"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2019, 2020, 2021 Caleb Bell
<Caleb.Andrew.Bell@gmail.com>
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
This module contains various vapor pressure estimation routines, dataframes
of fit coefficients, some compound-specific equations, some analytical fitting
routines, and sublimation pressure routines.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/chemicals/>`_.
.. contents:: :local:
Fit Correlations
----------------
.. autofunction:: chemicals.vapor_pressure.Antoine
.. autofunction:: chemicals.vapor_pressure.Wagner
.. autofunction:: chemicals.vapor_pressure.Wagner_original
.. autofunction:: chemicals.vapor_pressure.TRC_Antoine_extended
.. autofunction:: chemicals.vapor_pressure.Yaws_Psat
.. autofunction:: chemicals.vapor_pressure.TDE_PVExpansion
Fit Correlation Derivatives
---------------------------
.. autofunction:: chemicals.vapor_pressure.dAntoine_dT
.. autofunction:: chemicals.vapor_pressure.d2Antoine_dT2
.. autofunction:: chemicals.vapor_pressure.dWagner_dT
.. autofunction:: chemicals.vapor_pressure.d2Wagner_dT2
.. autofunction:: chemicals.vapor_pressure.dWagner_original_dT
.. autofunction:: chemicals.vapor_pressure.d2Wagner_original_dT2
.. autofunction:: chemicals.vapor_pressure.dTRC_Antoine_extended_dT
.. autofunction:: chemicals.vapor_pressure.d2TRC_Antoine_extended_dT2
.. autofunction:: chemicals.vapor_pressure.dYaws_Psat_dT
.. autofunction:: chemicals.vapor_pressure.d2Yaws_Psat_dT2
Jacobians (for fitting)
-----------------------
.. autofunction:: chemicals.vapor_pressure.Wagner_fitting_jacobian
.. autofunction:: chemicals.vapor_pressure.Wagner_original_fitting_jacobian
.. autofunction:: chemicals.vapor_pressure.Antoine_fitting_jacobian
.. autofunction:: chemicals.vapor_pressure.Yaws_Psat_fitting_jacobian
.. autofunction:: chemicals.vapor_pressure.TRC_Antoine_extended_fitting_jacobian
Vapor Pressure Estimation Correlations
--------------------------------------
.. autofunction:: chemicals.vapor_pressure.Lee_Kesler
.. autofunction:: chemicals.vapor_pressure.Ambrose_Walton
.. autofunction:: chemicals.vapor_pressure.boiling_critical_relation
.. autofunction:: chemicals.vapor_pressure.Sanjari
.. autofunction:: chemicals.vapor_pressure.Edalat
Sublimation Pressure Estimation Correlations
--------------------------------------------
.. autofunction:: chemicals.vapor_pressure.Psub_Clapeyron
Correlations for Specific Substances
------------------------------------
.. autofunction:: chemicals.vapor_pressure.Psat_IAPWS
.. autofunction:: chemicals.vapor_pressure.dPsat_IAPWS_dT
.. autofunction:: chemicals.vapor_pressure.Tsat_IAPWS
Analytical Fit Equations
------------------------
.. autofunction:: chemicals.vapor_pressure.Antoine_coeffs_from_point
.. autofunction:: chemicals.vapor_pressure.Antoine_AB_coeffs_from_point
.. autofunction:: chemicals.vapor_pressure.DIPPR101_ABC_coeffs_from_point
Fit Coefficients
----------------
All of these coefficients are lazy-loaded, so they must be accessed as an
attribute of this module.
.. data:: Psat_data_WagnerMcGarry
Coefficients for the Wagner 3,6 original model equation documented in
:obj:`Wagner_original` with data for 245 chemicals, from [1]_.
.. data:: Psat_data_WagnerPoling
Coefficients for the Wagner 2.5, 5 model equation documented in
:obj:`Wagner` in [2]_, with data for 104 chemicals.
.. data:: Psat_data_AntoinePoling
Standard Antoine equation coefficients, as documented in the function
:obj:`Antoine` and with data for 325 fluids from [2]_.
Coefficients were altered to be in units of Pa and Celcius.
.. data:: Psat_data_AntoineExtended
Data for 97 chemicals in [2]_ for the TRC extended Antoine model
:obj:`TRC_Antoine_extended`.
.. data:: Psat_data_Perrys2_8
A collection of 341 coefficient sets for :obj:`thermo.dippr.EQ101` from
the DIPPR database published openly in [4]_.
.. data:: Psat_data_VDI_PPDS_3
Coefficients for the Wagner equation :obj:`Wagner`, published
openly in [3]_.
.. data:: Psat_data_Alcock_elements
Coefficients for the DIPPR 101 equation :obj:`chemicals.dippr.EQ101`,
published in [5]_ and converted to provide base SI units (and use the
natural logarithm).
.. data:: Psub_data_Alcock_elements
Coefficients for the DIPPR 101 equation :obj:`chemicals.dippr.EQ101`,
published in [5]_ and converted to provide base SI units (and use the
natural logarithm). Note this is a sublimation pressure data set.
.. data:: Psub_data_Landolt_Antoine
Standard Antoine equation coefficients for sublimation pressure,
as documented in the function
:obj:`Antoine` and with data for ~1000 solids from [6]_, [7]_, and [8]_.
Coefficients were altered to be in units of Pa and Kelvin with the
exponential instead of base-10 power.
.. data:: Psat_data_Landolt_Antoine
Standard Antoine equation coefficients for vapor pressure,
as documented in the function
:obj:`Antoine` and with data for ~6000 liquids from [6]_, [7]_, and [8]_.
Coefficients were altered to be in units of Pa and Kelvin with the
exponential instead of base-10 power.
.. [1] 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.
.. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
.. [3] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
Berlin; New York:: Springer, 2010.
.. [4] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
Eighth Edition. McGraw-Hill Professional, 2007.
.. [5] Alcock, C. B., V. P. Itkin, and M. K. Horrigan. "Vapour Pressure
Equations for the Metallic Elements: 298-2500K." Canadian Metallurgical
Quarterly 23, no. 3 (July 1, 1984): 309-13.
https://doi.org/10.1179/cmq.1984.23.3.309.
.. [6] Hall, K. R. Vapor Pressure and Antoine Constants for Hydrocarbons,
and S, Se, Te, and Halogen Containing Organic Compounds. Springer, 1999.
.. [7] Dykyj, J., and K. R. Hall. "Vapor Pressure and Antoine Constants for
Oxygen Containing Organic Compounds". 2000.
.. [8] Hall, K. R. Vapor Pressure and Antoine Constants for Nitrogen
Containing Organic Compounds. Springer, 2001.
