def load_titer(self, titer):
"""
Load titer specification
Parameters
----------
titer : float
Titer for fermentors in g products / L effluent.
Notes
-----
Substrate concentration in bioreactor feed is adjusted to satisfy this
specification.
Warnings
--------
Changing the titer affects the productivity.
"""
f = self._titer_objective_function
feed = self.feed
feed.imol[self.products] = 0.
substates = feed.imass[self.substrates].sum()
self.titer = titer
try:
flx.aitken_secant(f, substates, ytol=1e-5)
except:
flx.IQ_interpolation(f, 1e-12, 0.50 * feed.F_mass, ytol=1e-5, maxiter=100)
self.reactor.tau = titer / self.productivity
def solve_payment(payment, loan, interest, years):
principal = initial_loan_principal(loan, interest)
payment = flx.aitken_secant(final_loan_principal,
payment, payment+10., 1., 10.,
args=(principal, interest, years),
maxiter=200, checkiter=False)
return payment
def solve_incentive(self):
"""
Return the required incentive (USD) to reach the break even point (NPV = 0)
through cash flow analysis.
"""
discount_factors = (1 + self.IRR)**self._get_duration_array()
sales_coefficients = np.ones_like(discount_factors)
start = self._start
sales_coefficients[:start] = 0
w0 = self._startup_time
sales_coefficients[self._start] = w0 * self.startup_VOCfrac + (1 - w0)
sales = self._sales
if not sales or np.isnan(sales): sales = 0.
taxable_cashflow, nontaxable_cashflow = self._taxable_and_nontaxable_cashflow_arrays(
)
args = (taxable_cashflow, nontaxable_cashflow, sales_coefficients,
discount_factors)
sales = flx.aitken_secant(self._NPV_with_sales,
sales,
1.0001 * sales + 1e-4,
xtol=1e-6,
ytol=10.,
maxiter=300,
args=args,
checkiter=False)
self._sales = sales
return sales
def solve_incentive(self, TEA=None):
"""
Return the required incentive (USD) to reach the break even point (NPV = 0)
through cash flow analysis.
Parameters
----------
TEA=None : TEA, optional
Incentive will be added using settings from given TEA. Defaults to
first TEA object in the `TEAs` attribute.
"""
IRR = self.IRR
if not TEA: TEA = self.TEAs[0]
discount_factors = (1. + IRR)**TEA._get_duration_array()
sales_coefficients = np.ones_like(discount_factors)
start = TEA._start
sales_coefficients[:start] = 0
w0 = TEA._startup_time
sales_coefficients[TEA._start] = w0 * TEA.startup_VOCfrac + (1 - w0)
sales = self._sales
if not sales or np.isnan(sales): sales = 0.
taxable_cashflow, nontaxable_cashflow = self.taxable_and_nontaxable_cashflow_arrays
args = (self.income_tax, taxable_cashflow, nontaxable_cashflow,
sales_coefficients, discount_factors)
sales = flx.aitken_secant(NPV_with_sales,
sales,
1.0001 * sales + 1e-4,
xtol=1e-6,
ytol=10.,
maxiter=300,
args=args,
checkiter=False)
self._sales = sales
return sales
def Tmax_Lakshmi_Prasad(MW):
"""
Returns the maximum temperature at which the Lakshmi Prasad method is
valid.
"""
T_max = flx.aitken_secant(Lakshmi_Prasad.function, 298.15, args=(MW, ))
return T_max - 10 # As an extra precaution
def solve_price(self, stream):
"""Return the price (USD/kg) of stream at the break even point (NPV = 0) through cash flow analysis.
Parameters
----------
stream : :class:`~thermosteam.Stream`
Stream with variable selling price.
"""
price2cost = self._price2cost(stream)
cashflow = self.cashflow
discount_factors = (1 + self.IRR)**self._duration_array
NPV = (cashflow / discount_factors).sum()
coefficients = np.ones_like(discount_factors)
start = self._start
coefficients[:start] = 0
w0 = self._startup_time
coefficients[self._start] = w0 * self.startup_VOCfrac + (1 - w0)
args = (NPV, coefficients, discount_factors)
sales = self._sales or stream.price * price2cost
sales = flx.aitken_secant(NPV_with_sales,
sales,
1.0001 * sales + 1e-4,
ytol=10.,
maxiter=200,
args=args,
checkiter=False)
self._sales = sales
if stream.sink:
return stream.price - sales / price2cost
elif stream.source:
return stream.price + sales / price2cost
else:
raise ValueError("stream must be either a feed or a product")
def solve_Px(self, z, T):
"""
Dew point given composition and temperature.
