"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2019, 2020 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 lookup functions and estimation methods for the
parameters molecular diameter `sigma` and the Stockmayer parameter `epsilon`.
These are used for diffusivity calculations. It also contains several
methods for computing the collision integral, another parameter used in the
Lennard-Jones model.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/chemicals/>`_.
.. contents:: :local:
Stockmayer Parameter
--------------------
.. autofunction:: chemicals.lennard_jones.Stockmayer
.. autofunction:: chemicals.lennard_jones.Stockmayer_methods
.. autodata:: chemicals.lennard_jones.Stockmayer_all_methods
Stockmayer Parameter Correlations
---------------------------------
.. autofunction:: chemicals.lennard_jones.epsilon_Flynn
.. autofunction:: chemicals.lennard_jones.epsilon_Bird_Stewart_Lightfoot_critical
.. autofunction:: chemicals.lennard_jones.epsilon_Bird_Stewart_Lightfoot_boiling
.. autofunction:: chemicals.lennard_jones.epsilon_Bird_Stewart_Lightfoot_melting
.. autofunction:: chemicals.lennard_jones.epsilon_Stiel_Thodos
.. autofunction:: chemicals.lennard_jones.epsilon_Tee_Gotoh_Steward_1
.. autofunction:: chemicals.lennard_jones.epsilon_Tee_Gotoh_Steward_2
Molecular Diameter
------------------
.. autofunction:: chemicals.lennard_jones.molecular_diameter
.. autofunction:: chemicals.lennard_jones.molecular_diameter_methods
.. autodata:: chemicals.lennard_jones.molecular_diameter_all_methods
Molecular Diameter Correlations
-------------------------------
.. autofunction:: chemicals.lennard_jones.sigma_Flynn
.. autofunction:: chemicals.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_2
.. autofunction:: chemicals.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_1
.. autofunction:: chemicals.lennard_jones.sigma_Bird_Stewart_Lightfoot_boiling
.. autofunction:: chemicals.lennard_jones.sigma_Bird_Stewart_Lightfoot_melting
.. autofunction:: chemicals.lennard_jones.sigma_Stiel_Thodos
.. autofunction:: chemicals.lennard_jones.sigma_Tee_Gotoh_Steward_1
.. autofunction:: chemicals.lennard_jones.sigma_Tee_Gotoh_Steward_2
.. autofunction:: chemicals.lennard_jones.sigma_Silva_Liu_Macedo
Utility Functions
-----------------
.. autofunction:: chemicals.lennard_jones.T_star
"""
__all__ = ['Stockmayer_all_methods', 'Stockmayer_methods', 'Stockmayer',
'molecular_diameter_all_methods', 'molecular_diameter', 'molecular_diameter_methods',
'sigma_Flynn',
'sigma_Bird_Stewart_Lightfoot_critical_2',
'sigma_Bird_Stewart_Lightfoot_critical_1',
'sigma_Bird_Stewart_Lightfoot_boiling',
'sigma_Bird_Stewart_Lightfoot_melting',
'sigma_Stiel_Thodos', 'sigma_Tee_Gotoh_Steward_1',
'sigma_Tee_Gotoh_Steward_2',
'sigma_Silva_Liu_Macedo', 'epsilon_Flynn',
'epsilon_Bird_Stewart_Lightfoot_critical',
'epsilon_Bird_Stewart_Lightfoot_boiling',
'epsilon_Bird_Stewart_Lightfoot_melting', 'epsilon_Stiel_Thodos',
'epsilon_Tee_Gotoh_Steward_1', 'epsilon_Tee_Gotoh_Steward_2',
'collision_integral_Neufeld_Janzen_Aziz',
'As_collision', 'Bs_collision', 'Cs_collision',
'collision_integral_Kim_Monroe', 'T_star']
from fluids.constants import k
from fluids.numerics import exp, log, sin
from chemicals import data_reader as dr
from chemicals.data_reader import (
data_source,
database_constant_lookup,
