WebSVN – svnkaklik – Path Comparison – / – /dokumenty/skolni/VAK Rev 720 and /dokumenty/skolni/VAK Rev 721

Ignore whitespace Rev 720 → Rev 721

/dokumenty/skolni/VAK/netesnosti/netesnosti.pdf
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 /dokumenty/skolni/VAK/netesnosti/netesnosti.tex
 ,21 → 54,21
 Po vyzkoušení předchozích hledacích metod jsme uzavřeli jehlový ventil na aparatuře a uvedli do provozu heliový hledač netěsností podle provozního postupu v přiložených deskách.
-Následně nastavili rozsah na nejmenší citlivost a začali zkoušet ofukovat aparaturu heliem z balonku. Po delší době jsme objevili netěsnost v oblasti příruby u Piraniho měrky.
+Následně nastavili rozsah na nejmenší citlivost a začali zkoušet ofukovat aparaturu heliem z balonku. Po chvíli jsme objevili netěsnost v oblasti příruby u Piraniho měrky.
 Urychlovací napětí potřebné k předání správné rychlosti jádrům helia, aby byla jejich dráha zakřivena na poloměr 40mm v magnetickém poli 150mT spočítáme podle Lorentzovy síly a dostředivého zrychlení, které se musejí rovnat.
-\begin{displaymath} F_d = \frac{m v^2}{r} = q v B. \end{displaymath}
+\begin{displaymath} q v B  =  \frac{m v^2}{r}  \end{displaymath}
-Po vyjádření $v$ dostáváme.
+Pro energii částice urychlené v elektrickém poli platí
-\begin{displaymath} v = \frac{r q B}{m}. \end{displaymath}
+\begin{displaymath} E = q U = \frac{m v^2}{2} . \end{displaymath}
-Dosadíme do vztahu pro kinetickou energii a máme.
+Do této rovnice dosadíme rychlost z předchozí rovnice a vyjádříme napětí.
-\begin{displaymath} E = \frac{(r q B)^2}{2 m}. \end{displaymath}
+\begin{displaymath} U = \frac{q r^2 B^2}{2 m} \end{displaymath}
-Po vyčíslení získáme energii \begin{displaymath} E = 2,8247 \times 10^{-16} [J] \end{displaymath} což odpovídá \begin{displaymath} E = 1763 [eV]. \end{displaymath}. A potřebné urychlovací napětí tedy je 1763 V.
+Dostaneme napětí potřebné k urychlení iontu helia, aby dopadl do otvoru detektoru v komoře heliového hledače. \begin{displaymath} U = 438 \textrm{V} \end{displaymath}.
 \section{Závěr}
 V praktiku jsme si tak vyzkoušeli několik zajímavých metod pro hledání netěsností v aparatuře.  Z nich některé mne překvapily svou jednoduchostí a přitom vysokou účinností, jako například hledání netěsnosti pomocí ethanolu.

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 /dokumenty/skolni/VAK/rotacni_vyveva/vyveva.tex
 ,22 → 33,16
 \section{Postup měření}
-$ S = - \frac{V}{t} \ln \frac{p}{p0} $
-$ q = C (p1 - p2) $
-$ C_{VM} = \frac{ \pi D^2}{4} \frac{D}{L} \left[ \frac{\pi}{128} \frac{D}{l_s} + \frac{1}{3} Z(D/l_s) \right]$
-\subsection{Sledujte čerpání uzavřeného objemu ROV}
+\subsection{Čerpání uzavřeného objemu Rotační olejovou vývěvou}
 Objem skleněné baňky (asi 11,8 l) jsme čerpali rotační olejovou vývěvou přes hadici. Zároveň jsme měřili tlak a hodnoty zapisovali do tabulky.