The structure of each dataframe is shown below:
.. ipython::
In [1]: import chemicals
In [2]: chemicals.vapor_pressure.Psat_data_WagnerMcGarry
In [3]: chemicals.vapor_pressure.Psat_data_WagnerPoling
In [4]: chemicals.vapor_pressure.Psat_data_AntoinePoling
In [5]: chemicals.vapor_pressure.Psat_data_AntoineExtended
In [6]: chemicals.vapor_pressure.Psat_data_Perrys2_8
In [7]: chemicals.vapor_pressure.Psat_data_VDI_PPDS_3
In [8]: chemicals.vapor_pressure.Psat_data_Alcock_elements
In [9]: chemicals.vapor_pressure.Psub_data_Alcock_elements
In [10]: chemicals.vapor_pressure.Psub_data_Landolt_Antoine
In [11]: chemicals.vapor_pressure.Psat_data_Landolt_Antoine
"""
__all__ = ['Antoine','dAntoine_dT', 'd2Antoine_dT2',
'Wagner_original', 'dWagner_original_dT', 'd2Wagner_original_dT2',
'Wagner', 'dWagner_dT', 'd2Wagner_dT2',
'TRC_Antoine_extended', 'dTRC_Antoine_extended_dT',
'd2TRC_Antoine_extended_dT2', 'dYaws_Psat_dT',
'boiling_critical_relation', 'Lee_Kesler', 'Ambrose_Walton',
'Edalat', 'Sanjari', 'Psat_IAPWS', 'dPsat_IAPWS_dT', 'Tsat_IAPWS',
'Psub_Clapeyron', 'Yaws_Psat', 'd2Yaws_Psat_dT2',
'Antoine_coeffs_from_point', 'Antoine_AB_coeffs_from_point',
'DIPPR101_ABC_coeffs_from_point', 'Wagner_original_fitting_jacobian',
'Wagner_fitting_jacobian', 'Yaws_Psat_fitting_jacobian',
'Antoine_fitting_jacobian', 'TRC_Antoine_extended_fitting_jacobian',
'TDE_PVExpansion']
from math import isinf, log10
from fluids.constants import R
from fluids.numerics import exp, log, sqrt, trunc_exp
from fluids.numerics import numpy as np
from chemicals.data_reader import data_source, register_df_source
from chemicals.utils import PY37, can_load_data, mark_numba_incompatible, os_path_join, source_path
folder = os_path_join(source_path, 'Vapor Pressure')
register_df_source(folder, 'Antoine Collection Poling.tsv')
register_df_source(folder, 'Table 2-8 Vapor Pressure of Inorganic and Organic Liquids.tsv')
register_df_source(folder, 'Alcock_Itkin_Horrigan_metalic_elements.tsv',
csv_kwargs={'dtype':{'Tmin': float, 'Tmax': float, 'A': float,
'B': float, 'C': float, 'D': float, 'E': float}})
register_df_source(folder, 'Landolt_antoine_sublimation_V20.tsv',
csv_kwargs={'dtype':{'Tmin': float, 'Tmax': float, 'A': float,
'B': float, 'C': float}})
register_df_source(folder, 'Landolt_antoine_V20.tsv',
csv_kwargs={'dtype':{'Tmin': float, 'Tmax': float, 'A': float,
'B': float, 'C': float}})
register_df_source(folder, 'Alcock_Itkin_Horrigan_metalic_elements_sublimation.tsv',
csv_kwargs={'dtype':{'Tmin': float, 'Tmax': float, 'A': float,
'B': float, 'C': float, 'D': float}}, index_col=1)
register_df_source(folder, 'Wagner Original McGarry.tsv', csv_kwargs={
'dtype': {'A': float, 'B': float, 'C': float, 'D': float,
'Pc': float, 'Tc': float, 'Tmin': float}})
register_df_source(folder, 'Wagner Collection Poling.tsv', csv_kwargs={
'dtype': {'A': float, 'B': float, 'C': float, 'D': float, 'Pc': float,
'Tc': float, 'Tmin': float, 'Tmax': float}})
register_df_source(folder, 'Antoine Extended Collection Poling.tsv', csv_kwargs={
'dtype':{'A': float, 'B': float, 'C': float, 'Tc': float, 'to': float,
'n': float, 'E': float, 'F': float, 'Tmin': float, 'Tmax': float}})
register_df_source(folder, 'VDI PPDS Boiling temperatures at different pressures.tsv', csv_kwargs={
'dtype':{'Tm': float, 'Tc': float, 'Pc': float, 'A': float,
'B': float, 'C': float, 'D': float}})
_vapor_pressure_dfs_loaded = False
@mark_numba_incompatible
def load_vapor_pressure_dfs():
global Psat_data_WagnerMcGarry, Psat_values_WagnerMcGarry, Psat_data_AntoinePoling, Psat_values_AntoinePoling
global Psat_data_WagnerPoling, Psat_values_WagnerPoling, Psat_data_AntoineExtended, Psat_values_AntoineExtended
global Psat_data_Perrys2_8, Psat_values_Perrys2_8, Psat_data_VDI_PPDS_3, Psat_values_VDI_PPDS_3
global Psat_data_Alcock_elements, Psat_values_Alcock_elements, Psub_data_Alcock_elements, Psub_values_Alcock_elements
global Psub_values_Landolt_Antoine, Psub_data_Landolt_Antoine
global Psat_values_Landolt_Antoine, Psat_data_Landolt_Antoine
global _vapor_pressure_dfs_loaded
if _vapor_pressure_dfs_loaded:
return
# 57463 bytes for df; 13720 bytes for numpy
Psat_data_WagnerMcGarry = data_source('Wagner Original McGarry.tsv')
Psat_values_WagnerMcGarry = np.array(Psat_data_WagnerMcGarry.values[:, 1:], dtype=float)
# 58216 bytes for df; 13000 bytes for numpy
Psat_data_AntoinePoling = data_source('Antoine Collection Poling.tsv')
Psat_values_AntoinePoling = np.array(Psat_data_AntoinePoling.values[:, 1:], dtype=float)
Psat_data_Alcock_elements = data_source('Alcock_Itkin_Horrigan_metalic_elements.tsv')
Psat_values_Alcock_elements = np.array(Psat_data_Alcock_elements.values[:, 1:], dtype=float)
Psub_data_Alcock_elements = data_source('Alcock_Itkin_Horrigan_metalic_elements_sublimation.tsv')
Psub_values_Alcock_elements = np.array(Psub_data_Alcock_elements.values[:, 1:], dtype=float)
# 20928 bytes for df; 7488 bytes for numpy
Psat_data_WagnerPoling = data_source('Wagner Collection Poling.tsv')
Psat_values_WagnerPoling = np.array(Psat_data_WagnerPoling.values[:, 1:], dtype=float)
# 21388 bytes for df; 7760 bytes for numpy
Psat_data_AntoineExtended = data_source('Antoine Extended Collection Poling.tsv')
Psat_values_AntoineExtended = np.array(Psat_data_AntoineExtended.values[:, 1:], dtype=float)
# 65740 bytes for df; 21760 bytes for numpy
Psat_data_Perrys2_8 = data_source('Table 2-8 Vapor Pressure of Inorganic and Organic Liquids.tsv')
Psat_values_Perrys2_8 = np.array(Psat_data_Perrys2_8.values[:, 1:], dtype=float)
# 52742 bytes for df; 15400 bytes for numpy
Psat_data_VDI_PPDS_3 = data_source('VDI PPDS Boiling temperatures at different pressures.tsv')
Psat_values_VDI_PPDS_3 = np.array(Psat_data_VDI_PPDS_3.values[:, 1:], dtype=float)
Psub_data_Landolt_Antoine = data_source('Landolt_antoine_sublimation_V20.tsv')
Psub_values_Landolt_Antoine = np.array(Psub_data_Landolt_Antoine.values[:, 1:], dtype=float)
Psat_data_Landolt_Antoine = data_source('Landolt_antoine_V20.tsv')
Psat_values_Landolt_Antoine = np.array(Psat_data_Landolt_Antoine.values[:, 1:], dtype=float)
_vapor_pressure_dfs_loaded = True
if PY37:
def __getattr__(name):
if name in ('Psat_data_WagnerMcGarry', 'Psat_values_WagnerMcGarry', 'Psat_data_AntoinePoling',
'Psat_values_AntoinePoling', 'Psat_data_WagnerPoling', 'Psat_values_WagnerPoling',
'Psat_data_AntoineExtended', 'Psat_values_AntoineExtended', 'Psat_data_Perrys2_8',
'Psat_values_Perrys2_8', 'Psat_data_VDI_PPDS_3', 'Psat_values_VDI_PPDS_3',
'Psat_data_Alcock_elements', 'Psat_values_Alcock_elements',
'Psub_data_Alcock_elements', 'Psub_values_Alcock_elements',
'Landolt_data_sublimation_Antoine', 'Psub_values_Landolt_Antoine',
'Psat_data_Landolt_Antoine', 'Psat_values_Landolt_Antoine'):
load_vapor_pressure_dfs()
return globals()[name]
raise AttributeError(f"module {__name__} has no attribute {name}")
else:
if can_load_data:
load_vapor_pressure_dfs()
[docs]def TDE_PVExpansion(T, a1, a2, a3, a4=0.0, a5=0.0, a6=0.0, a7=0.0, a8=0.0):
r'''Calculates vapor pressure or sublimation pressure of a chemical using
the PVExpansion equation for vapor pressure or sublimation pressure.
