"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2019 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 surface tension estimation routines, dataframes
of fit coefficients, fitting model equations, mixing rules, and
water-hydrocarbon interfacial tension estimation routines.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/chemicals/>`_.
.. contents:: :local:
Pure Component Correlations
---------------------------
.. autofunction:: chemicals.interface.Brock_Bird
.. autofunction:: chemicals.interface.Pitzer_sigma
.. autofunction:: chemicals.interface.Sastri_Rao
.. autofunction:: chemicals.interface.Zuo_Stenby
.. autofunction:: chemicals.interface.Hakim_Steinberg_Stiel
.. autofunction:: chemicals.interface.Miqueu
.. autofunction:: chemicals.interface.Aleem
.. autofunction:: chemicals.interface.Mersmann_Kind_sigma
.. autofunction:: chemicals.interface.sigma_Gharagheizi_1
.. autofunction:: chemicals.interface.sigma_Gharagheizi_2
Mixing Rules
------------
.. autofunction:: chemicals.interface.Winterfeld_Scriven_Davis
.. autofunction:: chemicals.interface.Weinaug_Katz
.. autofunction:: chemicals.interface.Diguilio_Teja
Correlations for Specific Substances
------------------------------------
.. autofunction:: chemicals.interface.sigma_IAPWS
Petroleum Correlations
----------------------
.. autofunction:: chemicals.interface.API10A32
Oil-Water Interfacial Tension Correlations
------------------------------------------
.. autofunction:: chemicals.interface.Meybodi_Daryasafar_Karimi
Fit Correlations
----------------
.. autofunction:: chemicals.interface.REFPROP_sigma
.. autofunction:: chemicals.interface.Somayajulu
.. autofunction:: chemicals.interface.Jasper
.. autofunction:: chemicals.interface.PPDS14
.. autofunction:: chemicals.interface.Watson_sigma
.. autofunction:: chemicals.interface.ISTExpansion
Fit Coefficients
----------------
All of these coefficients are lazy-loaded, so they must be accessed as an
attribute of this module.
.. data:: sigma_data_Mulero_Cachadina
Data from [5]_ with :obj:`REFPROP_sigma` coefficients.
.. data:: sigma_data_Jasper_Lange
Data as shown in [4]_ but originally in [3]_ with :obj:`Jasper` coefficients.
.. data:: sigma_data_Somayajulu
Data from [1]_ with :obj:`Somayajulu` coefficients.
.. data:: sigma_data_Somayajulu2
Data from [2]_ with :obj:`Somayajulu` coefficients. These should be
preferred over the original coefficients.
.. data:: sigma_data_VDI_PPDS_11
Data from [6]_ with :obj:`chemicals.dippr.EQ106` coefficients.
.. [1] Somayajulu, G. R. "A Generalized Equation for Surface Tension from
the Triple Point to the Critical Point." International Journal of
Thermophysics 9, no. 4 (July 1988): 559-66. doi:10.1007/BF00503154.
.. [2] Mulero, A., M. I. Parra, and I. Cachadina. "The Somayajulu
Correlation for the Surface Tension Revisited." Fluid Phase
Equilibria 339 (February 15, 2013): 81-88.
doi:10.1016/j.fluid.2012.11.038.
.. [3] Jasper, Joseph J. "The Surface Tension of Pure Liquid Compounds."
Journal of Physical and Chemical Reference Data 1, no. 4
(October 1, 1972): 841-1010. doi:10.1063/1.3253106.
.. [4] Speight, James. Lange's Handbook of Chemistry. 16 edition.
McGraw-Hill Professional, 2005.
.. [5] Mulero, A., I. Cachadiña, and M. I. Parra. “Recommended
Correlations for the Surface Tension of Common Fluids.” Journal of
Physical and Chemical Reference Data 41, no. 4 (December 1, 2012):
043105. doi:10.1063/1.4768782.
.. [6] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
Berlin; New York:: Springer, 2010.
The structure of each dataframe is shown below:
.. ipython::
In [1]: import chemicals
In [2]: chemicals.interface.sigma_data_Mulero_Cachadina
In [3]: chemicals.interface.sigma_data_Jasper_Lange
In [4]: chemicals.interface.sigma_data_Somayajulu
In [5]: chemicals.interface.sigma_data_Somayajulu2
In [6]: chemicals.interface.sigma_data_VDI_PPDS_11
"""
__all__ = ['REFPROP_sigma', 'Somayajulu', 'Jasper',
'Brock_Bird', 'Pitzer_sigma', 'Sastri_Rao', 'Zuo_Stenby',
'sigma_IAPWS', 'PPDS14', 'Watson_sigma',
'Mersmann_Kind_sigma', 'API10A32',
'Hakim_Steinberg_Stiel', 'Miqueu', 'Aleem',
'Winterfeld_Scriven_Davis', 'Diguilio_Teja', 'Weinaug_Katz',
'Meybodi_Daryasafar_Karimi', 'ISTExpansion', 'sigma_Gharagheizi_1',
'sigma_Gharagheizi_2']
from fluids.constants import N_A, k, root_two
from fluids.numerics import exp, log, sqrt
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, 'Interface')
register_df_source(folder, 'MuleroCachadinaParameters.tsv')
register_df_source(folder, 'Jasper-Lange.tsv')
register_df_source(folder, 'Somayajulu.tsv')
register_df_source(folder, 'SomayajuluRevised.tsv')
register_df_source(folder, 'VDI PPDS surface tensions.tsv')
_interface_dfs_loaded = False
@mark_numba_incompatible
def load_interface_dfs():
global _interface_dfs_loaded, sigma_data_Mulero_Cachadina, sigma_values_Mulero_Cachadina
global sigma_data_Jasper_Lange, sigma_values_Jasper_Lange
global sigma_data_Somayajulu, sigma_values_Somayajulu, sigma_data_Somayajulu2
global sigma_values_Somayajulu2, sigma_data_VDI_PPDS_11, sigma_values_VDI_PPDS_11
sigma_data_Mulero_Cachadina = data_source('MuleroCachadinaParameters.tsv')
sigma_values_Mulero_Cachadina = np.array(sigma_data_Mulero_Cachadina.values[:, 1:], dtype=float)
sigma_data_Jasper_Lange = data_source('Jasper-Lange.tsv')
sigma_values_Jasper_Lange = np.array(sigma_data_Jasper_Lange.values[:, 1:], dtype=float)
sigma_data_Somayajulu = data_source('Somayajulu.tsv')
sigma_values_Somayajulu = np.array(sigma_data_Somayajulu.values[:, 1:], dtype=float)
sigma_data_Somayajulu2 = data_source('SomayajuluRevised.tsv')
sigma_values_Somayajulu2 = np.array(sigma_data_Somayajulu2.values[:, 1:], dtype=float)
sigma_data_VDI_PPDS_11 = data_source('VDI PPDS surface tensions.tsv')
sigma_values_VDI_PPDS_11 = np.array(sigma_data_VDI_PPDS_11.values[:, 1:], dtype=float)
if PY37:
def __getattr__(name):
if name in ('sigma_data_Mulero_Cachadina', 'sigma_values_Mulero_Cachadina',
'sigma_data_Jasper_Lange', 'sigma_values_Jasper_Lange',
'sigma_data_Somayajulu', 'sigma_values_Somayajulu', 'sigma_data_Somayajulu2',
'sigma_values_Somayajulu2', 'sigma_data_VDI_PPDS_11', 'sigma_values_VDI_PPDS_11'
):
load_interface_dfs()
return globals()[name]
raise AttributeError(f"module {__name__} has no attribute {name}")
else:
if can_load_data:
load_interface_dfs()
[docs]def sigma_IAPWS(T):
r'''Calculate the surface tension of pure water as a function of .
