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# -*- coding: utf-8 -*- 

'''Chemical Engineering Design Library (ChEDL). Utilities for process modeling. 

Copyright (C) 2016, 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.''' 

 

from __future__ import division 

__all__ = ['MagalhaesLJ_data', 'Stockmayer_methods', 'Stockmayer', 

'molecular_diameter_methods', 'molecular_diameter', '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', 

'Neufeld_collision', 'collision_integral_Neufeld_Janzen_Aziz', 'As_collision', 

'Bs_collision', 'Cs_collision', 'collision_integral_Kim_Monroe', 'Tstar'] 

 

import os 

from thermo.utils import exp, log, sin 

from scipy.constants import k 

import pandas as pd 

 

 

folder = os.path.join(os.path.dirname(__file__), 'Viscosity') 

 

MagalhaesLJ_data = pd.read_csv(os.path.join(folder, 

'MagalhaesLJ.csv'), sep='\t', index_col=0) 

 

 

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' 

NONE = 'None' 

 

Stockmayer_methods = [MAGALHAES, TEEGOTOSTEWARD2, FLYNN, BSLC, TEEGOTOSTEWARD1, 

BSLB, BSLM, STIELTHODOS] 

 

 

def Stockmayer(Tm=None, Tb=None, Tc=None, Zc=None, omega=None, 

CASRN='', AvailableMethods=False, 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. 

 

Prefered 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 

 

Parameters 

---------- 

Tm : float, optional 

Melting temperature of fluid [K] 

Tb : float, optional 

Boiling temperature of fluid [K] 

Tc : float, optional 

Critical temperature, [K] 

Zc : float, optional 

Critical compressibility, [-] 

omega : float, optional 

Acentric factor of compound, [-] 

CASRN : string, optional 

CASRN [-] 

 

Returns 

------- 

epsilon_k : float 

Lennard-Jones depth of potential-energy minimum over k, [K] 

methods : list, only returned if AvailableMethods == True 

List of methods which can be used to obtain epsilon with the given 

inputs 

 

Other Parameters 

---------------- 

Method : string, optional 

A string for the method name to use, as defined by constants in 

Stockmayer_methods 

AvailableMethods : bool, optional 

If True, function will determine which methods can be used to obtain 

epsilon for the desired chemical, and will return methods instead of 

epsilon 

 

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. 

''' 

def list_methods(): 

methods = [] 

if CASRN in MagalhaesLJ_data.index: 

methods.append(MAGALHAES) 

if Tc and omega: 

methods.append(TEEGOTOSTEWARD2) 

if Tc: 

methods.append(FLYNN) 

methods.append(BSLC) 

methods.append(TEEGOTOSTEWARD1) 

if Tb: 

methods.append(BSLB) 

if Tm: 

methods.append(BSLM) 

if Tc and Zc: 

methods.append(STIELTHODOS) 

methods.append(NONE) 

return methods 

if AvailableMethods: 

return list_methods() 

if not Method: 

Method = list_methods()[0] 

 

if Method == FLYNN: 

epsilon = epsilon_Flynn(Tc) 

elif Method == BSLC: 

epsilon = epsilon_Bird_Stewart_Lightfoot_critical(Tc) 

elif Method == BSLB: 

epsilon = epsilon_Bird_Stewart_Lightfoot_boiling(Tb) 

elif Method == BSLM: 

epsilon = epsilon_Bird_Stewart_Lightfoot_melting(Tm) 

elif Method == STIELTHODOS: 

epsilon = epsilon_Stiel_Thodos(Tc, Zc) 

elif Method == TEEGOTOSTEWARD1: 

epsilon = epsilon_Tee_Gotoh_Steward_1(Tc) 

elif Method == TEEGOTOSTEWARD2: 

epsilon = epsilon_Tee_Gotoh_Steward_2(Tc, omega) 

 

elif Method == MAGALHAES: 

epsilon = float(MagalhaesLJ_data.at[CASRN, "epsilon"]) 

elif Method == NONE: 

epsilon = None 

else: 

raise Exception('Failure in in function') 

return epsilon 

 

 

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_methods = [MAGALHAES, TEEGOTOSTEWARD4, SILVALIUMACEDO, 

BSLC2, TEEGOTOSTEWARD3, STIELTHODOSMD, FLYNN, 

BSLC1, BSLB, BSLM] 

 

 

def molecular_diameter(Tc=None, Pc=None, Vc=None, Zc=None, omega=None, 

Vm=None, Vb=None, CASRN='', AvailableMethods=False, 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. 

