def _custom_set_poly_fit(self):
try:
Tmin, Tmax = self.poly_fit_Tmin, self.poly_fit_Tmax
poly_fit_coeffs = self.poly_fit_coeffs
v_Tmin = horner(poly_fit_coeffs, Tmin)
for T_trans in linspace(Tmin, Tmax, 25):
v, d1, d2 = horner_and_der2(poly_fit_coeffs, T_trans)
Psat = exp(v)
dPsat_dT = Psat * d1
d2Psat_dT2 = Psat * (d1 * d1 + d2)
A, B, C = Antoine_ABC = Antoine_coeffs_from_point(T_trans,
Psat,
dPsat_dT,
d2Psat_dT2,
base=e)
self.poly_fit_AB = list(
Antoine_AB_coeffs_from_point(T_trans,
Psat,
dPsat_dT,
base=e))
self.DIPPR101_ABC = list(
DIPPR101_ABC_coeffs_from_point(T_trans, Psat, dPsat_dT,
d2Psat_dT2))
B_OK = B > 0.0 # B is negated in this implementation, so the requirement is reversed
C_OK = -T_trans < C < 0.0
if B_OK and C_OK:
self.poly_fit_Antoine = Antoine_ABC
break
else:
continue
# Calculate the extrapolation values
v_Tmax = horner(poly_fit_coeffs, Tmax)
v, d1, d2 = horner_and_der2(poly_fit_coeffs, Tmax)
Psat = exp(v)
dPsat_dT = Psat * d1
d2Psat_dT2 = Psat * (d1 * d1 + d2)
# A, B, C = Antoine_ABC = Antoine_coeffs_from_point(T_trans, Psat, dPsat_dT, d2Psat_dT2, base=e)
self.poly_fit_AB_high = list(
Antoine_AB_coeffs_from_point(Tmax, Psat, dPsat_dT, base=e))
self.poly_fit_AB_high_ABC_compat = [
self.poly_fit_AB_high[0], -self.poly_fit_AB_high[1]
]
self.DIPPR101_ABC_high = list(
DIPPR101_ABC_coeffs_from_point(Tmax, Psat, dPsat_dT,
d2Psat_dT2))
except:
pass
def mul(self, T):
r'''Computes liquid viscosity at a specified temperature
of an organic compound using the Joback method as a function of
chemical structure only.
.. math::
\mu_{liq} = \text{MW} \exp\left( \frac{ \sum_i \mu_a - 597.82}{T}
+ \sum_i \mu_b - 11.202 \right)
Parameters
----------
T : float
Temperature, [K]
Returns
-------
mul : float
Liquid viscosity, [Pa*s]
Examples
--------
>>> J = Joback('CC(=O)C')
>>> J.mul(300)
0.0002940378347162687
'''
if self.calculated_mul_coeffs is None:
self.calculated_mul_coeffs = Joback.mul_coeffs(self.counts)
a, b = self.calculated_mul_coeffs
return self.MW*exp(a/T + b)
def EQ115(T, A, B, C=0, D=0, E=0):
r'''DIPPR Equation #115. No major uses; has been used as an alternate
liquid viscosity expression, and as a model for vapor pressure.
Only parameters A and B are required.
.. math::
Y = \exp\left(A + \frac{B}{T} + C\log T + D T^2 + \frac{E}{T^2}\right)
Parameters
----------
T : float
Temperature, [K]
A-E : float
Parameter for the equation; chemical and property specific [-]
Returns
-------
Y : float
Property [constant-specific]
Notes
-----
No coefficients found for this expression.
This function is not integrable for either dT or Y/T dT.
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
return exp(A + B / T + C * log(T) + D * T**2 + E / T**2)
def Lee_Kesler(T, Tc, Pc, omega):
r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
CSP relationship by [1]_; requires a chemical's critical temperature and
acentric factor.
The vapor pressure is given by:
.. math::
\ln P^{sat}_r = f^{(0)} + \omega f^{(1)}
.. math::
f^{(0)} = 5.92714-\frac{6.09648}{T_r}-1.28862\ln T_r + 0.169347T_r^6
.. math::
f^{(1)} = 15.2518-\frac{15.6875}{T_r} - 13.4721 \ln T_r + 0.43577T_r^6
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
Acentric factor [-]
Returns
-------
Psat : float
Vapor pressure at T [Pa]
Notes
-----
This equation appears in [1]_ in expanded form.
The reduced pressure form of the equation ensures predicted vapor pressure
cannot surpass the critical pressure.
Examples
--------
Example from [2]_; ethylbenzene at 347.2 K.
>>> Lee_Kesler(347.2, 617.1, 36E5, 0.299)
13078.694162949312
References
----------
.. [1] Lee, Byung Ik, and Michael G. Kesler. "A Generalized Thermodynamic
Correlation Based on Three-Parameter Corresponding States." AIChE Journal
21, no. 3 (1975): 510-527. doi:10.1002/aic.690210313.
.. [2] Reid, Robert C..; Prausnitz, John M.;; Poling, Bruce E.
The Properties of Gases and Liquids. McGraw-Hill Companies, 1987.
'''
Tr = T / Tc
logTr = log(Tr)
Tr6 = Tr * Tr
Tr6 *= Tr6 * Tr6
f0 = 5.92714 - 6.09648 / Tr - 1.28862 * logTr + 0.169347 * Tr6
f1 = 15.2518 - 15.6875 / Tr - 13.4721 * logTr + 0.43577 * Tr6
return exp(f0 + omega * f1) * Pc
def gibbs_excess_dgammas_dT(xs,
GE,
dGE_dT,
dG_dxs,
d2GE_dTdxs,
N,
T,
dgammas_dT=None):
if dgammas_dT is None:
dgammas_dT = [0.0] * N
xdx_totF0 = dGE_dT
for j in range(N):
xdx_totF0 -= xs[j] * d2GE_dTdxs[j]
xdx_totF1 = GE
for j in range(N):
xdx_totF1 -= xs[j] * dG_dxs[j]
T_inv = 1.0 / T
RT_inv = R_inv * T_inv
for i in range(N):
dG_dni = xdx_totF1 + dG_dxs[i]
dgammas_dT[i] = RT_inv * (d2GE_dTdxs[i] - dG_dni * T_inv +
xdx_totF0) * exp(dG_dni * RT_inv)
return dgammas_dT
def gammas(self):
r'''Calculate and return the activity coefficients of a liquid phase
using an activity coefficient model.
