r"""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 various volume/density estimation routines, dataframes
of fit coefficients, and mixing rules.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/chemicals/>`_.
.. contents:: :local:
Pure Low Pressure Liquid Correlations
-------------------------------------
.. autofunction:: chemicals.volume.Rackett
.. autofunction:: chemicals.volume.COSTALD
.. autofunction:: chemicals.volume.Yen_Woods_saturation
.. autofunction:: chemicals.volume.Yamada_Gunn
.. autofunction:: chemicals.volume.Townsend_Hales
.. autofunction:: chemicals.volume.Bhirud_normal
.. autofunction:: chemicals.volume.Campbell_Thodos
.. autofunction:: chemicals.volume.SNM0
Pure High Pressure Liquid Correlations
--------------------------------------
.. autofunction:: chemicals.volume.COSTALD_compressed
Liquid Mixing Rules
-------------------
.. autofunction:: chemicals.volume.Amgat
.. autofunction:: chemicals.volume.Rackett_mixture
.. autofunction:: chemicals.volume.COSTALD_mixture
Gas Correlations
----------------
Gas volumes are predicted with one of:
1) An equation of state
2) A virial coefficient model
3) The ideal gas law
Equations of state do much more than predict volume however. An implementation
of many of them can be found in `thermo <https://github.com/CalebBell/thermo>`_.
Virial functions are implemented in :obj:`chemicals.virial`.
.. autofunction:: chemicals.volume.ideal_gas
Pure Solid Correlations
-----------------------
Solid density does not depend on pressure significantly, and unless operating
in the geochemical or astronomical domain is normally neglected.
.. autofunction:: chemicals.volume.Goodman
Pure Component Liquid Fit Correlations
--------------------------------------
.. autofunction:: chemicals.volume.Rackett_fit
.. autofunction:: chemicals.volume.volume_VDI_PPDS
.. autofunction:: chemicals.volume.TDE_VDNS_rho
.. autofunction:: chemicals.volume.PPDS17
Pure Component High-Pressure Liquid Fit Correlations
----------------------------------------------------
.. autofunction:: chemicals.volume.Tait
.. autofunction:: chemicals.volume.Tait_molar
Pure Component Solid Fit Correlations
-------------------------------------
.. autofunction:: chemicals.volume.CRC_inorganic
Fit Coefficients
----------------
All of these coefficients are lazy-loaded, so they must be accessed as an
attribute of this module.
.. data:: rho_data_COSTALD
Coefficients for the :obj:`COSTALD` method from [3]_; 192 fluids have
coefficients published.
.. data:: rho_data_SNM0
Coefficients for the :obj:`SNM0` method for 73 fluids from [2]_.
.. data:: rho_data_Perry_8E_105_l
Coefficients for :obj:`chemicals.dippr.EQ105` from [1]_ for 344 fluids. Note
this is in terms of molar density; to obtain molar volume, invert the result!
.. data:: rho_data_VDI_PPDS_2
Coefficients in [5]_ developed by the PPDS using :obj:`chemicals.dippr.EQ116`
but in terms of mass density [kg/m^3]; Valid up to the critical temperature,
and extrapolates to very low temperatures well.
.. data:: rho_data_CRC_inorg_l
Single-temperature coefficient linear model in terms of mass density
for the density of inorganic liquids. Data is available for 177 fluids
normally valid over a
narrow range above the melting point, from [4]_; described in
:obj:`CRC_inorganic`.
.. data:: rho_data_CRC_inorg_l_const
Constant inorganic liquid molar volumes published in [4]_.
.. data:: rho_data_CRC_inorg_s_const
Constant solid densities molar volumes published in [4]_.
.. data:: rho_data_CRC_virial
Coefficients for a tempereture polynomial (T in Kelvin) for the second `B`
virial coefficient published in [4]_. The form of the equation is
:math:`B = (a_1 + t(a_2 + t(a_3 + t(a_4 + a_5 t)))) \times 10^{-6}` with
:math:`t = \frac{298.15}{T} - 1` and then `B` will be in units of m^3/mol.
.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
8E. McGraw-Hill Professional, 2007.
.. [2] Mchaweh, A., A. Alsaygh, Kh. Nasrifar, and M. Moshfeghian.
"A Simplified Method for Calculating Saturated Liquid Densities."
Fluid Phase Equilibria 224, no. 2 (October 1, 2004): 157-67.
doi:10.1016/j.fluid.2004.06.054
.. [3] Hankinson, Risdon W., and George H. Thomson. "A New Correlation for
Saturated Densities of Liquids and Their Mixtures." AIChE Journal
25, no. 4 (1979): 653-663. doi:10.1002/aic.690250412
.. [4] Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of
Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014.
