def raoult_bubble_temperature(model, mixture, pressure, psat=None):
""" Calculates the bubble temperature at a given pressure using Raoult's law
and a saturation pressure routine. Defaults to an approximate saturation
pressure calculated using the Clausius-Clapeyron equation.
Requires that the mixture have an overall composition, this is the assumed
composition of the liquid at the bubble point.
Returns a mixture with the vapor and liquid compositions at the bubble point
and all state values and departure functions completed.
This algorithm is unstable in the region close to the mixture critial point.
model: model that implements the Model interface
mixture: instance of Mixture
pressure: float in Pascals
psat: function, mixture -> saturation pressure in Pascals
"""
mixture.pressure = pressure
mixture.quality = 1.0
mixture.composition['liquid'] = mixture.composition['overall']
mixture.normalize()
if psat is None:
mixture.temperature = _rt_bt(mixture)
Ps = psatguess(mixture.critical['acentric'], mixture.reduced['temperature']) * mixture.critical['pressure']
else:
T0 = np.dot(tsatguess(mixture.critical['acentric'], mixture.reduced['pressure']) * mixture.critical['temperature'], mixture.composition['liquid'])
def temperature_loop(temperature):
mixture.temperature = temperature
Ps = psat(mixture)
return pressure - np.dot(Ps, mixture.composition['liquid'])
mixture.temperature = opt.newton(temperature_loop, T0)
Ps = psat(mixture)
mixture.composition['vapor'] = mixture.composition['liquid'] * Ps / mixture.pressure
return model(mixture)
def _rt_bt(mixture):
""" Calculate the bubble temperature using Raoult's Law"""
T0 = np.dot(tsatguess(mixture.critical['acentric'], mixture.reduced['pressure']) * mixture.critical['temperature'], mixture.composition['liquid'])
# Calculate bubble temperature
for count in range(MAX_ITER):
f0 = mixture.composition['liquid'] * mixture.critical['pressure'] * np.power(10, (7 / 3) * (mixture.critical['acentric'] + 1) * (1 - mixture.critical['temperature'] / T0))
T1 = T0 + (mixture.state['pressure'] - f0.sum()) / ((7 / 3) * np.log(10) * T0**-2 * np.sum(f0 * mixture.critical['temperature'] * (mixture.critical['acentric'] + 1)))
if np.allclose(T0, T1):
return T1
T0 = T1
raise SearchError("Bubble temperature failed to converge after {} iterations".format(MAX_ITER))
def two_loop_bubble_temperature(model, mixture, pressure, bounds=None):
""" Calculates the bubble temperature at a given pressure, uses Raoult's law
for an initial guess of the bubble temperature, unless otherwise specified.
Requires that the mixture have an overall composition, this is the assumed
composition of the liquid at the bubble point.
Returns a mixture with the vapor and liquid compositions at the bubble point
and all state values and departure functions completed.
This algorithm is unstable in the region close to the mixture critial point.
model: model that implements the Model interface
mixture: instance of Mixture
pressure: float in Pascals
bounds: tuple, boundary of temperature region to search, in Kelvin
"""
mixture.pressure = pressure
mixture.quality = 1.0
mixture.composition['liquid'] = mixture.composition['overall']
mixture.normalize()
def temperature_loop(temperature):
""" Uses fixed point iteration with Aitkens sequence acceleration to
find the vapor phase composition """
assert temperature > 0, "Temperature must be greater than zero"
mixture.temperature = temperature
lnphiL = model.logfug(mixture, 'liquid')
y0 = mixture.composition['liquid'] * psatguess(mixture.critical['acentric'], mixture.reduced['temperature']) / mixture.reduced['pressure']
for count in range(MAX_ITER):
mixture.composition['vapor'] = y0 / y0.sum()
lnphiV = model.logfug(mixture, 'vapor')
y1 = mixture.composition['liquid'] * np.exp(lnphiL - lnphiV)
mixture.composition['vapor'] = y1 / y1.sum()
lnphiV = model.logfug(mixture, 'vapor')
y2 = mixture.composition['liquid'] * np.exp(lnphiL - lnphiV)
d = y2 - 2.0 * y1 + y0
y = np.where(np.absolute(d) < 1e-16, y2, y2 - (y2 - y1)**2 / d)
if np.allclose(y, y2, atol=ERROR_TOL):
mixture.composition['vapor'] = y2 / y2.sum()
return 1 - y.sum()
y0 = y
raise SearchError("Temperature loop ({} K) failed to converge after {} iterations".format(temperature, MAX_ITER))
# Find bracket for temperature loop zero and solve for bubble temperature
if bounds is None:
# Use Raoult's Law to get an initial guess of the bubble temperature
try:
# This uses Newton's method and may fail spectacularly
t0 = _rt_bt(mixture)
except SearchError:
t0 = np.dot(tsatguess(mixture.critical['acentric'], mixture.reduced['pressure']) * mixture.critical['temperature'], mixture.composition['liquid'])
finally:
y0 = temperature_loop(t0)
dt = +0.2 * t0 if y0 > 0 else -0.2 * t0
bounds = bounds_search(temperature_loop, t0, dt)
mixture.temperature = bounds[0]
if np.all(mixture.reduced['temperature'] > Trcrit):
msg = "Temperature {} is too high, bubble point routine is unstable in this region".format(t0)
warnings.warn(msg, RuntimeWarning)
mixture.pressure = opt.brentq(temperature_loop, *bounds)
return model(mixture)