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\rhead{EP-DT-EO\\\today}
\lhead{Oskar M\aa rtensson\\osma0009@student.umu.se}
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\noindent
Department of Physics\hfill EP-DT-EO\\
Ume\aa{} University
\hfill \today
\title{Master Thesis}
\begin{center}
\vspace{15mm}
\textbf{{\Huge LHCb Upstream Tracker box}\\\vspace{3mm}{\Large Thermal studies and conceptual design}}
\vspace{30mm}
\includegraphics[width=0.3\textwidth]{cernlogo.jpg}\hspace{20mm}\includegraphics[width=0.3\textwidth]{UU.png}\\\Large{Master’s Thesis in Engineering Physics, 30hp}
\vspace{10mm}
\vspace{5mm}{\large
{\bf Student:} Oskar M\aa rtensson\\ {\bf E-mail:} osma0009@student.umu.se\\
}
\vspace{15mm}{\large
{\bf Supervisor:} Joao Carlos Batista Lopes\\ {\bf E-mail:} Joao.Carlos.Batista.Lopes@cern.ch}
\vspace{15mm}\\{\large
{\bf Examiner:} Michael Bradley \\{\bf E-mail:} michael.bradley@physics.umu.se}
\end{center}
\newpage
\section*{Abstract}
The LHC (Large Hadron Collider) will have a long shut down in the years of 2019 and 2020, referred to as LS2. During this stop the LHC injector complex will be upgraded to increase the luminosities, which will be the first step of the high luminosity LHC program (which will be realized during LS3 that takes place in 2024-2026). The LHCb experiment, whose main purpose is to study the \textit{CP}-violation, will during this long stop be upgraded in order to withstand a higher radiation dose, and to be able to read out the detector at a rate of 40MHz,compared to 1MHz at present. This change will improve the trigger efficiency significantly. One of the LHCb sub-detectors the Trigger Tracker (TT), will be replaced by a new sub-detector called UT. This report presents the early stage design (preparation for mock-up building) of the box that will be isolating the new UT detector from the surroundings and to ensure optimal detector operation. Methods to fulfill requirements such as light and gas tightness, Faraday-cage behavior and condensation free temperatures, without breaking the fragile beryllium beam pipe, are established.
\section*{Sammanfattning}
LHC (Large Hadron Collider) kommer under åren 2019-2020 att ha ett längre driftstopp. Under detta driftstopp så kommer LHC's injektionsanordningar att uppgraderas för att kunna sätta fler protoner i circulation i LHC, och därmed öka antalet partikelkollisioner per tidsenhet. Denna uppgradering kommer att vara första steget i "High Luminocity LHC"-programmet som kommer att realiseras år 2024-2026. LHCb-experimentet, vars främsta syfte är att studera \textit{CP}-brott, kommer också att uppgraderas under stoppet 2019-2020. Framför allt så ska avläsningsfrekvensen ökas från dagens 1MHz till 40MHz, och experimentet ska förberedas för de högre strålningsdoser som kommer att bli aktuella efter stoppet 2024-2026. En av LHCb's deldetektorer, TT detektorn, kommer att bytas ut mot en ny deldetektor som kallas UT. Den här rapporten presenterar den förberedande designen av den låda som ska isolera UT från dess omgivning och försäkra optimala förhållanden för detektorn. Kraven på den isolerande lådan och tillvägagångssätt för att uppfylla dessa krav presenteras.
\newpage
\section*{Nomenclature}
\subsection*{Acronyms}
\textbf{LHCb}\tab -Large Hadron Collider beauty\\
\textbf{ATLAS}\tab -A Torodial LHC Apparatus\\
\textbf{CMS}\tab -Compact Muon Solenoid\\
\textbf{ALICE}\tab -A Large Ion Collider Experiment\\
\textbf{LS}\tab \tab-Long stop\\
\textbf{TT}\tab -Trigger Tracker\\
\textbf{UT}\tab -Upstream Tracker\\
\textbf{RICH}\tab -Ring-imaging Cherenkov detector\\
\textbf{ECAL}\tab -Electromagnetic Calorimeter\\
\textbf{HCAL}\tab -Hadron Calorimeter\\
\textbf{FEA}\tab -Finite element analysis\\
\textbf{\textit{CP}-violation}\tab -Charge Parity violation\\
\textbf{TDR}\tab -Technical Design Report\\
\textbf{PUR}\tab -Polyurethane\\
\textbf{LEP}\tab -Large Electron–Positron collider\\
\textbf{IP}\tab \tab-Interaction Point
\subsection*{Parameters}
$A$\tab \tab -Area, [$m^2$]\\
$E$\tab \tab-Emitted power, [$W$]\\
$E_0$\tab \tab-Initial elasticity modulus, [$Pa$]\\
$H$\tab \tab-Hardness, [$Shore A$]\\
$I$\tab \tab-Current, [$A$]\\
$L$\tab \tab-Characteristic length, [$m$]\\
$R_e$\tab \tab -Electrical resistance, [$\Omega$]\\
$R_t$\tab \tab-Thermal resistance, [$K/W$]\\
$\Delta T$\tab -Temperature gradient, [$\degree C$] or [$K$]\\
$T_s$\tab \tab-Surface temperature, [$\degree C$] or [$K$]\\\
$T_\infty$ \tab-Ambient temperature, [$\degree C$] or [$K$]\\\
$U$\tab \tab-Voltage, [$V$]\\
$Nu_L$\tab -Nusselt number\\
$Ra_L$\tab -Rayleigh number\\
$Gr_L$\tab -Grashof number\\
$Pr$\tab \tab -Prandtl number\\
$h_c$\tab \tab -Convection heat transfer coefficient, [$W/m^2*K$]\\
$h_r$\tab \tab -Radiation heat transfer coefficient, [$W/m^2*K$]\\
$h_{comb}$ \tab -Combined heat transfer coefficient, [$W/m^2*K$]\\
$k$\tab \tab -Thermal conductivity, [$W/m*K$]\\
$q$\tab \tab-Heat flux [$W$]\\
$\nu$\tab \tab -Kinematic viscosity, [$m^2/s$]\\
$\alpha$\tab \tab -Thermal diffusivity, [$m^2/s$]\\
$\beta$\tab \tab -Volumetric thermal expansion coefficient, [$1/K$]\\
$\mu$\tab \tab-Viscosity, [$kg/s*m$]\\
$\lambda$\tab \tab-Wavelength, [m]\\
$\epsilon$\tab \tab-Emissivity\\
$c_p$\tab \tab-Specific heat at constant pressure, [$J/kg*K$]\\
\subsection*{Constants}
$g$\tab \tab-Gravitational acceleration,[$m/s^2$]\\
$c$\tab \tab-Speed of light [$m/s$]\\
$h$\tab \tab-Planck's constant [$eV*s$]\\
$\sigma$\tab \tab-Stefan-Boltzmann constant [$W/m^2K^4$]\\
\newpage
\vspace*{3cm}
\centerline{\Huge{\emph{Acknowledgments}}}
\vspace{1cm}
First of all, I would like to thank my supervisor Joao Carlos Batista Lopes, who has been helping me with all kind of stuffs during the whole year that I have spent on CERN. Everything from how to buy bus tickets, to how to use software's that I have never used before. He has introduced me to a lot of useful methods and techniques that i would probably not have tried if it were not for him, and for this I am truly grateful. \\
I would like to thank Francois Boyer, Olivier Jamet and Burkhard Schmidt for all their help and all the trust they have put in me during this project. I would also like to thank the rest of the EP-DT-EO group for making my time at CERN very pleasant and educational. I would like to thank Michael Bradley at Umeå University for the constructive comments he has given me on this project. Last but not least, I would like to thank my student colleague Kasper Hörnqvist for never turning my questions down and repeatedly providing me with useful insights.
\newpage
\tableofcontents
\newpage
\pagenumbering{arabic}
\section{Context}
The European Organization for Nuclear Research, CERN (\textit{Conseil Européen pour la Recherche Nucléaire}, in french) is a European research organization, founded in 1954, that operates the largest particle physics laboratory in the world. The organization has 21 member states. However, CERN cooperates with physicists, engineers and universities from far more countries than just the member states. The main purpose of CERN is to study high-energy physics in order to better understand the laws of physics and why our universe have become what it is today. The main site of CERN is located on the Swiss-French border outside the city of Geneva, and that is the place from where this project is executed.\cite{min}
\subsection{The Large Hadron Collider}
The Large Hadron Collider (LHC) is the worlds largest and most powerful particle collider. It was built between 1998 and 2008 and is the result of a collaboration between over 10 000 people from over 100 countries. LHC is a circular particle accelerator with a 27km circumference. It was built in the old tunnels made for the particle accelerator LEP, which was dismantled to make room for the newer LHC. In the LHC, protons are accelerated to speeds close to the speed of light and then smashed in to each other. There are several interaction points along the LHC, and each has its own set of detectors to record the collisions for further analysis. The LHC has multiple areas of use, for example investigation of supersymmetric theories and the nature of dark matter. \begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=.35\textwidth]{LHC.jpg}}}
\subfloat{{\includegraphics[width=.5\textwidth]{LHCairial.jpg}}}
\caption{Left: Sketch over the LHC with beam interaction points marked out. Right: photograph over the Geneva region with LHC underground path marked out.}
\label{lhc}
\end{figure}
\subsection{The LHCb experiment}
The LHCb experiment is carried out in a collaboration between 66 different institutes and involves about 700 scientists \cite{lhcb}. The main purpose of LHCb is to investigate composite particles containing b-quarks (also called \textit{beauty} quarks or \textit{bottom} quarks), or c-quarks (also called \textit{charm} quarks) and their decay. Compared to general purpose detectors like CMS or ATLAS, LHCb looks quite different. Unlike the general purpose detectors ATLAS and CMS, which are hermetic, LHCb uses a cone shape to take advantage of the fact that the probability distribution for B-meson production has a peak at low angles (angle between beam pipe and particle tracks).
