapter can be reproduced with the scripts available at \url{https://github.com/RemDelaporteMathurin/divHretention-Nucl.Fusion-2021}. +} -\section{\gls{iter} results} +\section{ITER results} \begin{figure*}[h!] \captionsetup[subfigure]{format=plain,singlelinecheck=true} % needed to center the subcaptions \centering - \begin{subfigure}{0.42\linewidth} + \begin{subfigure}{0.40\linewidth} \includegraphics[width=\linewidth]{Figures/Chapter4/ITER/inventory_along_inner_divertor.pdf} - \caption{IVT.} + \caption{Inner Vertical Target.} \end{subfigure}% \begin{subfigure}{0.58\linewidth} \includegraphics[width=\linewidth]{Figures/Chapter4/ITER/inventory_along_outer_divertor.pdf} - \caption{OVT.} + \caption{Outer Vertical Target.} \labfig{distrib outer target} \end{subfigure} \caption{Surface temperature, surface concentration and \gls{inventory} per unit thickness along the \gls{iter} \gls{divertor} with neutral pressures varying from \SI{2}{Pa} to \SI{11}{Pa}. The area corresponds to the 95\% confidence interval.} @@ -221,7 +230,7 @@ \section{\gls{iter} results} % local inventories -The inventory at the inner and outer \glspl{strike point} globally increases with the \gls{divertor} neutral pressure (see \reffig{local inventory neutral pressure}). +The inventory at the inner \gls{strike point} is constant from \SI{4}{Pa} whereas the inventory at the outer \gls{strike point} globally increases with the \gls{divertor} neutral pressure (see \reffig{local inventory neutral pressure}). The contribution of ions to the surface concentration at the inner strike point is around 50 \% and tends to decrease with increasing neutral pressure (see \reffig{ion contribution neutral pressure}). At low \gls{divertor} neutral pressure, the contribution of ions at the outer strike point is around 90 \% and tends to decrease with increasing neutral pressure. This can be explained by the fact that in both inner and outer targets, the integrated flux of ions decreases with increasing neutral pressure whereas the integrated flux of atoms increases, leading to a greater proportion of neutral particles. @@ -233,7 +242,7 @@ \section{\gls{iter} results} Past 300 discharges, the additional \gls{inventory} per discharge decreases with the number of discharges. The maximum is around \SI{5}{mg/discharge} between 30 and 100 discharges. -\section{\gls{west} results} +\section{WEST results} All the computations have been made for very long exposure times (\SI{e7}{s}) in order to better visualise trends. Even though cycling can have an effect on \gls{H} outgassing at the \gls{monoblock} plasma facing surface, it was shown in \refsec{influence of cycling} that the evolution of the \gls{monoblock} \gls{inventory} with the fluence was not affected. diff --git a/chapters/chapter5/He_transport_in_PFCs/Influence_on_H_transport.tex b/chapters/chapter5/He_transport_in_PFCs/Influence_on_H_transport.tex index 8761e67..4544d50 100644 --- a/chapters/chapter5/He_transport_in_PFCs/Influence_on_H_transport.tex +++ b/chapters/chapter5/He_transport_in_PFCs/Influence_on_H_transport.tex @@ -124,9 +124,9 @@ \subsection{Results} \item Initial state: the sample has some pre-existing defects %(proof from PAS that there are pre-existing defects before He implantation) \item \gls{He} implantation: all pre-existing defects are saturated with \gls{He} and bubbles are formed \item 1st \gls{D} implantation: \gls{D} can only be trapped around bubbles since defects are saturated with \gls{He} - \item 1st \gls{tds} (up to \SI{1250}{K}): \gls{D} is detrapped from bubbles is desorbed (\SI{550}{K} peak), \gls{He} dissociates from pre-existing defects + \item 1st \gls{tds} (up to \SI{1250}{K}): \gls{D} is detrapped from bubbles (\SI{550}{K} peak), \gls{He} dissociates