.. math::
\gamma_i = \exp\left(\frac{\frac{\partial n_i G^E}{\partial n_i }}{RT}\right)
Returns
-------
gammas : list[float]
Activity coefficients, [-]
Notes
-----
'''
try:
return self._gammas
except:
pass
# Matches the gamma formulation perfectly
GE = self.GE()
dG_dxs = self.dGE_dxs()
if self.scalar:
dG_dns = dxs_to_dn_partials(dG_dxs, self.xs, GE)
RT_inv = 1.0 / (R * self.T)
gammas = [exp(i * RT_inv) for i in dG_dns]
else:
gammas = gibbs_excess_gammas(self.xs, dG_dxs, GE, self.T)
self._gammas = gammas
return gammas
def Antoine_AB_coeffs_from_point(T, Psat, dPsat_dT, base=10.0):
r'''Calculates the antoine coefficients `A`, `B`, with `C` set to zero to
improve low-temperature or high-temperature extrapolation, from a known
vapor pressure and its first temperature derivative.
Parameters
----------
T : float
Temperature of fluid, [K]
Psat : float
Vapor pressure at specified `T` [Pa]
dPsat_dT : float
First temperature derivative of vapor pressure at specified `T` [Pa/K]
Base : float, optional
Base of logarithm; 10 by default
Returns
-------
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
Notes
-----
Coefficients are for calculating vapor pressure in Pascal. This is
primarily useful for interconverting vapor pressure models, not fitting
experimental data.
Derived with SymPy as follows:
>>> from sympy import * # doctest: +SKIP
>>> base, A, B, T = symbols('base, A, B, T') # doctest: +SKIP
>>> v = base**(A - B/T) # doctest: +SKIP
>>> d1, d2 = diff(v, T), diff(v, T, 2) # doctest: +SKIP
>>> vk, d1k = symbols('vk, d1k') # doctest: +SKIP
>>> solve([Eq(v, vk), Eq(d1, d1k)], [A, B]) # doctest: +SKIP
Examples
--------
Recalculate some coefficients from a calcualted value and its derivative:
>>> T = 178.01
>>> A, B = (27.358925161569008, 5445.569591293226)
>>> Psat = Antoine(T, A, B, C=0, base=exp(1))
>>> dPsat_dT = B*exp(1)**(A - B/T)*log(exp(1))/T**2
>>> Antoine_AB_coeffs_from_point(T, Psat, dPsat_dT, base=exp(1))
(27.35892516156901, 5445.569591293226)
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
'''
log_base_inv = 1.0 / log(base)
Psat_inv = 1.0 / Psat
A = log(Psat * exp(T * dPsat_dT * Psat_inv)) * log_base_inv
B = T * T * dPsat_dT * log_base_inv * Psat_inv
return (A, B)
def regular_solution_gammas(T,
xs,
Vs,
SPs,
lambda_coeffs,
N,
xsVs=None,
Hi_sums=None,
dGE_dxs=None,
gammas=None):
if xsVs is None:
xsVs = [0.0] * N
for i in range(N):
xsVs[i] = xs[i] * Vs[i]
xsVs_sum = 0.0
for i in range(N):
xsVs_sum += xsVs[i]
xsVs_sum_inv = 1.0 / xsVs_sum
if Hi_sums is None:
Hi_sums = [0.0] * N
Hi_sums = regular_solution_Hi_sums(SPs=SPs,
Vs=Vs,
xsVs=xsVs,
coeffs=lambda_coeffs,
N=N,
Hi_sums=Hi_sums)
GE = regular_solution_GE(SPs=SPs,
xsVs=xsVs,
coeffs=lambda_coeffs,
N=N,
xsVs_sum_inv=xsVs_sum_inv)
if dGE_dxs is None:
dGE_dxs = [0.0] * N
dG_dxs = regular_solution_dGE_dxs(Vs=Vs,
Hi_sums=Hi_sums,
N=N,
xsVs_sum_inv=xsVs_sum_inv,
GE=GE,
dGE_dxs=dGE_dxs)
xdx_totF = GE
for i in range(N):
xdx_totF -= xs[i] * dG_dxs[i]
if gammas is None:
gammas = [0.0] * N
for i in range(N):
gammas[i] = dG_dxs[i] + xdx_totF
RT_inv = 1.0 / (R * T)
for i in range(N):
gammas[i] *= RT_inv
for i in range(N):
gammas[i] = exp(gammas[i])
return gammas
def Henry_pressure_mixture(Hs, weights=None, zs=None):
r'''Mixing rule for Henry's law components. Applies a logarithmic average
to all solvent components and mole fractions. Optionally, weight factors
can be provided instead of using mole fractions - only specify one of them.
A common weight factor is using volume fractions of powers of them, or
using critical volumes.
Parameters
----------
Hs : list[float or None]
Henry's law constant between each gas and the solvent (None for other
solvents of gases without parameters available), [Pa]
weights : list[float], optional
Weight factors, [-]
zs : list[float]
Mole fractions of all species in phase, [-]
Returns
-------
H : value
Henry's law constant for the gas in the liquid phase, [-]
Notes
-----
The default weight factor formulation is from [1]_.
Examples
--------
>>> Henry_pressure_mixture([1072330.36341, 744479.751106, None], zs=[.48, .48, .04])
893492.1611602883
References
----------
.. [1] Gmehling, Jurgen. Chemical Thermodynamics: For Process Simulation.
Weinheim, Germany: Wiley-VCH, 2012.
'''
N = len(Hs)
if weights is None and zs is None:
raise ValueError("Weights or mole fractions are required")
if weights is None:
z_solvent = 0.0
for i in range(N):
if Hs[i] is not None:
z_solvent += zs[i]
# Default parameters - when weight specified only weight by that
z_solvent_inv = 1.0/z_solvent
weights = [0.0]*N
for i in range(N):
weights[i] = zs[i]*z_solvent_inv
num = 0.0
for i in range(N):
if Hs[i] is not None:
num += weights[i]*log(Hs[i])
H = exp(num)
return H
def Wagner_original(T, Tc, Pc, a, b, c, d):
r'''Calculates vapor pressure using the Wagner equation (3, 6 form).
Requires critical temperature and pressure as well as four coefficients
specific to each chemical.
.. math::
\ln P^{sat}= \ln P_c + \frac{a\tau + b \tau^{1.5} + c\tau^3 + d\tau^6}
{T_r}
.. math::
\tau = 1 - \frac{T}{T_c}
Parameters
----------
T : float
Temperature of fluid, [K]
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
a, b, c, d : floats
Parameters for wagner equation. Specific to each chemical. [-]
Returns
-------
Psat : float
Vapor pressure at T [Pa]
Notes
-----
Warning: Pc is often treated as adjustable constant.
Examples
--------
Methane, coefficients from [2]_, at 100 K.