Parameters `a1`, `a2`, `a3`, `a4`, `a5`, `a6`, `a7`, and `a8`
are chemical-dependent. Parameters
can be found in various sources; however units of the coefficients used
vary.
.. math::
\log P^{\text{sat}} = a_1 + \frac{a_2}{T} + a_3\ln(T) + a_4T + a_5T^2
+ \frac{a_6}{T^2} + a_7 T^6 + \frac{a_8}{T^4}
Parameters
----------
T : float
Temperature of fluid, [K]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
a3 : float
Regression parameter, [-]
a4 : float
Regression parameter, [-]
a5 : float
Regression parameter, [-]
a6 : float
Regression parameter, [-]
a7 : float
Regression parameter, [-]
a8 : float
Regression parameter, [-]
Returns
-------
Psat : float
Vapor pressure calculated with coefficients [Pa]
Notes
-----
Coefficients in [1]_ produce a vapor pressure in kPa; add log(1000) to
`a1` to make them produce vapor pressure in Pa.
Examples
--------
Coefficients for sublimation pressure from [1]_:
>>> TDE_PVExpansion(T=273.16, a1=23.7969+log(1000), a2=-11422, a3=0.177978)
4.06220657398e-05
References
----------
.. [1] "ThermoData Engine (TDE103a V10.1) User`s Guide."
https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-PhaseBoundaryLG/PVExpansion.htm
'''
T2 = T*T
return trunc_exp(a1 + a2/T + a3*log(T) + a4*T + a5*T2 + a6/T2 + a7*T2*T2*T2 + a8/(T2*T2))
[docs]def Yaws_Psat(T, A, B, C, D, E):
r'''Calculates vapor pressure of a chemical using the Yaws equation for
vapor pressure.
Parameters `A`, `B`, `C`, `D`, and `E` are chemical-dependent. Parameters
can be found in numerous sources; however units of the coefficients used
vary.
.. math::
\log_{10} P^{\text{sat}} = A + \frac{B}{T} + C\log_{10}(T) + DT + ET^2
Parameters
----------
T : float
Temperature of fluid, [K]
A : float
`A` parameter, [-]
B : float
`B` parameter, [K]
C : float
`C` parameter, [-]
D : float
`D` parameter, [1/K]
E : float
`E` parameter, [1/K^2]
Returns
-------
Psat : float
Vapor pressure calculated with coefficients [Pa]
Notes
-----
Assumes coefficients are for calculating vapor pressure in Pascal.
Coefficients should be consistent with input temperatures in Kelvin;
**Converting units in input coefficients:**
* **mmHg to Pa**: Add log10(101325/760)= 2.1249 to A.
* **kPa to Pa**: Add log_{10}(1000)= 3 to A
* **bar to Pa**: Add log_{10}(100000)= 5 to A
Examples
--------
Acetone, coefficients from [1]_, at 400 K and with the conversion of `A`
to obtain a result in Pa:
>>> Yaws_Psat(T=400.0, A=28.588 + log10(101325/760), B=-2469, C=-7.351, D=2.8025E-10, E=2.7361E-6)
708657.089106
Coefficients for benzene from [2]_ at 400 K; that source outputs vapor
pressure in kPa. That style of coefficients can be converted to `Pa`
by adding `3` to `A`.
>>> Yaws_Psat(T=400.0, A=39.7918+3, B=-2965.83, C=-12.073, D=0.0033269, E=1.58609e-6)
352443.191026
References
----------
.. [1] Yaws, Carl L. Chemical Properties Handbook: Physical, Thermodynamic,
Environmental, Transport, Safety, and Health Related Properties for
Organic and Inorganic Chemicals. McGraw-Hill, 2001.
.. [2] "ThermoData Engine (TDE103a V10.1) User`s Guide."
https://trc.nist.gov/TDE/Help/TDE103a/Eqns-Pure-PhaseBoundaryLG/Yaws-VaporPressure.htm.
'''
exponent = (A + B/T + C*log10(T) + T*(D + E*T))
if exponent > 308.0:
return 1e308
return 10.0**exponent
[docs]def dYaws_Psat_dT(T, A, B, C, D, E):
r'''Calculates the first temperature derivative of vapor pressure of a
chemical using the Yaws equation for vapor pressure.
Parameters `A`, `B`, `C`, `D`, and `E` are chemical-dependent. Parameters
can be found in numerous sources; however units of the coefficients used
vary.
.. math::
\frac{\partial P^{\text{sat}} }{\partial T} = 10^{A + \frac{B}{T}
+ \frac{C \log{\left(T \right)}}{\log{\left(10 \right)}} + D T
+ E T^{2}} \left(- \frac{B}{T^{2}} + \frac{C}{T \log{\left(10 \right)}}
+ D + 2 E T\right) \log{\left(10 \right)}
Parameters
----------
T : float
Temperature of fluid, [K]
A : float
`A` parameter, [-]
B : float
`B` parameter, [K]
C : float
`C` parameter, [-]
D : float
`D` parameter, [1/K]
E : float
`E` parameter, [1/K^2]
Returns
-------
dPsat_dT : float
First temperature derivative of vapor pressure calculated with
coefficients [Pa/K]
Examples
--------
Benzene:
>>> dYaws_Psat_dT(T=400.0, A=42.7918, B=-2965.83, C=-12.073, D=0.0033269, E=1.58609e-6)
8134.87548930
'''
x0 = 2.302585092994046
x1 = T*T
x2 = 1.0/T
x3 = C/x0
return 10.0**(A + B*x2 + D*T + E*x1 + x3*log(T))*x0*(-B/x1 + D + 2.0*E*T + x2*x3)
[docs]def d2Yaws_Psat_dT2(T, A, B, C, D, E):
r'''Calculates the second temperature derivative of vapor pressure of a
chemical using the Yaws equation for vapor pressure.
Parameters `A`, `B`, `C`, `D`, and `E` are chemical-dependent. Parameters
can be found in numerous sources; however units of the coefficients used
vary.
.. math::
\frac{\partial^2 P^{\text{sat}} }{\partial T^2} = 10^{A + \frac{B}{T}
+ \frac{C \log{\left(T \right)}}{\log{\left(10 \right)}} + D T
+ E T^{2}} \left(\frac{2 B}{T^{3}} - \frac{C}{T^{2} \log{\left(10
\right)}} + 2 E + \left(- \frac{B}{T^{2}} + \frac{C}{T \log{\left(10
\right)}} + D + 2 E T\right)^{2} \log{\left(10 \right)}\right)
\log{\left(10 \right)}
Parameters
----------
T : float
Temperature of fluid, [K]
A : float
`A` parameter, [-]
B : float
`B` parameter, [K]
C : float
`C` parameter, [-]
D : float
`D` parameter, [1/K]
E : float
`E` parameter, [1/K^2]
Returns
-------
d2Psat_dT2 : float
Second temperature derivative of vapor pressure calculated with
coefficients [Pa/K^2]
Examples
--------
Benzene:
>>> d2Yaws_Psat_dT2(T=400.0, A=42.7918, B=-2965.83, C=-12.073, D=0.0033269, E=1.58609e-6)
141.7181045862
'''
x0 = 2.302585092994046
x1 = 1.0/T
x2 = T*T
x3 = C/x0
x4 = 2.0*E
x5 = 1.0/x2
x6 = (-B*x5 + D + T*x4 + x1*x3)
return 10.0**(A + B*x1 + D*T + E*x2 + x3*log(T))*x0*(2.0*B*x1*x1*x1 + x0*x6*x6 - x3*x5 + x4)
[docs]def Yaws_Psat_fitting_jacobian(Ts, A, B, C, D, E):
r'''Compute and return the Jacobian of the property predicted by
the Yaws vapor pressure equation with respect to all the coefficients. This is
used in fitting parameters for chemicals.
