"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2019, 2020 Caleb Bell
<Caleb.Andrew.Bell@gmail.com>
Copyright (C) 2020 Yoel Rene Cortes-Pena
<yoelcortes@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 many heat capacity model equations, heat capacity estimation
equations, enthalpy and entropy integrals of those heat capacity equations,
enthalpy/entropy flash initialization routines, and many dataframes of
coefficients.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/chemicals/>`_.
.. contents:: :local:
Gas Heat Capacity Model Equations
---------------------------------
.. autofunction:: chemicals.heat_capacity.TRCCp
.. autofunction:: chemicals.heat_capacity.TRCCp_integral
.. autofunction:: chemicals.heat_capacity.TRCCp_integral_over_T
.. autofunction:: chemicals.heat_capacity.Shomate
.. autofunction:: chemicals.heat_capacity.Shomate_integral
.. autofunction:: chemicals.heat_capacity.Shomate_integral_over_T
.. autoclass:: chemicals.heat_capacity.ShomateRange
:members: calculate, calculate_integral, calculate_integral_over_T
.. autofunction:: chemicals.heat_capacity.Poling
.. autofunction:: chemicals.heat_capacity.Poling_integral
.. autofunction:: chemicals.heat_capacity.Poling_integral_over_T
.. autofunction:: chemicals.heat_capacity.PPDS2
Gas Heat Capacity Estimation Models
-----------------------------------
.. autofunction:: chemicals.heat_capacity.Lastovka_Shaw
.. autofunction:: chemicals.heat_capacity.Lastovka_Shaw_integral
.. autofunction:: chemicals.heat_capacity.Lastovka_Shaw_integral_over_T
.. autofunction:: chemicals.heat_capacity.Lastovka_Shaw_T_for_Hm
.. autofunction:: chemicals.heat_capacity.Lastovka_Shaw_T_for_Sm
.. autofunction:: chemicals.heat_capacity.Lastovka_Shaw_term_A
Gas Heat Capacity Theory
------------------------
.. autofunction:: chemicals.heat_capacity.Cpg_statistical_mechanics
.. autofunction:: chemicals.heat_capacity.Cpg_statistical_mechanics_integral
.. autofunction:: chemicals.heat_capacity.Cpg_statistical_mechanics_integral_over_T
.. autofunction:: chemicals.heat_capacity.vibration_frequency_cm_to_characteristic_temperature
Liquid Heat Capacity Model Equations
------------------------------------
.. autofunction:: chemicals.heat_capacity.Zabransky_quasi_polynomial
.. autofunction:: chemicals.heat_capacity.Zabransky_quasi_polynomial_integral
.. autofunction:: chemicals.heat_capacity.Zabransky_quasi_polynomial_integral_over_T
.. autofunction:: chemicals.heat_capacity.Zabransky_cubic
.. autofunction:: chemicals.heat_capacity.Zabransky_cubic_integral
.. autofunction:: chemicals.heat_capacity.Zabransky_cubic_integral_over_T
.. autoclass:: chemicals.heat_capacity.ZabranskySpline
:members: calculate, calculate_integral, calculate_integral_over_T
.. autoclass:: chemicals.heat_capacity.ZabranskyQuasipolynomial
:members: calculate, calculate_integral, calculate_integral_over_T
.. autofunction:: chemicals.heat_capacity.PPDS15
.. autofunction:: chemicals.heat_capacity.TDE_CSExpansion
Liquid Heat Capacity Estimation Models
--------------------------------------
.. autofunction:: chemicals.heat_capacity.Rowlinson_Poling
.. autofunction:: chemicals.heat_capacity.Rowlinson_Bondi
.. autofunction:: chemicals.heat_capacity.Dadgostar_Shaw
.. autofunction:: chemicals.heat_capacity.Dadgostar_Shaw_integral
.. autofunction:: chemicals.heat_capacity.Dadgostar_Shaw_integral_over_T
.. autofunction:: chemicals.heat_capacity.Dadgostar_Shaw_terms
Solid Heat Capacity Estimation Models
-------------------------------------
.. autofunction:: chemicals.heat_capacity.Perry_151
.. autofunction:: chemicals.heat_capacity.Lastovka_solid
.. autofunction:: chemicals.heat_capacity.Lastovka_solid_integral
.. autofunction:: chemicals.heat_capacity.Lastovka_solid_integral_over_T
Utility methods
---------------
.. autoclass:: chemicals.heat_capacity.PiecewiseHeatCapacity
Fit Coefficients
----------------
All of these coefficients are lazy-loaded, so they must be accessed as an
attribute of this module.
.. data:: Cp_data_Poling
Constains data for gases and liquids from [3]_.
Simple polynomials for gas heat capacity (not suitable for extrapolation) are available for 308 chemicals. Additionally, constant values in at 298.15 K are available for 348 gases. Constant values in at 298.15 K are available for 245 liquids.
.. data:: TRC_gas_data
A rigorous expression from [1]_ for modeling gas heat capacity.
Coefficients for 1961 chemicals are available.
.. data:: CRC_standard_data
Constant values tabulated in [4]_ at 298.15 K. Data is available for
533 gases. Data is available for 433 liquids. Data is available for 529
solids.
.. data:: Cp_dict_PerryI
Simple polynomials from [5]_ with vaious exponents selected for each expression.
Coefficients are in units of calories/mol/K. The full expression is
:math:`C_p = a + bT + c/T^2 + dT^2`. Data is available for 284 compounds.
Some compounds have gas data, some have liquid data, and have solid
(crystal structure) data, sometimes multiple coefficients for different
solid phases.
.. data:: zabransky_dicts
Complicated fits covering different cases and with different forms from [2]_.
.. data:: Cp_dict_characteristic_temperatures_adjusted_psi4_2022a
Theoretically calculated chatacteristic temperatures from vibrational
frequencies using psi4
.. data:: Cp_dict_characteristic_temperatures_psi4_2022a
Theoretically calculated chatacteristic temperatures from vibrational
frequencies using psi4, adjusted using a recommended coefficient
.. [1] Kabo, G. J., and G. N. Roganov. Thermodynamics of Organic Compounds
in the Gas State, Volume II: V. 2. College Station, Tex: CRC Press, 1994.
.. [2] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
.. [3] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
.. [4] Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of
Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014.
.. [5] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
Eighth Edition. McGraw-Hill Professional, 2007.
.. ipython::
In [1]: import chemicals
In [2]: chemicals.heat_capacity.Cp_data_Poling
In [3]: chemicals.heat_capacity.TRC_gas_data
In [4]: chemicals.heat_capacity.CRC_standard_data
In [5]: chemicals.heat_capacity.Cp_dict_PerryI['124-38-9'] # gas only
In [6]: chemicals.heat_capacity.Cp_dict_PerryI['7704-34-9'] # crystal and gas
In [7]: chemicals.heat_capacity.Cp_dict_PerryI['7440-57-5'] # crystal and liquid
In [8]: chemicals.heat_capacity.zabransky_dicts.keys()
"""
__all__ = ['heat_capacity_gas_methods',
'Poling', 'Poling_integral', 'Poling_integral_over_T',
'Lastovka_Shaw', 'Lastovka_Shaw_integral', 'Lastovka_Shaw_integral_over_T',
'Lastovka_Shaw_T_for_Hm', 'Lastovka_Shaw_T_for_Sm', 'Lastovka_Shaw_term_A',
'TRCCp', 'TRCCp_integral', 'TRCCp_integral_over_T',
'heat_capacity_liquid_methods', 'PPDS2', 'PPDS15',
'TDE_CSExpansion',
'Rowlinson_Poling', 'Rowlinson_Bondi', 'Dadgostar_Shaw',
'Zabransky_quasi_polynomial', 'Zabransky_quasi_polynomial_integral',
'Zabransky_quasi_polynomial_integral_over_T', 'Zabransky_cubic',
'Zabransky_cubic_integral', 'Zabransky_cubic_integral_over_T',
'Dadgostar_Shaw_integral', 'Dadgostar_Shaw_integral_over_T',
'Dadgostar_Shaw_terms',
'heat_capacity_solid_methods',
'Lastovka_solid', 'Lastovka_solid_integral',
'Lastovka_solid_integral_over_T', 'heat_capacity_solid_methods',
'ZabranskySpline', 'ZabranskyQuasipolynomial',
'PiecewiseHeatCapacity',
'Shomate_integral_over_T', 'Shomate_integral', 'Shomate',
'Cpg_statistical_mechanics', 'Cpg_statistical_mechanics_integral',
'Cpg_statistical_mechanics_integral_over_T',
'vibration_frequency_cm_to_characteristic_temperature',
]
import os
from cmath import exp as cexp
from cmath import log as clog
from math import expm1
from fluids.constants import R, c, h, k
from fluids.numerics import brenth, exp, log, polylog2, secant
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_uncacheable, os_path_join, source_path, to_num
### Methods introduced in this module
# Gases
TRCIG = 'TRC Thermodynamics of Organic Compounds in the Gas State (1994)'
POLING = 'Poling et al. (2001)'
POLING_CONST = 'Poling et al. (2001) constant'
CRCSTD = 'CRC Standard Thermodynamic Properties of Chemical Substances'
VDI_TABULAR = 'VDI Heat Atlas'
LASTOVKA_SHAW = 'Lastovka and Shaw (2013)'
heat_capacity_gas_methods = (
TRCIG, POLING, LASTOVKA_SHAW, CRCSTD, POLING_CONST, VDI_TABULAR
)
# Liquids
ZABRANSKY_SPLINE = 'Zabransky spline, averaged heat capacity'
ZABRANSKY_QUASIPOLYNOMIAL = 'Zabransky quasipolynomial, averaged heat capacity'
ZABRANSKY_SPLINE_C = 'Zabransky spline, constant-pressure'
ZABRANSKY_QUASIPOLYNOMIAL_C = 'Zabransky quasipolynomial, constant-pressure'
ZABRANSKY_SPLINE_SAT = 'Zabransky spline, saturation'
ZABRANSKY_QUASIPOLYNOMIAL_SAT = 'Zabransky quasipolynomial, saturation'
ROWLINSON_POLING = 'Rowlinson and Poling (2001)'
ROWLINSON_BONDI = 'Rowlinson and Bondi (1969)'
DADGOSTAR_SHAW = 'Dadgostar and Shaw (2011)'
heat_capacity_liquid_methods = (
ZABRANSKY_SPLINE, ZABRANSKY_QUASIPOLYNOMIAL, ZABRANSKY_SPLINE_C,
ZABRANSKY_QUASIPOLYNOMIAL_C, ZABRANSKY_SPLINE_SAT, ZABRANSKY_QUASIPOLYNOMIAL_SAT,
VDI_TABULAR, ROWLINSON_POLING, ROWLINSON_BONDI, DADGOSTAR_SHAW, POLING_CONST,
CRCSTD
)
# Solids
LASTOVKA_S = 'Lastovka, Fulem, Becerra and Shaw (2008)'
PERRY151 = "Perry's Table 2-151"
heat_capacity_solid_methods = (PERRY151, CRCSTD, LASTOVKA_S)
### Heat capacity classes
[docs]class ZabranskySpline:
r'''
Implementation of the cubic spline method presented in [1]_ for
calculating the heat capacity of a chemical.
Implements the enthalpy and entropy integrals as well.
.. math::
\frac{C}{R}=\sum_{j=0}^3 A_{j+1} \left(\frac{T}{100}\right)^j
Parameters
----------
coeffs : list[float]
Six coefficients for the equation, [-]
Tmin : float
Minimum temperature any experimental data was available at, [K]
Tmax : float
Maximum temperature any experimental data was available at, [K]
References
----------
.. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
'''
try:
IS_NUMBA # type: ignore # noqa: F821
except:
__slots__ = ('coeffs', 'Tmin', 'Tmax')
def __init__(self, coeffs, Tmin, Tmax):
self.coeffs = coeffs
self.Tmin = Tmin
self.Tmax = Tmax
[docs] def calculate(self, T):
r'''
Return heat capacity as a function of temperature.
Parameters
----------
T : float
Temperature, [K]
Returns
-------
Cp : float
Liquid heat capacity as T, [J/mol/K]
'''
return Zabransky_cubic(T, *self.coeffs)
[docs] def calculate_integral(self, Ta, Tb):
r'''
Return the enthalpy integral of heat capacity from `Ta` to `Tb`.