list_available_methods_from_df_dict,
register_df_source,
retrieve_any_from_df_dict,
retrieve_from_df_dict,
)
from chemicals.utils import PY37, can_load_data, mark_numba_incompatible, os_path_join, source_path
# Register data sources and lazy load them
folder = os_path_join(source_path, 'Viscosity')
register_df_source(folder, 'MagalhaesLJ.tsv')
register_df_source(folder, 'PolingLJ.tsv')
POLING = 'Poling et al. (2001)'
FLYNN = 'Flynn (1960)'
STIELTHODOS = 'Stiel and Thodos Tc, Zc (1962)'
MAGALHAES = 'Magalhães, Lito, Da Silva, and Silva (2013)'
TEEGOTOSTEWARD1 = 'Tee, Gotoh, and Stewart CSP with Tc (1966)'
TEEGOTOSTEWARD2 = 'Tee, Gotoh, and Stewart CSP with Tc, omega (1966)'
BSLC = 'Bird, Stewart, and Light (2002) critical relation'
BSLB = 'Bird, Stewart, and Light (2002) boiling relation'
BSLM = 'Bird, Stewart, and Light (2002) melting relation'
_LJ_data_loaded = False
def _load_LJ_data():
global _LJ_data_loaded, LJ_data_Magalhaes, LJ_data_Poling, LJ_sources
LJ_data_Magalhaes = data_source('MagalhaesLJ.tsv')
LJ_data_Poling = data_source('PolingLJ.tsv')
_LJ_data_loaded = True
LJ_sources = {
MAGALHAES: LJ_data_Magalhaes,
POLING: LJ_data_Poling,
}
if PY37:
def __getattr__(name):
if name in ('LJ_data_Magalhaes', 'LJ_data_Poling', 'LJ_sources'):
_load_LJ_data()
return globals()[name]
raise AttributeError(f"module {__name__} has no attribute {name}")
else:
if can_load_data:
_load_LJ_data()
Stockmayer_all_methods = (MAGALHAES, POLING, TEEGOTOSTEWARD2, STIELTHODOS, FLYNN, BSLC,
TEEGOTOSTEWARD1, BSLB, BSLM)
"""Tuple of method name keys. See the `Stockmayer` for the actual references"""
[docs]@mark_numba_incompatible
def Stockmayer_methods(CASRN=None, Tm=None, Tb=None, Tc=None, Zc=None, omega=None):
"""Return all methods available to obtain the Stockmayer parameter for the
desired chemical.
Parameters
----------
CASRN : str, optional
CASRN [-]
Tm : float, optional
Melting temperature of compound [K]
Tb : float, optional
Boiling temperature of compound [K]
Tc : float, optional
Critical temperature of compound, [K]
Zc : float, optional
Critical compressibility of compound, [-]
omega : float, optional
Acentric factor of compound, [-]
Returns
-------
methods : list[str]
Methods which can be used to obtain `Stockmayer` with the given inputs.
See Also
--------
Stockmayer
"""
if not _LJ_data_loaded: _load_LJ_data()
methods = list_available_methods_from_df_dict(LJ_sources, CASRN, 'Stockmayer')
if Tc:
if omega: methods.append(TEEGOTOSTEWARD2)
if Zc: methods.append(STIELTHODOS)
methods.append(FLYNN)
methods.append(BSLC)
methods.append(TEEGOTOSTEWARD1)
if Tb: methods.append(BSLB)
if Tm: methods.append(BSLM)
return methods
[docs]@mark_numba_incompatible
def Stockmayer(CASRN='', Tm=None, Tb=None, Tc=None, Zc=None, omega=None,
method=None):
r'''This function handles the retrieval or calculation a chemical's
Stockmayer parameter. Values are available from one source with lookup
based on CASRNs, or can be estimated from 7 CSP methods.
Will automatically select a data source to use if no method is provided;
returns None if the data is not available.
Preferred sources are 'Magalhães, Lito, Da Silva, and Silva (2013)' for
common chemicals which had valies listed in that source, and the CSP method
`Tee, Gotoh, and Stewart CSP with Tc, omega (1966)` for chemicals which
don't.