Parameters
----------
Ts : list[float]
Temperatures of the experimental data points, [K]
A : float
`A` parameter, [-]
B : float
`B` parameter, [K]
C : float
`C` parameter, [-]
D : float
`D` parameter, [1/K]
E : float
`E` parameter, [1/K^2]
Returns
-------
jac : list[list[float, 5], len(Ts)]
Matrix of derivatives of the equation with respect to the fitting
parameters, [various]
'''
x0 = 2.302585092994046
N = len(Ts)
# out = np.zeros((N, 5)) # numba: uncomment
out = [[0.0]*5 for _ in range(N)] # numba: delete
for i in range(N):
T = Ts[i]
r = out[i]
x1 = 1.0/T
x2 = T*T
x3 = log(T)
x4 = 10.0**(A + B*x1 + C*x3/x0 + D*T + E*x2)
x5 = x0*x4
r[0] = x5
r[1] = x1*x5
r[2] = x3*x4
r[3] = T*x5
r[4] = x2*x5
return out
[docs]def Antoine(T, A, B, C, base=10.0):
r'''Calculates vapor pressure of a chemical using the Antoine equation.
Parameters `A`, `B`, and `C` are chemical-dependent. Parameters can be
found in numerous sources; however units of the coefficients used vary.
Originally proposed by Antoine (1888) [2]_.
.. math::
\log_{\text{base}} P^{\text{sat}} = A - \frac{B}{T+C}
Parameters
----------
T : float
Temperature of fluid, [K]
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
base : float, optional
Optional base of logarithm; 10 by default
Returns
-------
Psat : float
Vapor pressure calculated with coefficients [Pa]
Notes
-----
Assumes coefficients are for calculating vapor pressure in Pascal.
Coefficients should be consistent with input temperatures in Kelvin;
however, if both the given temperature and units are specific to degrees
Celcius, the result will still be correct.
**Converting units in input coefficients:**
* **ln to log10**: Divide A and B by ln(10)=2.302585 to change
parameters for a ln equation to a log10 equation.
* **log10 to ln**: Multiply A and B by ln(10)=2.302585 to change
parameters for a log equation to a ln equation.
* **mmHg to Pa**: Add log10(101325/760)= 2.1249 to A.
* **kPa to Pa**: Add log_{base}(1000)= 6.908 to A for log(base)
* **bar to Pa**: Add log_{base}(100000)= 11.5129254 to A for log(base)
* **°C to K**: Subtract 273.15 from C only!
Note that if `C` is negative and `T` is less than `C`, the predicted vapor
pressure would be high and positive at those temperatures under `C`; and
a singularity would occur at `T` == `C`. This implementation is corrected
to return zero for the case of `T + C < 0.0`, which matches the intention
of the Antoine equation.
Examples
--------
Methane, coefficients from [1]_, at 100 K:
>>> Antoine(100.0, 8.7687, 395.744, -6.469)
34478.367349639906
Tetrafluoromethane, coefficients from [1]_, at 180 K
>>> Antoine(180, A=8.95894, B=510.595, C=-15.95)
702271.0518579542
Oxygen at 94.91 K, with coefficients from [3]_ in units of °C, mmHg, log10,
showing the conversion of coefficients A (mmHg to Pa) and C (°C to K)
>>> Antoine(94.91, 6.83706+2.1249, 339.2095, 268.70-273.15)
162978.88655572367
n-hexane with Antoine coefficients from the NIST webbook in units of K and
bar, calculating the vapor pressure in Pa at 200 K:
>>> Antoine(T=200, A=3.45604+5, B=1044.038, C=-53.893)
20.4329803671
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
.. [2] Antoine, C. 1888. Tensions des Vapeurs: Nouvelle Relation Entre les
Tensions et les Tempé. Compt.Rend. 107:681-684.
.. [3] Yaws, Carl L. The Yaws Handbook of Vapor Pressure: Antoine
Coefficients. 1 edition. Houston, Tex: Gulf Publishing Company, 2007.
'''
T_C = T + C
if T_C <= 0.0:
return 0.0
return base**(A - B/(T_C))
[docs]def dAntoine_dT(T, A, B, C, base=10.0):
r'''Calculates the first temperature derivative of vapor pressure of a
chemical using the Antoine equation.
Parameters `A`, `B`, and `C` are chemical-dependent.
.. math::
\frac{\partial P^{\text{sat}} }{\partial T} =
\frac{B \text{base}^{A - \frac{B}{C + T}} \log{\left(\text{base} \right)}}
{\left(C + T\right)^{2}}
Parameters
----------
T : float
Temperature of fluid, [K]
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
base : float, optional
Optional base of logarithm; 10 by default
Returns
-------
dPsat_dT : float
First temperature derivative of vapor pressure calculated with
coefficients [Pa/K]
Notes
-----
Examples
--------
Methane at 100 K:
>>> dAntoine_dT(100.0, 8.7687, 395.744, -6.469)
3591.4147747481
'''
T_C = T + C
if T_C <= 0.0:
return 0.0
den = 1.0/(T_C)
return B*base**(A - B*den)*log(base)*den*den
[docs]def d2Antoine_dT2(T, A, B, C, base=10.0):
r'''Calculates the second temperature derivative of vapor pressure of a
chemical using the Antoine equation.
Parameters `A`, `B`, and `C` are chemical-dependent.
.. math::
\frac{\partial^2 P^{\text{sat}} }{\partial T^2} =
\frac{B \text{base}^{A - \frac{B}{C + T}} \left(\frac{B \log{\left(
\text{base} \right)}}{C + T} - 2\right) \log{\left(\text{base}
\right)}}{\left(C + T\right)^{3}}
Parameters
----------
T : float
Temperature of fluid, [K]
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
base : float, optional
Optional base of logarithm; 10 by default
Returns
-------
d2Psat_dT2 : float
Second temperature derivative of vapor pressure calculated with
coefficients [Pa/K^2]
Notes
-----
Examples
--------
Methane at 100 K:
>>> d2Antoine_dT2(100.0, 8.7687, 395.744, -6.469)
297.30093799054
'''
T_C = T + C
if T_C <= 0.0:
return 0.0
den = 1.0/(T_C)
log_base = log(base)
x0 = B*den*log_base
return x0*base**(A - B*den)*(x0 - 2.0)*den*den
[docs]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))
(24.098947495155, 4346.793090994, -18.969684713118)
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
'''
x0 = 1.0/log(base)
x1 = Psat*d2Psat_dT2
dPsat_dT_2 = dPsat_dT*dPsat_dT
x3 = 1.0/(x1 - dPsat_dT_2)
x4 = dPsat_dT_2 + dPsat_dT_2
A = x0*log(Psat*exp(-x3*x4))
B = 4.0*Psat*dPsat_dT*dPsat_dT_2*x0/(Psat*Psat*d2Psat_dT2*d2Psat_dT2 + dPsat_dT_2*dPsat_dT_2 - x1*x4)
C = -x3*(2.0*Psat*dPsat_dT + T*x1 - T*dPsat_dT_2)
return (A, B, C)
[docs]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
# The expression from SymPy is as follows
#A = log(Psat*exp(T*dPsat_dT*Psat_inv))*log_base_inv
# However, that expression has overflows a lot
# Mathematical manipulation yields the following, which avoids overflows
A = (T*dPsat_dT*Psat_inv + log(Psat))*log_base_inv
B = T*T*dPsat_dT*log_base_inv*Psat_inv
return (A, B)
[docs]def DIPPR101_ABC_coeffs_from_point(T, Psat, dPsat_dT, d2Psat_dT2):
r'''Calculates the first three DIPPR101 coefficients `A`, `B`, and `C`
from a known vapor pressure and its first and second temperature derivative.
If the second derivative is infinity as is the case in some vapor pressure
models at the critical point, only the `A` and `C` coefficients are fit,
using the first derivative an the actual value of vapor pressure.