Parameters
----------
Ta : float
Initial temperature, [K]
Tb : float
Final temperature, [K]
Returns
-------
dH : float
Enthalpy difference between `Ta` and `Tb`, [J/mol]
'''
return (Zabransky_cubic_integral(Tb, *self.coeffs)
- Zabransky_cubic_integral(Ta, *self.coeffs))
[docs] def calculate_integral_over_T(self, Ta, Tb):
r'''
Return the entropy integral of heat capacity from `Ta` to `Tb`.
Parameters
----------
Ta : float
Initial temperature, [K]
Tb : float
Final temperature, [K]
Returns
-------
dS : float
Entropy difference between `Ta` and `Tb`, [J/mol/K]
'''
return (Zabransky_cubic_integral_over_T(Tb, *self.coeffs)
- Zabransky_cubic_integral_over_T(Ta, *self.coeffs))
force_calculate_integral_over_T = calculate_integral_over_T
force_calculate_integral = calculate_integral
force_calculate = calculate
try:
if IS_NUMBA: # type: ignore # noqa: F821
ZabranskySpline = jitclass([('coeffs', numba.types.UniTuple(numba.float64, 4)), # type: ignore # noqa: F821
('Tmin', numba.float64), # type: ignore # noqa: F821
('Tmax', numba.float64)])(ZabranskySpline) # type: ignore # noqa: F821
except:
pass
[docs]class ShomateRange:
r'''
Implementation of a range of the Shomate equation presented in [1]_ for
calculating the heat capacity of a chemical.
Implements the enthalpy and entropy integrals as well.
Parameters
----------
coeffs : list[float]
Six coefficients for the equation, [-]
Tmin : float
Minimum temperature any experimental data was available at, [K]
Tmax : float
Maximum temperature any experimental data was available at, [K]
References
----------
.. [1] Shen, V.K., Siderius, D.W., Krekelberg, W.P., and Hatch, H.W., Eds.,
NIST WebBook, NIST, http://doi.org/10.18434/T4M88Q
'''
try:
IS_NUMBA # type: ignore # noqa: F821
except:
__slots__ = ('coeffs', 'Tmin', 'Tmax')
def __init__(self, coeffs, Tmin, Tmax):
self.coeffs = coeffs
self.Tmin = Tmin
self.Tmax = Tmax
[docs] def calculate(self, T):
return Shomate(T, *self.coeffs)
try:
calculate.__doc__ = ZabranskySpline.calculate.__doc__
except:
pass
[docs] def calculate_integral(self, Ta, Tb):
return (Shomate_integral(Tb, *self.coeffs)
- Shomate_integral(Ta, *self.coeffs))
try:
calculate_integral.__doc__ = ZabranskySpline.calculate_integral.__doc__
except:
pass
[docs] def calculate_integral_over_T(self, Ta, Tb):
return (Shomate_integral_over_T(Tb, *self.coeffs)
- Shomate_integral_over_T(Ta, *self.coeffs))
try:
calculate_integral_over_T.__doc__ = ZabranskySpline.calculate_integral_over_T.__doc__
except:
pass
force_calculate_integral_over_T = calculate_integral_over_T
force_calculate_integral = calculate_integral
force_calculate = calculate
try:
if IS_NUMBA: # type: ignore # noqa: F821
ShomateRange = jitclass([('coeffs', numba.types.UniTuple(numba.float64, 5)), # type: ignore # noqa: F821
('Tmin', numba.float64), # type: ignore # noqa: F821
('Tmax', numba.float64)])(ShomateRange) # type: ignore # noqa: F821
except:
pass
[docs]class ZabranskyQuasipolynomial:
r'''
Quasi-polynomial object for calculating the heat capacity of a chemical.
Implements the enthalpy and entropy integrals as well.
.. math::
\frac{C}{R}=A_1\ln(1-T_r) + \frac{A_2}{1-T_r}
+ \sum_{j=0}^m A_{j+3} T_r^j
Parameters
----------
coeffs : list[float]
Six coefficients for the equation, [-]
Tc : float
Critical temperature of the chemical, as used in the formula, [K]
Tmin : float
Minimum temperature any experimental data was available at, [K]
Tmax : float
Maximum temperature any experimental data was available at, [K]
References
----------
.. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
'''
try:
IS_NUMBA # type: ignore # noqa: F821
except:
__slots__ = ('coeffs', 'Tc', 'Tmin', 'Tmax')
def __init__(self, coeffs, Tc, Tmin, Tmax):
self.coeffs = coeffs
self.Tc = Tc
self.Tmin = Tmin
self.Tmax = Tmax
[docs] def calculate(self, T):
r'''
Return the heat capacity as a function of temperature.
Parameters
----------
T : float
Temperature, [K]
Returns
-------
Cp : float
Liquid heat capacity as T, [J/mol/K]
'''
return Zabransky_quasi_polynomial(T, self.Tc, *self.coeffs)
[docs] def calculate_integral(self, Ta, Tb):
r'''
Return the enthalpy integral of heat capacity from `Ta` to `Tb`.
Parameters
----------
Ta : float
Initial temperature, [K]
Tb : float
Final temperature, [K]
Returns
-------
dH : float
Enthalpy difference between `Ta` and `Tb`, [J/mol]
'''
return (Zabransky_quasi_polynomial_integral(Tb, self.Tc, *self.coeffs)
- Zabransky_quasi_polynomial_integral(Ta, self.Tc, *self.coeffs))
[docs] def calculate_integral_over_T(self, Ta, Tb):
r'''
Return the entropy integral of heat capacity from `Ta` to `Tb`.
Parameters
----------
Ta : float
Initial temperature, [K]
Tb : float
Final temperature, [K]
Returns
-------
dS : float
Entropy difference between `Ta` and `Tb`, [J/mol/K]
'''
return (Zabransky_quasi_polynomial_integral_over_T(Tb, self.Tc, *self.coeffs)
- Zabransky_quasi_polynomial_integral_over_T(Ta, self.Tc, *self.coeffs))
try:
if IS_NUMBA: # type: ignore # noqa: F821
ZabranskyQuasipolynomial = jitclass([('coeffs', numba.types.UniTuple(numba.float64, 6)), # type: ignore # noqa: F821
('Tc', numba.float64), # type: ignore # noqa: F821
('Tmin', numba.float64), # type: ignore # noqa: F821
('Tmax', numba.float64)])(ZabranskyQuasipolynomial) # type: ignore # noqa: F821
except:
pass
[docs]class PiecewiseHeatCapacity:
r"""
Create a PiecewiseHeatCapacity object for calculating heat capacity and the
enthalpy and entropy integrals using piecewise models.
Parameters
----------
models : Iterable[HeatCapacity]
Piecewise heat capacity objects, [-]
"""
# Dev note - not possible to jitclass this as the model types are not explicit
__slots__ = ('models', 'Tmin', 'Tmax')
def __init__(self, models):
self.models = tuple(sorted(models, key=lambda x: x.Tmin))
self.Tmin = self.models[0].Tmin
self.Tmax = self.models[-1].Tmax
def __iter__(self):
return self.models.__iter__()
def calculate(self, T):
r'''
Return the heat capacity as a function of temperature.
Parameters
----------
T : float
Temperature, [K]
Raises
------
ValueError
If the temperature in not within the domain of any of the models
(i.e if Tmin <= T <= Tmax cannot be satisfied by any of the models).
See Also
--------
PiecewiseHeatCapacity.force_calculate
Returns
-------
Cp : float
Liquid heat capacity as T, [J/mol/K]
'''
if T >= self.Tmin:
for model in self.models:
if T <= model.Tmax: return model.calculate(T)
raise ValueError("no valid model at T=%g K" % T)
def force_calculate(self, T):
r'''
Return the heat capacity as a function of temperature.
Parameters
----------
T : float
Temperature, [K]
Notes
-----
This method extrapolates when temperature is not within the domain of
any of the models (i.e if Tmin <= T <= Tmax cannot be satisfied by any
of the models).
See Also
--------
PiecewiseHeatCapacity.calculate
Returns
-------
Cp : float
Liquid heat capacity as T, [J/mol/K]
'''
for model in self.models:
if T <= model.Tmax: break
return model.calculate(T)
def calculate_integral(self, Ta, Tb):
r'''
Return the enthalpy integral of heat capacity from `Ta` to `Tb`.
Parameters
----------
Ta : float
Initial temperature, [K]
Tb : float
Final temperature, [K]
Raises
------
ValueError
If the temperature in not within the domain of any of the models
(i.e if Tmin <= T <= Tmax cannot be satisfied by any of the models).
Notes
-----
Analytically integrates piecewise through all models.
See Also
--------
PiecewiseHeatCapacity.force_calculate_integral
Returns
-------
dH : float
Enthalpy difference between `Ta` and `Tb`, [J/mol]
'''
if Tb < Ta: return -self.calculate_integral(Tb, Ta)
if Ta < self.Tmin: raise ValueError("no valid model at T=%g K" % Ta)
elif Tb > self.Tmax: raise ValueError("no valid model at T=%g K" % Tb)
return self.force_calculate_integral(Ta, Tb)
def force_calculate_integral(self, Ta, Tb):
r'''
Return the enthalpy integral of heat capacity from `Ta` to `Tb`.
Parameters
----------
Ta : float
Initial temperature, [K]
Tb : float
Final temperature, [K]
Raises
------
ValueError
If the temperature in not within the domain of any of the models
(i.e if Tmin <= T <= Tmax cannot be satisfied by any of the models).
Notes
-----
Analytically integrates piecewise through all models and extrapolates
when temperature is not within the domain of any of the models
(i.e if Tmin <= T <= Tmax cannot be satisfied by any of the models).
See Also
--------
PiecewiseHeatCapacity.calculate_integral
Returns
-------
dH : float
Enthalpy difference between `Ta` and `Tb`, [J/mol]
'''
if Tb < Ta: return -self.force_calculate_integral(Tb, Ta)
integral = 0.
for model in self.models:
Tmax = model.Tmax
if Tb <= Tmax:
return integral + model.calculate_integral(Ta, Tb)
elif Ta < Tmax:
integral += model.calculate_integral(Ta, Tmax)
Ta = Tmax
return integral + model.calculate_integral(Ta, Tb)
def calculate_integral_over_T(self, Ta, Tb):
r'''
Return the entropy integral of heat capacity from `Ta` to `Tb`.
Parameters
----------
Ta : float
Initial temperature, [K]
Tb : float
Final temperature, [K]
Notes
-----
Analytically integrates piecewise through all models.
Raises
------
ValueError
If the temperature in not within the domain of any of the models
(i.e if Tmin <= T <= Tmax cannot be satisfied by any of the models).
See Also
--------
PiecewiseHeatCapacity.force_calculate_integral_over_T
Returns
-------
dS : float
Entropy difference between `Ta` and `Tb`, [J/mol/K]
'''
if Tb < Ta: return -self.calculate_integral_over_T(Tb, Ta)
if Ta < self.Tmin: raise ValueError("no valid model at T=%d K" % Ta)
elif Tb > self.Tmax: raise ValueError("no valid model at T=%d K" % Tb)
return self.force_calculate_integral_over_T(Ta, Tb)
def force_calculate_integral_over_T(self, Ta, Tb):
r'''
Return the entropy integral of heat capacity from `Ta` to `Tb`.
Parameters
----------
Ta : float
Initial temperature, [K]
Tb : float
Final temperature, [K]
Notes
-----
Analytically integrates piecewise through all models and extrapolates
when temperature is not within the domain of any of the models
(i.e if Tmin <= T <= Tmax cannot be satisfied by any of the models).