The experiment aims to answer the question; Why is there still a lot of matter left after the creation of universe, but no antimatter? A more complete description of so called B-physics and it's benefits can be found in Section \ref{Bphys}.
\subsubsection{Physics motivation}\label{Bphys}
Matter was born in the early universe and consisted out of a sort of plasma consisting of freely moving quarks and leptons. As the universe expanded and got colder, the more massive particles got converted by the weak interaction (the particles decay into lighter particle and emits a W or Z boson) into the lightest "generation" of matter, which is the only matter we can see stable on the Earth today \cite{particle}. In the high energy collisions in the LHC, the heavier generations of matter that existed in the early universe, can be reproduced and studied to a greater extent than the conditions on Earth normally allows.\\
One of the big questions that LHCb tries to answer is the origin of the matter- antimatter asymmetries in the universe , which physicists refer to as the \textit{CP}-violation (violation of the postulated \textit{CP}-symmetry). The \textit{CP}-symmetry is the product of \textbf{C}harge conjugation symmetry and \textbf{P}arity symmetry. This means that a systems behavior should not change if all electrical charges are inverted (charge conjugation) and all spatial coordinates are inverted (parity) at the same time. Particle transformations through the weak force have however been shown to violate this postulated symmetry in some cases. Investigation of these asymmetries is the key to understand why the universe today is so dominated by matter with respect to antimatter, as well as it might shed light on other phenomena that cannot be explained by the standard model. \\
\textit{CP}-violation was first observed in 1957 in the particles called K-mesons (also called "kaons"). A meson is a particle composed by one quark and one anti quark, held together by gluons (strong interaction force carrying particles). After the discovery of \textit{CP}-violation in the K-meson system, many theorists predicted that the same violation should be observed in other systems. The natural next step is to study B-mesons, since they contain the heavier b-quark from the third generation of matter (the lighter s-quark, or its antiquark, can be found in all K-mesons).
The b-quarks role in the standard model is shown in \autoref{SM}. The study of B-mesons was one of the main purposes for building LHCb, and it has been confirmed that the B-mesons do indeed show \textit{CP}-violation.\cite{cp1}\cite{cp2}
B-mesons are copiously produced in the LHC, in which protons are accelerated to an energy up to 7TeV, thus about 7000 times their rest mass.
Just like Einstein's rules of relativity becomes more pronounced at higher speeds, the unknown rules and equations that governs our fundamental particles are expected to be more pronounced at higher energy densities (such as in the early universe).\\
Another reason to increase the energies of the colliding particles is the ability to probe smaller distances. Since photons are limited to probe only distances larger than their own wavelength, given by de Broglie's relation
\begin{equation}\label{lamb}
\lambda=hc/E \hspace{2mm},
\end{equation}
smaller distances can be probed when higher energies are used. Thus by increasing the energies we are allowed to look "inside" things we before only could watch from the outside, or maybe not at all.
By increasing the luminosity, more collisions will be produced and there will be more particles to study. Increasing the luminosity will therefore increase statistical accuracy and rare decays will occur more frequently. Thus there is a pursuit of higher energy and higher luminosity in the particle beams, in order to be able to understand the physical laws we have yet to understand. \\\\
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=\textwidth]{SM.png}}}
\caption{A graphic presentation of the standard model. The b-quark marked in blue.}
\label{SM}
\end{figure}
\subsubsection{The LHCb Upstream Tracker (UT)}
The LHCb-detector is to be upgraded in 2019-2020, in order to improve the read out rate and to make the detector ready to operate at about 5 times higher luminosities. Many subdetectors are going to be either upgraded or replaced due to the new requirements and improved technologies available. One subdetector system that is going to be replaced is the Trigger Tracker (TT), which will be replaced by the new Upstream Tracker (UT). They both serve the same purpose, which is to reconstruct charged particle tracks with high spatial resolution. The TT was designed to withstand an integrated luminosity of approximately $10 fb^{-1}$ ($1 fb^{-1}$ is equal to approximately 100 trillion proton-proton collisions), which is not sufficient for the post-upgrade LHC. The new UT have been designed to withstand an integrated luminosity of at least $50 fb^{-1}$. Also, exchanging the old TT for the new UT will provide a trigger readout at 40MHz, compared to the old readout at 1MHz \cite{ut}. The position of the TT in LHCb is shown in \autoref{lhcbnuda}. A computer generated 3D-model of the LHCb-detector can be seen in \autoref{lhcb3d}
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=.9\textwidth]{lhcbsk.png}}}
\caption{The LHCb-detector as it looks today, with TT marked out by a purple arrow. \cite{tdr}}
\label{lhcbnuda}
\end{figure}
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=.9\textwidth]{LHCbDetectorlight1.jpg}}}
\caption{Computer generated 3D-model over the LHCb-detector.}
\label{lhcb3d}
\end{figure}
\newpage
\subsubsubsection{UT working principle}
The UT sensors are silicon sensors (more precisely; single sided silicon micro-strip sensors) located approximately 2.5m downstream from LHCb's interaction point, (upstream of LHCb dipole magnet). The silicon sensors are solid state detectors. When a charged particle with high energy travels through a n-type silicon plate it will ionize some of the atoms and create electron-hole pairs. Due to an electric field that is applied over the electrodes, the holes and electrons will drift away from each other. The holes will eventually reach the p-doped implant. This will create a hole overflow in the material, and thus create a small current. This current will then be seen as a signal that can be translated into coordinates for the particle track. This working principle is illustrated in \autoref{princ}.
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=\textwidth]{siliconstrip.JPG}}}
\caption{The working principle of a single sided silicon micro strip detector.}
\label{princ}
\end{figure}
In the UT, the silicon sensors are mounted on staves, together with cables and the other electronics that are necessary in order to make a readout from the sensors. The staves are then assembled in to four planes. The orientation and dimensions of the planes are shown in \autoref{utgeom}. \autoref{utstave} shows the stave construction.
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=0.7\textwidth]{UTStave.JPG}}}
\subfloat{{\includegraphics[width=0.23\textwidth]{UTStavemount2.JPG}}}
\caption{The UT staves \cite{tdr}}
\label{utstave}
\end{figure}
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[width=\textwidth]{UTgeom.JPG}
\captionof{figure}{The UT geometry looking downstream \cite{tdr}.}
\label{utgeom}
\end{figure}
\subsubsubsection{UT-box requirements}\label{req}
It is of great importance to isolate the detector from its surroundings in order to ensure adequate environment conditions. Light (photons) would produce signals in the detector, and thus the detector has to be operated in the dark. This is achieved by enclosing the detector in a box (from here on called the UT-box). The development of the UT-box will be the main focus in this thesis. The main requirements on the UT-box are the following;\\
\begin{enumerate}
\item\textbf{The force transmitted from the UT-box to the LHCb beam pipe has to be limited.}\\
The LHCb beam pipe is made of Beryllium, since it is almost transparent to the particles that should be studied. The use of a more dense metal could lead to undesired showering of particles coming from the interaction point. Beryllium is very fragile and will break if high pressure is applied. This is a critical requirement since a high pressure can cause the LHCb beam pipe to rupture and in that case, the LHC would stop. Not more than a few Newtons of force are allowed to be applied when the box is closed.
\item\textbf{The UT-box should be light tight.}\\
Photons will yield a signal from the UT-detector even though they are not to be detected by the UT.\newpage
\item\textbf{The UT-box should act as a Faraday-cage.}\\
External electromagnetic fields can produce noise and spoil the signals generated by particles of interest. Thus the UT-box should act as a Faraday-cage in order to prevent this from happening.
\item\textbf{The UT-box should provide good thermal insulation.}\\
The ambient temperature in the LHCb cavern is about $20\degree C$. On the other hand, the silicon sensors have to be kept at a temperature $\leq -5\degree C$ in order to operate properly. The UT-staves will be cooled using a $CO_2$ cooling system. This cooling system will have a capacity of 5kW, of which 500W are reserved for heat transfer through the UT-box. The rest is mostly needed for the heat that the detector electronics generate. However, if the UT-box is not well thermally insulated and let more heat flow through than the cooling system is able to transport away, the temperature will rise and cause the sensors to fail after irradiation.
\item\textbf{The UT-box should be light.}\\
The UT-box should be made of light materials with low radiation length.
Radiation length is a material property related to the energy loss of high energy, electromagnetic-interacting particles, when propagating through the material.
\item\textbf{The UT-box should be radiation resistant.}\\
The properties of the UT-box are not allowed to change due to the radiation exposure. The UT-box have to withstand a minimum dose of $5\times10^7 rad$.
\item\textbf{The UT-box should be gas tight.}\\
The UT-box will be continuously flushed with either dry air or dry $N_2$ gas in order to prevent moisture condensation on the detector itself and on the beam pipe. There will be a slight over pressure in the UT-box so that the dry gas will leak out, instead of humid gas leaking in through holes and gaps in the UT-box. Small leaks are expected but they have to be limited.
\item\textbf{The detector has to be accessible.}\\
To simplify future upgrades and reparations of the UT, it should be possible to easily remove the UT-box from the beam pipe, and to reinstall it without damaging anything.