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]
Returns
-------
A : float
DIPPR101 `A` parameter (same as Antoine `A`), [-]
B : float
DIPPR101 `B` parameter (same as Antoine `B`), [K]
C: float
DIPPR101 `C` parameter (NOT same as Antoine `C`, multiplied by log(T)),
[-]
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 = exp(A + B/T + C*log(T)) # 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
--------
Calculate the coefficients:
>>> T = 178.01
>>> Psat, dPsat_dT, d2Psat_dT2 = (0.03946094565666715, 0.006781441203850251, 0.0010801244983894853)
>>> DIPPR101_ABC_coeffs_from_point(T, Psat, dPsat_dT, d2Psat_dT2)
(72.47169926642, -6744.620564969, -7.2976291987890)
'''
if isinf(d2Psat_dT2):
# We cannot match the second derivative, so there will definitely be
# a discontinuity
# This means only two parameters can be obtained
# If the A and B parameters are matched, the vapor pressure will go down
# so the A and C paramters have to be matched.
return (-T*dPsat_dT*log(T)/Psat + log(Psat), 0.0, T*dPsat_dT/Psat)
x0 = Psat*Psat
x1 = 1.0/x0
x2 = Psat*dPsat_dT
x3 = -T*dPsat_dT*dPsat_dT
x4 = T*d2Psat_dT2
x5 = T*(Psat*(2.0*dPsat_dT + x4) + x3)
A = x1*(T*x2 + x0*log(Psat) - x5*log(T) - x5)
B = T*T*x1*(Psat*x4 + x2 + x3)
C = x1*x5
return (A, B, C)
[docs]def TRC_Antoine_extended(T, Tc, to, A, B, C, n, E, F):
r'''Calculates vapor pressure of a chemical using the TRC Extended Antoine
equation. Parameters are chemical dependent, and said to be from the
Thermodynamics Research Center (TRC) at Texas A&M. Coefficients for various
chemicals can be found in [1]_.
.. math::
\log_{10} P^{sat} = A - \frac{B}{T + C} + 0.43429x^n + Ex^8 + Fx^{12}
.. math::
x = \max \left(\frac{T-t_o-273.15}{T_c}, 0 \right)
Parameters
----------
T : float
Temperature of fluid, [K]
Tc : float
Critical temperature of fluid, [K]
to : float
Fit temperature-transition parameter, [K]
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
n : float
Fit parameter, [-]
E : float
Fit parameter, [-]
F : float
Fit parameter, [-]
Returns
-------
Psat : float
Vapor pressure calculated with coefficients [Pa]
Notes
-----
Assumes coefficients are for calculating vapor pressure in Pascal.
Coefficients should be consistent with input temperatures in Kelvin;
Examples
--------
Tetrafluoromethane, coefficients from [1]_, at 180 K:
>>> TRC_Antoine_extended(T=180.0, Tc=227.51, to=-120., A=8.95894,
... B=510.595, C=-15.95, n=2.41377, E=-93.74, F=7425.9)
706317.0898414153
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
'''
x = (T - to - 273.15)/Tc
if x < 0.0:
x = 0.0
x4 = x*x*x*x
T_C = T + C
if T_C <= 0.0:
return 0.0
return 10.**(A - B/T_C + 0.43429*x**n + x4*x4*(E + F*x4))
[docs]def dTRC_Antoine_extended_dT(T, Tc, to, A, B, C, n, E, F):
r'''Calculates the first temperature derivative of vapor pressure of a
chemical using the TRC Extended Antoine equation.
.. math::
\frac{\partial P^{\text{sat}} }{\partial T} =
10^{A - \frac{B}{C + T} + \frac{E \left(T - T_{ref} - to\right)^{8}}
{T_{c}^{8}} + \frac{F \left(T - T_{ref} - to\right)^{12}}{T_{c}^{12}}
+ f \left(\frac{T - T_{ref} - to}{T_{c}}\right)^{n}}
\left(\frac{B}{\left(C + T\right)^{2}} + \frac{8 E \left(T - T_{ref}
- to\right)^{7}}{T_{c}^{8}} + \frac{12 F \left(T - T_{ref} - to
\right)^{11}}{T_{c}^{12}} + \frac{f n \left(\frac{T - T_{ref} - to}
{T_{c}}\right)^{n}}{T - T_{ref} - to}\right) \log{\left(10 \right)}
.. math::
x = \max \left(\frac{T-t_o-273.15}{T_c}, 0 \right)
.. math::
T_{ref} = 273.15 \text{ K}
.. math::
f = 0.43429
Parameters
----------
T : float
Temperature of fluid, [K]
Tc : float
Critical temperature of fluid, [K]
to : float
Fit temperature-transition parameter, [K]
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
n : float
Fit parameter, [-]
E : float
Fit parameter, [-]
F : float
Fit parameter, [-]
Returns
-------
dPsat_dT : float
First temperature derivative of vapor pressure calculated with
coefficients [Pa/K]
Notes
-----
Examples
--------
Tetrafluoromethane at 180 K:
>>> dTRC_Antoine_extended_dT(T=180.0, Tc=227.51, to=-120., A=8.95894,
... B=510.595, C=-15.95, n=2.41377, E=-93.74, F=7425.9)
31219.6061263
'''
T_ref = 273.15
f = 0.43429
ln10 = 2.302585092994046
x = (T - to - 273.15)/Tc
if x < 0.0:
return dAntoine_dT(T, A, B, C, base=10.0)
x0 = C + T
x1 = -T + T_ref + to
x2 = E/Tc**8
x3 = F/Tc**12
x4 = f*(-x1/Tc)**n
return (-10**(A - B/x0 + x1**12*x3 + x1**8*x2 + x4)*(-B/x0**2 + n*x4/x1
+ 12.0*x1**11*x3 + 8.0*x1**7*x2)*ln10)
[docs]def d2TRC_Antoine_extended_dT2(T, Tc, to, A, B, C, n, E, F):
r'''Calculates the second temperature derivative of vapor pressure of a
chemical using the TRC Extended Antoine equation.
.. math::
\frac{\partial^2 P^{\text{sat}} }{\partial T^2} =
10^{A - \frac{B}{C + T} + \frac{E \left(- T + T_{ref} + to\right)^{8}}
{T_{c}^{8}} + \frac{F \left(- T + T_{ref} + to\right)^{12}}{T_{c}^{12}}
+ f \left(- \frac{- T + T_{ref} + to}{T_{c}}\right)^{n}} \left(
- \frac{2 B}{\left(C + T\right)^{3}} + \frac{56 E \left(- T + T_{ref}
+ to\right)^{6}}{T_{c}^{8}} + \frac{132 F \left(- T + T_{ref}
+ to\right)^{10}}{T_{c}^{12}} + \frac{f n^{2} \left(- \frac{- T
+ T_{ref} + to}{T_{c}}\right)^{n}}{\left(- T + T_{ref} + to\right)^{2}}
- \frac{f n \left(- \frac{- T + T_{ref} + to}{T_{c}}\right)^{n}}
{\left(- T + T_{ref} + to\right)^{2}} + \left(- \frac{B}{\left(C
+ T\right)^{2}} + \frac{8 E \left(- T + T_{ref} + to\right)^{7}}
{T_{c}^{8}} + \frac{12 F \left(- T + T_{ref} + to\right)^{11}}
{T_{c}^{12}} + \frac{f n \left(- \frac{- T + T_{ref} + to}
{T_{c}}\right)^{n}}{- T + T_{ref} + to}\right)^{2}
\log{\left(10 \right)}\right) \log{\left(10 \right)}
.. math::
x = \max \left(\frac{T-t_o-273.15}{T_c}, 0 \right)
.. math::
T_{ref} = 273.15 \text{ K}
.. math::
f = 0.43429
Parameters
----------
T : float
Temperature of fluid, [K]
Tc : float
Critical temperature of fluid, [K]
to : float
Fit temperature-transition parameter, [K]
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
n : float
Fit parameter, [-]
E : float
Fit parameter, [-]
F : float
Fit parameter, [-]
Returns
-------
d2Psat_dT2 : float
Second temperature derivative of vapor pressure calculated with
coefficients [Pa/K]
Notes
-----
Examples
--------
Tetrafluoromethane at 180 K:
>>> d2TRC_Antoine_extended_dT2(T=180.0, Tc=227.51, to=-120., A=8.95894,
... B=510.595, C=-15.95, n=2.41377, E=-93.74, F=7425.9)
1022.550368944
'''
T_ref = 273.15
f = 0.43429
x = (T - to - 273.15)/Tc
if x < 0.0:
return d2Antoine_dT2(T, A, B, C, base=10.0)
x0 = 2.302585092994046
x1 = C + T
x2 = -T + T_ref + to
x3 = E/Tc**8
x4 = F/Tc**12
x5 = f*(-x2/Tc)**n
x6 = x2**(-2)
x7 = n*x5
return (10**(A - B/x1 + x2**12*x4 + x2**8*x3 + x5)*x0*(-2.0*B/x1**3 + n**2*x5*x6
+ x0*(-B/x1**2 + 12.0*x2**11*x4 + 8.0*x2**7*x3 + x7/x2)**2
+ 132.0*x2**10*x4 + 56.0*x2**6*x3 - x6*x7))
[docs]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.