See Also
--------
PiecewiseHeatCapacity.calculate_integral_over_T
Returns
-------
dS : float
Entropy difference between `Ta` and `Tb`, [J/mol/K]
'''
if Tb < Ta: return -self.force_calculate_integral_over_T(Tb, Ta)
integral = 0.
for model in self.models:
Tmax = model.Tmax
if Tb <= Tmax:
return integral + model.calculate_integral_over_T(Ta, Tb)
elif Ta < Tmax:
integral += model.calculate_integral_over_T(Ta, Tmax)
Ta = Tmax
return integral + model.calculate_integral_over_T(Ta, Tb)
### Register data sources and lazy load them
folder = os_path_join(source_path, 'Heat Capacity')
register_df_source(folder, 'PolingDatabank.tsv')
register_df_source(folder, 'TRC Thermodynamics of Organic Compounds in the Gas State.tsv', csv_kwargs={
'dtype':{'Tmin': float, 'Tmax': float, 'a0': float, 'a1': float, 'a2': float,
'a3': float, 'a4': float, 'a5': float, 'a6': float, 'a7': float,
'I': float, 'J': float, 'Hfg': float}})
register_df_source(folder, 'CRC Standard Thermodynamic Properties of Chemical Substances.tsv')
_Cp_data_loaded = False
def _load_Cp_data():
global Cp_data_Poling, Cp_values_Poling, TRC_gas_data, TRC_gas_values
global CRC_standard_data, Cp_dict_PerryI
global WebBook_Shomate_liquids, WebBook_Shomate_gases, WebBook_Shomate_solids, WebBook_Shomate_coefficients
global zabransky_dict_sat_s, zabransky_dict_sat_p, zabransky_dict_const_s
global zabransky_dict_const_p, zabransky_dict_iso_s, zabransky_dict_iso_p
global type_to_zabransky_dict, zabransky_dicts, _Cp_data_loaded
global Cp_dict_characteristic_temperatures_adjusted_psi4_2022a, Cp_dict_characteristic_temperatures_psi4_2022a
global Cp_dict_JANAF_liquid, Cp_dict_JANAF_gas, Cp_dict_JANAF_solid
Cp_data_Poling = data_source('PolingDatabank.tsv')
TRC_gas_data = data_source('TRC Thermodynamics of Organic Compounds in the Gas State.tsv')
CRC_standard_data = data_source('CRC Standard Thermodynamic Properties of Chemical Substances.tsv')
TRC_gas_values = np.array(TRC_gas_data.values[:, 1:], dtype=float)
Cp_values_Poling = np.array(Cp_data_Poling.values[:, 1:], dtype=float)
# Read in a dict of heat capacities of irnorganic and elemental solids.
# These are in section 2, table 151 in:
# Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
# Eighth Edition. McGraw-Hill Professional, 2007.
# Formula:
# Cp(Cal/mol/K) = Const + Lin*T + Quadinv/T^2 + Quadinv*T^2
# Phases:
# c, gls, l, g.
zabransky_dict_sat_s = {}
zabransky_dict_sat_p = {}
zabransky_dict_const_s = {}
zabransky_dict_const_p = {}
zabransky_dict_iso_s = {}
zabransky_dict_iso_p = {}
# C means average heat capacity values, from less rigorous experiments
# sat means heat capacity along the saturation line
# p means constant-pressure values,
# second argument is whether or not it has a spline
type_to_zabransky_dict = {
('C', True): zabransky_dict_const_s,
('C', False): zabransky_dict_const_p,
('sat', True): zabransky_dict_sat_s,
('sat', False): zabransky_dict_sat_p,
('p', True): zabransky_dict_iso_s,
('p', False): zabransky_dict_iso_p
}
zabransky_dicts = {
ZABRANSKY_SPLINE: zabransky_dict_const_s,
ZABRANSKY_QUASIPOLYNOMIAL: zabransky_dict_const_p,
ZABRANSKY_SPLINE_C: zabransky_dict_iso_s,
ZABRANSKY_QUASIPOLYNOMIAL_C: zabransky_dict_iso_p,
ZABRANSKY_SPLINE_SAT: zabransky_dict_sat_s,
ZABRANSKY_QUASIPOLYNOMIAL_SAT: zabransky_dict_sat_p
}
with open(os.path.join(folder, 'Zabransky.tsv'), encoding='utf-8') as f:
next(f)
for line in f:
values = to_num(line.strip('\n').split('\t'))
(CAS, name, Type, uncertainty, Tmin, Tmax,
a1s, a2s, a3s, a4s, a1p, a2p, a3p, a4p, a5p, a6p, Tc) = values
spline = bool(a1s) # False if Quasypolynomial, True if spline
d = type_to_zabransky_dict[(Type, spline)]
if spline:
coeffs = (a1s, a2s, a3s, a4s)
if CAS not in d:
d[CAS] = [ZabranskySpline(coeffs, Tmin, Tmax)]
else:
d[CAS].append(ZabranskySpline(coeffs, Tmin, Tmax))
else:
# No duplicates for quasipolynomials
coeffs = (a1p, a2p, a3p, a4p, a5p, a6p)
d[CAS] = ZabranskyQuasipolynomial(coeffs, Tc, Tmin, Tmax)
for dct in (zabransky_dict_const_s, zabransky_dict_iso_s, zabransky_dict_sat_s):
for CAS in dct: dct[CAS] = PiecewiseHeatCapacity(dct[CAS])
# Used to generate data. Do not delete!
# Cp_dict_PerryI = {}
# with open(os.path.join(folder, 'Perrys Table 2-151.tsv'), encoding='utf-8') as f:
# '''Read in a dict of heat capacities of irnorganic and elemental solids.
# These are in section 2, table 151 in:
# Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
# Eighth Edition. McGraw-Hill Professional, 2007.
# Formula:
# Cp(Cal/mol/K) = Const + Lin*T + Quadinv/T^2 + Quadinv*T^2
# Phases: c, gls, l, g.
# '''
# next(f)
# for line in f:
# values = to_num(line.strip('\n').split('\t'))
# (CASRN, _formula, _phase, _subphase, Const, Lin, Quadinv, Quad, Tmin,
# Tmax, err) = values
# if Lin is None:
# Lin = 0
# if Quadinv is None:
# Quadinv = 0
# if Quad is None:
# Quad = 0
# if CASRN in _PerryI and CASRN:
# a = _PerryI[CASRN]
# a.update({_phase: {"Formula": _formula, "Phase": _phase,
# "Subphase": _subphase, "Const": Const,
# "Lin": Lin, "Quadinv": Quadinv, "Quad": Quad,
# "Tmin": Tmin, "Tmax": Tmax, "Error": err}})
# _PerryI[CASRN] = a
# else:
# _PerryI[CASRN] = {_phase: {"Formula": _formula, "Phase": _phase,
# "Subphase": _subphase, "Const": Const,
# "Lin": Lin, "Quadinv": Quadinv,
# "Quad": Quad, "Tmin": Tmin,
# "Tmax": Tmax, "Error": err}}
"""
Read in a dict of 2481 thermodynamic property sets of different phases from:
Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of
Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014.
Warning: 11 duplicated chemicals are present and currently clobbered.
"""
import json
with open(os.path.join(folder, 'Perrys Table 2-151.json')) as f:
Cp_dict_PerryI = json.loads(f.read())
with open(os.path.join(folder, 'psi4_unadjusted_characteristic_temperatures.json')) as f:
Cp_dict_characteristic_temperatures_psi4_2022a = json.loads(f.read())
with open(os.path.join(folder, 'psi4_adjusted_characteristic_temperatures.json')) as f:
Cp_dict_characteristic_temperatures_adjusted_psi4_2022a = json.loads(f.read())
with open(os.path.join(folder, 'JANAF_1998_liq_Cp.json')) as f:
Cp_dict_JANAF_liquid = json.loads(f.read())
with open(os.path.join(folder, 'JANAF_1998_gas_Cp.json')) as f:
Cp_dict_JANAF_gas = json.loads(f.read())
with open(os.path.join(folder, 'JANAF_1998_solid_Cp.json')) as f:
Cp_dict_JANAF_solid = json.loads(f.read())
with open(os.path.join(folder, 'webbook_shomate_coefficients.json')) as f:
WebBook_Shomate_coefficients = json.loads(f.read())
WebBook_Shomate_solids, WebBook_Shomate_liquids, WebBook_Shomate_gases = {}, {}, {}
for i, d in zip(range(3), [WebBook_Shomate_solids, WebBook_Shomate_liquids, WebBook_Shomate_gases]):
for CAS, phase_values in WebBook_Shomate_coefficients.items():
Cp_dat = phase_values[i]
if Cp_dat is not None:
if len(Cp_dat) == 1:
d[CAS] = ShomateRange(tuple(Cp_dat[0][2:]), Cp_dat[0][0], Cp_dat[0][1])
else:
phase_models = [ShomateRange(tuple(Cp_dat[i][2:]), Cp_dat[i][0], Cp_dat[i][1]) for i in range(len(Cp_dat))]
d[CAS] = PiecewiseHeatCapacity(phase_models)
_Cp_data_loaded = True
if PY37:
def __getattr__(name):
if name in ('Cp_data_Poling', 'Cp_values_Poling', 'TRC_gas_data', 'TRC_gas_values', 'CRC_standard_data',
'Cp_dict_PerryI', 'zabransky_dict_sat_s', 'zabransky_dict_sat_p',
'zabransky_dict_const_s', 'zabransky_dict_const_p', 'zabransky_dict_iso_s',
'zabransky_dict_iso_p', 'type_to_zabransky_dict', 'zabransky_dicts',
'WebBook_Shomate_liquids', 'WebBook_Shomate_gases', 'WebBook_Shomate_solids',
'WebBook_Shomate_coefficients',
'Cp_dict_JANAF_liquid', 'Cp_dict_JANAF_gas', 'Cp_dict_JANAF_solid'):
_load_Cp_data()
return globals()[name]
raise AttributeError(f"module {__name__} has no attribute {name}")
else:
if can_load_data:
_load_Cp_data()
### Heat capacities of gases
[docs]def Poling(T, a, b, c, d, e):
r"""
Return the ideal-gas molar heat capacity of a chemical using
polynomial regressed coefficients as described by Poling et. al. [1]_.
Parameters
----------
T : float
Temperature, [K]
a,b,c,d,e : float
Regressed coefficients.
Returns
-------
Cpgm : float
Gas molar heat capacity, [J/mol/K]
Notes
-----
The ideal gas heat capacity is given by:
.. math:: C_n = R*(a + bT + cT^2 + dT^3 + eT^4)
The data is based on the Poling data bank.
See Also
--------
Poling_integral
Poling_integral_over_T
Examples
--------
Compute the gas heat capacity of Methane at 300 K:
>>> Poling(T=300., a=4.568, b=-0.008975, c=3.631e-05, d=-3.407e-08, e=1.091e-11)
35.850973388425
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
"""
return R*(T*(T*(T*(T*e + d) + c) + b) + a)
[docs]def Poling_integral(T, a, b, c, d, e):
r"""
Return the integral of the ideal-gas constant-pressure heat capacity
of a chemical using polynomial regressed coefficients as described by
Poling et. al. [1]_.
Parameters
----------
T : float
Temperature, [K]
a,b,c,d,e : float
Regressed coefficients.
Returns
-------
H : float
Difference in enthalpy from 0 K, [J/mol]
Notes
-----
Integral was computed with SymPy.
See Also
--------
Poling
Poling_integral_over_T
Examples
--------
Compute the gas enthalpy of Methane at 300 K (with reference to 0 K):
>>> Poling_integral(T=300., a=4.568, b=-0.008975, c=3.631e-05, d=-3.407e-08, e=1.091e-11)
10223.67533722261
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
"""
return R*(((((0.2*e)*T + 0.25*d)*T + c*(1.0/3.))*T + 0.5*b)*T + a)*T
[docs]def Poling_integral_over_T(T, a, b, c, d, e):
r"""
Return the integral over temperature of the ideal-gas constant-pressure
heat capacity of a chemical using polynomial regressed coefficients as
described by Poling et. al. [1]_.
Parameters
----------
T : float
Temperature, [K]
a,b,c,d,e : float
Regressed coefficients.
Returns
-------
S : float
Difference in entropy from 0 K, [J/mol/K]
Notes
-----
Integral was computed with SymPy.
