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# -*- coding: utf-8 -*-#Nom du programme : VanDerWaals#Auteurs : Fran?§ois L??vrier, Emmanuel Baudin, Arnaud Raoux et la pr??pa agreg de Montrouge#Adresse : Departement de physique de l'Ecole Normale Superieure#24 rue Lhomond#75005 Paris#Contact : arnaud.raoux@ens.fr##Ann??e de cr??ation : 2016#Version : 1.10#Liste des modifications#v 1.00 : 2016-05-02 Premi?¨re version compl?¨te#v 1.10 : 2016-05-02 Mise ?? jour de la mise en page#LICENCE#Cette oeuvre, cr??ation, site ou texte est sous licence Creative Commons Attribution - Pas d'Utilisation Commerciale 4.0 International. Pour acc??der ?? une copie de cette licence, merci de vous rendre ?? l'adresse suivante ou envoyez un courrier ?? Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.#Description :#Ce programme repr??sente le diagramme PV en unit??s r??duite d'un fluide respectant l'??quation d'??tat de Van der Waals. les r??f??rences de temp??rature, pression et volume sont prises au point critique. Le diagramme trace les courbes isothermes ains que les courbes spinodales et courbes de saturation.#import des biblioth?¨ques pythonimport mathimport numpy as npimport scipyimport scipy.integrateimport matplotlib.pyplot as pltfrom matplotlib.widgets import Slider, Button, RadioButtons, CheckButtonsfrom matplotlib import animation# On charge les tableaux precalcules qui contiennent les donnees pour la courbe spinodale et la courbe de 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Creation de la figurefig, ax = plt.subplots()plt.subplots_adjust(left=0.15, bottom=0.25)# On trace la courbe spinodaleplt.plot(v_spin,p_spin,'C7--',lw=2)# On trace la courbe de saturationplt.plot(v_sat,p_sat,'C0--',lw=2)# LimitesVr_min=0.4Vr_max=3.0Pr_min=0.0Pr_max=2.5# Creation de l'axe des abscisses, ici le volume# On utilise le meme que pour la courbe de saturation, pour simplifier la recherche des points de croisementVr=np.array(v_sat)# Parametres de la fonction, avec des valeurs par defautT0 = 0.9Tr=T0# Equation de van der Waals en variables reduitesdef VdW_Pr(Tr,Vr): return((8.0*Tr/(3.0*Vr-1.0))-3/np.power(Vr,2.0))# Fonction de calcul de la spinodaledef spinodale(Tr): local_v_spin=[] local_p_spin=[] # On calcule la position des zeros de la derivee de p par rapport a v (tangentes horizontales) null_dpdv=np.roots([4.0*Tr,-9.0,6.0,-1.0]) # On ne garde que les valeurs v>1/3, etant donnee que c'est une limite inferieure pour le volume reduit null_dpdv=null_dpdv[np.where(null_dpdv>1./3.)] # On calcule la pression correspondant a chacun de ces points, et si elle est positive, on trace le point dans le plan (Vr,Pr) for v in null_dpdv: p=VdW_Pr(Tr,v) if (p>0.0): local_v_spin.append(v) local_p_spin.append(p) return(local_v_spin,local_p_spin)# Fonction de calcul de la courbe de saturation# Pour calculer la courbe de saturation, on suit la construction de Maxwell# On explore un domaine de pression "raisonnable" entre les deux pressions correspondant aux tangentes horizontalesdef saturation(Tr): # On calcule la pression reduite Pr=VdW_Pr(Tr,Vr) # On determine le domaine ou cette pression est inferieure a la pression de la courbe de saturation # On prendra garde, avant d'appeler cette fonction, a verifier que Tr<1 w=np.where(Pr<np.array(p_sat)) # On en tire les valeurs extremes, qui