.. [5] 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.volume.rho_data_COSTALD
In [3]: chemicals.volume.rho_data_SNM0
In [4]: chemicals.volume.rho_data_Perry_8E_105_l
In [5]: chemicals.volume.rho_data_VDI_PPDS_2
In [6]: chemicals.volume.rho_data_CRC_inorg_l
In [7]: chemicals.volume.rho_data_CRC_inorg_l_const
In [8]: chemicals.volume.rho_data_CRC_inorg_s_const
In [9]: chemicals.volume.rho_data_CRC_virial
"""
__all__ = ['volume_VDI_PPDS', 'Yen_Woods_saturation', 'Rackett', 'Yamada_Gunn', 'Townsend_Hales',
'Bhirud_normal', 'COSTALD', 'Campbell_Thodos', 'SNM0', 'CRC_inorganic',
'COSTALD_compressed', 'Amgat', 'Rackett_mixture', 'COSTALD_mixture',
'ideal_gas', 'Goodman', 'Rackett_fit', 'TDE_VDNS_rho', 'PPDS17',
'Tait', 'Tait_molar',
]
from fluids.constants import R, root_two
from fluids.numerics import exp, implementation_optimize_tck, log, np, splev, sqrt
from chemicals.data_reader import data_source, register_df_source
from chemicals.utils import PY37, can_load_data, mark_numba_incompatible, mixing_simple, os_path_join, source_path
folder = os_path_join(source_path, 'Density')
register_df_source(folder, 'COSTALD Parameters.tsv')
register_df_source(folder, 'Mchaweh SN0 deltas.tsv')
register_df_source(folder, 'Perry Parameters 105.tsv')
register_df_source(folder, 'CRC Liquid Inorganic Constant Densities.tsv')
register_df_source(folder, 'CRC Solid Inorganic Constant Densities.tsv')
register_df_source(folder, 'VDI PPDS Density of Saturated Liquids.tsv', csv_kwargs={
'dtype':{'rhoc': float}})
register_df_source(folder, 'CRC Inorganics densties of molten compounds and salts.tsv', csv_kwargs={
'dtype':{'rho': float}})
register_df_source(folder, 'CRC Virial polynomials.tsv', csv_kwargs={
'dtype':{'a1': float, 'a2': float, 'a3': float, 'a4': float, 'a5': float}})
_rho_data_loaded = False
@mark_numba_incompatible
def _load_rho_data():
global _rho_data_loaded, rho_data_COSTALD, rho_data_SNM0
global rho_data_Perry_8E_105_l, rho_values_Perry_8E_105_l
global rho_data_VDI_PPDS_2, rho_values_VDI_PPDS_2
global rho_data_CRC_inorg_l, rho_values_CRC_inorg_l
global rho_data_CRC_inorg_l_const, rho_data_CRC_inorg_s_const
global rho_data_CRC_virial, rho_values_CRC_virial
rho_data_COSTALD = data_source('COSTALD Parameters.tsv')
rho_data_SNM0 = data_source('Mchaweh SN0 deltas.tsv')
rho_data_Perry_8E_105_l = data_source('Perry Parameters 105.tsv')
rho_values_Perry_8E_105_l = np.array(rho_data_Perry_8E_105_l.values[:, 1:], dtype=float)
rho_data_VDI_PPDS_2 = data_source('VDI PPDS Density of Saturated Liquids.tsv')
rho_values_VDI_PPDS_2 = np.array(rho_data_VDI_PPDS_2.values[:, 1:], dtype=float)
rho_data_CRC_inorg_l = data_source('CRC Inorganics densties of molten compounds and salts.tsv')
rho_values_CRC_inorg_l = np.array(rho_data_CRC_inorg_l.values[:, 1:], dtype=float)
rho_data_CRC_inorg_l_const = data_source('CRC Liquid Inorganic Constant Densities.tsv')
rho_data_CRC_inorg_s_const = data_source('CRC Solid Inorganic Constant Densities.tsv')
rho_data_CRC_virial = data_source('CRC Virial polynomials.tsv')
rho_values_CRC_virial = np.array(rho_data_CRC_virial.values[:, 1:], dtype=float)
if PY37:
def __getattr__(name):
if name in ('rho_data_COSTALD', 'rho_data_SNM0', 'rho_data_Perry_8E_105_l',
'rho_values_Perry_8E_105_l', 'rho_data_VDI_PPDS_2',
'rho_values_VDI_PPDS_2', 'rho_data_CRC_inorg_l',
'rho_values_CRC_inorg_l', 'rho_data_CRC_inorg_l_const',
'rho_data_CRC_inorg_s_const', 'rho_data_CRC_virial',
'rho_values_CRC_virial'):
_load_rho_data()
return globals()[name]
raise AttributeError(f"module {__name__} has no attribute {name}")
else:
if can_load_data:
_load_rho_data()
[docs]def volume_VDI_PPDS(T, Tc, rhoc, a, b, c, d, MW=None):
r'''Calculates saturation liquid volume, using the critical properties
and fitted coefficients from [1]_. This is also known as the PPDS equation
10 or PPDS10.
.. math::
\rho_{mass} = \rho_{c} + a\tau^{0.35} + b \tau^{2/3} + c\tau + d\tau^{4/3}
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
rhoc : float
Critical density of fluid [kg/m^3]
a,b,c,d : float
Fitted coefficients [-]
MW : float, optional
Molecular weight of chemical [g/mol]
Returns
-------
Vs : float
Saturation liquid molar volume or density, [m^3/mol if MW given; kg/m^3 otherwise]
Examples
--------
Calculate density of nitrogen in kg/m3 at 300 K:
>>> volume_VDI_PPDS(300, 126.19, 313, 470.922, 493.251, -560.469, 389.611)
313.0
Calculate molar volume of nitrogen in m3/mol at 300 K:
>>> volume_VDI_PPDS(300, 126.19, 313, 470.922, 493.251, -560.469, 389.611, 28.01)
8.9488817891e-05
References
----------
.. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
Berlin; New York:: Springer, 2010.
'''
tau = 1. - T/Tc if T < Tc else 0.
tau_cbrt = tau**(1.0/3.)
rho = rhoc + a*tau**0.35 + b*tau_cbrt*tau_cbrt + tau*(c + d*tau_cbrt)
return rho if MW is None else 0.001 * MW / rho
[docs]def TDE_VDNS_rho(T, Tc, rhoc, a1, a2, a3, a4, MW=None):
r'''Calculates saturation liquid volume, using the critical properties
and fitted coefficients in the TDE VDNW form from [1]_.
.. math::
\rho_{mass} = \rho_{c} + a\tau^{0.35} + b \tau + c\tau^2 + d\tau^3
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
rhoc : float
Critical density of fluid [kg/m^3]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
a3 : float
Regression parameter, [-]
a4 : float
Regression parameter, [-]
MW : float, optional
Molecular weight of chemical [g/mol]
Returns
-------
Vs : float
Saturation liquid molar volume or density,
[m^3/mol if MW given; kg/m^3 otherwise]
Examples
--------
>>> TDE_VDNS_rho(T=400.0, Tc=772.999, rhoc=320.037, a1=795.092, a2=-169.132, a3=448.929, a4=-102.931)
947.4906064903
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User`s Guide."
https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-DensityLG/VDNSExpansion.htm.
'''
tau = 1.0 - T/Tc
rho = rhoc + a1*tau**0.35 + tau*(a2 + tau*(a3 + a4*tau))
return rho if MW is None else 0.001 * MW / rho
[docs]def PPDS17(T, Tc, a0, a1, a2, MW=None):
r'''Calculates saturation liquid volume, using the critical temperature
and fitted coefficients in the PPDS17 form in [1]_.