Examples
--------
>>> Stockmayer(CASRN='64-17-5')
1291.41
>>> Stockmayer('7727-37-9')
71.4
Parameters
----------
CASRN : str, optional
CASRN [-]
Tm : float, optional
Melting temperature of compound [K]
Tb : float, optional
Boiling temperature of compound [K]
Tc : float, optional
Critical temperature of compound, [K]
Zc : float, optional
Critical compressibility of compound, [-]
omega : float, optional
Acentric factor of compound, [-]
Returns
-------
epsilon_k : float
Lennard-Jones depth of potential-energy minimum over k, [K]
Other Parameters
----------------
method : string, optional
A string for the method name to use, as defined by constants in
Stockmayer_all_methods
Notes
-----
These values are somewhat rough, as they attempt to pigeonhole a chemical
into L-J behavior.
The tabulated data is from [2]_, for 322 chemicals.
References
----------
.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot.
Transport Phenomena, Revised 2nd Edition. New York:
John Wiley & Sons, Inc., 2006
.. [2] Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and
Carlos M. Silva. "Simple and Accurate Correlations for Diffusion
Coefficients of Solutes in Liquids and Supercritical Fluids over Wide
Ranges of Temperature and Density." The Journal of Supercritical Fluids
76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002.
'''
if dr.USE_CONSTANTS_DATABASE and method is None:
val, found = database_constant_lookup(CASRN, 'Stockmayer')
if found: return val
if not _LJ_data_loaded: _load_LJ_data()
if method is not None:
if method == FLYNN:
return epsilon_Flynn(Tc)
elif method == BSLC:
return epsilon_Bird_Stewart_Lightfoot_critical(Tc)
elif method == BSLB:
return epsilon_Bird_Stewart_Lightfoot_boiling(Tb)
elif method == BSLM:
return epsilon_Bird_Stewart_Lightfoot_melting(Tm)
elif method == STIELTHODOS:
return epsilon_Stiel_Thodos(Tc, Zc)
elif method == TEEGOTOSTEWARD1:
return epsilon_Tee_Gotoh_Steward_1(Tc)
elif method == TEEGOTOSTEWARD2:
return epsilon_Tee_Gotoh_Steward_2(Tc, omega)
else:
return retrieve_from_df_dict(LJ_sources, CASRN, 'Stockmayer', method)
else:
epsilon = retrieve_any_from_df_dict(LJ_sources, CASRN, 'Stockmayer')
if epsilon is not None: return epsilon
if Tc:
if omega: return epsilon_Tee_Gotoh_Steward_2(Tc, omega)
if Zc: return epsilon_Stiel_Thodos(Tc, Zc)
return epsilon_Flynn(Tc)
if Tb: return epsilon_Bird_Stewart_Lightfoot_boiling(Tb)
if Tm: return epsilon_Bird_Stewart_Lightfoot_melting(Tm)
TEEGOTOSTEWARD3 = 'Tee, Gotoh, and Stewart CSP with Tc, Pc (1966)'
TEEGOTOSTEWARD4 = 'Tee, Gotoh, and Stewart CSP with Tc, Pc, omega (1966)'
BSLC1 = 'Bird, Stewart, and Light (2002) critical relation with Vc'
BSLC2 = 'Bird, Stewart, and Light (2002) critical relation with Tc, Pc'
STIELTHODOSMD = 'Stiel and Thodos Vc, Zc (1962)'
SILVALIUMACEDO = 'Silva, Liu, and Macedo (1998) critical relation with Tc, Pc'
molecular_diameter_all_methods = (MAGALHAES, POLING, TEEGOTOSTEWARD4, SILVALIUMACEDO,
BSLC2, TEEGOTOSTEWARD3, STIELTHODOSMD, FLYNN,
BSLC1, BSLB, BSLM)
"""Tuple of method name keys. See the `molecular_diameter` for the actual references"""
[docs]@mark_numba_incompatible
def molecular_diameter_methods(CASRN=None, Tc=None, Pc=None, Vc=None, Zc=None,
omega=None, Vm=None, Vb=None):
"""Return all methods available to obtain the molecular diameter for the
desired chemical.
Parameters
----------
CASRN : str, optional
CASRN [-]
Tc : float, optional
Critical temperature, [K]
Pc : float, optional
Critical pressure, [Pa]
Vc : float, optional
Critical volume, [m^3/mol]
Zc : float, optional
Critical compressibility, [-]
omega : float, optional
Acentric factor of compound, [-]
Vm : float, optional
Molar volume of liquid at the melting point of the fluid [K]
Vb : float, optional
Molar volume of liquid at the boiling point of the fluid [K]
Returns
-------
methods : list[str]
Methods which can be used to obtain `molecular_diameter` with the given inputs.
