ycopin · November 15, 2020 20:18
diff --git a/Isothermes_vdWaals.svg b/Isothermes_vdWaals.svg
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diff --git a/vanDerWaals.py b/vanDerWaals.py
 #!/usr/bin/env python
 # Copyright: This document has been placed in the public domain.

 __author__ = "Yannick Copin <[email protected]>"

 import numpy as N
 import scipy.optimize as O
 import scipy.integrate as I
 import matplotlib.pyplot as P

 P.matplotlib.style.use('classic')

 def isotherme(v, t=1):
    """Equation des isothermes p=P/Pc--v=V/Vc en fonction de t=T/Tc"""

    return 8*t/(3*v - 1) - 3/v**2

 def dpdv(v, t=1):

    return -24*t/(3*v-1)**2 + 6/v**3

 def spinodale(v):
    """Equation de la spinodale: dp/dv=0"""

    #return 2*(3*v - 1)/v**3 - 3/v**2
    return 3/v**2 - 2/v**3

 def objfunc(p0, t, roots=False):

    # Solutions de p(v) = p0
    p0 = N.squeeze(p0)
    sols = N.roots([3*p0, -(p0+8*t), 9, -3])

    if not N.isreal(sols).all():
        sols = sols.real

    v1,v0,v2 = N.sort(sols)
    if roots:
        return v1,v2

    i1,err = I.quad(lambda v:isotherme(v,t=t) - p0, v1,v0)
    i2,err = I.quad(lambda v:isotherme(v,t=t) - p0, v0,v2)

    return i1 + i2

 def pSat(t):
    """Calcule de la pression de vapeur saturante a t"""

    if t > 1:
        raise ValueError("La pression de vapeur saturante est definie pour t<1")

    # Intersection de la spinodale avec l'isotherme
    sols = N.roots([4*t, -9, 6, -1])
    tmp,v1,v2 = N.sort(sols)
    p1,p2 = isotherme(N.array([v1,v2]), t=t)
    # zero de objfunc, pour p1 < p0 < p2
    try:
        psat = O.brentq(objfunc, p1, p2, args=(t,))
    except ValueError:
        psat = N.nan

    return psat

 def courbeSat(t=None):

    if t is None:
        t = N.linspace(0.85,1-1e-6)

    psats = N.array([ pSat(tt) for tt in t ])
    good = N.isfinite(psats)
    vsats = N.array([ objfunc(psat,tt, roots=True)
                      for psat,tt in zip(psats[good],t[good]) ])

    vs = N.concatenate((vsats[:,0],vsats[:,1][::-1]))
    ps = N.concatenate((psats[good],psats[good][::-1]))

    return vs,ps


 #fig,ax = P.subplots(1,1)
 fig = P.figure()
 ax = fig.add_subplot(1,1,1)

 ax.set_title("Isothermes du gaz de Van der Waals")
 ax.set_xlabel("V/Vc")
 ax.set_ylabel("P/Pc")

 v = N.linspace(0.4,5,1000)
 for t in N.linspace(0.8,1.2,21):
    ax.plot(v,isotherme(v,t), c='0.8', label="_")
 ax.plot(v,isotherme(v,0.9), label="T/Tc=0.9")
 ax.plot(v,isotherme(v,1), label="T/Tc=1")
 ax.plot(v,isotherme(v,1.1), label="T/Tc=1.1")
 ax.plot([1],[1],'go', label='_')
 ax.annotate('Point critique', (1,1), (5,5), textcoords='offset points')

 ax.plot(v,spinodale(v), 'k--', label='Spinodale')
 vsats,psats = courbeSat()
 ax.plot(vsats,psats, 'k-', label='Courbe de saturation')

 psat = pSat(0.9)
 v1,v2 = objfunc(psat,0.9,roots=True)
 ax.plot([v1,v2],[psat]*2, 'bo-', label='_')
 vs = N.linspace(v1,v2,100)
 ax.fill_between(vs,vs*0+psat,isotherme(vs,t=0.9),
                facecolor='b', edgecolor='none', alpha=0.2)

 ax.legend()
 #ax.semilogx()
 #xticks = [0.4,0.6,0.8,1,2,3,4,5]
 #ax.set(xticks=xticks, xticklabels=map(lambda v:'%.1f'%v,xticks))
 #ax.set_xlim(0.4,5)
 ax.set_xlim(0.4,3)
 ax.set_ylim(0,2.5)

 P.show()
	#!/usr/bin/env python
	# Copyright: This document has been placed in the public domain.