.. math::
\rho_{mass} = \frac{1}{a_0(a_1 + a_2\tau)^{\left(1 + \tau^{2/7} \right)}}
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
a0 : float
Regression parameter, [-]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
MW : float, optional
Molecular weight of chemical [g/mol]
Returns
-------
Vs : float
Saturation liquid molar volume or density,
[m^3/mol if MW given; kg/m^3 otherwise]
Examples
--------
Coefficients for the liquid density of benzene from [1]_ at 300 K:
>>> PPDS17(300, 562.05, a0=0.0115508, a1=0.281004, a2=-0.00635447)
871.520087707
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User`s Guide."
https://trc.nist.gov/TDE/TDE_Help/Eqns-Pure-DensityLG/PPDS17.htm.
'''
tau = 1.0 - T/Tc
rho = 1.0/(a0*(a1 + a2*tau)**(1.0 + tau**(2.0/7.)))
return rho if MW is None else 0.001 * MW / rho
### Critical-properties based
[docs]def Yen_Woods_saturation(T, Tc, Vc, Zc):
r'''Calculates saturation liquid volume, using the Yen and Woods [1]_ CSP
method and a chemical's critical properties.
The molar volume of a liquid is given by:
.. math::
Vc/Vs = 1 + A(1-T_r)^{1/3} + B(1-T_r)^{2/3} + D(1-T_r)^{4/3}
.. math::
D = 0.93-B
.. math::
A = 17.4425 - 214.578Z_c + 989.625Z_c^2 - 1522.06Z_c^3
.. math::
B = -3.28257 + 13.6377Z_c + 107.4844Z_c^2-384.211Z_c^3
\text{ if } Zc \le 0.26
.. math::
B = 60.2091 - 402.063Z_c + 501.0 Z_c^2 + 641.0 Z_c^3
\text{ if } Zc \ge 0.26
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Vc : float
Critical volume of fluid [m^3/mol]
Zc : float
Critical compressibility of fluid, [-]
Returns
-------
Vs : float
Saturation liquid volume, [m^3/mol]
Notes
-----
Original equation was in terms of density, but it is converted here.
No example has been found, nor are there points in the article. However,
it is believed correct. For compressed liquids with the Yen-Woods method,
see the `YenWoods_compressed` function.
Examples
--------
>>> Yen_Woods_saturation(300, 647.14, 55.45E-6, 0.245)
1.769533076529574e-05
References
----------
.. [1] Yen, Lewis C., and S. S. Woods. "A Generalized Equation for Computer
Calculation of Liquid Densities." AIChE Journal 12, no. 1 (1966):
95-99. doi:10.1002/aic.690120119
'''
Tr = T/Tc
A = Zc*(Zc*(989.625 - 1522.06*Zc) - 214.578) + 17.4425
if Zc <= 0.26:
B = Zc*(Zc*(107.4844 - 384.211*Zc) + 13.6377) - 3.28257
else:
B = Zc*(Zc*(641.0*Zc + 501.0) - 402.063) + 60.2091
D = 0.93 - B
tau_cbrt = (1.0 - Tr)**(1/3.)
Vm = Vc/(tau_cbrt*(A + tau_cbrt*(B + D*tau_cbrt*tau_cbrt)) + 1.0)
return Vm
[docs]def Rackett(T, Tc, Pc, Zc):
r'''Calculates saturation liquid volume, using Rackett CSP method and
critical properties.
The molar volume of a liquid is given by:
.. math::
V_s = \frac{RT_c}{P_c}{Z_c}^{[1+(1-{T/T_c})^{2/7} ]}
Units are all currently in m^3/mol - this can be changed to kg/m^3
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
Zc : float
Critical compressibility of fluid, [-]
Returns
-------
Vs : float
Saturation liquid volume, [m^3/mol]
Notes
-----
According to Reid et. al, underpredicts volume for compounds with Zc < 0.22
Examples
--------
Propane, example from the API Handbook
>>> from chemicals.utils import Vm_to_rho
>>> Vm_to_rho(Rackett(272.03889, 369.83, 4248000.0, 0.2763), 44.09562)
531.3221411755724
References
----------
.. [1] Rackett, Harold G. "Equation of State for Saturated Liquids."
Journal of Chemical & Engineering Data 15, no. 4 (1970): 514-517.
doi:10.1021/je60047a012
'''
return R*Tc/Pc*Zc**(1.0 + (1.0 - T/Tc)**(2.0/7.))
[docs]def Rackett_fit(T, Tc, rhoc, b, n, MW=None):
r'''Calculates saturation liquid volume, using the Rackett equation form
and a known or estimated critical temperature and density as well
as fit parameters `b` and `n`.
The density of a liquid is given by:
.. math::
\rho_{sat} = \rho_c b^{-\left(1 - \frac{T}{T_c}\right)^n}
The density is then converted to a specific volume by taking its inverse.
Note that the units of this equation in some sources are kg/m^3, g/mL in others,
and m^3/mol in others. If the units for the coefficients are in molar units,
do NOT provide `MW` or an incorrect value will be returned. If the units
are mass units and `MW` is not provided, the output will have the same
units as `rhoc`.
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
rhoc : float
Critical density of fluid, often a fit parameter only [kg/m^3]
b : float
Fit parameter, [-]
n : float
Fit parameter, [-]
MW : float, optional
Molecular weight, [g/mol]
Returns
-------
Vs : float
Saturation liquid volume, [m^3/mol if MW given; m^3/kg otherwise]
Notes
-----
Examples
--------
Input sample from NIST (naphthalene) (m^3/kg):
>>> Rackett_fit(T=400.0, Tc=748.402, rhoc=314.629, b=0.257033, n=0.280338)
0.00106174320755
Parameters in Yaws form (butane) (note the 1000 multiplier on `rhoc`, called `A`
in Yaws) (m^3/kg):
>>> Rackett_fit(T=298.15, Tc=425.18, rhoc=0.2283*1000, b=0.2724, n=0.2863)
0.00174520519958
Same Yaws point, with MW provided:
>>> Rackett_fit(T=298.15, Tc=425.18, rhoc=0.2283*1000, b=0.2724, n=0.2863, MW=58.123)
0.00010143656181
References
----------
.. [1] 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.