See Also
--------
molecular_diameter
"""
methods = list_available_methods_from_df_dict(LJ_sources, CASRN, 'molecular_diameter')
if Tc:
if Pc:
if omega: methods.append(TEEGOTOSTEWARD4)
methods.append(SILVALIUMACEDO)
methods.append(BSLC2)
methods.append(TEEGOTOSTEWARD3)
if Vc:
if Zc: methods.append(STIELTHODOSMD)
methods.append(FLYNN)
methods.append(BSLC1)
if Vb: methods.append(BSLB)
if Vm: methods.append(BSLM)
return methods
[docs]@mark_numba_incompatible
def molecular_diameter(CASRN=None, Tc=None, Pc=None, Vc=None, Zc=None, omega=None,
Vm=None, Vb=None, method=None):
r'''This function handles the retrieval or calculation a chemical's
L-J molecular diameter. Values are available from one source with lookup
based on CASRNs, or can be estimated from 9 CSP methods.
Will automatically select a data source to use if no method is provided;
returns None if the data is not available.
Preferred sources are 'Magalhães, Lito, Da Silva, and Silva (2013)' for
common chemicals which had valies listed in that source, and the CSP method
`Tee, Gotoh, and Stewart CSP with Tc, Pc, omega (1966)` for chemicals which
don't.
Examples
--------
>>> molecular_diameter(CASRN='64-17-5')
4.23738
>>> molecular_diameter('7727-37-9')
3.798
Parameters
----------
CASRN : str, optional
CASRN [-]
Tc : float, optional
Critical temperature, [K]
Pc : float, optional
Critical pressure, [Pa]
Vc : float, optional
Critical volume, [m^3/mol]
Zc : float, optional
Critical compressibility, [-]
omega : float, optional
Acentric factor of compound, [-]
Vm : float, optional
Molar volume of liquid at the melting point of the fluid [K]
Vb : float, optional
Molar volume of liquid at the boiling point of the fluid [K]
Returns
-------
sigma : float
Lennard-Jones molecular diameter, [Angstrom]
Other Parameters
----------------
method : string, optional
A string for the method name to use, as defined by constants in
molecular_diameter_all_methods
Notes
-----
These values are somewhat rough, as they attempt to pigeonhole a chemical
into L-J behavior.
The tabulated data is from [2]_, for 322 chemicals.
References
----------
.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot.
Transport Phenomena, Revised 2nd Edition. New York:
John Wiley & Sons, Inc., 2006
.. [2] Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and
Carlos M. Silva. "Simple and Accurate Correlations for Diffusion
Coefficients of Solutes in Liquids and Supercritical Fluids over Wide
Ranges of Temperature and Density." The Journal of Supercritical Fluids
76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002.
'''
if dr.USE_CONSTANTS_DATABASE and method is None:
val, found = database_constant_lookup(CASRN, 'molecular_diameter')
if found: return val
if not _LJ_data_loaded: _load_LJ_data()
if method is not None:
if method == FLYNN:
return sigma_Flynn(Vc)
elif method == BSLC1:
return sigma_Bird_Stewart_Lightfoot_critical_1(Vc)
elif method == BSLC2:
return sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc)
elif method == TEEGOTOSTEWARD3:
return sigma_Tee_Gotoh_Steward_1(Tc, Pc)
elif method == SILVALIUMACEDO:
return sigma_Silva_Liu_Macedo(Tc, Pc)
elif method == BSLB:
return sigma_Bird_Stewart_Lightfoot_boiling(Vb)
elif method == BSLM:
return sigma_Bird_Stewart_Lightfoot_melting(Vm)
elif method == STIELTHODOSMD:
return sigma_Stiel_Thodos(Vc, Zc)
elif method == TEEGOTOSTEWARD4:
return sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega)
else:
return retrieve_from_df_dict(LJ_sources, CASRN, 'molecular_diameter', method)
else:
epsilon = retrieve_any_from_df_dict(LJ_sources, CASRN, 'molecular_diameter')
if epsilon is not None: return epsilon
if Tc:
if Pc:
if omega: return sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega)
return sigma_Silva_Liu_Macedo(Tc, Pc)
if Vc:
if Zc: return sigma_Stiel_Thodos(Vc, Zc)
return sigma_Flynn(Vc)
if Vb: return sigma_Bird_Stewart_Lightfoot_boiling(Vb)
if Vm: return sigma_Bird_Stewart_Lightfoot_melting(Vm)
### Sigma Lennard-Jones
[docs]def sigma_Flynn(Vc):
r'''Calculates Lennard-Jones molecular diameter.