\newpage
\section{Design of the UT-box}
In this section, the design and the concepts implemented to ensure that the UT-box requirements are fulfilled, are presented.
\subsection{UT-Box geometry}
The dimensions and the position of the UT-box are presented in \autoref{geom1} and \autoref{geom2}.
In order to make the UT-detector accessible, the UT-box is constructed to be able to be split in half. The UT-box will be mounted on rails so that alignment of the UT-box is ensured when closing it. \autoref{splitbox} shows one half of the split UT-box, and the frame that will be attached to the rails. The rails can be seen in \autoref{geom2}. In \autoref{boxx}, the whole UT-box without frame, and without detector planes inside, can be seen.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{geom1.PNG}
\caption{Side view of the UT-box, dimensions in $[mm]$, \cite{struct}.}
\label{geom1}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{geom2.PNG}
\caption{Front view of the UT-box and rails, dimensions in $[mm]$, \cite{struct}.}
\label{geom2}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{conc1.PNG}
\captionof{figure}{Half UT-box, with rail mountings attached.}
\label{splitbox}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{conc2.jpg}
\captionof{figure}{Transparent view of empty UT-box.}
\label{boxx}
\end{minipage}
\end{figure}
\subsubsection{Interface to the beam pipe}
In order to reduce the force applied on the beam pipe to a minimum, but still ensure tightness, the area close to the beam pipe is made out of an elastomer. The elastomer can easily deform when compressed, and thus reduces the force transmitted to the beam pipe. This concept is illustrated in \autoref{lada}. The elastomer part closest to the beam pipe will from here on be called the UT-plug. \autoref{sass} investigates this requirement in depth.
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=0.6\textwidth]{lada.png}}}
\captionsetup{width=0.8\textwidth}
\caption{Concept picture of the UT-plug interfacing the LHCb beam pipe. The box dimensions are arbitrary in this picture}
\label{lada}
\end{figure}
\subsubsection{Gas tightness}
To reduce the risk of air getting in to the UT-box, the UT-box will be flushed with either dry air or dry $N_2$ gas at a slight over pressure, as mentioned before. To "lock in" the dry gas, the UT-plug will have a small arc-shape closest to the beam pipe. This will cause the slight over pressure to push the UT-plug towards the pipe and therefore achieve a better sealing. This concept can be seen in \autoref{lada} and \autoref{viz} among others. The arc shape closest to the beam pipe will also serve to redirect some of the force applied in the direction perpendicular to the beam pipe to be parallel to the beam pipe.
\subsubsection{Light tightness}
The critical part in the question of light tightness will be the area where the UT-box is split, especially around the beam pipe and therefore the UT-plug. To ensure light tightness of the UT-plug, two important concepts have been implemented. The first measure is to put a disc of an opaque material inside a cavity in the UT-plug. The photons cannot travel through the disc and instead have to travel around it, thus minimizing the risk of undesired photons interacting with the detector. This concept is illustrated in \autoref{light1}.
The second measure implemented in order to ensure light tightness is the angular displacement of the plugs split line on its inside compared to its outside. The risk of photons getting inside the UT-box, in case of inadequate closing, is significantly reduced by these measures. This concept is illustrated in \autoref{light2}.
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[height=0.4\textheight]{light1.PNG}
\caption{Photon blocking disc placed inside the UT-plug cavity.}
\label{light1}
\end{figure}
\begin{figure}
\centering
\captionsetup{width=0.8\textwidth}
\includegraphics[height=0.4\textheight]{light2.PNG}
\caption{Cross section view (top view) of the plugs split section and its angular displacement on the outside compared to the inside.}
\label{light2}
\end{figure}
\subsection{Material selection}
\subsubsection{Radiation hardness}
The critical point in terms of radiation hardness of the UT-box is the interface between the box and the LHCb beam pipe, the UT-plug. It is important that the properties of the elastomer in the UT-plug does not alter with radiation exposure. The UT-plug will have to withstand a gamma dose of $5\times 10^7$ rad (equivalent to $5\times 10^5$ Gy). Two elastomers have been shown to fulfill the radiation hardness requirements, Ethylene-propylene and Polyurethane. This is shown in \autoref{radhard}. Of those two elastomers, Polyurethane was chosen due to the slightly higher radiation dose required for moderate to severe damage, and due to previous experiences with the material. Polyurethane has been used for similar purposes at positions close to the beam pipe before and have been proven to perform well. In this report, each dimension of the plug is investigated from a thermal point of view, and the rigidity of the polyurethane is evaluated. This is important in order to be able to manufacture a plug that satisfies the requirements.\\\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=0.65\textwidth]{radhard.PNG}}}
\caption{Radiation hardness for some polymers \cite{rad}.}
\label{radhard}
\end{figure}
\subsubsection{Thermal properties}
The panels of the UT-box will be made out of a sandwich-structured composite. The sandwich-structured composite is a concept commonly used to achieve high bending stiffness with low density. Such a sandwich is composed by a thick lightweight core with a thin but stiff skin attached to each side. A low density is desired since it both increases the radiation length, and minimizes the stresses in the UT-box material.
In order to minimize the heat flux through the UT-box walls, it has been decided to use Airex R82.60 \cite{airexx} as core. Airex is a good thermal insulator with a thermal conductivity of $0.036 W/(mK)$, and is very light with a density of $60 kg/m^3$. The radiation length of Airex is about $0.25\% X_0$ for 2cm. Carbon fiber \cite{carbon}, are used for the skins, which provide the mechanical stability required for the UT-box.
\subsubsection{Electrical shielding}
In order to make the UT-box acting as a Faraday cage, one layer of a thin copper net (Dexmet 2CU6-100FA \cite{cu}), will be embedded in the carbon fiber skins of the UT-box panels. The copper net in all panels are connected together to make the Faraday cage. This concept have been tested and validated. The copper net is shown in \autoref{Cu}, a complete technical report on the panel production can be found in \aref{comprep}. The diameter of the holes in this copper net is 2.54mm, and thus electromagnetic radiation with a wavelength superior to this will to get blocked. 2.54mm wavelength is in the short end of the microwave-region.
\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{copper_net.jpg}
\caption{Copper net to assure electrical shielding.}
\label{Cu}
\end{figure}
\section{Thermal assessment}\label{tass}
The thermal analysis of the UT-box will include both analytical and numerical calculations.
The panels of the UT-box (the walls, roof and floor) are symmetrical in all directions except their normal directions. Thus the analysis can be reduced to one dimension and solved analytically.\\
The UT-plug on the other hand is not as symmetrical, and is not expected to insulate thermally as well as the panels. The UT-plug is therefore a critical area in terms of temperature. Because of this, a FEA (Finite Element Analysis) were conducted for the thermal aspects of the UT-plug. The finite element method is a numerical method to find approximate solutions to boundary value problems for partial differential equations. The problem domain is divided into smaller sub-domains called finite elements. The problem is then solved by minimizing an associated error function through variational methods.\\
Condensation shall be avoided in order to minimize the performance requirements of the detectors internal cooling system, and so that water does not start to accumulate in the experiment. The dew point in the experimental cavern is estimated to be at 12\degree C, thus a temperature superior to 12\degree C is desired all around the UT-box, including the UT-plug region.
\subsection{Heat transfer theory}\label{heat}
In this section I will explain the most important concept behind the calculations in my thermal analysis. A direct consequence of the second law of thermodynamics is that it is impossible to transfer heat from a cold body to a warmer body without inserting additional energy. Heat can however, and will, flow spontaneously from a warm body to a colder one.\\\\
\textit{\textbf{Second law of thermodynamics:} If no energy enters or leaves a closed system, the potential energy of the systems state will always be less than, or equal to that of the initial state.}\\\\
Heat can be transferred from one body or system to another in different ways, and in this section I will explain the different heat transfer modes that can occur, and how they relate to each other in a so-called "thermal circuit". The system will be in steady state and no time dependent analysis is therefore conducted.
\subsubsection{Conduction}\label{cond}
Heat transfer through conduction can be viewed as energy migrating from more energetic particles to less energetic particles through collisions between the particles. A well known empirical law that describes this kind of heat transfer is \textit{Fourier's law of conduction}, which is stated in it's differential form in as
\begin{equation}\label{Fdiff}
\vec{q''}=-k\nabla T\hspace{2mm} ,\\
\end{equation}
where $\vec{q''}$ is the heat flux density ($\vec{q''}\equiv \vec{q}/A$, where $\vec{q}$ is the heat flux and $A$ is the cross sectional area perpendicular to $\vec{q}$), $T$ is the temperature, and $k$ is the \textit{thermal conductivity}. The one dimensional version of the same law is
\begin{equation}\label{Fendim}
q''_x=-k\frac{dT}{dx}\hspace{2mm} .\\
\end{equation}
For all linear temperature distribution (in all solids at steady state), \autoref{Fendim} can be rewritten as
\begin{equation}\label{Fslut}
q''_x=-k\frac{\Delta T}{l}\hspace{2mm} ,\\
\end{equation}
where $\Delta T$ is the temperature difference between two points and $l$ is the distance between the same two points.
