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 implementations of most cubic equations of state for
pure components. This includes Peng-Robinson, SRK, Van der Waals, PRSV, TWU and
many other variants.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/thermo/>`_.
.. contents:: :local:
Base Class
==========
.. autoclass:: GCEOS
:members:
:undoc-members:
:special-members: __repr__
:show-inheritance:
:exclude-members: _P_zero_g_cheb_coeffs, _P_zero_l_cheb_coeffs,
main_derivatives_and_departures, derivatives_and_departures
Standard Peng-Robinson Family EOSs
==================================
Standard Peng Robinson
----------------------
.. autoclass:: PR
:show-inheritance:
:members: a_alpha_pure, a_alpha_and_derivatives_pure, d3a_alpha_dT3_pure,
solve_T, P_max_at_V, c1, c2, Zc
Peng Robinson (1978)
--------------------
.. autoclass:: PR78
:show-inheritance:
:members: low_omega_constants, high_omega_constants
Peng Robinson Stryjek-Vera
--------------------------
.. autoclass:: PRSV
:show-inheritance:
:members: solve_T, a_alpha_and_derivatives_pure, a_alpha_pure
Peng Robinson Stryjek-Vera 2
----------------------------
.. autoclass:: PRSV2
:show-inheritance:
:members: solve_T, a_alpha_and_derivatives_pure, a_alpha_pure
Peng Robinson Twu (1995)
------------------------
.. autoclass:: TWUPR
:show-inheritance:
:members: a_alpha_and_derivatives_pure, a_alpha_pure
Peng Robinson Polynomial alpha Function
---------------------------------------
.. autoclass:: PRTranslatedPoly
:show-inheritance:
:members: a_alpha_and_derivatives_pure, a_alpha_pure
Volume Translated Peng-Robinson Family EOSs
===========================================
Peng Robinson Translated
------------------------
.. autoclass:: PRTranslated
:show-inheritance:
:members: __init__
:exclude-members: __init__
Peng Robinson Translated Twu (1991)
-----------------------------------
.. autoclass:: PRTranslatedTwu
:show-inheritance:
:members: __init__
:exclude-members: __init__
Peng Robinson Translated-Consistent
-----------------------------------
.. autoclass:: PRTranslatedConsistent
:show-inheritance:
:members: __init__
:exclude-members: __init__
Peng Robinson Translated (Pina-Martinez, Privat, and Jaubert Variant)
---------------------------------------------------------------------
.. autoclass:: PRTranslatedPPJP
:show-inheritance:
:members: __init__
:exclude-members: __init__
Soave-Redlich-Kwong Family EOSs
===============================
Standard SRK
------------
.. autoclass:: SRK
:show-inheritance:
:members: c1, c2, epsilon, Zc, a_alpha_and_derivatives_pure, a_alpha_pure, P_max_at_V, solve_T
Twu SRK (1995)
--------------
.. autoclass:: TWUSRK
:show-inheritance:
:members: a_alpha_and_derivatives_pure, a_alpha_pure
API SRK
-------
.. autoclass:: APISRK
:show-inheritance:
:members: a_alpha_and_derivatives_pure, a_alpha_pure, solve_T
SRK Translated
--------------
.. autoclass:: SRKTranslated
:show-inheritance:
:members: __init__
:exclude-members: __init__
SRK Translated-Consistent
-------------------------
.. autoclass:: SRKTranslatedConsistent
:show-inheritance:
:members: __init__
:exclude-members: __init__
SRK Translated (Pina-Martinez, Privat, and Jaubert Variant)
-----------------------------------------------------------
.. autoclass:: SRKTranslatedPPJP
:show-inheritance:
:members: __init__
:exclude-members: __init__
MSRK Translated
---------------
.. autoclass:: MSRKTranslated
:show-inheritance:
:members: estimate_MN
Van der Waals Equations of State
================================
.. autoclass:: VDW
:show-inheritance:
:members: a_alpha_and_derivatives_pure, a_alpha_pure, solve_T, T_discriminant_zeros_analytical, P_discriminant_zeros_analytical, delta, epsilon, omega, Zc
Redlich-Kwong Equations of State
================================
.. autoclass:: RK
:show-inheritance:
:members: a_alpha_and_derivatives_pure, a_alpha_pure, solve_T, T_discriminant_zeros_analytical, epsilon, omega, Zc, c1, c2
Ideal Gas Equation of State
===========================
.. autoclass:: IG
:show-inheritance:
:members: volume_solutions, Zc, a, b, delta, epsilon, a_alpha_pure, a_alpha_and_derivatives_pure, solve_T
Lists of Equations of State
===========================
.. autodata:: eos_list
.. autodata:: eos_2P_list
Demonstrations of Concepts
==========================
Maximum Pressure at Constant Volume
-----------------------------------
Some equations of state show this behavior. At a liquid volume, if the
temperature is increased, the pressure should increase as well to create that
same volume. However in some cases this is not the case as can be demonstrated
for this hypothetical dodecane-like fluid:
.. plot:: plots/PR_maximum_pressure.py
Through experience, it is observed that this behavior is only shown for some
sets of critical constants. It was found that if the expression for
:math:`\frac{\partial P}{\partial T}_{V}` is set to zero, an analytical
expression can be determined for exactly what that maximum pressure is.
