From dd6e72f505274926c031897ff7b4f6eb39c91381 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Wed, 4 Nov 2020 08:47:09 -0600 Subject: [PATCH 01/19] Make a home for glow discharge model doc --- doc/.gitignore | 11 +++++ doc/Makefile | 31 ++++++++++++ doc/bibs/glow.bib | 10 ++++ doc/figures/glowfigcrop.pdf | Bin 0 -> 35824 bytes doc/gitinfo2.pm | 93 ++++++++++++++++++++++++++++++++++++ doc/glow.tex | 68 ++++++++++++++++++++++++++ doc/overview.tex | 11 +++++ doc/vc | 27 +++++++++++ doc/vc-git.awk | 89 ++++++++++++++++++++++++++++++++++ 9 files changed, 340 insertions(+) create mode 100644 doc/.gitignore create mode 100644 doc/Makefile create mode 100644 doc/bibs/glow.bib create mode 100644 doc/figures/glowfigcrop.pdf create mode 100755 doc/gitinfo2.pm create mode 100644 doc/glow.tex create mode 100644 doc/overview.tex create mode 100755 doc/vc create mode 100644 doc/vc-git.awk diff --git a/doc/.gitignore b/doc/.gitignore new file mode 100644 index 00000000..2664a243 --- /dev/null +++ b/doc/.gitignore @@ -0,0 +1,11 @@ +vc.tex +*~ +glow.aux +glow.fdb_latexmk +glow.fls +glow.log +glow.pdf +glow.bbl +glow.blg +glow.out +glow.toc diff --git a/doc/Makefile b/doc/Makefile new file mode 100644 index 00000000..847d01da --- /dev/null +++ b/doc/Makefile @@ -0,0 +1,31 @@ +PAPER := glow.pdf +TEX_SUFS := .aux .log .nav .out .snm .toc .vrb .fdb_latexmk .bbl .blg .fls +GITID := $(shell git describe --always 2> /dev/null) +TEXSRC := $(wildcard *.tex) +FIGURES := $(wildcard figures/*.pdf figures/*.jpg) +BIBS := $(wildcard bibs/*.bib) + +#$(info $$GITID is [${GITID}]) +$(info $$FIGURES is [${FIGURES}]) +$(info $$PAPER is [${PAPER}]) +$(info $$TEXSRC is [${TEXSRC}]) +$(info $$FIGURES is [${FIGURES}]) +$(info $$BIBS is [${BIBS}]) + + +%.pdf:%.tex $(TEXSRC) $(BIBS) $(FIGURES) gitinfo2.pm + echo "hello" +ifdef GITID + ./gitinfo2.pm +endif + latexmk -pdf $< + +all: $(PAPER) + +clean: + $(RM) $(foreach suf, ${TEX_SUFS}, $(PAPER:.pdf=${suf})) +ifdef GITID + $(RM) vc.tex +endif + $(RM) $(PAPER) + diff --git a/doc/bibs/glow.bib b/doc/bibs/glow.bib new file mode 100644 index 00000000..dd6aa8ea --- /dev/null +++ b/doc/bibs/glow.bib @@ -0,0 +1,10 @@ +@article{panneer2015computational, + title={Computational modeling of the effect of external electron injection into a direct-current microdischarge}, + author={Panneer Chelvam, Prem Kumar and Raja, Laxminarayan L}, + journal={Journal of Applied Physics}, + volume={118}, + number={24}, + pages={243301}, + year={2015}, + publisher={AIP Publishing LLC} +} diff --git a/doc/figures/glowfigcrop.pdf b/doc/figures/glowfigcrop.pdf new file mode 100644 index 0000000000000000000000000000000000000000..73a7838a46c42422a887ac56de51045d4664bd77 GIT binary patch literal 35824 zcmeFY1ys~s`zQ*6(h5@2NJ}@;C0znigMf4l1JWJR-Q6XiF!a!&0@96x3?NDh(jXn@ zH~7Bq_kI6!|M#A??pkN9v(|B`GrOMs)ZY84VSN5fo`Z)|0GkoEU3G!YPs>f~Xl92k zCPvGpYUyC@YC|gkbZKGJ($aFt+uFNYIs>2frmmLHEX^G)EU_gduw7i8ElnZVo;iFk zxKU(`WNA$;EDEj73KMdMbozz@PuQ^fk=vEX`eS=XeUz3L=R^jsbuF+JpC>J!rXJc)hly z<$7*vZAr^W%X7Qub5mza2Uh^>-&|3%w6HamarC4$1iH9sd3m2A|8ep^r8T-;AnWMh 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Barazzutti +# +# GitInfo2LatexMk - v0.2.1 +# Inspired by Brent Longborough's update-git.sh (part of gitinfo2 LaTeX package) +# +# The original update-git.sh is supposed to be "hooked" to some git events (such that +# Post-{commit,checkout,merge}). +# Although this approach is elegant, I find a bit too intrusive and complicated +# to maintain. +# +# This Perl variant makes sense for latexmk users. The only requirement is to add +# the following line into the .latexmkrc file that lays in the root of your +# LaTeX project (create it, if absent). +# +# do './gitinfo2.pm' +# +# That's it! Now it'd work! + +sub git_info_2 { + + # get file content as a string + my $get_file_content = sub { + my ($f)= @_; + + # do not separate the reads per line + local $/ = undef; + + open FILE, $f or return ""; + $string = ; + + close FILE; + return $string; + }; + + # compare two files` + my $cmp = sub { + my($a,$b) = @_; + + return $get_file_content->($a) ne $get_file_content->($b); + }; + + my $RELEASE_MATCHER = "[0-9]*.*"; + + if(%GI2TM_OPTIONS){ + if(exists $GI2TM_OPTIONS{"RELEASE_MATCHER"}){ + $RELEASE_MATCHER = $GI2TM_OPTIONS{"RELEASE_MATCHER"}; + } + } + + # When running in a sub-directories of the repo + my $REPOBASE = `git rev-parse --show-toplevel`; + chomp($REPOBASE); + my $GITDIR = "$REPOBASE/.git"; + + my $GIN = "$GITDIR/gitHeadInfo.gin"; + my $NGIN = "$GIN.new"; + + print "gitinfo2pm: saving git revision info at $GIN\n"; + +# if(length(`git status --porcelain`) == 0){ + # Get the first tag found in the history from the current HEAD + my $FIRSTTAG = `git describe --tags --always --dirty='-*'`; + chop($FIRSTTAG); + + # Get the first tag in history that looks like a Release + my $RELTAG = `git describe --tags --long --always --dirty='-*' --match '$RELEASE_MATCHER'`; + chop($RELTAG); + + # Hoover up the metadata + my $metadata =`git --no-pager log -1 --date=short --decorate=short --pretty=format:"shash={%h}, lhash={%H}, authname={%an}, authemail={%ae}, authsdate={%ad}, authidate={%ai}, authudate={%at}, commname={%an}, commemail={%ae}, commsdate={%ad}, commidate={%ai}, commudate={%at}, refnames={%d}, firsttagdescribe={$FIRSTTAG}, reltag={$RELTAG} " HEAD`; + + open(my $fh,'>',$NGIN); + print $fh "\\usepackage[".$metadata."]{gitexinfo}\n"; + close $fh; + +# }else{ +# print "GIT UNCLEAN\n"; +# } + + $cmp->($GIN,$NGIN ); + + if((-e $GIN || -e $NGIN) && $cmp->($GIN, $NGIN)) { + print "Status changed, request recompilation\n"; + $go_mode = 1; + unlink($GIN); + rename($NGIN, $GIN); + } else { + unlink($NGIN); + } +} + +git_info_2(); diff --git a/doc/glow.tex b/doc/glow.tex new file mode 100644 index 00000000..b01ee8a7 --- /dev/null +++ b/doc/glow.tex @@ -0,0 +1,68 @@ +\documentclass[11pt]{report} + +\usepackage{fullpage} +\usepackage{graphicx} +\usepackage{xcolor} +\usepackage{todonotes} +\usepackage[square,numbers]{natbib} +\usepackage[american,siunitx]{circuitikz} + +\usepackage[font=small,skip=2pt,justification=raggedright]{subcaption} +\usepackage[font=small,skip=2pt,justification=raggedright]{caption} + +\usepackage{amsmath,amssymb,stmaryrd} +\usepackage[makeroom]{cancel} + +\graphicspath{{./figures/}} + +% include version control watermark +\usepackage[mark,maxdepth=6,raisemark=2.\baselineskip]{gitinfo2} +\renewcommand{\gitMark}{\gitAbbrevHash{} (\gitAuthorDate)} +\renewcommand{\gitMarkPref}{[~revision~]} + +% custom colors +\definecolor{copper}{rgb}{0.72,0.45,0.20} +\definecolor{darksilver}{rgb}{0.67,0.67,0.67} +\definecolor{silver}{rgb}{0.75,0.75,0.75} +\definecolor{lightsilver}{rgb}{0.83,0.83,0.83} +\definecolor{mint}{rgb}{0.6,0.98,0.6} +\definecolor{darkgreen}{rgb}{0.0,0.5,0.0} + +% custom commands +\newcommand{\mbb}[1]{\mathbb{#1}} +\newcommand{\mbf}[1]{\mathbf{#1}} +\newcommand{\sbf}[1]{\boldsymbol{#1}} +\newcommand{\mcal}[1]{\mathcal{#1}} +\newcommand{\mfk}[1]{\mathfrak{#1}} +\newcommand{\pp}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\dd}[2]{\frac{d #1}{d #2}} +\newcommand{\rarrow}{\rightarrow} +\newcommand{\Rarrow}{\Rightarrow} +\newcommand{\LRarrow}{\Leftrightarrow} +\newcommand{\ud}{\,\mathrm{d}} +\newcommand{\jump}[1]{\llbracket #1 \rrbracket} +\newcommand{\avg}[1]{\{ #1 \}} + +% title foo +% TODO: make a snazzy title page +\title{Glow Discharge Modeling Document} +\author{% + The PECOS Development Team\\ + Oden Institute for Computational Engineering and Sciences\\ + The University of Texas at Austin +} +\date{} + + +\begin{document} +\maketitle +\tableofcontents + +\chapter*{Preface} +\input{overview} + + +\bibliographystyle{plainnat} +\bibliography{bibs/glow} + +\end{document} diff --git a/doc/overview.tex b/doc/overview.tex new file mode 100644 index 00000000..0c701b20 --- /dev/null +++ b/doc/overview.tex @@ -0,0 +1,11 @@ +This document describes the mathematical models and numerical methods +that will be used to simulate the glow discharge device. A schematic +of the device is shown in Figure~\ref{fig:glowSchematic}. +% +\begin{figure}[htp] +\begin{center} +\includegraphics[width=0.6\linewidth]{glowfigcrop.pdf} +\end{center} +\caption{Schematic of the glow discharge device (taken from the proposal).} +\label{fig:glowSchematic} +\end{figure} diff --git a/doc/vc b/doc/vc new file mode 100755 index 00000000..7ae6c8cc --- /dev/null +++ b/doc/vc @@ -0,0 +1,27 @@ +#!/bin/sh +# This is file 'vc' from the vc bundle for TeX. +# The original file can be found at CTAN:support/vc. +# This file is Public Domain. + +# Parse command line options. +full=0 +mod=0 +while [ -n "$(echo $1 | grep '-')" ]; do + case $1 in + -f ) full=1 ;; + -m ) mod=1 ;; + * ) echo 'usage: vc [-f] [-m]' + exit 1 + esac + shift +done + +MY_PATH="`dirname \"$0\"`" + +# English locale. +LC_ALL=C +git --no-pager log -1 HEAD --pretty=format:"Hash: %H%nAbr. Hash: %h%nParent Hashes: %P%nAbr. Parent Hashes: %p%nAuthor Name: %an%nAuthor Email: %ae%nAuthor Date: %ai%nCommitter Name: %cn%nCommitter Email: %ce%nCommitter Date: %ci%n" |gawk -v script=log -v full=$full -f $MY_PATH/vc-git.awk > vc.tex +if [ "$mod" = 1 ] +then + git status |gawk -v script=status -f vc-git.awk >> vc.tex +fi diff --git a/doc/vc-git.awk b/doc/vc-git.awk new file mode 100644 index 00000000..66b35263 --- /dev/null +++ b/doc/vc-git.awk @@ -0,0 +1,89 @@ +# This is file 'vc-git.awk' from the vc bundle for TeX. +# The original file can be found at CTAN:support/vc. +# This file is Public Domain. +BEGIN { + +### Process output of "git status". + if (script=="status") { + modified = 0 + } + +} + + + +### Process output of "git log". +script=="log" && /^Hash:/ { Hash = substr($0, 2+match($0, ":")) } +script=="log" && /^Abr. Hash:/ { AbrHash = substr($0, 2+match($0, ":")) } +script=="log" && /^Parent Hashes:/ { ParentHashes = substr($0, 2+match($0, ":")) } +script=="log" && /^Abr. Parent Hashes:/ { AbrParentHashes = substr($0, 2+match($0, ":")) } +script=="log" && /^Author Name:/ { AuthorName = substr($0, 2+match($0, ":")) } +script=="log" && /^Author Email:/ { AuthorEmail = substr($0, 2+match($0, ":")) } +script=="log" && /^Author Date:/ { AuthorDate = substr($0, 2+match($0, ":")) } +script=="log" && /^Committer Name:/ { CommitterName = substr($0, 2+match($0, ":")) } +script=="log" && /^Committer Email:/ { CommitterEmail = substr($0, 2+match($0, ":")) } +script=="log" && /^Committer Date:/ { CommitterDate = substr($0, 2+match($0, ":")) } + +### Process output of "git status". +### Changed index? +script=="status" && /^# Changes to be committed:/ { modified = 1 } +### Unstaged modifications? +script=="status" && /^# Changed but not updated:/ { modified = 2 } + + + +END { + +### Process output of "git log". + if (script=="log") { +### Standard encoding is UTF-8. + if (Encoding == "") Encoding = "UTF-8" +### Extract relevant information from variables. + LongDate = substr(AuthorDate, 1, 25) + DateRAW = substr(LongDate, 1, 10) + DateISO = DateRAW + DateTEX = DateISO + gsub("-", "/", DateTEX) + Time = substr(LongDate, 12, 14) +### Write file identification to vc.tex. + print "%%% This file has been generated by the vc bundle for TeX." + print "%%% Do not edit this file!" + print "%%%" +### Write Git specific macros. + print "%%% Define Git specific macros." + print "\\gdef\\GITHash{" Hash "}%" + print "\\gdef\\GITAbrHash{" AbrHash "}%" + print "\\gdef\\GITParentHashes{" ParentHashes "}%" + print "\\gdef\\GITAbrParentHashes{" AbrParentHashes "}%" + print "\\gdef\\GITAuthorName{" AuthorName "}%" + print "\\gdef\\GITAuthorEmail{" AuthorEmail "}%" + print "\\gdef\\GITAuthorDate{" AuthorDate "}%" + print "\\gdef\\GITCommitterName{" CommitterName "}%" + print "\\gdef\\GITCommitterEmail{" CommitterEmail "}%" + print "\\gdef\\GITCommitterDate{" CommitterDate "}%" +### Write generic version control macros. + print "%%% Define generic version control macros." + print "\\gdef\\VCRevision{\\GITAbrHash}%" + print "\\gdef\\VCAuthor{\\GITAuthorName}%" + print "\\gdef\\VCDateRAW{" DateRAW "}%" + print "\\gdef\\VCDateISO{" DateISO "}%" + print "\\gdef\\VCDateTEX{" DateTEX "}%" + print "\\gdef\\VCTime{" Time "}%" + print "\\gdef\\VCModifiedText{\\textcolor{red}{with local modifications!}}%" + print "%%% Assume clean working copy." + print "\\gdef\\VCModified{0}%" + print "\\gdef\\VCRevisionMod{\\VCRevision}%" + } + +### Process output of "git status". + if (script=="status") { + print "%%% Is working copy modified?" + print "\\gdef\\VCModified{" modified "}%" + if (modified==0) { + print "\\gdef\\VCRevisionMod{\\VCRevision}%" + } else { + print "\\gdef\\VCRevisionMod{\\VCRevision~\\VCModifiedText}%" + } + } + +} From 4fa2dd6381d99eeb876256c2e05d7cd5fc0187fa Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Wed, 4 Nov 2020 08:48:21 -0600 Subject: [PATCH 02/19] make build less chatty --- doc/Makefile | 8 -------- 1 file changed, 8 deletions(-) diff --git a/doc/Makefile b/doc/Makefile index 847d01da..ca320686 100644 --- a/doc/Makefile +++ b/doc/Makefile @@ -5,14 +5,6 @@ TEXSRC := $(wildcard *.tex) FIGURES := $(wildcard figures/*.pdf figures/*.jpg) BIBS := $(wildcard bibs/*.bib) -#$(info $$GITID is [${GITID}]) -$(info $$FIGURES is [${FIGURES}]) -$(info $$PAPER is [${PAPER}]) -$(info $$TEXSRC is [${TEXSRC}]) -$(info $$FIGURES is [${FIGURES}]) -$(info $$BIBS is [${BIBS}]) - - %.pdf:%.tex $(TEXSRC) $(BIBS) $(FIGURES) gitinfo2.pm echo "hello" ifdef GITID From a818709b6f153a14f3f357ef57f0a6c2af3631f6 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Wed, 4 Nov 2020 08:55:54 -0600 Subject: [PATCH 03/19] proposed chapter outline for glow discharge model doc --- doc/glow.tex | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/doc/glow.tex b/doc/glow.tex index b01ee8a7..c8d99b54 100644 --- a/doc/glow.tex +++ b/doc/glow.tex @@ -61,6 +61,16 @@ \chapter*{Preface} \input{overview} +\chapter{Governing PDEs} + +\chapter{Boundary Conditions} + +\chapter{Closures: Transport and Chemistry} + +\chapter{Supporting Models and Data} + +\chapter{Numerical Methods} + \bibliographystyle{plainnat} \bibliography{bibs/glow} From 97043a7fe4ea1359de03109ac21a5c94a4318a90 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Thu, 5 Nov 2020 21:34:17 -0600 Subject: [PATCH 04/19] Some notes on bolsig for glow discharge model doc. --- doc/Makefile | 2 +- doc/bolsig/eqns.tex | 112 ++++++++++++++++++++++++++++++++++++++++ doc/bolsig/overview.tex | 6 +++ doc/glow.tex | 2 + 4 files changed, 121 insertions(+), 1 deletion(-) create mode 100644 doc/bolsig/eqns.tex create mode 100644 doc/bolsig/overview.tex diff --git a/doc/Makefile b/doc/Makefile index ca320686..4fc7217e 100644 --- a/doc/Makefile +++ b/doc/Makefile @@ -1,7 +1,7 @@ PAPER := glow.pdf TEX_SUFS := .aux .log .nav .out .snm .toc .vrb .fdb_latexmk .bbl .blg .fls GITID := $(shell git describe --always 2> /dev/null) -TEXSRC := $(wildcard *.tex) +TEXSRC := $(wildcard *.tex bolsig/*.tex) FIGURES := $(wildcard figures/*.pdf figures/*.jpg) BIBS := $(wildcard bibs/*.bib) diff --git a/doc/bolsig/eqns.tex b/doc/bolsig/eqns.tex new file mode 100644 index 00000000..0fd33fa9 --- /dev/null +++ b/doc/bolsig/eqns.tex @@ -0,0 +1,112 @@ +\section{BOLSIG+ Governing Equations} +The electron Boltzmann equation may be written as follows: +% +\begin{equation} +\pp{f}{t} ++ +\mbf{v} \cdot \nabla f +- +\frac{e}{m} \mbf{E} \cdot \nabla_{\mbf{v}} f += +C[f], +\label{eqn:electron-boltzmann-full} +\end{equation} +% +where $f(\mbf{x}, \mbf{v}, t)$ denotes the electron distribution +function, $\mbf{v}$ is the velocity vector, $\nabla f$ is the spatial +gradient of $f$ (i.e., wrt $\mbf{x}$), $\nabla_{\mbf{v}} f$ is the +velocity gradient of $f$ (i.e., wrt $\mbf{v}$), $e$ is the absolute +value of the charge of an electron, $m$ is the mass of an electron, +$\mbf{E}$ is the electric field, and $C[f]$ denotes the collision +term. + +The development here is based upon the description of Hagelaar and +Pitchford~\cite{} (referred to as HP for short) and the BOLSIG+ User's +Manual~\cite{}. + +To begin, it is assumed that the electric field and collision +probabilities are spatially uniform. In this situation, HP asserts +that the distribution function $f$ is 1) ``symmetric in velocity space +around the electric field direction'' and 2) ``may vary only along the +field direction'' in physical space. The meaning of 1) is somewhat +unclear, but in the subsequent analysis, a spherical coordinate system +is used and no possible dependence on the azimuthal angle is included. +Thus, the combined import of these assumptions is to reduce the +dependence of $f$ from 6D plus time to 3D plus