\subsubsection{Convection}\label{conv}
Heat transfer caused by macroscopic movements of a fluid (advection) in combination with diffusion, is called heat transfer through convection. The heat transferred by convection is thus very dependent on the flow of the fluid involved. Since fluid dynamics is to great extent governed by the Navier-Stokes equations, which does not have a known general solution, practically all relations used to determine the \textit{convection heat transfer coefficient} $h_{c}$ is empirical. There are three modes of convective heat transfer. There is free convection due to buoyancy-driven fluid flow, forced convection due to external flow and convection with phase change (condensation or boiling). Typical $h_c$-values for each mode are shown in \autoref{convmodes}.
\begin{table}[H]
\centering
\captionsetup{justification=centering,margin=2cm}
\begin{tabular}{r|c|l}
Process&Phase& Typical $h_c$ [$W/(m^2*K)$]\\\hline
\multirow{2}{*}{}{Free convection}&Gas&2-25\\
&Liquid&50-1000\\\hline
\multirow{2}{*}{}{Forced convection}&Gas&25-250\\
&Liquid&100-20000\\\hline
Convection with phase change&both&2500-100000\\\hline
\end{tabular}
\caption{Typical heat transfer coefficient for each convective heat transfer mode\cite{t1}}\label{convmodes}
\end{table}
The most common way to calculate $h_{c}$ is to first calculate a parameter called the \textit{Nusselt number}
\begin{equation}\label{nuss}
Nu_L=\frac{\text{Convective heat transfer}}{\text{Conductive heat transfer}}=\frac{L \cdot h_{c}}{k}\hspace{2mm} , \\
\end{equation}
and from there calculate $h_{c}$ through
\begin{equation}\label{nusse}
h_{c}=\frac{k\cdot Nu_L}{L}\hspace{2mm},\\
\end{equation}
in which $L$ is the characteristic length.
The Nusselt number is the dimensionless ratio between heat transfer through convection, and heat transfer through conduction. There are many empirical relations for this number, which applies for different conditions. Since it is hard to determine an exact value for $Nu_L$ and $h$, it is common to instead calculate the average of those values over the whole contact surface between the fluid and the solid. In many of the empirical relations to calculate $Nu_L$, another dimensionless parameter is required, namely the \textit{Rayleigh number}. The Rayleigh number is the product of the \textit{Grashof number} and the \textit{Prandtl number}, as described by
\begin{equation}\label{Ra}
Ra_L=Gr_L \cdot Pr=\frac{g\beta(T_s-T_\infty)L^3}{\nu \alpha}\hspace{2mm}.\\
\end{equation}
In \autoref{Ra}, $g$ is the gravitational acceleration, $T_s$ is the surface temperature of the solid, $T_\infty$ is the ambient temperature, $\beta$ is the volumetric thermal expansion coefficient of the fluid, $\alpha$ is the thermal diffusivity and $\nu$ is the kinematic viscosity.\\
The Grashof number
\begin{equation}\label{gr}
Gr_L=\frac{g\beta(T_s-T_\infty)L^3}{\nu^2}\\
\end{equation}
can be interpreted as a measure of the ratio between buoyancy forces and viscous forces in the fluid, and the Prandtl number \begin{equation}\label{pr}
Pr=\frac{c_p\mu}{k}=\frac{\nu}{\alpha}
\end{equation} as the ratio between momentum and thermal diffusivities.
By calculating those dimensionless numbers, one can deduce the average $h_{cond}$ for many simple geometries. \newpage
\subsubsection{Radiation}\label{radi}
Thermal radiation is heat transported by photons that is emitted by all bodies of a non/zero temperature. Thus, unlike heat transfer through convection and conduction, this heat transfer mode does not require a material medium to be transported in. There is an upper limit to the magnitude of the emitted power from a surface of temperature $T_s$. This upper limit is described by the \textit{Stefan-Boltzmann law}
\begin{equation}\label{steffe}
E_b=\sigma T_s^4\hspace{2mm}.
\end{equation}
The constant $\sigma$ is the Stefan–Boltzmann constant. A body emitting the maximal possible heat, yielded by \autoref{steffe} is called a blackbody. In reality there is not a lot of real blackbodies. To make \autoref{steffe} apply to more realistic bodies, it is multiplied by a factor $\epsilon$. In the following equation, $\epsilon$ is a material property called \textit{emissivity} which has a value between 0 and 1, corresponding to how much of the maximal emissive power that is really emitted. \\
\begin{equation}\label{steffe2}
E=\epsilon \sigma T_s^4
\end{equation}
The absolute majority of all real bodies are not only able to emit thermal radiation, but are also able to absorb thermal radiation from it's surrounding. Let $G$ designate the total incident thermal radiation (i.e. \textit{irradiation}), and let $\gamma$ designate the fraction of incident thermal radiation that is absorbed by the body (i.e. \textit{absorptivity}). The net heat flux density is then given by
\begin{equation}\label{qradeq}
q''_{rad}=\frac{q_{rad}}{A}=\epsilon \sigma T_s^4-\gamma G\hspace{2mm}.
\end{equation}
\textit{Kirchoff's law of thermal radiation} states that $\gamma=\epsilon$ for all materials in thermal equilibrium (steady state).
From this,
\begin{equation}\label{qradeqq}
q''_{rad}=\epsilon\sigma (T_s^4-T_\infty^4)
\end{equation}
can be derived, and it applies to all cases with a large isothermal surrounding.
For the sake of putting up a thermal circuit, it is convenient to state the net heat flux as
\begin{equation}\label{hrad}
q_{rad}=h_rA(T_s-T_{\infty})\hspace{2mm}.
\end{equation}
By applying $a^2-b^2=(a-b)(a+b)$ on \autoref{qradeq} two times, and combining the result with \autoref{hrad}, the following expression for the \textit{radiation heat transfer coefficient} can be derived
\begin{equation}\label{hrad2}
h_r=\epsilon \sigma (T_s+T_\infty)(T_s^2+T_\infty^2)\hspace{2mm}.
\end{equation}
\newpage
\subsubsection{Thermal circuit concept}\label{tcirc}
The reciprocal of thermal conductance is called thermal resistance. This property is similar to electric resistance in the manner that it directs heat flow in a way similar to how electric resistance directs electric current. As an example, the relation between $U,I$ and $R_e$ described by Ohm's law \begin{equation}\label{ohm}
U = I \cdot R_e\hspace{2mm},
\end{equation}
is also a valid relation if the variables are replaced by $\Delta T, q$ and $R_t$, as in
\begin{equation}\label{ohmt}
\Delta T = q \cdot R_t\hspace{2mm}.
\end{equation}
A one dimensional temperature analysis through a solid of $n$ layers (with perfect thermal contact) can be viewed as $n$ thermal resistors connected in series (where $R_t=\frac{L}{A\cdot k}$), and the total resistance can then be yielded by
\begin{equation}\label{circuit}
R_{t_{tot}} = \sum_{i=1}^n R_{t_i} = \sum_{i=1}^n\frac{L}{A\cdot k_i}\hspace{2mm}.
\end{equation}
In case there is radiation or convection involved, then $k_i$ should be replaced by $h_{conv}$ or $h_{rad}$ respectively. If both convection and radiation occurs, then $k_i$ should be replaced by the \textit{combined heat transfer coefficient} stated in \cite{ashrae},\begin{equation}\label{htot}
h_{comb}=h_r+h_c\hspace{2mm}.
\end{equation} It should be noted that the $h$-coefficients are constructed to be independent of $L$ ( i.e. $L=1$ for all $h$) since it describes the behavior of a surface rather than that of a volume (which $k$ does for conduction).
\subsection{Analytical calculations}
To assist in future calculations similar to the ones that I have made, the calculations were saved as Mathcad files. I have made a few different tools, but to make this report concise, I will only present the tools that have been directly necessary to conduct the FEA. I have put the tools as appendices, but a short description of every tool is presented in this section.\\\\
\textbf{UT-box thermal analysis} (\aref{App:hcalc})\\
This tool assists in analytical calculations regarding temperature distributions in a box consisting of no more than three layers, with constant inner and surrounding temperature. This tool also provides the convection heat transfer coefficients ($h$) used in the thermal FEA. \\Since $h$ depends on the difference between the surface temperature and the ambient temperature, which is dependent on the heat flux, which in turn is dependent of $h$, an iterative method is used to solve $h$. An initial estimation of the surface temperatures is made. This initial estimation then gives a one dimensional circuit, from which a heat flux magnitude can be derived. This heat flux magnitude will then be used to recalculate the surface temperatures which were initially estimated. If the difference is large, a new estimation, based on the achieved results, is made. This process is then iterated until the temperatures match. The achieved surface temperatures and convection coefficients are stated in \autoref{toolres}. The calculated outside surface temperatures of the UT-box shows that condensation will not occur on the UT-box panels. However, those calculations does not show that there wont be condensation on the UT-plug, and thus a thermal FEA is conducted for the region closest to the beam pipe. \autoref{toolres} also shows that the calculated heat flux (a total of about 319W) is below the internal cooling systems limit at 500W.
\\\\
\begin{table}[H]
\centering
\captionsetup{justification=centering,margin=2cm}
\begin{tabular}{|r|cc|cc|c|}\hline
Wall&$T_{in} [\degree C]$&$T_{out} [\degree C]$&$h_{in} [W/(m^2*K)]$&$h_{out} [W/(m^2*K)]$&Q $[W]$ \\\hline
Vertical&0.1&15.7&1.080&1.018&266.6\\
Top&-0.9&15.6&0.894&0.841&24.7\\
Bottom&0.2&16.3&2.796&3.025&27.5\\\hline
\end{tabular}
\caption{Summary of most important results from \aref{App:hcalc}.}\label{toolres}
\end{table}
\textbf{Cavity convection analysis} (\aref{cavitycalc})\\
This tool serves to check if convection can be neglected in an air volume enclosed
by two parallel plates of different temperatures. \\\\
\textbf{Heat transfer comparison between two vertical plates} (\aref{radcalc})\\
This is a tool to compare the magnitudes of heat transfer through radiation, conduction and through natural convection, between two equal parallel plates with air in between. This can justify the neglecting of a heat transfer mode, if it is shown to be of insignificant magnitude compared to the other modes, and thereby simplify calculations.\\
\subsection{Numerical calculations (FEA) on the UT-plug}
This section presents a thermal FEA (parametric study) made on the polyurethane interface between the LHCb beam pipe and the UT-box, the UT-plug. The study is conducted in the \textit{ANSYS 16.1} software package.