Parameters
----------
z : ndarray
Molar composition.
T : float
Temperature (K).
Returns
-------
P : float
Dew point pressure (Pa).
x : ndarray
Liquid phase molar composition.
Examples
--------
>>> import thermosteam as tmo
>>> import numpy as np
>>> chemicals = tmo.Chemicals(['Water', 'Ethanol'], cache=True)
>>> tmo.settings.set_thermo(chemicals)
>>> DP = tmo.equilibrium.DewPoint(chemicals)
>>> DP.solve_Px(z=np.array([0.5, 0.5]), T=352.28)
(82444.29876, array([0.853, 0.147]))
"""
z_norm = z / z.sum()
Psats = array([i(T) for i in self.Psats], dtype=float)
z_over_Psats = z / Psats
args = (T, z_norm, z_over_Psats)
self.T = T
f = self._P_error
P_guess = self.P or self._P_ideal(z_over_Psats)
try:
P = flx.aitken_secant(f,
P_guess,
P_guess - 10,
1e-3,
5e-12,
args,
checkiter=False)
except (InfeasibleRegion, DomainError):
Pmin = self.Pmin
Pmax = self.Pmax
P = flx.IQ_interpolation(f,
Pmin,
Pmax,
f(Pmin, *args),
f(Pmax, *args),
P_guess,
1e-3,
5e-12,
args,
checkiter=False,
checkbounds=False)
self.x = fn.normalize(self.x)
return P, self.x.copy()
def solve_sales(self):
"""
Return the required additional sales (USD) to reach the breakeven
point (NPV = 0) through cash flow analysis.
"""
discount_factors = (1 + self.IRR)**self._get_duration_array()
sales_coefficients = np.ones_like(discount_factors)
start = self._start
sales_coefficients[:start] = 0
w0 = self._startup_time
sales_coefficients[self._start] = w0 * self.startup_VOCfrac + (1 - w0)
sales = self._sales
if not np.isfinite(sales): sales = 0.
taxable_cashflow, nontaxable_cashflow = self._taxable_and_nontaxable_cashflow_arrays(
)
args = (taxable_cashflow, nontaxable_cashflow, sales_coefficients,
discount_factors, self._fill_tax_and_incentives)
x0 = sales
x1 = 1.01 * sales + 10
f = NPV_with_sales
try:
sales = flx.aitken_secant(f,
x0,
x1,
xtol=10,
ytol=1000.,
maxiter=1000,
args=args,
checkiter=True)
except Exception as e:
y0 = f(x0, *args)
y1 = f(x1, *args)
if y0 == y1 and y0 < 0.:
x0 = -y1 / self._years
else:
raise e
sales = flx.aitken_secant(f,
x0,
x1,
xtol=10,
ytol=1000.,
maxiter=1000,
args=args,
checkiter=True)
self._sales = sales
return sales
def solve_Tx(self, z, P):
"""
Dew point given composition and pressure.
Parameters
----------
z : ndarray
Molar composition.
P : float
Pressure [Pa].
Returns
-------
T : float
Dew point temperature [K].
x : numpy.ndarray
Liquid phase molar composition.
Examples
--------
>>> import thermosteam as tmo
>>> import numpy as np
>>> chemicals = tmo.Chemicals(['Water', 'Ethanol'], cache=True)
>>> tmo.settings.set_thermo(chemicals)
>>> DP = tmo.equilibrium.DewPoint(chemicals)
>>> DP.solve_Tx(z=np.array([0.5, 0.5]), P=101325)
(357.451847, array([0.849, 0.151]))
"""
f = self._T_error
z_norm = z / z.sum()
zP = z * P
args = (P, z_norm, zP)
self.P = P
T_guess = self.T or self._T_ideal(zP)
try:
T = flx.aitken_secant(f,
T_guess,
T_guess + 1e-3,
1e-9,
5e-12,
args,
checkiter=False)
except (InfeasibleRegion, DomainError):
Tmin = self.Tmin
Tmax = self.Tmax
T = flx.IQ_interpolation(f,
Tmin,
Tmax,
f(Tmin, *args),
f(Tmax, *args),
T_guess,
1e-9,
5e-12,
args,
checkiter=False,
checkbounds=False)
self.x = fn.normalize(self.x)
return T, self.x.copy()
def _run(self):
steam = self.ins[1]
steam_mol = steam.F_mol
steam_mol = flx.aitken_secant(self.pressure_objective_function,
steam_mol,
steam_mol + 0.1,
1e-4,
1e-4,
checkroot=False)
def solve_IRR(self):
"""Return the IRR at the break even point (NPV = 0) through cash flow analysis."""