 \begin{table}[htbp]
 \caption{Průběhy tlaku v recipientu při čerpání rotační olejovou vývěvou}
 \begin{center}
-\begin{tabular}{|r|r|r|}
+\begin{tabular}{|c|c|c|}
 \hline
-\multicolumn{1}{|l|}{} & \multicolumn{1}{l|}{Bez proplachování} & \multicolumn{1}{l|}{S proplachováním} \\ \hline
-\multicolumn{1}{|l|}{[Pa]} & \multicolumn{1}{l|}{[s]} & \multicolumn{1}{l|}{[s]} \\ \hline
+\multicolumn{1}{|c|}{} & \multicolumn{1}{l|}{Bez proplachování} & \multicolumn{1}{l|}{S proplachováním} \\ \hline
+[Pa] & [s] & [s] \\ \hline
 ,00E+005 & 0 & 38 \\ \hline
 ,00E+004 & 6 & 53 \\ \hline
 ,00E+004 & 26 & 60 \\ \hline
 ,11 → 53,11
 ,00E+002 & 145 & 152 \\ \hline
 ,00E+002 & 166 & 189 \\ \hline
 ,00E+002 & 182 & 300 \\ \hline
-,00E+001 & 204 & \multicolumn{1}{l|}{} \\ \hline
-,00E+001 & 232 & \multicolumn{1}{l|}{} \\ \hline
-& 282 & \multicolumn{1}{l|}{} \\ \hline
-& 380 & \multicolumn{1}{l|}{} \\ \hline
-& 720 & \multicolumn{1}{l|}{} \\ \hline
+,00E+001 & 204 &  \\ \hline
+,00E+001 & 232 &  \\ \hline
+& 282 &  \\ \hline
+& 380 &  \\ \hline
+& 720 &  \\ \hline
 \end{tabular}
 \end{center}
 \label{}
 ,25 → 74,218
 \subsection{Efektivní čerpací rychlost}
+Pro odčerpané množství plynu při konstantním objemu platí vztah
+\begin{displaymath} S p = - \frac{d(p V)}{dt} =  - V \frac{dp}{dt} ,      \end{displaymath}
+ze kterého po separaci proměnných plyne
+\begin{displaymath} \ln p = \ln p_0 - \frac{ S (t_2 - t_1)}{V} ,  \end{displaymath}
+Interpolací naměřených hodnot tímto výrazem dostáváme čerpací rychlosti.
+\begin{displaymath} S_1 = 0,42 \ \textrm{l/s} \qquad S_2 = 0,36 \ \textrm{l/s} ,  \end{displaymath}
+kde $ S_1 $ je čerpací rychlost bez proplachování a $ S_2 $ je čerpací rychlost s proplachováním.
+\subsection{Měření čerpací rychlosti}
+Při čerpání rotační vývěvou jsme pootevřeli jehlový ventil oddělující mikrobyretu, tak aby se tlak v aparatuře ustálil na hodnotě mezi 5 a 20 Pa. Potom jsme měřili průtok plynu mikrobyretou. Měření jsme několikrát opakovali.
+Proud plynu je stejný v mikrobyretě i v rotační vývěvě. Proto při konstantním tlaku platí
+\begin{displaymath} S_{EF} p = - \frac{d(p V)}{dt} =  p_A \frac{\Delta V}{\Delta t} . \end{displaymath}
+Pro efektivní čerpací rychlost tedy platí
+\begin{displaymath} S_{EF} =  \frac{p_A}{p} \frac{\Delta V}{\Delta t} . \end{displaymath}
+Ze vzorce uvedeného na mikrobyretě jsme vypočítali
+\begin{displaymath} \frac{\Delta V}{\Delta t} = 4,75 \cdot 10^{-2} \cdot \frac{l}{t} \, \textrm{[cm; s; cm}^3 \textrm{/s]} , \end{displaymath}
+kde $l$ je délka o kterou se posunula hladina oleje v mikrobyretě za čas $t$. Za $p_A$ jsme dosadili $10^5$\,Pa. Výsledné hodnoty jsou uvedené v tabulce \ref{mikrobyreta}.