time. Specifically, +letting $z$ denote the direction of the electric field, $v$ denote the +velocity magnitude, and $\theta$ denote the polar angle between the +electric field direction and the velocity direction, we have $f = f(z, +v, \theta, t)$. In this situation,~\ref{eqn:electron-boltzmann-full} +simplifies to +% +\begin{equation} +\pp{f}{t} ++ +v \cos \theta \pp{f}{z} +- +\frac{e}{m} E \left( \cos \theta \pp{f}{v} - \frac{\sin \theta}{v} \pp{f}{\theta} \right) += +C[f]. +\label{eqn:electron-boltzmann-reduced} +\end{equation} +% + +In preparation for subsequent development, let $\xi = \cos \theta$. +Then,~\ref{eqn:electron-boltzmann-reduced} becomes +% +\begin{equation} +\pp{f}{t} ++ +v \xi \pp{f}{z} +- +\frac{e}{m} E \left( \xi \pp{f}{v} + \frac{(1-\xi^2)}{v} \pp{f}{\xi} \right) += +C[f]. +\label{eqn:electron-boltzmann-xi} +\end{equation} +% +To continue, the following two-term expansion is introduced: +% +\begin{equation} +f(v, \xi, z, t) = f_0(v, z, t) + \xi f_1(v, z, t). +\label{eqn:two-term} +\end{equation} +% +To derive governing equations for $f_0$ and $f_1$, we apply the +Galerkin method in the $\xi$ coordinate. Specifically, +substitute~\eqref{eqn:two-term} into +~\eqref{eqn:electron-boltzmann-xi} and integrate against $1$ and +$\xi$. Rewriting the result in terms of a rescaled energy variable +$\epsilon \equiv (v^2 / \gamma^2)$, where $\gamma^2 = 2e / m$, gives +% +\begin{gather} +\pp{f_0}{t} ++ +\frac{\gamma}{3} \sqrt{\epsilon} \pp{f_1}{z} +- +\frac{\gamma}{3} \frac{E}{\sqrt{\epsilon}} \pp{}{\epsilon} \left( \epsilon f_1 \right) += +C_0 +\label{eqn:f0}\\ +% +\pp{f_1}{t} ++ +\gamma \sqrt{\epsilon} \pp{f_0}{z} +- +\gamma E \sqrt{\epsilon} \pp{f_0}{\epsilon} += +C_1 +\label{eqn:f1} +\end{gather} +% +where +% +\begin{gather*} +C_0 = \frac{1}{2} \int_{-1}^{1} C[f_0 + \xi f_1] \, d\xi\\ +C_1 = \frac{3}{2} \int_{-1}^{1} \xi C[f_0 + \xi f_1] \, d\xi. +\end{gather*} +% +Note that~\eqref{eqn:f0} and~\eqref{eqn:f1} are equivalent to HP (5) +and (6), respectively. Although, to match the forms in HP more +precisely, we must examine the collision terms in more detail. +\todo[inline]{Collisions} diff --git a/doc/bolsig/overview.tex b/doc/bolsig/overview.tex new file mode 100644 index 00000000..34aaa43e --- /dev/null +++ b/doc/bolsig/overview.tex @@ -0,0 +1,6 @@ +This chapter documents the models and data that support determination +of transport coefficients and reaction rates. For electron transport +properties and electron-impact reactions, the coefficients and rates +are determined using the BOLSIG+ +code\footnote{http://www.bolsig.laplace.univ-tlse.fr/}, which solves +an approximate form of the electron Boltzmann equation. diff --git a/doc/glow.tex b/doc/glow.tex index c8d99b54..e17cde14 100644 --- a/doc/glow.tex +++ b/doc/glow.tex @@ -68,6 +68,8 @@ \chapter{Boundary Conditions} \chapter{Closures: Transport and Chemistry} \chapter{Supporting Models and Data} +\input{bolsig/overview} +\input{bolsig/eqns} \chapter{Numerical Methods} From a2b88917dd89df779af68ccb41657f84449c9e67 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Wed, 2 Dec 2020 11:45:55 -0600 Subject: [PATCH 05/19] Some documentation for the glow discharge model --- doc/Makefile | 2 +- doc/fluid/bcs.tex | 1 + doc/fluid/closure.tex | 3 ++ doc/fluid/pdes.tex | 87 +++++++++++++++++++++++++++++++++++++++++++ doc/glow.tex | 9 ++--- 5 files changed, 96 insertions(+), 6 deletions(-) create mode 100644 doc/fluid/bcs.tex create mode 100644 doc/fluid/closure.tex create mode 100644 doc/fluid/pdes.tex diff --git a/doc/Makefile b/doc/Makefile index 4fc7217e..014ef84c 100644 --- a/doc/Makefile +++ b/doc/Makefile @@ -1,7 +1,7 @@ PAPER := glow.pdf TEX_SUFS := .aux .log .nav .out .snm .toc .vrb .fdb_latexmk .bbl .blg .fls GITID := $(shell git describe --always 2> /dev/null) -TEXSRC := $(wildcard *.tex bolsig/*.tex) +TEXSRC := $(wildcard *.tex bolsig/*.tex fluid/*.tex) FIGURES := $(wildcard figures/*.pdf figures/*.jpg) BIBS := $(wildcard bibs/*.bib) diff --git a/doc/fluid/bcs.tex b/doc/fluid/bcs.tex new file mode 100644 index 00000000..d540158e --- /dev/null +++ b/doc/fluid/bcs.tex @@ -0,0 +1 @@ +\section{Boundary Conditions} diff --git a/doc/fluid/closure.tex b/doc/fluid/closure.tex new file mode 100644 index 00000000..641c07d3 --- /dev/null +++ b/doc/fluid/closure.tex @@ -0,0 +1,3 @@ +\section{Fluid Closure Models: Transport and Chemistry} \label{sec:fluid-closures} +This section details the transport and chemistry models necessary to +close the PDEs in the fluid model. diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex new file mode 100644 index 00000000..9e14b95c --- /dev/null +++ b/doc/fluid/pdes.tex @@ -0,0 +1,87 @@ +The fluid model takes the form of a set of coupled PDEs governing the +species densities, temperatures, and the electric potential. We +document a 1-D version of the model here, including the PDEs, boundary +conditions, and transport and chemistry closure models. Extension to +2-D is necessary to model the full device, including the edge effects, +but this is not expected to be necessary for the envisioned model +validation relevant to the torch simulations. + +\section{PDEs} +We require PDEs for the species densities, temperatures, and electric potential. + +\subsection{Species Continuity} +The species transport equations are as follows: +% +\begin{equation} +\pp{n_{\alpha}}{t} + \pp{F_{\alpha}}{x} = \dot{\omega}_{\alpha}, +\label{eqn:species} +\end{equation} +% +where $t$ is time, $x$ is the coordinate in the electrode-normal +direction, $n_{\alpha}$ denotes the number density of species +$\alpha$, $F_{\alpha}$ is the corresponding flux, and +$\dot{\omega}_{\alpha}$ is the net production rate of species $\alpha$ +due to chemical reactions. The flux is closed as follows: +% +\begin{equation*} +F_{\alpha} = z_{\alpha} \mu_{\alpha} n_{\alpha} E - D_{\alpha} \pp{n_{\alpha}}{x}, +\end{equation*} +% +where $z_{\alpha}$ is the charge number, $\mu_{\alpha}$ is the +mobility, $D_{\alpha}$ is the diffusion coefficient, and $E$ is the +electric field. The transport and chemistry closures are discussed in +\S\ref{sec:fluid-closures}. The electric field is determined from the +potential, $E = -\pp{\phi}{x}$, which is governed by a Poisson +equation (\S\ref{sec:pdes-poisson}). + +For a mixture of $N_s$ species, $N_s - 1$ species densities are +evolved according to~\eqref{eqn:species}. The density of the dominant +background species (e.g., argon) is determined from the pressure, +which is assumed known, and the ideal gas law: +% +\begin{equation*} +p = \sum_{\alpha = 1}^{N_s} n_{\alpha} k_B T_{\alpha}, +\end{equation*} +% +where $k_B$ is the Boltzmann constant and $T_{\alpha}$ is the +temperature of species $\alpha$. As discussed in +\S\ref{sec:pdes-energy}, a two temperature model is adopted where the +heavy species share a single temperature $T_g$, such that the ideal +gas law becomes +% +\begin{equation*} +p = n_e k_B T_e + k_B T_g \sum_{\alpha \neq e} n_{\alpha}. +\end{equation*} +% + + +\subsection{Energy Equations} \label{sec:pdes-energy} +A two-temperature model is adopted. We assume a common temperature +$T_g$ for the heavy species and a separate temperature for the +electrons. These temperatures are governed by the gas and electron +energy equations, respectively: +% +\begin{gather*} +TODO: put in equations +\end{gather*} +% + +\subsection{Electromagnetics} \label{sec:pdes-poisson} +A quasi-static model is used for the electromagnetics. In this case, +Maxwell's equations reduce to a Poisson equation governing the +electric potential $\phi$: +% +\begin{equation*} +-\pp{\phi}{x} = \frac{\rho_c}{\epsilon_0} +\end{equation*} +% +where $\rho_c$ is the charge density and $\epsilon_0$ is the +permittivity of free space. The charge density is given from the +species densities: +% +\begin{equation*} +\rho_c = \sum_{\alpha=1}^{N_s} z_{\alpha} q_e n_{\alpha}, +\end{equation*} +% +where $q_e$ is the elementary charge (the magnitude of the charge of +an electron). diff --git a/doc/glow.tex b/doc/glow.tex index e17cde14..0057bd68 100644 --- a/doc/glow.tex +++ b/doc/glow.tex @@ -61,11 +61,10 @@ \chapter*{Preface} \input{overview} -\chapter{Governing PDEs} - -\chapter{Boundary Conditions} - -\chapter{Closures: Transport and Chemistry} +\chapter{Fluid Model} +\input{fluid/pdes} +\input{fluid/bcs} +\input{fluid/closure} \chapter{Supporting Models and Data} \input{bolsig/overview} From 52f8689cec4d3fdea4f6d426610911931700cb69 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Mon, 14 Dec 2020 10:41:37 -0600 Subject: [PATCH 06/19] WIP: hybrid fluid-kinetic model eqns for glow discharge --- doc/Makefile | 2 +- doc/glow.tex | 7 +- doc/hybrid/eqns.tex | 291 ++++++++++++++++++++++++++++++++++++++++ doc/hybrid/overview.tex | 5 + 4 files changed, 303 insertions(+), 2 deletions(-) create mode 100644 doc/hybrid/eqns.tex create mode 100644 doc/hybrid/overview.tex diff --git a/doc/Makefile b/doc/Makefile index 014ef84c..371ca0b7 100644 --- a/doc/Makefile +++ b/doc/Makefile @@ -1,7 +1,7 @@ PAPER := glow.pdf TEX_SUFS := .aux .log .nav .out .snm .toc .vrb .fdb_latexmk .bbl .blg .fls GITID := $(shell git describe --always 2> /dev/null) -TEXSRC := $(wildcard *.tex bolsig/*.tex fluid/*.tex) +TEXSRC := $(wildcard *.tex bolsig/*.tex fluid/*.tex hybrid/*.tex) FIGURES := $(wildcard figures/*.pdf figures/*.jpg) BIBS := $(wildcard bibs/*.bib) diff --git a/doc/glow.tex b/doc/glow.tex index 0057bd68..2b07c177 100644 --- a/doc/glow.tex +++ b/doc/glow.tex @@ -12,6 +12,7 @@ \usepackage{amsmath,amssymb,stmaryrd} \usepackage[makeroom]{cancel} +\usepackage{mhchem} \graphicspath{{./figures/}} @@ -61,7 +62,7 @@ \chapter*{Preface} \input{overview} -\chapter{Fluid Model} +\chapter{Fluid Model} \label{sec:fluid} \input{fluid/pdes} \input{fluid/bcs} \input{fluid/closure} @@ -70,6 +71,10 @@ \chapter{Supporting Models and Data} \input{bolsig/overview} \input{bolsig/eqns} +\chapter{Hybrid Fluid/Kinetic Model} +\input{hybrid/overview} +\input{hybrid/eqns} + \chapter{Numerical Methods} diff --git a/doc/hybrid/eqns.tex b/doc/hybrid/eqns.tex new file mode 100644 index 00000000..4813237e --- /dev/null +++ b/doc/hybrid/eqns.tex @@ -0,0 +1,291 @@ +\section{PDEs} +We require PDEs for the heavy species densities, the electric +potential, and the electron distribution function. In this initial +work, the background gas temperature $T_g$ is taken as given and thus +no gas energy equation is necessary. + +\subsection{Species Continuity} +The species transport equations are as follows: +% +\begin{equation} +\pp{n_{\alpha}}{t} + \pp{F_{\alpha}}{x} = \dot{\omega}_{\alpha}, +\label{eqn:species} +\end{equation} +% +where $t$ is time, $x$ is the coordinate in the electrode-normal +direction, $n_{\alpha}$ denotes the number density of species +$\alpha$, $F_{\alpha}$ is the corresponding flux, and +$\dot{\omega}_{\alpha}$ is the net production rate of species $\alpha$ +due to chemical reactions. The flux is closed as follows: +% +\begin{equation*} +F_{\alpha} = \mu_{\alpha} n_{\alpha} E - D_{\alpha} \pp{n_{\alpha}}{x}, +\end{equation*} +% +where $z_{\alpha}$ is the charge number, $\mu_{\alpha}$ is the +mobility, $D_{\alpha}$ is the diffusion coefficient, and $E$ is the +electric field. The transport closures are as given in +\S\ref{sec:fluid-closures}. The electric field is determined from the +potential, $E = -\pp{\phi}{x}$, which is governed by a Poisson +equation (\S\ref{sec:hybrid-poisson}). Finally, the heavy species +production due to chemical reactions are discussed in +\S\ref{sec:hybrid-chemistry}. + +For a mixture of $N_s$ species, $N_s - 2$ species densities are +evolved according to~\eqref{eqn:species}. The electron density is +obtained from the electron distribution function, which is governed by +the electron Boltzmann equation (\S\ref{sec:hybrid-ebolt}). +Further,the density of the dominant background species (e.g., argon) +is determined from the pressure, which is assumed known, and the ideal +gas law: +% +\begin{equation*} +p = \sum_{\alpha = 1}^{N_s} n_{\alpha} k_B T_{\alpha}, +\end{equation*} +% +where $k_B$ is the Boltzmann constant and $T_{\alpha}$ is the +temperature of species $\alpha$. A two temperature model is adopted +where the heavy species share a single temperature $T_g$, which is +different from the electron temperature, such that the ideal gas law +becomes +% +\begin{equation*} +p = n_e k_B T_e + k_B T_g \sum_{\alpha \neq e} n_{\alpha}. +\end{equation*} +% +The gas temperature is assumed known. The electron temperture is +governed by the electron Boltzmann equation (\S\ref{sec:ebolt}). + +\subsection{Electromagnetics} \label{sec:hybrid-poisson} +A quasi-static model is used for the electromagnetics. In this case, +Maxwell's equations reduce to a Poisson equation governing the +electric potential $\phi$: +% +\begin{equation*} +-\pp{\phi}{x} = \frac{\rho_c}{\epsilon_0} +\end{equation*} +% +where $\rho_c$ is the charge density and $\epsilon_0$ is the +permittivity of free space. The charge density is given from the +species densities: +% +\begin{equation*} +\rho_c = \sum_{\alpha=1}^{N_s} z_{\alpha} q_e n_{\alpha}, +\end{equation*} +% +where $q_e$ is the elementary charge (the magnitude of the charge of +an electron). + +The electron number density $n_e$ is determined from the electron +distribution function, +% +\begin{equation*} +n_e(x,t) = \int_{\mbb{R}^3} f_e(x, \mbf{v}_e, t) \, d^3 \mbf{v}_e +\end{equation*} +% +which is governed by the electron Boltzmann equation. + +\subsection{Electron Boltzmann Equation} \label{sec:hybrid-ebolt} +Let $f_e$ denote the electron distribution function. In the 1-D glow +discharge model, the distribution function depends on space $x \in +(0,h)$, velocity $\mbf{v} \in \mathbb{R}^3$, and time $t \in +\mathbb{R}^+$. Since $f_e$ depends only on $x$ and $\mbf{E} = E +\hat{x}$, the electron Boltzmann equation simplifies to +% +\begin{equation*} +\pp{f_e}{t} + v_x \pp{f_e}{x} - \frac{q_e}{m_e} E \pp{f_e}{v_x} = S_e + C_e, +\end{equation*} +% +where $S_e+C_e$ denotes the collision term, with $S_e$ representing elastic +collisions and $C_e$ representing reactive collisions. + +For $S_e$, we consider only two-body interactions. Then, suppressing +the dependence on $x$ and $t$, $S_e$ can be written as +% +\begin{equation*} +S_e(\mbf{v}_e) += +\sum_{\alpha=1}^{N_s} +\int \left( +f_e(\mbf{v}_e^{\prime}) f_{\alpha}(\mbf{v}_{\alpha}^{\prime} +- +f_e(\mbf{v}_e) f_{\alpha}(\mbf{v}_{\alpha}) +\right) \, +W_{e \alpha} +d^3 \mbf{v}_{\alpha} d^3 \mbf{v}_e^{\prime} d^3 \mbf{v}_{\alpha}^{\prime}, +\end{equation*} +% +where $W_{e \alpha}$ is the transition probability associated with the +nonreactive collision between and electron and species $\alpha$. In +practice, we will begin by considering only non-reactive collisions +between electrons and the background gas (argon), such that $S_e$ becomes +% +\begin{equation*} +S_e (\mbf{v}_e) += +\int \left( +f_e(\mbf{v}_e^{\prime}) f_{b}(\mbf{v}_{b}^{\prime}) +- +f_e(\mbf{v}_e) f_{b}(\mbf{v}_{b}) +\right) \, +W_{e b} +d^3 \mbf{v}_{b} d^3 \mbf{v}_e^{\prime} d^3 \mbf{v}_{b}^{\prime}, +\end{equation*} +% +where subscript $b$ denotes the background species. The background +species distribution function $f_b$ is assumed given by the Maxwellian +consistent with its number density $n_b$ and the gas temperature +$T_g$. Thus, +% +\begin{equation*} +f_b(x, \mbf{v}, t) += +n_b(x,t) +\left( \frac{m}{2 \pi k_B T_g} \right)^{3/2} +\exp \left( \frac{-m_b \|\mbf{v}\|^2}{2 k_B T_g} \right). +\end{equation*} +% +The integral in $S_e$ may be simplified by recognizing that many +interactions considered in the integral are impossible due to +conservation of momentum and energy. Specifically, for elastic +collisions, conservation of momentum and energy require that +% +\begin{equation*} +m_e \mbf{v}_e + m_b \mbf{v}_b += +m_e \mbf{v}_e^{\prime} + m_b \mbf{v}_b^{\prime} +\quad +\frac{1}{2} m_e \mbf{v}_e \cdot \mbf{v}_e + +\frac{1}{2} m_b \mbf{v}_b \cdot \mbf{v}_b += +\frac{1}{2} m_e \mbf{v}_e^{\prime} \cdot \mbf{v}_e^{\prime} + +\frac{1}{2} m_b \mbf{v}_b^{\prime} \cdot \mbf{v}_b^{\prime} +\end{equation*} +% +Then, given the direction of the post-collision velocity +$\hat{\mbf{e}}$, the post-collision velocities may be written as +% +\begin{equation*} +PUT IN +\end{equation*} +% +and the nine-dimensional integral above may be re-written as a +five-dimensional integral: +% +\begin{equation*} +\int_{\mbb{R}^3} \int_{S^2} +\left( +f_e(\mbf{v}_e^{\prime}) f_{b}(\mbf{v}_{b}^{\prime}) +- +f_e(\mbf{v}_e) f_{b}(\mbf{v}_{b}) +\right) \, +\|\mbf{v}_e - \mbf{v}_b\| \sigma_{eb}(\|\mbf{v}_e - \mbf{v}_b\|, \hat{\mbf{e}}) +d\hat{\mbf{e}} \, d^3 \mbf{v}_{b}, +\end{equation*} +% +where $\sigma_{eb}$ denotes the differential cross section. + +To write the contribution of reactive collisions, we follow the +formulation of Ern and Giovangigli~\cite{}. The full reaction term is +written as follows: +% +\begin{equation*} +C_e = \sum_{r=1}^{N_r} C_e^r, +\end{equation*} +% +where $N_r$ is the number of reactions and $C_e^r$ is the contributin +of the $r$th reaction. To formulate $C_e^r$, consider the following +reaction: +% +\begin{equation*} +\sum_{j \in R^r} \chi_j +\ce{<=>} +\sum_{k \in P^r} \chi_k, +\end{equation*} +% +where $\chi_j$ denotes the $j$th species, $R^r$ is the set of +reactants, and $P^r$ is the set of products. For reactions that +include the same species more than once, $R^r$ and $P^r$ include that +species multiple times. For example, the set of products of the argon +ionization reaction $\ce{Ar + e <=> Ar^+ + e + e}$ is given by $P^r = +\{\ce{Ar^+}, \ce{e}, \ce{e}\}$. Further, for a given species $i$, let +$R^r_i$ and $P^r_i$ denote the set of reactants and products for +reaction $r$ with species $i$ removed only once. Thus, for the argon +ionization reaction example, $P^r_e = \{\ce{Ar^+}, \ce{e}\}$. + +Equivalently, one may write the generic chemical reaction as +% +\begin{equation*} +\sum_{j =1}^{N_s} \nu^{r}_j\chi_j +\ce{<=>} +\sum_{k=1}^{N_s} \nu^{\prime,r}_k \chi_k, +\end{equation*} +% +where $\nu^r_{j}$ and $\nu^{\prime,r}_j$ denote the stoichiometric +coefficients of the $j$th species in the reactants and products, +respectively. Using all of this notation, the contribution of the +$r$th reaction to the reactive collision term is given by +% +\begin{align*} +C_e^r +=& +\nu_e^r \int \left( \prod_{k \in P^r} \beta_k f_k - \prod_{j \in R^r} \beta_j f_j \right) +\frac{W_{R^r P^r}}{\prod_{j \in R^r} \beta_j} \, +\prod_{j \in R^r_e} d^3 \mbf{v}_j \, \prod_{k \in P^r} d^3 \mbf{v}_k\\ +&+ +\nu_e^{\prime,r} \int \left( \prod_{j \in R^r} \beta_j f_j - \prod_{k \in P^r} \beta_k f_k \right) +\frac{W_{R^r P^r}}{\prod_{j \in R^r} \beta_j} \, +\prod_{j \in R^r} d^3 \mbf{v}_j \, \prod_{k \in P^r_e} d^3 \mbf{v}_k, +\end{align*} +% +where $W_{R^r P^r}$ is the transition probability for the forward +reaction, +% +\begin{equation*} +\beta_j = \frac{h^3_{\mathrm{P}}}{a_j m_j^3}, +\end{equation*} +% +and $a_j$ is the degeneracy. + +For binary reactions (e.g., $\chi_0 + \chi_1 \ce{<=>} \chi_2 + +\chi_3$), this may be rewritten in terms of the differential collision +cross section. For example, the contribution of such a reaction to +species 0 is given by +% +\begin{equation*} +C_0 = \int_{\mbb{R}^3} \int_{S^2} \left( \frac{\beta_2 \beta_3}{\beta_0 \beta_1} f_2 f_3 - f_0 f_1\right) g \sigma(g, \hat{\mbf{e}}) d\hat{\mbf{e}} d^3 \mbf{v}_1 +\end{equation*} +% + +Formally, reactions involving three +particles (e.g., direct ionization and recombination: $\ce{Ar + e <=> + Ar^+ + e + e}$ are supported by this framework, but this introduces +complexity into the formulation of $W$. While this can likely be +overcome, +\todo{Alexeev introduces a collision cross section for such reactions, but I don't understand his notation.} + we may also avoid it by decomposing such three particle +interactions into two steps. For instance, instead of including +direct impact ionization in the mechanism, we have +% +\begin{gather*} +\ce{Ar + e <=> Ar^{\dagger} + e} \\ +\ce{Ar^{\dagger} -> Ar^+ + e}, +\end{gather*} +% +where $Ar^{\dagger}$ is an intermediate species. For these reactions, +letting $\ce{A}$ be species 1, $\ce{Ar^{\dagger}}$ be species 2, and +$\ce{Ar^+}$ be species 3, the reactive collision terms in the electron +Boltzmann equation reads +% +\begin{align*} +C_e(\mbf{v}_e) +&= +\int_{\mbb{R}^3} \int_{S^2} +\left( \frac{\beta_2}{\beta_1} f_e(\mbf{v}^{\prime}_e) f_2(\mbf{v}_2) - f_e(\mbf{v}_e) f_1(\mbf{v}_1) \right) W_{10,20} d^3 \mbf{v}_1 d^3 \mbf{v}^{\prime}_e d^3 \mbf{v}_2 +\\ +&+\int_{\mbb{R}^3} \int_{S^2} +\left( f_e(\mbf{v}^{\prime}_e) f_1(\mbf{v}_1) - \frac{\beta_2}{\beta_1} f_e(\mbf{v}_e) f_2(\mbf{v}_2) \right) W_{10,20} d^3 \mbf{v}^{\prime}_e d^3 \mbf{v}_1 d^3 \mbf{v}_2 ++ +\int_{S^2} \frac{1}{\tau} f_2(\mbf{v}_2) d\hat{\mbf{e}}. +\end{align*} +% diff --git a/doc/hybrid/overview.tex b/doc/hybrid/overview.tex new file mode 100644 index 00000000..c92bb311 --- /dev/null +++ b/doc/hybrid/overview.tex @@ -0,0 +1,5 @@ +This section details a hybrid fluid-kinetic model. In particular, the +heavy species are represented with a fluid model, as in +\S\ref{sec:fluid}, but the electron number density and energy +equations are omitted. These equations are replaced by the electron +Boltzmann equation. From 8d361f63f9c2c2f3f7fbbee1a800c1aaade692c4 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Mon, 14 Dec 2020 10:54:12 -0600 Subject: [PATCH 07/19] fix typo --- doc/hybrid/eqns.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/hybrid/eqns.tex b/doc/hybrid/eqns.tex index 4813237e..a5ff8822 100644 --- a/doc/hybrid/eqns.tex +++ b/doc/hybrid/eqns.tex @@ -62,7 +62,7 @@ \subsection{Electromagnetics} \label{sec:hybrid-poisson} electric potential $\phi$: % \begin{equation*} --\pp{\phi}{x} = \frac{\rho_c}{\epsilon_0} +-\pp{^2 \phi}{x^2} = \frac{\rho_c}{\epsilon_0} \end{equation*} % where $\rho_c$ is the charge density and $\epsilon_0$ is the From 1f9f5efc14c0f7e13cb8e0b86cb5096d5610e6ed Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Wed, 16 Dec 2020 14:21:32 -0600 Subject: [PATCH 08/19] Some work on closure section of the glow discharge. --- doc/fluid/closure.tex | 89 +++++++++++++++++++++++++++++++++++++++++++ doc/fluid/pdes.tex | 2 +- doc/glow.tex | 1 + 3 files changed, 91 insertions(+), 1 deletion(-) diff --git a/doc/fluid/closure.tex b/doc/fluid/closure.tex index 641c07d3..bd5b63c3 100644 --- a/doc/fluid/closure.tex +++ b/doc/fluid/closure.tex @@ -1,3 +1,92 @@ \section{Fluid Closure Models: Transport and Chemistry} \label{sec:fluid-closures} This section details the transport and chemistry models necessary to close the PDEs in the fluid model. + +\subsection{Chemistry} +The chemical source terms $\dot{\omega}_{\alpha}$ are determined by +specifying a chemical mechanism and associated rate parameters. This +section gives the form of the source terms appearing in the species +transport equations for a generic mechanism as well as example +mechanisms for argon plasma. + +\subsubsection{General Form} +Assume that we have a set of $N_r$ chemical reactions, each taking the +following form: +% +\begin{equation*} +\sum_{\alpha=1}^{N_s} \nu^{\prime}_{\alpha r} X_{\alpha} +\ce{<=>} +\sum_{\alpha=1}^{N_s} \nu^{\prime\prime}_{\alpha r} X_{\alpha} +\quad +\textrm{for} \,\, r = 1, \ldots, N_r. +\end{equation*} +% +Then, +% +\begin{equation*} +\dot{\omega}_{\alpha} += +\sum_{r=1}^{N_r} (\nu^{\prime \prime}_{\alpha r} - \nu^{\prime}_{\alpha r}) G_r +\end{equation*} +% +where $M_{\alpha}$ is the molar mass of species $\alpha$ and $G_r$ is +the rate of progress of reaction $r$. The rate of progress is given +by +% +\begin{equation*} +G_r += +k_{f,r} \prod_{s=1}^{N_s} n_s^{\nu^{\prime}_{sr}} +- +k_{b,r} \prod_{s=1}^{N_s} n_s^{\nu^{\prime \prime}_{sr}}, +\end{equation*} +% +where $k_{f,r}$ and $k_{b,r}$ are the forward and backward rate +coefficients. These rate coefficients are functions of temperature +that must be provided as part of the mechanism. + +\subsubsection{Simple argon mechanism} + +\subsubsection{Advanced argon mechanism} + + +\subsection{Transport} +To close the fluid model transport equation, we require transport +properties, namely the diffusion coefficient and thermal conductivity +for all species and the mobility for charged species. + +\subsubsection{Simple} +Some authors~\cite{} specify these quantities as constant for a given +background number density. Data from~\cite{} is shown in +Table~\ref{tbl:simpleTransport}. +% +\begin{table}[htp] +\caption{Transport property values from~\cite{}.} +\begin{center} +\begin{tabular}{|c|ccc|} +\hline +Name & Symbol & Units & Value \\ +\hline +Electron (\ce{e}) diffusivity & $n_{\ce{Ar}} D_{\ce{e}}$ & \si{(cm.s)^{-1}} & $3.86 \times 10^{22}$ \\ +Positive ion (\ce{Ar^+}) diffusivity & $n_{\ce{Ar}} D_{\ce{Ar^+}}$ & \si{(cm.s)^{-1}} & $2.07 \times 10^{18}$ \\ +Metastable atom (\ce{Ar^*}) diffusivity & $n_{\ce{Ar}} D_{\ce{Ar^{\ast}}}$ & \si{(cm.s)^{-1}} & $2.42 \times 10^{18}$ \\ +\hline +Electron mobility & $n_{\ce{Ar}} \mu_{\ce{e}}$ & \si{(V.cm.s)^{-1}} & $9.66 \times 10^{21}$ \\ +Positive ion mobility & $n_{\ce{Ar}} \mu_{\ce{Ar^+}}$ & \si{(V.cm.s)^{-1}} & $4.65 \times 10^{19}$ \\ +\hline +\end{tabular} +\end{center} +\label{tbl:simpleTransport} +\end{table} +% +The required thermal conductivities may be computed from the +diffusivity as follows: +% +\begin{equation*} +\kappa_{\alpha} = \frac{5}{2} n_{\alpha} k_B D_{\alpha}. +\end{equation*} +% +\todo[inline]{NB: this is from Panneer Chelvam and Raja. Some authors use 3/2 instead of 5/2. Why? Is it an error or a different modeling choice?} + + +\subsubsection{Advanced} diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex index 9e14b95c..7a8840b4 100644 --- a/doc/fluid/pdes.tex +++ b/doc/fluid/pdes.tex @@ -72,7 +72,7 @@ \subsection{Electromagnetics} \label{sec:pdes-poisson} electric potential $\phi$: % \begin{equation*} --\pp{\phi}{x} = \frac{\rho_c}{\epsilon_0} +-\pp{^2\phi}{x^2} = \frac{\rho_c}{\epsilon_0} \end{equation*} % where $\rho_c$ is the charge density and $\epsilon_0$ is the diff --git a/doc/glow.tex b/doc/glow.tex index 2b07c177..a7f0f153 100644 --- a/doc/glow.tex +++ b/doc/glow.tex @@ -4,6 +4,7 @@ \usepackage{graphicx} \usepackage{xcolor} \usepackage{todonotes} +\usepackage{siunitx} \usepackage[square,numbers]{natbib} \usepackage[american,siunitx]{circuitikz} From 0fbd143779b03b6e400421f824cc5de7c173054c Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Wed, 16 Dec 2020 17:41:39 -0600 Subject: [PATCH 09/19] add electron energy equation --- doc/fluid/pdes.tex | 23 ++++++++++++++++++++--- 1 file changed, 20 insertions(+), 3 deletions(-) diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex index 7a8840b4..e3b55e20 100644 --- a/doc/fluid/pdes.tex +++ b/doc/fluid/pdes.tex @@ -58,13 +58,30 @@ \subsection{Species Continuity} \subsection{Energy Equations} \label{sec:pdes-energy} A two-temperature model is adopted. We assume a common temperature $T_g$ for the heavy species and a separate temperature for the -electrons. These temperatures are governed by the gas and electron -energy equations, respectively: +electrons. The electron energy equation is given by % \begin{gather*} -TODO: put in equations +\pp{}{t} \left( \frac{3}{2} k_B n_e T_e \right) ++ +\pp{}{x} \left( \frac{5}{2} k_B T_e F_e - \kappa_e \pp{T_e}{x} \right) += +\underbrace{- q_e F_e E}_{\textrm{Joule heating}} +- +\underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} +- +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E_j G_j}_{\textrm{Inelastic}}, \end{gather*} % +where $m_e$ and $m_b$ denote the molecular mass of electron and the +background species, respectively, $\kappa_e$ is the electron thermal +conductivity, $\Delta E_j$ is the energy lost per electron in the +collisional process represented by reaction $j$, and $G_j$ is the rate +of progress of reaction $j$. The thermal conductivity $\kappa_e$ and +chemistry related quantities $\Delta E_j$ and $G_j$ are described more +in \S\ref{sec:fluid-closures}. + +The gas energy equation is given by\ldots + \subsection{Electromagnetics} \label{sec:pdes-poisson} A quasi-static model is used for the electromagnetics. In this case, From fc21a15271f7bb388ae4c9cfc9c827aa2a8d3598 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Thu, 17 Dec 2020 11:34:00 -0600 Subject: [PATCH 10/19] More details on fluid model --- doc/fluid/bcs.tex | 75 ++++++++++++++++++++++++++++++++++++++++++- doc/fluid/closure.tex | 35 ++++++++++++++++++-- doc/fluid/pdes.tex | 23 +++++++++++-- 3 files changed, 128 insertions(+), 5 deletions(-) diff --git a/doc/fluid/bcs.tex b/doc/fluid/bcs.tex index d540158e..54a0f535 100644 --- a/doc/fluid/bcs.tex +++ b/doc/fluid/bcs.tex @@ -1 +1,74 @@ -\section{Boundary Conditions} +\section{Boundary Conditions} \label{sec:fluid-bcs} + +\subsection{Species Continuity} +We impose conditions on the species fluxes at the walls. The electron +flux $F_e$ is given by +% +\begin{equation*} +F_e \mbf{n}_x += +\frac{1}{4} n_e \left( \frac{8 k_B T_e}{\pi m_e} \right)^{1/2} +- +\sum_{\alpha=1}^{N_s} \gamma_{\alpha} F_{\alpha} \mbf{n}_x +\end{equation*} +% +where $\mbf{n}_x$ denotes the outward pointing unit normal (in 1D, +$\mbf{n}_x =-1$ on the 'left' and $\mbf{n}_x = 1$ on the 'right') and +$\gamma_{\alpha}$ is the secondary electron emission coefficient for +species $\alpha$. \todo[inline]{We need values for the + $\gamma_{\alpha}$ parameters!} + +The ion flux, $F_i$, is given by +% +\begin{equation*} +F_i \mbf{n}_x += +\frac{1}{4} n_i \left( \frac{8 k_B T_g}{\pi m_i} \right)^{1/2} ++ +n_i \max(0, \mu_i E \mbf{n}_x), +\end{equation*} +% +\todo[inline]{Check inconsistency in refs for ion BC with Raja} + +Finally, the all neutral species fluxes ars given by +% +\begin{equation*} +F_{\alpha} \mbf{n}_x += +\frac{1}{4} n_{\alpha} \left( \frac{8 k_B T_g}{\pi m_{\alpha}} \right)^{1/2} +\end{equation*} +% + +\subsection{Electric Potential} +Dirichlet conditions on the electric potential are imposed on both +sides. For instance, assuming $x=0$ is ground and $x=L$ has an +imposed fluctuating voltage: +% +\begin{equation*} +\phi(0,t) = 0, \quad \phi(L,t) = V_0 \sin(2 \pi f t), +\end{equation*} +% +where $V_0$ is the voltage amplitude and $f$ is the frequency. + +\subsection{Energy} +For the electron energy equation, the total surface energy flux at the +wall is given by +% +\begin{equation*} +Q_{e,wall} \mbf{n}_x +\equiv +\left. \left( \frac{5}{2} k_B T_e F_e - \kappa_e \pp{T_e}{x} \right)\right|_{wall} \mbf{n}_x += +\frac{5}{2} k_B T_e F_e \mbf{n}_x, +\end{equation*} +% +where $F_e \mbf{n}_x$ is taken from the species continuity boundary +condition. + +Isothermal wall conditions are imposed on the gas temperature: +% +\begin{equation*} +T_g(0,t) = T_g(L,t) = T_{g,wall}, +\end{equation*} +% +where $T_{g,wall}$ is a specified value. diff --git a/doc/fluid/closure.tex b/doc/fluid/closure.tex index bd5b63c3..15678e12 100644 --- a/doc/fluid/closure.tex +++ b/doc/fluid/closure.tex @@ -45,10 +45,41 @@ \subsubsection{General Form} coefficients. These rate coefficients are functions of temperature that must be provided as part of the mechanism. -\subsubsection{Simple argon mechanism} +\subsubsection{Simple argon mechanisms} +The simplest possible argon mechanism that has been used~\cite{} +includes only irreversible ionization of ground state argon: +% +\begin{equation*} +\ce{Ar + e -> Ar^+ + 2e}. +\end{equation*} +% +Denoting this reaction 0, the energy loss per electron in $\Delta E_0 += 15.7 \si{eV}$, and the rate of progress is +% +\begin{equation*} +G_0 = k_i(T_e) n_b n_e, +\end{equation*} +% +where $n_b$ is the argon number density. Thus, +% +\begin{equation*} +\dot{\omega}_e = (2 - 1) G_0 = k_i n_b n_e. +\end{equation*} +% +Liu et al.~\cite{} uses +% +\begin{equation*} +k_i(T_e) = A \exp(-C/T_e), +\end{equation*} +% +where $A = 1.235 \times 10^{-7} \si{cm^3.s^{-1}}$ and $C = +-18.687\si{eV}$.\todo{Ref isn't clear only units for C, but has to be eV, right?