IRR = self._IRR
if not IRR or np.isnan(IRR) or IRR < 0.: IRR = self.IRR
if not IRR or np.isnan(IRR) or IRR < 0.: IRR = 0.10
args = (self.cashflow_array, self._get_duration_array())
self._IRR = flx.aitken_secant(NPV_at_IRR,
IRR, 1.0001 * IRR + 1e-3, xtol=1e-6, ytol=10.,
maxiter=200, args=args, checkiter=False)
return self._IRR
def correct_mass_balance(self, variable=None):
"""
Make sure mass is not created or destroyed by varying the
reactant stoichiometric coefficient.
"""
if variable:
index = self.chemicals.get_index(variable)
else:
index = self._X_index
stoichiometry_by_wt = self._get_stoichiometry_by_wt()
def f(x):
stoichiometry_by_wt[index] = x
return stoichiometry_by_wt.sum()
flx.aitken_secant(f, 1)
if self._basis == 'mol':
self._stoichiometry[:] = stoichiometry_by_wt / self.MWs
self._rescale()
def adjust_F402_V():
H2O_molfrac = D404_P.outs[0].get_molar_composition('H2O')
V0 = H2O_molfrac
F402.V = aitken_secant(f=purity_at_V,
x0=V0,
x1=V0 + 0.001,
xtol=0.001,
ytol=0.001,
maxiter=50,
args=())
def _run(self):
feed, steam = self._ins
steam_mol = steam.F_mol
mixed = self.outs[0]
steam_mol = aitken_secant(self._P_at_flow,
steam_mol, steam_mol+0.1,
1e-4, 1e-4,
args=(self.P, steam, mixed, feed))
mixed.P = self.P
hu = self.heat_utilities[0]
hu(steam.Hvap, mixed.T)
def solve_IRR(self):
"""Return the IRR at the break even point (NPV = 0) through cash flow analysis."""
IRR = self._IRR
args = (self.cashflow, self._duration_array)
IRR = flx.aitken_secant(NPV_at_IRR,
IRR,
1.0001 * IRR + 1e-3,
xtol=1e-6,
ytol=10.,
maxiter=200,
args=args,
checkiter=False)
self._IRR = IRR
return IRR
def solve_IRR(self):
"""Return the IRR at the break even point (NPV = 0) through cash flow analysis."""
IRR = self._IRR
args = (tuple([i.cashflow for i in self.TEAs]),
tuple([i._duration_array for i in self.TEAs]))
self._IRR = flx.aitken_secant(sum_NPV_at_IRR,
IRR,
1.0001 * IRR + 1e-3,
xtol=1e-6,
ytol=10.,
maxiter=200,
args=args,
checkiter=False)
return self._IRR
def solve_Py(self, z, T):
"""
Bubble point at given composition and temperature.
Parameters
----------
z : ndarray
Molar composotion.
T : float
Temperature [K].
Returns
-------
P : float
Bubble point pressure [Pa].
y : ndarray
Vapor phase molar composition.
