From de1d4b2fd8b41b0e21e3b6b7c3d0b8f80f3ceeb6 Mon Sep 17 00:00:00 2001 From: guyotv Date: Sun, 27 Nov 2022 21:02:32 +0100 Subject: [PATCH] Deux exos sur les cycles --- Annexe-Exercices/Annexe-Exercices.tex | 115 +++ Annexe-Exercices/Annexe-Exercices.tex.bak | 114 +++ Annexe-Exercices/Images/cycle1.eps | 281 ++++++ Annexe-Exercices/Images/cycle1.svg | 202 ++++ Annexe-Exercices/Images/cycle1.tex | 949 +++++++++++++++++ Annexe-Exercices/Images/cycle1.tex.bak | 507 ++++++++++ Annexe-Exercices/Images/cycle2.eps | 321 ++++++ Annexe-Exercices/Images/cycle2.eps_tex | 63 ++ Annexe-Exercices/Images/cycle2.eps_tex.bak | 63 ++ Annexe-Exercices/Images/cycle2.svg | 198 ++++ CoursMecaniqueOS.pdf | Bin 12866431 -> 12882517 bytes CoursMecaniqueOSDF.pdf | Bin 12866431 -> 12882517 bytes SolutionsOS.tex | 84 +- SolutionsOS.tex.bak | 1064 +++++++++++++++++++- 14 files changed, 3908 insertions(+), 53 deletions(-) create mode 100644 Annexe-Exercices/Images/cycle1.eps create mode 100644 Annexe-Exercices/Images/cycle1.svg create mode 100644 Annexe-Exercices/Images/cycle1.tex create mode 100644 Annexe-Exercices/Images/cycle1.tex.bak create mode 100644 Annexe-Exercices/Images/cycle2.eps create mode 100644 Annexe-Exercices/Images/cycle2.eps_tex create mode 100644 Annexe-Exercices/Images/cycle2.eps_tex.bak create mode 100644 Annexe-Exercices/Images/cycle2.svg diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index b2c986b..06aeb39 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2769,6 +2769,121 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + On fait passer \SI{32}{\gram} de méthane (\(CH_4\)) de l'état (\SI{1}{\bar} ; \SI{60}{\celsius}) à l'état (\SI{5}{\bar} ; \SI{60}{\celsius}) par une compression isotherme, puis, par une compression adiabatique, à l'état (\SI{30}{\bar} ; ?). Calculez our chaque transformations~: + \begin{enumerate} + \item le travail dépensé, + \item la chaleur échangée avec le milieu extérieur et + \item la variation d'énergie interne. + \end{enumerate} + \begin{solos} + Comme la masse atomique du carbone vaut \SI{12}{uma} et celle de l'hydrogène \SI{1}{uma}, celle de la molécule de méthane (\(CH_4\)) vaut \SI{16}{uma}. Ainsi, sa masse molaire vaut \SI{16}{\gram\per\mol}. Si on a \SI{32}{\gram} de ce gaz, on a donc \SI{2}{\mol} de \(CH_4\). + + \subsubsection*{Compression isotherme} + Avec une température de 273 + 60 = \SI{333}{\kelvin}, on peut alors calculer les volumes~: + \begin{align*} + V_1&=\frac{n\cdot R\cdot T}{p_1}=\frac{2\cdot 8,31\cdot 333}{10^5}=\SI{0,055}{\metre\cubed}\\ + V_2&=\frac{n\cdot R\cdot T}{p_2}=\frac{2\cdot 8,31\cdot 333}{5\cdot 10^5}=\SI{0,011}{\metre\cubed} + \end{align*} + Le travail isotherme se calcule alors aisément par~: + \begin{align*} + A&=n\cdot R\cdot T\cdot ln(\frac{V_2}{V_1})\\ + &=2\cdot 8,31\cdot 333\cdot ln(\frac{0,011}{0,055}=\SI{-8907}{\joule} + \end{align*} + Comme la compression est isotherme, on a aussi~: + \[\Delta U = 0\;\text{et}\;Q=A=\SI{-8907}{\joule}\] + + \subsubsection*{Compression adiabatique} + La compression adiabatique se fait de