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Jari Tapani Kolehmainen added file pressureloss.eps
over 8 years ago
Commit id: b14d2ed3a05a7b5184fb567893292442f5ae018e
deletions | additions
diff --git a/pressureloss.eps b/pressureloss.eps
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diff --git a/results.tex b/results.tex
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\section{Results}
\subsection{Overview}
We simulated a slugging fluidized bed with various work function values and two different superficial velocities, namely $0.2V_t$ and $0.3V_t$ where $V_t$ is the terminal velocity of $150 \micro \meter$ polyethylene particle with density of $910 \mathrm{g} / \mathrm{cm}^3$. The geometry was a long pipe with funnel attached at the top of pipe. The pipe was $640$ particle diameters tall and $20$ particle diameters wide. Particles had initially a zero charge and $36410$ particles were evenly distributed through out the fluidized bed.
We imposed an uniform superficial velocity boundary condition at the grate, and a constant pressure boundary condition at the outlet. The mesh was a hexahedra butterfly mesh with each cell having roughly three particle diameter witdth. This setup is was chosen to obtain grid independent CFDEM simulation.
The maximum particle charge was estimated from the charge transfer equation by setting the local electricfield term equal to the work function difference at the wall. This gave maximum charge $q_{eq}$
\begin{equation}
q_{eq} = \frac{2 \pi \varepsilon_0 }{\delta_c e}\Delta \phi r_i^2,
\label{eq:r1}
\end{equation} where $\delta_c$ is the electron cut-off distance, $\Delta \phi$ the work function difference between the particle and the wall which was varied between different simulation cases, and $r_i$ particle radius.
In this study we studies monodisperse suspensions of particles that had $r_i=150 \micro \meter$ and density similar to polyethylene $910 \kilogram / \meter^3$.
We characterized the strength of the electrostatic effects by looking at the ratio of the electricfield at the contact of two particles with charge $q_{eq}$ and the gravitational force. More formally we have
\begin{equation}
e/g = \frac{q_{eq} E}{mg},
\label{eq:r2}
\end{equation} that becomes effectively a function of work function via Eq. \eqref{eq:r1}. The cut-off radius of the electrostatic effects was chosen such that the electricfield coming from a particle with charge $q_{eq}$ would be neglected when the force between two particles with charge $q_{eq}$ was less than 10 percent from the gravitational force. The particles typically had charge less than $q_{eq}$, hence the 10 percent characterizes the largest possible cut-off error. The cut-off radius varied between $0.5$ to $7$ particle diameters in our simulations depending on the imposed work function difference. Six different work function cases were investigated, namely $e/g=0$, $e/g=0.1$, $e/g=0.5$, $e/g=1.0$, $e/g=1.5$, and $e/g=3.0$.
The tribocharging is generally a slow phenomenon, and takes minutes to hours to reach a steady state. To overcome this issue we accelerated the charge transfer by employing a coefficient similar to Eq. \eqref{eq:e11} that multiplied the charge transfer by $50$.
\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{snapshot.eps}
\caption{Snapshot taken at $1.8 \, \mathrm{s}$ of eight different fluidization cases. Cases (A)-(D) have superficial velocity of $0.2V_t$ while cases (E)-(H) have superficial velocity of $0.3V_t$. The background is colored by the solids volume fraction, and particles are colored based on the vertical velocity. Every $20$:th particle is shown in the figure from total of $36410$ particles.}
\label{fig:c0}
\end{figure}
Fig. \ref{fig:c0} shows snapshots taken from the simulations. From the figure one can observe that the amount of particles sticking to the wall (green particles that have zero velocity) increases with increasing charge. Also the bed homogenuity increases with increasing charge, and there are no visible slugs in the highly charged cases. Interestingly, the
\subsection{Tribocharging Rate}
We computed the average charge on a particle at given time instance. As predicted by the Eq. \eqref{eq:e9}, the charge increased by an exponential law, and approached the equilibrium charge. The charge evolution of the fluidized bed is shown in Fig. \ref{fig:c1}. The scaled total charge for all the three presented cases is very similar in the beginning of the simulations. However, once the particles charge the charging rate respect to equilibrium charge slows down. The charging rate decrease seems to be larger with larger charge.