The purpose of this study is to determine a geometry for the UT-plug that ensures that no condensation occurs on the surfaces which are in contact with air. \autoref{model}, \autoref{param1} and \autoref{param2} provide a list of parameters which describes the geometric and material properties of the UT-plug. Values that were considered as constants in the computation are shown in \autoref{model}. The analyzed geometries of the UT-plug were selected based on geometric constraints, UT-operating requirements and material limitations. The parameters are varied one at the time in order to understand the link between the temperature in the outer face of the UT-plug and each specific parameter. The mounting of the interface in the UT-box is illustrated in \autoref{lada}.
\autoref{viz} shows a sketch of the UT-plug cross-section and illustrates the parameters that were considered in the iterative FEA thermal analysis.
\subsubsection{Boundary conditions and model parameters}
\begin{table}[H]
\centering
\captionsetup{justification=centering,margin=2cm}
\begin{tabular}{|r|c|l|}\hline
Property&Value&Unit\\\hline
Airex conductivity (K3)&0.036&$W/(m*K)$\\
Polymer emissivity&0.95&\\
Outside temperature&20&\degree C\\
Inside temperature&-5&\degree C\\
Outside convection coefficient&1.018&$W/(m^2*K)$\\
Inside convection coefficient&1.080&$W/(m^2*K)$\\
Mesh element size&0.001&\\
Element type& All solid&\\\hline
\end{tabular}
\caption{Non-altered model properties. In-depth calculations of the convection coefficients can be found in \aref{App:hcalc}.}\label{model}
\end{table}
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=\textwidth]{viz.PNG}}}
\caption{Parameters visualized.}
\label{viz}
\end{figure}
\newpage
The following assumptions and simplifications have been made:\\
\vspace{1mm}\\
\textbf{Symmetry planes:} XY,XZ\\
\textbf{Surface contacts:}\\ Assumed perfect contact between the materials and the air in contact with solids.\\
\textbf{Perfect insulated surfaces:}\\ Surfaces in face-to-face contact with either the LHCb beam pipe or the wall of the UT-box.\\
\textbf{Convection in cavity:}\\
According to analytical calculations, convection in the cavity can be neglected as long as L4<6.8mm. The calculations performed can be seen in \aref{cavitycalc}.\\
\textbf{Radiation in cavity:}\\
Taking into account the temperature difference between the cavity’s surfaces (worst case scenario) it can be concluded that the heat transfer by radiation can be neglected with respect to the heat transfer by conduction (see detailed calculation in \aref{radcalc}). \autoref{radi} shows that the highest temperature difference is 4.2 \degree C which corresponds to a heat transfer of 1.0$W/m^2$. This values of heat transfer is insignificant with respect to the value computed by conduction, 157$W/m^2$. This is a typical behavior for all configurations.
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=0.5\textwidth]{radi.png}}}
\subfloat{{\includegraphics[width=0.5\textwidth]{radii.png}}}
\caption{Temperatures without radiation in the cavity.}
\label{radi}
\end{figure}
\subsubsection{Results}
A variety of configurations were used. For each configuration, a thermal assessment was performed in which some parameters were kept constant whilst other parameters were varied.
In this report the two configurations that are most important for the final conclusions are presented. The first configuration does not include an Airex divider in between the polyurethane parts, whilst the second does.
\subsubsubsection{Homogeneous polyurethane plug}
\autoref{surf} shows a typical temperature distribution on the UT-plug $(a)$ and more specifically on its outer surface $(b)$, for the standard configuration without Airex, whose parameter values are shown in\autoref{param1}. \autoref{surf} $(b)$ shows that the highest temperature on the outer surface is located at the inner radius and the lowest temperature is located at the outer radius. \autoref{h1} to \autoref{k2} show plots of the temperature extremes on the UT-plug’s outer surface. The blue line shows the minimum temperature of the outer surface, and the red line shows the maximum temperature of the outer surface. \autoref{h1} shows the temperatures dependency with the parameter H1.
\begin{table}[H]
\captionsetup{justification=centering,margin=2cm}
\centering
\begin{tabular}{|r|c|c|c|l|}\hline
Parameter&Standard value&Tested range&Unit&Figure\\\hline
H1&40&30-70&mm&\autoref{h1}\\
H2&10&5-15&mm&\autoref{h2}\\
L1&20&20-40&mm&\autoref{l1}\\
L2&0&0-20&mm&\autoref{l2}\\
L3&14&10-18&mm&\autoref{l3}\\
L4&2&1-4&mm&\autoref{l4}\\
K1&0.4&0.05-0.7&W/(m*K)&\autoref{k1}\\
K2&0.036&0.03-0.18&W/(m*K)&\autoref{k2}\\\hline
\end{tabular}
\caption{Standard values and tested range for all parameters, for the non-Airex configuration.}\label{param1}
\end{table}
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=\textwidth]{typical.JPG}}}
\caption{Typical temperature distribution on all bodies (a) and outer surface (b), for the non-Airex standard configuration.}
\label{surf}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{H1.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs H1, for the non-Airex standard configuration.}
\label{h1}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{H2.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs H2, for the non-Airex standard configuration.}
\label{h2}
\end{minipage}
\end{figure}
Based on the results plotted in \autoref{h1}, it can be concluded that H1 should be maximized in order to increase the minimum temperature of the outer surface. The range of the minimum outside temperature however, is very small for H1 and thus H1 is regarded insignificant for the thermal aspects. \autoref{h2} suggests that H2 should be minimized and \autoref{l1} suggests that L1 should be maximized.
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{L1d.PNG}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs L1, for the non-Airex standard configuration.}
\label{l1}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{L2d.PNG}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs L2, for the non-Airex standard configuration.}
\label{l2}
\end{minipage}
\end{figure}
\autoref{l2} suggest that introducing an Airex piece between two polymer parts, could increase the outer temperature. According to the figure, a 3mm thick Airex piece can be sufficient to achieve an outer temperature greater than 12\degree C. The thicker the Airex piece, the better.
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{L3m.PNG}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs L3, for the non-Airex standard configuration.}
\label{l3}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{L4.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs L4, for the non-Airex standard configuration.}
\label{l4}
\end{minipage}
\end{figure}
\autoref{l3} shows that the minimum outside temperature has a maxima when L3 is about 11mm.
The relationship between L4 and outer temperatures is shown in \autoref{l4}. The dependency is not very strong, for the minimum temperature. However, all outside temperatures increases as L4 increases. Thus L4 should be maximized.
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{K1.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs K1, for the non-Airex standard configuration.}
\label{k1}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{K2.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs K2, for the non-Airex standard configuration.}
\label{k2}
\end{minipage}
\end{figure}
The dependence of the thermal conductivity of the polymer is shown in \autoref{k1} and the dependence of the thermal conductivity of the disc inside the cavity is shown in \autoref{k2}. The temperature range is greater in \autoref{k1} than \autoref{k2}, thus it is evident that the thermal conductivity of the polymer has a bigger impact on the outer surface temperatures, than the thermal conductivity of the disc. The thermal error for the standard configuration without Airex is shown in \autoref{terror}. The thermal error in \textit{ANSYS 16.1} is the percentage error in the energy norm, which is calculated through a method described in \cite{errort}, by using discontinuities in the heat flux field. The magnitude of the error indicates that the computed results are good to at least the second decimal.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{terror-NA.png}
\caption{Thermal error for the standard configuration without Airex.}
\label{terror}
\end{figure}
A summary of the conclusion on each parameter and the range of the minimum temperature related to it is presented in \autoref{concNA}.
By looking at \autoref{concNA} it can be concluded that there are three parameters that can be altered to achieve a minimum outside temperature superior to 12\degree C. However, K1 will be difficult to regulate since the material used needs to fulfill other requirements such as radiation hardness. L1 will most probably be limited to a maximum of 20mm due to mechanical properties of the UT-box. This leaves L2, the width of the dividing Airex, as the easiest way to ensure an outside temperature superior to 12\degree C. The average heat flux through the outer surface for this standard configuration is calculated to $117[W/m^2]$.
\begin{table}[H]
\centering
\captionsetup{justification=centering,margin=2cm}
\begin{tabular}{|r|c|l|}\hline
Parameter&Should be&Minimum outside temperature range $ [ \degree C ] $ \\\hline
K1&Minimized&14.77 - 9.58 = 5.19\\
L2&Maximized&15.02 - 11.00 = 4.02\\
L1&Maximized&12.03 - 10.10 = 1.93\\
H2&Minimized&10.81 - 9.85 = 0.96\\
L3&Minimized, max value about 11mm&10.11 - 10.02 = 0.09\\
K2&Minimized&10.10 - 10.05 = 0.05\\
H1&Maximized&10.11 - 10.08 = 0.03\\
L4&Maximized&10.11 - 10.08 = 0.03\\\hline
\end{tabular}
\caption{Summary of suggested approaches for each parameter in the standard configuration without Airex, ordered by relevance.}\label{concNA}
\end{table}
\subsubsubsection{Heterogeneous plug}
According to \autoref{l2}, a 3mm Airex piece can be sufficient to ensure that no condensation occurs on the outside. To further investigate the behavior of a plug divided by an Airex piece, a new standard configuration was set up. This configuration is presented in \autoref{param2}. L2 is set to 6mm and is not altered since it is already evident that thicker is better, and 6mm is regarded to be close to the upper limit of the thickness when regarding mechanical problems that might occur. Most critical is the attachment of the polyurethane to the Airex. The attachment might not be rigid enough if the Airex is too wide, since this implies thinner polyurethane parts.