Examples
--------
>>> import thermosteam as tmo
>>> import numpy as np
>>> chemicals = tmo.Chemicals(['Water', 'Ethanol'], cache=True)
>>> tmo.settings.set_thermo(chemicals)
>>> BP = tmo.equilibrium.BubblePoint(chemicals)
>>> BP.solve_Py(z=np.array([0.703, 0.297]), T=352.28)
(91830.9798, array([0.419, 0.581]))
"""
self.T = T
Psat = array([i(T) for i in self.Psats])
z_norm = z / z.sum()
z_Psat_gamma_pcf = z * Psat * self.gamma(z_norm, T) * self.pcf(z_norm, T)
f = self._P_error
args = (T, z_Psat_gamma_pcf)
P_guess = self.P or self._P_ideal(z_Psat_gamma_pcf)
try:
P = flx.aitken_secant(f, P_guess, P_guess-1, 1e-3, 1e-9,
args, checkiter=False)
except (InfeasibleRegion, DomainError):
Pmin = self.Pmin; Pmax = self.Pmax
P = flx.IQ_interpolation(f, Pmin, Pmax,
f(Pmin, *args), f(Pmax, *args),
P_guess, 1e-3, 5e-12, args,
checkiter=False, checkbounds=False)
self.y = fn.normalize(self.y)
return P, self.y.copy()
def solve_Ty(self, z, P):
"""
Bubble point at given composition and pressure.
Parameters
----------
z : ndarray
Molar composotion.
P : float
Pressure [Pa].
Returns
-------
T : float
Bubble point temperature [K].
y : ndarray
Vapor phase molar composition.
Examples
--------
>>> import thermosteam as tmo
>>> import numpy as np
>>> chemicals = tmo.Chemicals(['Water', 'Ethanol'], cache=True)
>>> tmo.settings.set_thermo(chemicals)
>>> BP = tmo.equilibrium.BubblePoint(chemicals)
>>> BP.solve_Ty(z=np.array([0.6, 0.4]), P=101325)
(353.7543, array([0.381, 0.619]))
"""
self.P = P
f = self._T_error
z_norm = z / z.sum()
z_over_P = z_norm/P
args = (P, z_over_P, z_norm)
T_guess = self.T or self._T_ideal(z_over_P)
try:
T = flx.aitken_secant(f, T_guess, T_guess + 1e-3,
1e-9, 5e-12, args,
checkiter=False)
except (InfeasibleRegion, DomainError):
Tmin = self.Tmin; Tmax = self.Tmax
T = flx.IQ_interpolation(f, Tmin, Tmax,
f(Tmin, *args), f(Tmax, *args),
T_guess, 1e-9, 5e-12, args,
checkiter=False, checkbounds=False)
self.y = fn.normalize(self.y)
return T, self.y.copy()
def _run(self):
ins = self._ins
steam = ins[1]
steam_mol = steam.molnet
mixed = self.outs[0]
index_water = mixed.index('7732-18-5')
steam_mol = aitken_secant(self._P_at_flow,
steam_mol,
steam_mol + 0.1,
1e-4,
1e-4,
args=(self.P, steam.mol, index_water, mixed,
ins))
mixed.P = self.P
hu = self._heat_utilities[0]
hu.ID = 'Low pressure steam'
hu.flow = steam_mol
hu.cost = steam_mol * bst.HeatUtility.heating_agents[
'Low pressure steam'].price_kmol
def solve_price(self, stream, TEA=None):
"""Return the price (USD/kg) of stream at the break even point (NPV = 0) through cash flow analysis.
Parameters
----------
stream : :class:`~thermosteam.Stream`
Stream with variable selling price.
TEA : TEA, optional
Stream should belong here.
"""
TEAs = self.TEAs
if not TEA:
for TEA in TEAs:
if stream in TEA.system.feeds or stream in TEA.system.products:
break
price2cost = TEA._price2cost(stream)
IRR = self.IRR
NPV = self.NPV
discount_factors = (1. + IRR)**TEA._duration_array
coefficients = np.ones_like(discount_factors)
start = TEA._start
coefficients[:start] = 0
w0 = TEA._startup_time
coefficients[TEA._start] = w0 * TEA.startup_VOCfrac + (1 - w0)
args = (NPV, coefficients, discount_factors)
sales = self._sales or stream.price * price2cost
sales = flx.aitken_secant(NPV_with_sales,
sales,
1.0001 * sales + 1e-4,
ytol=10.,
maxiter=200,
args=args,
checkiter=False)
self._sales = sales
if stream.sink:
return stream.price - sales / price2cost
elif stream.source:
return stream.price + sales / price2cost
else:
raise ValueError("stream must be either a feed or a product")
def solve_Ty(self, z, P):
"""
Bubble point at given composition and pressure.
