From 6b3be17dde5d639951ce495521fd1f03ef4d8d86 Mon Sep 17 00:00:00 2001 From: guyotv Date: Sun, 26 Jan 2025 11:37:56 +0100 Subject: [PATCH] Report exo Diesel --- Annexe-Exercices/Annexe-Exercices.tex | 90 ++++- Annexe-Exercices/Annexe-Exercices.tex.bak | 99 +++++- Annexe-Exercices/Images/Diesel.ps_tex | 70 ++++ Annexe-Exercices/Images/Diesel.svg | 392 ++++++++++++++++++++++ Annexe-Exercices/Images/Lemoteur.svg | 28 +- CoursMecaniqueOSDF.out | 6 +- CoursMecaniqueOSDF.pdf | Bin 13487232 -> 13498846 bytes SolutionsOS.tex | 75 ++++- 8 files changed, 746 insertions(+), 14 deletions(-) create mode 100644 Annexe-Exercices/Images/Diesel.ps_tex create mode 100644 Annexe-Exercices/Images/Diesel.svg diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index c89bab9..611dcae 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2939,6 +2939,94 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + Un cycle Diesel se déroule en principe de la façon suivante. Un gaz parfait diatomique est initialement dans l'état \(p_1=\SI{1}{\bar}\), \(V_1=\SI{1000}{\centi\metre\cubed}\) et \(T_1=\SI{333,3}{\kelvin}\) et on lui fait subir quatre transformations successives~: + \begin{description} + \item[Compression adiabatique] jusqu'à l'état \(p_2=\,?\), \(V_2=\SI{70}{\centi\metre\cubed}\) et \(T_2=\SI{965,6}{\kelvin}\). + \item[Échauffement à pression constante] jusqu'à l'état \(p_3=p_2\), \(V_3=\SI{140}{\centi\metre\cubed}\) et \(T_3=\,?\). + \item[Détente adiabatique] jusqu'à l'état \(p_4=\,?\), \(V_3=V_1\) et \(T_3=\SI{879,6}{\kelvin}\). + \item[Refroidissement à volume constant] jusqu'à l'état initial. + + \smallskip + On demande de~: + \begin{enumerate} + \item Calculer nR, \(p_2\), \(T_3\) et \(p_4\). + \item Esquisser le diagramme PV du cycle. + \item Calculer pour chaque transformation \(\Delta U\), \(A\) et \(Q\). + \item Établir le bilan du cycle. + \item Calculer le rendement. + \end{enumerate} + \end{description} + \begin{solos} + Voici l'analyse du cycle. + \begin{enumerate} + \item Les grandeurs manquantes. + \begin{align*} + nR&=\frac{p_1\cdot V_1}{T_1}=\frac{10^5\cdot 1000\cdot 10^{-6}}{333,3}=0,3\\ + p_2&=\frac{nR\cdot T_2}{V_2}=\frac{0,3\cdot 965,6}{70\cdot 10^{-6}}=\SI{41,4}{\bar}\\ + T_3&=\frac{p_3\cdot V_3}{nR}=\frac{41,4\cdot 10^5\cdot 140\dot 10^{-6}}{0,3}\\ + &=\SI{1931,2}{\kelvin}\\ + p_4&=\frac{nR\cdot T_4}{V_4}=\frac{0,3\cdot 879,6}{1000\cdot 10^{-6}}=\SI{2,6}{\bar} + \end{align*} + \item Le diagramme PV est celui de la figure \ref{exos:diesel}. + + \begin{figure} + \def\svgwidth{7cm} + \begin{center} + \input{Annexe-Exercices/Images/Diesel.ps_tex} + \end{center} + \caption{Le diagramme PV\label{exos:diesel}} + \end{figure} + + \item Les grandeurs des transformations. + \begin{description} + \item[1-2] Adiabatique + \begin{align*} + \Delta