See Also
--------
Poling
Poling_integral
Examples
--------
Compute the gas entropy of Methane at 300 K (with reference to 0 K):
>>> Poling_integral_over_T(T=300., a=4.568, b=-0.008975, c=3.631e-05, d=-3.407e-08, e=1.091e-11)
205.46526328058
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
"""
return R*(((((0.25*e)*T + d*(1.0/3.))*T + 0.5*c)*T + b)*T + a*log(T))
[docs]def PPDS2(T, Ts, C_low, C_inf, a1, a2, a3, a4, a5):
r'''Calculates the ideal-gas heat capacity using the [1]_
emperical (parameter-regressed) method, called the PPDS 2 equation for
heat capacity.
.. math::
\frac{C_p^0}{R} = C_{low} + (C_\inf - C_{low})y^2\left(1 + (y-1)
\left[\sum_{i=0}^4 a_i y^i\right]\right)
.. math::
y = \frac{T}{T + T_s}
Parameters
----------
T : float
Temperature of fluid [K]
Ts : float
Fit temperature; no physical meaning [K]
C_low : float
Fit parameter equal to Cp/R at a low temperature, [-]
C_inf : float
Fit parameter equal to Cp/R at a high temperature, [-]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
a3 : float
Regression parameter, [-]
a4 : float
Regression parameter, [-]
a5 : float
Regression parameter, [-]
Returns
-------
Cpgm : float
Gas molar heat capacity, [J/mol/K]
Notes
-----
Examples
--------
n-pentane at 350 K from [1]_
>>> PPDS2(T=350.0, Ts=462.493, C_low=4.54115, C_inf=9.96847, a1=-103.419, a2=695.484, a3=-2006.1, a4=2476.84, a5=-1186.47)
136.46338956689
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User`s Guide."
https://trc.nist.gov/TDE/TDE_Help/Eqns-Pure-Cp0/PPDS2Cp0.htm.
'''
y = T/(T + Ts)
tot = a1 + y*(a2 + y*(a3 + y*(a4 + a5*y)))
main = C_low + (C_inf - C_low)*y*y*(1.0 + (y - 1.0)*tot)
return R*main
[docs]def PPDS15(T, Tc, a0, a1, a2, a3, a4, a5):
r'''Calculates the saturation liquid heat capacity using the [1]_
emperical (parameter-regressed) method, called the PPDS 15 equation for
heat capacity.
.. math::
\frac{C_{p,l}}{R} = \frac{a_0}{\tau} + a_1 + a_2\tau + a_3\tau^2
+ a_4\tau^3 + a_5\tau^4
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, [-]
a3 : float
Regression parameter, [-]
a4 : float
Regression parameter, [-]
a5 : float
Regression parameter, [-]
Returns
-------
Cplm : float
Liquid molar saturation heat capacity, [J/mol/K]
Notes
-----
Examples
--------
Benzene at 400 K from [1]_
>>> PPDS15(T=400.0, Tc=562.05, a0=0.198892, a1=24.1389, a2=-20.2301, a3=5.72481, a4=4.43613e-7, a5=-3.10751e-7)
161.8983143509
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User`s Guide."
https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-CsatL/PPDS15-Csat.htm.
'''
tau = 1.0 - T/Tc
poly_term = a1 + tau*(a2 + tau*(a3 + tau*(a4 + a5*tau)))
return R*(a0/tau + poly_term)
[docs]def TDE_CSExpansion(T, Tc, b, a1, a2=0.0, a3=0.0, a4=0.0):
r'''Calculates the saturation liquid heat capacity using the [1]_
CSExpansion method from NIST's TDE:
.. math::
C_{p,l}= \frac{b}{\tau} + a_1 + a_2T + a_3 T^2 + a_4 T^3
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
b : float
Regression parameter, [-]
a1 : float
Regression parameter, [-]
a2 : float
Regression parameter, [-]
a3 : float
Regression parameter, [-]
a4 : float
Regression parameter, [-]
Returns
-------
Cplm : float
Liquid molar saturation heat capacity, [J/mol/K]
Notes
-----
Examples
--------
2-methylquinoline at 550 K from [1]_
>>> TDE_CSExpansion(550.0, 778.0, 0.626549, 120.705, 0.255987, 0.000381027, -3.03077e-7)
328.472042686
References
----------
.. [1] "ThermoData Engine (TDE103b V10.1) User`s Guide."
https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-CsatL/CSExpansion.htm
'''
tau = 1.0 - T/Tc
return b/tau + a1 + T*(a2 + T*(a3 + a4*T))
[docs]def Lastovka_Shaw_term_A(similarity_variable, cyclic_aliphatic):
"""
Return Term A in Lastovka-Shaw equation.
Parameters
----------
similarity_variable : float
Similarity variable as defined in [1]_, [mol/g]
cyclic_aliphatic: bool, optional
Whether or not chemical is cyclic aliphatic, [-]
Returns
-------
term_A : float
Term A in Lastovka-Shaw equation, [J/g]
See Also
--------
Lastovka_Shaw
Lastovka_Shaw_integral
Lastovka_Shaw_integral_over_T
References
----------
.. [1] Lastovka, Vaclav, and John M. Shaw. "Predictive Correlations for
Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions."
Fluid Phase Equilibria 356 (October 25, 2013): 338-370.
doi:10.1016/j.fluid.2013.07.023.
"""
a = similarity_variable
if cyclic_aliphatic:
A1 = -0.1793547
A2 = 3.86944439
term_A = A1 + A2*a
else:
# A1 = 0.58
A2 = 1.25
A1_minus_A2 = -0.67 # (A1 - A2)
A3 = 0.17338003 # 803 instead of 8003 in another paper
# A4 = 0.014
A4_inv = 71.42857142857143 # 1 / A4
term_A = A2 + A1_minus_A2/(1. + exp((a-A3)*A4_inv)) # One reference says exp((a-A3)/A4)
# Personal communication confirms the change
return term_A
[docs]def Lastovka_Shaw(T, similarity_variable, cyclic_aliphatic=False, MW=None, term_A=None):
r'''Calculate ideal-gas constant-pressure heat capacity with the similarity
variable concept and method as shown in [1]_.
.. math::
term_A = A1 + A2*a \text{ if cyclic aliphatic}
.. math::
term_A = \left(A_2 + \frac{A_1 - A_2}{1 + \exp(\frac{\alpha-A_3}{A_4})}\right) \text{ if not cyclic aliphatic}
.. math::
C_p^0 = term_A
+ (B_{11} + B_{12}\alpha)\left(-\frac{(C_{11} + C_{12}\alpha)}{T}\right)^2
\frac{\exp(-(C_{11} + C_{12}\alpha)/T)}{[1-\exp(-(C_{11}+C_{12}\alpha)/T)]^2}
+ (B_{21} + B_{22}\alpha)\left(-\frac{(C_{21} + C_{22}\alpha)}{T}\right)^2
\frac{\exp(-(C_{21} + C_{22}\alpha)/T)}{[1-\exp(-(C_{21}+C_{22}\alpha)/T)]^2}
Parameters
----------
T : float
Temperature of gas [K]
similarity_variable : float
Similarity variable as defined in [1]_, [mol/g]
cyclic_aliphatic: bool, optional
Whether or not chemical is cyclic aliphatic, [-]
MW : float, optional
Molecular weight, [g/mol]
term_A : float, optional
Term A in Lastovka-Shaw equation, [J/g]
Returns
-------
Cpg : float
Gas constant-pressure heat capacity, J/mol/K if MW given;
J/kg/K otherwise
Notes
-----
Original model is in terms of J/g/K.
A1 = -0.1793547 \text{ if cyclic aliphatic}
A1 = 0.58 \text{ if not cyclic aliphatic}
A2 = 3.86944439 \text{ if cyclic aliphatic}
A2 = 1.25 \text{ if not cyclic aliphatic}
A3 = 0.17338003
A4 = 0.014
B11 = 0.73917383
B12 = 8.88308889
C11 = 1188.28051
C12 = 1813.04613
B21 = 0.0483019
B22 = 4.35656721
C21 = 2897.01927
C22 = 5987.80407
Examples
--------
Estimate the heat capacity of n-decane gas in J/kg/K:
>>> Lastovka_Shaw(1000.0, 0.22491)
3730.2807601773725
Estimate the heat capacity of n-decane gas in J/mol/K:
>>> Lastovka_Shaw(1000.0, 0.22491, MW=142.28)
530.7443465580366
References
----------
.. [1] Lastovka, Vaclav, and John M. Shaw. "Predictive Correlations for
Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions."
Fluid Phase Equilibria 356 (October 25, 2013): 338-370.
doi:10.1016/j.fluid.2013.07.023.
'''
a = similarity_variable
if term_A is None: term_A = Lastovka_Shaw_term_A(a, cyclic_aliphatic)
T_inv = 1.0/T
B11 = 0.73917383
B12 = 8.88308889
C11 = 1188.28051
C12 = 1813.04613
B21 = 0.0483019
B22 = 4.35656721
C21 = 2897.01927
C22 = 5987.80407
C11_C12a_T =(C11+C12*a)*T_inv
expm_C11_C12a_T = exp(-C11_C12a_T)
x1 = 1.0/(1.0 - expm_C11_C12a_T)
C21_C22a_T = (C21+C22*a)*T_inv
expm_C21_C22a_T = exp(-C21_C22a_T)
x2 = 1.0/(1.0 - expm_C21_C22a_T)
Cp = term_A + (B11 + B12*a)*(C11_C12a_T*C11_C12a_T)*expm_C11_C12a_T*x1*x1
Cp += (B21 + B22*a)*(C21_C22a_T*C21_C22a_T)*expm_C21_C22a_T*x2*x2
return Cp*1000. if MW is None else Cp*MW
[docs]def Lastovka_Shaw_integral(T, similarity_variable, cyclic_aliphatic=False,
MW=None, term_A=None):
r'''Calculate the integral of ideal-gas constant-pressure heat capacity
with the similarity variable concept and method as shown in [1]_.
Parameters
----------
T : float
Temperature of gas [K]
cyclic_aliphatic: bool, optional
Whether or not chemical is cyclic aliphatic, [-]
MW : float, optional
Molecular weight, [g/mol]
term_A : float, optional
Term A in Lastovka-Shaw equation, [J/g]
Returns
-------
H : float
Difference in enthalpy from 0 K, J/mol if MW given; J/kg otherwise
Notes
-----
Original model is in terms of J/g/K.
Integral was computed with SymPy.
See Also
--------
Lastovka_Shaw
Lastovka_Shaw_integral_over_T
Examples
--------
>>> Lastovka_Shaw_integral(300.0, 0.1333)
5283095.816018478
References
----------
.. [1] Lastovka, Vaclav, and John M. Shaw. "Predictive Correlations for
Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions."
Fluid Phase Equilibria 356 (October 25, 2013): 338-370.
doi:10.1016/j.fluid.2013.07.023.
'''
a = similarity_variable
if term_A is None: term_A = Lastovka_Shaw_term_A(a, cyclic_aliphatic)
B11 = 0.73917383
B12 = 8.88308889
C11 = 1188.28051
C12 = 1813.04613
B21 = 0.0483019
B22 = 4.35656721
C21 = 2897.01927
C22 = 5987.80407
x1 = -C11 - C12*a
x2 = -C21 - C22*a
T_inv = 1.0/T
H = (T*term_A - (B11 + B12*a)*(x1*x1)/(x1 - x1*exp(x1*T_inv))
- (B21 + B22*a)*(x2*x2)/(x2 - x2*exp(x2*T_inv)))
return H*1000. if MW is None else H*MW
[docs]def Lastovka_Shaw_integral_over_T(T, similarity_variable, cyclic_aliphatic=False,
MW=None, term_A=None):
r'''Calculate the integral over temperature of ideal-gas constant-pressure
heat capacity with the similarity variable concept and method as shown in
[1]_.
Parameters
----------
T : float
Temperature of gas [K]
similarity_variable : float
Similarity variable as defined in [1]_, [mol/g]
cyclic_aliphatic: bool, optional
Whether or not chemical is cyclic aliphatic, [-]
MW : float, optional
Molecular weight, [g/mol]
term_A : float, optional
Term A in Lastovka-Shaw equation, [J/g]
Returns
-------
S : float
Difference in entropy from 0 K, [J/mol/K if MW given; J/kg/K otherwise]
Notes
-----
Original model is in terms of J/g/K. Note that the model is for predicting
mass heat capacity, not molar heat capacity like most other methods!
Integral was computed with SymPy.