+\begin{table}[htbp]
+\caption{Čerpací rychlost změřená pomocí mikrobyrerty}
+\begin{center}
+\begin{tabular}{|c|c|c|c|}
+\hline
+$p$ [Pa] & $l$ [cm] & $t$ [s] & $S_{EF}$ [l/s] \\ \hline
+& 13 & 65 & 0,190 \\
+& 13 & 48,5 & 0,182 \\
+& 13 & 27,3 & 0,226 \\
+& 13 & 14,7 & 0,210 \\
+\hline
+\end{tabular}
+\end{center}
+\label{mikrobyreta}
+\end{table}
 \subsection{Vodivost hadice k ROV}
 Výpočtem jsme zjistili vodivost gumové hadice (průměru 19,51mm a délky 730mm)
+Pro kruhový průřez a viskózně molekulární je pak vodivost dána Knudsenovým empirickým vztahem
+\begin{displaymath} C_{VM} =  C_V + Z \cdot C_{M,DT} , \end{displaymath}
+kde pro vzduch při $20\,^\circ$C platí
+\begin{displaymath} C_V = 1,365 \cdot \frac{D^4}{L} \cdot \frac{p_1 + p_2}{2} \ \textrm{[l/s; cm; Pa]} , \qquad C_{M,DT} = 12,1 \cdot \frac{D^3}{L} \ \textrm{[l/s; cm]} , \qquad Z = \frac{2 + 2,507 \cdot \frac{D}{l_S}}{2 + 3,095 \cdot \frac{D}{l_S}} .  \end{displaymath}
+Efektivní čerpací rychlost vývěvy jsme spočítali ze vztahu
+\begin{displaymath} S_{EF} =  \frac{C_{VM} S}{C_{VM} + S} , \end{displaymath}
+kam jsme za $S$ dosadili čerpací rychlost $S_1 = 0,42 \ \textrm{l/s}$ vývěvy bez proplachovaní vypočtenou v předchozí úloze. Výsledné vodivosti a efektivní čerpací rychlosti pro čtyři různé tlaky jsou uvedeny v tabulce \ref{vodivost} (za $p_1+p_2$ jsme dosadili $2p$).
+\begin{table}[htbp]
+\caption{Vodivost hadice mezi ROV a recipientem}
+\begin{center}
+\begin{tabular}{|c|c|c|c|}
+\hline
+$p$ [Pa] & $l_S$ [mm] & $C_{VM}$ [l/s] & $S_{EF}$ [l/s] \\ \hline
+& 1,3 & 2,4 & 0,36 \\
+& 0,94 & 2,9 & 0,37\\
+& 0,66 & 3,7 & 0,38\\
+& 0,33 & 6,4 & 0,39\\
+\hline
+\end{tabular}
+\end{center}
+\label{vodivost}
+\end{table}
+Z této tabulky je patrné, že efektivní čerpací rychlost není za hadicí nijak výrazně závislá na tlaku a zároveň tato konkrétní hadice připojená k ROV snižuje čerpací rychlost o zhruba 20 \%.
 \subsection{Termočlánkový vakuometr}
-Kalibraci termočlánkového vakuometru jsme prováděli po jeho přeroubovaní na přírubu rozbočovače u skleněné baňky následně postupně měnili tlak. A hodnoty z obou vakuometrů McLeod i termočlánkový vakuometr zapisovali do tabulky.
+Kalibraci termočlánkového vakuometru jsme prováděli po jeho přešroubovaní na přírubu rozbočovače u skleněné baňky následně postupně měnili tlak. A hodnoty z obou vakuometrů McLeod i termočlánkový vakuometr zapisovali do tabulky.
-Výsledkem je graf, který by se dal považovat za kalibrační křivku termočlánkového vakuometru. Pro ilustraci je zde uvedena i ideální lineární odezva vakuometru.
+Tlak z Mcleodova vakuometru jsme určili podle přiloženého vzorce
+\begin{displaymath} p =  \frac{133,3 \cdot l h}{1100 - l} \,\textrm{[Pa; mm]} . \end{displaymath}
+Hodnoty jsou uvedené v tabulce \ref{cejchovani} a grafu.