.. [2] Yaws, Carl L. "Liquid Density of the Elements: A Comprehensive
Tabulation for All the Important Elements from Ag to Zr." Chemical
Engineering 114, no. 12 (2007): 44-47.
'''
rho_calc = rhoc*b**(-(1.0 - T/Tc)**n)
if MW is not None:
return 1e-3*MW/rho_calc
return 1.0/rho_calc
[docs]def Yamada_Gunn(T, Tc, Pc, omega):
r'''Calculates saturation liquid volume, using Yamada and Gunn CSP method
and a chemical's critical properties and acentric factor.
The molar volume of a liquid is given by:
.. math::
V_s = \frac{RT_c}{P_c}{(0.29056-0.08775\omega)}^{[1+(1-{T/T_c})^{2/7}]}
Units are in m^3/mol.
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
-------
Vs : float
saturation liquid volume, [m^3/mol]
Notes
-----
This equation is an improvement on the Rackett equation.
This is often presented as the Rackett equation.
The acentric factor is used here, instead of the critical compressibility
A variant using a reference fluid also exists
Examples
--------
>>> Yamada_Gunn(300, 647.14, 22048320.0, 0.245)
2.188284384699659e-05
References
----------
.. [1] Gunn, R. D., and Tomoyoshi Yamada. "A Corresponding States
Correlation of Saturated Liquid Volumes." AIChE Journal 17, no. 6
(1971): 1341-45. doi:10.1002/aic.690170613
.. [2] Yamada, Tomoyoshi, and Robert D. Gunn. "Saturated Liquid Molar
Volumes. Rackett Equation." Journal of Chemical & Engineering Data 18,
no. 2 (1973): 234-36. doi:10.1021/je60057a006
'''
return R*Tc/Pc*(0.29056 - 0.08775*omega)**(1.0 + (1.0 - T/Tc)**(2.0/7.))
[docs]def Townsend_Hales(T, Tc, Vc, omega):
r'''Calculates saturation liquid density, using the Townsend and Hales
CSP method as modified from the original Riedel equation. Uses
chemical critical volume and temperature, as well as acentric factor
The density of a liquid is given by:
.. math::
Vs = V_c/\left(1+0.85(1-T_r)+(1.692+0.986\omega)(1-T_r)^{1/3}\right)
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
-------
Vs : float
Saturation liquid volume, [m^3/mol]
Notes
-----
The requirement for critical volume and acentric factor requires all data.
Examples
--------
>>> Townsend_Hales(300, 647.14, 55.95E-6, 0.3449)
1.8007361992619923e-05
References
----------
.. [1] Hales, J. L, and R Townsend. "Liquid Densities from 293 to 490 K of
Nine Aromatic Hydrocarbons." The Journal of Chemical Thermodynamics
4, no. 5 (1972): 763-72. doi:10.1016/0021-9614(72)90050-X
'''
Tr = T/Tc
return Vc/(1.0 + 0.85*(1.0-Tr) + (1.692 + 0.986*omega)*(1.0-Tr)**(1.0/3.))
Bhirud_normal_Trs = [0.98, 0.982, 0.984, 0.986, 0.988, 0.99, 0.992, 0.994,
0.996, 0.998, 0.999, 1]
Bhirud_normal_lnU0s = [-1.6198, -1.604, -1.59, -1.578, -1.564, -1.548, -1.533,
-1.515, -1.489, -1.454, -1.425, -1.243]
Bhirud_normal_lnU1 = [-0.4626, -0.459, -0.451, -0.441, -0.428, -0.412, -0.392,
-0.367, -0.337, -0.302, -0.283, -0.2629]
Bhirud_normal_lnU0_tck = implementation_optimize_tck([[0.98, 0.98, 0.98, 0.98, 0.984, 0.986, 0.988, 0.99, 0.992, 0.994, 0.996, 0.998, 1.0, 1.0, 1.0, 1.0],
[-1.6198000000000001, -1.6090032590995371, -1.5930934818009272, -1.5785097772986094, -1.5645133351302174, -1.547436882180519, -1.5337391361477044, -1.5156065732286643, -1.4938345709376377, -1.4430551430207872, -1.4529816189930713, -1.243, 0.0, 0.0, 0.0, 0.0],
3])
Bhirud_normal_lnU1_tck = implementation_optimize_tck([[0.98, 0.98, 0.98, 0.98, 0.984, 0.986, 0.988, 0.99, 0.992, 0.994, 0.996, 0.998, 1.0, 1.0, 1.0, 1.0],
[-0.4626000000000001, -0.4624223420145324, -0.4543553159709354, -0.4416003593769303, -0.4284319856698448, -0.41267169794369013, -0.39288122255539465, -0.3678034118347314, -0.3379051301056794, -0.30257606774255136, -0.2767190885302606, -0.2629, 0.0, 0.0, 0.0, 0.0],
3])
[docs]def Bhirud_normal(T, Tc, Pc, omega):
r'''Calculates saturation liquid density using the Bhirud [1]_ CSP method.
Uses Critical temperature and pressure and acentric factor.
The density of a liquid is given by:
.. math::
\ln \frac{P_c}{\rho RT} = \ln U^{(0)} + \omega\ln U^{(1)}
.. math::
\ln U^{(0)} = 1.396 44 - 24.076T_r+ 102.615T_r^2
-255.719T_r^3+355.805T_r^4-256.671T_r^5 + 75.1088T_r^6
.. math::
\ln U^{(1)} = 13.4412 - 135.7437 T_r + 533.380T_r^2-
1091.453T_r^3+1231.43T_r^4 - 728.227T_r^5 + 176.737T_r^6
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
Acentric factor for fluid, [-]
Returns
-------
Vm : float
Saturated liquid molar volume, [mol/m^3]
Notes
-----
Claimed inadequate by others.
An interpolation table for ln U values are used from Tr = 0.98 - 1.000.
Has terrible behavior at low reduced temperatures.