Uses critical volume. CSP method by [1]_ as reported in [2]_.
.. math::
\sigma = 0.561(V_c^{1/3})^{5/4}
Parameters
----------
Vc : float
Critical volume of fluid [m^3/mol]
Returns
-------
sigma : float
Lennard-Jones molecular diameter, [Angstrom]
Notes
-----
Vc is originally in units of mL/mol.
Examples
--------
>>> sigma_Flynn(0.000268)
5.2506948422196285
References
----------
.. [1] Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).
.. [2] Stiel, L. I., and George Thodos. "Lennard-Jones Force Constants
Predicted from Critical Properties." Journal of Chemical & Engineering
Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023
'''
Vc = Vc*1E6 # m^3/mol to cm^3/mol
sigma = 0.561*(Vc**(1/3.))**1.2
return sigma
[docs]def sigma_Stiel_Thodos(Vc, Zc):
r'''Calculates Lennard-Jones molecular diameter.
Uses critical volume and compressibility. CSP method by [1]_.
.. math::
\sigma = 0.1866 V_c^{1/3} Z_c^{-6/5}
Parameters
----------
Vc : float
Critical volume of fluid [m^3/mol]
Zc : float
Critical compressibility of fluid, [-]
Returns
-------
sigma : float
Lennard-Jones molecular diameter, [Angstrom]
Notes
-----
Vc is originally in units of mL/mol.
Examples
--------
Monofluorobenzene
>>> sigma_Stiel_Thodos(0.000271, 0.265)
5.94300853971033
References
----------
.. [1] Stiel, L. I., and George Thodos. "Lennard-Jones Force Constants
Predicted from Critical Properties." Journal of Chemical & Engineering
Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023
'''
Vc = Vc*1E6
sigma = 0.1866*Vc**(1/3.0)*Zc**(-1.2)
return sigma
[docs]def sigma_Tee_Gotoh_Steward_1(Tc, Pc):
r'''Calculates Lennard-Jones molecular diameter.
Uses critical temperature and pressure. CSP method by [1]_.
.. math::
\sigma = 2.3647 \left(\frac{T_c}{P_c}\right)^{1/3}
Parameters
----------
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
Returns
-------
sigma : float
Lennard-Jones molecular diameter, [Angstrom]
Notes
-----
Original units of Pc are atm. Further regressions with other parameters
were performed in [1]_ but are not included here, except for
`sigma_Tee_Gotoh_Steward_2`.
Examples
--------
>>> sigma_Tee_Gotoh_Steward_1(560.1, 4550000)
5.48402779790962
References
----------
.. [1] Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. "Molecular
Parameters for Normal Fluids. Lennard-Jones 12-6 Potential." Industrial
& Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63.
doi:10.1021/i160019a011
'''
Pc = Pc/101325.
sigma = 2.3647*(Tc/Pc)**(1/3.)
return sigma
[docs]def sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega):
r'''Calculates Lennard-Jones molecular diameter.
Uses critical temperature, pressure, and acentric factor. CSP method by
[1]_.
.. math::
\sigma = (2.3551 - 0.0874\omega)\left(\frac{T_c}{P_c}\right)^{1/3}
Parameters
----------
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
Acentric factor for fluid, [-]
Returns
-------
sigma : float
Lennard-Jones molecular diameter, [Angstrom]
Notes
-----
Original units of Pc are atm. Further regressions with other parameters
were performed in [1]_ but are not included here, except for
`sigma_Tee_Gotoh_Steward_1`.
