%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Jacobs Landscape Poster
% LaTeX Template
% Version 1.0 (29/03/13)
%
% Created by:
% Computational Physics and Biophysics Group, Jacobs University
% https://teamwork.jacobs-university.de:8443/confluence/display/CoPandBiG/LaTeX+Poster
% 
% Further modified by:
% Nathaniel Johnston (nathaniel@njohnston.ca)
%
% This template has been downloaded from:
% http://www.LaTeXTemplates.com
%
% License:
% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%----------------------------------------------------------------------------------------
%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%----------------------------------------------------------------------------------------

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%----------------------------------------------------------------------------------------
%	TITLE SECTION 
%----------------------------------------------------------------------------------------

\title{ {Modeling thermal transport in gas-particle flows using machine learning} } % Poster title

\author{S. Beetham$^a$, A. Lattanzi$^a$ \& J. Capecelatro$^{a,b}$} % Author(s)

\institute{Departments of $^a$Mechanical Engineering and $^b$Aerospace Engineering, University of Michigan} % Institution(s)

%----------------------------------------------------------------------------------------
\usebackgroundtemplate{\includegraphics[width=1.25\paperwidth]{figures/CITbackground}}

\begin{document}


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\begin{frame}[t] % The whole poster is enclosed in one beamer frame
\centering
\begin{columns}[t] % The whole poster consists of three major columns, the second of which is split into two columns twice - the [t] option aligns each column's content to the top

%\begin{column}{\sepwid}\end{column} % Empty spacer column
\begin{column}{\sepwid}\end{column} 
\begin{column}{\onecolwid} % The first column


%----------------------------------------------------------------------------------------
%	OBJECTIVES
%----------------------------------------------------------------------------------------
\begin{alertblock}{Particle-scale interactions impact macro-scale thermal transport}
\centering
A motivating example: 
\includegraphics[width = 0.95\onecolwid]{figures/Motivation}

\end{alertblock}
\vspace{0.5em}
\centering
\begin{block}{Industrial applications rely on idealized, Euler-Euler models$^{\dagger}$}
\begin{columns}
\begin{column}{0.5\onecolwid}
\vspace{0.5em}
\begin{small}
\begin{align*}
\frac{\text{d} \langle \theta_f \rangle}{\text{d} \hat{x}} &= - \frac{6 \langle \varepsilon_p \rangle k_f \langle \text{Nu} \rangle}{d_p \rho_f \langle \varepsilon_f \rangle u_f c_{p,f}} \left(\langle \theta_f \rangle - \langle \theta_s \rangle \right) \\
\frac{\text{d} \langle \theta_s \rangle}{\text{d} \hat{x}} &=  \frac{12 k_f \langle \text{Nu} \rangle}{d_p \rho_p u_p c_{p,p}} \left( \langle \theta_f \rangle - \langle \theta_s \rangle \right) \\
\end{align*}
$\langle \theta_f \rangle$, $\langle \theta_p \rangle$: Average gas, particle temperatures
\end{small}
\end{column}
\begin{column}{0.35\onecolwid}
\hspace{1em} \StickyNoteTwo[10cm]{\centering \textcolor{white}{\large These models \emph{neglect} important multiphase effects like clustering.}}[10cm]
\end{column}
\end{columns}
\vspace{0.5em}
\begin{centering}
\begin{tabular}{c c}
{\small example E-E solution} & {\small example clustered case}\\
 \includegraphics[height=0.35\onecolwid]{figures/ODE_0} &
 \includegraphics[height=0.35\onecolwid]{figures/LeiHT.png} \\
 \end{tabular} \\
$\dagger$ \begin{scriptsize} Zhou \& Subramaniam (ISU) \end{scriptsize}
\end{centering}
\end{block}