See Also
--------
Lastovka_Shaw
Lastovka_Shaw_integral
Examples
--------
>>> Lastovka_Shaw_integral_over_T(300.0, 0.1333)
3609.791928945323
References
----------
.. [1] Lastovka, Vaclav, and John M. Shaw. "Predictive Correlations for
Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions."
Fluid Phase Equilibria 356 (October 25, 2013): 338-370.
doi:10.1016/j.fluid.2013.07.023.
'''
a = similarity_variable
if term_A is None: term_A = Lastovka_Shaw_term_A(a, cyclic_aliphatic)
T_inv = 1.0/T
a2 = a*a
B11 = 0.73917383
B12 = 8.88308889
C11 = 1188.28051
C12 = 1813.04613
B21 = 0.0483019
B22 = 4.35656721
C21 = 2897.01927
C22 = 5987.80407
S = (term_A*clog(T) + (-B11 - B12*a)*clog(cexp((-C11 - C12*a)*T_inv) - 1.)
+ (-B11*C11 - B11*C12*a - B12*C11*a - B12*C12*a2)/(T*cexp((-C11
- C12*a)*T_inv) - T) - (B11*C11 + B11*C12*a + B12*C11*a + B12*C12*a2)*T_inv)
S += ((-B21 - B22*a)*clog(cexp((-C21 - C22*a)*T_inv) - 1.) + (-B21*C21 - B21*C22*a
- B22*C21*a - B22*C22*a2)/(T*cexp((-C21 - C22*a)*T_inv) - T) - (B21*C21
+ B21*C22*a + B22*C21*a + B22*C22*a2)*T_inv)
# There is a non-real component, but it is only a function of similariy
# variable and so will always cancel out.
return S.real*1000. if MW is None else S.real*MW
def Lastovka_Shaw_T_for_Hm_err(T, MW, similarity_variable, H_ref, Hm, cyclic_aliphatic, term_A):
H1 = Lastovka_Shaw_integral(T, similarity_variable, cyclic_aliphatic, MW, term_A)
dH = H1 - H_ref
err = (dH - Hm)
return err
[docs]@mark_numba_uncacheable
def Lastovka_Shaw_T_for_Hm(Hm, MW, similarity_variable, T_ref=298.15,
factor=1.0, cyclic_aliphatic=None, term_A=None):
r'''Uses the Lastovka-Shaw ideal-gas heat capacity correlation to solve for
the temperature which has a specified `Hm`, as is required in PH flashes,
as shown in [1]_.
Parameters
----------
Hm : float
Molar enthalpy spec, [J/mol]
MW : float
Molecular weight of the pure compound or mixture average, [g/mol]
similarity_variable : float
Similarity variable as defined in [1]_, [mol/g]
T_ref : float, optional
Reference enthlapy temperature, [K]
factor : float, optional
A factor to increase or decrease the predicted value of the
method, [-]
cyclic_aliphatic: bool, optional
Whether or not chemical is cyclic aliphatic, [-]
term_A : float, optional
Term A in Lastovka-Shaw equation, [J/g]
Returns
-------
T : float
Temperature of gas to meet the molar enthalpy spec, [K]
Notes
-----
See Also
--------
Lastovka_Shaw
Lastovka_Shaw_integral
Lastovka_Shaw_integral_over_T
Examples
--------
>>> Lastovka_Shaw_T_for_Hm(Hm=55000, MW=80.0, similarity_variable=0.23)
600.0943429567602
References
----------
.. [1] Lastovka, Vaclav, and John M. Shaw. "Predictive Correlations for
Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions."
Fluid Phase Equilibria 356 (October 25, 2013): 338-370.
doi:10.1016/j.fluid.2013.07.023.
'''
Hm /= factor
a = similarity_variable
if term_A is None: term_A = Lastovka_Shaw_term_A(a, cyclic_aliphatic)
H_ref = Lastovka_Shaw_integral(T_ref, similarity_variable, cyclic_aliphatic, MW, term_A)
args = (MW, a, H_ref, Hm, cyclic_aliphatic, term_A)
try:
return secant(Lastovka_Shaw_T_for_Hm_err, 500.0, ytol=1e-4, args=args)
except:
try:
return brenth(Lastovka_Shaw_T_for_Hm_err, 1e-3, 1e5, args)
except:
if Lastovka_Shaw_T_for_Hm_err(1e-11, *args) > 0:
raise ValueError("For gas only enthalpy spec to be correct, "
"model requires negative temperature")
raise ValueError("Could not converge")
def Lastovka_Shaw_T_for_Sm_err(T, MW, similarity_variable, S_ref, Sm, cyclic_aliphatic, term_A):
S1 = Lastovka_Shaw_integral_over_T(T, similarity_variable, cyclic_aliphatic, MW, term_A)
dS = S1 - S_ref
err = (dS - Sm)
return err
[docs]@mark_numba_uncacheable
def Lastovka_Shaw_T_for_Sm(Sm, MW, similarity_variable, T_ref=298.15,
factor=1.0, cyclic_aliphatic=None, term_A=None):
r'''Uses the Lastovka-Shaw ideal-gas heat capacity correlation to solve for
the temperature which has a specified `Sm`, as is required in PS flashes,
as shown in [1]_.
Parameters
----------
Sm : float
Molar entropy spec, [J/mol/K]
MW : float
Molecular weight of the pure compound or mixture average, [g/mol]
similarity_variable : float
Similarity variable as defined in [1]_, [mol/g]
T_ref : float, optional
Reference enthlapy temperature, [K]
factor : float, optional
A factor to increase or decrease the predicted value of the
method, [-]
cyclic_aliphatic: bool, optional
Whether or not chemical is cyclic aliphatic, [-]
term_A : float, optional
Term A in Lastovka-Shaw equation, [J/g]
Returns
-------
T : float
Temperature of gas to meet the molar entropy spec, [K]
Notes
-----
See Also
--------
Lastovka_Shaw
Lastovka_Shaw_integral
Lastovka_Shaw_integral_over_T
Examples
--------
>>> Lastovka_Shaw_T_for_Sm(Sm=112.80, MW=72.151, similarity_variable=0.2356)
603.4298291570276
References
----------
.. [1] Lastovka, Vaclav, and John M. Shaw. "Predictive Correlations for
Ideal Gas Heat Capacities of Pure Hydrocarbons and Petroleum Fractions."
Fluid Phase Equilibria 356 (October 25, 2013): 338-370.
doi:10.1016/j.fluid.2013.07.023.
'''
Sm /= factor
a = similarity_variable
if term_A is None: term_A = Lastovka_Shaw_term_A(a, cyclic_aliphatic)
S_ref = Lastovka_Shaw_integral_over_T(T_ref, a, cyclic_aliphatic, MW, term_A)
args = (MW, a, S_ref, Sm, cyclic_aliphatic, term_A)
try:
return secant(Lastovka_Shaw_T_for_Sm_err, 500, ytol=1e-4, high=10000,
args=args)
except:
try:
return brenth(Lastovka_Shaw_T_for_Sm_err, 1e-3, 1e5,
args=args)
except:
if Lastovka_Shaw_T_for_Sm_err(1e-11, *args) > 0.0:
raise ValueError("For gas only entropy spec to be correct, "
"model requires negative temperature")
raise ValueError("Could not converge")
[docs]def TRCCp(T, a0, a1, a2, a3, a4, a5, a6, a7):
r'''Calculates ideal gas heat capacity using the model developed in [1]_.
The ideal gas heat capacity is given by:
.. math::
C_p = R\left(a_0 + (a_1/T^2) \exp(-a_2/T) + a_3 y^2
+ (a_4 - a_5/(T-a_7)^2 )y^j \right)
.. math::
y = \frac{T-a_7}{T+a_6} \text{ for } T > a_7 \text{ otherwise } 0
Parameters
----------
T : float
Temperature [K]
a1-a7 : float
Coefficients
Returns
-------
Cp : float
Ideal gas heat capacity , [J/mol/K]
Notes
-----
j is set to 8. Analytical integrals are available for this expression.
Examples
--------
>>> TRCCp(300, 4.0, 7.65E5, 720., 3.565, -0.052, -1.55E6, 52., 201.)
42.065271080974654
References
----------
.. [1] Kabo, G. J., and G. N. Roganov. Thermodynamics of Organic Compounds
in the Gas State, Volume II: V. 2. College Station, Tex: CRC Press, 1994.
'''
if T <= a7:
y = 0.
else:
y = (T - a7)/(T + a6)
T_inv = 1.0/T
y2 = y*y
T_m_a7 = T - a7
if T == a7:
# When T_m_a7 approaches 0, T = a7
Cp = R*(a0 + (a1*T_inv*T_inv)*exp(-a2*T_inv) + y2*(a3 + (a4)*y2*y2*y2))
else:
Cp = R*(a0 + (a1*T_inv*T_inv)*exp(-a2*T_inv) + y2*(a3 + (a4 - a5/(T_m_a7*T_m_a7))*y2*y2*y2))
return Cp
[docs]def TRCCp_integral(T, a0, a1, a2, a3, a4, a5, a6, a7, I=0):
r'''Integrates ideal gas heat capacity using the model developed in [1]_.
Best used as a delta only.
The difference in enthalpy with respect to 0 K is given by:
.. math::
\frac{H(T) - H^{ref}}{RT} = a_0 + a_1x(a_2)/(a_2T) + I/T + h(T)/T
.. math::
h(T) = (a_5 + a_7)\left[(2a_3 + 8a_4)\ln(1-y)+ \left\{a_3\left(1 +
\frac{1}{1-y}\right) + a_4\left(7 + \frac{1}{1-y}\right)\right\}y
+ a_4\left\{3y^2 + (5/3)y^3 + y^4 + (3/5)y^5 + (1/3)y^6\right\}
+ (1/7)\left\{a_4 - \frac{a_5}{(a_6+a_7)^2}\right\}y^7\right]
.. math::
h(T) = 0 \text{ for } T \le a_7
y = \frac{T-a_7}{T+a_6} \text{ for } T > a_7 \text{ otherwise } 0
Parameters
----------
T : float
Temperature [K]
a1-a7 : float
Coefficients
I : float, optional
Integral offset
Returns
-------
H-H(0) : float
Difference in enthalpy from 0 K , [J/mol]
Notes
-----
Analytical integral as provided in [1]_ and verified with numerical
integration.
Examples
--------
>>> TRCCp_integral(298.15, 4.0, 7.65E5, 720., 3.565, -0.052, -1.55E6, 52.,
... 201., 1.2)
10802.536262068483
References
----------
.. [1] Kabo, G. J., and G. N. Roganov. Thermodynamics of Organic Compounds
in the Gas State, Volume II: V. 2. College Station, Tex: CRC Press, 1994.
'''
if T <= a7:
y = 0.
else:
y = (T - a7)/(T + a6)
y2 = y*y
y4 = y2*y2
if T <= a7:
h = 0.0
else:
first = a6 + a7
one_m_y = 1.0 - y
second = (2.*a3 + 8.*a4)*log(one_m_y)
third = (a3*(1. + 1./(one_m_y)) + a4*(7. + 1./(one_m_y)))*y
fourth = a4*(3.*y2 + 5./3.*y*y2 + y4 + 0.6*y4*y + 1/3.*y4*y2)
fifth = 1/7.*(a4 - a5/(first*first))*y4*y2*y
h = first*(second + third + fourth + fifth)
return (a0 + a1*exp(-a2/T)/(a2*T) + I/T + h/T)*R*T
[docs]def TRCCp_integral_over_T(T, a0, a1, a2, a3, a4, a5, a6, a7, J=0):
r'''Integrates ideal gas heat capacity over T using the model developed in
[1]_. Best used as a delta only.