} -\subsubsection{Advanced argon mechanism} +This may be sufficient for code development purposes, but a more +complex mechanism will be required for realistic simulations. +\subsubsection{Advanced argon mechanism} +More complex argon plasma chemistry models are detailed in~\cite{}. \subsection{Transport} To close the fluid model transport equation, we require transport diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex index e3b55e20..ff11dde7 100644 --- a/doc/fluid/pdes.tex +++ b/doc/fluid/pdes.tex @@ -80,8 +80,27 @@ \subsection{Energy Equations} \label{sec:pdes-energy} chemistry related quantities $\Delta E_j$ and $G_j$ are described more in \S\ref{sec:fluid-closures}. -The gas energy equation is given by\ldots - +The gas energy equation is given by +% +\begin{gather*} +\pp{}{t} \left( \sum_{\alpha \in h} C_{v,\alpha} n_\alpha T_g \right) ++ +\pp{}{x} \left( \sum_{\alpha \in h} C_{p,\alpha} F_\alpha T_g - \sum_{\alpha \in h} \kappa_\alpha \pp{T_g}{x} \right) += +\underbrace{C_{\textrm{inc}} \sum_{\alpha \in h} q_e z_{\alpha} F_\alpha E}_{\textrm{Joule heating}} ++ +\underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} ++ +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E_j G_j}_{\textrm{Inelastic}}, +\end{gather*} +% +where the notation $\sum_{\alpha \in h}$ denotes the summation over +the heavy species, $C_{v,\alpha}$ and $C_{p,\alpha}$ are the specific +heat at constant volume and pressure, respectively, for species +$\alpha$. Finally, $C_{\textrm{inc}}$ is a constant that may be used +to account for incomplete conversion of the kinetic energy gained by +the electrons into heat due to the fact that the ion mean free path +can be of the same order of the sheath thickness~\cite{}. \subsection{Electromagnetics} \label{sec:pdes-poisson} A quasi-static model is used for the electromagnetics. In this case, From 5073ce6c106153c13652977f75163a96c4b38ab0 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Thu, 22 Apr 2021 21:41:00 -0500 Subject: [PATCH 11/19] Some notes on collisions --- doc/glow.tex | 1 + doc/hybrid/collisions.tex | 217 ++++++++++++++++++++++++++++++++++++++ doc/hybrid/eqns.tex | 1 + 3 files changed, 219 insertions(+) create mode 100644 doc/hybrid/collisions.tex diff --git a/doc/glow.tex b/doc/glow.tex index a7f0f153..09efea0a 100644 --- a/doc/glow.tex +++ b/doc/glow.tex @@ -75,6 +75,7 @@ \chapter{Supporting Models and Data} \chapter{Hybrid Fluid/Kinetic Model} \input{hybrid/overview} \input{hybrid/eqns} +\input{hybrid/collisions} \chapter{Numerical Methods} diff --git a/doc/hybrid/collisions.tex b/doc/hybrid/collisions.tex new file mode 100644 index 00000000..de5dfbe5 --- /dev/null +++ b/doc/hybrid/collisions.tex @@ -0,0 +1,217 @@ +\subsection{Collision Notes} \label{sec:col-notes} +NOTE: +\emph{This is a place for making notes about the collision integrals. + Eventually it will replace or be combined with the previous section.} + +We consider three classes of collision: +% +\begin{enumerate} +\item Electron-neutral elastic collisions, e.g., $\ce{ Ar + e -> Ar + e}$; +\item Electron-neutral excitation reactions, e.g., $\ce{ Ar + e <=> Ar^{\ast} + e}$; +\item Electron-neutral ionization reactions, e.g., $\ce{ Ar^{\ast} + e <=> Ar^+ + e + e}$. +\end{enumerate} +% + +\subsubsection{Electron-Neutral Elastic Scattering} +Let $()_e$ denote electron quantities---e.g., $f_e$ is the electron +distribution function, $m_e$ is the mass of an electron---and let +$()_N$ denote the analogous quantities for the neutral particle. +Then, the net rate of production of particles (per unit volume in +state space) is given by +% +\begin{equation} +Q_{EL}(\mbf{v}_e) += +\int_{\mbb{R}^3} \int_{\mbb{R}^3}\int_{\mbb{R}^3} +\left( +f_e(\mbf{v}_e^{\prime}) f_N(\mbf{v}_N^{\prime}) +- +f_e(\mbf{v}_e) f_N(\mbf{v}_N) +\right) W_{EL}(\mbf{v}_e, \mbf{v}_N; \mbf{v}_e^{\prime}, \mbf{v}_N^{\prime}) +\, +d^3\mbf{v}_N \, d^3\mbf{v}_e^{\prime} \, d^3\mbf{v}_N^{\prime} +\label{eqn:elastic-collision-9d} +\end{equation} +% +where $\mbf{v}_e$ and $\mbf{v}_N$ denote the ``pre-collision'' +velocities, $\mbf{v}_e^{\prime}$ and $\mbf{v}_N^{\prime}$ denote the +``post-collision'' velocities, and $W_{EL}$ denotes the elastic +collision transition probability, which encodes both constraints +(conservation of momentum and energy) \emph{and} information about the +relative likelihood of allowable transitions. + +The nine dimensional integral in~\eqref{eqn:elastic-collision-9d} can +be reduced to a five dimensional integral by imposing conservation of +momentum and energy. To do so in the simplest possible way, let +$()_0$ denote the quantities associated with the center of mass of the +two-particle system. Specifically, the pre-collision position and +velocity of the center of mass are given by +% +\begin{equation*} +\mbf{x}_0 = \frac{m_N \mbf{x}_N + m_e \mbf{x}_e}{m_N + m_e}, +\quad +\mbf{v}_0 = \frac{m_N \mbf{v}_N + m_e \mbf{v}_e}{m_N + m_e} +\end{equation*} +% +Conservation of momentum requires that +% +\begin{gather*} +m_N \mbf{v}_N + m_e \mbf{v}_e = m_N \mbf{v}_N^{\prime} + m_e \mbf{v}_e^{\prime}\\ +(m_N +m_e) \mbf{v}_0 = (m_N + m_e) \mbf{v}_0^{\prime}. +\end{gather*} +% +That is, the velocity of the center of mass is not altered by the +collision. Thus, letting $\mbf{u}$ denote velocities measured in a +frame attached to the center of mass, one may write +% +\begin{equation*} +m_N \mbf{u}_N^{\prime} + m_e \mbf{u}_e^{\prime} += +m_N \mbf{v}_N^{\prime} + m_e \mbf{v}_e^{\prime} - (m_N + m_e) \mbf{v}_0 += 0. +\end{equation*} +% +Thus, conservation of momentum implies the following relationship +between the post-collision relative velocities: +% +\begin{equation*} +\mbf{u}_N^{\prime} = -\,\frac{m_e}{m_N} \mbf{u}_e^{\prime}. +\end{equation*} +% +To continue, let $\mbf{\hat{e}}$ a unit vector in the direction of the +relative velocity between the post-collision particles. Then, the +relative post-collision velocity may be written as +% +\begin{equation*} +\mbf{u}_N^{\prime} - \mbf{u}_e^{\prime} = g^{\prime} \mbf{\hat{e}}, +\end{equation*} +% +where $g'$ denotes the the magnitude of the post-collision relative +velocity. To continue, we use conservation of energy to determine +$g^{\prime}$. + +In the frame moving with the center of mass, conservation of energy requires that +% +\begin{equation*} +\frac{1}{2} m_N |\mbf{u}_N|^2 + \frac{1}{2} m_e |\mbf{u}_e|^2 += +\frac{1}{2} m_N |\mbf{u}_N^{\prime}|^2 + \frac{1}{2} m_e |\mbf{u}_e^{\prime}|^2. +\end{equation*} +% +In order to solve for $g^{\prime}$, we express the post-collision +velocities in terms of $g^{\prime}$ and $\mbf{\hat{e}}$. By +definitions and previous results, one can show that +% +\begin{gather*} +\mbf{u}_N^{\prime} = \frac{m_e}{m_N + m_e} g^{\prime} \mbf{\hat{e}},\\ +\mbf{u}_e^{\prime} = \frac{-m_N}{m_N + m_e} g^{\prime} \mbf{\hat{e}}. +\end{gather*} +% +Substituting into conservation of energy gives +% +\begin{equation*} +\frac{1}{2} m_N |\mbf{u}_N|^2 + \frac{1}{2} m_e |\mbf{u}_e|^2 += +\frac{1}{2} m_N \frac{m_e^2}{(m_N+m_e)^2} (g^{\prime})^2 ++ +\frac{1}{2} m_e \frac{m_N^2}{(m_N+m_e)^2} (g^{\prime})^2 += +\frac{1}{2} \frac{m_e m_N}{m_e + m_N} (g^{\prime})^2. +\end{equation*} +% +Thus, +% +\begin{equation*} +g^{\prime} = \left( \frac{m_e + m_N}{m_e m_N} \left( m_N |\mbf{u}_N|^2 + m_e |\mbf{u}_e|^2 \right) \right)^{1/2}. +\end{equation*} +% +This expression can be further simplified by introducing the +pre-collision relative speed $g$: +% +\begin{equation*} +g = | \mbf{u}_N - \mbf{u}_e | +\end{equation*} +% +Then, +\begin{equation*} +\frac{1}{2} m_N |\mbf{u}_N|^2 + \frac{1}{2} m_e |\mbf{u}_e|^2 += +\frac{1}{2} \frac{m_e m_N}{m_e + m_N} g^2, +\end{equation*} +% +which implies that $g^{\prime} = g$. + +Thus, given the pre-collision velocities $\mbf{v}_N$ and $\mbf{v}_e$ +and the post-collision relative velocity direction $\mbf{\hat{e}}$, +the post-collision velocities are completely determined: +% +\begin{gather*} +\mbf{v}_N^{\prime} += +\mbf{v}_0 + \mbf{u}_N^{\prime} += +m_N \mbf{v}_N + m_e \mbf{v}_e + \frac{m_e}{m_N+m_e} g \mbf{\hat{e}},\\ +% +\mbf{v}_N^{\prime} += +\mbf{v}_0 + \mbf{u}_N^{\prime} += +m_N \mbf{v}_N + m_e \mbf{v}_e - \frac{m_N}{m_N+m_e} g \mbf{\hat{e}}. +\end{gather*} +% +Using these constraints, the collision integral +in~\eqref{eqn:elastic-collision-9d} may be rewritten as follows: +% +\begin{equation} +Q_{EL}(\mbf{v}_e) += +\int_{\mbb{R}^3} \int_{S^2} +\left( +f_e(\mbf{v}_e^{\prime}) f_N(\mbf{v}_N^{\prime}) +- +f_e(\mbf{v}_e) f_N(\mbf{v}_N) +\right) B(\mbf{v}_e, \mbf{v}_N; \mbf{\hat{e}}) +\, +d\mbf{\hat{e}} \, d^3\mbf{v}_N, +\label{eqn:elastic-collision-5d} +\end{equation} +% +where we integrate only over transitions allowed by conservation of +momentum and energy and $B(\mbf{v}_e, \mbf{v}_N; \mbf{\hat{e}})$ has +taken the place of $W_{EL}$ for these allowable transitions. It is +common to formulate $B$ as follows +% +\begin{equation*} +B(\mbf{v}_e, \mbf{v}_N; \mbf{\hat{e}}) = g \sigma(g,\mbf{\hat{e}}), +\end{equation*} +% +where $\sigma$ is the differential collision cross section. +Substituting this into~\eqref{eqn:elastic-collision-5d} gives the +typical form: +% +\begin{equation} +Q_{EL}(\mbf{v}_e) += +\int_{\mbb{R}^3} \int_{S^2} +\left( +f_e(\mbf{v}_e^{\prime}) f_N(\mbf{v}_N^{\prime}) +- +f_e(\mbf{v}_e) f_N(\mbf{v}_N) +\right) g \sigma(g, \mbf{\hat{e}}) +\, +d\mbf{\hat{e}} \, d^3\mbf{v}_N, +\label{eqn:elastic-collision-5d} +\end{equation} +% +where, when evaluating the integral (e.g., using quadrature), the +quantities $\mbf{v}_e^{\prime}$, $\mbf{v}_N^{\prime}$, and $g$ are +determined from $\mbf{v}_e$, $\mbf{v}_N$, and $\mbf{\hat{e}}$ as +discussed previously. + + + +\subsubsection{Electron-Neutral Excitation Reactions} + + +\subsubsection{Electron-Neutral Ionization Reactions} + diff --git a/doc/hybrid/eqns.tex b/doc/hybrid/eqns.tex index a5ff8822..adda1bba 100644 --- a/doc/hybrid/eqns.tex +++ b/doc/hybrid/eqns.tex @@ -289,3 +289,4 @@ \subsection{Electron Boltzmann Equation} \label{sec:hybrid-ebolt} \int_{S^2} \frac{1}{\tau} f_2(\mbf{v}_2) d\hat{\mbf{e}}. \end{align*} % + From 32409d47b6900aa3808296c4f0441eda460fa75d Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Fri, 23 Apr 2021 09:27:27 -0500 Subject: [PATCH 12/19] more notes on collision integrals --- doc/hybrid/collisions.tex | 142 +++++++++++++++++++++++++++++++++++++- 1 file changed, 140 insertions(+), 2 deletions(-) diff --git a/doc/hybrid/collisions.tex b/doc/hybrid/collisions.tex index de5dfbe5..04cfd1fd 100644 --- a/doc/hybrid/collisions.tex +++ b/doc/hybrid/collisions.tex @@ -12,7 +12,7 @@ \subsection{Collision Notes} \label{sec:col-notes} \end{enumerate} % -\subsubsection{Electron-Neutral Elastic Scattering} +\subsubsection{Electron-Neutral Elastic Scattering} \label{sec:en-scattering} Let $()_e$ denote electron quantities---e.g., $f_e$ is the electron distribution function, $m_e$ is the mass of an electron---and let $()_N$ denote the analogous quantities for the neutral particle. @@ -210,8 +210,146 @@ \subsubsection{Electron-Neutral Elastic Scattering} -\subsubsection{Electron-Neutral Excitation Reactions} +\subsubsection{Electron-Neutral Excitation Reactions}\label{sec:en-excitation} +Consider the following reaction: +% +\begin{equation*} +\ce{A + e <=> B + e}, +\end{equation*} +% +which has the form of, for example, electron impact excitation of +argon ($\ce{Ar + e <=> Ar^{\ast} + e}$). Two particle reactive +collisions are treated by many authors~\cite{}, but special care must +be taken here since the electron appears on both sides of the +reaction. To be clear, we first consider the forward and reverse +reactions separately. + +\paragraph{Forward} +The forward process is $\ce{A + e -> B + e}$. At a given point in +velocity space $\mbf{v}_e$, this leads to both destruction and +creation of electrons. The destruction term is given by integrating +over all interactions that consume electrons at $\mbf{v}_e$, weighted +by the transition probability: +% +\begin{equation*} +Q_{f, D}(\mbf{v}_e) += +-\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} +f_e(\mbf{v}_e) f_A(\mbf{v}_A) W_f(\mbf{v}_0, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime}) +\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}, +\end{equation*} +% +where $W_f(\mbf{v}_0, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime})$ is +the ``transition probability'' for the forward process taking an +$\ce{A}$ particle at $\mbf{v}_A$ and an electron at $\mbf{v}_e$ and +producing a $\ce{B}$ particle at $\mbf{v}_B$ and an electron at +$\mbf{v}_e^{\prime}$. + +Similarly, the creation term is given by integrating over all +interactions that produce electrons at $\mbf{v}_e$: +% +\begin{equation*} +Q_{f, C}(\mbf{v}_e) += +\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} +f_e(\mbf{v}_e^{\prime}) f_A(\mbf{v}_A) W_f(\mbf{v}_0, \mbf{v}_e^{\prime}; \mbf{v}_B, \mbf{v}_e) +\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}. +\end{equation*} +% + +\paragraph{Backward} +Analogously, the backward process ($\ce{B + e -> A + e}$) also leads +to both destruction and creation of electrons at a given velocity: +% +\begin{gather*} +Q_{b, D}(\mbf{v}_e) += +-\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} +f_e(\mbf{v}_e) f_B(\mbf{v}_B) W_b(\mbf{v}_B, \mbf{v}_e; \mbf{v}_A, \mbf{v}_e^{\prime}) +\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}, \\ +% +Q_{b, C}(\mbf{v}_e) += +\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} +f_e(\mbf{v}_e^{\prime}) f_B(\mbf{v}_B) W_b(\mbf{v}_B, \mbf{v}_e^{\prime}; \mbf{v}_A, \mbf{v}_e) +\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}, +\end{gather*} +% +where $W_b$ denotes the transition probability of the backward process. + +\paragraph{Detailed Baland and Net Creation} +Following~\cite{}, we observe that, to satisfy detailed balance, the +foward and backward transition probablities are related as follows: +% +\begin{equation*} +W_b(\mbf{v}_B, \mbf{v}_e^{\prime}; \mbf{v}_A, \mbf{v}_e) += +\alpha W_f(\mbf{v}_A, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime}) +\end{equation*} +% +where $\alpha$ is a constant given by\todo{figure this out}. + +Thus, the forward and backward expressions above may be combined to +for the net production rate as follows: +% +\begin{align*} +Q(\mbf{v}_e) = & \int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} +\left[ \alpha f_e(\mbf{v}_e^{\prime}) f_B(\mbf{v}_B) - f_e(\mbf{v}_e) f_A(\mbf{v}_A) \right] +W_f(\mbf{v}_0, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime}) +\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime} \\ +% +& + +\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} +\left[ f_e(\mbf{v}_e^{\prime}) f_A(\mbf{v}_A) - \alpha f_e(\mbf{v}_e) f_B(\mbf{v}_B) \right] +W_f(\mbf{v}_A, \mbf{v}_e^{\prime}; \mbf{v}_B, \mbf{v}_e) +\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}. +\end{align*} +% +As in~\S\ref{sec:en-scattering} the nine dimensional integrals above +may be reduced to five dimensional integrals by imposing conservation +of mass, momentum, and energy. The first term on the right hand side +is particularly similar because, as in the scattering case, we may +take the ``pre-collision'' velocities $\mbf{v}_A$ and $\mbf{v}_e$ as +given. Then, conservation of momentum and energy require that +% +\begin{gather*} +\mbf{v}_B += +m_A \mbf{v}_A + m_e \mbf{v}_e + \frac{m_e}{m_B+m_e} g^{\prime} \mbf{\hat{e}},\\ +% +\mbf{v}_e^{\prime} += +m_A \mbf{v}_A + m_e \mbf{v}_e - \frac{m_B}{m_B+m_e} g^{\prime} \mbf{\hat{e}}, +\end{gather*} +% +where $\mbf{\hat{e}}$ the direction of the ``post-collision'' relative +velocity and $g^{\prime}$ is ``post-collision'' relative speed, given +by +% +\begin{equation*} +g^{\prime} = \left( g^2 - \frac{2 m_A m_e}{m_A + m_e} \Delta E \right)^2, +\end{equation*} +% +where $g$ is the precollision relative speed (i.e., $g = |\mbf{v}_A - +\mbf{v}_e|$) and $\Delta E$ is the change in internal energy +associated with the reaction. Thus, +% +\begin{align*} +\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} & +\left[ \alpha f_e(\mbf{v}_e^{\prime}) f_B(\mbf{v}_B) - f_e(\mbf{v}_e) f_A(\mbf{v}_A) \right] +W_f(\mbf{v}_0, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime}) +\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime} += \\ +& \int_{\mbb{R}^3}\int_{S^2} +\left[ \alpha f_e(\mbf{v}_e^{\prime}) f_B(\mbf{v}_B) - f_e(\mbf{v}_e) f_A(\mbf{v}_A) \right] +g \sigma(g, \mbf{\hat{e}}) +\, d\mbf{\hat{e}} \, d^3\mbf{v}_A, +\end{align*} +% +where $\sigma$ is the forward reaction differential cross section. +In principle, one may reduce the second 9D integral to 5D in an +analogous way, but the details are more complicated. \todo{finish} \subsubsection{Electron-Neutral Ionization Reactions} From 342ac6907635023670fa52d99124690e9ec89d10 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Fri, 28 May 2021 14:25:39 -0500 Subject: [PATCH 13/19] Update glow discharge modeling doc to include total energy --- doc/fluid/pdes.tex | 156 +++++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 152 insertions(+), 4 deletions(-) diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex index ff11dde7..9e07c1a2 100644 --- a/doc/fluid/pdes.tex +++ b/doc/fluid/pdes.tex @@ -69,12 +69,12 @@ \subsection{Energy Equations} \label{sec:pdes-energy} - \underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} - -\underbrace{q_e \sum_{j=1}^{N_r} \Delta E_j G_j}_{\textrm{Inelastic}}, +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}}, \end{gather*} % where $m_e$ and $m_b$ denote the molecular mass of electron and the background species, respectively, $\kappa_e$ is the electron thermal -conductivity, $\Delta E_j$ is the energy lost per electron in the +conductivity, $\Delta E^e_j$ is the energy lost per electron in the collisional process represented by reaction $j$, and $G_j$ is the rate of progress of reaction $j$. The thermal conductivity $\kappa_e$ and chemistry related quantities $\Delta E_j$ and $G_j$ are described more @@ -90,8 +90,8 @@ \subsection{Energy Equations} \label{sec:pdes-energy} \underbrace{C_{\textrm{inc}} \sum_{\alpha \in h} q_e z_{\alpha} F_\alpha E}_{\textrm{Joule heating}} + \underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} -+ -\underbrace{q_e \sum_{j=1}^{N_r} \Delta E_j G_j}_{\textrm{Inelastic}}, +- +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E^g_j G_j}_{\textrm{Inelastic}}, \end{gather*} % where the notation $\sum_{\alpha \in h}$ denotes the summation over @@ -102,6 +102,154 @@ \subsection{Energy Equations} \label{sec:pdes-energy} the electrons into heat due to the fact that the ion mean free path can be of the same order of the sheath thickness~\cite{}. +\subsection{Total Energy} \label{sec:pdes-energy} +Rather than solving the electron energy and heavy species thermal +energy equations, it is also possible to formulate the model in terms +of the total energy. Specifically, let $n e_t$ denote the total +energy per unit volume: +% +\begin{equation*} +n e_t += +n_e e_e + \sum_{\alpha \in h} n_{\alpha} e_{\alpha} += +n_e \frac{3}{2} k_B T_e + \sum_{\alpha \in h} n_{\alpha} \left( \frac{3}{2} k_B T_g + \varepsilon_{\alpha} \right), +\end{equation*} +% +where $\varepsilon_{\alpha}$ denotes the internal energy of species +$\alpha$ (i.e., energy not associated with thermal motion). To begin, +we assume that the heavy species are monatomic, such that the internal +energy is associated with electronic excitation. Since these excited +states are explicitly tracked as separate species, $\varepsilon_{\alpha}$ +is a given constant number, not a function requiring additional state +information or assumptions. +% +\todo{How will this go with more complex gases, e.g., diatomics, where + there are more ways to store internal energy other than electronic + excitation (vibration, rotation)? Still track explicitly or + introduce addtional temperatures or assume some kind of + equilibrium?