>>> Wagner_original(100.0, 190.53, 4596420., a=-6.00435, b=1.1885,
... c=-0.834082, d=-1.22833)
34520.44601450499
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
.. [2] McGarry, Jack. "Correlation and Prediction of the Vapor Pressures of
Pure Liquids over Large Pressure Ranges." Industrial & Engineering
Chemistry Process Design and Development 22, no. 2 (April 1, 1983):
313-22. doi:10.1021/i200021a023.
'''
Tr = T / Tc
tau = 1.0 - Tr
tau2 = tau * tau
tau_Tr = tau / Tr
return Pc * exp(((d * tau2 * tau + c) * tau2 + a + b * sqrt(tau)) * tau_Tr)
def calculate_derivative(self, T, method, order=1):
r'''Method to calculate a derivative of a vapor pressure with respect to
temperature, of a given order using a specified method. If the method
is BESTFIT, an anlytical derivative is used; otherwise SciPy's
derivative function, with a delta of 1E-6 K and a number of points
equal to 2*order + 1.
If the calculation does not succeed, returns the actual error
encountered.
Parameters
----------
T : float
Temperature at which to calculate the derivative, [K]
method : str
Method for which to find the derivative
order : int
Order of the derivative, >= 1
Returns
-------
derivative : float
Calculated derivative property, [`units/K^order`]
'''
if order == 1 and method == BESTFIT:
# if T < self.poly_fit_Tmin:
# return self.poly_fit_Tmin_slope*exp(
# (T - self.poly_fit_Tmin)*self.poly_fit_Tmin_slope
# + self.poly_fit_Tmin_value)
# elif T > self.poly_fit_Tmax:
# return self.poly_fit_Tmax_slope*exp((T - self.poly_fit_Tmax)
# *self.poly_fit_Tmax_slope
# + self.poly_fit_Tmax_value)
# else:
v, der = horner_and_der(self.poly_fit_coeffs, T)
return der * exp(v)
elif order == 2 and method == BESTFIT:
v, der, der2 = horner_and_der2(self.poly_fit_coeffs, T)
return (der * der + der2) * exp(v)
return super(VaporPressure,
self).calculate_derivative(T, method, order)
def gibbs_excess_gammas(xs, dG_dxs, GE, T, gammas=None):
xdx_totF = GE
N = len(xs)
for i in range(N):
xdx_totF -= xs[i] * dG_dxs[i]
RT_inv = R_inv / T
if gammas is None:
gammas = [0.0] * N
for i in range(N):
gammas[i] = exp((dG_dxs[i] + xdx_totF) * RT_inv)
return gammas
def Psub_Clapeyron(T, Tt, Pt, Hsub_t):
r'''Calculates sublimation pressure of a solid at arbitrary temperatures
using an approximate themodynamic identity - the Clapeyron equation as
described in [1]_ and [2]_.
Requires a chemical's triple temperature, triple pressure, and triple
enthalpy of sublimation.
The sublimation pressure of a chemical at `T` is given by:
.. math::
\ln \frac{P}{P_{tp}} = -\frac{\Delta H_{sub}}{R}
\left(\frac{1}{T}-\frac{1}{T_{tp}} \right)
Parameters
----------
T : float
Temperature of solid [K]
Tt : float
Triple temperature of solid [K]
Pt : float
Truple pressure of solid [Pa]
Hsub_t : float
Enthalpy of fusion at the triple point of the chemical, [J/mol]
Returns
-------
Psub : float
Sublimation pressure, [Pa]
Notes
-----
Does not seem to capture the decrease in sublimation pressure quickly
enough.
Examples
--------
>>> Psub_Clapeyron(250, Tt=273.15, Pt=611.0, Hsub_t=51100.0)
76.06457150831804
>>> Psub_Clapeyron(300, Tt=273.15, Pt=611.0, Hsub_t=51100.0)
4577.282832876156
References
----------
.. [1] Goodman, B. T., W. V. Wilding, J. L. Oscarson, and R. L. Rowley.
"Use of the DIPPR Database for the Development of QSPR Correlations:
Solid Vapor Pressure and Heat of Sublimation of Organic Compounds."
International Journal of Thermophysics 25, no. 2 (March 1, 2004):
337-50. https://doi.org/10.1023/B:IJOT.0000028471.77933.80.
.. [2] Feistel, Rainer, and Wolfgang Wagner. "Sublimation Pressure and
Sublimation Enthalpy of H2O Ice Ih between 0 and 273.16K." Geochimica et
Cosmochimica Acta 71, no. 1 (January 1, 2007): 36-45.
https://doi.org/10.1016/j.gca.2006.08.034.
'''
return Pt * exp(Hsub_t * (T - Tt) / (R * T * Tt))
def solubility_eutectic(T, Tm, Hm, Cpl=0, Cps=0, gamma=1):
r'''Returns the maximum solubility of a solute in a solvent.
.. math::
\ln x_i^L \gamma_i^L = \frac{\Delta H_{m,i}}{RT}\left(
1 - \frac{T}{T_{m,i}}\right) - \frac{\Delta C_{p,i}(T_{m,i}-T)}{RT}
+ \frac{\Delta C_{p,i}}{R}\ln\frac{T_m}{T}
.. math::
\Delta C_{p,i} = C_{p,i}^L - C_{p,i}^S
Parameters
----------
T : float
Temperature of the system [K]
Tm : float
Melting temperature of the solute [K]
Hm : float
Heat of melting at the melting temperature of the solute [J/mol]
Cpl : float, optional
Molar heat capacity of the solute as a liquid [J/mol/K]
Cpls: float, optional
Molar heat capacity of the solute as a solid [J/mol/K]
gamma : float, optional
Activity coefficient of the solute as a liquid [-]
Returns
-------
x : float
Mole fraction of solute at maximum solubility [-]
Notes
-----
gamma is of the solute in liquid phase
Examples
--------
From [1]_, matching example
>>> solubility_eutectic(T=260., Tm=278.68, Hm=9952., Cpl=0, Cps=0, gamma=3.0176)
0.2434007130748
References
----------
.. [1] Gmehling, Jurgen. Chemical Thermodynamics: For Process Simulation.
Weinheim, Germany: Wiley-VCH, 2012.
'''
dCp = Cpl - Cps
x = exp(-R_inv * ((Hm * (1.0 - T / Tm) - dCp *
(Tm - T)) / T + dCp * log(Tm / T))) / gamma
return x
def Wagner(T, Tc, Pc, a, b, c, d):
r'''Calculates vapor pressure using the Wagner equation (2.5, 5 form).
Requires critical temperature and pressure as well as four coefficients
specific to each chemical.