The difference in ideal-gas entropy with respect to 0 K is given by:
.. math::
\frac{S^\circ}{R} = J + a_0\ln T + \frac{a_1}{a_2^2}\left(1
+ \frac{a_2}{T}\right)x(a_2) + s(T)
s(T) = \left[\left\{a_3 + \left(\frac{a_4 a_7^2 - a_5}{a_6^2}\right)
\left(\frac{a_7}{a_6}\right)^4\right\}\left(\frac{a_7}{a_6}\right)^2
\ln z + (a_3 + a_4)\ln\left(\frac{T+a_6}{a_6+a_7}\right)
+\sum_{i=1}^7 \left\{\left(\frac{a_4 a_7^2 - a_5}{a_6^2}\right)\left(
\frac{-a_7}{a_6}\right)^{6-i} - a_4\right\}\frac{y^i}{i}
- \left\{\frac{a_3}{a_6}(a_6 + a_7) + \frac{a_5 y^6}{7a_7(a_6+a_7)}
\right\}y\right]
.. math::
s(T) = 0 \text{ for } T \le a_7
.. math::
z = \frac{T}{T+a_6} \cdot \frac{a_7 + a_6}{a_7}
.. math::
y = \frac{T-a_7}{T+a_6} \text{ for } T > a_7 \text{ otherwise } 0
Parameters
----------
T : float
Temperature [K]
a1-a7 : float
Coefficients
J : float, optional
Integral offset
Returns
-------
S-S(0) : float
Difference in entropy from 0 K , [J/mol/K]
Notes
-----
Analytical integral as provided in [1]_ and verified with numerical
integration.
Examples
--------
>>> TRCCp_integral_over_T(300, 4.0, 124000, 245, 50.539, -49.469,
... 220440000, 560, 78)
213.80156219151888
References
----------
.. [1] Kabo, G. J., and G. N. Roganov. Thermodynamics of Organic Compounds
in the Gas State, Volume II: V. 2. College Station, Tex: CRC Press, 1994.
'''
# Possible optimizations: pre-cache as much as possible.
# If this were replaced by a cache, much of this would not need to be computed.
if T <= a7:
y = 0.
else:
y = (T - a7)/(T + a6)
a6_inv = 1.0/a6
x3 = a7 + a6
z = T*x3/(a7*(T + a6))
if T <= a7:
s = 0.
else:
a72 = a7*a7
a62_inv = a6_inv*a6_inv
a7_a6 = a7*a6_inv # a7/a6
a7_a6_2 = a7_a6*a7_a6
a7_a6_4 = a7_a6_2*a7_a6_2
x1 = (a4*a72 - a5)*a62_inv # part of third, sum
first = (a3 + ((a4*a72 - a5)*a62_inv)*a7_a6_4)*a7_a6_2*log(z)
second = (a3 + a4)*log((T + a6)/(x3))
third = 0.0
y_pow = 1.0
a7_a6_pow = a7_a6_2*a7_a6_4
na7_a6_inv = -1.0/a7_a6
for i in range(1, 8):
y_pow = y_pow*y
a7_a6_pow *= na7_a6_inv
third += (x1*a7_a6_pow - a4)*y_pow/i
fourth = -(a3*a6_inv*x3 + a5*y_pow/(7.0*y*a7*x3))*y
s = first + second + third + fourth
x2 = a2/T
return R*(J + a0*log(T) + a1/(a2*a2)*(1. + x2)*exp(-x2) + s)
[docs]def Shomate(T, A, B, C, D, E):
r'''Calculates heat capacity using the Shomate polynomial model [1]_.
The heat capacity is given by:
.. math::
C_p = A + BT + CT^2 + DT^3 + \frac{E}{T^2}
Parameters
----------
T : float
Temperature [K]
A : float
Parameter, [J/(mol*K)]
B : float
Parameter, [J/(mol*K^2)]
C : float
Parameter, [J/(mol*K^3)]
D : float
Parameter, [J/(mol*K^4)]
E : float
Parameter, [J*K/(mol)]
Returns
-------
Cp : float
Heat capacity , [J/mol/K]
Notes
-----
Analytical integrals are available for this expression. In some sources
such as [1]_, the equation is written with temperature in units of
kilokelvin. The coefficients can be easily adjusted to be in the proper
SI form.
Examples
--------
Coefficients for water vapor from [1]_:
>>> water_low_gas_coeffs = [30.09200, 6.832514/1e3, 6.793435/1e6, -2.534480/1e9, 0.082139*1e6]
>>> Shomate(500, *water_low_gas_coeffs)
35.21836175
References
----------
.. [1] Shen, V.K., Siderius, D.W., Krekelberg, W.P., and Hatch, H.W., Eds.,
NIST WebBook, NIST, http://doi.org/10.18434/T4M88Q
'''
return A + T*(B + T*(C + D*T)) + E/(T*T)
[docs]def Shomate_integral(T, A, B, C, D, E):
r'''Calculates the enthalpy integral using the Shomate polynomial model [1]_.
The difference in enthalpy with respect to 0 K is given by:
.. math::
{H(T) - H^{0}} = A T + \frac{B T^{2}}{2} + \frac{C T^{3}}{3}
+ \frac{D T^{4}}{4} - \frac{E}{T}
Parameters
----------
T : float
Temperature [K]
A : float
Parameter, [J/(mol*K)]
B : float
Parameter, [J/(mol*K^2)]
C : float
Parameter, [J/(mol*K^3)]
D : float
Parameter, [J/(mol*K^4)]
E : float
Parameter, [J*K/(mol)]
Returns
-------
H-H(0) : float
Difference in enthalpy from 0 K , [J/mol]
Notes
-----
Examples
--------
Coefficients for water vapor from [1]_:
>>> water_low_gas_coeffs = [30.09200, 6.832514/1e3, 6.793435/1e6, -2.534480/1e9, 0.082139*1e6]
>>> Shomate_integral(500, *water_low_gas_coeffs)
15979.2447
References
----------
.. [1] Shen, V.K., Siderius, D.W., Krekelberg, W.P., and Hatch, H.W., Eds.,
NIST WebBook, NIST, http://doi.org/10.18434/T4M88Q
'''
return T*(A + T*(B*0.5 + T*(C*(1.0/3.0) + D*T*0.25))) - E/T
[docs]def Shomate_integral_over_T(T, A, B, C, D, E):
r'''Integrates the heat capacity over T using the model developed in
[1]_.
The difference in entropy with respect to 0 K is given by:
.. math::
s(T) = A \log{\left(T \right)} + B T + \frac{C T^{2}}{2}
+ \frac{D T^{3}}{3} - \frac{E}{2 T^{2}}
Parameters
----------
T : float
Temperature [K]
A : float
Parameter, [J/(mol*K)]
B : float
Parameter, [J/(mol*K^2)]
C : float
Parameter, [J/(mol*K^3)]
D : float
Parameter, [J/(mol*K^4)]
E : float
Parameter, [J*K/(mol)]
Returns
-------
S-S(0) : float
Difference in entropy from 0 K , [J/mol/K]
Notes
-----
Examples
--------
Coefficients for water vapor from [1]_:
>>> water_low_gas_coeffs = [30.09200, 6.832514/1e3, 6.793435/1e6, -2.534480/1e9, 0.082139*1e6]
>>> Shomate_integral_over_T(500, *water_low_gas_coeffs)
191.00554
References
----------
.. [1] Shen, V.K., Siderius, D.W., Krekelberg, W.P., and Hatch, H.W., Eds.,
NIST WebBook, NIST, http://doi.org/10.18434/T4M88Q
'''
T2 = T*T
return A*log(T) + B*T + 0.5*C*T2 + D*T*T2*(1.0/3.0) - E/T2*0.5
### Heat capacities of liquids
[docs]def Rowlinson_Poling(T, Tc, omega, Cpgm):
r'''Calculate liquid constant-pressure heat capacity with the [1]_ CSP method.
This equation is not terrible accurate.
The heat capacity of a liquid is given by:
.. math::
\frac{Cp^{L} - Cp^{g}}{R} = 1.586 + \frac{0.49}{1-T_r} +
\omega\left[ 4.2775 + \frac{6.3(1-T_r)^{1/3}}{T_r} + \frac{0.4355}{1-T_r}\right]
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
omega : float
Acentric factor for fluid, [-]
Cpgm : float
Constant-pressure gas heat capacity, [J/mol/K]
Returns
-------
Cplm : float
Liquid constant-pressure heat capacity, [J/mol/K]
Notes
-----
Poling compared 212 substances, and found error at 298K larger than 10%
for 18 of them, mostly associating. Of the other 194 compounds, AARD is 2.5%.
Examples
--------
>>> Rowlinson_Poling(350.0, 435.5, 0.203, 91.21)
143.80196224081436
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
'''
Tr = T/Tc
one_minus_Tr = 1. - Tr
Cplm = Cpgm+ R*(1.586 + 0.49/one_minus_Tr + omega*(4.2775 + 6.3*one_minus_Tr**(1/3.)/Tr + 0.4355/one_minus_Tr))
return Cplm
[docs]def Rowlinson_Bondi(T, Tc, omega, Cpgm):
r'''Calculate liquid constant-pressure heat capacity with the CSP method
shown in [1]_.
The heat capacity of a liquid is given by:
.. math::
\frac{Cp^L - Cp^{ig}}{R} = 1.45 + 0.45(1-T_r)^{-1} + 0.25\omega
[17.11 + 25.2(1-T_r)^{1/3}T_r^{-1} + 1.742(1-T_r)^{-1}]
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
omega : float
Acentric factor for fluid, [-]
Cpgm : float
Constant-pressure gas heat capacity, [J/mol/K]
Returns
-------
Cplm : float
Liquid constant-pressure heat capacity, [J/mol/K]
Notes
-----
Less accurate than `Rowlinson_Poling`.
Examples
--------
>>> Rowlinson_Bondi(T=373.28, Tc=535.55, omega=0.323, Cpgm=119.342)
175.3976263003074
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
.. [2] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
Berlin; New York:: Springer, 2010.
.. [3] J.S. Rowlinson, Liquids and Liquid Mixtures, 2nd Ed.,
Butterworth, London (1969).
'''
Tr = T/Tc
one_minus_Tr = 1. - Tr
Cplm = Cpgm + R*(1.45 + 0.45/(one_minus_Tr) + 0.25*omega*(17.11 + 25.2*(one_minus_Tr)**(1/3.)/Tr + 1.742/one_minus_Tr))
return Cplm
[docs]def Dadgostar_Shaw_terms(similarity_variable):
"""
Return terms for the computation of Dadgostar-Shaw heat capacity equation.
Parameters
----------
similarity_variable : float
Similarity variable, [mol/g]
Returns
-------
first : float
First term, [-]
second : float
Second term, [-]
third : float
Third term, [-]
See Also
--------
Dadgostar_Shaw
"""
a = similarity_variable
a2 = a*a
a11 = -0.3416
a12 = 2.2671
a21 = 0.1064
a22 = -0.3874
a31 = -9.8231E-05
a32 = 4.182E-04
# Didn't seem to improve the comparison; sum of errors on some
# points included went from 65.5 to 286.
# Author probably used more precision in their calculation.
# constant = 3*R*(theta/T)**2*exp(theta/T)/(exp(theta/T)-1)**2
constant = 24.5
return (constant * (a11*a + a12*a2),
a21*a + a22*a2,
a31*a + a32*a2)
[docs]def Dadgostar_Shaw(T, similarity_variable, MW=None, terms=None):
r'''Calculate liquid constant-pressure heat capacity with the similarity
variable concept and method as shown in [1]_.
.. math::
C_{p} = 24.5(a_{11}\alpha + a_{12}\alpha^2)+ (a_{21}\alpha
+ a_{22}\alpha^2)T +(a_{31}\alpha + a_{32}\alpha^2)T^2
Parameters
----------
T : float
Temperature of liquid [K]
similarity_variable : float
similarity variable as defined in [1]_, [mol/g]
MW : float, optional
Molecular weight of the pure compound or mixture average, [g/mol]
terms : float, optional
Terms in Dadgostar-Shaw equation as computed by :obj:`Dadgostar_Shaw_terms`
Returns
-------
Cpl : float
Liquid constant-pressure heat capacity, J/mol/K if MW given;
J/kg/K otherwise
Notes
-----
Many restrictions on its use.
Original model is in terms of J/g/K. Note that the model is for predicting
mass heat capacity, not molar heat capacity like most other methods!
a11 = -0.3416; a12 = 2.2671; a21 = 0.1064; a22 = -0.3874l;
a31 = -9.8231E-05; a32 = 4.182E-04
Examples
--------
>>> Dadgostar_Shaw(355.6, 0.139)
1802.5291501191516
References
----------
.. [1] Dadgostar, Nafiseh, and John M. Shaw. "A Predictive Correlation for
the Constant-Pressure Specific Heat Capacity of Pure and Ill-Defined
Liquid Hydrocarbons." Fluid Phase Equilibria 313 (January 15, 2012):
211-226. doi:10.1016/j.fluid.2011.09.015.