} +% +Further, for monatomics, $C_v = \frac{3}{2} k_B$. As before, the +heavy species have been assumed to have the same temperature $T_g$. +Finally, we have assumed that the bulk motion of the fluid is +negligible, such that there is no contribution from the kinetic energy +associated with the bulk velocity. +% + +This energy is governed by the following PDE: +% +\begin{equation} +\pp{ne_t}{t} = +\underbrace{-\,\pp{q}{x}}_{\mathrm{transport}} + +\underbrace{\sum_{\alpha} q_e z_{\alpha} F_{\alpha} E}_{\textrm{Joule heating}} + +\underbrace{\dot{Q}_{rad}}_{\textrm{Radiative heating}}, +\label{eqn:totalE} +\end{equation} +% +where the heat flux $q$ is given by +% +\begin{equation} +q += +\frac{5}{2} k_B T_e F_e - \kappa_e \pp{T_e}{x} + +\sum_{\alpha \in h} \left\{ \left( \frac{5}{2} k_B T_g + \varepsilon_{\alpha} \right) F_{\alpha} - \kappa_{\alpha} \pp{T_g}{x} \right\}, +\label{eqn:totalE_flux} +\end{equation} +% +and the radiative heating is given by +% +\begin{equation} +\dot{Q}_{rad} = I dunno yet +\label{eqn:totalE_rad} +\end{equation} +\todo{Fill this in, but... we can get going assuming it is zero} +% + +\subsubsection{Total Energy Sanity Check} +As a check on our total energy equation, we derive it from previously +introduced equations. We have +% +\begin{align*} +\pp{ne_t}{t} &= +\underbrace{\pp{}{t} \left( n_e \frac{3}{2} k_B T_e \right)}_{\textrm{Electron energy}} + +\underbrace{\pp{}{t} \left( \sum_{\alpha \in h} \frac{3}{2} k_B n_\alpha T_g \right)}_{\textrm{Heavy energy}} + +\underbrace{\sum_{\alpha \in h} \pp{n_{\alpha}}{t} \varepsilon_{\alpha}}_{\textrm{Species continuity}}, +\end{align*} +% +where the underbraces denote the previous equations that each term is +associated with. Collecting the transport terms from these equations, +it is straightforward to show that $q$ defined +in~\eqref{eqn:totalE_flux} is consistent. Similarly, collecting the +Joule heating terms from the electron and heavy species energy +equations, and setting $C_{\mathrm{inc}} = 1$ gives the Joule heating +term in~\eqref{eqn:totalE}. Clearly, we may allow $C_{\mathrm{inc}} +\neq 1$ in~\eqref{eqn:totalE} by a simple modification. + +From the remaining source terms we have +% +\begin{equation*} +- +\underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} +- +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}}, ++ +\underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} +- +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E^g_j G_j}_{\textrm{Inelastic}}, ++ +\sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha}. +\end{equation*} +% +As expected, the elastic terms simply exchange thermal energy between +the electrons and heavies and thus cancel each other. The remaining +(inelastic) terms represent exchanges between thermal and internal +energy: +% +\begin{equation*} +- +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}}, +- +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E^g_j G_j}_{\textrm{Inelastic}}, ++ +\sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha}. +\end{equation*} +% +To the extent that these do not balance, it must be because the +mismatch is radiated away. For instance, we can imagine a +de-excitation event where part of the excess energy is radiated away +rather than being converted into thermal energy. This must be +accounted for in the radiation model $\dot{Q}_{rad}$. + +If we assert that $\dot{Q}_{rad} = 0$, this implies that +% +\begin{equation*} +\sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha} += +q_e \sum_{j=1}^{N_r} ( \Delta E^e_j + \Delta E^g_j) G_j +\end{equation*} +% +To assess this, we rewrite $\dot{\omega}_{\alpha}$ in terms of the +contribution from each reaction: + +\begin{equation*} +\sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha} += +\sum_{j=1}^{N_r} \left\{ \sum_{\alpha} \left( \nu_{\alpha j}^{\prime \prime} - \nu_{\alpha j}^{\prime} \right) \varepsilon_{\alpha} \right\} G_j, +\end{equation*} +% +where $\nu_{\alpha j}^{\prime \prime}$ and $\nu_{\alpha j}^{\prime}$ +are stoichiometric coefficients (see~\ref{sec:fluid-closures}). Thus, +for $\dot{Q}_{rad} = 0$, it is sufficient to have +% +\begin{equation*} +\sum_{\alpha} \left( \nu_{\alpha j}^{\prime \prime} - \nu_{\alpha j}^{\prime} \right) \varepsilon_{\alpha} += +\Delta E^e_j + \Delta E^g_j +\end{equation*} +% +for all reactions $j$. + \subsection{Electromagnetics} \label{sec:pdes-poisson} A quasi-static model is used for the electromagnetics. In this case, Maxwell's equations reduce to a Poisson equation governing the From a576e595b46a455c9aeae094f5b8a96e322888cf Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Thu, 3 Jun 2021 08:42:32 -0500 Subject: [PATCH 14/19] Add subsection on pressure constraint --- doc/fluid/pdes.tex | 103 +++++++++++++++++++++++++++++++++------------ 1 file changed, 75 insertions(+), 28 deletions(-) diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex index 9e07c1a2..7021e694 100644 --- a/doc/fluid/pdes.tex +++ b/doc/fluid/pdes.tex @@ -48,18 +48,18 @@ \subsection{Species Continuity} \S\ref{sec:pdes-energy}, a two temperature model is adopted where the heavy species share a single temperature $T_g$, such that the ideal gas law becomes -% + \begin{equation*} p = n_e k_B T_e + k_B T_g \sum_{\alpha \neq e} n_{\alpha}. \end{equation*} -% + \subsection{Energy Equations} \label{sec:pdes-energy} A two-temperature model is adopted. We assume a common temperature $T_g$ for the heavy species and a separate temperature for the electrons. The electron energy equation is given by -% + \begin{gather*} \pp{}{t} \left( \frac{3}{2} k_B n_e T_e \right) + @@ -71,7 +71,7 @@ \subsection{Energy Equations} \label{sec:pdes-energy} - \underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}}, \end{gather*} -% + where $m_e$ and $m_b$ denote the molecular mass of electron and the background species, respectively, $\kappa_e$ is the electron thermal conductivity, $\Delta E^e_j$ is the energy lost per electron in the @@ -81,7 +81,7 @@ \subsection{Energy Equations} \label{sec:pdes-energy} in \S\ref{sec:fluid-closures}. The gas energy equation is given by -% + \begin{gather*} \pp{}{t} \left( \sum_{\alpha \in h} C_{v,\alpha} n_\alpha T_g \right) + @@ -93,7 +93,7 @@ \subsection{Energy Equations} \label{sec:pdes-energy} - \underbrace{q_e \sum_{j=1}^{N_r} \Delta E^g_j G_j}_{\textrm{Inelastic}}, \end{gather*} -% + where the notation $\sum_{\alpha \in h}$ denotes the summation over the heavy species, $C_{v,\alpha}$ and $C_{p,\alpha}$ are the specific heat at constant volume and pressure, respectively, for species @@ -107,7 +107,7 @@ \subsection{Total Energy} \label{sec:pdes-energy} energy equations, it is also possible to formulate the model in terms of the total energy. Specifically, let $n e_t$ denote the total energy per unit volume: -% + \begin{equation*} n e_t = @@ -115,7 +115,7 @@ \subsection{Total Energy} \label{sec:pdes-energy} = n_e \frac{3}{2} k_B T_e + \sum_{\alpha \in h} n_{\alpha} \left( \frac{3}{2} k_B T_g + \varepsilon_{\alpha} \right), \end{equation*} -% + where $\varepsilon_{\alpha}$ denotes the internal energy of species $\alpha$ (i.e., energy not associated with thermal motion). To begin, we assume that the heavy species are monatomic, such that the internal @@ -123,22 +123,22 @@ \subsection{Total Energy} \label{sec:pdes-energy} states are explicitly tracked as separate species, $\varepsilon_{\alpha}$ is a given constant number, not a function requiring additional state information or assumptions. -% + \todo{How will this go with more complex gases, e.g., diatomics, where there are more ways to store internal energy other than electronic excitation (vibration, rotation)? Still track explicitly or introduce addtional temperatures or assume some kind of equilibrium?} -% + Further, for monatomics, $C_v = \frac{3}{2} k_B$. As before, the heavy species have been assumed to have the same temperature $T_g$. Finally, we have assumed that the bulk motion of the fluid is negligible, such that there is no contribution from the kinetic energy associated with the bulk velocity. -% + This energy is governed by the following PDE: -% + \begin{equation} \pp{ne_t}{t} = \underbrace{-\,\pp{q}{x}}_{\mathrm{transport}} + @@ -146,9 +146,9 @@ \subsection{Total Energy} \label{sec:pdes-energy} \underbrace{\dot{Q}_{rad}}_{\textrm{Radiative heating}}, \label{eqn:totalE} \end{equation} -% + where the heat flux $q$ is given by -% + \begin{equation} q = @@ -156,27 +156,27 @@ \subsection{Total Energy} \label{sec:pdes-energy} \sum_{\alpha \in h} \left\{ \left( \frac{5}{2} k_B T_g + \varepsilon_{\alpha} \right) F_{\alpha} - \kappa_{\alpha} \pp{T_g}{x} \right\}, \label{eqn:totalE_flux} \end{equation} -% + and the radiative heating is given by -% + \begin{equation} \dot{Q}_{rad} = I dunno yet \label{eqn:totalE_rad} \end{equation} \todo{Fill this in, but... we can get going assuming it is zero} -% + \subsubsection{Total Energy Sanity Check} As a check on our total energy equation, we derive it from previously introduced equations. We have -% + \begin{align*} \pp{ne_t}{t} &= \underbrace{\pp{}{t} \left( n_e \frac{3}{2} k_B T_e \right)}_{\textrm{Electron energy}} + \underbrace{\pp{}{t} \left( \sum_{\alpha \in h} \frac{3}{2} k_B n_\alpha T_g \right)}_{\textrm{Heavy energy}} + \underbrace{\sum_{\alpha \in h} \pp{n_{\alpha}}{t} \varepsilon_{\alpha}}_{\textrm{Species continuity}}, \end{align*} -% + where the underbraces denote the previous equations that each term is associated with. Collecting the transport terms from these equations, it is straightforward to show that $q$ defined @@ -187,7 +187,7 @@ \subsubsection{Total Energy Sanity Check} \neq 1$ in~\eqref{eqn:totalE} by a simple modification. From the remaining source terms we have -% + \begin{equation*} - \underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} @@ -200,12 +200,12 @@ \subsubsection{Total Energy Sanity Check} + \sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha}. \end{equation*} -% + As expected, the elastic terms simply exchange thermal energy between the electrons and heavies and thus cancel each other. The remaining (inelastic) terms represent exchanges between thermal and internal energy: -% + \begin{equation*} - \underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}}, @@ -214,7 +214,7 @@ \subsubsection{Total Energy Sanity Check} + \sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha}. \end{equation*} -% + To the extent that these do not balance, it must be because the mismatch is radiated away. For instance, we can imagine a de-excitation event where part of the excess energy is radiated away @@ -222,13 +222,13 @@ \subsubsection{Total Energy Sanity Check} accounted for in the radiation model $\dot{Q}_{rad}$. If we assert that $\dot{Q}_{rad} = 0$, this implies that -% + \begin{equation*} \sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha} = q_e \sum_{j=1}^{N_r} ( \Delta E^e_j + \Delta E^g_j) G_j \end{equation*} -% + To assess this, we rewrite $\dot{\omega}_{\alpha}$ in terms of the contribution from each reaction: @@ -237,19 +237,66 @@ \subsubsection{Total Energy Sanity Check} = \sum_{j=1}^{N_r} \left\{ \sum_{\alpha} \left( \nu_{\alpha j}^{\prime \prime} - \nu_{\alpha j}^{\prime} \right) \varepsilon_{\alpha} \right\} G_j, \end{equation*} -% + where $\nu_{\alpha j}^{\prime \prime}$ and $\nu_{\alpha j}^{\prime}$ are stoichiometric coefficients (see~\ref{sec:fluid-closures}). Thus, for $\dot{Q}_{rad} = 0$, it is sufficient to have -% + \begin{equation*} \sum_{\alpha} \left( \nu_{\alpha j}^{\prime \prime} - \nu_{\alpha j}^{\prime} \right) \varepsilon_{\alpha} = \Delta E^e_j + \Delta E^g_j \end{equation*} -% + for all reactions $j$. +\subsection{Pressure Constraint} +As noted previously, the pressure is assumed constant, which imposes +an additional constraint on the system. There are potentially +multiple methods to implement this constraint. Here, we assume that +the background gas (argon) is locally added or removed in order to +maintain the fixed pressure. This results in both mass and energy +source terms in the governing PDEs. Letting $n_b$ denote the number +density of the background species, then +% +\begin{equation*} +\pp{n_b}{t} = S, +\end{equation*} +% +where $s$ is determined by requiring constant pressure. For a mixture +of monatomic gases, +% +\begin{equation*} +ne_t = \frac{3}{2} p + \sum_{\alpha} n_{\alpha} \varepsilon_{\alpha}. +\end{equation*} +% +Thus, since the pressure is fixed, +% +\begin{equation*} +\pp{ne_t}{t} += \frac{3}{2} \pp{p}{t} + \sum_{\alpha} \pp{n_{\alpha}}{t} \varepsilon_{\alpha} += \sum_{\alpha} \pp{n_{\alpha}}{t} \varepsilon_{\alpha}. +\end{equation*} +% +Using the total energy equation from before, plus a source term +imposed by addition of the background species at the rate $S$ with the +local temperature $T_g$, this imposes a constraint that determines +$S$. Specifically, +% +\begin{equation*} +\sum_{\alpha} \pp{n_{\alpha}}{t} \varepsilon_{\alpha} += +-\,\pp{q}{x} ++ +\sum_{\alpha} q_e z_{\alpha} F_{\alpha} E ++ +\dot{Q}_{rad} ++ +\frac{3}{2} S k_B T_g. +\end{equation*} +% +This is an algebraic constraint that can be used to eliminate $S$. + \subsection{Electromagnetics} \label{sec:pdes-poisson} A quasi-static model is used for the electromagnetics. In this case, Maxwell's equations reduce to a Poisson equation governing the From ab7ba94b750a2e88a5d285b7aa1aea03b15f6d10 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Thu, 3 Jun 2021 11:00:57 -0500 Subject: [PATCH 15/19] Start on an eqn summary --- doc/fluid/pdes.tex | 38 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 38 insertions(+) diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex index 7021e694..4f96ec7f 100644 --- a/doc/fluid/pdes.tex +++ b/doc/fluid/pdes.tex @@ -316,3 +316,41 @@ \subsection{Electromagnetics} \label{sec:pdes-poisson} % where $q_e$ is the elementary charge (the magnitude of the charge of an electron). + + +\subsection{Equation Summary} +The state vector consists of the species densities ($n_e$ and +$n_{\alpha}$ for $\alpha \in h$) and the electron energy ($n_e e_e$). +The evolution equations are +% +\begin{gather*} +\pp{n_b}{t} = S, \quad \textrm{where ``b'' indicates the background specie}\\ +\pp{n_{\alpha}}{t} + \pp{F_{\alpha}}{x} = \dot{\omega}_{\alpha}, \quad \textrm{for all non-background species}\\ +\pp{}{t} \left( \frac{3}{2} k_B n_e T_e \right) ++ +\pp{}{x} \left( \frac{5}{2} k_B T_e F_e - \kappa_e \pp{T_e}{x} \right) += +- q_e F_e E +- +\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b} +- +q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j. +\end{gather*} +% +These equations are unclosed because of the precence of the fields $S$ +and $E = - \nabla \phi$. These fields are determined by two +additional constraints: +% +\begin{gather*} +\sum_{\alpha} \pp{n_{\alpha}}{t} \varepsilon_{\alpha} += +-\,\pp{q}{x} ++ +\sum_{\alpha} q_e z_{\alpha} F_{\alpha} E ++ +\dot{Q}_{rad} ++ +\frac{3}{2} S k_B T_g \\ +% +-\pp{^2\phi}{x^2} = \frac{1}{\epsilon_0} \sum_{\alpha=1}^{N_s} z_{\alpha} q_e n_{\alpha}. +\end{gather*} From 26d23ac2becc1ffece1ac522ce6b6b4d16126248 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Thu, 3 Jun 2021 14:00:49 -0500 Subject: [PATCH 16/19] some cleanup in fluid model pdes section --- doc/fluid/pdes.tex | 216 +++++++++++++++++++++++++++------------------ 1 file changed, 129 insertions(+), 87 deletions(-) diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex index 4f96ec7f..4a974ca8 100644 --- a/doc/fluid/pdes.tex +++ b/doc/fluid/pdes.tex @@ -7,9 +7,10 @@ validation relevant to the torch simulations. \section{PDEs} -We require PDEs for the species densities, temperatures, and electric potential. +We require equations governing for the species densities, +temperatures, and electric potential. -\subsection{Species Continuity} +\subsection{Species Continuity} \label{sec:species-continuity} The species transport equations are as follows: % \begin{equation} @@ -35,9 +36,23 @@ \subsection{Species Continuity} equation (\S\ref{sec:pdes-poisson}). For a mixture of $N_s$ species, $N_s - 1$ species densities are -evolved according to~\eqref{eqn:species}. The density of the dominant -background species (e.g., argon) is determined from the pressure, -which is assumed known, and the ideal gas law: +evolved according to~\eqref{eqn:species}. Alternatively, the density +of the dominant background species (e.g., argon) is determined either +by assuming it to be constant or by assuming that the pressure is +constant. We denote these as the ``constant $n_b$'' and ``constant +$p$'' cases. In the latter case, the background specie density, +denoted $n_b$, is evolved according to +% +\begin{equation*} +\pp{n_b}{t} = S, +\end{equation*} +% +where the field $S$ is determined by imposing the constraint that the +pressure remains constant, as detailed in~\S\ref{sec:pconstraint}. + +\subsection{Ideal Gas Law} +In either case (constant $n_b$ or constant $p$), the ideal gas law is +used to relate the pressure to the species densities and temperatures: % \begin{equation*} p = \sum_{\alpha = 1}^{N_s} n_{\alpha} k_B T_{\alpha}, @@ -45,21 +60,27 @@ \subsection{Species Continuity} % where $k_B$ is the Boltzmann constant and $T_{\alpha}$ is the temperature of species $\alpha$. As discussed in -\S\ref{sec:pdes-energy}, a two temperature model is adopted where the +\S\ref{sec:pdes-energy}, a two temperature model is adopted here. In this model, the heavy species share a single temperature $T_g$, such that the ideal gas law becomes - +% \begin{equation*} -p = n_e k_B T_e + k_B T_g \sum_{\alpha \neq e} n_{\alpha}. +p = n_e k_B T_e + k_B T_g \sum_{\alpha \in h} n_{\alpha}, \end{equation*} - - +% +where $\alpha \in h$ indicates that $\alpha$ ranges over the heavy +species only (i.e., all species except the electrons). \subsection{Energy Equations} \label{sec:pdes-energy} A two-temperature model is adopted. We assume a common temperature $T_g$ for the heavy species and a separate temperature for the -electrons. The electron energy equation is given by +electrons. In this section, we formulate three energy equations. Any +combination of two of these is sufficient to describe the system. +\subsubsection{Electron Energy} +Letting $n_e e_e = \frac{3}{2} k_B n_e T_e$, the electron energy +equation is given by +% \begin{gather*} \pp{}{t} \left( \frac{3}{2} k_B n_e T_e \right) + @@ -71,7 +92,7 @@ \subsection{Energy Equations} \label{sec:pdes-energy} - \underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}}, \end{gather*} - +% where $m_e$ and $m_b$ denote the molecular mass of electron and the background species, respectively, $\kappa_e$ is the electron thermal conductivity, $\Delta E^e_j$ is the energy lost per electron in the @@ -80,34 +101,44 @@ \subsection{Energy Equations} \label{sec:pdes-energy} chemistry related quantities $\Delta E_j$ and $G_j$ are described more in \S\ref{sec:fluid-closures}. -The gas energy equation is given by - -\begin{gather*} +\subsubsection{Heavy Species Energy} +The heavy species energy equation is given by +% +\begin{align*} \pp{}{t} \left( \sum_{\alpha \in h} C_{v,\alpha} n_\alpha T_g \right) -+ ++& \pp{}{x} \left( \sum_{\alpha \in h} C_{p,\alpha} F_\alpha T_g - \sum_{\alpha \in h} \kappa_\alpha \pp{T_g}{x} \right) -= += \\ +& \underbrace{C_{\textrm{inc}} \sum_{\alpha \in h} q_e z_{\alpha} F_\alpha E}_{\textrm{Joule heating}} + \underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} - \underbrace{q_e \sum_{j=1}^{N_r} \Delta E^g_j G_j}_{\textrm{Inelastic}}, -\end{gather*} - +\end{align*} +% where the notation $\sum_{\alpha \in h}$ denotes the summation over the heavy species, $C_{v,\alpha}$ and $C_{p,\alpha}$ are the specific heat at constant volume and pressure, respectively, for species -$\alpha$. Finally, $C_{\textrm{inc}}$ is a constant that may be used -to account for incomplete conversion of the kinetic energy gained by -the electrons into heat due to the fact that the ion mean free path -can be of the same order of the sheath thickness~\cite{}. - -\subsection{Total Energy} \label{sec:pdes-energy} +$\alpha$. From here forward, we assume that only monatomic species +are included, such that $C_{v,\alpha} = \frac{3}{2} k_B$ and +$C_{p,\alpha} = \frac{5}{2} k_B$. This assumption is true for the +argon-related species we include in year 1 ($\ce{Ar}$, +$\ce{Ar^{\ast}}$ and $\ce{Ar^+}$), but will need to be relaxed to +model air (or even more complex argon chemistry). Similarly to +$\Delta E^e_j$, the constants $\Delta E^g_j$ represent the energy lost +per heavy species particle in reaction $j$. Finally, +$C_{\textrm{inc}}$ is a constant that may be used to account for +incomplete conversion of the kinetic energy gained (by the ions from +the electric field) into heat due to the fact that the ion mean free +path can be of the same order of the sheath thickness~\cite{}. + +\subsubsection{Total Energy} Rather than solving the electron energy and heavy species thermal energy equations, it is also possible to formulate the model in terms of the total energy. Specifically, let $n e_t$ denote the total energy per unit volume: - +% \begin{equation*} n e_t = @@ -115,30 +146,20 @@ \subsection{Total Energy} \label{sec:pdes-energy} = n_e \frac{3}{2} k_B T_e + \sum_{\alpha \in h} n_{\alpha} \left( \frac{3}{2} k_B T_g + \varepsilon_{\alpha} \right), \end{equation*} - +% where $\varepsilon_{\alpha}$ denotes the internal energy of species $\alpha$ (i.e., energy not associated with thermal motion). To begin, we assume that the heavy species are monatomic, such that the internal -energy is associated with electronic excitation. Since these excited -states are explicitly tracked as separate species, $\varepsilon_{\alpha}$ -is a given constant number, not a function requiring additional state -information or assumptions. - -\todo{How will this go with more complex gases, e.g., diatomics, where - there are more ways to store internal energy other than electronic - excitation (vibration, rotation)? Still track explicitly or - introduce addtional temperatures or assume some kind of - equilibrium?} - -Further, for monatomics, $C_v = \frac{3}{2} k_B$. As before, the -heavy species have been assumed to have the same temperature $T_g$. -Finally, we have assumed that the bulk motion of the fluid is -negligible, such that there is no contribution from the kinetic energy -associated with the bulk velocity. - - -This energy is governed by the following PDE: - +energy is associated only with electronic excitation or ionization. +That is for excited state species, $\varepsilon_{\alpha}$ is the +excitation energy, and for ions, $\varepsilon_{\alpha}$ is the +ionization energy. For electrons and ground state argon, +$\varepsilon_{\alpha} = 0$. Finally, we have assumed that the bulk +motion of the fluid is negligible, such that there is no contribution +from the kinetic energy associated with the bulk velocity. + +The total energy is governed by the following PDE: +% \begin{equation} \pp{ne_t}{t} = \underbrace{-\,\pp{q}{x}}_{\mathrm{transport}} + @@ -146,9 +167,9 @@ \subsection{Total Energy} \label{sec:pdes-energy} \underbrace{\dot{Q}_{rad}}_{\textrm{Radiative heating}}, \label{eqn:totalE} \end{equation} - +% where the heat flux $q$ is given by - +% \begin{equation} q = @@ -156,27 +177,23 @@ \subsection{Total Energy} \label{sec:pdes-energy} \sum_{\alpha \in h} \left\{ \left( \frac{5}{2} k_B T_g + \varepsilon_{\alpha} \right) F_{\alpha} - \kappa_{\alpha} \pp{T_g}{x} \right\}, \label{eqn:totalE_flux} \end{equation} - -and the radiative heating is given by - -\begin{equation} -\dot{Q}_{rad} = I dunno yet -\label{eqn:totalE_rad} -\end{equation} -\todo{Fill this in, but... we can get going assuming it is zero} +% +and the radiative heating is modeled in $\dot{Q}_{rad}$. To start, we +set $\dot{Q}_{rad} = 0$, but this assumption will be replaced once the +initial implementation is complete. \subsubsection{Total Energy Sanity Check} As a check on our total energy equation, we derive it from previously introduced equations. We have - +% \begin{align*} \pp{ne_t}{t} &= \underbrace{\pp{}{t} \left( n_e \frac{3}{2} k_B T_e \right)}_{\textrm{Electron energy}} + \underbrace{\pp{}{t} \left( \sum_{\alpha \in h} \frac{3}{2} k_B n_\alpha T_g \right)}_{\textrm{Heavy energy}} + \underbrace{\sum_{\alpha \in h} \pp{n_{\alpha}}{t} \varepsilon_{\alpha}}_{\textrm{Species continuity}}, \end{align*} - +% where the underbraces denote the previous equations that each term is associated with. Collecting the transport terms from these equations, it is straightforward to show that $q$ defined @@ -187,7 +204,7 @@ \subsubsection{Total Energy Sanity Check} \neq 1$ in~\eqref{eqn:totalE} by a simple modification. From the remaining source terms we have - +% \begin{equation*} - \underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} @@ -200,12 +217,12 @@ \subsubsection{Total Energy Sanity Check} + \sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha}. \end{equation*} - +% As expected, the elastic terms simply exchange thermal energy between the electrons and heavies and thus cancel each other. The remaining (inelastic) terms represent exchanges between thermal and internal energy: - +% \begin{equation*} - \underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}}, @@ -214,57 +231,57 @@ \subsubsection{Total Energy Sanity Check} + \sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha}. \end{equation*} - +% To the extent that these do not balance, it must be because the mismatch is radiated away. For instance, we can imagine a de-excitation event where part of the excess energy is radiated away rather than being converted into thermal energy. This must be -accounted for in the radiation model $\dot{Q}_{rad}$. - -If we assert that $\dot{Q}_{rad} = 0$, this implies that - +accounted for in the radiation model $\dot{Q}_{rad}$. If we assert +that $\dot{Q}_{rad} = 0$, this implies that +% \begin{equation*} \sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha} = q_e \sum_{j=1}^{N_r} ( \Delta E^e_j + \Delta E^g_j) G_j \end{equation*} - +% To assess this, we rewrite $\dot{\omega}_{\alpha}$ in terms of the contribution from each reaction: - +% \begin{equation*} \sum_{\alpha} \dot{\omega}_{\alpha} \varepsilon_{\alpha} = \sum_{j=1}^{N_r} \left\{ \sum_{\alpha} \left( \nu_{\alpha j}^{\prime \prime} - \nu_{\alpha j}^{\prime} \right) \varepsilon_{\alpha} \right\} G_j, \end{equation*} - +% where $\nu_{\alpha j}^{\prime \prime}$ and $\nu_{\alpha j}^{\prime}$ -are stoichiometric coefficients (see~\ref{sec:fluid-closures}). Thus, +are stoichiometric coefficients (see~\S\ref{sec:fluid-closures}). Thus, for $\dot{Q}_{rad} = 0$, it is sufficient to have - +% \begin{equation*} \sum_{\alpha} \left( \nu_{\alpha j}^{\prime \prime} - \nu_{\alpha j}^{\prime} \right) \varepsilon_{\alpha} = \Delta E^e_j + \Delta E^g_j \end{equation*} - +% for all reactions $j$. -\subsection{Pressure Constraint} -As noted previously, the pressure is assumed constant, which imposes -an additional constraint on the system. There are potentially -multiple methods to implement this constraint. Here, we assume that -the background gas (argon) is locally added or removed in order to -maintain the fixed pressure. This results in both mass and energy -source terms in the governing PDEs. Letting $n_b$ denote the number +\subsection{Pressure Constraint} \label{sec:pconstraint} +As noted in~\S\ref{sec:species-continuity}, in the ``constant $p$'' +mode, we assume constant pressure is maintained by adding or +subtracting the background species. Letting $n_b$ denote the number density of the background species, then % \begin{equation*} \pp{n_b}{t} = S, \end{equation*} % -where $s$ is determined by requiring constant pressure. For a mixture -of monatomic gases, +where $S$ is determined by requiring constant pressure. Assuming this +mass is introduced at the local gas temperature, this also introduces +a source term on the righthand side of the heavy species and total +energy equations. This source term is given by $\frac{3}{2} S k_B +T_g$ (again assuming only monatomic species). Then, to determine $S$, +we may use the relationship between the total energy and the pressure. Specifically, % \begin{equation*} ne_t = \frac{3}{2} p + \sum_{\alpha} n_{\alpha} \varepsilon_{\alpha}. @@ -278,10 +295,9 @@ \subsection{Pressure Constraint} = \sum_{\alpha} \pp{n_{\alpha}}{t} \varepsilon_{\alpha}. \end{equation*} % -Using the total energy equation from before, plus a source term -imposed by addition of the background species at the rate $S$ with the -local temperature $T_g$, this imposes a constraint that determines -$S$. Specifically, +Using the total energy equation~\eqref{eqn:totalE}, with the addition of the +source term imposed by addition of the background specie, this result +imposes a constraint that determines $S$: % \begin{equation*} \sum_{\alpha} \pp{n_{\alpha}}{t} \varepsilon_{\alpha} @@ -295,7 +311,31 @@ \subsection{Pressure Constraint} \frac{3}{2} S k_B T_g. \end{equation*} % -This is an algebraic constraint that can be used to eliminate $S$. +Thus, +% +\begin{equation*} +S += +\frac{ +-\sum_{\alpha} \pp{n_{\alpha}}{t} \varepsilon_{\alpha} +- +\pp{q}{x} ++ +\sum_{\alpha} q_e z_{\alpha} F_{\alpha} E ++ +\dot{Q}_{rad} +} +{\frac{3}{2} k_B T_g} +\end{equation*} +% +Thus, using the ideal gas law to determine $T_g$ from the species +densities and electron energy, +% +\begin{equation*} +\frac{3}{2} k_B T_g = \frac{ \frac{3}{2} p - n_e e_e }{\sum_{\alpha \in h} n_{\alpha}}, +\end{equation*} +% +$S$ may be determined given the state and its spatial derivatives. \subsection{Electromagnetics} \label{sec:pdes-poisson} A quasi-static model is used for the electromagnetics. In this case, @@ -354,3 +394,5 @@ \subsection{Equation Summary} % -\pp{^2\phi}{x^2} = \frac{1}{\epsilon_0} \sum_{\alpha=1}^{N_s} z_{\alpha} q_e n_{\alpha}. \end{gather*} +% + From ab173c18a26ff078a39b75341a8bed2aebfc1248 Mon Sep 17 00:00:00 2001 From: Violeta Karyofylli Date: Thu, 3 Jun 2021 16:24:05 -0500 Subject: [PATCH 17/19] Fixing coflicts after merging. --- doc/appendix.tex | 77 ++++++++++++++++++++++++++++++++++++++++++++++ doc/fluid/pdes.tex | 21 +++++++++++-- doc/glow.tex | 1 + 3 files changed, 96 insertions(+), 3 deletions(-) create mode 100644 doc/appendix.tex diff --git a/doc/appendix.tex b/doc/appendix.tex new file mode 100644 index 00000000..5ce0e5fd --- /dev/null +++ b/doc/appendix.tex @@ -0,0 +1,77 @@ +\appendix +\chapter{Non-dimensionalization} + +For the non-dimensionalization of the system of equations, the following dimensionless variables are used +% +\begin{align*} +t' = \frac{t}{\tau}, \quad \phi' = \frac{\phi}{V_0}, \quad x' = \frac{x}{L}, \quad n_{i,e}' = \frac{n_{i,e}}{n_{p0}}, \quad n_{\alpha}' = \frac{n_{\alpha}}{n_{p0}}, \quad \left(n_{e}e_{e}\right)' = \frac{n_{e}e_{e}}{n_{p0} e_0}, \quad ne_t' = \frac{ne_t}{n_{Ar} e_0}. +\end{align*} +% +The Poisson equation governing the +electric potential $\phi$ takes the following form: +% +\begin{align*} +-\pp{^2\phi}{x^2} = \frac{\rho_c}{\epsilon_0} = \frac{q_e}{\epsilon_0} \left(n_i - n_e\right) \quad &\Rightarrow \quad \pp{^2\phi}{x^2} = - \frac{q_e}{\epsilon_0} \left(n_i - n_e\right) \quad \Rightarrow \quad \frac{V_0}{L^2}\pp{^2\phi'}{x'^2} = - \frac{q_e}{\epsilon_0} \left(n_i' - n_e'\right) n_{p0} \\ +\quad &\Rightarrow \quad \pp{^2\phi'}{x'^2} = \color{red}\boxed{\color{black} - \frac{q_e n_{p0} L^2}{\epsilon_0 V_0}} \color{black}\left(n_i' - n_e'\right) . +\end{align*} +% +The species transport equations can be non-dimensionalized as follows: +% +\begin{align*} +\pp{n_{\alpha}}{t} + \pp{}{x} \left( - z_{\alpha} \mu_{\alpha} n_{\alpha} \pp{\phi}{x} - D_{\alpha} \pp{n_{\alpha}}{x}\right)= k n_{Ar} n_{e} \ &\Rightarrow \ \frac{1}{\tau}\pp{n_{\alpha}'}{t'} + \frac{1}{L}\pp{}{x'} \left( - z_{\alpha} \mu_{\alpha} n_{\alpha}' \frac{V_0}{L}\pp{\phi'}{x'} - D_{\alpha} \frac{1}{L} \pp{n_{\alpha}'}{x'}\right) = k n_{Ar} n_{e}' \\ +\ &\Rightarrow \ \pp{n_{\alpha}'}{t'} - \pp{}{x'} \left( z_{\alpha} \color{red}\boxed{\color{black} \mu_{\alpha} \frac{\tau V_0}{L^2} } \color{black} n_{\alpha}'\pp{\phi'}{x'} + \color{red}\boxed{\color{black} D_{\alpha} \frac{\tau}{L^2}} \color{black} \pp{n_{\alpha}'}{x'}\right) = \color{red}\boxed{\color{black} k n_{Ar} \tau } \color{black} n_{e}' . +\label{eqn:species} +\end{align*} +% +The electron energy equation can be non-dimensionalized as below: +% +\begin{align*} +\pp{}{t} \left( n_e e_e \right) ++ +\pp{}{x} \left( - \frac{5}{3} D_e \pp{\left(n_e e_e\right)}{x} \right. \\ +\left. - \frac{5}{3} z_e \mu_e \pp{\phi}{x} \left(n_e e_e\right) \right) +&= +\underbrace{q_e \left(-z_{e} \mu_{e} n_{e} \pp{\phi}{x} - D_{e} \pp{n_{e}}{x} \right) \pp{\phi}{x}}_{\textrm{Joule heating}} \\ +&- +\underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} \\ +&- +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}} \ \Rightarrow +\end{align*} +\begin{align*} +{\color{blue}\frac{n_{p0} e_0}{\tau}} \pp{}{t'} \left( n_e e_e \right)' ++ +{\color{blue}\frac{1}{L}} \pp{}{x} \left( - \frac{5}{3} D_e {\color{blue}\frac{n_{p0} e_0}{L}} \pp{\left(n_e e_e\right)'}{x'} \right. \\ +\left. - \frac{5}{3} z_e \mu_e {\color{blue}\frac{V_{0}}{L}} \pp{\phi'}{x'} {\color{blue}n_{p0} e_0} \left(n_e e_e\right)' \right) +&= +\underbrace{q_e \left(-z_{e} \mu_{e} {\color{blue}\frac{V_{0} n_{p0}}{L}} n_{e}' \pp{\phi'}{x'} - D_{e} {\color{blue}\frac{n_{p0}}{L}} \pp{n_{e}'}{x'} \right) {\color{blue}\frac{V_{0}}{L}} \pp{\phi'}{x'}}_{\textrm{Joule heating}} \\ +&- +\underbrace{\frac{3}{2} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} \\ +&- +\underbrace{q_e \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}} \ \Rightarrow +\end{align*} +\begin{align*} +\pp{}{t'} \left( n_e e_e \right)' ++ +\pp{}{x} \left( - \frac{5}{3} {\color{red}\boxed{\color{black} D_e \frac{\tau}{L^2}}} \color{black}\pp{\left(n_e e_e\right)'}{x'} \right. \\ +\left. - \frac{5}{3} z_e {\color{red}\boxed{\color{black} \mu_e \frac{\tau V_{0}}{L^2}}} \color{black}\pp{\phi'}{x'} \left(n_e e_e\right)' \right) +&= +\underbrace{q_e \left(-z_{e} {\color{red}\boxed{\color{black} \mu_{e} \frac{V_{0} \tau}{L^2}}} \color{black}n_{e}' \pp{\phi'}{x'} - {\color{red}\boxed{\color{black} D_{e} \frac{\tau}{L^2}}} \color{black} \pp{n_{e}'}{x'} \right) {\color{red}\boxed{\color{black} V_{0}}} \color{black} \pp{\phi'}{x'}}_{\textrm{Joule heating}} \\ +&- +\underbrace{\frac{3}{2} {\color{blue}\frac{\tau}{n_{p0} e_0}} k_B n_e \frac{2 m_e}{m_b} \left( T_e - T_g \right) \bar{\nu}_{e,b}}_{\textrm{Elastic}} \\ +&- +\underbrace{q_e {\color{blue}\frac{\tau}{n_{p0} e_0}} \sum_{j=1}^{N_r} \Delta E^e_j G_j}_{\textrm{Inelastic}} \ \Rightarrow +\end{align*} +% +The non-dimensionalization of the ideal gas law is presented below: +% +\begin{align*} +p = n_e k_B T_e + k_B T_g \sum_{\alpha \in h} n_{\alpha} \quad \Rightarrow \quad {\color{blue} n_{Ar} k_B T_0} \ p' = {\color{blue} n_{p0} e_0} \frac{2}{3} \left(n_e e_e\right)' + {\color{blue} n_{p0} e_0} \frac{2}{3} \left(k_B T_g \sum_{\alpha \in h} n_{\alpha} \right)'. +\end{align*} +% +\begin{align*} +\pp{ne_t}{t} = &- \frac{5}{3} \left( D_{e}\pp{n_e e_e}{x} + z_{e}\mu_{e} \pp{V}{x} n_e e_e \right) \\ + &- \frac{5}{3} \sum_{\alpha \in h} \left\{ \left( D_{a}\pp{\left( \frac{3}{2} k_B n_a T_g\right)}{x} + z_{a} \mu_{a} \pp{V}{x} \left( \frac{3}{2} k_B n_a T_g\right) \right) \right\} \\ + &+ \sum_{\alpha \in h} \left\{ \varepsilon_{\alpha} \left( - z_{\alpha} \mu_{\alpha} n_{\alpha} \pp{V}{x} - D_{\alpha} \pp{n_{\alpha}}{x} \right) \right\} + +\underbrace{\sum_{\alpha} q_e z_{\alpha} F_{\alpha} E}_{\textrm{Joule heating}} +\end{align*} +% \ No newline at end of file diff --git a/doc/fluid/pdes.tex b/doc/fluid/pdes.tex index 4a974ca8..7654945c 100644 --- a/doc/fluid/pdes.tex +++ b/doc/fluid/pdes.tex @@ -65,7 +65,7 @@ \subsection{Ideal Gas Law} gas law becomes % \begin{equation*} -p = n_e k_B T_e + k_B T_g \sum_{\alpha \in h} n_{\alpha}, +p = n_e k_B T_e + k_B T_g \sum_{\alpha \in h} n_{\alpha}. \end{equation*} % where $\alpha \in h$ indicates that $\alpha$ ranges over the heavy @@ -168,7 +168,7 @@ \subsubsection{Total Energy} \label{eqn:totalE} \end{equation} % -where the heat flux $q$ is given by +where the heat flux $q$ is given by % \begin{equation} q @@ -181,7 +181,22 @@ \subsubsection{Total Energy} and the radiative heating is modeled in $\dot{Q}_{rad}$. To start, we set $\dot{Q}_{rad} = 0$, but this assumption will be replaced once the initial implementation is complete. - +% +The heat flux $q$ in~\eqref{eqn:totalE_flux} can be simplified as +\begin{align*} +q += +&+ \frac{5}{2} k_B T_e F_e - \kappa_e \pp{T_e}{x} + +\sum_{\alpha \in h} \left\{ \left( \frac{5}{2} k_B T_g + \varepsilon_{\alpha} \right) F_{\alpha} - \kappa_{\alpha} \pp{T_g}{x} \right\} \\ += &- \frac{5}{3} \left( D_{e}\pp{\left( \frac{3}{2} k_B n_e T_e\right)}{x} + z_{e}\mu_{e} \pp{V}{x} \left( \frac{3}{2} k_B n_e T_e\right) \right) \\ + &- \frac{5}{3} \sum_{\alpha \in h} \left\{ \left( D_{a}\pp{\left( \frac{3}{2} k_B n_a T_g\right)}{x} + z_{a} \mu_{a} \pp{V}{x} \left( \frac{3}{2} k_B n_a T_g\right) \right) \right\} \\ + &+ \sum_{\alpha \in h} \left\{ \varepsilon_{\alpha} \left( - z_{\alpha} \mu_{\alpha} n_{\alpha} \pp{V}{x} - D_{\alpha} \pp{n_{\alpha}}{x} \right) \right\} \\ += &- \frac{5}{3} \left( D_{e}\pp{n_e e_e}{x} + z_{e}\mu_{e} \pp{V}{x} n_e e_e \right) \\ + &- \frac{5}{3} \sum_{\alpha \in h} \left\{ \left( D_{a}\pp{\left( \frac{3}{2} k_B n_a T_g\right)}{x} + z_{a} \mu_{a} \pp{V}{x} \left( \frac{3}{2} k_B n_a T_g\right) \right) \right\} \\ + &+ \sum_{\alpha \in h} \left\{ \varepsilon_{\alpha} \left( - z_{\alpha} \mu_{\alpha} n_{\alpha} \pp{V}{x} - D_{\alpha} \pp{n_{\alpha}}{x} \right) \right\} +\label{eqn:totalE_fluxSimplified} +\end{align*} +% \subsubsection{Total Energy Sanity Check} As a check on our total energy equation, we derive it from previously diff --git a/doc/glow.tex b/doc/glow.tex index 09efea0a..f3c7261d 100644 --- a/doc/glow.tex +++ b/doc/glow.tex @@ -79,6 +79,7 @@ \chapter{Hybrid Fluid/Kinetic Model} \chapter{Numerical Methods} +\input{appendix} \bibliographystyle{plainnat} \bibliography{bibs/glow} From 493b159c2f7564f85f40e32a79fe03a6d59a5523 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Wed, 25 Jun 2025 14:15:58 -0500 Subject: [PATCH 18/19] rm hybrid section This is not supported as part of the current code. Eliminate it to avoid possible confusion. --- doc/glow.tex | 6 +- doc/hybrid/collisions.tex | 355 -------------------------------------- doc/hybrid/eqns.tex | 292 ------------------------------- doc/hybrid/overview.tex | 5 - 4 files changed, 1 insertion(+), 657 deletions(-) delete mode 100644 doc/hybrid/collisions.tex delete mode 100644 doc/hybrid/eqns.tex delete mode 100644 doc/hybrid/overview.tex diff --git a/doc/glow.tex b/doc/glow.tex index f3c7261d..61144b1e 100644 --- a/doc/glow.tex +++ b/doc/glow.tex @@ -72,12 +72,8 @@ \chapter{Supporting Models and Data} \input{bolsig/overview} \input{bolsig/eqns} -\chapter{Hybrid Fluid/Kinetic Model} -\input{hybrid/overview} -\input{hybrid/eqns} -\input{hybrid/collisions} - \chapter{Numerical Methods} +\todo{Write me} \input{appendix} diff --git a/doc/hybrid/collisions.tex b/doc/hybrid/collisions.tex deleted file mode 100644 index 04cfd1fd..00000000 --- a/doc/hybrid/collisions.tex +++ /dev/null @@ -1,355 +0,0 @@ -\subsection{Collision Notes} \label{sec:col-notes} -NOTE: -\emph{This is a place for making notes about the collision integrals. - Eventually it will replace or be combined with the previous section.} - -We consider three classes of collision: -% -\begin{enumerate} -\item Electron-neutral elastic collisions, e.g., $\ce{ Ar + e -> Ar + e}$; -\item Electron-neutral excitation reactions, e.g., $\ce{ Ar + e <=> Ar^{\ast} + e}$; -\item Electron-neutral ionization reactions, e.g., $\ce{ Ar^{\ast} + e <=> Ar^+ + e + e}$. -\end{enumerate} -% - -\subsubsection{Electron-Neutral Elastic Scattering} \label{sec:en-scattering} -Let $()_e$ denote electron quantities---e.g., $f_e$ is the electron -distribution function, $m_e$ is the mass of an electron---and let -$()_N$ denote the analogous quantities for the neutral particle. -Then, the net rate of production of particles (per unit volume in -state space) is given by -% -\begin{equation} -Q_{EL}(\mbf{v}_e) -= -\int_{\mbb{R}^3} \int_{\mbb{R}^3}\int_{\mbb{R}^3} -\left( -f_e(\mbf{v}_e^{\prime}) f_N(\mbf{v}_N^{\prime}) -- -f_e(\mbf{v}_e) f_N(\mbf{v}_N) -\right) W_{EL}(\mbf{v}_e, \mbf{v}_N; \mbf{v}_e^{\prime}, \mbf{v}_N^{\prime}) -\, -d^3\mbf{v}_N \, d^3\mbf{v}_e^{\prime} \, d^3\mbf{v}_N^{\prime} -\label{eqn:elastic-collision-9d} -\end{equation} -% -where $\mbf{v}_e$ and $\mbf{v}_N$ denote the ``pre-collision'' -velocities, $\mbf{v}_e^{\prime}$ and $\mbf{v}_N^{\prime}$ denote the -``post-collision'' velocities, and $W_{EL}$ denotes the elastic -collision transition probability, which encodes both constraints -(conservation of momentum and energy) \emph{and} information about the -relative likelihood of allowable transitions. - -The nine dimensional integral in~\eqref{eqn:elastic-collision-9d} can -be reduced to a five dimensional integral by imposing conservation of -momentum and energy. To do so in the simplest possible way, let -$()_0$ denote the quantities associated with the center of mass of the -two-particle system. Specifically, the pre-collision position and -velocity of the center of mass are given by -% -\begin{equation*} -\mbf{x}_0 = \frac{m_N \mbf{x}_N + m_e \mbf{x}_e}{m_N + m_e}, -\quad -\mbf{v}_0 = \frac{m_N \mbf{v}_N + m_e \mbf{v}_e}{m_N + m_e} -\end{equation*} -% -Conservation of momentum requires that -% -\begin{gather*} -m_N \mbf{v}_N + m_e \mbf{v}_e = m_N \mbf{v}_N^{\prime} + m_e \mbf{v}_e^{\prime}\\ -(m_N +m_e) \mbf{v}_0 = (m_N + m_e) \mbf{v}_0^{\prime}. -\end{gather*} -% -That is, the velocity of the center of mass is not altered by the -collision. Thus, letting $\mbf{u}$ denote velocities measured in a -frame attached to the center of mass, one may write -% -\begin{equation*} -m_N \mbf{u}_N^{\prime} + m_e \mbf{u}_e^{\prime} -= -m_N \mbf{v}_N^{\prime} + m_e \mbf{v}_e^{\prime} - (m_N + m_e) \mbf{v}_0 -= 0. -\end{equation*} -% -Thus, conservation of momentum implies the following relationship -between the post-collision relative velocities: -% -\begin{equation*} -\mbf{u}_N^{\prime} = -\,\frac{m_e}{m_N} \mbf{u}_e^{\prime}. -\end{equation*} -% -To continue, let $\mbf{\hat{e}}$ a unit vector in the direction of the -relative velocity between the post-collision particles. Then, the -relative post-collision velocity may be written as -% -\begin{equation*} -\mbf{u}_N^{\prime} - \mbf{u}_e^{\prime} = g^{\prime} \mbf{\hat{e}}, -\end{equation*} -% -where $g'$ denotes the the magnitude of the post-collision relative -velocity. To continue, we use conservation of energy to determine -$g^{\prime}$. - -In the frame moving with the center of mass, conservation of energy requires that -% -\begin{equation*} -\frac{1}{2} m_N |\mbf{u}_N|^2 + \frac{1}{2} m_e |\mbf{u}_e|^2 -= -\frac{1}{2} m_N |\mbf{u}_N^{\prime}|^2 + \frac{1}{2} m_e |\mbf{u}_e^{\prime}|^2. -\end{equation*} -% -In order to solve for $g^{\prime}$, we express the post-collision -velocities in terms of $g^{\prime}$ and $\mbf{\hat{e}}$. By -definitions and previous results, one can show that -% -\begin{gather*} -\mbf{u}_N^{\prime} = \frac{m_e}{m_N + m_e} g^{\prime} \mbf{\hat{e}},\\ -\mbf{u}_e^{\prime} = \frac{-m_N}{m_N + m_e} g^{\prime} \mbf{\hat{e}}. -\end{gather*} -% -Substituting into conservation of energy gives -% -\begin{equation*} -\frac{1}{2} m_N |\mbf{u}_N|^2 + \frac{1}{2} m_e |\mbf{u}_e|^2 -= -\frac{1}{2} m_N \frac{m_e^2}{(m_N+m_e)^2} (g^{\prime})^2 -+ -\frac{1}{2} m_e \frac{m_N^2}{(m_N+m_e)^2} (g^{\prime})^2 -= -\frac{1}{2} \frac{m_e m_N}{m_e + m_N} (g^{\prime})^2. -\end{equation*} -% -Thus, -% -\begin{equation*} -g^{\prime} = \left( \frac{m_e + m_N}{m_e m_N} \left( m_N |\mbf{u}_N|^2 + m_e |\mbf{u}_e|^2 \right) \right)^{1/2}. -\end{equation*} -% -This expression can be further simplified by introducing the -pre-collision relative speed $g$: -% -\begin{equation*} -g = | \mbf{u}_N - \mbf{u}_e | -\end{equation*} -% -Then, -\begin{equation*} -\frac{1}{2} m_N |\mbf{u}_N|^2 + \frac{1}{2} m_e |\mbf{u}_e|^2 -= -\frac{1}{2} \frac{m_e m_N}{m_e + m_N} g^2, -\end{equation*} -% -which implies that $g^{\prime} = g$. - -Thus, given the pre-collision velocities $\mbf{v}_N$ and $\mbf{v}_e$ -and the post-collision relative velocity direction $\mbf{\hat{e}}$, -the post-collision velocities are completely determined: -% -\begin{gather*} -\mbf{v}_N^{\prime} -= -\mbf{v}_0 + \mbf{u}_N^{\prime} -= -m_N \mbf{v}_N + m_e \mbf{v}_e + \frac{m_e}{m_N+m_e} g \mbf{\hat{e}},\\ -% -\mbf{v}_N^{\prime} -= -\mbf{v}_0 + \mbf{u}_N^{\prime} -= -m_N \mbf{v}_N + m_e \mbf{v}_e - \frac{m_N}{m_N+m_e} g \mbf{\hat{e}}. -\end{gather*} -% -Using these constraints, the collision integral -in~\eqref{eqn:elastic-collision-9d} may be rewritten as follows: -% -\begin{equation} -Q_{EL}(\mbf{v}_e) -= -\int_{\mbb{R}^3} \int_{S^2} -\left( -f_e(\mbf{v}_e^{\prime}) f_N(\mbf{v}_N^{\prime}) -- -f_e(\mbf{v}_e) f_N(\mbf{v}_N) -\right) B(\mbf{v}_e, \mbf{v}_N; \mbf{\hat{e}}) -\, -d\mbf{\hat{e}} \, d^3\mbf{v}_N, -\label{eqn:elastic-collision-5d} -\end{equation} -% -where we integrate only over transitions allowed by conservation of -momentum and energy and $B(\mbf{v}_e, \mbf{v}_N; \mbf{\hat{e}})$ has -taken the place of $W_{EL}$ for these allowable transitions. It is -common to formulate $B$ as follows -% -\begin{equation*} -B(\mbf{v}_e, \mbf{v}_N; \mbf{\hat{e}}) = g \sigma(g,\mbf{\hat{e}}), -\end{equation*} -% -where $\sigma$ is the differential collision cross section. -Substituting this into~\eqref{eqn:elastic-collision-5d} gives the -typical form: -% -\begin{equation} -Q_{EL}(\mbf{v}_e) -= -\int_{\mbb{R}^3} \int_{S^2} -\left( -f_e(\mbf{v}_e^{\prime}) f_N(\mbf{v}_N^{\prime}) -- -f_e(\mbf{v}_e) f_N(\mbf{v}_N) -\right) g \sigma(g, \mbf{\hat{e}}) -\, -d\mbf{\hat{e}} \, d^3\mbf{v}_N, -\label{eqn:elastic-collision-5d} -\end{equation} -% -where, when evaluating the integral (e.g., using quadrature), the -quantities $\mbf{v}_e^{\prime}$, $\mbf{v}_N^{\prime}$, and $g$ are -determined from $\mbf{v}_e$, $\mbf{v}_N$, and $\mbf{\hat{e}}$ as -discussed previously. - - - -\subsubsection{Electron-Neutral Excitation Reactions}\label{sec:en-excitation} -Consider the following reaction: -% -\begin{equation*} -\ce{A + e <=> B + e}, -\end{equation*} -% -which has the form of, for example, electron impact excitation of -argon ($\ce{Ar + e <=> Ar^{\ast} + e}$). Two particle reactive -collisions are treated by many authors~\cite{}, but special care must -be taken here since the electron appears on both sides of the -reaction. To be clear, we first consider the forward and reverse -reactions separately. - -\paragraph{Forward} -The forward process is $\ce{A + e -> B + e}$. At a given point in -velocity space $\mbf{v}_e$, this leads to both destruction and -creation of electrons. The destruction term is given by integrating -over all interactions that consume electrons at $\mbf{v}_e$, weighted -by the transition probability: -% -\begin{equation*} -Q_{f, D}(\mbf{v}_e) -= --\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} -f_e(\mbf{v}_e) f_A(\mbf{v}_A) W_f(\mbf{v}_0, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime}) -\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}, -\end{equation*} -% -where $W_f(\mbf{v}_0, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime})$ is -the ``transition probability'' for the forward process taking an -$\ce{A}$ particle at $\mbf{v}_A$ and an electron at $\mbf{v}_e$ and -producing a $\ce{B}$ particle at $\mbf{v}_B$ and an