Examples
--------
Pentane
>>> Bhirud_normal(280.0, 469.7, 33.7E5, 0.252)
0.00011249657842514176
References
----------
.. [1] Bhirud, Vasant L. "Saturated Liquid Densities of Normal Fluids."
AIChE Journal 24, no. 6 (November 1, 1978): 1127-31.
doi:10.1002/aic.690240630
'''
Tr = T/Tc
if Tr <= 0.98:
lnU0 = Tr*(Tr*(Tr*(Tr*(Tr*(75.1088*Tr - 256.671) + 355.805) - 255.719)
+ 102.615) - 24.076) + 1.39644
lnU1 = Tr*(Tr*(Tr*(Tr*(Tr*(176.737*Tr - 728.227) + 1231.43) - 1091.453)
+ 533.38) - 135.7437) + 13.4412
elif Tr > 1.0:
raise ValueError('Critical phase, correlation does not apply')
else:
lnU0 = float(splev(Tr, Bhirud_normal_lnU0_tck))
lnU1 = float(splev(Tr, Bhirud_normal_lnU1_tck))
Unonpolar = exp(lnU0 + omega*lnU1)
Vm = Unonpolar*R*T/Pc
return Vm
[docs]def COSTALD(T, Tc, Vc, omega):
r'''Calculate saturation liquid density using the COSTALD CSP method.
A popular and accurate estimation method. If possible, fit parameters are
used; alternatively critical properties work well.
The density of a liquid is given by:
.. math::
V_s=V^*V^{(0)}[1-\omega_{SRK}V^{(\delta)}]
.. math::
V^{(0)}=1-1.52816(1-T_r)^{1/3}+1.43907(1-T_r)^{2/3}
- 0.81446(1-T_r)+0.190454(1-T_r)^{4/3}
.. math::
V^{(\delta)}=\frac{-0.296123+0.386914T_r-0.0427258T_r^2-0.0480645T_r^3}
{T_r-1.00001}
Units are that of critical or fit constant volume.
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Vc : float
Critical volume of fluid [m^3/mol].
This parameter is alternatively a fit parameter
omega : float
(ideally SRK) Acentric factor for fluid, [-]
This parameter is alternatively a fit parameter.
Returns
-------
Vs : float
Saturation liquid volume
Notes
-----
196 constants are fit to this function in [1]_.
Range: 0.25 < Tr < 0.95, often said to be to 1.0
This function has been checked with the API handbook example problem.
Examples
--------
Propane, from an example in the API Handbook:
>>> from chemicals.utils import Vm_to_rho
>>> Vm_to_rho(COSTALD(272.03889, 369.83333, 0.20008161E-3, 0.1532), 44.097)
530.3009967969844
References
----------
.. [1] Hankinson, Risdon W., and George H. Thomson. "A New Correlation for
Saturated Densities of Liquids and Their Mixtures." AIChE Journal
25, no. 4 (1979): 653-663. doi:10.1002/aic.690250412
'''
if T > Tc:
T = Tc
Tr = T/Tc
tau = 1.0 - Tr
tau_cbrt = (tau)**(1.0/3.)
V_delta = (-0.296123 + Tr*(Tr*(-0.0480645*Tr - 0.0427258) + 0.386914))/(Tr - 1.00001)
V_0 = tau_cbrt*(tau_cbrt*(tau_cbrt*(0.190454*tau_cbrt - 0.81446) + 1.43907) - 1.52816) + 1.0
return Vc*V_0*(1.0 - omega*V_delta)
[docs]def Campbell_Thodos(T, Tb, Tc, Pc, MW, dipole=0.0, has_hydroxyl=False):
r'''Calculate saturation liquid density using the Campbell-Thodos [1]_
CSP method.
An old and uncommon estimation method.
.. math::
V_s = \frac{RT_c}{P_c}{Z_{RA}}^{[1+(1-T_r)^{2/7}]}
.. math::
Z_{RA} = \alpha + \beta(1-T_r)
.. math::
\alpha = 0.3883-0.0179s
.. math::
s = T_{br} \frac{\ln P_c}{(1-T_{br})}
.. math::
\beta = 0.00318s-0.0211+0.625\Lambda^{1.35}
.. math::
\Lambda = \frac{P_c^{1/3}} { MW^{1/2} T_c^{5/6}}
For polar compounds:
.. math::
\theta = P_c \mu^2/T_c^2
.. math::
\alpha = 0.3883 - 0.0179s - 130540\theta^{2.41}
.. math::
\beta = 0.00318s - 0.0211 + 0.625\Lambda^{1.35} + 9.74\times
10^6 \theta^{3.38}
Polar Combounds with hydroxyl groups (water, alcohols)
.. math::
\alpha = \left[0.690T_{br} -0.3342 + \frac{5.79\times 10^{-10}}
{T_{br}^{32.75}}\right] P_c^{0.145}
.. math::
\beta = 0.00318s - 0.0211 + 0.625 \Lambda^{1.35} + 5.90\Theta^{0.835}
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 of the fluid [g/mol]
dipole : float, optional
Dipole moment of the fluid [debye]
has_hydroxyl : bool, optional
Swith to use the hydroxyl variant for polar fluids
Returns
-------
Vs : float
Saturation liquid volume, [m^3/mol]
Notes
-----
If a dipole is provided, the polar chemical method is used.
The paper is an excellent read.
Pc is internally converted to atm.
Examples
--------
Ammonia, from [1]_.
>>> Campbell_Thodos(T=405.45, Tb=239.82, Tc=405.45, Pc=111.7*101325, MW=17.03, dipole=1.47)
7.347366126245e-05
References
----------
.. [1] Campbell, Scott W., and George Thodos. "Prediction of Saturated
Liquid Densities and Critical Volumes for Polar and Nonpolar
Substances." Journal of Chemical & Engineering Data 30, no. 1
(January 1, 1985): 102-11. doi:10.1021/je00039a032.