'''
first, second, third = terms or Dadgostar_Shaw_terms(similarity_variable)
Cp = (first + second*T + third*T**2)
return Cp*1000. if MW is None else Cp*MW
[docs]def Dadgostar_Shaw_integral(T, similarity_variable, MW=None, terms=None):
r'''Calculate the integral of liquid constant-pressure heat capacity
with the similarity variable concept and method as shown in [1]_.
Parameters
----------
T : float
Temperature of gas [K]
similarity_variable : float
similarity variable as defined in [1]_, [mol/g]
MW : float, optional
Molecular weight of the pure compound or mixture average, [g/mol]
terms : float, optional
Terms in Dadgostar-Shaw equation as computed by :obj:`Dadgostar_Shaw_terms`
Returns
-------
H : float
Difference in enthalpy from 0 K, J/mol if MW given; J/kg otherwise
Notes
-----
Original model is in terms of J/g/K. Note that the model is for predicting
mass heat capacity, not molar heat capacity like most other methods!
Integral was computed with SymPy.
See Also
--------
Dadgostar_Shaw
Dadgostar_Shaw_integral_over_T
Examples
--------
>>> Dadgostar_Shaw_integral(300.0, 0.1333)
238908.15142664989
References
----------
.. [1] Dadgostar, Nafiseh, and John M. Shaw. "A Predictive Correlation for
the Constant-Pressure Specific Heat Capacity of Pure and Ill-Defined
Liquid Hydrocarbons." Fluid Phase Equilibria 313 (January 15, 2012):
211-226. doi:10.1016/j.fluid.2011.09.015.
'''
T2 = T*T
first, second, third = terms or Dadgostar_Shaw_terms(similarity_variable)
H = T2*T/3.*third + T2*0.5*second + T*first
return H*1000. if MW is None else H*MW
[docs]def Dadgostar_Shaw_integral_over_T(T, similarity_variable, MW=None, terms=None):
r'''Calculate the integral of liquid constant-pressure heat capacity
with the similarity variable concept and method as shown in [1]_.
Parameters
----------
T : float
Temperature of gas [K]
similarity_variable : float
similarity variable as defined in [1]_, [mol/g]
MW : float, optional
Molecular weight of the pure compound or mixture average, [g/mol]
terms : float, optional
Terms in Dadgostar-Shaw equation as computed by :obj:`Dadgostar_Shaw_terms`
Returns
-------
S : float
Difference in entropy from 0 K, J/mol/K if MW given; J/kg/K otherwise
Notes
-----
Original model is in terms of J/g/K. Note that the model is for predicting
mass heat capacity, not molar heat capacity like most other methods!
Integral was computed with SymPy.
See Also
--------
Dadgostar_Shaw
Dadgostar_Shaw_integral
Examples
--------
>>> Dadgostar_Shaw_integral_over_T(300.0, 0.1333)
1201.1409113147918
References
----------
.. [1] Dadgostar, Nafiseh, and John M. Shaw. "A Predictive Correlation for
the Constant-Pressure Specific Heat Capacity of Pure and Ill-Defined
Liquid Hydrocarbons." Fluid Phase Equilibria 313 (January 15, 2012):
211-226. doi:10.1016/j.fluid.2011.09.015.
'''
first, second, third = terms or Dadgostar_Shaw_terms(similarity_variable)
S = T*T*0.5*third + T*second + first*log(T)
return S*1000. if MW is None else S*MW
[docs]def Zabransky_quasi_polynomial(T, Tc, a1, a2, a3, a4, a5, a6):
r'''Calculates liquid heat capacity using the model developed in [1]_.
.. math::
\frac{C}{R}=A_1\ln(1-T_r) + \frac{A_2}{1-T_r}
+ \sum_{j=0}^m A_{j+3} T_r^j
Parameters
----------
T : float
Temperature [K]
Tc : float
Critical temperature of fluid, [K]
a1-a6 : float
Coefficients
Returns
-------
Cp : float
Liquid heat capacity, [J/mol/K]
Notes
-----
Used only for isobaric heat capacities, not saturation heat capacities.
Designed for reasonable extrapolation behavior caused by using the reduced
critical temperature. Used by the authors of [1]_ when critical temperature
was available for the fluid.
Analytical integrals are available for this expression.
Examples
--------
>>> Zabransky_quasi_polynomial(330, 591.79, -3.12743, 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
165.472878778683
References
----------
.. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
'''
Tr = T/Tc
if Tr >= 1.0:
Tr = (1-1e-15) # TODO limitd
return R*(a1*log(1.0-Tr) + a2/(1.0-Tr) + a3 + Tr*(Tr*(Tr*a6 + a5) + a4))
[docs]def Zabransky_quasi_polynomial_integral(T, Tc, a1, a2, a3, a4, a5, a6):
r'''Calculates the integral of liquid heat capacity using the
quasi-polynomial model developed in [1]_.
Parameters
----------
T : float
Temperature [K]
a1-a6 : float
Coefficients
Returns
-------
H : float
Difference in enthalpy from 0 K, [J/mol]
Notes
-----
The analytical integral was derived with SymPy; it is a simple polynomial
plus some logarithms.
Examples
--------
>>> H2 = Zabransky_quasi_polynomial_integral(300, 591.79, -3.12743,
... 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
>>> H1 = Zabransky_quasi_polynomial_integral(200, 591.79, -3.12743,
... 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
>>> H2 - H1
14662.031376528757
References
----------
.. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
'''
Tc2 = Tc*Tc
Tc3 = Tc2*Tc
term = T - Tc
return R*(T*(T*(T*(T*a6/(4.*Tc3) + a5/(3.*Tc2)) + a4/(2.*Tc)) - a1 + a3)
+ T*a1*log(1. - T/Tc) - 0.5*Tc*(a1 + a2)*log(term*term))
[docs]def Zabransky_quasi_polynomial_integral_over_T(T, Tc, a1, a2, a3, a4, a5, a6):
r'''Calculates the integral of liquid heat capacity over T using the
quasi-polynomial model developed in [1]_.
Parameters
----------
T : float
Temperature [K]
a1-a6 : float
Coefficients
Returns
-------
S : float
Difference in entropy from 0 K, [J/mol/K]
Notes
-----
The analytical integral was derived with Sympy. It requires the
Polylog(2,x) function, which is unimplemented in SciPy. A very accurate
numerical approximation was implemented as :obj:`fluids.numerics.polylog2`.
Relatively slow due to the use of that special function.
Examples
--------
>>> S2 = Zabransky_quasi_polynomial_integral_over_T(300, 591.79, -3.12743,
... 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
>>> S1 = Zabransky_quasi_polynomial_integral_over_T(200, 591.79, -3.12743,
... 0.0857315, 13.7282, 1.28971, 6.42297, 4.10989)
>>> S2 - S1
59.16999297436473
References
----------
.. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
'''
term = T - Tc
logT = log(T)
Tc2 = Tc*Tc
Tc3 = Tc2*Tc
return R*(a3*logT -a1*polylog2(T/Tc) - a2*(-logT + 0.5*log(term*term))
+ T*(T*(T*a6/(3.*Tc3) + a5/(2.*Tc2)) + a4/Tc))
[docs]def Zabransky_cubic(T, a1, a2, a3, a4):
r'''Calculates liquid heat capacity using the model developed in [1]_.
.. math::
\frac{C}{R}=\sum_{j=0}^3 A_{j+1} \left(\frac{T}{100 \text{K}}\right)^j
Parameters
----------
T : float
Temperature [K]
a1 : float
Coefficient, [-]
a2 : float
Coefficient, [-]
a3 : float
Coefficient, [-]
a4 : float
Coefficient, [-]
Returns
-------
Cp : float
Liquid heat capacity, [J/mol/K]
Notes
-----
Most often form used in [1]_.
Analytical integrals are available for this expression.
Examples
--------
>>> Zabransky_cubic(298.15, 20.9634, -10.1344, 2.8253, -0.256738)
75.31465144297
References
----------
.. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
'''
T = T*1e-2
return R*(((a4*T + a3)*T + a2)*T + a1)
[docs]def Zabransky_cubic_integral(T, a1, a2, a3, a4):
r'''Calculates the integral of liquid heat capacity using the model
developed in [1]_.
Parameters
----------
T : float
Temperature [K]
a1 : float
Coefficient, [-]
a2 : float
Coefficient, [-]
a3 : float
Coefficient, [-]
a4 : float
Coefficient, [-]
Returns
-------
H : float
Difference in enthalpy from 0 K, [J/mol]
Notes
-----
The analytical integral was derived with Sympy; it is a simple polynomial.
Examples
--------
>>> Zabransky_cubic_integral(298.15, 20.9634, -10.1344, 2.8253, -0.256738)
31051.690370364
References
----------
.. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
'''
T = T*1e-2
return 100.0*R*T*(T*(T*(T*a4*0.25 + a3*(1.0/3.)) + a2*0.5) + a1)
[docs]def Zabransky_cubic_integral_over_T(T, a1, a2, a3, a4):
r'''Calculates the integral of liquid heat capacity over T using the model
developed in [1]_.
Parameters
----------
T : float
Temperature [K]
a1 : float
Coefficient, [-]
a2 : float
Coefficient, [-]
a3 : float
Coefficient, [-]
a4 : float
Coefficient, [-]
Returns
-------
S : float
Difference in entropy from 0 K, [J/mol/K]
Notes
-----
The analytical integral was derived with Sympy; it is a simple polynomial,
plus a logarithm
Examples
--------
>>> Zabransky_cubic_integral_over_T(298.15, 20.9634, -10.1344, 2.8253,
... -0.256738)
24.732465342840
References
----------
.. [1] Zabransky, M., V. Ruzicka Jr, V. Majer, and Eugene S. Domalski.
Heat Capacity of Liquids: Critical Review and Recommended Values.
2 Volume Set. Washington, D.C.: Amer Inst of Physics, 1996.
'''
T = T*1e-2
return R*(T*(T*(T*a4/3.0 + 0.5*a3) + a2) + a1*log(T))
### Solid
[docs]def Perry_151(T, a, b, c, d):
r"""
Return the solid molar heat capacity of a chemical using the Perry 151 method,
as described in [1]_.
Parameters
----------
a,b,c,d : float
Regressed coefficients.
Returns
-------
Cps : float
Solid constant-pressure heat capacity, [J/mol/K]
Notes
-----
The solid heat capacity is given by:
.. math:: C_n = 4.184 (a + bT + \frac{c}{T^2} + dT^2)
Coefficients are listed in section 2, table 151 of [1]_. Note that the
original model was in a Calorie basis, but has been translated to Joules.
Examples
--------
Heat capacity of solid aluminum at 300 K:
>>> Perry_151(300, 4.8, 0.00322, 0., 0.)
24.124944
References
----------
.. [1] Green, Don, and Robert Perry.
Perry's Chemical Engineers' Handbook,
Eighth Edition. McGraw-Hill Professional, 2007.
"""
T2 = T*T
return (a + b*T + c/T2 + d*T2) * 4.184
[docs]def Lastovka_solid(T, similarity_variable, MW=None):
r'''Calculate solid constant-pressure heat capacity with the similarity
variable concept and method as shown in [1]_.
.. math::
C_p = 3(A_1\alpha + A_2\alpha^2)R\left(\frac{\theta}{T}\right)^2
\frac{\exp(\theta/T)}{[\exp(\theta/T)-1]^2}
+ (C_1\alpha + C_2\alpha^2)T + (D_1\alpha + D_2\alpha^2)T^2
Parameters
----------
T : float
Temperature of solid [K]
similarity_variable : float
similarity variable as defined in [1]_, [mol/g]
MW : float, optional
Molecular weight of the pure compound or mixture average, [g/mol]
Returns
-------
Cps : float
Solid constant-pressure heat capacity, J/mol/K if MW given; J/kg/K
otherwise
Notes
-----
Many restrictions on its use. Trained on data with MW from 12.24 g/mol
to 402.4 g/mol, C mass fractions from 61.3% to 95.2%,
H mass fractions from 3.73% to 15.2%, N mass fractions from 0 to 15.4%,
O mass fractions from 0 to 18.8%, and S mass fractions from 0 to 29.6%.