+\begin{table}[htbp]
+\caption{Korekční hodnoty mezi McLeodovým a termočlánkovým vakuometrem}
 \begin{center}
+\begin{tabular}{|c|c|c|c|}
+\hline
+$l$ [mm] & $h$ [mm] & $p$ [Pa] & Počet dílků \\ \hline
+,0&2,0&1,2&30 \\
+,0&3,8&3,2&28 \\
+,0&5,3&5,2&26 \\
+,5&5,5&5,7&24 \\
+,5&6,0&7,0&22 \\
+,5&7,5&9,6&20 \\
+,0&9,0&13,2&18 \\
+,5&11,3&17,3&15 \\
+,5&11,7&19,4&13 \\
+,0&15,4&34,2&10 \\
+,5&21,8&63,4&7 \\
+,3&27,3&96,1&6 \\
+\hline
+\end{tabular}
+\end{center}
+\label{cejchovani}
+\end{table}
+Výsledkem je graf, který by se dal považovat za kalibrační křivku termočlánkového vakuometru. Pro ilustraci je zde uvedena i ideální lineární odezva vakuometru.
+\begin{center}
 \begin{figure}
 \includegraphics[width=150mm]{kalibrace.png}
-\caption{Průběhy tlaku v recipientu při čerpání rotační olejovou vývěvou}
+\caption{Korekce termočlánkového vakuometru na skutečný tlak}
 \end{figure}
 \end{center}
-\subsection{Vodivost trubice}
+\subsection{Vodivost kovové trubice}
+Mezi recipient a mikrobyretu byla ve skutečnosti umístěna kovová trubice o průměru $D = 8,5$\,mm a délce $L = 100$\,cm. Mezi mikrobyretu a trubici jsme ještě připojili již ocejchovaný termočlánkový vakuometr, který byl předtím umístěn na recipientu. Pomocí tohoto termočlánkového vakuometru a McLeodova vakuometru umístěného na recipientu, můžeme nyní měřit tlakový spád na trubici. Nyní jsme nastavili jehlový ventil tak, aby tlak vzduchu v recipientu byl v mezi 5 a 50\,Pa. Měření jsme opakovali pro 7 různých hodnot. Proud plynu je v každém místě stejný, proto bude platit
+\begin{displaymath} C (p_1-p_2) = p_A \frac{\Delta V}{\Delta t} , \end{displaymath}
+Odkud pro vodivost trubice platí (molekulární proudění)
+\begin{displaymath} C = \frac{p_A}{(p_1-p_2)} \frac{\Delta V}{\Delta t} , \end{displaymath}
+Kde $p_A = 10^5$\,Pa je atmosférický tlak, $p_1$ je tlak mezi mikrobyretou a kovovou trubicí měřený termočlánkovým vakuometrem, $p_2$ je tlak v recipientu měřený McLeodovým vakuometrem a podíl $\frac{\Delta V}{\Delta t}$ jsme určili pomocí mikrobyrety. Hodnoty a výsledky jsou uvedeny v tabulce \ref{trubice}, kde $N$ je počet dílků na termočlánkovém vakuometru a $C$ je vodivost kovové trubice.