	__author__ = "Yannick Copin <[email protected]>"

	import numpy as N
	import scipy.optimize as O
	import scipy.integrate as I
	import matplotlib.pyplot as P

	P.matplotlib.style.use('classic')

	def isotherme(v, t=1):
	"""Equation des isothermes p=P/Pc--v=V/Vc en fonction de t=T/Tc"""

	return 8t/(3v - 1) - 3/v**2

	def dpdv(v, t=1):

	return -24t/(3v-1)2 + 6/v3

	def spinodale(v):
	"""Equation de la spinodale: dp/dv=0"""

	#return 2(3v - 1)/v3 - 3/v2
	return 3/v2 - 2/v3

	def objfunc(p0, t, roots=False):

	# Solutions de p(v) = p0
	p0 = N.squeeze(p0)
	sols = N.roots([3p0, -(p0+8t), 9, -3])

	if not N.isreal(sols).all():
	sols = sols.real

	v1,v0,v2 = N.sort(sols)
	if roots:
	return v1,v2

	i1,err = I.quad(lambda v:isotherme(v,t=t) - p0, v1,v0)
	i2,err = I.quad(lambda v:isotherme(v,t=t) - p0, v0,v2)

	return i1 + i2

	def pSat(t):
	"""Calcule de la pression de vapeur saturante a t"""

	if t > 1:
	raise ValueError("La pression de vapeur saturante est definie pour t<1")

	# Intersection de la spinodale avec l'isotherme
	sols = N.roots([4*t, -9, 6, -1])
	tmp,v1,v2 = N.sort(sols)
	p1,p2 = isotherme(N.array([v1,v2]), t=t)
	# zero de objfunc, pour p1 < p0 < p2
	try:
	psat = O.brentq(objfunc, p1, p2, args=(t,))
	except ValueError:
	psat = N.nan

	return psat

	def courbeSat(t=None):

	if t is None:
	t = N.linspace(0.85,1-1e-6)

	psats = N.array([ pSat(tt) for tt in t ])
	good = N.isfinite(psats)
	vsats = N.array([ objfunc(psat,tt, roots=True)
	for psat,tt in zip(psats[good],t[good]) ])

	vs = N.concatenate((vsats[:,0],vsats[:,1][::-1]))
	ps = N.concatenate((psats[good],psats[good][::-1]))

	return vs,ps


	#fig,ax = P.subplots(1,1)
	fig = P.figure()
	ax = fig.add_subplot(1,1,1)

	ax.set_title("Isothermes du gaz de Van der Waals")
	ax.set_xlabel("V/Vc")
	ax.set_ylabel("P/Pc")

	v = N.linspace(0.4,5,1000)
	for t in N.linspace(0.8,1.2,21):
	ax.plot(v,isotherme(v,t), c='0.8', label="_")
	ax.plot(v,isotherme(v,0.9), label="T/Tc=0.9")
	ax.plot(v,isotherme(v,1), label="T/Tc=1")
	ax.plot(v,isotherme(v,1.1), label="T/Tc=1.1")
	ax.plot([1],[1],'go', label='_')
	ax.annotate('Point critique', (1,1), (5,5), textcoords='offset points')

	ax.plot(v,spinodale(v), 'k--', label='Spinodale')
	vsats,psats = courbeSat()
	ax.plot(vsats,psats, 'k-', label='Courbe de saturation')

	psat = pSat(0.9)
	v1,v2 = objfunc(psat,0.9,roots=True)
	ax.plot([v1,v2],[psat]*2, 'bo-', label='_')
	vs = N.linspace(v1,v2,100)
	ax.fill_between(vs,vs*0+psat,isotherme(vs,t=0.9),
	facecolor='b', edgecolor='none', alpha=0.2)

	ax.legend()
	#ax.semilogx()
	#xticks = [0.4,0.6,0.8,1,2,3,4,5]
	#ax.set(xticks=xticks, xticklabels=map(lambda v:'%.1f'%v,xticks))
	#ax.set_xlim(0.4,5)
	ax.set_xlim(0.4,3)
	ax.set_ylim(0,2.5)

	P.show()