sont donc les volumes reduits correspondant aux croisements de l'isotherme avec la courbe de saturation V0=Vr[w[0][0]] V1=Vr[w[0][len(w[0])-1]] # On calcule les pressions correspondantes P0=VdW_Pr(Tr,V0) P1=VdW_Pr(Tr,V1) return([V0,V1],[P0,P1])# Creation de la fonction a tracerif Tr > 1: Pr = VdW_Pr(Tr,Vr) l, = plt.plot(Vr,Pr, lw=4, color='C0')else : local_v_sat,local_p_sat=saturation(Tr) Vr_temp = np.concatenate((Vr[Vr<local_v_sat[0]], Vr[Vr>local_v_sat[1]])) Pr = VdW_Pr(Tr, Vr_temp) l, = plt.plot(Vr_temp,Pr, lw=4, color='C0') Vinstabl = np.linspace(local_v_sat[0], local_v_sat[1], 500) Pinstabl = VdW_Pr(Tr, Vinstabl) linstabl, = plt.plot(Vinstabl, Pinstabl, 'C3')plt.legend(["Courbe spinodale","Courbe de saturation","Isotherme"])# Calcul et affichage du point courant sur la courbe spinodalelocal_v_spin,local_p_spin=spinodale(Tr)lspin,=plt.plot(local_v_spin,local_p_spin,'C7o')# Calcul et affichage du point courant sur la courbe de saturationlocal_v_sat,local_p_sat=saturation(Tr)lsat,=plt.plot(local_v_sat,local_p_sat,'C0o',ls='-',lw=4)# Specification des limites des axes (xmin,xmax,ymin,ymax)plt.axis([Vr_min,Vr_max,Pr_min,Pr_max])# Affichage du point critiqueplt.plot(1.0,1.0,'C2o')# Affichage des axesplt.xlabel(r"Volume reduit $V_r$",fontsize=20)plt.ylabel(r"Pression reduite $P_r$"+"\n",fontsize=20)plt.title('Transition liquide-vapeur pour un fluide de Van der Waals')for tick in ax.xaxis.get_major_ticks():tick.label.set_fontsize(15)for tick in ax.yaxis.get_major_ticks():tick.label.set_fontsize(15)# Creation des barres de modification amplitude et frequenceaxcolor = 'white'axT = plt.axes([0.25, 0.07, 0.65, 0.03], facecolor=axcolor)sT = Slider(axT, 'Temperature reduite $T_r$', 0.85,1.15, valinit=T0) # Remarquer la valeur initiale f0# Fonction de mise a jour du graphiquedef update(val): Tr = sT.val # On recupere la valeur de la barre sfreq comme frequence if Tr > 1: Pr = VdW_Pr(Tr,Vr) l.set_xdata(Vr) l.set_ydata(Pr) lspin.set_visible(False) lsat.set_visible(False) linstabl.set_visible(False) else : local_v_sat,local_p_sat=saturation(Tr) Vr_temp = np.concatenate((Vr[Vr<local_v_sat[0]], Vr[Vr>local_v_sat[1]])) Pr = VdW_Pr(Tr, Vr_temp) l.set_xdata(Vr_temp) # On met a jour l'objet 'l' avec ces nouvelles valeurs l.set_ydata(Pr) local_v_spin,local_p_spin=spinodale(Tr) lspin.set_xdata(local_v_spin) lspin.set_ydata(local_p_spin) local_v_sat,local_p_sat=saturation(Tr) lsat.set_xdata(local_v_sat) lsat.set_ydata(local_p_sat) lspin.set_visible(True) lsat.set_visible(True) linstabl.set_visible(True) Vinstabl = np.linspace(local_v_sat[0], local_v_sat[1], 500) Pinstabl = VdW_Pr(Tr, Vinstabl) linstabl.set_xdata(Vinstabl) linstabl.set_ydata(Pinstabl) #Pr=VdW_Pr(Tr,Vr)sT.on_changed(update) # lorsque la barre sT est modifiee, on applique la fonction update# Creation du bouton de "reset"resetax = plt.axes([0.8, 0.015, 0.1, 0.04])button = Button(resetax, 'Reset', color=axcolor, hovercolor='0.975')# Definition de la fonction de "reset" (valeurs par defaut)def reset(event): sT.reset() # La methode .reset() appliquee a la barre sT lui redonne sa valeur valinit, soit T0button.on_clicked(reset) # Lorsqu'on clique sur "reset", on applique la fonction reset definie au dessusplt.show() # On provoque l'affichage a l'ecran ................
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