U&=\frac{i}{2}\cdot \Delta (pV)\\ + &=\frac{5}{2}\cdot (41,4\cdot 10^5\cdot 70\cdot 10^{-6}\\ + &-10^5\cdot 1000\cdot 10^{-6})=\SI{474,5}{\joule}\\ + A&=-\Delta U=\SI{-474,5}{\joule}\\ + Q&=0 + \end{align*} + \item[2-3] Isobare + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T\\ + &=\frac{5}{2}\cdot 0,3\cdot (1931,2-965,6)=\SI{724,2}{\joule}\\ + A&=p_2\cdot (V_3-V_2)\\ + &=41,4\cdot 10^5\cdot (140\cdot 10^{-6}-70\cdot 10^{-6})\\ + &=\SI{289,7}{\joule}\\ + Q&=\Delta U+A=724,2+289,7=\SI{1013,9}{\joule} + \end{align*} + \item[3-4] Adiabatique + \begin{align*} + \Delta U&=\frac{i}{2}\cdot \Delta (pV)\\ + &=\frac{5}{2}\cdot (p_4\cdot V_4-p_3\cdot V_3)\\ + &=\SI{789,3}{\joule}\\ + A&=-\Delta U=\SI{-789,3}{\joule}\\ + Q&=0 + \end{align*} + \item[4-1] Isochore + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T=\frac{i}{2}\cdot nR (T_1-T_4)\\ + &=\frac{5}{2}\cdot 0,3\cdot (333,3-879,6)\\ + &=\SI{-409,7}{\joule}\\ + A&=0\\ + Q&=\Delta U=\SI{-409,7}{\joule} + \end{align*} + \end{description} + \item Le bilan du cycle. + + Chaleur reçue par le moteur~: \[\sum Q>0=\SI{1014}{\joule}\]. + + Chaleur rejetée par le moteur~: \[\sum q<0=\SI{-409,7}{\joule}\]. + + Travail fourni par le moteur~: \[\sum A=\SI{604,5}{\joule}\]. + \item Le rendement. + \[\eta=\frac{A}{\sum Q>0}=\frac{604,5}{1014}=60\%\] + \end{enumerate} + \end{solos} +\end{exos} + \begin{exos} \begin{figure}[t] \centering @@ -2947,7 +3035,7 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \def\svgwidth{0.9\columnwidth} \input{Annexe-Exercices/Images/Lemoteur.ps_tex} \end{figure} - On donne le diagramme PV d'un moteur thermique qui fonctionne avec un gaz diatomique dans le sens A → B → C. + On donne le diagramme PV (figure \ref{lemoteur}) d'un moteur thermique qui fonctionne avec un gaz diatomique dans le sens A → B → C. \begin{enumerate} \item Calculez les caractéristiques manquantes. \item Déterminez \(\Delta U\), \(A\) et \(Q\) pour chaque étape. diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index f36c193..611dcae 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2939,6 +2939,94 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + Un cycle Diesel se déroule en principe de la façon suivante. Un gaz parfait diatomique est initialement dans l'état \(p_1=\SI{1}{\bar}\), \(V_1=\SI{1000}{\centi\metre\cubed}\) et \(T_1=\SI{333,3}{\kelvin}\) et on lui fait subir quatre transformations successives~: + \begin{description} + \item[Compression adiabatique] jusqu'à l'état \(p_2=\,?\), \(V_2=\SI{70}{\centi\metre\cubed}\) et \(T_2=\SI{965,6}{\kelvin}\). + \item[Échauffement à pression constante] jusqu'à l'état \(p_3=p_2\), \(V_3=\SI{140}{\centi\metre\cubed}\) et \(T_3=\,?\). + \item[Détente adiabatique] jusqu'à l'état \(p_4=\,?