\vspace{0.5em}
\begin{alertblock}{Improving industrial-scale models}
\centering
\vspace{-0.5em}
\begin{columns}
\begin{column}{0.05\onecolwid}
\end{column}
\begin{column}{0.95\onecolwid}
\begin{itemize}
\hands We use an \emph{Euler-Lagrange} framework to study the effect of clustering on thermal development length.
\hands This highly-resolved data will be used by \emph{sparse regression} to learn improved models. 
\end{itemize}
\end{column}
\end{columns}
%\StickyNote[5cm]{\large \textit{Q: How can we distill high fidelity data into tractable turbulence models for challenging flow conditions?}}[26cm]
%\vspace{0.02em}
\end{alertblock}


%----------------------------------------------------------------------------------------
%	INTRODUCTION
%----------------------------------------------------------------------------------------

\end{column} % End of the first column
\begin{column}{\sepwid}\end{column} % Empty spacer column

\begin{column}{1.2\onecolwid}
\begin{block}{A novel, two-step approach}
\centering
\includegraphics[scale = 1.25]{figures/Two-Step}
\end{block}
\vspace{1.25em}

\begin{alertblock}{Phase-averaged, thermal equation} 
\centering
\begin{small}
Spatial and temporal averages denoted as $\langle (\cdot ) \rangle$. \\
Phase-averaging denoted as $\widetilde{(\cdot)} = \langle \varepsilon_f (\cdot) \rangle / \langle \varepsilon_f \rangle$. \\
Fluctuations from mean quantities denoted as $(\cdot)^{\prime}$. \\
\vspace{0.1em}
\begin{align*}
&\frac{\text{d}\widetilde{T}_f }{\text{d}x} = -\frac{6 \kappa_f}{C_{p,f} \rho_f d_p^2 \langle \varepsilon_f \rangle} \Big[ 
\underbrace{\frac{\langle \varepsilon_p \rangle \langle {\text{Nu}} \rangle}{\widetilde{u_f}} \left( \widetilde{T}_f - \widetilde{T}_p \right)}_{\text{Term 1}}  + 
\underbrace{ \frac{\langle \varepsilon_p^{\prime} \text{Nu}^{\prime } \rangle}{\widetilde{u_f}} \left(  \widetilde{T}_f - \widetilde{T}_p \right)}_{\text{Term 2}} + \\
&\underbrace{\frac{\langle \text{Nu} \rangle}{\widetilde{u_f}} \left( \langle \varepsilon_p ^{\prime } T_f^{ \prime} \rangle - \langle \varepsilon_p ^{\prime } T_p^{\prime } \rangle \right)}_{\text{Term 3}} +
\underbrace{\frac{\langle \varepsilon_p \rangle}{\widetilde{u_f}} \left( \langle \text{Nu}^{ \prime} T_f^{ \prime} \rangle - \langle \text{Nu}^{\prime } T_p^{ \prime} \rangle \right)}_{\text{Term 4}} 
\Big] 
- \underbrace{\frac{1}{\widetilde{u_f}} \frac{\text{d}}{\text{d}x} \widetilde{u_f^{\prime } T_f^{\prime}}}_{\text{Term 5}}
\end{align*}
\end{small}
\end{alertblock} 

\vspace{1.25em}
\begin{block}{Several relevant dimensionless groups arise...}
...and are useful for establishing the modeling parameter space.
\begin{columns}
\begin{column}{0.01\onecolwid}
\end{column}
\begin{column}{0.99\onecolwid}
\begin{itemize}
\item Particle volume fraction, $\langle \varepsilon_p \rangle$ 
\item Particle Reynolds number, Re$_p = \tau_p g d_p/\nu_f$, where $\tau_p = \rho_p d_p^2/(18\rho_f\nu_f)$
\item Prandtl number, Pr$ = C_{p,f} \nu_f \rho_f /\kappa_f$, the ratio of momentum and thermal diffusivities.
\item P\'{e}clet number, Pe$ = d_p u_{\text{bulk}}/\alpha_f$
\end{itemize}
\end{column}
\end{columns}
\end{block}
\vspace{1.25em}

\begin{tikzpicture}
\node[draw, fill = Mblue,text width=0.98\textwidth] at (0,0) {\centering \tiny \color{white} This work is supported by an NSF Graduate Fellowship and computations were carried out on the Great Lakes high-performance cluster.};
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\vspace{1em}