Recommended for organic compounds with low mass fractions of hetero-atoms
and especially when molar mass exceeds 200 g/mol. This model does not show
and effects of phase transition but should not be used passed the triple
point.
Original model is in terms of J/g/K. Note that the model s for predicting
mass heat capacity, not molar heat capacity like most other methods!
A1 = 0.013183
A2 = 0.249381
:math:`\theta` = 151.8675
C1 = 0.026526
C2 = -0.024942
D1 = 0.000025
D2 = -0.000123
Examples
--------
>>> Lastovka_solid(300, 0.2139)
1682.0637469909211
References
----------
.. [1] Laštovka, Václav, Michal Fulem, Mildred Becerra, and John M. Shaw.
"A Similarity Variable for Estimating the Heat Capacity of Solid Organic
Compounds: Part II. Application: Heat Capacity Calculation for
Ill-Defined Organic Solids." Fluid Phase Equilibria 268, no. 1-2
(June 25, 2008): 134-41. doi:10.1016/j.fluid.2008.03.018.
'''
A1 = 0.013183
A2 = 0.249381
theta = 151.8675
C1 = 0.026526
C2 = -0.024942
D1 = 0.000025
D2 = -0.000123
theta_div_T = theta/T
exp_term = exp(theta_div_T)
a = similarity_variable
Cp = a*(3.0*(A1 + A2*a)*R*(theta_div_T)**2*exp_term/(exp_term-1)**2
+ (C1 + C2*a)*T
+ (D1 + D2*a)*T**2)
return Cp*1000. if MW is None else Cp*MW
[docs]def Lastovka_solid_integral(T, similarity_variable, MW=None):
r'''Integrates solid constant-pressure heat capacity with the similarity
variable concept and method as shown in [1]_.
uses an explicit form as derived with Sympy.
Parameters
----------
T : float
Temperature of solid [K]
similarity_variable : float
similarity variable as defined in [1]_, [mol/g]
MW : float, optional
Molecular weight of the pure compound or mixture average, [g/mol]
Returns
-------
H : float
Difference in enthalpy from 0 K, J/mol if MW given; J/kg otherwise
Notes
-----
Original model is in terms of J/g/K. Note that the model is for predicting
mass heat capacity, not molar heat capacity like most other methods!
See Also
--------
Lastovka_solid
Examples
--------
>>> Lastovka_solid_integral(300, 0.2139)
283246.1519409122
References
----------
.. [1] Laštovka, Václav, Michal Fulem, Mildred Becerra, and John M. Shaw.
"A Similarity Variable for Estimating the Heat Capacity of Solid Organic
Compounds: Part II. Application: Heat Capacity Calculation for
Ill-Defined Organic Solids." Fluid Phase Equilibria 268, no. 1-2
(June 25, 2008): 134-41. doi:10.1016/j.fluid.2008.03.018.
'''
A1 = 0.013183
A2 = 0.249381
theta = 151.8675
C1 = 0.026526
C2 = -0.024942
D1 = 0.000025
D2 = -0.000123
a = similarity_variable
T2 = T*T
H = a*(T*T2*(D1 + D2*a)/3.
+ 0.5*T2*(C1 + C2*a)
+ 3.0*R*theta*(A1 + A2*a)/(exp(theta/T) - 1.))
return H*1000. if MW is None else H*MW
[docs]def Lastovka_solid_integral_over_T(T, similarity_variable, MW=None):
r'''Integrates over T solid constant-pressure heat capacity with the
similarity variable concept and method as shown in [1]_.
uses an explicit form as derived with Sympy.
Parameters
----------
T : float
Temperature of solid [K]
similarity_variable : float
similarity variable as defined in [1]_, [mol/g]
MW : float, optional
Molecular weight of the pure compound or mixture average, [g/mol]
Returns
-------
S : float
Difference in entropy from 0 K, J/mol/K if MW given; J/kg/K otherwise
Notes
-----
Original model is in terms of J/g/K. Note that the model is for predicting
mass heat capacity, not molar heat capacity like most other methods!
See Also
--------
Lastovka_solid
Examples
--------
>>> Lastovka_solid_integral_over_T(300, 0.2139)
1947.5537561495564
References
----------
.. [1] Laštovka, Václav, Michal Fulem, Mildred Becerra, and John M. Shaw.
"A Similarity Variable for Estimating the Heat Capacity of Solid Organic
Compounds: Part II. Application: Heat Capacity Calculation for
Ill-Defined Organic Solids." Fluid Phase Equilibria 268, no. 1-2
(June 25, 2008): 134-41. doi:10.1016/j.fluid.2008.03.018.
'''
A1 = 0.013183
A2 = 0.249381
theta = 151.8675
C1 = 0.026526
C2 = -0.024942
D1 = 0.000025
D2 = -0.000123
a = similarity_variable
exp_theta_T = exp(theta/T)
A_term = (A1 + A2*a)
S = a*(-3.0*R*A_term*log(exp_theta_T - 1.)
+ 0.5*T**2*(D1 + D2*a)
+ T*(C1 + C2*a)
+ 3.0*R*theta*A_term*(1/(T*exp_theta_T - T) + 1/T))
return S*1000. if MW is None else S*MW
[docs]def Cpg_statistical_mechanics(T, thetas, linear=False):
r'''Calculates the ideal-gas heat capacity using of a molecule using
its characteristic temperatures, themselves calculated from each of the
frequencies of vibration of the molecule. These can be obtained from
spectra or quantum mechanical calculations.
.. math::
\frac{C_p^{0}}{R} = \frac{C_p^{0}}{R} \text{rotational}
+ \frac{C_p^{0}}{R} \text{translational}
+ \frac{C_p^{0}}{R} \text{vibrational}
.. math::
\frac{C_p^{0}}{R} \text{rotational} = 2.5
.. math::
\frac{C_p^{0}}{R} \text{translational} = 1 \text{ if linear else } 1.5
.. math::
\frac{C_p^{0}}{R} \text{vibrational} = \sum_{i=1}^{3n_A-6+\delta}
\left(\frac{\theta_i}{T}\right)^2
\left[\frac{\exp(\theta_i/T)}
{\left(\exp(\theta_i/T)-1\right)^2}\right]
In the above equation, :math:`delta` is 1 if the molecule is linear
otherwise 0.
Parameters
----------
T : float
Temperature of fluid [K]
thetas : list[float]
Characteristic temperatures, [K]
Returns
-------
Cpgm : float
Gas molar heat capacity at specified temperature, [J/mol/K]
Notes
-----
This equation implies that there is a maximum heat capacity for an ideal
gas, and all diatomic or larger gases
Monoatomic gases have a simple heat capacity of 2.5R, the lower limit for
ideal gas heat capacity. This function does not cover that type of a gas.
The specific gases that are monoatomic are helium, neon, argon, krypton, xenon, radon.
At very low temperatures hydrogen behaves like a monoatomic gas as well.
Examples
--------
Sample calculation in [1]_ for ammonia:
>>> thetas = [1360, 2330, 2330, 4800, 4880, 4880]
>>> Cpg_statistical_mechanics(300.0, thetas)
35.55983440173097
References
----------
.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
Eighth Edition. McGraw-Hill Professional, 2007.
'''
# delta == 1 for linear molecules, 0 for nonlinear
# linear molecules have 1 R
# nonlinear have 1.5 R
# https://github.com/psi4/psi4/blob/7636787c5a3adac3b85493d3922663b7a5fcdea8/psi4/driver/qcdb/vib.py#L935
# 2.5 is always a fixed translational heat capacity
Cp_R = 2.5 + (1.0 if linear else 1.5)
tmp = 0.0
# The limit for this term as T approaches 0 is zero
# The limit for this term as T approaches infinity is 1 per term!
if T > 0.0:
for j in range(len(thetas)):
t = thetas[j]
r = t/T
if r > 120.0:
# The term is too small, nothing to add, no need to compute and a computational error will occur
continue
exponential = exp(r)
if exponential == 1.0:
tmp += 1.0
else:
den = expm1(r)
val = r*r*(exponential/(den*den))
if val > 1.0:
val = 1.0
tmp += val
Cp_R += tmp
return Cp_R*R
[docs]def Cpg_statistical_mechanics_integral(T, thetas, linear=False):
r'''Calculates the integral of ideal-gas heat capacity using of a molecule
using its characteristic temperatures.
.. math::
\int {C_p^{0}} = 2.5RT + RT \text{ if linear else } 1.5RT
+ \int C_p^{0}\text{vibrational}
.. math::
\int {C_p^{0}} \text{vibrational} = R\sum_{i=1}^{3n_A-6+\delta}
\frac{\theta_i}{\exp(\theta_i/T)-1}
Parameters
----------
T : float
Temperature of fluid [K]
thetas : list[float]
Characteristic temperatures, [K]
Returns
-------
H : float
Integrated gas molar heat capacity at specified temperature, [J/mol]
Notes
-----
Examples
--------
>>> thetas = [1360, 2330, 2330, 4800, 4880, 4880]
>>> Cpg_statistical_mechanics_integral(300.0, thetas)
10116.6053294
'''
H_R = (2.5 + (1.0 if linear else 1.5))*T
if T > 0.0:
for j in range(len(thetas)):
t = thetas[j]
r = t/T
if r < 120.0:
# The term can be too small, nothing to add, no need to compute and a computational error will occur
# Also important to use expm1 for accuracy at higher temperatures
# At high temperatures the contribution to enthalpy is R*T
H_R += t/expm1(r)
return H_R*R
[docs]def Cpg_statistical_mechanics_integral_over_T(T, thetas, linear=False):
r'''Calculates the integral over T of ideal-gas heat capacity using of a
molecule using its characteristic temperatures.
.. math::
\int \frac{C_p^{0}}{T} = 2.5R\log(T) + 1R\log(T)
\text{ if linear else } 1.5R\log(T)
+ \int \frac{C_p^{0}}{T}\text{vibrational}
.. math::
\int \frac{C_p^{0}}{T} \text{vibrational} = \sum_{i=1}^{3n_A-6+\delta}
\frac{\theta_i}{T\exp(\theta_i/T)-T} - \log(\exp(\theta_i/T)-1)
+ \theta_i/T
Parameters
----------
T : float
Temperature of fluid [K]
thetas : list[float]
Characteristic temperatures, [K]
Returns
-------
S : float
Entropy integral of gas molar heat capacity at specified temperature,
[J/mol/K]
Notes
-----
Examples
--------
>>> thetas = [1360, 2330, 2330, 4800, 4880, 4880]
>>> Cpg_statistical_mechanics_integral_over_T(300.0, thetas)
190.25658088
'''
S_R = (2.5 + (1.0 if linear else 1.5))*log(T)
for j in range(len(thetas)):
t = thetas[j]
r = t/T
if r < 36.87869878248:
# The term is always zero above 37 ish, polished to find exactly where the transition happens
# but it could technically be libm dependent, but should be good!
factor = expm1(r)
S_R += r/factor - log(factor) + r
return S_R*R
[docs]def vibration_frequency_cm_to_characteristic_temperature(frequency, scale=1):
r'''Convert a vibrational frequency in units of 1/cm to a characteristic
temperature for use in calculating heat capacity.
.. math::
\theta = \frac{100\cdot h\cdot c\cdot \text{scale}}{k}
Parameters
----------
frequency : float
Vibrational frequency, [1/cm]
scale : float
A scale factor used to adjust the frequency for differences in
experimental vs. calculated values, [-]
Returns
-------
theta : float
Characteristic temperature [K]
Notes
-----
In the equation, `k` is Boltzmann's constant, `c` is the speed of light,
and `h` is the Planck constant.
A scale factor for the MP2/6-31G** method recommended by NIST is 0.9365.
Using this scale factor will not improve results in all cases however.
Examples
--------
>>> vibration_frequency_cm_to_characteristic_temperature(667)
959.6641613636505
'''
frequency *= 100.0*scale # convert to 1/m
hz = frequency*c
return h*hz/k