\documentclass[preprint,12pt]{article}

\usepackage{graphicx}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{epstopdf}
\usepackage[colorlinks]{hyperref}
\usepackage{verbatim}
\usepackage{color}
\usepackage{lineno}
\usepackage{tikz}
\usetikzlibrary{3d}
\usepackage[T1]{fontenc}
\usepackage[cp1250]{inputenc}
\usepackage[normalem]{ulem}


\newcommand{\EQ}{\begin{equation}}
\newcommand{\EN}{\end{equation}}
\newcommand{\EQA}{\begin{eqnarray}}
\newcommand{\ENA}{\end{eqnarray}}
\newcommand{\eq}[1]{(\ref{#1})}
\newcommand{\fig}[1]{fig.~(\ref{#1})}
\newcommand{\Eq}[1]{Eq.~(\ref{#1})}
\newcommand{\Eqs}[2]{Eqs.~(\ref{#1}) and~(\ref{#2})}
\newcommand{\Eqss}[2]{Eqs.~(\ref{#1}) - (\ref{#2})}
\newcommand{\eqs}[2]{(\ref{#1}) and~(\ref{#2})}
\newcommand{\Fig}[1]{Fig.~\ref{#1}}
\newcommand{\Tab}[1]{Table~\ref{#1}}
\newcommand{\Figs}[2]{Figs.~\ref{#1} and \ref{#2}}
\newcommand{\Hig}[1]{{\textbf{ #1 }}}
\newcommand{\Com}[1]{{\color{green}NILS: #1 }}
\newcommand{\ComK}[1]{{\color{blue}KEN: #1 }}
\newcommand{\ComTRJ}[1]{{\color{cyan}TRJ: #1 }}
\newcommand{\ComAQ}[1]{{\color{brown}AQ: #1 }}
\newcommand{\Nu}{\mbox{\rm Nu}}
\newcommand{\Sh}{\mbox{\rm Sh}}
\newcommand{\CD}{C_D}
\newcommand{\Ma}{\rm Ma}
\newcommand{\Rey}{\mbox{\rm Re}}
\newcommand{\Pran}{\mbox{\rm Pr}}
\newcommand{\Sc}{\mbox{\rm Sc}}
\newcommand{\Si}{\,{\rm Si}}
\newcommand{\Ox}{\,{\rm O}}

\begin{document}


\title{Notes}

\begin{abstract}
%% Text of abstract
\end{abstract}


\section{Introduction}
In the kinetmatic regime (droplets are much smaller than the mean free pat
of the molecules, the effluent flux is given by
\EQ
\mathcal{F}=Nu/4
\EN
%where $N=C_{SiO2}-C_s$ is the molar consentration of the condensing gas,
where $N=C_{\Si\Ox_2}-C_s$ is the molar consentration of the condensing gas,
$C_{SiO2}$ is the molar concentration of SiO$_2$ gas, $C_{s}$ is
saturation concentration of the same gas and $u$
is the mean velocity of the gas molecules, which can be found from
the mean of the Maxwell-Botzmann distribution:
\EQ
u=\sqrt{8k_BT/(\pi m_{SiO2})},
\EN
where $k_B$ is Boltzmann's constant, $T$ is fluid temperature and $m_{SiO2}$ is
the molecular mass of SiO$_2$.
Hence, the units of $\mathcal{F}$ is mol/m$^2$/s. The total molar rate of
condensation on the surface of a particle with radius $r$ is then given by
\EQ
F=\mathcal{F}4\pi r^2.
\EN
From this it is clear that the volumetric change of the droplet becomes
\EQ
\frac{dV}{dt}=F M_{SiO2}/\rho_{ms},
\EN
where $M_{SiO2}=60$~kg/kmol is the molar mass of SiO$_2$ and
$\rho_{ms}\sim 2300$~kg/m$_3$ is the true density of a
microsilica particle/droplet.
For a spherical particle we know that
$dV=4\pi r^2dr$, such that
\EQA
\frac{dr}{dt}
&=&\frac{dV}{dt}\frac{1}{4\pi r^2}
=\mathcal{F}4\pi r^2 \frac{M_{SiO2}}{4\pi r^2\rho_{ms}} \nonumber \\
&=&(C_{SiO2}-C_s) \sqrt{8k_BT/(\pi m_{SiO2})}\frac{M_{SiO2}}{4\rho_{ms}} \nonumber \\
&=&A(C_{SiO2}-C_s) \sqrt{T},
\ENA
where $A=0.122$.



%\vspace{1cm}
%\bibliographystyle{elsarticle-num}
%\bibliography{references}

\end{document}