Examples
--------
>>> sigma_Tee_Gotoh_Steward_2(560.1, 4550000, 0.245)
5.412104867264477
References
----------
.. [1] Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. "Molecular Parameters
for Normal Fluids. Lennard-Jones 12-6 Potential." Industrial
& Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63.
doi:10.1021/i160019a011
'''
Pc = Pc/101325.
sigma = (2.3551-0.0874*omega)*(Tc/Pc)**(1/3.)
return sigma
[docs]def sigma_Silva_Liu_Macedo(Tc, Pc):
r'''Calculates Lennard-Jones molecular diameter.
Uses critical temperature and pressure. CSP method by [1]_.
.. math::
\sigma_{LJ}^3 = 0.17791 + 11.779 \left( \frac{T_c}{P_c}\right)
- 0.049029\left( \frac{T_c}{P_c}\right)^2
Parameters
----------
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
Returns
-------
sigma : float
Lennard-Jones molecular diameter, [Angstrom]
Notes
-----
Pc is originally in bar. An excellent paper. None is
returned if the polynomial returns a negative number, as in the case of
1029.13 K and 3.83 bar.
Examples
--------
>>> sigma_Silva_Liu_Macedo(560.1, 4550000)
5.164483998730177
References
----------
.. [1] Silva, Carlos M., Hongqin Liu, and Eugenia A. Macedo. "Models for
Self-Diffusion Coefficients of Dense Fluids, Including Hydrogen-Bonding
Substances." Chemical Engineering Science 53, no. 13 (July 1, 1998):
2423-29. doi:10.1016/S0009-2509(98)00037-2
'''
Pc = Pc/1E5 # Pa to bar
term = 0.17791 + 11.779*(Tc/Pc) - 0.049029 * (Tc/Pc)**2
if term < 0:
sigma = None
else:
sigma = (term)**(1/3.)
return sigma
### epsilon Lennard-Jones
[docs]def epsilon_Flynn(Tc):
r'''Calculates Lennard-Jones depth of potential-energy minimum.
Uses critical temperature. CSP method by [1]_ as reported in [2]_.
.. math::
\epsilon/k = 1.77 T_c^{5/6}
Parameters
----------
Tc : float
Critical temperature of fluid [K]
Returns
-------
epsilon_k : float
Lennard-Jones depth of potential-energy minimum over k, [K]
Notes
-----
Examples
--------
>>> epsilon_Flynn(560.1)
345.2984087011443
References
----------
.. [1] Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).
.. [2] Stiel, L. I., and George Thodos. "Lennard-Jones Force Constants
Predicted from Critical Properties." Journal of Chemical & Engineering
Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023
'''
epsilon_k = 1.77*Tc**(5/6.)
return epsilon_k
[docs]def epsilon_Stiel_Thodos(Tc, Zc):
r'''Calculates Lennard-Jones depth of potential-energy minimum.
Uses Critical temperature and critical compressibility. CSP method by [1]_.
.. math::
\epsilon/k = 65.3 T_c Z_c^{3.6}
Parameters
----------
Tc : float
Critical temperature of fluid [K]
Zc : float
Critical compressibility of fluid, [-]
Returns
-------
epsilon_k : float
Lennard-Jones depth of potential-energy minimum over k, [K]
Notes
-----
Examples
--------
Fluorobenzene
>>> epsilon_Stiel_Thodos(358.5, 0.265)
196.3755830305783
References
----------
.. [1] Stiel, L. I., and George Thodos. "Lennard-Jones Force Constants
Predicted from Critical Properties." Journal of Chemical & Engineering
Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023
'''
epsilon_k = 65.3*Tc*Zc**3.6
return epsilon_k
[docs]def epsilon_Tee_Gotoh_Steward_1(Tc):
r'''Calculates Lennard-Jones depth of potential-energy minimum.
Uses Critical temperature. CSP method by [1]_.
.. math::
\epsilon/k = 0.7740T_c
Parameters
----------
Tc : float
Critical temperature of fluid [K]
Returns
-------
epsilon_k : float
Lennard-Jones depth of potential-energy minimum over k, [K]
Notes
-----
Further regressions with other parameters were performed in [1]_ but are
not included here, except for `epsilon_Tee_Gotoh_Steward_2`.