\), \(V_3=V_1\) et \(T_3=\SI{879,6}{\kelvin}\). + \item[Refroidissement à volume constant] jusqu'à l'état initial. + + \smallskip + On demande de~: + \begin{enumerate} + \item Calculer nR, \(p_2\), \(T_3\) et \(p_4\). + \item Esquisser le diagramme PV du cycle. + \item Calculer pour chaque transformation \(\Delta U\), \(A\) et \(Q\). + \item Établir le bilan du cycle. + \item Calculer le rendement. + \end{enumerate} + \end{description} + \begin{solos} + Voici l'analyse du cycle. + \begin{enumerate} + \item Les grandeurs manquantes. + \begin{align*} + nR&=\frac{p_1\cdot V_1}{T_1}=\frac{10^5\cdot 1000\cdot 10^{-6}}{333,3}=0,3\\ + p_2&=\frac{nR\cdot T_2}{V_2}=\frac{0,3\cdot 965,6}{70\cdot 10^{-6}}=\SI{41,4}{\bar}\\ + T_3&=\frac{p_3\cdot V_3}{nR}=\frac{41,4\cdot 10^5\cdot 140\dot 10^{-6}}{0,3}\\ + &=\SI{1931,2}{\kelvin}\\ + p_4&=\frac{nR\cdot T_4}{V_4}=\frac{0,3\cdot 879,6}{1000\cdot 10^{-6}}=\SI{2,6}{\bar} + \end{align*} + \item Le diagramme PV est celui de la figure \ref{exos:diesel}. + + \begin{figure} + \def\svgwidth{7cm} + \begin{center} + \input{Annexe-Exercices/Images/Diesel.ps_tex} + \end{center} + \caption{Le diagramme PV\label{exos:diesel}} + \end{figure} + + \item Les grandeurs des transformations. + \begin{description} + \item[1-2] Adiabatique + \begin{align*} + \Delta U&=\frac{i}{2}\cdot \Delta (pV)\\ + &=\frac{5}{2}\cdot (41,4\cdot 10^5\cdot 70\cdot 10^{-6}\\ + &-10^5\cdot 1000\cdot 10^{-6})=\SI{474,5}{\joule}\\ + A&=-\Delta U=\SI{-474,5}{\joule}\\ + Q&=0 + \end{align*} + \item[2-3] Isobare + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T\\ + &=\frac{5}{2}\cdot 0,3\cdot (1931,2-965,6)=\SI{724,2}{\joule}\\ + A&=p_2\cdot (V_3-V_2)\\ + &=41,4\cdot 10^5\cdot (140\cdot 10^{-6}-70\cdot 10^{-6})\\ + &=\SI{289,7}{\joule}\\ + Q&=\Delta U+A=724,2+289,7=\SI{1013,9}{\joule} + \end{align*} + \item[3-4] Adiabatique + \begin{align*} + \Delta U&=\frac{i}{2}\cdot \Delta (pV)\\ + &=\frac{5}{2}\cdot (p_4\cdot V_4-p_3\cdot V_3)\\ + &=\SI{789,3}{\joule}\\ + A&=-\Delta U=\SI{-789,3}{\joule}\\ + Q&=0 + \end{align*} + \item[4-1] Isochore + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T=\frac{i}{2}\cdot nR (T_1-T_4)\\ + &=\frac{5}{2}\cdot 0,3\cdot (333,3-879,6)\\ + &=\SI{-409,7}{\joule}\\ + A&=0\\ + Q&=\Delta U=\SI{-409,7}{\joule} + \end{align*} + \end{description} + \item Le bilan du cycle. + + Chaleur reçue par le moteur~: \[\sum Q>0=\SI{1014}{\joule}\]. + + Chaleur rejetée par le moteur~: \[\sum q<0=\SI{-409,7}{\joule}\]. + + Travail fourni par le moteur~: \[\sum A=\SI{604,5}{\joule}\]. + \item Le rendement. + \[\eta=\frac{A}{\sum Q>0}=\frac{604,5}{1014}=60\%\] + \end{enumerate} + \end{solos} +\end{exos} + \begin{exos} \begin{figure}[t] \centering @@ -2947,7 +3035,7 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \def\svgwidth{0.9\columnwidth} \input{Annexe-Exercices/Images/Lemoteur.ps_tex} \end{figure} - On