temperature. Assumes the 2011 IAPWS [1]_ formulation.
.. math::
\sigma = B\tau^\mu(1+b\tau)\\
.. math::
\tau = 1-T/T_c\\
.. math::
B = 0.2358 \text{N/m}\\
.. math::
b = -0.625\\
.. math::
\mu = 1.256
Parameters
----------
T : float
Temperature of liquid [K]
Returns
-------
sigma : float
Air-liquid surface tension, [N/m]
Notes
-----
This function is valid from the triple temperature to the critical
temperature. No effects for pressure are included in the formulation.
Test values are from IAPWS 2010 book.
The equation is valid from the triple point to the critical point,
647.096 K; but [1]_ also recommends its use down to -25°C.
If a value larger than the critical temperature is input, 0.0 is returned.
Examples
--------
>>> sigma_IAPWS(300.)
0.0716859625271
>>> sigma_IAPWS(450.)
0.0428914991565
>>> sigma_IAPWS(600.)
0.0083756108728
References
----------
.. [1] IAPWS. 2014. Revised Release on Surface Tension of Ordinary Water
Substance
'''
tau = 1. - T*(1.0/647.096)
if tau < 0.0:
tau = 0.0
return 0.2358*tau**1.256*(1.0 - 0.625*tau)
### Regressed coefficient-based functions
[docs]def REFPROP_sigma(T, Tc, sigma0, n0, sigma1=0.0, n1=0.0, sigma2=0.0, n2=0.0):
r'''Calculates air-liquid surface tension using the REFPROP_sigma [1]_
regression-based method. Relatively recent, and most accurate.
.. math::
\sigma(T)=\sigma_0\left(1-\frac{T}{T_c}\right)^{n_0}+
\sigma_1\left(1-\frac{T}{T_c}\right)^{n_1}+
\sigma_2\left(1-\frac{T}{T_c}\right)^{n_2}
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
sigma0 : float
First emperical coefficient of a fluid
n0 : float
First emperical exponent of a fluid
sigma1 : float, optional
Second emperical coefficient of a fluid.
n1 : float, optional
Second emperical exponent of a fluid.
sigma1 : float, optional
Third emperical coefficient of a fluid.
n2 : float, optional
Third emperical exponent of a fluid.
Returns
-------
sigma : float
Liquid surface tension, [N/m]
Notes
-----
Function as implemented in [1]_. No example necessary; results match
literature values perfectly.
Form of function returns imaginary results when T > Tc; 0 is returned
if this is the case.
When fitting parameters to this function, it is easy to end up with a
fit that returns negative surface tension near but not quite at the
critical point.
Examples
--------
Parameters for water at 298.15 K
>>> REFPROP_sigma(298.15, 647.096, -0.1306, 2.471, 0.2151, 1.233)
0.07205503890847453
References
----------
.. [1] Diky, Vladimir, Robert D. Chirico, Chris D. Muzny, Andrei F.
Kazakov, Kenneth Kroenlein, Joseph W. Magee, Ilmutdin Abdulagatov, and
Michael Frenkel. "ThermoData Engine (TDE): Software Implementation of
the Dynamic Data Evaluation Concept." Journal of Chemical Information
and Modeling 53, no. 12 (2013): 3418-30. doi:10.1021/ci4005699.
'''
Tr = T/Tc
tau = 1.0 - Tr
if tau <= 0.0:
return 0.0
if n1 == 0.0 and sigma1 != 0.0:
sigma1 = 0.0
if n2 == 0.0 and sigma2 != 0.0:
sigma2 = 0.0
sigma = sigma0*(tau)**n0 + sigma1*(tau)**n1 + sigma2*(tau)**n2
return sigma
[docs]def PPDS14(T, Tc, a0, a1, a2):
r'''Calculates air-liquid surface tension using the [1]_
emperical (parameter-regressed) method, called the PPDS 14 equation for
surface tension.
.. math::
\sigma = a_0 \tau^{a_1}(1 + a_2 \tau)
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
a0 : float
Regression parameter, [N/m]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
Returns
-------
sigma : float
Liquid surface tension, [N/m]
Notes
-----
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
If this model is fit with `a0` and `a2` as positive values, it is
guaranteed to predict only positive values of `sigma` right up to the
critical point. However, `a2` is often fit to be a negative value.
Examples
--------
Benzene at 280 K from [1]_
>>> PPDS14(T=280, Tc=562.05, a0=0.0786269, a1=1.28646, a2=-0.112304)
0.030559764256249854
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User`s Guide."
https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-SurfaceTension/PPDS14.htm.