\begin{table}[H]
\captionsetup{justification=centering,margin=2cm}
\centering
\begin{tabular}{|r|c|c|c|l|}\hline
Parameter&Standard value&Tested range&Unit&Figure\\\hline
H1&65&50-70&mm&\autoref{ah1}\\
H2&10&5-15&mm&\autoref{ah2}\\
L1&20&20-40&mm&\autoref{al1}\\
L2&6¬ altered&mm&--\\
L3&12&8-16&mm&\autoref{al3}\\
L4&2&0.5-5&mm&\autoref{al4}\\
K1&0.4&0.05-0.7&W/(m*K)&\autoref{ak1}\\
K2&0.036&0.03-0.06&W/(m*K)&\autoref{ak2}\\
K3&0.036&0.036-0.1&W/(m*K)&\autoref{ak3}\\\hline
\end{tabular}
\caption{Standard values and tested range for all parameters, for the configuration with Airex.}\label{param2}
\end{table}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{aH1.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs H1, for the standard configuration with an Airex divider.}
\label{ah1}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{aH2m.PNG}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs H2, for the standard configuration with an Airex divider.}
\label{ah2}
\end{minipage}
\end{figure}
The behavior of the minimum outside temperature in \autoref{ah1} and \autoref{ah2} is similar to the behavior in \autoref{h1} and \autoref{h2} respectively, with the difference that H2 now have maxima for the maximum temperature at about 9.5mm. However, the minimum temperature is decreasing with increasing H2 throughout the whole computed range. Thus H2 should be minimized. In \autoref{al1}, L1 is plotted against the outside temperature extremes and in \autoref{al3}, L3 is.
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{aL1.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs L1, for the standard configuration with an Airex divider.}
\label{al1}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{aL3m.PNG}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs L3, for the standard configuration with an Airex divider.}
\label{al3}
\end{minipage}
\end{figure}
The minimum temperature range in \autoref{l1} is almost half of the range presented in \autoref{al1}. Thus the L1 dependence is smaller for the Airex configuration compared to the non-Airex configuration. In \autoref{al3} it is shown that both the outside maximum and minimum temperatures have a maxima at about 13mm. Thus this value should be targeted for L3.
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{aL4.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs L4, for the standard configuration with an Airex divider.}
\label{al4}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{aK1.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs K1, for the standard configuration with an Airex divider.}
\label{ak1}
\end{minipage}
\end{figure}
The L4 dependence shown in \autoref{al4} is similar to the one shown in \autoref{l4}, but with a larger range. Thus L4 should still be maximized. \autoref{ak1} shows the new dependence of K1 and \autoref{ak2} the dependence of K2. As expected, the thermal conductivity of the polymer still has a bigger impact on the outside temperatures than the thermal conductivity of the disc in the cavity.
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{aK2.png}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs K2, for the standard configuration with an Airex divider.}
\label{ak2}
\end{minipage}
\hspace{.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=\textwidth]{aK3d.PNG}
\captionof{figure}{Maximum and minimum temperatures on the outer surface vs K3, for the standard configuration with an Airex divider.}
\label{ak3}
\end{minipage}
\end{figure}
In case it is not possible to put Airex between the polymer parts, K3 have also been investigated in this standard configuration. \autoref{ak3} suggest that Airex could be replaced by a material with a thermal conductivity below approximately 0.075 $W/(m*K)$, without causing condensation. A typical thermal error for the standard configuration is shown in \autoref{terrorA}. The error is similar to the one presented in \autoref{terror} and the results are also here rounded to a two decimal digit. The average heat flux through the outer surface for this standard configuration is calculated to $96[W/m^2]$.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{terror-A.png}
\caption{Thermal error for the standard configuration with Airex.}
\label{terrorA}
\end{figure}
\subsubsection{Conclusions}
In \autoref{distr} and \autoref{distr2} a comparison between the temperature distributions corresponding to the two standard configurations are presented. The configuration without Airex will according to this analysis cause condensation, whilst the configuration with Airex will not. \autoref{distr} shows all bodies and \autoref{distr2} shows the outer surface of the plug.
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[width=\textwidth]{distrib.png}
\caption{Temperature distribution on all bodies for the standard configuration with Airex $(A)$, and without Airex $(B)$}
\label{distr}
\end{figure}
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[width=\textwidth]{distriout.PNG}
\caption{Temperature distribution on the outer surface for the standard configuration with Airex $(A)$, and without Airex $(B)$}
\label{distr2}
\end{figure}
A summary of the conclusion on each parameter and the range of the minimum temperature related to it, are presented in \autoref{concNA} (without Airex) and \autoref{concA} (with Airex).
\begin{table}[H]
\centering
\captionsetup{justification=centering,margin=2cm}
\begin{tabular}{|r|c|l|}\hline
Parameter&Should be&Minimum outside temperature range $ [ \degree C ] $ \\\hline
K3&Minimized&15.82 - 11.54 = 4.28\\
K1&Minimized&14.85 - 13.22 = 1.63\\
L1&Maximized&14.27 - 13.26 = 1.01\\
H2&Minimized&13.66 - 13.03 = 0.63\\
K2&Minimized&13.33 - 13.03 = 0.30\\
L4&Maximized&13.37 - 13.16 = 0.21\\
L3&Maximized, max value about 13mm&13.25 - 13.11 = 0.14\\
H1&Maximized&13.25 - 13.24 = 0.01\\\hline
\end{tabular}
\caption{Summary of suggested approaches for each parameter in the standard configuration with Airex, ordered by relevance.}\label{concA}
\end{table}
It can be seen in \autoref{concA} that the top 3 relevant parameters for the outside temperature are K3, K1 and L1, of which none is easy to regulate due to mechanical and radiation requirements. The fourth most relevant parameter only have a minimum outside temperature range of 0.631\degree C. Thus one can conclude that an Airex piece (or any other material with a thermal conductivity of less than 0.075$W/(m*K)$ ), in between the polymer parts will be the easiest way to avoid condensation on the outside without making the plug very thick. The calculated heat fluxes suggest that the total heat flux through the plug and the rest of the UT-box, will not exceed the 500W of cooling that the cooling plant can provide. The results are visualized in \autoref{sumry}
\begin{figure}[H]
\centering
\captionsetup{width=0.8\textwidth}
\includegraphics[width=\textwidth]{summary.PNG}
\caption{Visualization of the resulting temperature extremes presented in \autoref{concA} and \autoref{concNA}. Configuration with Airex in red and configuration without Airex in blue.}
\label{sumry}
\end{figure}
\subsubsection{Decided dimensions}\label{dims}
Taking into account the results of this thermal analysis, mechanical analysis and geometric constraints, dimensions for the UT-plug were set as indicated in the figures below. The plug will be manufactured in three separate parts, of which two are identical polyurethane parts. The polyurethane part is shown in \autoref{plug} and the central Airex part in \autoref{divi}. The assembly of the suggested parts are shown in \autoref{asse}. The polyurethane parts were manufactured according to the process described in \aref{polyrep}
\begin{figure}[H]
.\centering \subfloat{{\includegraphics[height=0.45\textheight]{plug.pdf}}}
\caption{Drawing of the Polymer part.}
\label{plug}
\centering \subfloat{{\includegraphics[height=0.45\textheight]{middle.pdf}}}
\caption{Drawing of the middle part.}
\label{divi}
\end{figure}
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[height=0.45\textheight]{Assem7.pdf}}}
\caption{Drawing of the assembly of the polymer part and the middle part into a complete plug}
\label{asse}
\end{figure}
\section{Force evaluation}\label{sass}
To assure that the fragile beryllium beam pipe will not break from the pressure applied by the plug, a mechanical simulation and rigidity measurements were performed on the plug.
\subsection{Simulation}\label{mechsim}
The plug simulated have the geometries stated in Section \ref{dims}.
\subsubsection{Model}
To decrease the execution time for the mechanical simulation, only half of one side of the plug is simulated. The results should thus be doubled in order to achieve the total force applied by one half plug. The value of the friction coefficient between the beryllium pipe and the polyurethane plug is not known but estimated to be somewhere in the region 0.1-0.8 \cite{fric}.
The displacement of the plug will not be more than 2mm in case of adequate closing. However, in case the plug accidentally get displaced more, a displacement larger than 2mm should be investigated. Thus a 4mm displacement towards the beam pipe is applied. The displacement is ramped up.