 

Prefered 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 

 

Parameters 

---------- 

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] 

CASRN : string, optional 

CASRN [-] 

 

Returns 

------- 

sigma : float 

Lennard-Jones molecular diameter, [Angstrom] 

methods : list, only returned if AvailableMethods == True 

List of methods which can be used to obtain epsilon with the given 

inputs 

 

Other Parameters 

---------------- 

Method : string, optional 

A string for the method name to use, as defined by constants in 

molecular_diameter_methods 

AvailableMethods : bool, optional 

If True, function will determine which methods can be used to obtain 

sigma for the desired chemical, and will return methods instead of 

sigma 

 

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. 

''' 

def list_methods(): 

methods = [] 

if CASRN in MagalhaesLJ_data.index: 

methods.append(MAGALHAES) 

if Tc and Pc and omega: 

methods.append(TEEGOTOSTEWARD4) 

if Tc and Pc: 

methods.append(SILVALIUMACEDO) 

methods.append(BSLC2) 

methods.append(TEEGOTOSTEWARD3) 

if Vc and Zc: 

methods.append(STIELTHODOSMD) 

if Vc: 

methods.append(FLYNN) 

methods.append(BSLC1) 

if Vb: 

methods.append(BSLB) 

if Vm: 

methods.append(BSLM) 

methods.append(NONE) 

return methods 

 

if AvailableMethods: 

return list_methods() 

if not Method: 

Method = list_methods()[0] 

if Method == FLYNN: 

sigma = sigma_Flynn(Vc) 

elif Method == BSLC1: 

sigma = sigma_Bird_Stewart_Lightfoot_critical_1(Vc) 

elif Method == BSLC2: 

sigma = sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc) 

elif Method == TEEGOTOSTEWARD3: 

sigma = sigma_Tee_Gotoh_Steward_1(Tc, Pc) 

elif Method == SILVALIUMACEDO: 

sigma = sigma_Silva_Liu_Macedo(Tc, Pc) 

elif Method == BSLB: 

sigma = sigma_Bird_Stewart_Lightfoot_boiling(Vb) 

elif Method == BSLM: 

sigma = sigma_Bird_Stewart_Lightfoot_melting(Vm) 

elif Method == STIELTHODOSMD: 

sigma = sigma_Stiel_Thodos(Vc, Zc) 

elif Method == TEEGOTOSTEWARD4: 

sigma = sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega) 

elif Method == MAGALHAES: 

sigma = float(MagalhaesLJ_data.at[CASRN, "sigma"]) 

elif Method == NONE: 

sigma = None 

else: 

raise Exception('Failure in in function') 

return sigma 

 

 

### Sigma Lennard-Jones 

 

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 

 

 

def sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc): 

r'''Calculates Lennard-Jones molecular diameter. 

Uses critical temperature and pressure. CSP method by [1]_. 

 

.. math:: 

\sigma = 2.44(T_c/P_c)^{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 critical pressure are atmospheres. 

 

Examples 

-------- 

>>> sigma_Bird_Stewart_Lightfoot_critical_2(560.1, 4550000) 

5.658657684653222 

 

References 

---------- 

.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. 

Transport Phenomena, Revised 2nd Edition. New York: 

John Wiley & Sons, Inc., 2006 

''' 

Pc = Pc/101325. 

sigma = 2.44*(Tc/Pc)**(1/3.0) 

return sigma 

 

 

def sigma_Bird_Stewart_Lightfoot_critical_1(Vc): 

r'''Calculates Lennard-Jones molecular diameter. 

Uses critical volume. CSP method by [1]_. 

 

.. math:: 

\sigma = 0.841 V_c^{1/3} 

 

Parameters 

---------- 

Vc : float 

Critical volume of fluid [m^3/mol] 

 

Returns 

------- 

sigma : float 

Lennard-Jones molecular diameter, [Angstrom] 

 

Notes 

----- 

Original units of Vc are mL/mol. 

 

Examples 

-------- 

>>> sigma_Bird_Stewart_Lightfoot_critical_1(0.000268) 

5.422184116631474 

 

References 

---------- 

.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. 

Transport Phenomena, Revised 2nd Edition. New York: 

John Wiley & Sons, Inc., 2006 

''' 

Vc = Vc*1E6 # m^3/mol to cm^3/mol 

sigma = 0.841*Vc**(1/3.0) 

return sigma 

 

 

def sigma_Bird_Stewart_Lightfoot_boiling(Vb): 

r'''Calculates Lennard-Jones molecular diameter. 