This is also called the PPDS1 equation [3]_.
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.
.. [3] PPDS2 Temperature-Dependent Equation Forms. National Engineering
Laboratory, 2004
https://web.archive.org/web/20050510061545/http://www.ppds.co.uk/library/pdf/PPDS_EquationForms.pdf
'''
Tr = T/Tc
if Tr > 1.0:
Tr = 1.0
tau = 1.0 - Tr
tau2 = tau*tau
try:
tau_Tr = tau/Tr
except:
# T = 0; Tr = 0
return 0.0
return Pc*exp(((d*tau2*tau + c)*tau2 + a + b*sqrt(tau))*tau_Tr)
[docs]def dWagner_original_dT(T, Tc, Pc, a, b, c, d):
r'''Calculates first temperature derivative of vapor pressure using the
Wagner equation (3, 6 form).
Requires critical temperature and pressure as well as four coefficients
specific to each chemical.
.. math::
\frac{\partial P^{\text{sat}} }{\partial T} =
P_{c} \left(\frac{T_{c} \left(- \frac{a}{T_{c}} - \frac{1.5 b
\tau^{0.5}}{T_{c}} - \frac{3 c \tau^{2}}{T_{c}} - \frac{6 d \tau^{5}}
{T_{c}}\right)}{T} - \frac{T_{c} \left(a \tau + b \tau^{1.5}
+ c \tau^{3} + d \tau^{6}\right)}{T^{2}}\right) e^{\frac{T_{c} \left(a
\tau + b \tau^{1.5} + c \tau^{3} + d \tau^{6}\right)}{T}}
.. 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
-------
dPsat_dT : float
First temperature derivative of vapor pressure at T [Pa/K]
Notes
-----
Examples
--------
Methane at 100 K.
>>> dWagner_original_dT(100.0, 190.53, 4596420., a=-6.00435, b=1.1885,
... c=-0.834082, d=-1.22833)
3593.70783283
'''
Tr = T/Tc
tau = 1.0 - Tr
tau2 = tau*tau
tau3 = tau2*tau
try:
T_inv = 1.0/T
Tr_inv = Tc*T_inv
except:
# T = 0; Tr = 0
return 0.0
tau_rt = sqrt(tau)
y0 = a*tau + b*tau*tau_rt + tau3*(c + d*tau3)
exp_term = exp(Tr_inv*y0)
if exp_term == 0.0:
# Avoid underflowing to nan
return 0.0
dPsat_dT = Pc*(T_inv*((-a - 1.5*b*tau_rt - tau2*(3.0*c + 6.0*d*tau3))
- Tc*y0*T_inv)
*exp_term)
return dPsat_dT
[docs]def d2Wagner_original_dT2(T, Tc, Pc, a, b, c, d):
r'''Calculates second temperature derivative of vapor pressure using the
Wagner equation (3, 6 form).
Requires critical temperature and pressure as well as four coefficients
specific to each chemical.
.. math::
\frac{\partial^2 P^{\text{sat}} }{\partial T^2} =
\frac{P_{c} \left(\frac{\frac{0.75 b}{\tau^{0.5}} + 6 c \tau + 30 d
\tau^{4}}{T_{c}} + \frac{2 \left(a + 1.5 b \tau^{0.5} + 3 c \tau^{2}
+ 6 d \tau^{5}\right)}{T} + \frac{36 \left(\frac{a}{6} + 0.25 b
\tau^{0.5} + \frac{c \tau^{2}}{2} + d \tau^{5} - \frac{T_{c} \left(
- a \tau - b \tau^{1.5} - c \tau^{3} - d \tau^{6}\right)}{6 T}
\right)^{2}}{T} - \frac{2 T_{c} \left(- a \tau - b \tau^{1.5}
- c \tau^{3} - d \tau^{6}\right)}{T^{2}}\right) e^{- \frac{T_{c}
\left(- a \tau - b \tau^{1.5} - c \tau^{3} - d \tau^{6}\right)}{T}}}{T}
.. 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
-------
d2Psat_dT2 : float
Second temperature derivative of vapor pressure at T [Pa/K^2]
Notes
-----
This second derivative is infinity at T == Tc.
Examples
--------
Methane at 100 K.
>>> d2Wagner_original_dT2(100.0, 190.53, 4596420., a=-6.00435, b=1.1885,
... c=-0.834082, d=-1.22833)
296.87593368224
'''
Tr = T/Tc
tau = 1.0 - Tr
tau2 = tau*tau
tau3 = tau2*tau
try:
T_inv = 1.0/T
# Tr_inv = Tc*T_inv
except:
# T = 0; Tr = 0
return 0.0
# y0 = a*tau + b*tau*tau_rt + tau3*(c + d*tau3)
#
# tau_rt = sqrt(tau)
tau_rt = sqrt(tau)
x1 = Tc*tau*(a + b*tau_rt + tau2*(c + d*tau3))
x2 = T_inv*x1
exp_term = exp(x2)
if exp_term == 0.0:
# Avoid underflowing to nan
return 0.0
x4 = b*tau_rt
x5 = c*tau2
x6 = d*tau2*tau3
x7 = (a*(1.0/6.0) + x2*(1.0/6.0) + 0.25*x4 + 0.5*x5 + x6)
return (Pc*T_inv*T_inv*(2.0*(a + 1.5*x4 + 3.0*x5 + 6.0*x6) + 36.0*x7*x7
+ (0.75*b/tau_rt + 6.0*c*tau + 30.0*d*tau2*tau2)*Tr
+ 2.0*x1*T_inv)*exp_term)
[docs]def Wagner_original_fitting_jacobian(Ts, Tc, Pc, a, b, c, d):
r'''Calculates the jacobian of the Wagner (3, 6) vapor pressure equation
for use in fitting these parameters when experimental values are known.
Requires critical temperature and pressure as well as four coefficients
specific to each chemical.
Parameters
----------
Ts : list[float]
Temperatures of fluid data points, [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
-------
jac : list[list[float, 4], len(Ts)]
Matrix of derivatives of the equation with respect to the fitting
parameters, [various]
'''
N = len(Ts)
# out = np.zeros((N, 4)) # numba: uncomment
out = [[0.0]*4 for _ in range(N)] # numba: delete
for i in range(N):
Tr = Ts[i]/Tc
tau = 1.0 - Tr
x2 = tau*sqrt(tau)
x3 = x2*x2
x4 = x3*x3
x1 = 1.0/Tr
x5 = Pc*x1*exp(x1*(a*tau + b*x2 + c*x3 + d*x4))
row = out[i]
row[0] = tau*x5
row[1] = x2*x5
row[2] = x3*x5
row[3] = x4*x5
return out
[docs]def Wagner_fitting_jacobian(Ts, Tc, Pc, a, b, c, d):
r'''Calculates the jacobian of the Wagner (2.5, 5) vapor pressure equation
for use in fitting these parameters when experimental values are known.
Requires critical temperature and pressure as well as four coefficients
specific to each chemical.