Parameters
----------
z : ndarray
Molar composition.
P : float
Pressure [Pa].
Returns
-------
T : float
Bubble point temperature [K].
y : ndarray
Vapor phase molar composition.
Examples
--------
>>> import thermosteam as tmo
>>> import numpy as np
>>> chemicals = tmo.Chemicals(['Water', 'Ethanol'], cache=True)
>>> tmo.settings.set_thermo(chemicals)
>>> BP = tmo.equilibrium.BubblePoint(chemicals)
>>> BP.solve_Ty(z=np.array([0.6, 0.4]), P=101325)
(353.7543, array([0.381, 0.619]))
"""
positives = z > 0.
N = positives.sum()
if N == 0:
raise ValueError('no components present')
if N == 1:
T = self.chemicals[fn.first_true_index(positives)].Tsat(P)
y = z.copy()
else:
if P > self.Pmax: P = self.Pmax
elif P < self.Pmin: P = self.Pmin
f = self._T_error
z_norm = z / z.sum()
z_over_P = z / P
T_guess, y = self._Ty_ideal(z_over_P)
args = (P, z_over_P, z_norm, y)
try:
T = flx.aitken_secant(f,
T_guess,
T_guess + 1e-3,
1e-9,
5e-12,
args,
checkiter=False)
except (InfeasibleRegion, DomainError):
Tmin = self.Tmin
Tmax = self.Tmax
T = flx.IQ_interpolation(f,
Tmin,
Tmax,
f(Tmin, *args),
f(Tmax, *args),
T_guess,
1e-9,
5e-12,
args,
checkiter=False,
checkbounds=False)
return T, fn.normalize(y)
def solve_Tx(self, z, P):
"""
Dew point given composition and pressure.
Parameters
----------
z : ndarray
Molar composition.
P : float
Pressure [Pa].
Returns
-------
T : float
Dew point temperature [K].
x : numpy.ndarray
Liquid phase molar composition.
Examples
--------
>>> import thermosteam as tmo
>>> import numpy as np
>>> chemicals = tmo.Chemicals(['Water', 'Ethanol'], cache=True)
>>> tmo.settings.set_thermo(chemicals)
>>> DP = tmo.equilibrium.DewPoint(chemicals)
>>> DP.solve_Tx(z=np.array([0.5, 0.5]), P=101325)
(357.451847, array([0.849, 0.151]))
"""
positives = z > 0.
N = positives.sum()
if N == 0:
raise ValueError('no positive components present')
if N == 1:
T = self.chemicals[fn.first_true_index(positives)].Tsat(P)
x = z.copy()
else:
if P > self.Pmax: P = self.Pmax
elif P < self.Pmin: P = self.Pmin
f = self._T_error
z_norm = z / z.sum()
zP = z * P
T_guess, x = self._Tx_ideal(zP)
args = (P, z_norm, zP, x)
try:
T = flx.aitken_secant(f,
T_guess,
T_guess + 1e-3,
1e-9,
5e-12,
args,
checkiter=False)
except (InfeasibleRegion, DomainError):
Tmin = self.Tmin
Tmax = self.Tmax
T = flx.IQ_interpolation(f,
Tmin,
Tmax,
f(Tmin, *args),
f(Tmax, *args),
T_guess,
1e-9,
5e-12,
args,
checkiter=False,
checkbounds=False)
return T, fn.normalize(x)
def solve_Px(self, z, T):
"""
Dew point given composition and temperature.
Parameters
----------
z : ndarray
Molar composition.
T : float
Temperature (K).
Returns
-------
P : float
Dew point pressure (Pa).
x : ndarray
Liquid phase molar composition.
Examples
--------
>>> import thermosteam as tmo
>>> import numpy as np
>>> chemicals = tmo.Chemicals(['Water', 'Ethanol'], cache=True)
>>> tmo.settings.set_thermo(chemicals)
>>> DP = tmo.equilibrium.DewPoint(chemicals)
>>> DP.solve_Px(z=np.array([0.5, 0.5]), T=352.28)
(82444.29876, array([0.853, 0.147]))
"""
positives = z > 0.