Examples
--------
>>> epsilon_Tee_Gotoh_Steward_1(560.1)
433.5174
References
----------
.. [1] Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. "Molecular Parameters
for Normal Fluids. Lennard-Jones 12-6 Potential." Industrial &
Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63.
doi:10.1021/i160019a011
'''
epsilon_k = 0.7740*Tc
return epsilon_k
[docs]def epsilon_Tee_Gotoh_Steward_2(Tc, omega):
r'''Calculates Lennard-Jones depth of potential-energy minimum.
Uses critical temperature and acentric factor. CSP method by [1]_.
.. math::
\epsilon/k = (0.7915 + 0.1693 \omega)T_c
Parameters
----------
Tc : float
Critical temperature of fluid [K]
omega : float
Acentric factor for fluid, [-]
Returns
-------
epsilon_k : float
Lennard-Jones depth of potential-energy minimum over k, [K]
Notes
-----
Further regressions with other parameters were performed in [1]_ but are
not included here, except for `epsilon_Tee_Gotoh_Steward_1`.
Examples
--------
>>> epsilon_Tee_Gotoh_Steward_2(560.1, 0.245)
466.55125785
References
----------
.. [1] Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. "Molecular Parameters
for Normal Fluids. Lennard-Jones 12-6 Potential." Industrial &
Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63.
doi:10.1021/i160019a011
'''
epsilon_k = (0.7915 + 0.1693*omega)*Tc
return epsilon_k
### Collision Integral
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
As_collision = {(1, 1): -1.10367290,
(1, 2): 1.35555540,
(1, 3): 1.06771150,
(1, 4): 0.80959899,
(1, 5): 0.74128322,
(1, 6): 0.80998324,
(1, 7): 0.81808091,
(2, 2): -0.92032979,
(2, 3): 2.59557990,
(2, 4): 1.60427450,
(2, 5): 0.82064641,
(2, 6): 0.79413652,
(3, 3): 1.26304910,
(3, 4): 2.21146360,
(3, 5): 1.50498090,
(4, 4): 2.62223930
}
Bs_collision = {
(1, 1): [2.6431984,0.0060432255,-0.15158773,0.054237938,-0.0090468682,0.0006174200700],
(1, 2): [-0.44668594,0.42734391,-0.16036459,0.031461648,-0.0032587575,0.0001386025700],
(1, 3): [-0.1394539,0.17696362,-0.026252211,-0.0043814141,0.00167521,-0.0001438280100],
(1, 4): [0.1293817,0.059760309,0.0071109469,-0.0063851124,0.0010498938,-0.0000581492570],
(1, 5): [0.1778885,0.027398438,0.0076254248,-0.0031650182,0.0003278652,-0.0000092890016],
(1, 6): [0.073071217,0.034607908,-0.0011457199,0.000281986,-0.0002006054,0.0000214464830],
(1, 7): [0.044232851,0.029750283,-0.0022011682,0.0006326412,-0.0001755553,0.0000142557040],
(2, 2): [2.3508044,0.50110649,-0.47193769,0.15806367,-0.026367184,0.0018120118000],
(2, 3): [-1.8569443,0.96985775,-0.39888526,0.090063692,-0.010918991,0.0005664679700],
(2, 4): [-0.67406115,0.42671907,-0.10177069,0.0006185714,0.0031225358,-0.0003520605100],
(2, 5): [0.23195128,0.12233793,0.013891578,-0.020903423,0.0046715462,-0.0003520430300],
(2, 6): [0.23766123,0.077125802,0.013060901,-0.010982362,0.0018034505,-0.0000959825710],
(3, 3): [-0.36104243,0.68116214,-0.36401583,0.10500196,-0.016400134,0.0010880886000],
(3, 4): [-1.4743107,0.64918549,-0.24075196,0.051820149,-0.0060565396,0.0002981232600],
(3, 5): [-0.64335529,0.3261704,-0.082126072,0.0059682011,0.0010269488,-0.0001595725200],
(4, 4): [-1.9158462,1.016638,-0.43355278,0.10496591,-0.013951104,0.0008004853400]
}
Cs_collision = {
(1, 1): [1.6690746, -0.6914589, 0.15502132, -0.020642189, 0.001540207700, -0.000049729535],
(1, 2): [-0.47499422, 0.14482036, -0.032158368, 0.0044357933, -0.00034138118, 0.000011259742],
(1, 3): [-0.25258689, 0.059709197, -0.013332695, 0.0019619285, -0.000160630760, 0.0000055804557],