from pre-existing defects \item 2nd \gls{D} implantation: \gls{D} is trapped around bubbles and in the non-saturated defects - \item 2nd \gls{tds} (up to \SI{1350}{K}): \gls{D} is detrapped from bubbles is desorbed (\SI{550}{K} peak) and from non-saturated defects (peaks 400K, 450K and 500K) + \gls{He} trapped in deeper traps dissociate (because the \gls{tds} goes to higher temperatures) + \item 2nd \gls{tds} (up to \SI{1350}{K}): \gls{D} is detrapped from bubbles (\SI{550}{K} peak) and from non-saturated defects (peaks 400K, 450K and 500K) + \gls{He} trapped in deeper traps dissociate (because the \gls{tds} goes to higher temperatures) \item 3rd to 5th \gls{D} implantations: \gls{D} is trapped around bubbles and in pre-existing defects \item 3rd to 5th \gls{tds}: \gls{D} is detrapped from bubbles and pre-existing defects (now more available than at the 2nd \gls{tds}) \end{itemize} diff --git a/chapters/chapter5/He_transport_in_PFCs/indirect_implantation.tex b/chapters/chapter5/He_transport_in_PFCs/indirect_implantation.tex index 0342e69..38dbfd7 100644 --- a/chapters/chapter5/He_transport_in_PFCs/indirect_implantation.tex +++ b/chapters/chapter5/He_transport_in_PFCs/indirect_implantation.tex @@ -6,7 +6,7 @@ \subsection{Neutron induced transmutation} In combination with the \gls{paramak} code \sidecite{shimwell_paramak_2021} used for creating the \gls{monoblock} geometry, a neutronics simulation was run to assess the total quantity of helium generation in a \gls{monoblock} under neutron irradiation with the \gls{openmc} code \sidecite{romano_openmc_2015}, a modern open-source Monte-Carlo neutron and photon transport code. \Gls{openmc} simulates the transport of neutroncs by modelling their paths from their birth until their deaths. -Neutron interactions with matter (reflexion, absorption, fission...) are simulated using a probabilistic approach where each reaction has a corresponding cross-section (taken from the \gls{endf}). +Neutron interactions with matter (reflexion, absorption, fission...) are simulated using a probabilistic approach where each reaction has a corresponding cross-section (taken from the \gls{endf} \sidecite{brown_endfb-viii0_2018}). In this simulation, a neutron source was placed above the \gls{monoblock} and the total helium production was tallied via the $(n,X\alpha)$ reaction rate (MT reaction number 207). The neutron source corresponds to a \SI{500}{MW} DT neutron source, which gives a neutron generation rate of \SI{1.8e20}{neutrons.s^{-1}} (based on the energy produced by the DT fusion reaction). @@ -97,7 +97,7 @@ \subsection{Comparison to direct implantation} \begin{equation} \Gamma = \varphi_\mathrm{imp} \, f(x) \end{equation} -where $\varphi_\mathrm{imp}$ is the implanted helium flux and $f(x)$ is a gaussian distribution centered on $R_p=\SI{1.5}{nm}$ and a width $\sigma=\SI{1.0}{nm}$. +where $\varphi_\mathrm{imp} = \SI{5e25}{m^{-2}.s^{-1}}$ is the implanted helium flux and $f(x)$ is a gaussian distribution centered on $R_p=\SI{1.5}{nm}$ and a width $\sigma=\SI{1.0}{nm}$, which correspond to typical implantation parameters for helium exposure in tokamaks. When comparing the production of helium from indirect sources with the quantity helium implanted from the plasma $\Gamma$, it appears that the indirect sources are negligible in the exposed region (see \reffig{comparison helium generation}). Indirect helium production may become dominant in bulk regions - though may not necessarily be enough to produce helium bubbles. diff --git a/chapters/chapter5/He_transport_in_PFCs/model.tex b/chapters/chapter5/He_transport_in_PFCs/model.tex index 23fb7be..38a1e2c 100644 --- a/chapters/chapter5/He_transport_in_PFCs/model.tex +++ b/chapters/chapter5/He_transport_in_PFCs/model.tex @@ -47,16 +47,20 @@ \subsection{Helium clustering model} \end{equation} where $\rho = \SI{6.3e28}{m^{-3}}$ is the atomic density of W in $\si{m^{-3}}$ and $E_b$ is the binding energy for the equilibrium \ce{AB -> A + B} in \si{eV}. -The reaction term $R_i$ is the coupling term between concentrations and is expressed as: +Considering only the absorption of $\mathrm{He}_1$ by other clusters (see \reffig{clustering sketch}): +\begin{equation} + \ce{He_xV_y + He_1 <=>[k^+][k^-] He_{x+1}V_y} +\end{equation} + +The reaction term $R_i$ is the coupling term between concentrations and is expressed as: +% [k^+_{1,x}][k^_{x+1}] \begin{equation} R_i= k^+_{i,i-1} c_1 c_{i-1} - k_{i, 1}^+ c_i c_{1} + k_{i+1}^- c_{i+1} - k_i^- c_i \labeq{reaction term} \end{equation} - - -In \refeq{reaction term}, $c_i$ is the concentration of clusters of size $i$ in \si{m^{-3}}. -The first term corresponds to the reactions producing clusters of size $i$. +where $c_i$ is the concentration of clusters of size $i$ in \si{m^{-3}}. +In \refeq{reaction term}, the first term corresponds to the reactions producing clusters of size $i$. The second one corresponds to the ones reacting with clusters of size $i$. The third term accounts for bigger clusters dissociating. Finally, the last term corresponds to clusters of size $i$ dissociating. @@ -80,10 +84,10 @@ \subsection{Grouped approach} A grouped approach proposed by Faney et al.\ \sidecite{faney_spatially_2014} for reducing the number of equations will therefore be employed. This technique consists in grouping clusters from an arbitrary size $N$ in a single equation while explicitly accounting for smaller clusters. To do so, the dissociation of large clusters is neglected (i.e.\ $k_i^- = 0$ for $i>N$). -This assumption is valid since the activation energy for \gls{trap mutation} events is lower than that of He or \gls{vacancy} emission\sidecite{boisse_modeling_2014}. +This assumption is valid since the activation energy for \gls{trap mutation} events is lower than that of He or \gls{vacancy} emission \sidecite{boisse_modeling_2014}. Dissociation of large clusters by \gls{vacancy} or He emission is therefore negligible. Moreover, clusters containing more than six \gls{He} atoms are assumed to be immobile (i.e.\ $D_i = 0$ for $i>6$). -This assumption is motivated by \gls{dft} and \gls{md} results suggesting that the \gls{self-trapping} energy is below the binding energy of one He atom in a pure He cluster for clusters containing more than five He atoms \sidecite{boisse_modeling_2014}. +This assumption is motivated by \gls{dft} and \gls{md} results suggesting that the \gls{self-trapping} energy is below the binding energy of one He atom in a pure He cluster for clusters containing more than five He atoms \cite{boisse_modeling_2014}. For smaller clusters ($\mathrm{He}_1$, $\mathrm{He}_2$,$\ldots$, $\mathrm{He}_6$) the diffusion coefficient and the dissociation by He emission energy vary with the number of \gls{He} atoms in the cluster (see \reftab{clusters properties}). @@ -173,7 +177,7 @@ \subsection{Grouped approach} \langle r_b \rangle = r_{\mathrm{He}_0 \mathrm{V}_1} + \left(\frac{3}{4 \pi} \frac{a_0^3}{2} \right)^{1/3} \frac{1}{c_b}\sum\limits_{i=N+1}^\infty c_i(\frac{i}{4})^{1/3} - \left(\frac{3}{4 \pi} \frac{a_0^3}{2} \right)^{1/3} \end{equation} -Assuming $c_i$ follows a narrow gaussian distribution, $\frac{1}{c_b}\sum\limits_{i=N+1}^\infty c_i(\frac{i}{4})^{1/3} \approx \left( \frac{1}{c_b}\sum\limits_{i=N+1}^\infty c_i\frac{i}{4} \right)^{1/3} $ (see \reffig{sum of powers approximation}). +Assuming $c_i$ follows a narrow gaussian distribution \cite{faney_spatially_2015}, $\frac{1}{c_b}\sum\limits_{i=N+1}^\infty c_i(\frac{i}{4})^{1/3} \approx \left( \frac{1}{c_b}\sum\limits_{i=N+1}^\infty c_i\frac{i}{4} \right)^{1/3} $ (see \reffig{sum of powers approximation}). % is assumed to be only dependent on $\langle i_b \rangle$. \begin{figure} @@ -188,7 +192,7 @@ \subsection{Grouped approach} \end{figure} -The final expression of the bubble radius is therefore: +The final expression of the bubble mean radius is therefore: \begin{equation} \langle r_b \rangle = r_{\mathrm{He}_0 \mathrm{V}_1} + \left(\frac{3}{4 \pi} \frac{a_0^3}{2} \frac{\langle i_b \rangle}{4} \right)^{1/3} - \left(\frac{3}{4 \pi} \frac{a_0^3}{2} \right)^{1/3} \end{equation} diff --git a/chapters/chapter5/He_transport_in_PFCs/verification_validation.tex b/chapters/chapter5/He_transport_in_PFCs/verification_validation.tex index f0c7668..a52ebec 100644 --- a/chapters/chapter5/He_transport_in_PFCs/verification_validation.tex +++ b/chapters/chapter5/He_transport_in_PFCs/verification_validation.tex @@ -7,7 +7,7 @@ \subsection{Code comparison: Tendril case} \labsec{tendril case} \labfig{tendril profiles} \end{figure*} -The current implementation was compared to literature results\cite{faney_spatially_2015}. +The current implementation was compared to literature results \cite{faney_spatially_2015}. Helium exposure in a \gls{tendril} was simulated in 1D. The helium flux is \SI{1e22}{m^{-2}.s^{-1}} and the fluence was \SI{5e25}{m^{-2}}. @@ -81,5 +81,6 @@ \subsection{Comparison with experiments} The bubble radius $\langle r_b \rangle$ is however overestimated by an order of magnitude compared to experimental measurements. This could imply that the current model linking the He content to the bubble radius is overestimated and that a more accurate one is needed. -Bursting in over-pressurised bubbles close to the surface would also reduce the bubble size. +The model parametetrisation could also have an impact on these results (see \reffig{parametric study dissociation energies}). +Bursting in over-pressurised bubbles close to the surface would also reduce the bubble size. Finally, it would be worth investigating this further to determine the impact of initial defects. diff --git a/chapters/codeposition_model.tex b/chapters/codeposition_model.tex deleted file mode 100644 index 6eadbeb..0000000 --- a/chapters/codeposition_model.tex +++ /dev/null @@ -1,3 +0,0 @@ -\setchapterstyle{lines} -\labpage{Co-deposition model} -\chapter{Co-deposition model} \ No newline at end of file diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index 96ae5e0..6319e39 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -38,7 +38,7 @@ \section*{Limitations} For instance, at the time the simulations in \refsec{influence of exposure conditions} were run, the surface concentration could not be inhomogeneous \emph{and} directly dependent on the inhomogeneous surface temperature due to the heat flux. Therefore, the choice that was made was to impose a homogeneous surface temperature instead (and a homogeneous surface concentration). Even though the required feature was added a few months later, re-running all the \gls{festim} simulations was too much time-consuming given the time constraints. -Many of these development drawbacks could have been alleviated if \gls{festim} had been open-sourced, as external community experts such as \gls{fenics} developers could have more easily contributed to its development or bug fixes. +% Many of these development drawbacks could have been alleviated if \gls{festim} had been open-sourced, as external community experts such as \gls{fenics} developers could have more easily contributed to its development or bug fixes. This study also has physical limitations inherent to the assumptions that have been made. While some of these assumptions are conservative (i.e.