.. math::
\ln P^{sat}= \ln P_c + \frac{a\tau + b \tau^{1.5} + c\tau^{2.5}
+ d\tau^5} {T_r}
.. math::
\tau = 1 - \frac{T}{T_c}
Parameters
----------
T : float
Temperature of fluid, [K]
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
a, b, c, d : floats
Parameters for wagner equation. Specific to each chemical. [-]
Returns
-------
Psat : float
Vapor pressure at T [Pa]
Notes
-----
Warning: Pc is often treated as adjustable constant.
Examples
--------
Methane, coefficients from [2]_, at 100 K.
>>> Wagner(100., 190.551, 4599200, -6.02242, 1.26652, -0.5707, -1.366)
34415.00476263708
References
----------
.. [1] Wagner, W. "New Vapour Pressure Measurements for Argon and Nitrogen and
a New Method for Establishing Rational Vapour Pressure Equations."
Cryogenics 13, no. 8 (August 1973): 470-82. doi:10.1016/0011-2275(73)90003-9
.. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
'''
Tr = T / Tc
tau = 1.0 - T / Tc
return Pc * exp((a * tau + b * tau**1.5 + c * tau**2.5 + d * tau**5) / Tr)
def solve_prop_poly_fit(self, goal):
poly_fit_Tmin, poly_fit_Tmax = self.poly_fit_Tmin, self.poly_fit_Tmax
poly_fit_Tmin_slope, poly_fit_Tmax_slope = self.poly_fit_Tmin_slope, self.poly_fit_Tmax_slope
poly_fit_Tmin_value, poly_fit_Tmax_value = self.poly_fit_Tmin_value, self.poly_fit_Tmax_value
coeffs = self.poly_fit_coeffs
T_low = log(goal * exp(poly_fit_Tmin * poly_fit_Tmin_slope -
poly_fit_Tmin_value)) / poly_fit_Tmin_slope
if T_low <= poly_fit_Tmin:
return T_low
T_high = log(goal * exp(poly_fit_Tmax * poly_fit_Tmax_slope -
poly_fit_Tmax_value)) / poly_fit_Tmax_slope
if T_high >= poly_fit_Tmax:
return T_high
else:
lnPGoal = log(goal)
def to_solve(T):
# dPsat and Psat are both in log basis
dPsat = Psat = 0.0
for c in coeffs:
dPsat = T * dPsat + Psat
Psat = T * Psat + c
return Psat - lnPGoal, dPsat
# Guess with the two extrapolations from the linear fits
# By definition both guesses are in the range of they would have been returned
if T_low > poly_fit_Tmax:
T_low = poly_fit_Tmax
if T_high < poly_fit_Tmin:
T_high = poly_fit_Tmin
T = newton(to_solve,
0.5 * (T_low + T_high),
fprime=True,
low=poly_fit_Tmin,
high=poly_fit_Tmax)
return T
def Watson_sigma(T, Tc, a1, a2, a3=0.0, a4=0.0, a5=0.0):
r'''Calculates air-water surface tension using the Watson [1]_
emperical (parameter-regressed) method developed by NIST.
.. math::
\sigma = \exp\left[a_{1} + \ln(1 - T_r)\left(
a_2 + a_3T_r + a_4T_r^2 + a_5T_r^3 \right)\right]
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
a3 : float
Regression parameter, [-]
a4 : float
Regression parameter, [-]
a5 : float
Regression parameter, [-]
Returns
-------
sigma : float
Liquid surface tension, [N/m]
Notes
-----
This expression is also used for enthalpy of vaporization in [1]_.
The coefficients from NIST TDE for enthalpy of vaporization are kJ/mol.
Examples
--------
Isooctane at 350 K from [1]_:
>>> Watson_sigma(T=350.0, Tc=543.836, a1=-3.02417, a2=1.21792, a3=-5.26877e-9, a4=5.62659e-9, a5=-2.27553e-9)
0.0138340926605649
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User’s Guide."
https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-SurfaceTension/HVPExpansion-SurfaceTension.htm
'''
Tr = T / Tc
l = log(1.0 - Tr)
return exp(a1 + l * (a2 + Tr * (a3 + Tr * (a4 + a5 * Tr))))
def boiling_critical_relation(T, Tb, Tc, Pc):
r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
CSP relationship as in [1]_; requires a chemical's critical temperature
and pressure as well as boiling point.
The vapor pressure is given by:
.. math::
\ln P^{sat}_r = h\left( 1 - \frac{1}{T_r}\right)
.. math::
h = T_{br} \frac{\ln(P_c/101325)}{1-T_{br}}
Parameters
----------
T : float
Temperature of fluid [K]
Tb : float
Boiling temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
Returns
-------
Psat : float
Vapor pressure at T [Pa]
Notes
-----
Units are Pa. Formulation makes intuitive sense; a logarithmic form of
interpolation.
Examples
--------
Example as in [1]_ for ethylbenzene
>>> boiling_critical_relation(347.2, 409.3, 617.1, 36E5)
15209.467273093938
References
----------
.. [1] Reid, Robert C..; Prausnitz, John M.;; Poling, Bruce E.
The Properties of Gases and Liquids. McGraw-Hill Companies, 1987.
'''
Tbr = Tb / Tc
Tr = T / Tc
h = Tbr * log(Pc / 101325.) / (1 - Tbr)
return exp(h * (1 - 1 / Tr)) * Pc
def Henry_pressure(T, A, B=0.0, C=0.0, D=0.0, E=0.0, F=0.0):
r'''Calculates Henry's law constant as a function of temperature according
to the SI units of `Pa` and using a common temperature dependence as used
in many process simulation applications.
Only the `A` parameter is required - which has no temperature dependence
when used by itself.
As the model is exponential, a sufficiently high temperature may cause an
OverflowError.
A negative temperature (or just low, if fit poorly) may cause a math domain
error.
.. math::
H_{12} = \exp\left(A_{12} + \frac{B_{12}}{T} + C_{12}\ln(T) + D_{12}T
+ \frac{E_{12}}{T^2} \right)
Parameters
----------
T : float
Temperature, [K]
A-F : float
Parameter for the equation; chemical and property specific [-]
Returns
-------
H12 : float
Henry's constant [Pa]
Notes
-----
Add 11.51292 to the `A` constant if it is said to provide units of `bar`,
so that it provides units of `Pa` instead.
The `F` parameter is not often included in models. It is rare to fit
all parameters.
Examples
--------
Random test example.
>>> Henry_pressure(300.0, A=15.0, B=300.0, C=.04, D=1e-3, E=1e2, F=1e-5)
37105004.47898146
References
----------
.. [1] Gmehling, Jurgen. Chemical Thermodynamics: For Process Simulation.