donne le diagramme PV d'un moteur thermique qui fonctionne avec un gaz diatomique dans le sens A → B → C. + On donne le diagramme PV (figure \ref{lemoteur}) d'un moteur thermique qui fonctionne avec un gaz diatomique dans le sens A → B → C. \begin{enumerate} \item Calculez les caractéristiques manquantes. \item Déterminez \(\Delta U\), \(A\) et \(Q\) pour chaque étape. @@ -2994,7 +3082,14 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r &=\SI{-5600}{\joule} \end{align*} \end{description} - \item + \item On peut calculer le travail total de deux manières~: + \begin{align*} + A_{tot}&=\sum A=0+3218,9-1600=\SI{1618,9}{\joule}\\ + A_{tot}&=\sum Q=4000+3218,9-5600\\ + &=\SI{1618,9}{\joule} + \end{align*} + \item La puissance est simplement~: + \[P=\frac{A}{t}=\frac{3000\cdot 1618,9}{60}=\SI{80985}{\watt}\] \end{enumerate} \end{solos} \end{exos} diff --git a/Annexe-Exercices/Images/Diesel.ps_tex b/Annexe-Exercices/Images/Diesel.ps_tex new file mode 100644 index 0000000..fdcb425 --- /dev/null +++ b/Annexe-Exercices/Images/Diesel.ps_tex @@ -0,0 +1,70 @@ +%% Creator: Inkscape 1.2.2 (b0a8486541, 2022-12-01), www.inkscape.org +%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010 +%% Accompanies image file 'Diesel.ps' (pdf, eps, ps) +%% +%% To include the image in your LaTeX document, write +%% \input{.pdf_tex} +%% instead of +%% \includegraphics{.pdf} +%% To scale the image, write +%% \def\svgwidth{} +%% \input{.pdf_tex} +%% instead of +%% \includegraphics[width=]{.pdf} +%% +%% Images 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zN#(ZNlfD*zQ5wV?XFjTqz#YS_q|VOgsje7z1hB{?6)Xu39X+;0xpa0U=xo6o+7guc zKw&ELw2QzyU}4t=rrMo7Y(YSG$O7A9ns%ihSQG^i%mZdAAp_MoE#M@au)zLnbr`pT z;IJYDW?)>6`imC)bqamYDpzRVVYh7zU?=2{ejnFlB6x!jQVBX%;-&L7FnU6?L(~u0 z`nvTb|IyisTh!_{f6|10LZ8}XY9;pemtd58ZNf$^uBZA){$t?C^+CNid95aclIJVI zjL`OY4BT%;!ElGhWbUud$!O5r`|3FF7?0z;ql>gVe+&{ZswKGA$Ko~_^N{}ln(py+=TAr|+_p=7J&?N?jE_>vdV6cit3)MHw2MlZv6KTN^ z&A`G;gf2XeK{^Tr!%r0hBYn5+)MVh)PK1iAZhb|B2->cRjR^lxs``lrmGv)FEQZYa9n_{?&A^RL|Hr35)Xht&NfRTli}2Lh9nsJ!Ah*V5`u{> z1q;-(pbyMs9ff=l0tr;33pfi_nK#4M8J7uf02sPRA<jQq(CQC$H0qbVT>%b}2B{)mF7SOX#)6~1b@DteZM_t_i40mBBkTS_)11g$Xgh2?*D z!zZCa77@$`3CwbH5d56VeH}A6!CeQuMhPV6)Q%3)d<5r|d zBe>1(iCawO#}a~f>Zg1`=%Vt@MJhaX=0 zz(+pxzW4tBw)lbP-uJ%ee*7ms^aMLgxSV)pOtWZ6EoaFFp7F0mB0; AtpET3 diff --git a/SolutionsOS.tex b/SolutionsOS.tex index 2cc69e1..efab2d3 100644 --- a/SolutionsOS.tex +++ b/SolutionsOS.tex @@ -1173,6 +1173,75 @@ \end{Solution OS} \begin{Solution OS}{38} + Voici l'analyse du cycle. + \begin{enumerate} + \item Les grandeurs manquantes. + \begin{align*} + nR&=\frac{p_1\cdot V_1}{T_1}=\frac{10^5\cdot 1000\cdot 10^{-6}}{333,3}=0,3\\ + p_2&=\frac{nR\cdot T_2}{V_2}=\frac{0,3\cdot 965,6}{70\cdot 10^{-6}}=\SI{41,4}{\bar}\\ + T_3&=\frac{p_3\cdot