electron at -$\mbf{v}_e^{\prime}$. - -Similarly, the creation term is given by integrating over all -interactions that produce electrons at $\mbf{v}_e$: -% -\begin{equation*} -Q_{f, C}(\mbf{v}_e) -= -\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} -f_e(\mbf{v}_e^{\prime}) f_A(\mbf{v}_A) W_f(\mbf{v}_0, \mbf{v}_e^{\prime}; \mbf{v}_B, \mbf{v}_e) -\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}. -\end{equation*} -% - -\paragraph{Backward} -Analogously, the backward process ($\ce{B + e -> A + e}$) also leads -to both destruction and creation of electrons at a given velocity: -% -\begin{gather*} -Q_{b, D}(\mbf{v}_e) -= --\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} -f_e(\mbf{v}_e) f_B(\mbf{v}_B) W_b(\mbf{v}_B, \mbf{v}_e; \mbf{v}_A, \mbf{v}_e^{\prime}) -\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}, \\ -% -Q_{b, C}(\mbf{v}_e) -= -\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} -f_e(\mbf{v}_e^{\prime}) f_B(\mbf{v}_B) W_b(\mbf{v}_B, \mbf{v}_e^{\prime}; \mbf{v}_A, \mbf{v}_e) -\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}, -\end{gather*} -% -where $W_b$ denotes the transition probability of the backward process. - -\paragraph{Detailed Baland and Net Creation} -Following~\cite{}, we observe that, to satisfy detailed balance, the -foward and backward transition probablities are related as follows: -% -\begin{equation*} -W_b(\mbf{v}_B, \mbf{v}_e^{\prime}; \mbf{v}_A, \mbf{v}_e) -= -\alpha W_f(\mbf{v}_A, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime}) -\end{equation*} -% -where $\alpha$ is a constant given by\todo{figure this out}. - -Thus, the forward and backward expressions above may be combined to -for the net production rate as follows: -% -\begin{align*} -Q(\mbf{v}_e) = & \int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} -\left[ \alpha f_e(\mbf{v}_e^{\prime}) f_B(\mbf{v}_B) - f_e(\mbf{v}_e) f_A(\mbf{v}_A) \right] -W_f(\mbf{v}_0, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime}) -\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime} \\ -% -& + -\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} -\left[ f_e(\mbf{v}_e^{\prime}) f_A(\mbf{v}_A) - \alpha f_e(\mbf{v}_e) f_B(\mbf{v}_B) \right] -W_f(\mbf{v}_A, \mbf{v}_e^{\prime}; \mbf{v}_B, \mbf{v}_e) -\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime}. -\end{align*} -% -As in~\S\ref{sec:en-scattering} the nine dimensional integrals above -may be reduced to five dimensional integrals by imposing conservation -of mass, momentum, and energy. The first term on the right hand side -is particularly similar because, as in the scattering case, we may -take the ``pre-collision'' velocities $\mbf{v}_A$ and $\mbf{v}_e$ as -given. Then, conservation of momentum and energy require that -% -\begin{gather*} -\mbf{v}_B -= -m_A \mbf{v}_A + m_e \mbf{v}_e + \frac{m_e}{m_B+m_e} g^{\prime} \mbf{\hat{e}},\\ -% -\mbf{v}_e^{\prime} -= -m_A \mbf{v}_A + m_e \mbf{v}_e - \frac{m_B}{m_B+m_e} g^{\prime} \mbf{\hat{e}}, -\end{gather*} -% -where $\mbf{\hat{e}}$ the direction of the ``post-collision'' relative -velocity and $g^{\prime}$ is ``post-collision'' relative speed, given -by -% -\begin{equation*} -g^{\prime} = \left( g^2 - \frac{2 m_A m_e}{m_A + m_e} \Delta E \right)^2, -\end{equation*} -% -where $g$ is the precollision relative speed (i.e., $g = |\mbf{v}_A - -\mbf{v}_e|$) and $\Delta E$ is the change in internal energy -associated with the reaction. Thus, -% -\begin{align*} -\int_{\mbb{R}^3}\int_{\mbb{R}^3}\int_{\mbb{R}^3} & -\left[ \alpha f_e(\mbf{v}_e^{\prime}) f_B(\mbf{v}_B) - f_e(\mbf{v}_e) f_A(\mbf{v}_A) \right] -W_f(\mbf{v}_0, \mbf{v}_e; \mbf{v}_B, \mbf{v}_e^{\prime}) -\, d^3\mbf{v}_A \, d^3\mbf{v}_B \, d^3 \mbf{v}_e^{\prime} -= \\ -& \int_{\mbb{R}^3}\int_{S^2} -\left[ \alpha f_e(\mbf{v}_e^{\prime}) f_B(\mbf{v}_B) - f_e(\mbf{v}_e) f_A(\mbf{v}_A) \right] -g \sigma(g, \mbf{\hat{e}}) -\, d\mbf{\hat{e}} \, d^3\mbf{v}_A, -\end{align*} -% -where $\sigma$ is the forward reaction differential cross section. - -In principle, one may reduce the second 9D integral to 5D in an -analogous way, but the details are more complicated. \todo{finish} - -\subsubsection{Electron-Neutral Ionization Reactions} - diff --git a/doc/hybrid/eqns.tex b/doc/hybrid/eqns.tex deleted file mode 100644 index adda1bba..00000000 --- a/doc/hybrid/eqns.tex +++ /dev/null @@ -1,292 +0,0 @@ -\section{PDEs} -We require PDEs for the heavy species densities, the electric -potential, and the electron distribution function. In this initial -work, the background gas temperature $T_g$ is taken as given and thus -no gas energy equation is necessary. - -\subsection{Species Continuity} -The species transport equations are as follows: -% -\begin{equation} -\pp{n_{\alpha}}{t} + \pp{F_{\alpha}}{x} = \dot{\omega}_{\alpha}, -\label{eqn:species} -\end{equation} -% -where $t$ is time, $x$ is the coordinate in the electrode-normal -direction, $n_{\alpha}$ denotes the number density of species -$\alpha$, $F_{\alpha}$ is the corresponding flux, and -$\dot{\omega}_{\alpha}$ is the net production rate of species $\alpha$ -due to chemical reactions. The flux is closed as follows: -% -\begin{equation*} -F_{\alpha} = \mu_{\alpha} n_{\alpha} E - D_{\alpha} \pp{n_{\alpha}}{x}, -\end{equation*} -% -where $z_{\alpha}$ is the charge number, $\mu_{\alpha}$ is the -mobility, $D_{\alpha}$ is the diffusion coefficient, and $E$ is the -electric field. The transport closures are as given in -\S\ref{sec:fluid-closures}. The electric field is determined from the -potential, $E = -\pp{\phi}{x}$, which is governed by a Poisson -equation (\S\ref{sec:hybrid-poisson}). Finally, the heavy species -production due to chemical reactions are discussed in -\S\ref{sec:hybrid-chemistry}. - -For a mixture of $N_s$ species, $N_s - 2$ species densities are -evolved according to~\eqref{eqn:species}. The electron density is -obtained from the electron distribution function, which is governed by -the electron Boltzmann equation (\S\ref{sec:hybrid-ebolt}). -Further,the density of the dominant background species (e.g., argon) -is determined from the pressure, which is assumed known, and the ideal -gas law: -% -\begin{equation*} -p = \sum_{\alpha = 1}^{N_s} n_{\alpha} k_B T_{\alpha}, -\end{equation*} -% -where $k_B$ is the Boltzmann constant and $T_{\alpha}$ is the -temperature of species $\alpha$. A two temperature model is adopted -where the heavy species share a single temperature $T_g$, which is -different from the electron temperature, such that the ideal gas law -becomes -% -\begin{equation*} -p = n_e k_B T_e + k_B T_g \sum_{\alpha \neq e} n_{\alpha}. -\end{equation*} -% -The gas temperature is assumed known. The electron temperture is -governed by the electron Boltzmann equation (\S\ref{sec:ebolt}). - -\subsection{Electromagnetics} \label{sec:hybrid-poisson} -A quasi-static model is used for the electromagnetics. In this case, -Maxwell's equations reduce to a Poisson equation governing the -electric potential $\phi$: -% -\begin{equation*} --\pp{^2 \phi}{x^2} = \frac{\rho_c}{\epsilon_0} -\end{equation*} -% -where $\rho_c$ is the charge density and $\epsilon_0$ is the -permittivity of free space. The charge density is given from the -species densities: -% -\begin{equation*} -\rho_c = \sum_{\alpha=1}^{N_s} z_{\alpha} q_e n_{\alpha}, -\end{equation*} -% -where $q_e$ is the elementary charge (the magnitude of the charge of -an electron). - -The electron number density $n_e$ is determined from the electron -distribution function, -% -\begin{equation*} -n_e(x,t) = \int_{\mbb{R}^3} f_e(x, \mbf{v}_e, t) \, d^3 \mbf{v}_e -\end{equation*} -% -which is governed by the electron Boltzmann equation. - -\subsection{Electron Boltzmann Equation} \label{sec:hybrid-ebolt} -Let $f_e$ denote the electron distribution function. In the 1-D glow -discharge model, the distribution function depends on space $x \in -(0,h)$, velocity $\mbf{v} \in \mathbb{R}^3$, and time $t \in -\mathbb{R}^+$. Since $f_e$ depends only on $x$ and $\mbf{E} = E -\hat{x}$, the electron Boltzmann equation simplifies to -% -\begin{equation*} -\pp{f_e}{t} + v_x \pp{f_e}{x} - \frac{q_e}{m_e} E \pp{f_e}{v_x} = S_e + C_e, -\end{equation*} -% -where $S_e+C_e$ denotes the collision term, with $S_e$ representing elastic -collisions and $C_e$ representing reactive collisions. - -For $S_e$, we consider only two-body interactions. Then, suppressing -the dependence on $x$ and $t$, $S_e$ can be written as -% -\begin{equation*} -S_e(\mbf{v}_e) -= -\sum_{\alpha=1}^{N_s} -\int \left( -f_e(\mbf{v}_e^{\prime}) f_{\alpha}(\mbf{v}_{\alpha}^{\prime} -- -f_e(\mbf{v}_e) f_{\alpha}(\mbf{v}_{\alpha}) -\right) \, -W_{e \alpha} -d^3 \mbf{v}_{\alpha} d^3 \mbf{v}_e^{\prime} d^3 \mbf{v}_{\alpha}^{\prime}, -\end{equation*} -% -where $W_{e \alpha}$ is the transition probability associated with the -nonreactive collision between and electron and species $\alpha$. In -practice, we will begin by considering only non-reactive collisions -between electrons and the background gas (argon), such that $S_e$ becomes -% -\begin{equation*} -S_e (\mbf{v}_e) -= -\int \left( -f_e(\mbf{v}_e^{\prime}) f_{b}(\mbf{v}_{b}^{\prime}) -- -f_e(\mbf{v}_e) f_{b}(\mbf{v}_{b}) -\right) \, -W_{e b} -d^3 \mbf{v}_{b} d^3 \mbf{v}_e^{\prime} d^3 \mbf{v}_{b}^{\prime}, -\end{equation*} -% -where subscript $b$ denotes the background species. The background -species distribution function $f_b$ is assumed given by the Maxwellian -consistent with its number density $n_b$ and the gas temperature -$T_g$. Thus, -% -\begin{equation*} -f_b(x, \mbf{v}, t) -= -n_b(x,t) -\left( \frac{m}{2 \pi k_B T_g} \right)^{3/2} -\exp \left( \frac{-m_b \|\mbf{v}\|^2}{2 k_B T_g} \right). -\end{equation*} -% -The integral in $S_e$ may be simplified by recognizing that many -interactions considered in the integral are impossible due to -conservation of momentum and energy. Specifically, for elastic -collisions, conservation of momentum and energy require that -% -\begin{equation*} -m_e \mbf{v}_e + m_b \mbf{v}_b -= -m_e \mbf{v}_e^{\prime} + m_b \mbf{v}_b^{\prime} -\quad -\frac{1}{2} m_e \mbf{v}_e \cdot \mbf{v}_e + -\frac{1}{2} m_b \mbf{v}_b \cdot \mbf{v}_b -= -\frac{1}{2} m_e \mbf{v}_e^{\prime} \cdot \mbf{v}_e^{\prime} + -\frac{1}{2} m_b \mbf{v}_b^{\prime} \cdot \mbf{v}_b^{\prime} -\end{equation*} -% -Then, given the direction of the post-collision velocity -$\hat{\mbf{e}}$, the post-collision velocities may be written as -% -\begin{equation*} -PUT IN -\end{equation*} -% -and the nine-dimensional integral above may be re-written as a -five-dimensional integral: -% -\begin{equation*} -\int_{\mbb{R}^3} \int_{S^2} -\left( -f_e(\mbf{v}_e^{\prime}) f_{b}(\mbf{v}_{b}^{\prime}) -- -f_e(\mbf{v}_e) f_{b}(\mbf{v}_{b}) -\right) \, -\|\mbf{v}_e - \mbf{v}_b\| \sigma_{eb}(\|\mbf{v}_e - \mbf{v}_b\|, \hat{\mbf{e}}) -d\hat{\mbf{e}} \, d^3 \mbf{v}_{b}, -\end{equation*} -% -where $\sigma_{eb}$ denotes the differential cross section. - -To write the contribution of reactive collisions, we follow the -formulation of Ern and Giovangigli~\cite{}. The full reaction term is -written as follows: -% -\begin{equation*} -C_e = \sum_{r=1}^{N_r} C_e^r, -\end{equation*} -% -where $N_r$ is the number of reactions and $C_e^r$ is the contributin -of the $r$th reaction. To formulate $C_e^r$, consider the following -reaction: -% -\begin{equation*} -\sum_{j \in R^r} \chi_j -\ce{<=>} -\sum_{k \in P^r} \chi_k, -\end{equation*} -% -where $\chi_j$ denotes the $j$th species, $R^r$ is the set of -reactants, and $P^r$ is the set of products. For reactions that -include the same species more than once, $R^r$ and $P^r$ include that -species multiple times. For example, the set of products of the argon -ionization reaction $\ce{Ar + e <=> Ar^+ + e + e}$ is given by $P^r = -\{\ce{Ar^+}, \ce{e}, \ce{e}\}$. Further, for a given species $i$, let -$R^r_i$ and $P^r_i$ denote the set of reactants and products for -reaction $r$ with species $i$ removed only once. Thus, for the argon -ionization reaction example, $P^r_e = \{\ce{Ar^+}, \ce{e}\}$. - -Equivalently, one may write the generic chemical reaction as -% -\begin{equation*} -\sum_{j =1}^{N_s} \nu^{r}_j\chi_j -\ce{<=>} -\sum_{k=1}^{N_s} \nu^{\prime,r}_k \chi_k, -\end{equation*} -% -where $\nu^r_{j}$ and $\nu^{\prime,r}_j$ denote the stoichiometric -coefficients of the $j$th species in the reactants and products, -respectively. Using all of this notation, the contribution of the -$r$th reaction to the reactive collision term is given by -% -\begin{align*} -C_e^r -=& -\nu_e^r \int \left( \prod_{k \in P^r} \beta_k f_k - \prod_{j \in R^r} \beta_j f_j \right) -\frac{W_{R^r P^r}}{\prod_{j \in R^r} \beta_j} \, -\prod_{j \in R^r_e} d^3 \mbf{v}_j \, \prod_{k \in P^r} d^3 \mbf{v}_k\\ -&+ -\nu_e^{\prime,r} \int \left( \prod_{j \in R^r} \beta_j f_j - \prod_{k \in P^r} \beta_k f_k \right) -\frac{W_{R^r P^r}}{\prod_{j \in R^r} \beta_j} \, -\prod_{j \in R^r} d^3 \mbf{v}_j \, \prod_{k \in P^r_e} d^3 \mbf{v}_k, -\end{align*} -% -where $W_{R^r P^r}$ is the transition probability for the forward -reaction, -% -\begin{equation*} -\beta_j = \frac{h^3_{\mathrm{P}}}{a_j m_j^3}, -\end{equation*} -% -and $a_j$ is the degeneracy. - -For binary reactions (e.g., $\chi_0 + \chi_1 \ce{<=>} \chi_2 + -\chi_3$), this may be rewritten in terms of the differential collision -cross section. For example, the contribution of such a reaction to -species 0 is given by -% -\begin{equation*} -C_0 = \int_{\mbb{R}^3} \int_{S^2} \left( \frac{\beta_2 \beta_3}{\beta_0 \beta_1} f_2 f_3 - f_0 f_1\right) g \sigma(g, \hat{\mbf{e}}) d\hat{\mbf{e}} d^3 \mbf{v}_1 -\end{equation*} -% - -Formally, reactions involving three -particles (e.g., direct ionization and recombination: $\ce{Ar + e <=> - Ar^+ + e + e}$ are supported by this framework, but this introduces -complexity into the formulation of $W$. While this can likely be -overcome, -\todo{Alexeev introduces a collision cross section for such reactions, but I don't understand his notation.} - we may also avoid it by decomposing such three particle -interactions into two steps. For instance, instead of including -direct impact ionization in the mechanism, we have -% -\begin{gather*} -\ce{Ar + e <=> Ar^{\dagger} + e} \\ -\ce{Ar^{\dagger} -> Ar^+ + e}, -\end{gather*} -% -where $Ar^{\dagger}$ is an intermediate species. For these reactions, -letting $\ce{A}$ be species 1, $\ce{Ar^{\dagger}}$ be species 2, and -$\ce{Ar^+}$ be species 3, the reactive collision terms in the electron -Boltzmann equation reads -% -\begin{align*} -C_e(\mbf{v}_e) -&= -\int_{\mbb{R}^3} \int_{S^2} -\left( \frac{\beta_2}{\beta_1} f_e(\mbf{v}^{\prime}_e) f_2(\mbf{v}_2) - f_e(\mbf{v}_e) f_1(\mbf{v}_1) \right) W_{10,20} d^3 \mbf{v}_1 d^3 \mbf{v}^{\prime}_e d^3 \mbf{v}_2 -\\ -&+\int_{\mbb{R}^3} \int_{S^2} -\left( f_e(\mbf{v}^{\prime}_e) f_1(\mbf{v}_1) - \frac{\beta_2}{\beta_1} f_e(\mbf{v}_e) f_2(\mbf{v}_2) \right) W_{10,20} d^3 \mbf{v}^{\prime}_e d^3 \mbf{v}_1 d^3 \mbf{v}_2 -+ -\int_{S^2} \frac{1}{\tau} f_2(\mbf{v}_2) d\hat{\mbf{e}}. -\end{align*} -% - diff --git a/doc/hybrid/overview.tex b/doc/hybrid/overview.tex deleted file mode 100644 index c92bb311..00000000 --- a/doc/hybrid/overview.tex +++ /dev/null @@ -1,5 +0,0 @@ -This section details a hybrid fluid-kinetic model. In particular, the -heavy species are represented with a fluid model, as in -\S\ref{sec:fluid}, but the electron number density and energy -equations are omitted. These equations are replaced by the electron -Boltzmann equation. From 3e34c4194c43bf458994d1f71224d6e501102842 Mon Sep 17 00:00:00 2001 From: "Todd A. Oliver" Date: Wed, 25 Jun 2025 14:28:22 -0500 Subject: [PATCH 19/19] Add minimal README --- README.md | 14 ++++++++++++++ 1 file changed, 14 insertions(+) create mode 100644 README.md diff --git a/README.md b/README.md new file mode 100644 index 00000000..9ffa51cc --- /dev/null +++ b/README.md @@ -0,0 +1,14 @@ +## Overview + +`glowDischargeSolver` is a collection of python functions supporting +the approximation of plasma systems in "glow discharge" regime. The +model takes the form of a fluid model that uses the drift-diffusion +approximation so that the state variables consist of a set of species +number densities and an electron temperature. For more details on the +physical model, see the LaTeX documentation in the `doc` directory. + +These equations are discretized in space using Chebyshev collation and +in time using fully implicit schemes, either backward Euler or +Crank-Nicolson. Further, to support the time-periodic case, +time-periodic-shooting is implemented to accelerate converged to the +time-periodic solution.