Weinheim, Germany: Wiley-VCH, 2012.
'''
T_inv = 1.0/T
return exp(A + T_inv*(B + E*T_inv) + C*log(T) + T*(D + F*T))
def dgammas_dT(self):
r'''Calculate and return the temperature derivatives of activity
coefficients of a liquid phase using an activity coefficient model.
.. math::
\frac{\partial \gamma_i}{\partial T} =
\left(\frac{\frac{\partial^2 n G^E}{\partial T \partial n_i}}{RT} -
\frac{{\frac{\partial n_i G^E}{\partial n_i }}}{RT^2}\right)
\exp\left(\frac{\frac{\partial n_i G^E}{\partial n_i }}{RT}\right)
Returns
-------
dgammas_dT : list[float]
Temperature derivatives of activity coefficients, [1/K]
Notes
-----
'''
r'''
from sympy import *
R, T = symbols('R, T')
f = symbols('f', cls=Function)
diff(exp(f(T)/(R*T)), T)
'''
try:
return self._dgammas_dT
except AttributeError:
pass
d2nGE_dTdns = self.d2nGE_dTdns()
dG_dxs = self.dGE_dxs()
GE = self.GE()
dG_dns = dxs_to_dn_partials(dG_dxs, self.xs, GE)
T_inv = 1.0/self.T
RT_inv = R_inv*T_inv
self._dgammas_dT = dgammas_dT = []
for i in self.cmps:
x1 = dG_dns[i]*T_inv
dgammas_dT.append(RT_inv*(d2nGE_dTdns[i] - x1)*exp(dG_dns[i]*RT_inv))
return dgammas_dT
def EQ101(T, A, B, C, D, E):
r'''DIPPR Equation # 101. Used in calculating vapor pressure, sublimation
pressure, and liquid viscosity.
All 5 parameters are required. E is often an integer. As the model is
exponential, a sufficiently high temperature will cause an OverflowError.
A negative temperature (or just low, if fit poorly) may cause a math domain
error.
.. math::
Y = \exp\left(A + \frac{B}{T} + C\cdot \ln T + D \cdot T^E\right)
Parameters
----------
T : float
Temperature, [K]
A-E : float
Parameter for the equation; chemical and property specific [-]
Returns
-------
Y : float
Property [constant-specific]
Notes
-----
This function is not integrable for either dT or Y/T dT.
Examples
--------
Water vapor pressure; DIPPR coefficients normally listed in Pa.
>>> EQ101(300, 73.649, -7258.2, -7.3037, 4.1653E-6, 2)
3537.44834545549
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
return exp(A + B / T + C * log(T) + D * T**E)
def interaction_exp(T, N, A, B, C, D, E, F, lambdas=None):
if lambdas is None:
lambdas = [[0.0] * N for i in range(N)] # numba: delete
# lambdas = zeros((N, N)) # numba: uncomment
# # 87% of the time of this routine is the exponential.
T2 = T * T
Tinv = 1.0 / T
T2inv = Tinv * Tinv
logT = log(T)
for i in range(N):
Ai = A[i]
Bi = B[i]
Ci = C[i]
Di = D[i]
Ei = E[i]
Fi = F[i]
lambdais = lambdas[i]
# Might be more efficient to pass over this matrix later,
# and compute all the exps
# Spoiler: it was not.
# Also - it was tested the impact of using fewer terms
# there was very little, to no impact from that
# the exp is the huge time sink.
for j in range(N):
lambdais[j] = exp(Ai[j] + Bi[j] * Tinv + Ci[j] * logT + Di[j] * T +
Ei[j] * T2inv + Fi[j] * T2)
# lambdas[i][j] = exp(A[i][j] + B[i][j]*Tinv
# + C[i][j]*logT + D[i][j]*T
# + E[i][j]*T2inv + F[i][j]*T2)
# 135 µs ± 1.09 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each) # without out specified numba
# 129 µs ± 2.45 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each) # with out specified numba
# 118 µs ± 2.67 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each) # without out specified numba 1 term
# 115 µs ± 1.77 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each) # with out specified numba 1 term
return lambdas
def calculate(self, T, method):
r'''Method to calculate sublimation pressure of a fluid at temperature
`T` with a given method.
This method has no exception handling; see :obj:`T_dependent_property <thermo.utils.TDependentProperty.T_dependent_property>`
for that.
Parameters
----------
T : float
Temperature at calculate sublimation pressure, [K]
method : str
Name of the method to use
Returns
-------
Psub : float
Sublimation pressure at T, [pa]
'''
if method == BESTFIT:
if T < self.poly_fit_Tmin:
Psub = (T - self.poly_fit_Tmin
) * self.poly_fit_Tmin_slope + self.poly_fit_Tmin_value
elif T > self.poly_fit_Tmax:
Psub = (T - self.poly_fit_Tmax
) * self.poly_fit_Tmax_slope + self.poly_fit_Tmax_value
else:
Psub = horner(self.poly_fit_coeffs, T)
Psub = exp(Psub)
elif method == PSUB_CLAPEYRON:
Psub = max(
Psub_Clapeyron(T, Tt=self.Tt, Pt=self.Pt, Hsub_t=self.Hsub_t),
1e-200)
elif method in self.tabular_data:
Psub = self.interpolate(T, method)
return Psub
def Edalat(T, Tc, Pc, omega):
r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
CSP relationship by [1]_. Requires a chemical's critical temperature,
pressure, and acentric factor. Claimed to have a higher accuracy than the
Lee-Kesler CSP relationship.
The vapor pressure of a chemical at `T` is given by:
.. math::
\ln(P^{sat}/P_c) = \frac{a\tau + b\tau^{1.5} + c\tau^3 + d\tau^6}
{1-\tau}
.. math::
a = -6.1559 - 4.0855\omega
.. math::
b = 1.5737 - 1.0540\omega - 4.4365\times 10^{-3} d
.. math::
c = -0.8747 - 7.8874\omega
.. math::
d = \frac{1}{-0.4893 - 0.9912\omega + 3.1551\omega^2}
.. math::
\tau = 1 - \frac{T}{T_c}
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
Acentric factor [-]
Returns
-------
Psat : float
Vapor pressure, [Pa]
Notes
-----
[1]_ found an average error of 6.06% on 94 compounds and 1106 data points.
Examples
--------
>>> Edalat(347.2, 617.1, 36E5, 0.299)
13461.273080743307
References
----------
.. [1] Edalat, M., R. B. Bozar-Jomehri, and G. A. Mansoori. "Generalized
Equation Predicts Vapor Pressure of Hydrocarbons." Oil and Gas Journal;
91:5 (February 1, 1993).