\end{column}

\begin{column}{\sepwid}\end{column} % Empty spacer column

\begin{column}{1.5\onecolwid}

\begin{alertblock}{Multiphase physics affect thermal development length}
\centering
\begin{tabular}{c c c} 
\multicolumn{3}{c}{Particle volume fraction $\langle \varepsilon_p \rangle$} \\
0.001 & 0.0255 & 0.05\\
\includegraphics[height= 0.22\textwidth]{figures/DevLengthCase1} & 
\includegraphics[height = 0.22\textwidth]{figures/DevLengthCase2} & 
\includegraphics[height = 0.22\textwidth]{figures/DevLengthCase3} 
\end{tabular}
\begin{tabular}{c c c c} 
E-E Model for $T_p$: & \ModelTp & E-E Model for $T_f$: & \ModelTg \\ 
E-L average of $T_p$: & \ELTp & E-L average of $T_f$: & \ELTg \\ 
\end{tabular}\\
Shaded regions denote the square root of the variance of the EL data. 
\end{alertblock}

\vspace{0.5em}

\begin{block}{Which terms are dominate and require modeling?}
\centering
\begin{tabular}{c c c} 
\multicolumn{3}{c}{Particle volume fraction $\langle \varepsilon_p \rangle$} \\
0.001 & 0.0255 & 0.05\\
\includegraphics[height= 0.22\textwidth]{figures/BalanceCase1} & 
\includegraphics[height = 0.22\textwidth]{figures/BalanceCase2} & 
\includegraphics[height = 0.22\textwidth]{figures/BalanceCase3} 
\end{tabular}
\begin{tabular}{c c c}
\TermOne Term 1:  $\frac{\langle \varepsilon_p \rangle \langle {\text{Nu}} \rangle}{\widetilde{u_f}} \left( \widetilde{T}_f - \widetilde{T}_p \right) $ &\hspace{2em} &
\TermThree Term 3: $\frac{\langle \text{Nu} \rangle}{\widetilde{u_f}} \left( \langle \varepsilon_p ^{\prime } T_f^{ \prime} \rangle - \langle \varepsilon_p ^{\prime } T_p^{\prime } \rangle \right)$ \\
Closed (no model required) & & Fluid temperature \emph{seen} by particles
\end{tabular}
\end{block}
\vspace{0.5em}
%\begin{alertblock}{Additional features}
%\end{alertblock}

\begin{alertblock}{Using machine learning to develop improved models} 
\centering
\vspace{0.25em}
\begin{columns}
\begin{column}{0.5\onecolwid}
\begin{columns}
\begin{column}{0.01\onecolwid}
\end{column}
\begin{column}{0.6\onecolwid}
Sparse regression postulates that any quantity can be modeled as 
{\large $$\mathcal{D} = \mathcal{T}\hat{\beta}$$}
\vspace{-0.5em}
\begin{itemize}
\item $\mathcal{D}$ is the quantity of interest
\item $\mathcal{T}$ is a matrix of (potentially nonlinear) trial functions
\item $\hat{\beta}$ is an optimal set of coefficients 
\end{itemize} 
\end{column}
\end{columns}
\end{column}
\begin{column}{0.5\onecolwid}
Ideal coefficients are found by solving,\\
\vspace{0.25em}
\includegraphics[scale = 0.65]{figures/SparseRegression2}
\end{column}
\end{columns}
\StickyNote[2.5cm]{\large \centering Sparse regression results in a \textit{compact, algebraic} model!}[36cm] \\
This methodology will be used to model dominant terms.
\end{alertblock}

%\begin{block}{References}

%\nocite{*} % Insert publications even if they are not cited in the poster
%\tiny{\bibliographystyle{unsrt}
%\bibliography{sample}\vspace{0.75in}}
%\end{block}

%----------------------------------------------------------------------------------------
%	ACKNOWLEDGEMENTS
%----------------------------------------------------------------------------------------


\end{column}

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\end{document}