l'état 2 à l'état 3. Comme vu précédemment, le volume de l'état 2 est \(V_2=\SI{0,011}{\metre\cubed}\). + + Des propriétés de la transformation adiabatique, on tire alors~: + \begin{align*} + p_2\cdot V_2&=p_3\cdot V_3\;\Rightarrow\; 5\cdot 0,011^{4/3}=30\cdot V_3^{4/3}\\ + V_3^{4/3}&=4,1\cdot 10^{-4}\;\Rightarrow\;V_3=\SI{2,87e-3}{\metre\cubed} + \end{align*} + Par la loi des gaz parfaits, on en tire que~: + \[T_3=\frac{p_3\cdot V_3}{n\cdot R}=\SI{518}{\kelvin}=\SI{245}{\celsius}\] + Le travail est alors~: + \begin{align*} + A&=-\frac{i}{2}\cdot (p_3\cdot V_3-p_2\cdot V_2)\\ + &=-\frac{6}{2}\cdot (30\cdot 10^5\cdot 2,87\cdot 10^{-3}-5\cdot 10^5\cdot 0,011)\\ + &=\SI{-9330}{\joule} + \end{align*} + Avec pour échange de chaleur et variation d'énergie interne~: + \[Q=0\;\text{et}\;\Delta U=-A=\SI{9330}{\joule}\] + \end{solos} +\end{exos} + +\begin{exos} + Une machine thermique cyclique travaille avec un gaz parfait monoatomique qui subit quatre transformations. Le tableau suivant présente les échange d'énergie au cours du cycle. + \begin{center} + \begin{tabular}{|c|c|c|c|} + \hline + Transformation & \(\Delta U\) & \(A\) & \(Q\) \\ + & J & J & J \\\hline + 1 & 0 & -1109 & ? \\ + 2 & 7200 & ? & 0 \\ + 3 & ? & 4436 & 4436 \\ + 4 & ? & 7200 & ? \\ + \hline + \end{tabular} + \end{center} + \smallskip + \begin{enumerate} + \item Complétez le tableau en justifiant vos résultats. + + \item Calculez le rendement. + \item Le gaz se trouvant initialement dans l'état : \(V_0=\SI{16}{\deci\metre\cubed}\), \(p_0=\SI{1e5}{\pascal}\) et \(T_0=\SI{400}{\kelvin}\), calculez le produit \(n\cdot R\). + \item Déterminez les températures à la fin des étapes 1, 2 et 3. + \item Faites un diagramme de bilan du cycle. + \end{enumerate} + + \begin{solos} + Le tableau complété est le suivant : + \begin{center} + \begin{tabular}{|c|c|c|c|} + \hline + Transformation & \(\Delta U\) & \(A\) & \(Q\) \\ + & J & J & J \\\hline + isotherme & 0 & -1109 & -1109 \\ + adiabatique & 7200 & -7200 & 0 \\ + isotherme & 0 & 4436 & 4436 \\ + adiabatique & -7200 & 7200 & 0 \\ + \hline + \(\sum\) & 0 & 3327 & \\ + \hline + \end{tabular} + \end{center} + \smallskip + \begin{enumerate} + \item Les trois premières lignes sont justifiées par le premier principe. Pour la dernière, on a utilisé le fait que la somme des variations des énergies internes sur un cycle fermé est nulle, puisqu'on se retrouve dans l'état initial. Ainsi : + \[\sum \Delta U = 0 + 7200 + 0 -7200=0\] + \item le rendement est donné par : + \[\eta = \frac{\sum A}{\sum Q_+}=\frac{3327}{4436}=75\%\] + \item La loi des gaz parfaits donne : + \[n\cdot R=\frac{p_0\cdot V_0}{T_0}=\frac{10^5\cdot 16\cdot 10^{-3}}{400}=4\] + \item La température à l'état initial \(T_0=\SI{400}{\kelvin}\) est donnée. La première transformation étant isotherme, on a que \(T_1=\SI{400}{\kelvin}\). Pour la seconde transformation, on peut écrire : + \begin{align*} + \Delta U&=\frac{i}{2}\cdot n\cdot R\cdot \Delta T\\ + 7200&=\frac{3}{2}\cdot 4\cdot \Delta T\\ + \Delta T&=1200=T_2-T_1=T_2-400\\ + &\Rightarrow\;T_2=\SI{1600}{\kelvin} + \end{align*} + Et comme la troisième est isotherme, on a : \(T_3=\SI{1600}{\kelvin}\). \item Le diagramme de bilan est donné à la figure \ref{exos:cycle1}. + \end{enumerate} + + \begin{figure} + \def\svgwidth{7cm} + \begin{center} + %\input{Annexe-Exercices/Images/cycle2.eps_tex} + \includegraphics[scale=0.9]{cycle2.eps} + \end{center} + \caption{Bilan du cycle\label{exos:cycle1}} + \end{figure} + + \end{solos} +\end{exos} + %\begin{exos} % Un énoncé de test. % \begin{solos} diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index b2c986b..679b122 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2769,6 +2769,120 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + On fait passer \SI{32}{\gram} de méthane (\(CH_4\)) de l'état (\SI{1}{\bar} ; \SI{60}{\celsius}) à l'état (\SI{5}{\bar} ; \SI{60}{\celsius}) par une compression isotherme, puis, par une compression adiabatique, à l'état (\SI{30}{\bar} ; ?). Calculez our chaque transformations~: + \begin{enumerate} + \item le travail dépensé, + \item la chaleur échangée avec le milieu extérieur et + \item la variation d'énergie interne. + \end{enumerate} + \begin{solos} + Comme la masse atomique du carbone vaut \SI{12}{uma} et celle de l'hydrogène \SI{1}{uma}, celle de la molécule de méthane (\(CH_4\)) vaut \SI{16}{uma}. Ainsi, sa masse molaire vaut \SI{16}{\gram\per\mol}. Si on a \SI{32}{\gram} de ce gaz, on a donc \SI{2}{\mol} de \(CH_4\). + + \subsubsection*{Compression isotherme} + Avec une température de 273 + 60 = \SI{333}{\kelvin}, on peut alors calculer les volumes~: + \begin{align*} + V_1&=\frac{n\cdot R\cdot T}{p_1}=\frac{2\cdot 8,31\cdot 333}{10^5}=\SI{0,055}{\metre\cubed}\\ + V_2&=\frac{n\cdot R\cdot T}{p_2}=\frac{2\cdot 8,31\cdot 333}{5\cdot 10^5}=\SI{0,011}{\metre\cubed} + \end{align*} + Le travail isotherme se calcule alors aisément par~: + \begin{align*} + A&=n\cdot R\cdot T\cdot ln(\frac{V_2}{V_1})\\ + &=2\cdot 8,31\cdot 333\cdot ln(\frac{0,011}{0,055}=\SI{-8907}{\joule} + \end{align*} + Comme la compression est isotherme, on a aussi~: + \[\Delta U = 0\;\text{et}\;Q=A=\SI{-8907}{\joule}\] + + \subsubsection*{Compression adiabatique} + La compression adiabatique se fait de l'état 2 à l'état 3. Comme vu précédemment, le volume de l'état 2 est \(V_2=\SI{0,011}{\metre\cubed}\). + + Des propriétés de la transformation adiabatique, on tire alors~: + \begin{align*} + p_2\cdot V_2&=p_3\cdot V_3\;\Rightarrow\; 5\cdot 0,011^{4/3}=30\cdot V_3^{4/3}\\ + V_3^{4/3}&=4,1\cdot 10^{-4}\;\Rightarrow\;V_3=\SI{2,87e-3}{\metre\cubed} + \end{align*} + Par la loi des gaz parfaits, on en tire que~: + \[T_3=\frac{p_3\cdot V_3}{n\cdot R}=\SI{518}{\kelvin}=\SI{245}{\celsius}\] + Le travail est alors~: + \begin{align*} + A&=-\frac{i}{2}\cdot (p_3\cdot V_3-p_2\cdot V_2)\\ + &=-\frac{6}{2}\cdot (30\cdot 10^5\cdot 2,87\cdot 10^{-3}-5\cdot 10^5\cdot 0,011)\\ + &=\SI{-9330}{\joule} + \end{align*} + Avec pour échange de chaleur et variation d'énergie interne~: + \[Q=0\;\text{et}\;\Delta U=-A=\SI{9330}{\joule}\] + \end{solos} +\end{exos} + +\begin{exos} + Une