.. [2] Frenkel, Michael, Robert D. Chirico, Vladimir Diky, Xinjian Yan,
Qian Dong, and Chris Muzny. "ThermoData Engine (TDE): Software
Implementation of the Dynamic Data Evaluation Concept." Journal of
Chemical Information and Modeling 45, no. 4 (July 1, 2005): 816-38.
https://doi.org/10.1021/ci050067b.
'''
tau = 1.0 - T/Tc
if tau <= 0.0:
return 0.0
return a0*tau**a1*(1.0 + a2*tau)
[docs]def Watson_sigma(T, Tc, a1, a2, a3=0.0, a4=0.0, a5=0.0):
r'''Calculates air-liquid surface tension using the Watson [1]_
emperical (parameter-regressed) method developed by NIST.
.. math::
\sigma = \exp\left[a_{1} + \ln(1 - T_r)\left(
a_2 + a_3T_r + a_4T_r^2 + a_5T_r^3 \right)\right]
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
a3 : float
Regression parameter, [-]
a4 : float
Regression parameter, [-]
a5 : float
Regression parameter, [-]
Returns
-------
sigma : float
Liquid surface tension, [N/m]
Notes
-----
This expression is also used for enthalpy of vaporization in [1]_.
The coefficients from NIST TDE for enthalpy of vaporization are kJ/mol.
This model is coded to return 0 values at Tr >= 1. It is otherwise not
possible to evaluate this expression at Tr = 1, as log(0) is undefined
(although the limit shows the expression converges to 0).
This equation does not have any term forcing it to become near-zero
at the critical point, but it cannot be fit so as to produce negative
values.
Examples
--------
Isooctane at 350 K from [1]_:
>>> Watson_sigma(T=350.0, Tc=543.836, a1=-3.02417, a2=1.21792, a3=-5.26877e-9, a4=5.62659e-9, a5=-2.27553e-9)
0.0138340926605649
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User`s Guide."
https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-SurfaceTension/HVPExpansion-SurfaceTension.htm
'''
Tr = T/Tc
if Tr >= 1.0:
return 0.0
l = log(1.0 - Tr)
return exp(a1 + l*(a2 + Tr*(a3 + Tr*(a4 + a5*Tr))))
[docs]def ISTExpansion(T, Tc, a1, a2, a3=0.0, a4=0.0, a5=0.0):
r'''Calculates air-liquid surface tension using the IST expansion [1]_
emperical (parameter-regressed) method developed by NIST.
.. math::
\sigma = \sum_i a_i\left(1 - \frac{T}{T_c} \right)^i
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
a3 : float
Regression parameter, [-]
a4 : float
Regression parameter, [-]
a5 : float
Regression parameter, [-]
Returns
-------
sigma : float
Liquid surface tension, [N/m]
Notes
-----
This equation hsa a term term forcing it to become zero
at the critical point, but it can easily be fit so as to produce negative
values at any reduced temperature.
Examples
--------
Diethyl phthalate at 400 K from [1]_:
>>> ISTExpansion(T=400.0, Tc=776.0, a1=0.037545, a2=0.0363288)
0.02672100905515996
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User`s Guide."
https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-SurfaceTension/ISTExpansion-SurfaceTension.htm
'''
tau = 1.0 - T/Tc
if tau <= 0.0:
return 0.0
return tau*(a1 + tau*(a2 + tau*(a3 + tau*(a4 + a5*tau))))
[docs]def Somayajulu(T, Tc, A, B, C):
r'''Calculates air-liquid surface tension using the [1]_
emperical (parameter-regressed) method. Well regressed, no recent data.
.. math::
\sigma=aX^{5/4}+bX^{9/4}+cX^{13/4}
.. math::
X=(T_c-T)/T_c
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
A : float
Regression parameter
B : float
Regression parameter
C : float
Regression parameter
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
Presently untested, but matches expected values. Internal units are mN/m.
Form of function returns imaginary results when T > Tc; 0.0 is returned
if this is the case. Function is claimed valid from the triple to the
critical point. Results can be evaluated beneath the triple point.
This function can be accidentally fit to return negative
values of surface tension.
Examples
--------
Water at 300 K
>>> Somayajulu(300, 647.126, 232.713514, -140.18645, -4.890098)
0.07166386387996758
References
----------
.. [1] Somayajulu, G. R. "A Generalized Equation for Surface Tension from
the Triple Point to the Critical Point." International Journal of
Thermophysics 9, no. 4 (July 1988): 559-66. doi:10.1007/BF00503154.
'''
if T >= Tc:
return 0.0
X = (Tc-T)/Tc
return X*sqrt(sqrt(X))*(A + X*(B + C*X))*1e-3
[docs]def Jasper(T, a, b):
r'''Calculates surface tension of a fluid given two parameters, a linear
fit in Celcius from [1]_ with data reprinted in [2]_.
.. math::
\sigma = a - bT
Parameters
----------
T : float
Temperature of fluid, [K]
a : float
Parameter for equation. Chemical specific.
b : float
Parameter for equation. Chemical specific.
Returns
-------
sigma : float
Surface tension [N/m]
Notes
-----
Internal units are mN/m, and degrees Celcius.
This function has been checked against several references.
As this is a linear model, negative values of surface tension will
eventually arise. 0 is returned in these cases.
Examples
--------
>>> Jasper(298.15, 24, 0.0773)
0.0220675
References
----------
.. [1] Jasper, Joseph J. "The Surface Tension of Pure Liquid Compounds."
Journal of Physical and Chemical Reference Data 1, no. 4
(October 1, 1972): 841-1010. doi:10.1063/1.3253106.
.. [2] Speight, James. Lange's Handbook of Chemistry. 16 edition.
McGraw-Hill Professional, 2005.
'''
sigma = (a - b*(T-273.15))*1e-3
if sigma < 0.0:
return 0.0
return sigma
### CSP methods
[docs]def Brock_Bird(T, Tb, Tc, Pc):
r'''Calculates air-liquid surface tension using the [1]_
emperical method. Old and tested.