The Mooney-Rivlin model was the model used in this simulation with the parameters stated in \autoref{props}. The initial elasticity modulus was based on the material hardness and calculated through
\begin{equation}
E_0=6894.76\cdot(11.427\cdot H-0.4445\cdot H^2+0.0071\cdot H^3)\hspace{2mm},
\end{equation}
\label{propeq} where \textit{H} is the hardness. The Poisson ratio were calculated to $0.501$, which means that there is no global volume change under loads.
\begin{table}[H]
\centering
\captionsetup{justification=centering,margin=2cm}
\begin{tabular}{|r|c|c|c|l|}\hline
Property&\multicolumn{3}{c|}{Value}&Unit\\\hline
Material hardness&40&60&80&Shore A\\
Initial elasticity modulus&1.38$\cdot10^6$&4.27$\cdot10^6$&1.18$\cdot10^7$&Pa\\
Initial shear modulus&$4.60\cdot10^5$&$1.42\cdot10^6$&$3.92\cdot10^6$&Pa\\
Mooney coefficients&$C_{10}=1.84\cdot10^5$&$C_{10}=5.69\cdot10^5$&$C_{10}=1.57\cdot10^6$&Pa\\
&$C_{01}=4.60\cdot10^4$&$C_{01}=1.42\cdot10^5$&$C_{01}=3.91\cdot10^5$&Pa\\\hline
\end{tabular}
\caption{Material properties used in the mechanical simulations.}\label{props}
\end{table}
The geometry and the boundary conditions are shown in \autoref{mechgeo} and the results are shown in \autoref{mechsimres}-\autoref{simmus}. \autoref{mechsimres} shows the force dependence on the friction coefficient, with 4mm displacement and \textit{Shore A 40} as hardness. \autoref{mechsimres2} shows the force dependence on displacement, with 0.45 as friction coefficient and \textit{Shore A 40} as hardness.
\begin{figure}[H]
\centering
\captionsetup{width=0.8\textwidth}
\includegraphics[height=0.4\textheight]{bundrs.PNG}
\caption{Geometry and boundary conditions for the mechanical simulation. The reaction force are probed on the beige pipe section.}
\label{mechgeo}
\end{figure}
\subsubsection{Results}
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering
\includegraphics[height=0.35\textheight]{simres.png}
\caption{Simulation results for \textit{Shore A 40}. Reaction force plotted against friction coefficient. Blue line and scale shows the total reaction force, the red line and scale shows the reaction force in the Z-direction (perpendicular to the beam pipe). }
\label{mechsimres}
\captionsetup{width=0.7\textwidth}
\centering \subfloat{{\includegraphics[height=0.35\textheight]{simres2.PNG}}}
\caption{Simulation results for \textit{Shore A 40} with 0.45 as friction coefficient. Reaction force plotted against time (since displacement of 4mm are ramped up over 1s). Time can be converted to displacement by multiplying the time value by 4mm.}
\label{mechsimres2}
\end{figure}
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=0.9\textwidth]{displ.PNG}}}
\caption{Calculated deformation on the UT-plug for \textit{Shore A 40}. Deformation in X-direction $(A)$, and total deformation $(B)$.}
\label{deform}
\end{figure}
\autoref{mechsimres2} suggests that the force applied on the beam pipe is about 3N at 2mm displacement (force have to be doubled since only one side is simulated) for \textit{Shore A 40}. This magnitude of the force is acceptable. In \autoref{deform} $(A)$. it can be seen that the deformation in the X-direction (parallel to the beam pipe) is below 4.5mm. The total deformation is shown in \autoref{deform} $(B)$, and is calculated to be about 5mm. \autoref{simmus} suggest that there is a strong hardness dependence and that the total reaction force could be as high as 58.2N at 4mm displacement for a plug with the hardness \textit{Shore A 80}.
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering \subfloat{{\includegraphics[width=0.8\textwidth]{simus.eps}}}
\caption{Simulation results for all hardness's. The total reaction force (i.e. for both plug halves) plotted against displacement.}
\label{simmus}
\end{figure}
\subsection{Testing}
\subsubsection{Setup}
To perform the compression tests, the tensile testing machine \textit{Tinius Olsen H5KT} \cite{data} were used. This machine measures both displacement and applied force. The results are analyzed in MATLAB R2014a.
To be able to use this tensile testing machine for a compression test, a sample holder has been manufactured. The sample holder has been manufactured in two separate parts. One part that holds the actual sample (\autoref{holder}), and second part that is used to attach the holder to the tensile testing machine (\autoref{backt}). The part shown in \autoref{backt} can thus be replaced in the future in order to use the holder in another machine or setup. \autoref{holderass} shows the assembly of the two holder parts. Both holder parts were 3D-printed. The test was performed by attaching the sample holder to the compression machine and compress it against an acrylic pipe. The acrylic beam pipe was used to simulate the LHCb beam pipe. The diameter of the acrylic beam pipe is 25mm.
\begin{figure}[H]
\centering \subfloat{{\includegraphics[height=0.45\textheight]{holder.pdf}}}
\caption{Drawing of the main part of the sample holder.}
\label{holder}
\end{figure}
\begin{figure}[H]
\centering \subfloat{{\includegraphics[height=0.45\textheight]{backT.pdf}}}
\caption{Drawing of the back piece to the sample holder.}
\label{backt}
\centering \subfloat{{\includegraphics[height=0.45\textheight]{holderassembly.pdf}}}
\caption{Drawing of the assembled sample holder.}
\label{holderass}
\end{figure}
\vspace{10mm}
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=\textwidth]{shorez.PNG}}}
\caption{Plug parts used in compression test.}
\label{pluugs}
\end{figure}
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=\textwidth]{assmbl.PNG}}}
\caption{Plug assembly process}
\label{assmbl}
\end{figure}
The tested plugs are shown in \autoref{pluugs} and the assembly process is shown in \autoref{assmbl}. The assembled setup, during a compression test, is shown in \autoref{comtest}.
\begin{figure}[H]
\centering \subfloat{{\includegraphics[height=0.9\textheight]{comtest.jpg}}}
\caption{A compression test being performed}
\label{comtest}
\end{figure}
\newpage
\subsubsection{Results}
Each plug were tested 10 times. The measurements are started at the smallest visible distance above the beam pipe. This means that there is a negative force applied (since the machine then have to hold the sample up). To make sure that the compression tests starts at the same position relative to the beam pipe, for all measurements, the data before the first registered positive force are erased, and the measured displacement at this point is subtracted from the displacement values. The results from the compression test are shown in \autoref{rezz}. \autoref{rezz2} shows the order in which the tests on the \textit{Shore A 80} plug were performed.
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=0.55\textwidth]{nyrez.PNG}}}
\caption{Compression test results.}
\label{rezz}
\end{figure}
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=0.6\textwidth]{Shore80TestsNumbered.eps}}}
\caption{Numbered compression test results for the \textit{Shore A 80} plug.}
\label{rezz2}
\end{figure}
\subsection{Conclusion}
\begin{figure}[H]
\centering \subfloat{{\includegraphics[width=\textwidth]{AllsimsHalf.eps}}}
\caption{All calculated and measured forces plotted against the displacement}
\label{allst}
\end{figure}
\autoref{allst} shows that there is a large difference between calculated and measured forces. This could be a consequence of inadequate estimations of the material properties, or it might be the case that the Mooney-Rivlin model is not suited for polyurethane of such shape. The Mooney-Rivlin model is one of the most basic models for hyper-elastic materials and is known to overestimate the reaction force, but a deviation of this magnitude was not expected. However, it should be noted that the \textit{Shore A 60} test is similar to the \textit{Shore A 40} calculations.\\
\autoref{rezz} shows that the largest measured force at a 2mm displacement is about 3N, which is acceptable. The spread of the shore 80 measurements suggest that hardness of the plug implies a greater dependence on previous compressions.
The fact that some of the measured forces for the \textit{Shore A 80} plug is lower than for the \textit{Shore A 60} plug indicates that the shore number is not to be trusted completely.
Other geometries than the one stated in Section \ref{dims} have been tested and the maximum force that have been measured at 2mm displacement is about 6.5N. The maximum force measured for 4mm compression is about 17N. This underpins the conclusion that the shore numbers should not be trusted completely and suggests that forces of this magnitude should still be considered possible, and that the calculations for \textit{Shore A 80} should be considered the worst case scenario.
In order to be concise in the manufacturing process of the UT-plugs, and to assure that the force applied on the beam pipe will not brake it, a mixing ratio of the Polyurethane ingredients, and a curing cycle should be defined and carefully tested.\\
There seems to be a correlation between H1 and the applied force, but since the geometries tested does not have all other dimensions in common, it cannot be confirmed. An increased H1 seem to imply an increased force (and it increases the mass), thus it could be desired to minimize H1. Manufacturing difficulties puts a limit on how small H1 could be. It should be investigated how small H1 could be without causing problems in the manufacturing process.
The observed displacement in the beam pipe direction are of similar shape to the displacement shown in \autoref{deform}. A linear regression has been made on all simulation- and test results. The reaction force $F$ at a $x$ mm displacement can thus be approximated through
\begin{equation}
F(x)=x*P(1)+P(2)
\end{equation}
where the coefficients $P(1)$ and $P(2)$ are fetched from \autoref{regress}. \autoref{valtab} shows the calculated and measured forces for some displacements.