Uses molar volume of liquid at boiling. CSP method by [1]_. 

 

.. math:: 

\sigma = 1.166V_{b,liq}^{1/3} 

 

Parameters 

---------- 

Vb : float 

Boiling molar volume of liquid [m^3/mol] 

 

Returns 

------- 

sigma : float 

Lennard-Jones collision integral, [Angstrom] 

 

Notes 

----- 

Original units of Vb are mL/mol. 

 

Examples 

-------- 

>>> sigma_Bird_Stewart_Lightfoot_boiling(0.0001015) 

5.439018856944655 

 

References 

---------- 

.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. 

Transport Phenomena, Revised 2nd Edition. New York: 

John Wiley & Sons, Inc., 2006 

''' 

Vb = Vb*1E6 

sigma = 1.166*Vb**(1/3.0) 

return sigma 

 

 

def sigma_Bird_Stewart_Lightfoot_melting(Vm): 

r'''Calculates Lennard-Jones molecular diameter. 

Uses molar volume of a liquid at its melting point. CSP method by [1]_. 

 

.. math:: 

\sigma = 1.222 V_{m,sol}^{1/3} 

 

Parameters 

---------- 

Vm : float 

Melting molar volume of a liquid at its melting point [m^3/mol] 

 

Returns 

------- 

sigma : float 

Lennard-Jones molecular diameter, [Angstrom] 

 

Notes 

----- 

Original units of Vm are mL/mol. 

 

Examples 

-------- 

>>> sigma_Bird_Stewart_Lightfoot_melting(8.8e-05) 

5.435407341351406 

 

References 

---------- 

.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. 

Transport Phenomena, Revised 2nd Edition. New York: 

John Wiley & Sons, Inc., 2006 

''' 

Vm = Vm*1E6 

sigma = 1.222*Vm**(1/3.) 

return sigma 

 

 

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 

 

 

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 

 

 

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 

 

 

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 

 

 

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 

 

 

def epsilon_Bird_Stewart_Lightfoot_critical(Tc): 

r'''Calculates Lennard-Jones depth of potential-energy minimum. 

Uses critical temperature. CSP method by [1]_. 

 

.. math:: 

\epsilon/k = 0.77T_c 

 

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_Bird_Stewart_Lightfoot_critical(560.1) 

431.27700000000004 

 

References 

---------- 

.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. 

Transport Phenomena, Revised 2nd Edition. New York: 

John Wiley & Sons, Inc., 2006 

''' 

epsilon_k = 0.77*Tc 

return epsilon_k 

 

 

def epsilon_Bird_Stewart_Lightfoot_boiling(Tb): 

r'''Calculates Lennard-Jones depth of potential-energy minimum. 

Uses boiling temperature. CSP method by [1]_. 

 

.. math:: 

\epsilon/k = 1.15 T_b 

 

Parameters 

---------- 

Tb : float 

Boiling temperature [K] 

 

Returns 

------- 

epsilon_k : float 

Lennard-Jones depth of potential-energy minimum over k, [K] 

 

Notes 

----- 

 

Examples 

-------- 

>>> epsilon_Bird_Stewart_Lightfoot_boiling(357.85) 

411.5275 

 

References 

---------- 

.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. 

Transport Phenomena, Revised 2nd Edition. New York: 

John Wiley & Sons, Inc., 2006 

''' 

epsilon_k = 1.15*Tb 

return epsilon_k 

 

 

def epsilon_Bird_Stewart_Lightfoot_melting(Tm): 

r'''Calculates Lennard-Jones depth of potential-energy minimum. 

Uses melting temperature. CSP method by [1]_. 

 

.. math:: 

\epsilon/k = 1.92T_m 

 

Parameters 

---------- 

Tm : float 

Melting temperature [K] 

 

Returns 

------- 

epsilon_k : float 

Lennard-Jones depth of potential-energy minimum over k, [K] 

 

Notes 

----- 

 

Examples 

-------- 

>>> epsilon_Bird_Stewart_Lightfoot_melting(231.15) 

443.808 

 

References 

---------- 

.. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. 