N = positives.sum()
if N == 0:
raise ValueError('no positive components present')
if N == 1:
P = self.chemicals[fn.first_true_index(z)].Psat(T)
x = z.copy()
else:
if T > self.Tmax: T = self.Tmax
elif T < self.Tmin: T = self.Tmin
z_norm = z / z.sum()
Psats = np.array([i(T) for i in self.Psats], dtype=float)
z_over_Psats = z / Psats
P_guess, x = self._Px_ideal(z_over_Psats)
args = (T, z_norm, z_over_Psats, x)
f = self._P_error
try:
P = flx.aitken_secant(f,
P_guess,
P_guess - 10,
1e-3,
5e-12,
args,
checkiter=False)
except (InfeasibleRegion, DomainError):
Pmin = self.Pmin
Pmax = self.Pmax
P = flx.IQ_interpolation(f,
Pmin,
Pmax,
f(Pmin, *args),
f(Pmax, *args),
P_guess,
1e-3,
5e-12,
args,
checkiter=False,
checkbounds=False)
return P, fn.normalize(x)
x0, x1 = [-5, 5]
f = lambda x: x**3 - 40 + 2*x
p = flx.Profiler(f)
x_brentq = opt.brentq(p, x0, x1, xtol=1e-8)
p.archive('[Scipy] Brent-Q') # Save/archive results with given name
x_brenth = opt.brenth(p, x0, x1)
p.archive('[Scipy] Brent-H')
x_IQ = flx.IQ_interpolation(p, x0, x1)
p.archive('IQ-interpolation')
x_false_position = flx.false_position(p, x0, x1)
p.archive('False position')
p.plot(r'$f(x) = 0 = x^3 + 2 \cdot x - 40$ where $-5 < x < 5$')
p = flx.Profiler(f)
x_guess = -5
x_aitken_secant = flx.aitken_secant(p, x_guess)
p.archive('Aitken')
x_secant = flx.secant(p, x_guess)
p.archive('Secant')
x_newton = opt.newton(p, x_guess)
p.archive('[Scipy] Newton')
p.plot(r'$f(x) = 0 = x^3 + 2 \cdot x - 40$')
# Note that x = 40/x^2 - 2/x is the same
# objective function as x**3 - 40 + 2*x = 0
f = lambda x: 40/x**2 - 2/x
p = flx.Profiler(f)
x_guess = 5.
x_wegstein = flx.wegstein(p, x_guess)
p.archive('Wegstein')
x_aitken = flx.aitken(p, x_guess)
def solve_Py(self, z, T):
"""
Bubble point at given composition and temperature.
Parameters
----------
z : ndarray
Molar composition.
T : float
Temperature [K].
Returns
-------
P : float
Bubble point pressure [Pa].
y : ndarray
Vapor phase molar composition.
Examples
--------
>>> import thermosteam as tmo
>>> import numpy as np
>>> chemicals = tmo.Chemicals(['Water', 'Ethanol'], cache=True)
>>> tmo.settings.set_thermo(chemicals)
>>> BP = tmo.equilibrium.BubblePoint(chemicals)
>>> BP.solve_Py(z=np.array([0.703, 0.297]), T=352.28)
(91830.9798, array([0.419, 0.581]))
"""
positives = z > 0.
N = positives.sum()
if N == 0:
raise ValueError('no components present')
if N == 1:
P = self.chemicals[fn.first_true_index(positives)].Psat(T)
y = z.copy()
else:
if T > self.Tmax: T = self.Tmax
elif T < self.Tmin: T = self.Tmin
Psat = np.array([i(T) for i in self.Psats])
z_norm = z / z.sum()
z_Psat_gamma_pcf = z * Psat * self.gamma(z_norm, T) * self.pcf(
z_norm, T)
f = self._P_error
P_guess, y = self._Py_ideal(z_Psat_gamma_pcf)
args = (T, z_Psat_gamma_pcf, y)
try:
P = flx.aitken_secant(f,
P_guess,
P_guess - 1,
1e-3,
1e-9,
args,
checkiter=False)
except (InfeasibleRegion, DomainError):
Pmin = self.Pmin
Pmax = self.Pmax
P = flx.IQ_interpolation(f,
Pmin,
Pmax,
f(Pmin, *args),
f(Pmax, *args),
P_guess,
1e-3,
5e-12,
args,
checkiter=False,
checkbounds=False)
return P, fn.normalize(y)