'''
Tc_inv = 1.0/Tc
Tr = T * Tc_inv
Tbr = Tb * Tc_inv
Pc = Pc/101325.
s = Tbr*log(Pc)/(1.0 - Tbr)
Lambda = Pc**(1.0/3.)/(MW**0.5*Tc**(5/6.))
beta = 0.00318*s - 0.0211 + 0.625*Lambda**(1.35)
if dipole is None:
alpha = 0.3883 - 0.0179*s
else:
theta = Pc*dipole*dipole/(Tc*Tc)
beta += 9.74E6 * theta**3.38
if has_hydroxyl:
beta += 5.90*theta**0.835
alpha = (0.69*Tbr - 0.3342 + 5.79E-10*Tbr**-32.75)*Pc**0.145
else:
alpha = 0.3883 - 0.0179*s - 130540 * theta**2.41
Zra = alpha + beta*(1.0 - Tr)
p = 1.0 if T == Tc else (1.0 + (1.0 - Tr)**(2.0/7.))
Vs = R*Tc/(Pc*101325.0)*Zra**p
return Vs
[docs]def SNM0(T, Tc, Vc, omega, delta_SRK=None):
r'''Calculates saturated liquid density using the Mchaweh, Moshfeghian
model [1]_. Designed for simple calculations.
.. math::
V_s = V_c/(1+1.169\tau^{1/3}+1.818\tau^{2/3}-2.658\tau+2.161\tau^{4/3}
.. math::
\tau = 1-\frac{(T/T_c)}{\alpha_{SRK}}
.. math::
\alpha_{SRK} = [1 + m(1-\sqrt{T/T_C}]^2
.. math::
m = 0.480+1.574\omega-0.176\omega^2
If the fit parameter `delta_SRK` is provided, the following is used:
.. math::
V_s = V_C/(1+1.169\tau^{1/3}+1.818\tau^{2/3}-2.658\tau+2.161\tau^{4/3})
/\left[1+\delta_{SRK}(\alpha_{SRK}-1)^{1/3}\right]
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, [-]
delta_SRK : float, optional
Fitting parameter [-]
Returns
-------
Vs : float
Saturation liquid volume, [m^3/mol]
Notes
-----
73 fit parameters have been gathered from the article.
Examples
--------
Argon, without the fit parameter and with it. Tabulated result in Perry's
is 3.4613e-05. The fit increases the error on this occasion.
>>> SNM0(121, 150.8, 7.49e-05, -0.004)
3.440225640273e-05
>>> SNM0(121, 150.8, 7.49e-05, -0.004, -0.03259620)
3.493288100008e-05
References
----------
.. [1] Mchaweh, A., A. Alsaygh, Kh. Nasrifar, and M. Moshfeghian.
"A Simplified Method for Calculating Saturated Liquid Densities."
Fluid Phase Equilibria 224, no. 2 (October 1, 2004): 157-67.
doi:10.1016/j.fluid.2004.06.054
'''
Tr = T/Tc
m = 0.480 + 1.574*omega - 0.176*omega*omega
x0 = (1.0 + m*(1.0 - sqrt(Tr)))
alpha_SRK = x0*x0
tau = 1. - Tr/alpha_SRK
tau_cbrt = tau**(1/3.)
rho0 = 1. + tau_cbrt*(1.169 + 1.818*tau_cbrt) - 2.658*tau + 2.161*tau*tau_cbrt
V0 = 1./rho0
if delta_SRK is None:
return Vc*V0
else:
return Vc*V0/(1. + delta_SRK*(alpha_SRK - 1.0)**(1.0/3.0))
[docs]def CRC_inorganic(T, rho0, k, Tm, MW=None):
r'''Calculates liquid density of a molten element or salt at temperature
above the melting point. Some coefficients are given nearly up to the
boiling point.
The mass density of the inorganic liquid is given by:
.. math::
\rho = \rho_{0} - k(T-T_m)
Parameters
----------
T : float
Temperature of the liquid, [K]
rho0 : float
Mass density of the liquid at Tm, [kg/m^3]
k : float
Linear temperature dependence of the mass density, [kg/m^3/K]
Tm : float
The normal melting point, used in the correlation [K]
MW : float, optional
Molecular weight of chemical [g/mol]
Returns
-------
rho : float
Mass density of molten metal or salt, [m^3/mol if MW given; kg/m^3 otherwise]
Notes
-----
[1]_ has units of g/mL. While the individual densities could have been
converted to molar units, the temperature coefficient could only be
converted by refitting to calculated data. To maintain compatibility with
the form of the equations, this was not performed.
This linear form is useful only in small temperature ranges.
Coefficients for one compound could be used to predict the temperature
dependence of density of a similar compound.
Examples
--------
>>> CRC_inorganic(300, 2370.0, 2.687, 239.08)
2206.30796
References
----------
.. [1] Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of
Chemistry and Physics, 95E. [Boca Raton, FL]: CRC press, 2014.
'''
rho = rho0 - k*(T-Tm)
return rho if MW is None else 0.001 * MW / rho
[docs]def COSTALD_compressed(T, P, Psat, Tc, Pc, omega, Vs):
r'''Calculates compressed-liquid volume, using the COSTALD [1]_ CSP
method and a chemical's critical properties.
The molar volume of a liquid is given by:
.. math::
V = V_s\left( 1 - C \ln \frac{B + P}{B + P^{sat}}\right)
.. math::
\frac{B}{P_c} = -1 + a\tau^{1/3} + b\tau^{2/3} + d\tau + e\tau^{4/3}
.. math::
e = \exp(f + g\omega_{SRK} + h \omega_{SRK}^2)
.. math::
C = j + k \omega_{SRK}
Parameters
----------
T : float
Temperature of fluid [K]
P : float
Pressure of fluid [Pa]
Psat : float
Saturation pressure of the fluid [Pa]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
(ideally SRK) Acentric factor for fluid, [-]
This parameter is alternatively a fit parameter.
Vs : float
Saturation liquid volume, [m^3/mol]
Returns
-------
V_dense : float
High-pressure liquid volume, [m^3/mol]
Notes
-----
Original equation was in terms of density, but it is converted here.
The example is from DIPPR, and exactly correct.
This is DIPPR Procedure 4C: Method for Estimating the Density of Pure
Organic Liquids under Pressure.