+\begin{table}[htbp]
+\caption{Vodivost kovové trubice určená pomocí mikrobyrety a rozdílu tlaků}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|}
+\hline
+\multicolumn{1}{|l|}{$N$} & \multicolumn{1}{l|}{$p_1$ [Pa]} & \multicolumn{1}{l|}{$p_2$ [Pa]} & \multicolumn{1}{l|}{$\frac{dV}{dt}$ [cm3/s]} & \multicolumn{1}{l|}{$C$ [l/s]} \\ \hline
+&5,2&1,7& $2,72 \cdot 10^3$ &0,078 \\
+&8,3&3,4& $6,64 \cdot 10^3$ &0,135 \\
+&17,3&6,2& $1,93 \cdot 10^2$ &0,173 \\
+&19,4&8,6& $2,87 \cdot 10^2$ &0,266 \\
+&34,2&15,9& $4,94 \cdot 10^2$ &0,271 \\
+&53,7&31,3&  $7,35 \cdot 10^2$ &0,327 \\
+&96,1&71,2&$1,47 \cdot 10^1$ &0,590 \\
+\hline
+\end{tabular}
+\end{center}
+\label{trubice}
+\end{table}
+Pro výpočet vodivosti trubice použijeme vzorec
+\begin{displaymath} C_{VM} =  C_V + Z \cdot C_{M,DT} , \end{displaymath}
+kde pro vzduch při $20\,^\circ$C platí
+\begin{displaymath} C_V = 1,365 \cdot \frac{D^4}{L} \cdot \frac{p_1 + p_2}{2} \ \textrm{[l/s; cm; Pa]} , \qquad C_{M,DT} = 12,1 \cdot \frac{D^3}{L} \ \textrm{[l/s; cm]} , \qquad Z = \frac{2 + 2,507 \cdot \frac{D}{l_S}}{2 + 3,095 \cdot \frac{D}{l_S}} .  \end{displaymath}
+Pro střední volnou dráhu částic vzduchu při $20\, ^\circ $C platí
+\begin{displaymath} l_S = 6,6 \cdot 10^{-3} \frac{2}{p_1+p_2} \ \textrm{[m; Pa]}. \end{displaymath}
+Výsledky po dosazení hodnot pro příslušné tlaky jsou uvedeny v tabulce \ref{trubice2}.
+\begin{table}[htbp]
+\caption{Výpočet vodivosti kovové trubice}
+\begin{center}
+\begin{tabular}{|r|r|r|r|}
+\hline
+\multicolumn{1}{|l|}{$p_1$ [Pa]} & \multicolumn{1}{l|}{$p_2$ [Pa]} & \multicolumn{1}{l|}{$l_S$ [mm]} & \multicolumn{1}{l|}{$C_{VM}$ [l/s]} \\ \hline
+,2 & 1,7 & 1,9 & 0,085 \\
+,3 & 3,4 & 1,1 & 0,102 \\
+,3&6,2& 0,56 &0,144 \\
+,4&8,6& 0,47 &0,160 \\
+,2&15,9& 0,26 &0,239 \\
+,7&31,3&  0,16 &0,363 \\
+,1&71,2&0,08 &0,656 \\
+\hline
+\end{tabular}
+\end{center}
+\label{trubice2}
+\end{table}
+\begin{table}[htbp]
+\caption{Srovnání vypočtených a naměřených hodnot}
+\begin{center}
+\begin{tabular}{|c|c|c|c|}
+\hline
+\multicolumn{1}{|l|}{$C$ [l/s]} & \multicolumn{1}{l|}{$C_{VM}$ [l/s]} & \multicolumn{1}{|l|}{Absolutní chyba [l/s]} & \multicolumn{1}{l|}{Relativní chyba} \\ \hline
+,078&0,085&-0,01&9 \% \\
+,135&0,102&0,03&33 \% \\
+,173&0,144&0,03&20 \% \\
+,266&0,160&0,11&66 \% \\
+,271&0,239&0,03&14 \% \\
+,327&0,363&-0,04&10 \% \\
+,590&0,656&-0,07&10 \% \\
+\hline
+\end{tabular}
+\end{center}
+\label{porovnani}
+\end{table}
 \section{Závěr}
-V úloze byly dobře demonstrovány možnosti rotačních olejových vývěv a i nejjednodušší způsoby měření tlaků a průtoků ve vakuové technice.
+V úloze byly dobře demonstrovány možnosti rotačních olejových vývěv a i nejjednodušší způsoby měření tlaků a průtoků ve vakuové technice.
+výpočty jsme pak ověřili že není snadné přesně vypočítat vodivost reálné trubice, jelikož rozdíly vypočítaných a naměřených hodnot dosahují desítek procent.
 \end{document}

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