.. math::
\sigma = P_c^{2/3}T_c^{1/3}Q(1-T_r)^{11/9}
.. math::
Q = 0.1196 \left[ 1 + \frac{T_{br}\ln (P_c/1.01325)}{1-T_{br}}\right]-0.279
Parameters
----------
T : float
Temperature of fluid [K]
Tb : float
Boiling temperature of the fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
Numerous arrangements of this equation are available.
This is DIPPR Procedure 7A: Method for the Surface Tension of Pure,
Nonpolar, Nonhydrocarbon Liquids
The exact equation is not in the original paper.
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
p-dichloribenzene at 412.15 K, from DIPPR; value differs due to a slight
difference in method.
>>> Brock_Bird(412.15, 447.3, 685, 3.952E6)
0.02208448325192495
Chlorobenzene from Poling, as compared with a % error value at 293 K.
>>> Brock_Bird(293.15, 404.75, 633.0, 4530000.0)
0.032985686413713036
References
----------
.. [1] Brock, James R., and R. Byron Bird. "Surface Tension and the
Principle of Corresponding States." AIChE Journal 1, no. 2
(June 1, 1955): 174-77. doi:10.1002/aic.690010208
'''
if T >= Tc:
return 0.0
Tc_inv = 1.0/Tc
Tbr = Tb*Tc_inv
Tr = T*Tc_inv
Pc = Pc*1e-5 # Convert to bar
Q = 0.1196*(1.0 + Tbr*log(Pc*(1.0/1.01325))/(1.0 - Tbr)) - 0.279
sigma = (Pc)**(2.0/3.0)*Tc**(1.0/3.0)*Q*(1.0 - Tr)**(11.0/9.0)
sigma = sigma*1e-3 # convert to N/m
return sigma
[docs]def Pitzer_sigma(T, Tc, Pc, omega):
r'''Calculates air-liquid surface tension using the correlation derived
by [1]_ from the works of [2]_ and [3]_. Based on critical property CSP
methods.
.. math::
\sigma = P_c^{2/3}T_c^{1/3}\frac{1.86 + 1.18\omega}{19.05}
\left[ \frac{3.75 + 0.91 \omega}{0.291 - 0.08 \omega}\right]^{2/3} (1-T_r)^{11/9}
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 for fluid, [-]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
The source of this equation has not been reviewed.
Internal units of presure are bar, surface tension of mN/m.
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
Chlorobenzene from Poling, as compared with a % error value at 293 K.
>>> Pitzer_sigma(293., 633.0, 4530000.0, 0.249)
0.03458453513446388
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
.. [2] Curl, R. F., and Kenneth Pitzer. "Volumetric and Thermodynamic
Properties of Fluids-Enthalpy, Free Energy, and Entropy." Industrial &
Engineering Chemistry 50, no. 2 (February 1, 1958): 265-74.
doi:10.1021/ie50578a047
.. [3] Pitzer, K. S.: Thermodynamics, 3d ed., New York, McGraw-Hill,
1995, p. 521.
'''
if T >= Tc:
return 0.0
Tr = T/Tc
Pc = Pc*1e-5 # Convert to bar
sigma = Pc**(2.0/3.0)*Tc**(1.0/3.0)*(1.86 + 1.18*omega)*(1.0/19.05)*(
(3.75 + 0.91*omega)/(0.291 - 0.08*omega))**(2.0/3.0)*(1.0 - Tr)**(11.0/9.0)
return sigma*1e-3 # N/m, please
[docs]def Sastri_Rao(T, Tb, Tc, Pc, chemicaltype=None):
r'''Calculates air-liquid surface tension using the correlation derived by
[1]_ based on critical property CSP methods and chemical classes.
.. math::
\sigma = K P_c^xT_b^y T_c^z\left[\frac{1-T_r}{1-T_{br}}\right]^m
Parameters
----------
T : float
Temperature of fluid [K]
Tb : float
Boiling temperature of the fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
The source of this equation has not been reviewed.
Internal units of presure are bar, surface tension of mN/m.
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
Chlorobenzene from Poling, as compared with a % error value at 293 K.
>>> Sastri_Rao(293.15, 404.75, 633.0, 4530000.0)
0.03234567739694441
References
----------
.. [1] Sastri, S. R. S., and K. K. Rao. "A Simple Method to Predict
Surface Tension of Organic Liquids." The Chemical Engineering Journal
and the Biochemical Engineering Journal 59, no. 2 (October 1995): 181-86.
doi:10.1016/0923-0467(94)02946-6.
'''
if T >= Tc:
return 0.0
if chemicaltype == 'alcohol':
k, x, y, z, m = 2.28, 0.25, 0.175, 0, 0.8
elif chemicaltype == 'acid':
k, x, y, z, m = 0.125, 0.50, -1.5, 1.85, 11.0/9.0
else:
k, x, y, z, m = 0.158, 0.50, -1.5, 1.85, 11.0/9.0
Tr = T/Tc
Tbr = Tb/Tc
Pc = Pc*1E-5 # Convert to bar
sigma = k*Pc**x*Tb**y*Tc**z*((1.0 - Tr)/(1.0 - Tbr))**m
sigma = sigma*1e-3 # N/m
return sigma
[docs]def Zuo_Stenby(T, Tc, Pc, omega):
r'''Calculates air-liquid surface tension using the reference fluids
methods of [1]_.
.. math::
\sigma^{(1)} = 40.520(1-T_r)^{1.287}
.. math::
\sigma^{(2)} = 52.095(1-T_r)^{1.21548}
.. math::
\sigma_r = \sigma_r^{(1)}+ \frac{\omega - \omega^{(1)}}
{\omega^{(2)}-\omega^{(1)}} (\sigma_r^{(2)}-\sigma_r^{(1)})
.. math::
\sigma = T_c^{1/3}P_c^{2/3}[\exp{(\sigma_r)} -1]
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 for fluid, [-]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
Presently untested. Have not personally checked the sources.
The reference values for methane and n-octane are from the DIPPR database.