Parameters
----------
Ts : list[float]
Temperatures of fluid data points, [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
-------
jac : list[list[float, 4], len(Ts)]
Matrix of derivatives of the equation with respect to the fitting
parameters, [various]
'''
N = len(Ts)
# out = np.zeros((N, 4)) # numba: uncomment
out = [[0.0]*4 for _ in range(N)] # numba: delete
for i in range(N):
Tr = Ts[i]/Tc
tau = 1.0 - Tr
x2 = tau*sqrt(tau)
x3 = tau*x2
x4 = x3*x3
x1 = 1.0/Tr
x5 = Pc*x1*exp(x1*(a*tau + b*x2 + c*x3 + d*x4))
row = out[i]
row[0] = tau*x5
row[1] = x2*x5
row[2] = x3*x5
row[3] = x4*x5
return out
[docs]def Antoine_fitting_jacobian(Ts, A, B, C, base=10.0):
r'''Calculates the jacobian of the Antoine vapor pressure equation
for use in fitting these parameters when experimental values are known.
Requires three coefficients specific to each chemical.
Parameters
----------
Ts : list[float]
Temperatures of fluid data points, [K]
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
base : float, optional
Optional base of logarithm; 10 by default, [-]
Returns
-------
jac : list[list[float, 3], len(Ts)]
Matrix of derivatives of the equation with respect to the fitting
parameters, [various]
'''
N = len(Ts)
# out = np.zeros((N, 3)) # numba: uncomment
out = [[0.0]*3 for _ in range(N)] # numba: delete
ln_base = log(base)
for i in range(N):
row = out[i]
x0 = C + Ts[i]
if x0 <= 0.0:
row[0] = 0.0
row[1] = 0.0
row[2] = 0.0
else:
x1 = 1.0/x0
x2 = base**(A - B*x1)*ln_base
row[0] = x2
row[1] = -x1*x2
row[2] = B*x2*x1*x1
return out
[docs]def TRC_Antoine_extended_fitting_jacobian(Ts, Tc, to, A, B, C, n, E, F):
r'''Calculates the jacobian of the TRC Antoine extended vapor pressure
equation for use in fitting these parameters when experimental values are
known.
Requires 7 coefficients specific to each chemical.
Parameters
----------
Ts : list[float]
Temperatures of fluid data points, [K]
Tc : float
Critical temperature of fluid, [K]
to : float
Fit temperature-transition parameter, [K]
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
n : float
Fit parameter, [-]
E : float
Fit parameter, [-]
F : float
Fit parameter, [-]
Returns
-------
jac : list[list[float, 7], len(Ts)]
Matrix of derivatives of the equation with respect to the fitting
parameters, [various]
'''
N = len(Ts)
# out = np.zeros((N, 7)) # numba: uncomment
out = [[0.0]*7 for _ in range(N)] # numba: delete
ln_base = 2.302585092994046 #log(10)
c0 = 273.15
c1 = 0.43429
Tc_inv = 1.0/Tc
Tc_inv2 = Tc_inv*Tc_inv
Tc_inv4 = Tc_inv2*Tc_inv2
Tc_n8 = Tc_inv4*Tc_inv4
Tc_n12 = Tc_n8*Tc_inv4
v0 = - to - c0
for i in range(N):
row = out[i]
T = Ts[i]
x = (T + v0)*Tc_inv
if x < 0.0:
x0 = C + T
if x0 > 0.0:
x1 = 1.0/x0
x2 = 10.0**(A - B*x1)*ln_base
x1x2 = x1*x2
row[1] = x2
row[2] = -x1x2
row[3] = B*x1x2*x1
else:
x1 = -T + c0 + to
x1_2 = x1*x1
x1_4 = x1_2*x1_2
x1_8 = x1_4*x1_4
x2 = -x1*Tc_inv
x3 = c1*x2**n
x5 = Tc_n8*(E + F*x1_4*Tc_inv4)
x6 = C + T
x7 = 1.0/x6
x9 = 10.0**(A - B*x7 + x3 + x5*x1_8)*ln_base
row[0] = x9*(4.0*F*Tc_n12*x1_8*x1_2*x1 + n*x3/x1 + 8.0*x1_4*x1_2*x1*x5)
row[1] = x9
row[2] = -x7*x9
row[3] = B*x9*x7*x7
row[4] = x3*x9*log(x2)
row[5] = Tc_n8*x1_8*x9
row[6] = Tc_n12*x1_8*x1_4*x9
return out
[docs]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.
This is also called the PPDS16 equation [3]_.
Examples
--------
Methane, coefficients from [2]_, at 100 K.
>>> Wagner(100., 190.551, 4599200, -6.02242, 1.26652, -0.5707, -1.366)
34415.004762637
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.
.. [3] PPDS2 Temperature-Dependent Equation Forms. National Engineering
Laboratory, 2004
https://web.archive.org/web/20050510061545/http://www.ppds.co.uk/library/pdf/PPDS_EquationForms.pdf
'''
Tr = T/Tc
if Tr > 1.0:
Tr = 1.0
tau = 1.0 - Tr
tau_rt = sqrt(tau)
tau15 = tau*tau_rt
tau25 = tau*tau15
return Pc*exp((a + b*tau_rt + tau15*(c + d*tau25))*tau/Tr)
[docs]def dWagner_dT(T, Tc, Pc, a, b, c, d):
r'''Calculates the first temperature derivative of 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::
\frac{\partial P^{\text{sat}} }{\partial T} =
P_{c} \left(\frac{T_{c} \left(- \frac{a}{T_{c}} - \frac{1.5 b
\tau^{0.5}}{T_{c}} - \frac{2.5 c \tau^{1.5}}{T_{c}} - \frac{5 d
\tau^{4}}{T_{c}}\right)}{T} - \frac{T_{c} \left(a \tau + b \tau^{1.5}
+ c \tau^{2.5} + d \tau^{5}\right)}{T^{2}}\right) e^{\frac{T_{c}
\left(a \tau + b \tau^{1.5} + c \tau^{2.5} + d \tau^{5}\right)}{T}}
.. 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
-------
dPsat_dT : float
First temperature derivative of vapor pressure at T [Pa/K]
Notes
-----
Examples
--------
Methane at 100 K.
>>> dWagner_dT(100., 190.551, 4599200, -6.02242, 1.26652, -0.5707, -1.366)
3587.2910498076
'''
tau = 1.0 - T/Tc
tau2 = tau*tau
try:
x0 = 1.0/T
except:
# T = 0
return 0.0
tau_rt = sqrt(tau)
x1 = tau*tau_rt
x2 = Tc*x0*(a*tau + b*x1 + tau2*(c*tau_rt + d*tau2*tau))
exp_term = exp(x2)
if exp_term == 0.0:
# Avoid nan issues
return 0.0
return -Pc*x0*(a + 1.5*b*tau_rt + 2.5*c*x1 + 5.0*d*tau2*tau2 + x2)*exp_term
[docs]def d2Wagner_dT2(T, Tc, Pc, a, b, c, d):
r'''Calculates the second temperature derivative of 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::
\frac{\partial^2 P^{\text{sat}} }{\partial T^2} =
\frac{P_{c} \left(\frac{\frac{0.75 b}{\tau^{0.5}} + 3.75 c \tau^{0.5}
+ 20 d \tau^{3}}{T_{c}} + \frac{2 \left(a + 1.5 b \tau^{0.5}
+ 2.5 c \tau^{1.5} + 5 d \tau^{4}\right)}{T} + \frac{25 \left(
\frac{a}{5} + 0.3 b \tau^{0.5} + 0.5 c \tau^{1.5} + d \tau^{4}
- \frac{T_{c} \left(- a \tau - b \tau^{1.5} - c \tau^{2.5}
- d \tau^{5}\right)}{5 T}\right)^{2}}{T} - \frac{2 T_{c} \left(- a
\tau - b \tau^{1.5} - c \tau^{2.5} - d \tau^{5}\right)}{T^{2}}\right)
e^{- \frac{T_{c} \left(- a \tau - b \tau^{1.5} - c \tau^{2.5}
- d \tau^{5}\right)}{T}}}{T}
.. 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
-------
d2Psat_dT2 : float
Second temperature derivative of vapor pressure at T [Pa/K^2]
Notes
-----
This second derivative is infinity at T == Tc.
Examples
--------
Methane at 100 K.