V_3}{nR}=\frac{41,4\cdot 10^5\cdot 140\dot 10^{-6}}{0,3}\\ + &=\SI{1931,2}{\kelvin}\\ + p_4&=\frac{nR\cdot T_4}{V_4}=\frac{0,3\cdot 879,6}{1000\cdot 10^{-6}}=\SI{2,6}{\bar} + \end{align*} + \item Le diagramme PV est celui de la figure \ref{exos:diesel}. + + \begin{figure} + \def\svgwidth{7cm} + \begin{center} + \input{Annexe-Exercices/Images/Diesel.ps_tex} + \end{center} + \caption{Le diagramme PV\label{exos:diesel}} + \end{figure} + + \item Les grandeurs des transformations. + \begin{description} + \item[1-2] Adiabatique + \begin{align*} + \Delta U&=\frac{i}{2}\cdot \Delta (pV)\\ + &=\frac{5}{2}\cdot (41,4\cdot 10^5\cdot 70\cdot 10^{-6}\\ + &-10^5\cdot 1000\cdot 10^{-6})=\SI{474,5}{\joule}\\ + A&=-\Delta U=\SI{-474,5}{\joule}\\ + Q&=0 + \end{align*} + \item[2-3] Isobare + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T\\ + &=\frac{5}{2}\cdot 0,3\cdot (1931,2-965,6)=\SI{724,2}{\joule}\\ + A&=p_2\cdot (V_3-V_2)\\ + &=41,4\cdot 10^5\cdot (140\cdot 10^{-6}-70\cdot 10^{-6})\\ + &=\SI{289,7}{\joule}\\ + Q&=\Delta U+A=724,2+289,7=\SI{1013,9}{\joule} + \end{align*} + \item[3-4] Adiabatique + \begin{align*} + \Delta U&=\frac{i}{2}\cdot \Delta (pV)\\ + &=\frac{5}{2}\cdot (p_4\cdot V_4-p_3\cdot V_3)\\ + &=\SI{789,3}{\joule}\\ + A&=-\Delta U=\SI{-789,3}{\joule}\\ + Q&=0 + \end{align*} + \item[4-1] Isochore + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T=\frac{i}{2}\cdot nR (T_1-T_4)\\ + &=\frac{5}{2}\cdot 0,3\cdot (333,3-879,6)\\ + &=\SI{-409,7}{\joule}\\ + A&=0\\ + Q&=\Delta U=\SI{-409,7}{\joule} + \end{align*} + \end{description} + \item Le bilan du cycle. + + Chaleur reçue par le moteur~: \[\sum Q>0=\SI{1014}{\joule}\]. + + Chaleur rejetée par le moteur~: \[\sum q<0=\SI{-409,7}{\joule}\]. + + Travail fourni par le moteur~: \[\sum A=\SI{604,5}{\joule}\]. + \item Le rendement. + \[\eta=\frac{A}{\sum Q>0}=\frac{604,5}{1014}=60\%\] + \end{enumerate} + +\end{Solution OS} +\begin{Solution OS}{39} Au préalable, calculons nR. \begin{equation*} nR=\frac{p_B\cdot V_B}{T_B}=\frac{10\cdot 10^5\cdot 2\cdot 10^{-3}}{1500}=1,33 @@ -1221,7 +1290,7 @@ \end{enumerate} \end{Solution OS} -\begin{Solution OS}{39} +\begin{Solution OS}{40} Le diagramme PV du cycle est présenté à la figure \ref{climatiseur}. \begin{figure}[t] \centering @@ -1341,7 +1410,7 @@ \end{enumerate} \end{Solution OS} -\begin{Solution OS}{40} +\begin{Solution OS}{41} Procédons simplement au calcul de l'incertitude absolue. \smallskip @@ -1389,7 +1458,7 @@ Cette expression est légèrement différente de la précédente. Mais, le second terme est négligeable en raison de la présence de l'incertitude sur le temps au carré. On voit ainsi qu'il est nécessaire de faire attention aux ordres de grandeurs. \end{Solution OS} -\begin{Solution OS}{41} +\begin{Solution OS}{42} \dots \end{Solution OS}