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
Chlorobenzene
>>> Zuo_Stenby(293., 633.0, 4530000.0, 0.249)
0.03345569011871088
References
----------
.. [1] Zuo, You-Xiang, and Erling H. Stenby. "Corresponding-States and
Parachor Models for the Calculation of Interfacial Tensions." The
Canadian Journal of Chemical Engineering 75, no. 6 (December 1, 1997):
1130-37. doi:10.1002/cjce.5450750617
'''
if T >= Tc:
return 0.0
Tc_1, Pc_1, omega_1 = 190.56, 4599000.0*1e-5, 0.012
Tc_2, Pc_2, omega_2 = 568.7, 2490000.0*1e-5, 0.4
Pc = Pc*1e-5
Tr = T/Tc
ST_1 = 40.520*(1.0 - Tr)**1.287 # Methane
ST_2 = 52.095*(1.0 - Tr)**1.21548 # n-octane
ST_r_1 = log(1.0 + 0.013537770442486932*ST_1) # Constant from 1/(Tc_1**(1.0/3.0)*Pc_1**(2.0/3.0))
# ST_r_1 = log(1.0 + ST_1/(Tc_1**(1.0/3.0)*Pc_1**(2.0/3.0)))
ST_r_2 = log(1.0 + 0.014154874587259097*ST_2) # Constant from /(Tc_2**(1.0/3.0)*Pc_2**(2.0/3.0))
sigma_r = ST_r_1 + (omega-omega_1)*(ST_r_2-ST_r_1)*2.5773195876288657
# sigma_r = ST_r_1 + (omega-omega_1)/(omega_2 - omega_1)*(ST_r_2-ST_r_1)
sigma = Tc**(1.0/3.0)*Pc**(2.0/3.0)*(exp(sigma_r) - 1.0)
sigma = sigma*1e-3 # N/m, please
return sigma
[docs]def Hakim_Steinberg_Stiel(T, Tc, Pc, omega, StielPolar=0.0):
r'''Calculates air-liquid surface tension using the reference fluids methods
of [1]_.
.. math::
\sigma = 4.60104\times 10^{-7} P_c^{2/3}T_c^{1/3}Q_p \left(\frac{1-T_r}{0.4}\right)^m
.. math::
Q_p = 0.1574+0.359\omega-1.769\chi-13.69\chi^2-0.51\omega^2+1.298\omega\chi
.. math::
m = 1.21+0.5385\omega-14.61\chi-32.07\chi^2-1.65\omega^2+22.03\omega\chi
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 for fluid, [-]
StielPolar : float, optional
Stiel Polar Factor, [-]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
Original equation for m and Q are used. Internal units are atm and mN/m.
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
1-butanol, as compared to value in CRC Handbook of 0.02493.
>>> Hakim_Steinberg_Stiel(298.15, 563.0, 4414000.0, 0.59, StielPolar=-0.07872)
0.02190790257519
References
----------
.. [1] Hakim, D. I., David Steinberg, and L. I. Stiel. "Generalized
Relationship for the Surface Tension of Polar Fluids." Industrial &
Engineering Chemistry Fundamentals 10, no. 1 (February 1, 1971): 174-75.
doi:10.1021/i160037a032.
'''
if T >= Tc:
return 0.0
omega2 = omega*omega
StielPolar2 = StielPolar*StielPolar
Q = (0.1574 + 0.359*omega - 1.769*StielPolar - 13.69*StielPolar2
- 0.510*omega2 + 1.298*StielPolar*omega)
m = (1.210 + 0.5385*omega - 14.61*StielPolar - 32.07*StielPolar2
- 1.656*omega2 + 22.03*StielPolar*omega)
Tr = T/Tc
Pc = Pc*(1.0/101325.0)
sigma = Pc**(2.0/3.)*Tc**(1.0/3.0)*Q*(2.5*(1.0 - Tr))**m
sigma = sigma*1e-3 # convert to N/m
return sigma
[docs]def Miqueu(T, Tc, Vc, omega):
r'''Calculates air-liquid surface tension using the methods of [1]_.
.. math::
\sigma = k T_c \left( \frac{N_a}{V_c}\right)^{2/3}
(4.35 + 4.14 \omega)t^{1.26}(1+0.19t^{0.5} - 0.487t)
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Vc : float
Critical volume of fluid [m^3/mol]
omega : float
Acentric factor for fluid, [-]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
Uses Avogadro's constant and the Boltsman constant.
Internal units of volume are mL/mol and mN/m. However, either a typo
is in the article or author's work, or my value of k is off by 10; this is
corrected nonetheless.
Created with 31 normal fluids, none polar or hydrogen bonded. Has an
AARD of 3.5%.
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
Bromotrifluoromethane, 2.45 mN/m
>>> Miqueu(300., 340.1, 0.000199, 0.1687)
0.003474100774091376
References
----------
.. [1] Miqueu, C, D Broseta, J Satherley, B Mendiboure, J Lachaise, and
A Graciaa. "An Extended Scaled Equation for the Temperature Dependence
of the Surface Tension of Pure Compounds Inferred from an Analysis of
Experimental Data." Fluid Phase Equilibria 172, no. 2 (July 5, 2000):
169-82. doi:10.1016/S0378-3812(00)00384-8.
'''
if T >= Tc:
return 0.0
Vc = Vc*1E6
t = 1. - T/Tc
sigma = k*Tc*(N_A/Vc)**(2.0/3.0)*(4.35 + 4.14*omega)*t**1.26*(1.0 + 0.19*sqrt(t) - 0.25*t)*10000.0
return sigma
[docs]def Aleem(T, MW, Tb, rhol, Hvap_Tb, Cpl):
r'''Calculates vapor-liquid surface tension using the correlation derived by
[1]_ based on critical property CSP methods.
.. math::
\sigma = \phi \frac{MW^{1/3}} {6N_A^{1/3}}\rho_l^{2/3}\left[H_{vap}
+ C_{p,l}(T_b-T)\right]
.. math::
\phi = 1 - 0.0047MW + 6.8\times 10^{-6} MW^2
Parameters
----------
T : float
Temperature of fluid [K]
MW : float
Molecular weight [g/mol]
Tb : float
Boiling temperature of the fluid [K]
rhol : float
Liquid density at T and P [kg/m^3]
Hvap_Tb : float
Mass enthalpy of vaporization at the normal boiling point [kg/m^3]
Cpl : float
Liquid heat capacity of the chemical at T [J/kg/K]
Returns
-------
sigma : float
Liquid-vapor surface tension [N/m]
Notes
-----
Internal units of molecuar weight are kg/mol. This model is dimensionally
consistent.