'''
tau = 1. - T / Tc
a = -6.1559 - 4.0855 * omega
c = -0.8747 - 7.8874 * omega
d = 1. / (-0.4893 - 0.9912 * omega + 3.1551 * omega * omega)
b = 1.5737 - 1.0540 * omega - 4.4365E-3 * d
tau_15 = tau**1.5
tau3 = tau_15 * tau_15
lnPr = (a * tau + b * tau_15 + c * tau3 + d * tau3 * tau3) / (1. - tau)
return exp(lnPr) * Pc
def Sanjari(T, Tc, Pc, omega):
r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
CSP relationship by [1]_. Requires a chemical's critical temperature,
pressure, and acentric factor. Although developed for refrigerants,
this model should have some general predictive ability.
The vapor pressure of a chemical at `T` is given by:
.. math::
P^{sat} = P_c\exp(f^{(0)} + \omega f^{(1)} + \omega^2 f^{(2)})
.. math::
f^{(0)} = a_1 + \frac{a_2}{T_r} + a_3\ln T_r + a_4 T_r^{1.9}
.. math::
f^{(1)} = a_5 + \frac{a_6}{T_r} + a_7\ln T_r + a_8 T_r^{1.9}
.. math::
f^{(2)} = a_9 + \frac{a_{10}}{T_r} + a_{11}\ln T_r + a_{12} T_r^{1.9}
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
Acentric factor [-]
Returns
-------
Psat : float
Vapor pressure, [Pa]
Notes
-----
a[1-12] are as follows:
6.83377, -5.76051, 0.90654, -1.16906,
5.32034, -28.1460, -58.0352, 23.57466,
18.19967, 16.33839, 65.6995, -35.9739.
For a claimed fluid not included in the regression, R128, the claimed AARD
was 0.428%. A re-calculation using 200 data points from 125.45 K to
343.90225 K evenly spaced by 1.09775 K as generated by NIST Webbook April
2016 produced an AARD of 0.644%. It is likely that the author's regression
used more precision in its coefficients than was shown here. Nevertheless,
the function is reproduced as shown in [1]_.
For Tc=808 K, Pc=1100000 Pa, omega=1.1571, this function actually declines
after 770 K.
Examples
--------
>>> Sanjari(347.2, 617.1, 36E5, 0.299)
13651.916109552523
References
----------
.. [1] Sanjari, Ehsan, Mehrdad Honarmand, Hamidreza Badihi, and Ali
Ghaheri. "An Accurate Generalized Model for Predict Vapor Pressure of
Refrigerants." International Journal of Refrigeration 36, no. 4
(June 2013): 1327-32. doi:10.1016/j.ijrefrig.2013.01.007.
'''
Tr = T / Tc
Tr_inv = 1.0 / Tr
log_Tr = log(Tr)
Tr_19 = Tr**1.9
f0 = 6.83377 + -5.76051 * Tr_inv + 0.90654 * log_Tr + -1.16906 * Tr_19
f1 = 5.32034 + -28.1460 * Tr_inv + -58.0352 * log_Tr + 23.57466 * Tr_19
f2 = 18.19967 + 16.33839 * Tr_inv + 65.6995 * log_Tr + -35.9739 * Tr_19
return Pc * exp(f0 + omega * f1 + omega * omega * f2)
def Ambrose_Walton(T, Tc, Pc, omega):
r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
CSP relationship by [1]_; requires a chemical's critical temperature and
acentric factor.
The vapor pressure is given by:
.. math::
\ln P_r=f^{(0)}+\omega f^{(1)}+\omega^2f^{(2)}
.. math::
f^{(0)}=\frac{-5.97616\tau + 1.29874\tau^{1.5}- 0.60394\tau^{2.5}
-1.06841\tau^5}{T_r}
.. math::
f^{(1)}=\frac{-5.03365\tau + 1.11505\tau^{1.5}- 5.41217\tau^{2.5}
-7.46628\tau^5}{T_r}
.. math::
f^{(2)}=\frac{-0.64771\tau + 2.41539\tau^{1.5}- 4.26979\tau^{2.5}
+3.25259\tau^5}{T_r}
.. math::
\tau = 1-T_{r}
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
Acentric factor [-]
Returns
-------
Psat : float
Vapor pressure at T [Pa]
Notes
-----
Somewhat more accurate than the :obj:`Lee_Kesler` formulation.
Examples
--------
Example from [2]_; ethylbenzene at 347.25 K.
>>> Ambrose_Walton(347.25, 617.15, 36.09E5, 0.304)
13278.878504306222
References
----------
.. [1] Ambrose, D., and J. Walton. "Vapour Pressures up to Their Critical
Temperatures of Normal Alkanes and 1-Alkanols." Pure and Applied
Chemistry 61, no. 8 (1989): 1395-1403. doi:10.1351/pac198961081395.
.. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
'''
Tr = T / Tc
tau = 1.0 - Tr
tau15 = tau**1.5
tau25 = tau * tau15
tau5 = tau25 * tau25
f0 = (-5.97616 * tau + 1.29874 * tau15 - 0.60394 * tau25 - 1.06841 * tau5)
f1 = (-5.03365 * tau + 1.11505 * tau15 - 5.41217 * tau25 - 7.46628 * tau5)
f2 = (-0.64771 * tau + 2.41539 * tau15 - 4.26979 * tau25 + 3.25259 * tau5)
return Pc * exp((f0 + omega * (f1 + f2 * omega)) / Tr)
def mu(self):
try:
return self._mu
except AttributeError:
pass
phase_fractions = self.phase_fractions
phase_count = len(phase_fractions)
result = self.result
if phase_count == 1:
self._mu = mu = self.phases[0].mu()
return mu
elif self.state == 'l' or self.result.gas is None:
# Multiple liquids - either a bulk liquid, or a result with no gases
method = self.settings.mu_LL
if method == AS_ONE_LIQUID:
mu = self.correlations.ViscosityLiquidMixture.mixture_property(
self.T, self.P, self.zs, self.ws())
else:
mus = [i.mu() for i in self.phases]
if method in mole_methods:
betas = self.phase_fractions
elif method in mass_methods:
betas = self.betas_mass
elif method in volume_methods:
betas = self.betas_volume
mu = 0.0
if method in linear_methods:
for i in range(len(self.phase_fractions)):
mu += betas[i] * mus[i]
elif method in prop_power_methods:
exponent = self.settings.mu_LL_power_exponent
for i in range(len(self.phase_fractions)):
mu += betas[i] * mus[i]**exponent
mu = mu**(1.0 / exponent)
elif method in log_prop_methods:
for i in range(len(self.phase_fractions)):
mu += betas[i] * log(mus[i])
mu = exp(mu)
self._mu = mu
return mu
method = self.settings.mu_VL
if method == AS_ONE_LIQUID:
self._mu = mu = self.correlations.ViscosityLiquidMixture.mixture_property(
self.T, self.P, self.zs, self.ws())
return mu
elif method == AS_ONE_GAS:
self._mu = mu = self.correlations.ViscosityGasMixture.mixture_property(
self.T, self.P, self.zs, self.ws())
return mu
mug = result.gas.mu()
if phase_count == 2:
mul = result.liquids[0].mu()
else:
mul = result.liquid_bulk.mu()
if method in MU_VL_CORRELATIONS_SET:
x = result.betas_mass[0]
rhog = result.gas.rho_mass()
if phase_count == 2:
rhol = result.liquids[0].rho_mass()
else:
rhol = result.liquid_bulk.rho_mass()
mu = gas_liquid_viscosity(x, mul, mug, rhol, rhog, Method=method)
else:
mus = [mug, mul]
if method in mole_methods:
VF = self.result.beta_gas
betas = [VF, 1.0 - VF]
elif method in mass_methods:
betas = self.betas_mass_states[:2]
elif method in volume_methods:
betas = self.betas_volume_states[:2]
if method in linear_methods:
mu = betas[0] * mus[0] + betas[1] * mus[1]
elif method in prop_power_methods:
exponent = self.settings.mu_LL_power_exponent
mu = (betas[0] * mus[0]**exponent +
betas[1] * mus[1]**exponent)**(1.0 / exponent)
elif method in log_prop_methods:
mu = exp(betas[0] * log(mus[0]) + betas[1] * log(mus[1]))
self._mu = mu
return mu
def calculate(self, T, method):
r'''Method to calculate vapor pressure of a fluid at temperature `T`
with a given method.