Examples
--------
>>> COSTALD_compressed(303., 9.8E7, 85857.9, 466.7, 3640000.0, 0.281, 0.000105047)
9.287482879788505e-05
References
----------
.. [1] Thomson, G. H., K. R. Brobst, and R. W. Hankinson. "An Improved
Correlation for Densities of Compressed Liquids and Liquid Mixtures."
AIChE Journal 28, no. 4 (July 1, 1982): 671-76. doi:10.1002/aic.690280420
'''
a = -9.070217
b = 62.45326
d = -135.1102
f = 4.79594
g = 0.250047
h = 1.14188
j = 0.0861488
k = 0.0344483
e = exp(f + omega*(g + h*omega))
C = j + k*omega
tau = 1.0 - T/Tc
tau13 = tau**(1.0/3.0)
B = Pc*(-1.0 + a*tau13 + b*tau13*tau13 + d*tau + e*tau*tau13)
return Vs*(1.0 - C*log((B + P)/(B + Psat)))
[docs]def Tait(P, P_ref, rho_ref, B, C):
r'''Calculates compressed-liquid mass density using the Tait
model [1]_ and fit coefficients
`B` and `C` and the reference (usually saturation) liquid
density. `B` and `C` are normally temperature dependent but it
is assumed they are constant (or calculated earlier) in this
function
The mass density of the compressed liquid is given by:
.. math::
\rho = \frac{\rho_{ref}}{1 - C \ln \frac{B+P}{B + P_{ref}}}
Parameters
----------
P : float
Pressure of fluid [Pa]
P_ref : float
Pressure of the fluid at the reference density; normally
saturation at higher pressures and either 1 atm or 1 MPa
at low enough temperatures the saturation pressure
stops being an important factor, [Pa]
rho_ref : float
The mass density of the fluid at the reference condition, [kg/m^3]
B : float
Fit coefficient, [Pa]
C : float
Fit coefficient, [-]
Returns
-------
rho : float
High-pressure liquid mass density, [kg/m^3]
Notes
-----
If `P` is set to be lower than `P_ref`, it is adjusted to
have the same value as `P_ref` (saturation condition).
If B becomes negative and higher than
`P_ref`, the logarithm will become undefined.
Examples
--------
Coefficients for methanol from the CRC Handbook [1]_ at 300 K and
1E8 Pa.
>>> B_coeffs = [649718000.0, -4345830.0, 13072.2, -20.029200000000003, 0.0120566]
>>> C_coeffs = [0.115068, -5.322e-05]
>>> T = 300
>>> Tait(P=1e8, P_ref=101325, rho_ref=784.85, B=np.polyval(B_coeffs[::-1], T), C=np.polyval(C_coeffs[::-1], T))
853.744916
References
----------
.. [1] Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of
Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014.
'''
if P < P_ref:
# The model is not fit on pressures below saturation and cannot extrapolate there
return rho_ref
return rho_ref/(1.0 - C*log((B+P)/(B+P_ref) ))
[docs]def Tait_molar(P, P_ref, V_ref, B, C):
r'''Calculates compressed-liquid volume using the Tait
model [1]_ and fit coefficients
`B` and `C` and the reference (usually saturation) liquid
density. `B` and `C` are normally temperature dependent but it
is assumed they are constant (or calculated earlier) in this
function
The molar volume of the compressed liquid is given by:
.. math::
V_m = V_{ref}\cdot \left({1 - C \ln \frac{B+P}{B + P_{ref}}}\right)
Parameters
----------
P : float
Pressure of fluid [Pa]
P_ref : float
Pressure of the fluid at the reference density; normally
saturation at higher pressures and either 1 atm or 1 MPa
at low enough temperatures the saturation pressure
stops being an important factor, [Pa]
V_ref : float
The molar volume of the fluid at the reference condition, [m^3/mol]
B : float
Fit coefficient, [Pa]
C : float
Fit coefficient, [-]
Returns
-------
V_dense : float
High-pressure liquid volume, [m^3/mol]
Notes
-----
If `P` is set to be lower than `P_ref`, it is adjusted to
have the same value as `P_ref` (saturation condition).
Examples
--------
Coefficients for methanol from the CRC Handbook [1]_ at 300 K and
1E8 Pa.
>>> Tait_molar(P=1e8, P_ref=101325.0, V_ref=4.0825e-05, B=79337060.0, C=0.099102)
3.75305e-05
References
----------
.. [1] Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of
Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014.
'''
if P < P_ref:
P = P_ref
return V_ref*(1.0 - C*log((B + P)/(B + P_ref) ))
### Liquid Mixtures
[docs]def Amgat(xs, Vms):
r'''Calculate mixture liquid density using the Amgat mixing rule.
Highly inacurate, but easy to use. Assumes idea liquids with
no excess volume. Average molecular weight should be used with it to obtain
density.
.. math::
V_{mix} = \sum_i x_i V_i
or in terms of density:
.. math::
\rho_{mix} = \sum\frac{x_i}{\rho_i}
Parameters
----------
xs : array
Mole fractions of each component, []
Vms : array
Molar volumes of each fluids at conditions [m^3/mol]
Returns
-------
Vm : float
Mixture liquid volume [m^3/mol]
Notes
-----
Units are that of the given volumes.
It has been suggested to use this equation with weight fractions,
but the results have been less accurate.
Examples
--------
>>> Amgat([0.5, 0.5], [4.057e-05, 5.861e-05])
4.9590000000000005e-05
'''
return mixing_simple(xs, Vms)
[docs]def Rackett_mixture(T, xs, MWs, Tcs, Pcs, Zrs):
r'''Calculate mixture liquid density using the Rackett-derived mixing rule
as shown in [2]_.
.. math::
V_m = \sum_i\frac{x_i T_{ci}}{MW_i P_{ci}} Z_{R,m}^{(1 + (1 - T_r)^{2/7})} R \sum_i x_i MW_i
Parameters
----------
T : float
Temperature of liquid [K]
xs: list
Mole fractions of each component, []
MWs : list
Molecular weights of each component [g/mol]
Tcs : list
Critical temperatures of each component [K]
Pcs : list
Critical pressures of each component [Pa]
Zrs : list
Rackett parameters of each component []
Returns
-------
Vm : float
Mixture liquid volume [m^3/mol]
Notes
-----
Model for pure compounds in [1]_ forms the basis for this model, shown in
[2]_. Molecular weights are used as weighing by such has been found to
provide higher accuracy in [2]_. The model can also be used without
molecular weights, but results are somewhat different.