This model does not use the critical temperature. After it predicts a
surface tension of 0 at a sufficiently high temperature, it returns
negative results. The temperature at which this occurs (the "predicted"
critical temperature) can be calculated as follows:
.. math::
\sigma = 0 \to T_{c,predicted} \text{ at } T_b + \frac{H_{vap}}{Cp_l}
To handle this case, if Tc is larger than T, 0 is returned as the model would return complex
numbers.
Because of its dependence on density, it has the potential to model the
effect of pressure on surface tension.
Claims AAD of 4.3%. Developed for normal alkanes. Total of 472 data points.
Behaves worse for higher alkanes. Behaves very poorly overall.
Examples
--------
Methane at 90 K
>>> Aleem(T=90, MW=16.04246, Tb=111.6, rhol=458.7, Hvap_Tb=510870.,
... Cpl=2465.)
0.01669970230131523
References
----------
.. [1] Aleem, W., N. Mellon, S. Sufian, M. I. A. Mutalib, and D. Subbarao.
"A Model for the Estimation of Surface Tension of Pure Hydrocarbon
Liquids." Petroleum Science and Technology 33, no. 23-24 (December 17,
2015): 1908-15. doi:10.1080/10916466.2015.1110593.
'''
MW = MW*1e-3 # Use kg/mol for consistency with the other units
sphericity = 1. - MW*(0.0047 - 6.8E-6*MW)
res = sphericity*MW**(1.0/3.0)/(6.*N_A**(1.0/3.0))*rhol**(2.0/3.)*(Hvap_Tb + Cpl*(Tb-T))
if res < 0.0:
return 0.0
return res
[docs]def Mersmann_Kind_sigma(T, Tm, Tb, Tc, Pc, n_associated=1):
r'''Estimates the surface tension of organic liquid substances
according to the method of [1]_.
.. math::
\sigma^* = \frac{\sigma n_{ass}^{1/3}} {(kT_c)^{1/3} T_{rm}P_c^{2/3}}
.. math::
\sigma^* = \left(\frac{T_b - T_m}{T_m}\right)^{1/3}
\left[6.25(1-T_r) + 31.3(1-T_r)^{4/3}\right]
Parameters
----------
T : float
Temperature of the fluid [K]
Tm : float
Melting temperature [K]
Tb : float
Boiling temperature of the fluid [K]
Tc : float
Critical temperature of the fluid [K]
Pc : float
Critical pressure of the fluid [Pa]
n_associated : float
Number of associated molecules in a cluster (2 for alcohols, 1
otherwise), [-]
Returns
-------
sigma : float
Liquid-vapor surface tension [N/m]
Notes
-----
In the equation, all quantities must be in SI units. `k` is the boltzman
constant.
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
MTBE at STP (the actual value is 0.0181):
>>> Mersmann_Kind_sigma(298.15, 164.15, 328.25, 497.1, 3430000.0)
0.016744311449290426
References
----------
.. [1] Mersmann, Alfons, and Matthias Kind. "Prediction of Mechanical and
Thermal Properties of Pure Liquids, of Critical Data, and of Vapor
Pressure." Industrial & Engineering Chemistry Research, January 31,
2017. https://doi.org/10.1021/acs.iecr.6b04323.
'''
if T >= Tc:
return 0.0
Tr = T/Tc
sigma_star = ((Tb - Tm)/Tm)**(1.0/3.)*(6.25*(1. - Tr) + 31.3*(1. - Tr)**(4.0/3.))
sigma = sigma_star*(k*Tc)**(1.0/3.0)*(Tm/Tc)*Pc**(2.0/3.0)*n_associated**(-1.0/3.0)
return sigma
[docs]def sigma_Gharagheizi_1(T, Tc, MW, omega):
r'''Calculates air-liquid surface tension using the
equation 4 derived in [1]_ by gene expression programming.
.. math::
\sigma = 8.948226\times 10^{-4}\left[\frac{A^2}{MW}\sqrt{\frac{A\omega}{MW}}
\right]^{0.5}
.. math::
A = (T_{c} - T - \omega)
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
MW : float
Molecular weight [g/mol]
omega : float
Acentric factor for fluid, [-]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
This equation may fail before the critical point. In this case it returns 0.0
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
Methane at 93 K, point from [1]_'s supporting material:
>>> sigma_Gharagheizi_1(T=95, Tc=190.564, MW=16.04, omega=0.012)
0.0110389739
References
----------
.. [1] Gharagheizi, Farhad, Ali Eslamimanesh, Mehdi Sattari, Amir H. Mohammadi,
and Dominique Richon. "Development of Corresponding States Model for
Estimation of the Surface Tension of Chemical Compounds." AIChE Journal 59,
no. 2 (2013): 613-21. https://doi.org/10.1002/aic.13824.
'''
# Equation 4
A = (Tc - T - omega)
if A < 0.0:
return 0.0
sigma = 8.948226e-4*sqrt(A*A/MW*sqrt(A*omega/MW))
return sigma
[docs]def sigma_Gharagheizi_2(T, Tb, Tc, Pc, Vc):
r'''Calculates air-liquid surface tension using the
equation 6 derived in [1]_ by gene expression programming.
.. math::
\frac{\sigma}{\text{N}/\text{m}} = 10^{-4}\left(\frac{P_c}{\text{bar}}\right)^{2/3}
\left(\frac{T_c}{\text{K}}\right)^{1/3}(1-T_r)^{11/9}
\left[7.728729T_{br} + 2.476318\left(T_{br}^3 + \frac{V_{c}}{\text{m}^3 /\text{kmol}}\right)
\right]
Parameters
----------
T : float
Temperature of fluid [K]
Tb : float
Boiling temperature of the fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
MW : float
Molecular weight [g/mol]
Vc : float
Critical volume of fluid [m^3/mol]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
This expression gives does converge to 0 at the critical point.