\ represent a worst-case scenario) and do not jeopardise the key findings, others were made as a response to uncertainties. diff --git a/chapters/demo_monoblock.tex b/chapters/demo_monoblock.tex index c272db8..b59f748 100644 --- a/chapters/demo_monoblock.tex +++ b/chapters/demo_monoblock.tex @@ -40,8 +40,8 @@ \section{Methodology} \begin{subequations} \begin{align} - -\lambda \vec{\nabla} T \cdot \vec{n} &=\varphi_\mathrm{heat} \quad &\text { on } \Gamma_\mathrm{top}\\ - -\lambda \vec{\nabla} T\cdot \vec{n} &= -h \cdot \left(T_\mathrm{coolant} - T\right)\quad &\text { on } \Gamma_\mathrm{coolant} + -\lambda \nabla T \cdot \mathbf{n} &=\varphi_\mathrm{heat} \quad &\text { on } \Gamma_\mathrm{top}\\ + -\lambda \nabla T\cdot \mathbf{n} &= -h \cdot \left(T_\mathrm{coolant} - T\right)\quad &\text { on } \Gamma_\mathrm{coolant} \end{align} \labeq{bc thermal DEMO monoblock} \end{subequations} @@ -50,9 +50,9 @@ \section{Methodology} \begin{subequations} \begin{align} c_\mathrm{m} &= \frac{\varphi_\mathrm{imp} R_p}{D} \quad &\text { on } \Gamma_\mathrm{top}\\ - -D \vec{\nabla} c_\mathrm{m} \cdot \vec{n} &= K_\mathrm{CuCrZr} \cdot c_\mathrm{m}^{2} \quad &\text { on } \Gamma_\mathrm{coolant} \\ + -D \nabla c_\mathrm{m} \cdot \mathbf{n} &= K_\mathrm{CuCrZr} \cdot c_\mathrm{m}^{2} \quad &\text { on } \Gamma_\mathrm{coolant} \\ c_\mathrm{m} &= 0 \quad &\text { on } \Gamma_\mathrm{toroidal} \text{ and } \Gamma_\mathrm{pipe} \\ - c_\mathrm{m} &= 0 \quad \text{or} \quad -D \vec{\nabla} c_\mathrm{m} \cdot \vec{n} = 0 &\text { on } \Gamma_\mathrm{poloidal} + c_\mathrm{m} &= 0 \quad \text{or} \quad -D \nabla c_\mathrm{m} \cdot \mathbf{n} = 0 &\text { on } \Gamma_\mathrm{poloidal} \end{align} \labeq{bc H transport DEMO monoblock} \end{subequations} diff --git a/glossary.tex b/glossary.tex index 88beb5e..1818b5c 100644 --- a/glossary.tex +++ b/glossary.tex @@ -136,7 +136,7 @@ } \newglossaryentry{retention}{ name=retention, - description={Local concentration of soluted species (hydrogen or helium) including the mobile species and the trapped ones} + description={Local concentration of soluted species (hydrogen or helium) including the mobile and trapped species} } \newglossaryentry{inventory}{ name=inventory, @@ -148,13 +148,13 @@ } % Glossary entries (used in text with e.g. \acrfull{fpsLabel} or \acrshort{fpsLabel}) -\newacronym{H}{H}{Hydrogen} -\newacronym{He}{He}{Helium} -\newacronym{D}{D}{Deuterium} -\newacronym{T}{T}{Tritium} -\newacronym{Li}{Li}{Lithium} -\newacronym{W}{W}{Tungsten} -\newacronym{Be}{Be}{Beryllium} +\newacronym{H}{H}{hydrogen} +\newacronym{He}{He}{helium} +\newacronym{D}{D}{deuterium} +\newacronym{T}{T}{tritium} +\newacronym{Li}{Li}{lithium} +\newacronym{W}{W}{tungsten} +\newacronym{Be}{Be}{beryllium} \newacronym[longplural={Plasma Facing Units}, shortplural={PFUs}]{pfuLabel}{PFU}{Plasma Facing Unit} @@ -170,7 +170,7 @@ \newacronym{hcll}{HCLL}{Helium-Cooled-Liquid-Lead} \newacronym{dcll}{DCLL}{Dual-Coolant-Lithium-Lead} \newacronym{tbr}{TBR}{Tritium breeding ratio} -\newacronym{fpy}{FPY}{Full power year} +\newacronym{fpy}{FPY}{Full Power Year} \newglossaryentry{tokamak}{ @@ -218,7 +218,7 @@ } \newacronym{bcc}{bcc}{Body-Centered Cubic (type of metallic lattice)} \newglossaryentry{lattice}{ - name=Lattice, + name=lattice, description={Three-dimensional crystalline structure of metals. The lattice is how the atoms are ordered within a metal} } diff --git a/main.tex b/main.tex index e6ecd7e..cc50544 100644 --- a/main.tex +++ b/main.tex @@ -176,7 +176,7 @@ Hydrogen transport in tokamaks\\ } \subtitle{ - Estimation of the ITER divertor tritium inventory \\ + Estimation of the ITER divertor tritium inventory and influence of helium exposure \\ \noindent\rule{8cm}{0.4pt} }