This method has no exception handling; see :obj:`thermo.utils.TDependentProperty.T_dependent_property`
for that.
Parameters
----------
T : float
Temperature at calculate vapor pressure, [K]
method : str
Name of the method to use
Returns
-------
Psat : float
Vapor pressure at T, [pa]
'''
if method == BESTFIT:
if T < self.poly_fit_Tmin:
Psat = (T - self.poly_fit_Tmin
) * self.poly_fit_Tmin_slope + self.poly_fit_Tmin_value
elif T > self.poly_fit_Tmax:
Psat = (T - self.poly_fit_Tmax
) * self.poly_fit_Tmax_slope + self.poly_fit_Tmax_value
else:
Psat = horner(self.poly_fit_coeffs, T)
Psat = exp(Psat)
elif method == BEST_FIT_AB:
if T < self.poly_fit_Tmax:
return self.calculate(T, BESTFIT)
A, B = self.poly_fit_AB_high_ABC_compat
return exp(A + B / T)
elif method == BEST_FIT_ABC:
if T < self.poly_fit_Tmax:
return self.calculate(T, BESTFIT)
A, B, C = self.DIPPR101_ABC_high
return exp(A + B / T + C * log(T))
elif method == WAGNER_MCGARRY:
Psat = Wagner_original(T, self.WAGNER_MCGARRY_Tc,
self.WAGNER_MCGARRY_Pc,
*self.WAGNER_MCGARRY_coefs)
elif method == WAGNER_POLING:
Psat = Wagner(T, self.WAGNER_POLING_Tc, self.WAGNER_POLING_Pc,
*self.WAGNER_POLING_coefs)
elif method == ANTOINE_EXTENDED_POLING:
Psat = TRC_Antoine_extended(T, *self.ANTOINE_EXTENDED_POLING_coefs)
elif method == ANTOINE_POLING:
A, B, C = self.ANTOINE_POLING_coefs
Psat = Antoine(T, A, B, C, base=10.0)
elif method == DIPPR_PERRY_8E:
Psat = EQ101(T, *self.Perrys2_8_coeffs)
elif method == VDI_PPDS:
Psat = Wagner(T, self.VDI_PPDS_Tc, self.VDI_PPDS_Pc,
*self.VDI_PPDS_coeffs)
elif method == COOLPROP:
Psat = PropsSI('P', 'T', T, 'Q', 0, self.CASRN)
elif method == BOILING_CRITICAL:
Psat = boiling_critical_relation(T, self.Tb, self.Tc, self.Pc)
elif method == LEE_KESLER_PSAT:
Psat = Lee_Kesler(T, self.Tc, self.Pc, self.omega)
elif method == AMBROSE_WALTON:
Psat = Ambrose_Walton(T, self.Tc, self.Pc, self.omega)
elif method == SANJARI:
Psat = Sanjari(T, self.Tc, self.Pc, self.omega)
elif method == EDALAT:
Psat = Edalat(T, self.Tc, self.Pc, self.omega)
elif method == EOS:
Psat = self.eos[0].Psat(T)
elif method == BESTFIT:
Psat = exp(horner(self.poly_fit_coeffs, T))
else:
return self._base_calculate(T, method)
return Psat
def collision_integral_Neufeld_Janzen_Aziz(T_star, l=1, s=1):
r'''Calculates Lennard-Jones collision integral for any of 16 values of
(l,j) for the wide range of 0.3 < T_star < 100. Values are accurate to
0.1 % of actual values, but the calculation of actual values is
computationally intensive and so these simplifications are used, developed
in [1]_.
.. math::
\Omega_D = \frac{A}{T^{*B}} + \frac{C}{\exp(DT^*)} +
\frac{E}{\exp(FT^{*})} + \frac{G}{\exp(HT^*)} + RT^{*B}\sin(ST^{*W}-P)
Parameters
----------
Tstar : float
Reduced temperature of the fluid [-]
l : int
term
s : int
term
Returns
-------
Omega : float
Collision integral of A and B
Notes
-----
Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5),
(1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4),
(3, 5), and (4, 4).
.. math::
T^* = \frac{k_b T}{\epsilon}
Results are very similar to those of the more modern formulation,
`collision_integral_Kim_Monroe`.
Calculations begin to yield overflow errors in some values of (l, 2) after
T_star = 75, beginning with (1, 7). Also susceptible are (1, 5) and (1, 6).