If Tc is larger than T, 0 is returned as the model would return complex
numbers.
Examples
--------
Methane at 93 K, point from [1]_'s supporting material:
>>> sigma_Gharagheizi_2(T=95, Tb=111.66, Tc=190.564, Pc=45.99e5, Vc=0.0986e-3)
0.01674894057
References
----------
.. [1] Gharagheizi, Farhad, Ali Eslamimanesh, Mehdi Sattari, Amir H. Mohammadi,
and Dominique Richon. "Development of Corresponding States Model for
Estimation of the Surface Tension of Chemical Compounds." AIChE Journal 59,
no. 2 (2013): 613-21. https://doi.org/10.1002/aic.13824.
'''
# Equation 6
if T >= Tc:
return 0.0
Pc *= 1e-5 # Pc, Pa to bar
Tr = T/Tc
Tbr = Tb/Tc
Tbr2 = Tbr*Tbr
Vc *= 1e3 # m^3/mol to m^3/kmol
sigma = 1e-4*Pc**(2.0/3.0)*Tc**(1.0/3.0)*(1.0 - Tr)**(11/9.)
sigma *= (7.728729*Tbr + 2.476318*(Tbr*Tbr2 + Vc))
return sigma
[docs]def API10A32(T, Tc, K_W):
r'''Calculates the interfacial tension between
a liquid petroleum fraction and air, using the oil's pseudocritical
temperature and Watson K Characterization factor.
.. math::
\sigma = \frac{673.7\left[\frac{\left(T_c - T\right)}{T_c}\right]^{1.232}}{K_W}
Parameters
----------
T : float
Liquid temperature, [K]
Tc : float
Pseudocritical temperature (or critical temperature if using
the equation with a pure component), [K]
K_W : float
Watson characterization factor
Returns
-------
sigma : float
Air-liquid surface tension, [N/m]
Notes
-----
[1]_ cautions that this should not be applied to coal liquids,
and that it will give higher errors at pressures above 500 psi.
[1]_ claims this has an average error of 10.7%.
This function converges to zero at `Tc`. If Tc is larger than T,
0 is returned as the model would return complex numbers.
Examples
--------
Sample problem in Comments on Procedure 10A3.2.1 of [1]_;
>>> from fluids.core import F2K, R2K
>>> API10A32(T=F2K(60), Tc=R2K(1334), K_W=12.4)
29.577333312096968
References
----------
.. [1] API Technical Data Book: General Properties & Characterization.
American Petroleum Institute, 7E, 2005.
'''
if T >= Tc:
return 0.0
return 673.7*((Tc-T)/Tc)**1.232/K_W
### Surface Tension Mixtures
[docs]def Winterfeld_Scriven_Davis(xs, sigmas, rhoms):
r'''Calculates surface tension of a liquid mixture according to
mixing rules in [1]_ and also in [2]_.
.. math::
\sigma_M = \sum_i \sum_j \frac{1}{V_L^{L2}}\left(x_i V_i \right)
\left( x_jV_j\right)\sqrt{\sigma_i\cdot \sigma_j}
Parameters
----------
xs : array-like
Mole fractions of all components, [-]
sigmas : array-like
Surface tensions of all components, [N/m]
rhoms : array-like
Molar densities of all components, [mol/m^3]
Returns
-------
sigma : float
Air-liquid surface tension of mixture, [N/m]
Notes
-----
DIPPR Procedure 7C: Method for the Surface Tension of Nonaqueous Liquid
Mixtures
Becomes less accurate as liquid-liquid critical solution temperature is
approached. DIPPR Evaluation: 3-4% AARD, from 107 nonaqueous binary
systems, 1284 points. Internally, densities are converted to kmol/m^3. The
Amgat function is used to obtain liquid mixture density in this equation.
Raises a ZeroDivisionError if either molar volume are zero, and a
ValueError if a surface tensions of a pure component is negative.
Examples
--------
>>> Winterfeld_Scriven_Davis([0.1606, 0.8394], [0.01547, 0.02877],
... [8610., 15530.])
0.02496738845043982
References
----------
.. [1] Winterfeld, P. H., L. E. Scriven, and H. T. Davis. "An Approximate
Theory of Interfacial Tensions of Multicomponent Systems: Applications
to Binary Liquid-Vapor Tensions." AIChE Journal 24, no. 6
(November 1, 1978): 1010-14. doi:10.1002/aic.690240610.
.. [2] Danner, Ronald P, and Design Institute for Physical Property Data.
Manual for Predicting Chemical Process Design Data. New York, N.Y, 1982.
'''
N = len(xs)
Vms = [0.0]*N
rho = 0.0
for i in range(N):
Vms[i] = 1e3/rhoms[i]
rho += xs[i]*Vms[i]
# rho = 1./rho
rho = root_two/rho # factor out rt2
# For speed, transform the Vms array to contain
# xs[i]*Vms[i]*sigmas_05[i]*rho
tot = 0.0
for i in range(N):
val = sqrt(sigmas[i])*xs[i]*rho*Vms[i]
Vms[i] = val
tot += val*val
tot *= 0.5
for i in range(N):
# Symmetric - can be slightly optimized
temp = 0.0
for j in range(i):
temp += Vms[j]
tot += Vms[i]*temp
return tot
[docs]def Diguilio_Teja(T, xs, sigmas_Tb, Tbs, Tcs):
r'''Calculates surface tension of a liquid mixture according to
mixing rules in [1]_.
.. math::
\sigma = 1.002855(T^*)^{1.118091} \frac{T}{T_b} \sigma_r
.. math::
T^* = \frac{(T_c/T)-1}{(T_c/T_b)-1}
.. math::
\sigma_r = \sum x_i \sigma_i
.. math::
T_b = \sum x_i T_{b,i}
.. math::
T_c = \sum x_i T_{c,i}
Parameters
----------
T : float
Temperature of fluid [K]
xs : array-like
Mole fractions of all components
sigmas_Tb : array-like
Surface tensions of all components at the boiling point, [N/m]
Tbs : array-like
Boiling temperatures of all components, [K]
Tcs : array-like
Critical temperatures of all components, [K]
Returns
-------
sigma : float
Air-liquid surface tension of mixture, [N/m]
Notes
-----
Simple model, however it has 0 citations. Gives similar results to the
`Winterfeld_Scriven_Davis` model.