\begin{table}[H]
\centering
\captionsetup{justification=centering,margin=2cm}
\begin{tabular}{r|c|c|c|c|c|c|}
&\multicolumn{3}{c|}{Simulation}&\multicolumn{3}{c|}{Test}\\
Hardness [Shore A]&40&60&80&40&60&80\\\hline\hline\hline
P(1)&1.57&5.28&14.51&0.31&1.44&1.52\\
P(2)&-0.27&-0.96&-2.44&-0.16&-0.11&-0.46\\\hline
\end{tabular}
\caption{Linear regression coefficients for the simulation- and test results. }\label{regress}
\end{table}
\begin{table}[H]
\centering
\captionsetup{justification=centering,margin=2cm}
\begin{tabular}{r|c|c|c|c|c|c|}
&\multicolumn{3}{c|}{Simulation}&\multicolumn{3}{c|}{Test}\\
Hardness [Shore A]&40&60&80&40&60&80\\\hline\hline\hline
Displacement [mm]&\multicolumn{6}{c|}{Force [N]}\\\hline
0.5&0.6&2.0&5.4&0.2-0.3&0.5-0.8&0.3-0.7\\
1&1.3&4.2&11.2&0.2-0.4&1.2&0.6-1.7\\
1.5&2.1&6.4&18&0.3-0.5&1.7-1.8&1.0-2.4\\
2&2.9&9.0&24.8&0.5-0.7&2.3-2.7&1.8-3.2\\
2.5&3.8&11.8&32.4&0.6-0.8&3.2-3.5&2.6-4.2\\
3&4.7&14.6&40.4&0.8-1.0&4.0-4.2&3.2-4.8\\
3.5&5.7&17.5&48.4&1.0-1.2&4.8-5.2&3.8-5.0\\
4&6.8&21.1&58.2&1.2-1.3&6.3-6.5&4.5-5.7\\\hline
\end{tabular}
\caption{Simulation and measurement results for some displacements.}\label{valtab}
\end{table}
\section{Conclusions}
The aim of this thesis is to establish concepts and methods so that the requirements of the UT-box can be fulfilled. The most important conclusions, and therefore the main results of this thesis, are stated below.
\begin{itemize}
\item \textbf{A heterogeneous UT-plug has been calculated to perform better thermally than a homogeneous polyurethane plug.}\\
Through analytical and numerical calculations it has become evident that the heterogeneous plug is the best approach to avoid condensation on the UT-plug.\\
\item \textbf{It has been confirmed that condensation will not occur on the outside of the panels.}\\
\item \textbf{The Carbon-Airex composite panel concept has been validated.}\\
The sandwich structured composites, that will make the UT-walls, have been confirmed to be stable. Stable in the way that the carbon fiber skins and the Airex core are properly attached to each other by the cocuring process.\\
\item \textbf{The copper net concept to ensure Faraday-cage behavior has been validated.}\\
The electrical conductance in a copper net embedded in the carbon fiber skins has been measured and proven to be good.\\
\item \textbf{A possible method for panel connection has been validated.}\\
The method to attach a 90 \degree angle piece in the joints between the panels, in order to connect the copper nets together has been validated.\\
\item \textbf{Radiation resistance of the UT-plug has been assured.}\\
\item \textbf{A UT-plug manufacturing method has been validated.}\\
The UT-plug can be manufactured by 3D-printing a mold (high resolution is required). The method have however shown difficulties in production of small plugs.\\
\item \textbf{A mixing ratio and curing cycle for the Polyurethane in the UT-plug have to be defined.} \\In order to ensure that all UT-plugs have the same material properties, and to eliminate the risk of rupture in the beam pipe, a mixing ratio and a curing cycle for the polyurethane part have to be defined and tested.\\
\end{itemize}
Those conclusions will serve as a basis for the construction of a mock-up of the UT-box. The mock-up will serve to test how everything works out when put together and to determine which areas might require special attention or further development. If the mock-up do not show the need to change any method or concept, they will be implemented in the LHCb upgrade taking place in 2019-2020.
\newpage
\begin{thebibliography}{11}
\bibitem{min}
Oskar Mårtensson, \textit{Cooling system design for the Micromegas detector},
October 2015\\
Available at: \\\url{https://edms.cern.ch/document/1538222}\\
\textit{07/12/2015,09:39}
\bibitem{lhcb}
\url{http://home.web.cern.ch/about/experiments/lhcb}\\
\textit{21/09/2015,13:24}
\bibitem{particle}
Frank Close, \textit{Particle physics- A very short introduction}, 2012, Oxford University Press\\
\bibitem{cp1}
John Ellis, \textit{Why does CP violation matter to the universe}, September 1999\\
Available at: \\\url{http://cerncourier.com/cws/article/cern/28092}\\
\textit{11/10/2015,18:56}
\bibitem{cp2}
Alfio Lazzaro, \textit{CP Violation in B mesons}, March 2007\\
Available at: \\\url{http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-12440.pdf}\\
\textit{11/10/2015,20:44}
\bibitem{ut}
Jason Andrews, \textit{Upgrade of the Upstream Tracker at LHCb}, April 2015\\
Available at: \\\url{https://cds.cern.ch/record/2013250/files/LHCb-TALK-2015-081.pdf}\\
\textit{29/10/2015,09:35}
\bibitem{tdr}
LHCb collaborations \textit{LHCb tracker upgrade TDR}, February 2014\\
Available at:\\ \url{https://cds.cern.ch/record/1647400/files/LHCB-TDR-015.pdf}\\
\textit{21/09/2015,13:36}
\bibitem{struct}
Joao Carlos Batista Lopes and Oskar Mårtensson, \textit{UT BOX STRUCTURAL AND THERMAL ASSESSMENT}, April 2015\\
Available at: \\\url{https://edms.cern.ch/document/1507417}\\
\textit{11/12/2015,10:41}
\bibitem{rad}
{M.H. Van de Voorde and C.Restat, \textit{Selection guide to organic materials for nuclear engineering}, May 1972\\
Available at: \\\url{https://cds.cern.ch/record/186329/files/CERN-72-07.pdf}\\
\textit{07/12/2015,09:39}
\bibitem{airexx}
\url{http://www.sicomin.com/datasheets/product-pdf1037.pdf}\\
\textit{11/12/2015,11:18}, (Data sheet for Airex R82)
\bibitem{carbon} \url{http://www.aerospares.hu/files/loctite/SynSkin_HC_9837_1.pdf}\\
\textit{19/01/2016,13:00}, (Data sheet for carbon fiber)
\bibitem{cu}
\url{http://www.dexmet.com/1_pdf/Lightning\%20Strike\%20Brochure.pdf}\\
\textit{01/12/2015,14:15}, (Data sheet for copper net)
\bibitem{t1}
Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera \& David P. Dewitt, \textit{Fundamentals of heat and mass transfer}
6th Edition, 2007, R.R. Donnelley
\bibitem{ashrae}
ASHRAE, \textit{ASHRAE Handbook - Fundamentals}, 2015\\
Available at: \\\url{http://shop.iccsafe.org/media/wysiwyg/material/8950P203-sample.pdf}\\
\textit{04/12/2015,14:31}
\bibitem{errort}
H. C. Huang and R. W. Lewis, \textit{Adaptive Analysis for Heat Flow Problems Using Error Estimation Techniques.}, April 1989.\\
Proc. Sixth Int. Conf. for Numerical Methods in Thermal Problems, U.K. July 1989,
Pineridge Press, SWansea (1989)
\bibitem{t2}
Yunus A. Cengel, \textit{ Introduction to Thermodynamics and Heat Transfer}, 2nd Edition, 2008, McGraw-Hill Publishing Company
\bibitem{fric}
\url{http://www.gallaghercorp.com/images/kinetic-coefficient-graphs.gif}\\
\textit{14/12/2015,16:02}
\bibitem{data}
\url{http://www.tiniusolsen.com/pdf/B140D.pdf}\\
\textit{01/12/2015,14:15}, (Data sheet for tensile tester)}
\end{thebibliography}
\newpage
\begin{appendices}
\includepdf[scale=0.68,clip,trim=0cm 0cm 0cm 0cm,frame,pages={1},pagecommand={\section{: UT-box thermal analysis}\label{App:hcalc}}]{Thermal_calculations_on_UT-box-NEW.pdf} \includepdf[scale=0.75,clip,frame,trim=0cm 0cm 0cm 0cm,pages={2-},pagecommand={}]{Thermal_calculations_on_UT-box-NEW.pdf}
\newpage
\includepdf[scale=0.68,clip,trim=0cm 0cm 0cm 0cm,frame,pages={1},pagecommand={\section{: Cavity convection analysis}\label{cavitycalc}}]{CavityConvectionCheck.pdf} \includepdf[scale=0.75,clip,frame,trim=0cm 0cm 0cm 0cm,pages={2-},pagecommand={}]{CavityConvectionCheck.pdf}
\newpage
\includepdf[scale=0.68,clip,trim=0cm 0cm 0cm 0cm,frame,pages={1},pagecommand={\section{: Heat transfer comparison between two vertical plates}\label{radcalc}}]{Heat_transfer_comparison-_Parallel_vertical_plates.pdf} \includepdf[scale=0.75,clip,frame,trim=0cm 0cm 0cm 0cm,pages={2-},pagecommand={}]{Heat_transfer_comparison-_Parallel_vertical_plates.pdf}
\newpage
\includepdf[scale=0.68,clip,trim=0cm 0cm 0cm 0cm,frame,pages={1},pagecommand={\section{: Composite manufacturing report}\label{comprep}}]{comporepp.pdf} \includepdf[scale=0.68,clip,frame,trim=0cm 0cm 0cm 0cm,pages={2-},pagecommand={}]{comporepp.pdf}
\newpage
\includepdf[scale=0.68,clip,trim=0cm 0cm 0cm 0cm,frame,pages={1},pagecommand={\section{: Polymer manufacturing report}\label{polyrep}}]{polllyrep.pdf} \includepdf[scale=0.68,clip,frame,trim=0cm 0cm 0cm 0cm,pages={2-},pagecommand={}]{polllyrep.pdf}
\newpage
\includepdf[scale=0.68,clip,trim=0cm 0cm 0cm 0cm,frame,pages={1},pagecommand={\section{: Conceptual design report, UT PLUG – INTERFACE BETWEEN UT AND LHCB BEAM PIPE}\label{polyrep}}]{inteface.pdf} \includepdf[scale=0.68,clip,frame,trim=0cm 0cm 0cm 0cm,pages={2-},pagecommand={}]{inteface.pdf}
\end{appendices}
\end{enumerate}
\end{document}