Examples
--------
>>> collision_integral_Neufeld_Janzen_Aziz(100, 1, 1)
0.516717697672334
References
----------
.. [1] Neufeld, Philip D., A. R. Janzen, and R. A. Aziz. "Empirical
Equations to Calculate 16 of the Transport Collision Integrals
Omega(l, S)* for the Lennard-Jones (12-6) Potential." The Journal of
Chemical Physics 57, no. 3 (August 1, 1972): 1100-1102.
doi:10.1063/1.1678363
'''
if l == 1 and s == 1:
A, B, C, D, E, F, G, H = 1.06036, 0.1561, 0.193, 0.47635, 1.03587, 1.52996, 1.76474, 3.89411
elif l == 1 and s == 2:
A, B, C, D, E, F, G, H = 1.0022, 0.1553, 0.16105, 0.72751, 0.86125, 2.06848, 1.95162, 4.84492
elif l == 1 and s == 3:
A, B, C, D, E, F, R, S, W, P = 0.96573, 0.15611, 0.44067, 1.5242, 2.38981, 5.08063, -0.0005373, 19.2866, -1.30775, 6.58711
elif l == 1 and s == 4:
A, B, C, D, E, F, R, S, W, P = 0.93447, 0.15578, 0.39478, 1.85761, 2.45988, 6.15727, 0.0004246, 12.988, -1.36399, 3.3329
elif l == 1 and s == 5:
A, B, C, D, E, F, R, S, W, P = 0.90972, 0.15565, 0.35967, 2.18528, 2.45169, 7.17936, -0.0003814, 9.38191, 0.14025, 9.93802
elif l == 1 and s == 6:
A, B, C, D, E, F, R, S, W, P = 0.88928, 0.15562, 0.33305, 2.51303, 2.36298, 8.1169, -0.0004649, 9.86928, 0.12851, 9.82414
elif l == 1 and s == 7:
A, B, C, D, E, F, R, S, W, P = 0.87208, 0.15568, 0.36583, 3.01399, 2.70659, 9.9231, -0.0004902, 10.2274, 0.12306, 9.97712
elif l == 2 and s == 2:
A, B, C, D, E, F, R, S, W, P = 1.16145, 0.14874, 0.52487, 0.7732, 2.16178, 2.43787, -0.0006435, 18.0323, -0.7683, 7.27371
elif l == 2 and s == 3:
A, B, C, D, E, F, R, S, W, P = 1.11521, 0.14796, 0.44844, 0.99548, 2.30009, 3.06031, 0.0004565, 38.5868, -0.69403, 2.56375
elif l == 2 and s == 4:
A, B, C, D, E, F, R, S, W, P = 1.08228, 0.14807, 0.47128, 1.31596, 2.42738, 3.90018, -0.0005623, 3.08449, 0.28271, 3.22871
elif l == 2 and s == 5:
A, B, C, D, E, F, R, S, W, P = 1.05581, 0.14822, 0.51203, 1.67007, 2.57317, 4.85939, -0.000712, 4.7121, 0.2173, 4.7353
elif l == 2 and s == 6:
A, B, C, D, E, F, R, S, W, P = 1.03358, 0.14834, 0.53928, 2.01942, 2.7235, 5.84817, -0.0008576, 7.66012, 0.15493, 7.6011
elif l == 3 and s == 3:
A, B, C, D, E, F, G, H, R, S, W, P = 1.05567, 0.1498, 0.30887, 0.86437, 1.35766, 2.44123, 1.2903, 5.55734, 0.0002339, 57.7757, -1.0898, 6.9475
elif l == 3 and s == 4:
A, B, C, D, E, F, R, S, W, P = 1.02621, 0.1505, 0.55381, 1.4007, 2.06176, 4.26234, 0.0005227, 11.3331, -0.8209, 3.87185
elif l == 3 and s == 5:
A, B, C, D, E, F, R, S, W, P = 0.99958, 0.15029, 0.50441, 1.64304, 2.06947, 4.87712, -0.0005184, 3.45031, 0.26821, 3.73348
elif l == 4 and s == 4:
A, B, C, D, E, F, R, S, W, P = 1.12007, 0.14578, 0.53347, 1.11986, 2.28803, 3.27567, 0.0007427, 21.048, -0.28759, 6.69149
else:
raise ValueError('Input values of l and s are not supported')
omega = A / T_star**B + C / exp(D * T_star) + E / exp(F * T_star)
if (l == 1 and (s == 1 or s == 2)) or (s == 3 and l == 3):
omega += G / exp(H * T_star)
if not (l == 1 and (s == 1 or s == 2)):
omega += R * T_star**B * sin(S * T_star**W - P)
return omega
def Antoine_coeffs_from_point(T, Psat, dPsat_dT, d2Psat_dT2, base=10.0):
r'''Calculates the antoine coefficients `A`, `B`, and `C` from a known
vapor pressure and its first and second temperature derivative.
Parameters
----------
T : float
Temperature of fluid, [K]
Psat : float
Vapor pressure at specified `T` [Pa]
dPsat_dT : float
First temperature derivative of vapor pressure at specified `T` [Pa/K]
d2Psat_dT2 : float
Second temperature derivative of vapor pressure at specified `T` [Pa/K^2]
Base : float, optional
Base of logarithm; 10 by default
Returns
-------
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
Notes
-----
Coefficients are for calculating vapor pressure in Pascal. This is
primarily useful for interconverting vapor pressure models, not fitting
experimental data.
Derived with SymPy as follows:
>>> from sympy import * # doctest: +SKIP
>>> base, A, B, C, T = symbols('base, A, B, C, T') # doctest: +SKIP
>>> v = base**(A - B/(T + C)) # doctest: +SKIP
>>> d1, d2 = diff(v, T), diff(v, T, 2) # doctest: +SKIP
>>> vk, d1k, d2k = symbols('vk, d1k, d2k') # doctest: +SKIP
>>> solve([Eq(v, vk), Eq(d1, d1k), Eq(d2, d2k)], [A, B, C]) # doctest: +SKIP
Examples
--------
Recalculate some coefficients from a calcualted value and its derivative:
>>> T = 178.01
>>> A, B, C = (24.0989474955895, 4346.793091137991, -18.96968471040141)
>>> Psat = Antoine(T, A, B, C, base=exp(1))
>>> dPsat_dT, d2Psat_dT2 = (0.006781441203850251, 0.0010801244983894853) # precomputed
>>> Antoine_coeffs_from_point(T, Psat, dPsat_dT, d2Psat_dT2, base=exp(1))
(24.098947495155453, 4346.793090994682, -18.969684713118813)
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
'''
A = log(Psat * exp(2 * dPsat_dT**2 /
(dPsat_dT**2 - d2Psat_dT2 * Psat))) / log(base)
B = 4 * dPsat_dT**3 * Psat / (
(dPsat_dT**4 - 2 * dPsat_dT**2 * d2Psat_dT2 * Psat +
d2Psat_dT2**2 * Psat**2) * log(base))
C = (-T * dPsat_dT**2 + T * d2Psat_dT2 * Psat +
2 * dPsat_dT * Psat) / (dPsat_dT**2 - d2Psat_dT2 * Psat)
return (A, B, C)