Raises a ValueError if temperature is greater than the mixture's critical
temperature or if the given temperature is negative, or if the mixture's
boiling temperature is higher than its critical temperature.
[1]_ claims a 4.63 percent average absolute error on 21 binary and 4
ternary non-aqueous systems. [1]_ also considered Van der Waals mixing
rules for `Tc`, but found it provided a higher error of 5.58%
Examples
--------
>>> Diguilio_Teja(T=298.15, xs=[0.1606, 0.8394],
... sigmas_Tb=[0.01424, 0.02530], Tbs=[309.21, 312.95], Tcs=[469.7, 508.0])
0.025716823875045505
References
----------
.. [1] Diguilio, Ralph, and Amyn S. Teja. "Correlation and Prediction of
the Surface Tensions of Mixtures." The Chemical Engineering Journal 38,
no. 3 (July 1988): 205-8. doi:10.1016/0300-9467(88)80079-0.
'''
Tc, Tb, sigmar = 0.0, 0.0, 0.0
for i in range(len(xs)):
Tc += Tcs[i]*xs[i]
Tb += Tbs[i]*xs[i]
sigmar += sigmas_Tb[i]*xs[i]
if T > Tc:
raise ValueError('T > Tc according to Kays rule - model is not valid in this range.')
Tst = (Tc/T - 1.)/(Tc/Tb - 1.0)
return 1.002855*Tst**1.118091*(T/Tb)*sigmar
[docs]def Weinaug_Katz(parachors, Vml, Vmg, xs, ys):
r'''Calculates surface tension of a liquid mixture according to
mixing rules in [1]_ and also in [2]_. This is based on the
Parachor concept. This is called the Macleod-Sugden model in some places.
.. math::
\sigma_M = \left[\sum_i P_i\left( \frac{x_i}{V_{m,l}}
- \frac{y_i}{V_{m,g}}\right) \right]^4
Parameters
----------
parachors : list[float]
Parachors of each component, [N^0.25*m^2.75/mol]
Vml : float
Liquid mixture molar volume, [m^3/mol]
Vmg : float
Gas mixture molar volume; this can be set to zero at
low pressures, [m^3/mol]
xs : list[float]
Mole fractions of all components in liquid phase, [-]
xs : list[float]
Mole fractions of all components in gas phase, [-]
Returns
-------
sigma : float
Air-liquid surface tension of mixture, [N/m]
Notes
-----
This expression is efficient and does not require pure component
surface tensions. Its accuracy is dubious.
Examples
--------
>>> Weinaug_Katz([5.1e-5, 7.2e-5], Vml=0.000125, Vmg=0.02011, xs=[.4, .6], ys=[.6, .4])
0.06547479150776776
Neglect the vapor phase density by setting `Vmg` to a high value:
>>> Weinaug_Katz([5.1e-5, 7.2e-5], Vml=0.000125, Vmg=1e100, xs=[.4, .6], ys=[.6, .4])
0.06701752894095361
References
----------
.. [1] Weinaug, Charles F., and Donald L. Katz. "Surface Tensions of
Methane-Propane Mixtures." Industrial & Engineering Chemistry 35,
no. 2 (February 1, 1943): 239-246. https://doi.org/10.1021/ie50398a028.
.. [2] Pedersen, Karen Schou, Aage Fredenslund, and Per Thomassen.
Properties of Oils and Natural Gases. Vol. 5. Gulf Pub Co, 1989.
'''
tot = 0.0
rhoml = 1.0/Vml
rhomg = 1.0/Vmg
for i in range(len(parachors)):
tot += parachors[i]*(xs[i]*rhoml - ys[i]*rhomg)
tot *= tot
tot *= tot # fourth power it
return tot
### Water-hydrocarbon interfacial tensions
[docs]def Meybodi_Daryasafar_Karimi(rho_water, rho_oil, T, Tc):
r'''Calculates the interfacial tension between water and a hydrocabon
liquid according to the correlation of [1]_.
.. math::
\gamma_{hw} = \left(\frac{A_1 + A_2 \Delta \rho + A_3\Delta\rho^2
+ A_4\Delta\rho^3} {A_5 + A_6\frac{T^{A_7}}{T_{c,h}} + A_8T^{A_9}}
\right)^{A_{10}}
Parameters
----------
rho_water : float
The density of the aqueous phase, [kg/m^3]
rho_oil : float
The density of the hydrocarbon phase, [kg/m^3]
T : float
Temperature of the fluid, [K]
Tc : float
Critical temperature of the hydrocarbon mixture, [K]
Returns
-------
sigma : float
Hydrocarbon-water surface tension [N/m]
Notes
-----
Internal units of the equation are g/mL and mN/m.
Examples
--------
>>> Meybodi_Daryasafar_Karimi(980, 760, 580, 914)
0.02893598143089256
References
----------
.. [1] Kalantari Meybodi, Mahdi, Amin Daryasafar, and Masoud Karimi.
"Determination of Hydrocarbon-Water Interfacial Tension Using a New
Empirical Correlation." Fluid Phase Equilibria 415 (May 15, 2016):
42-50. doi:10.1016/j.fluid.2016.01.037.
'''
A1 = -1.3687340042E-1
A2 = -3.0391828884E-1
A3 = 5.6225871072E-1
A4 = -3.3074367079E-1
A5 = -3.0050179309E0
A6 = 5.8914210205E-5
A7 = -4.1388901263E0
A8 = 3.0084299030E0
A9 = -3.8203072876E-3
# A10 = 3.5000000000E0
drho = abs(rho_water - rho_oil)*1e-3 # Correlation in units of g/mL
sigma = ((A1 + drho*(A2 + drho*(A3 + A4*drho)))
/(A5 + A6*T**A7/Tc + A8*T**A9))
return sigma*sigma*sigma*sqrt(sigma)*1e-3 # mN/m to N/m