(1, 4): [-0.045055948, -0.022642753, 0.0056672308, -0.0006570876, 0.000040733113, -0.0000010820157],
(1, 5): [0.0013668724, -0.041730962, 0.010378923, -0.0013492954, 0.000096963599, -0.0000030307552],
(1, 6): [-0.071180849, -0.012738119, 0.0038582834, -0.0004706043, 0.000030466929, -0.00000085305576],
(1, 7): [-0.089417548, -0.0051856424, 0.0021882143, -0.0002487447, 0.000013745859, -0.00000030285365],
(2, 2): [1.6330213, -0.69795156, 0.16096572, -0.02210944, 0.0017031434, -0.000056699986],
(2, 3): [-1.4586197, 0.52947262, -0.11946363, 0.016264589, -0.0012354315, 0.000040366357],
(2, 4): [-0.62774499, 0.20700644, -0.04760169, 0.0067153792, -0.00052706167, 0.000017705708],
(2, 5): [0.039184885, -0.057316906, 0.012794497, -0.0015336449, 0.00010241454, -0.0000029975563],
(2, 6): [0.050470266, -0.062621672, 0.014326724, -0.0017806541, 0.00012353365, -0.0000037501381],
(3, 3): [-0.33227158, 0.079723851, -0.015470355, 0.0018686705, -0.00012179945, 0.0000032594587],
(3, 4): [-1.1942554, 0.43000688, -0.097525871, 0.013399366, -0.0010283777, 0.000033956674],
(3, 5): [-0.60014514, 0.19764859, -0.045212434, 0.0063650284, -0.00049991689, 0.000016833944],
(4, 4): [-1.4676253, 0.53048161, -0.11909781, 0.016123847, -0.0012174905, 0.0000395451]
}
def collision_integral_Kim_Monroe(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 < 400. Values are accurate to
0.007 % of actual values, but the calculation of actual values is
computationally intensive and so these simplifications are used, developed
in [1]_.
.. math::
\Omega^{(l,s)*} = A^{(l,s)} + \sum_{k=1}^6 \left[ \frac{B_k^{(l,s)}}
{(T^*)^k} + C_k^{(l,s)} (\ln T^*)^k \right]
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}
Examples
--------
>>> collision_integral_Kim_Monroe(400, 1, 1)
0.4141818082392228
References
----------
.. [1] Kim, Sun Ung, and Charles W. Monroe. "High-Accuracy Calculations of
Sixteen Collision Integrals for Lennard-Jones (12-6) Gases and Their
Interpolation to Parameterize Neon, Argon, and Krypton." Journal of
Computational Physics 273 (September 15, 2014): 358-73.
doi:10.1016/j.jcp.2014.05.018.
'''
if (l, s) not in As_collision:
raise ValueError('Input values of l and s are not supported')
omega = As_collision[(l, s)]
Bs = Bs_collision[(l, s)]
Cs = Cs_collision[(l, s)]
for ki in range(6):
omega += Bs[ki] / T_star ** (ki + 1) + Cs[ki] * log(T_star) ** (ki + 1)
return omega
### Misc
[docs]def T_star(T, epsilon_k=None, epsilon=None):
r'''This function calculates the parameter `T_star` as needed in performing
collision integral calculations.
.. math::
T^* = \frac{kT}{\epsilon}
Examples
--------
>>> T_star(T=318.2, epsilon_k=308.43)
1.0316765554582887
Parameters
----------
epsilon_k : float, optional
Lennard-Jones depth of potential-energy minimum over k, [K]
epsilon : float, optional
Lennard-Jones depth of potential-energy minimum [J]
Returns
-------
T_star : float
Dimentionless temperature for calculating collision integral, [-]
Notes
-----
Tabulated values are normally listed as epsilon/k. k is the Boltzman
constant, with units of J/K.
References
----------
.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot.
Transport Phenomena, Revised 2nd Edition. New York:
John Wiley & Sons, Inc., 2006
'''
if epsilon_k:
_Tstar = T/(epsilon_k)
elif epsilon:
_Tstar = k*T/epsilon
else:
raise ValueError('Either epsilon/k or epsilon must be provided')
return _Tstar