As with the Rackett model, critical compressibilities may be used if
Rackett parameters have not been regressed.
Critical mixture temperature, and compressibility are all obtained with
simple mixing rules.
Examples
--------
Calculation in [2]_ for methanol and water mixture. Result matches example.
>>> Rackett_mixture(T=298., xs=[0.4576, 0.5424], MWs=[32.04, 18.01], Tcs=[512.58, 647.29], Pcs=[8.096E6, 2.209E7], Zrs=[0.2332, 0.2374])
2.6252894930056885e-05
References
----------
.. [1] Rackett, Harold G. "Equation of State for Saturated Liquids."
Journal of Chemical & Engineering Data 15, no. 4 (1970): 514-517.
doi:10.1021/je60047a012
.. [2] Danner, Ronald P, and Design Institute for Physical Property Data.
Manual for Predicting Chemical Process Design Data. New York, N.Y, 1982.
'''
bigsum, Tc, Zr, MW = 0.0, 0.0, 0.0, 0.0
# Fastest for numba and PyPy and CPython
for i in range(len(xs)):
x0 = Tcs[i]*xs[i]
Tc += x0
Zr += Zrs[i]*xs[i]
MW += MWs[i]*xs[i]
bigsum += x0/(Pcs[i]*MWs[i])
Tr = T/Tc
return (R*bigsum*Zr**(1.0 + (1.0 - Tr)**(2.0/7.0)))*MW
[docs]def COSTALD_mixture(xs, T, Tcs, Vcs, omegas):
r'''Calculate mixture liquid density using the COSTALD CSP method.
A popular and accurate estimation method. If possible, fit parameters are
used; alternatively critical properties work well.
The mixing rules giving parameters for the pure component COSTALD
equation are:
.. math::
T_{cm} = \frac{\sum_i\sum_j x_i x_j (V_{ij}T_{cij})}{V_m}
.. math::
V_m = 0.25\left[ \sum x_i V_i + 3(\sum x_i V_i^{2/3})(\sum_i x_i V_i^{1/3})\right]
.. math::
V_{ij}T_{cij} = (V_iT_{ci}V_{j}T_{cj})^{0.5}
.. math::
\omega = \sum_i z_i \omega_i
Parameters
----------
xs: list
Mole fractions of each component
T : float
Temperature of fluid [K]
Tcs : list
Critical temperature of fluids [K]
Vcs : list
Critical volumes of fluids [m^3/mol].
This parameter is alternatively a fit parameter
omegas : list
(ideally SRK) Acentric factor of all fluids, [-]
This parameter is alternatively a fit parameter.
Returns
-------
Vs : float
Saturation liquid mixture volume
Notes
-----
Range: 0.25 < Tr < 0.95, often said to be to 1.0
No example has been found.
Units are that of critical or fit constant volume.
Examples
--------
>>> COSTALD_mixture([0.4576, 0.5424], 298., [512.58, 647.29], [0.000117, 5.6e-05], [0.559,0.344])
2.7065887732713534e-05
References
----------
.. [1] Hankinson, Risdon W., and George H. Thomson. "A New Correlation for
Saturated Densities of Liquids and Their Mixtures." AIChE Journal
25, no. 4 (1979): 653-663. doi:10.1002/aic.690250412
'''
N = len(xs)
sum1, sum2, sum3, omega = 0.0, 0.0, 0.0, 0.0
for i in range(N):
sum1 += xs[i]*Vcs[i]
p = Vcs[i]**(1.0/3.)
v = xs[i]*p
sum2 += v
sum3 += v*p
omega += xs[i]*omegas[i]
Vm = 0.25*(sum1 + 3.0*sum2*sum3)
Vm_inv_root = root_two*(Vm)**-0.5
vec = [0.0]*N
for i in range(N):
vec[i] = (Tcs[i]*Vcs[i])**0.5*xs[i]*Vm_inv_root
Tcm = 0.0
for i in range(N):
for j in range(i):
Tcm += vec[i]*vec[j]
Tcm += 0.5*vec[i]*vec[i]
return COSTALD(T, Tcm, Vm, omega)
### Gases
[docs]def ideal_gas(T, P):
r'''Calculates ideal gas molar volume.
The molar volume of an ideal gas is given by:
.. math::
V = \frac{RT}{P}
Parameters
----------
T : float
Temperature of fluid [K]
P : float
Pressure of fluid [Pa]
Returns
-------
V : float
Gas volume, [m^3/mol]
Examples
--------
>>> ideal_gas(298.15, 101325.)
0.024465403697038125
'''
return R*T/P
### Solids
[docs]def Goodman(T, Tt, Vml):
r'''Calculates solid density at T using the simple relationship
by a member of the DIPPR.
The molar volume of a solid is given by:
.. math::
\frac{1}{V_m} = \left( 1.28 - 0.16 \frac{T}{T_t}\right)
\frac{1}{{Vm}_L(T_t)}
Parameters
----------
T : float
Temperature of fluid [K]
Tt : float
Triple temperature of fluid [K]
Vml : float
Liquid molar volume of the organic liquid at the triple point,
[m^3/mol]
Returns
-------
Vms : float
Solid molar volume, [m^3/mol]
Notes
-----
Works to the next solid transition temperature or to approximately 0.3Tt.
Examples
--------
Decane at 200 K:
>>> Goodman(200, 243.225, 0.00023585)
0.0002053665090860923
References
----------
.. [1] Goodman, Benjamin T., W. Vincent Wilding, John L. Oscarson, and
Richard L. Rowley. "A Note on the Relationship between Organic Solid
Density and Liquid Density at the Triple Point." Journal of Chemical &
Engineering Data 49, no. 6 (2004): 1512-14. doi:10.1021/je034220e.
'''
return Vml/(1.28 - 0.16*(T/Tt))