>>> d2Wagner_dT2(100., 190.551, 4599200, -6.02242, 1.26652, -0.5707, -1.366)
296.7091513877
'''
tau = 1.0 - T/Tc
tau_rt = sqrt(tau)
tau2 = tau*tau
try:
T_inv = 1.0/T
except:
# T = 0
return 0.0
x1 = tau*tau_rt
x2 = Tc*(a*tau + b*x1 + tau*(c*x1 + d*tau2*tau2))
x3 = T_inv*x2
x5 = b*tau_rt
x6 = c*x1
x7 = d*tau2*tau2
exp_term = exp(x3)
if exp_term == 0.0:
# Avoid nan issues
return 0.0
return (Pc*T_inv*(2.0*T_inv*(a + 1.5*x5 + 2.5*x6 + 5.0*x7) + 25.0*T_inv*(a*(1.0/5.0) + x3*(1.0/5.0)
+ 0.3*x5 + 0.5*x6 + x7)**2 + (0.75*b/tau_rt + 3.75*c*tau_rt
+ 20.0*d*tau*tau2)/Tc + 2.0*x2*T_inv*T_inv)*exp_term)
[docs]def Psat_IAPWS(T):
r'''Calculates vapor pressure of water using the IAPWS explicit equation.
.. math::
P^{sat} = 10^6 \left[ \frac{2C}{-B + \sqrt{B^2 - 4AC}} \right]^4
.. math::
A = \nu^2 + n_1 \nu + n_2
.. math::
B = n_3 \nu^2 + n_4\nu + n_5
.. math::
C = n_6\nu^2 + n_7\nu + n_8
.. math::
\nu = T + \frac{n_9}{T - n_{10}}
Parameters
----------
T : float
Temperature of water, [K]
Returns
-------
Psat : float
Vapor pressure at T [Pa]
Notes
-----
This formulation is quite efficient, and can also be solved backward.
The range of validity of this equation is 273.15 K < T < 647.096 K, the
IAPWS critical point.
Extrapolation to lower temperatures is very poor. The function continues to
decrease until a pressure of 5.7 mPa is reached at 159.77353993926621 K;
under that pressure the vapor pressure increases, which is obviously wrong.
Examples
--------
>>> Psat_IAPWS(300.)
3536.58941301301
References
----------
.. [1] Kretzschmar, Hans-Joachim, and Wolfgang Wagner. International Steam
Tables: Properties of Water and Steam Based on the Industrial
Formulation IAPWS-IF97. Springer, 2019.
'''
v = T - 0.23855557567849/(T - 0.65017534844798E3)
v2 = v*v
A = v2 + 0.11670521452767E4*v - 0.72421316703206E6
B = -0.17073846940092E2*v2 + 0.12020824702470E5*v - 0.32325550322333E7
C = 0.14915108613530E2*v2 - 0.48232657361591E4*v + 0.40511340542057E6
x = ((C + C)/(sqrt(B*B - 4.0*A*C) - B))
x2 = x*x
P = 1E6*x2*x2
return P
[docs]def dPsat_IAPWS_dT(T):
r'''Calculates the first temperature dervative of vapor pressure of water
using the IAPWS explicit equation. This was derived with SymPy, using the
CSE method.
Parameters
----------
T : float
Temperature of water, [K]
Returns
-------
dPsat_dT : float
Temperature dervative of vapor pressure at T [Pa/K]
Notes
-----
The derivative of this is useful when solving for water dew point.
Examples
--------
>>> dPsat_IAPWS_dT(300.)
207.88388134164282
References
----------
.. [1] Kretzschmar, Hans-Joachim, and Wolfgang Wagner. International Steam
Tables: Properties of Water and Steam Based on the Industrial
Formulation IAPWS-IF97. Springer, 2019.
'''
# 1 power, four divisions
x0 = 1.0/(T - 650.175348447980014)
x1 = T - 0.238555575678490006*x0
x2 = x1*x1
x3 = -0.0119059642846224972*T + 0.00284023416392566131*x0 + 0.0000368171193891888733*x2 + 1.0
x4 = 0.00371867596455583913*T
x5 = 0.000887110885486382334*x0
x6 = 5.28184261979789455e-6*x2
x7 = x4 - x5 - x6 - 1.0
x8 = 4668.20858110680001*T - 1113.62718545319967*x0 + 4.0*x2 - 2896852.66812823992
x9 = sqrt(x7*x7 - 9.56991643658934639e-14*x8*(-4823.26573615910002*T + 1150.61693433977007*x0 + 14.9151086135300002*x2 + 405113.405420569994))
x10 = -x4 + x5 + x6 + x9 + 1.0
x10_inv = 1.0/x10
x11 = 0.00153804662447919488*T - 1.0
x11 = 1.0/(x11*x11)
x12 = x1*(1.12864813714895499e-6*x11 + 2.0)
x20 = x10_inv*x10_inv*x10_inv*x10_inv
return (-3946.77829948379076*x3*x3*x3*(2.68752888216023447e-8*x11 - 0.000147268477556755493*x12
+ 0.0476238571384899889 + x3*(-8.39415340011308276e-9*x11 + 0.0000211273704791915782*x12
- 0.0148747038582233565 + 2.0*(x7*(4.19707670005654138e-9*x11
- 0.0000105636852395957891*x12 + 0.00743735192911167825) + x8*(2.60482114645230015e-16*x11
- 1.42736343074136086e-12*x12 + 4.6158250046507185e-10) + (0.00263438245944447809*x11
+ 4.0*x12 + 4668.20858110680001)*(4.6158250046507185e-10*T - 1.10113079121562103e-10*x0
- 1.42736343074136086e-12*x2 - 3.8769014372169964e-8))/x9)*x10_inv)*x20)
[docs]def Tsat_IAPWS(P):
r'''Calculates the saturation temperature of water using the IAPWS explicit
equation.
.. math::
T_s = \frac{n_{10} + D - \left[(n_{10}+D)^2 - 4(n_9 + n_{10}D) \right]^{0.5}}{2}
.. math:
D = \frac{2G}{-F - (F^2 - 4EG)^{0.5}}
.. math::
E = \beta^2 + n_3 \beta + n_6
.. math::
F = n_1 \beta^2 + n_4\beta + n_7
.. math::
G = n_2\beta^2 + n_5\beta + n_8
.. math::
\beta = \left(P_{sat} \right)^{0.25}
Parameters
----------
Psat : float
Vapor pressure at T [Pa]
Returns
-------
T : float
Temperature of water along the saturation curve at `Psat`, [K]
Notes
-----
The range of validity of this equation is 273.15 K < T < 647.096 K, the
IAPWS critical point.
The coefficients `n1` to `n10` are (0.11670521452767E4, -0.72421316703206E6,
-0.17073846940092E2, 0.12020824702470E5, -0.32325550322333E7, 0.14915108613530E2,
-0.48232657361591E4, 0.40511340542057E6, -0.23855557567849, 0.65017534844798E3)
Examples
--------
>>> Tsat_IAPWS(1E5)
372.75591861133773
References
----------
.. [1] Kretzschmar, Hans-Joachim, and Wolfgang Wagner. International Steam
Tables: Properties of Water and Steam Based on the Industrial
Formulation IAPWS-IF97. Springer, 2019.
'''
B = sqrt(sqrt(P*1E-6))
E = B*(B + -0.17073846940092E2) + 0.14915108613530E2
F = B*(0.11670521452767E4*B + 0.12020824702470E5) + -0.48232657361591E4
G = B*(-0.72421316703206E6*B + -0.32325550322333E7) + 0.40511340542057E6
D = 2.0*G/(-F - sqrt(F*F - 4.0*E*G))
n10 = 0.65017534844798E3
x0 = (n10 + D)
T = (n10 + D - sqrt(x0*x0 - 4.0*(-0.23855557567849+n10*D)))*0.5
return T
### CSP Methods
[docs]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*(1.0 - 1.0/Tr))*Pc
[docs]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
[docs]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*sqrt(tau)
tau25 = tau*tau15
tau5 = tau25*tau25
if omega < 0.0:
# The omega term is based on positive omegas only, prevent it from
# going negative
omega = 0.0
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*trunc_exp((f0 + omega*(f1 + f2*omega))/Tr)
[docs]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)
[docs]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
### Sublimation Pressure
[docs]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))