One likely explanation for this is that the highly charged particles form a layer at the wall. This layer repels other charged particles ultimately decreasing the collision rate and collision velocity of these particles with the wall. Hence, in the highly charged cases this mechanism will cause increasing decrease in the charging rate as the particle charge increases.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{charge.eps}
\caption{Charging rate of the fluidized bed with various rations of electrostatic forces to gravity.}
\label{fig:c1}
\end{figure}
Fig. \ref{fig:cc1} shows charging of the system with two different work function differences, and two different superficial velocities. The higher superficial velocity case charges slightly faster, but the difference is small. Hence, the superficial velocity had no major effects on the particle charging rate.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{charge2.eps}
\caption{Charging rate of the fluidized bed with various two ratios of electrostatic forces to gravity, and varying superficial velocity.}
\label{fig:cc1}
\end{figure}
Fig. \ref{fig:cc2} shows the probability density functions of charge of the system with three different work function differences. As the work function values increase the distribution shifts to larger levels of charge. Interestingly, the density function also becomes wider, ans shows two separate peaks for the $e/g=3$ case. To inspect this behavior further we computed charge distributions from particles that are in one diameter radius from wall, and the particles that are in the inner region. These results are shown in the Fig. \ref{fig:cc3}. As can be seen from the figure, the particles at the wall have different density function than the particles at the inner region. The difference between the density functions increases with increasing charge leading to two separate peaks in the case $e/g=3$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{chargehist.eps}
\caption{Probability distribution of charge.}
\label{fig:cc2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{chargehist2.eps}
\caption{Probability distribution of charge respect to walls and interior.}
\label{fig:cc3}
\end{figure}
\subsection{Particle Volume and Charge Distribution}
The effect of charging to particle radial distribution and to radial charge distribution was investigated. These distributions were computed from time averages taken after the bed was reached to at least $70\%$ percent from the equilibrium charge. The particle radial distribution is shown in Fig. \ref{fig:c2}. The charged particles tend to move at the wall due to attractive electrostatic force between the wall and the particles. Interestingly there is a low solid volume fraction area next to the wall. Possible cause for this could be the repeling force between the particles at the wall and particles approaching the wall which would also explain the decreases in the charging rate in Fig. \ref{fig:c1}. Fig. \ref{fig:w2} shows fraction of the particles that is at the wall. We can see that the fraction increases to certain limit until the $e/g=1$ case, and stays after that effectively constant. Furthermore, the higher superficial velocity has larger fraction of particles at the wall probably due to particles being able to also stick at the higher parts of the fluidized bed.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{vofdist.eps}
\caption{Radial particle distribution.}
\label{fig:c2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{wall.eps}
\caption{Particle fraction at the wall.}
\label{fig:w2}
\end{figure}
Fig. \ref{fig:c3} shows the radial charge distribution for particle. The particles at the wall had the heighest charge, but interestingly the the particle charge decreased sharply after the wall, and started again slowly increasing. The possible cause for this behavior is that highly charged particles at the wall repel other particles with high charge. Hence it is more likely to find a particles with lower charge next to the wall layer than another particle with high charge. Authors believe that the gap between the interior and the wall might be effected by the high acceleration factor, since with low acceleration factor the particles would have more time to collide with the wall before they form the protective layer at the wall.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{chargedist.eps}
\caption{Radial average particle charge distribution.}
\label{fig:c3}
\end{figure}
\subsection{Particle Velocity Distribution}
%We also computed the average coordination number of particles. The results are shown in Fig. \ref{fig:c4}. The coordination was a decreasing curve for the non- or slightly charged cases, but had non monotonic behavior for the case $e/g=1.0$(??).
%\begin{figure}
%\centering
%\includegraphics[width=\textwidth]{coordination.eps}
%\caption{Coordination numbers of various cases. Blue line shows the early case after $0.3 \second$ where substantial charging has not happened, and the late case after $1.8 \second$. }
%\label{fig:c4}
%\end{figure}
%\begin{figure}
%\centering
%\includegraphics[width=\textwidth]{coordination2.eps}
%\caption{Coordination numbers of various cases divided to coordination number of particles at the wall and particles in the interior of the bed. }
%\label{fig:e4}
%\end{figure}
We computed the average particle velocity at different distances from particle center. The results are shown in Fig. \ref{fig:c5}. The lowly charged cases show particles moving up in the center of the bed, and falling down at the wall that is consistent with the uncharged case. However, when the charge on the particles increases the center of bed has a falling slug, and the intermediate particles next to the wall move up. This behavior could be caused by the reduced volume fraction next to the wall. Once the charge is increased even more, the bed returns to similar velocity distribution that the lowly charged cases have. It is also visible from the Fig. \ref{fig:c5} that the particles form a static layer at the wall around $e/g=1$ as the wall particle velocity goes effectively to zero.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{veldist.eps}
\caption{Time averaged vertical velocity distribution at different distances from the center of the bed. Various work function values and superficial velocities are shown. }
\label{fig:c5}
\end{figure}
\subsection{Effects on Bed Height and to Pressure Loss}
Fig. \ref{fig:c7} shows bed heights at different times that contain $95\%$ of the particles. The particles fall quicly to the hydrodynamical pseudo steady state. In the higher superficial gas velocity case, the bed height seems to be effected only slightly due to particle charging. However, the bed height oscillations are reduced clearly with the increasing charge. This is consistent with the prementioned observation of particles sticking at the wall and with the observation of Hassani \cite{hassani_numerical_2013} that monodispersed charged particles tend to produce more homogenous fluidization than uncharged particles.
For the lower superficial gas velocity case we also observe the decrease in the bed height oscillations, but also the bed height is decreased with at first by increased particle charge, and then expanded again by increasing charge. This behavior can be explained by two mechanism: particle repelling force and particles sticking to the wall. When particles stick to the wall, the volume fraction in the center of the bed is decreased, and in-turn the particle drag is reduced. The reduced drag causes bed to collapse. When the particle charge is increased even further the repelling force of particles cause particles to stay farther away from each other. This will effectively cause the bed to expans and explains the increase seen form $e/g=1$ to $e/g=3$. The first case where particle sticking is observed is the case with $e/g=1$, and we would expect a sudden drop at this point in the bed height that is visible in the lower superficial velocity case, but not with the higher superficial gas velocity.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{bed_height_oscil.eps}
\caption{Figure showing instantenous bed heights that contain $95\%$ of the particles. }
\label{fig:c7}
\end{figure}
The time averaged gas pressure curve at the center of the bed is shown in Fig. \ref{fig:c6}. The pressure loss over the bed decreases with with increasing particle charge.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{pressureloss.eps}
\caption{Time averaged pressure curves from bed bottom to the outlet with varying electrostatic force. }
\label{fig:c6}
\end{figure}
%\begin{figure}
%\centering
%\includegraphics[width=\textwidth]{pressure_intro.eps}
%\caption{Illustration of bed height determination from pressure curve. }
%\label{fig:c7}
%\end{figure}