machine thermique cyclique travaille avec un gaz parfait monoatomique qui subit quatre transformations. Le tableau suivant présente les échange d'énergie au cours du cycle. + \begin{center} + \begin{tabular}{|c|c|c|c|} + \hline + Transformation & \(\Delta U\) & \(A\) & \(Q\) \\ + & J & J & J \\\hline + 1 & 0 & -1109 & ? \\ + 2 & 7200 & ? & 0 \\ + 3 & ? & 4436 & 4436 \\ + 4 & ? & 7200 & ? \\ + \hline + \end{tabular} + \end{center} + \smallskip + \begin{enumerate} + \item Complétez le tableau en justifiant vos résultats. + + \item Calculez le rendement. + \item Le gaz se trouvant initialement dans l'état : \(V_0=\SI{16}{\deci\metre\cubed}\), \(p_0=\SI{1e5}{\pascal}\) et \(T_0=\SI{400}{\kelvin}\), calculez le produit \(n\cdot R\). + \item Déterminez les températures à la fin des étapes 1, 2 et 3. + \item Faites un diagramme de bilan du cycle. + \end{enumerate} + + \begin{solos} + Le tableau complété est le suivant : + \begin{center} + \begin{tabular}{|c|c|c|c|} + \hline + Transformation & \(\Delta U\) & \(A\) & \(Q\) \\ + & J & J & J \\\hline + isotherme & 0 & -1109 & -1109 \\ + adiabatique & 7200 & -7200 & 0 \\ + isotherme & 0 & 4436 & 4436 \\ + adiabatique & -7200 & 7200 & 0 \\ + \hline + \(\sum\) & 0 & 3327 & \\ + \hline + \end{tabular} + \end{center} + \smallskip + \begin{enumerate} + \item Les trois premières lignes sont justifiées par le premier principe. Pour la dernière, on a utilisé le fait que la somme des variations des énergies internes sur un cycle fermé est nulle, puisqu'on se retrouve dans l'état initial. Ainsi : + \[\sum \Delta U = 0 + 7200 + 0 -7200=0\] + \item le rendement est donné par : + \[\eta = \frac{\sum A}{\sum Q_+}=\frac{3327}{4436}=75\%\] + \item La loi des gaz parfaits donne : + \[n\cdot R=\frac{p_0\cdot V_0}{T_0}=\frac{10^5\cdot 16\cdot 10^{-3}}{400}=4\] + \item La température à l'état initial \(T_0=\SI{400}{\kelvin}\) est donnée. La première transformation étant isotherme, on a que \(T_1=\SI{400}{\kelvin}\). Pour la seconde transformation, on peut écrire : + \begin{align*} + \Delta U&=\frac{i}{2}\cdot n\cdot R\cdot \Delta T\\ + 7200&=\frac{3}{2}\cdot 4\cdot \Delta T\\ + \Delta T&=1200=T_2-T_1=T_2-400\\ + &\Rightarrow\;T_2=\SI{1600}{\kelvin} + \end{align*} + Et comme la troisième est isotherme, on a : \(T_3=\SI{1600}{\kelvin}\). \item Le diagramme de bilan est donné à la figure \ref{exos:cycle1}. + \end{enumerate} + + \begin{figure} + \def\svgwidth{7cm} + \begin{center} + \input{Annexe-Exercices/Images/cycle2.eps_tex} + \end{center} + \caption{Bilan du cycle\label{exos:cycle1}} + \end{figure} + + \end{solos} +\end{exos} + %\begin{exos} % Un énoncé de test. % \begin{solos} diff --git a/Annexe-Exercices/Images/cycle1.eps b/Annexe-Exercices/Images/cycle1.eps new file mode 100644 index 0000000..6c2140c --- /dev/null +++ b/Annexe-Exercices/Images/cycle1.eps @@ -0,0 +1,281 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: cairo 1.16.0 (https://cairographics.org) +%%CreationDate: Sun Nov 13 22:39:13 2022 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