Transport Phenomena, Revised 2nd Edition. New York: 

John Wiley & Sons, Inc., 2006 

''' 

epsilon_k = 1.92*Tm 

return epsilon_k 

 

 

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 

 

 

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 

 

 

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 

 

Neufeld_collision = { 

(1, 1): [1.06036, 0.1561, 0.193, 0.47635, 1.03587, 1.52996, 1.76474, 3.89411, None, None, None, None], 

(1, 2): [1.0022, 0.1553, 0.16105, 0.72751, 0.86125, 2.06848, 1.95162, 4.84492, None, None, None, None], 

(1, 3): [0.96573, 0.15611, 0.44067, 1.5242, 2.38981, 5.08063, None, None, -0.0005373, 19.2866, -1.30775, 6.58711], 

(1, 4): [0.93447, 0.15578, 0.39478, 1.85761, 2.45988, 6.15727, None, None, 0.0004246, 12.988, -1.36399, 3.3329], 

(1, 5): [0.90972, 0.15565, 0.35967, 2.18528, 2.45169, 7.17936, None, None, -0.0003814, 9.38191, 0.14025, 9.93802], 

(1, 6): [0.88928, 0.15562, 0.33305, 2.51303, 2.36298, 8.1169, None, None, -0.0004649, 9.86928, 0.12851, 9.82414], 

(1, 7): [0.87208, 0.15568, 0.36583, 3.01399, 2.70659, 9.9231, None, None, -0.0004902, 10.2274, 0.12306, 9.97712], 

(2, 2): [1.16145, 0.14874, 0.52487, 0.7732, 2.16178, 2.43787, None, None, -0.0006435, 18.0323, -0.7683, 7.27371], 

(2, 3): [1.11521, 0.14796, 0.44844, 0.99548, 2.30009, 3.06031, None, None, 0.0004565, 38.5868, -0.69403, 2.56375], 

(2, 4): [1.08228, 0.14807, 0.47128, 1.31596, 2.42738, 3.90018, None, None, -0.0005623, 3.08449, 0.28271, 3.22871], 

(2, 5): [1.05581, 0.14822, 0.51203, 1.67007, 2.57317, 4.85939, None, None, -0.000712, 4.7121, 0.2173, 4.7353], 

(2, 6): [1.03358, 0.14834, 0.53928, 2.01942, 2.7235, 5.84817, None, None, -0.0008576, 7.66012, 0.15493, 7.6011], 

(3, 3): [1.05567, 0.1498, 0.30887, 0.86437, 1.35766, 2.44123, 1.2903, 5.55734, 0.0002339, 57.7757, -1.0898, 6.9475], 

(3, 4): [1.02621, 0.1505, 0.55381, 1.4007, 2.06176, 4.26234, None, None, 0.0005227, 11.3331, -0.8209, 3.87185], 

(3, 5): [0.99958, 0.15029, 0.50441, 1.64304, 2.06947, 4.87712, None, None, -0.0005184, 3.45031, 0.26821, 3.73348], 

(4, 4): [1.12007, 0.14578, 0.53347, 1.11986, 2.28803, 3.27567, None, None, 0.0007427, 21.048, -0.28759, 6.69149] 

} 

 

 

def collision_integral_Neufeld_Janzen_Aziz(Tstar, 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 < Tstar < 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 

Tstar = 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, s) not in Neufeld_collision: 

raise Exception('Input values of l and s are not supported') 

A, B, C, D, E, F, G, H, R, S, W, P = Neufeld_collision[(l, s)] 

omega = A/Tstar**B + C/exp(D*Tstar) + E/exp(F*Tstar) 

if (l, s) in [(1, 1), (1, 2), (3, 3)]: 

omega += G/exp(H*Tstar) 

if (l, s) not in [(1, 1), (1, 2)]: 

omega += R*Tstar**B*sin(S*Tstar**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(Tstar, 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 < Tstar < 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 Exception('Input values of l and s are not supported') 

omega = As_collision[(l, s)] 

for ki in range(6): 

Bs = Bs_collision[(l, s)] 

Cs = Cs_collision[(l, s)] 

omega += Bs[ki]/Tstar**(ki+1) + Cs[ki]*log(Tstar)**(ki+1) 

return omega 

 

 

### Misc 

 

 

def Tstar(T, epsilon_k=None, epsilon=None): 

r'''This function calculates the parameter `Tstar` as needed in performing 

collision integral calculations. 

 

.. math:: 

T^* = \frac{kT}{\epsilon} 

 

Examples 

-------- 

>>> Tstar(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 

------- 

Tstar : 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 Exception('Either epsilon/k or epsilon must be provided') 

return _Tstar