Some EOSs implement this function as `P_max_at_V`; those that don't, and fluids
where there is no maximum pressure, will have that method but it will return None.
Debug Plots to Understand EOSs
------------------------------
The :obj:`GCEOS.volume_errors` method shows the relative error in the volume
solution. `mpmath` is requried for this functionality. It is not likely there
is an error here but many problems have been found in the past.
.. plot:: plots/PRTC_volume_error.py
The :obj:`GCEOS.PT_surface_special` method shows some of the special curves of
the EOS.
.. plot:: plots/PRTC_PT_surface_special.py
The :obj:`GCEOS.a_alpha_plot` method shows the alpha function curve. The
following sample shows the SRK's default alpha function for methane.
.. plot:: plots/SRK_a_alpha.py
If this doesn't look healthy, that is because it is not. There are strict
thermodynamic consistency requirements that we know of today:
* The alpha function must be positive and continuous
* The first derivative must be negative and continuous
* The second derivative must be positive and continuous
* The third derivative must be negative
The first criterial and second criteria fail here.
There are two methods to review the saturation properties solution.
The more general way is to review saturation properties as a plot:
.. plot:: plots/SRK_H_dep.py
.. plot:: plots/SRK_fugacity.py
The second plot is more detailed, and is focused on the direct calculation of
vapor pressure without using an iterative solution. It shows the relative
error of the fit, which normally way below where it would present any issue -
only 10-100x more error than it is possible to get with floating point numbers
at all.
.. plot:: plots/SRK_Psat_error.py
'''
__all__ = ['GCEOS', 'PR', 'SRK', 'PR78', 'PRSV', 'PRSV2', 'VDW', 'RK',
'APISRK', 'TWUPR', 'TWUSRK', 'eos_list', 'eos_2P_list',
'IG', 'PRTranslatedPPJP', 'SRKTranslatedPPJP',
'PRTranslatedConsistent', 'SRKTranslatedConsistent', 'MSRKTranslated',
'SRKTranslated', 'PRTranslated', 'PRTranslatedCoqueletChapoyRichon',
'PRTranslatedTwu', 'PRTranslatedPoly',
'main_derivatives_and_departures',
'main_derivatives_and_departures_VDW',
'eos_lnphi'
]
from cmath import log as clog
from math import isinf, isnan, log1p, log10
from chemicals.flash_basic import Wilson_K_value
from chemicals.utils import hash_any_primitive, object_data
from fluids.constants import R, mmHg
from fluids.numerics import (
NoSolutionError,
bisect,
brenth,
catanh,
chebder,
chebval,
derivative,
exp,
horner,
inf,
is_micropython,
isclose,
linspace,
log,
logspace,
newton,
polyder,
roots_cubic,
roots_quartic,
secant,
sqrt,
)
from fluids.numerics import numpy as np
from thermo import serialize
from thermo.eos_alpha_functions import Mathias_Copeman_poly_a_alpha, Poly_a_alpha, Soave_1979_a_alpha, Twu91_a_alpha, TwuPR95_a_alpha, TwuSRK95_a_alpha
from thermo.eos_volume import volume_solutions_halley, volume_solutions_ideal, volume_solutions_mpmath, volume_solutions_mpmath_float, volume_solutions_NR
R2 = R*R
R_2 = 0.5*R
R_inv = 1.0/R
R_inv2 = R_inv*R_inv
def main_derivatives_and_departures(T, P, V, b, delta, epsilon, a_alpha,
da_alpha_dT, d2a_alpha_dT2):
epsilon2 = epsilon + epsilon
x0 = 1.0/(V - b)
x1 = 1.0/(V*(V + delta) + epsilon)
x3 = R*T
x4 = x0*x0
x5 = V + V + delta
x6 = x1*x1
x7 = a_alpha*x6
x8 = P*V
x9 = delta*delta
x10 = x9 - epsilon2 - epsilon2
try:
x11 = 1.0/sqrt(x10)
except:
# Needed for ideal gas model
x11 = 0.0
x11_half = 0.5*x11
# hard_input = x11*x5
# arg2 = (hard_input + 1.0)/(hard_input - 1.0)
# fancy = 0.25*log(arg2*arg2)
# x12 = 2.*x11*fancy # Possible to use a catan, but then a complex division and sq root is needed too
hard_input = x11*x5
hard_term = catanh(hard_input).real # numba: delete
# hard_term = 0.25*log1p(4.*hard_input/((1.0-hard_input)*(1.0-hard_input))) # numba: uncomment
x12 = 2.*x11*hard_term # Possible to use a catan, but then a complex division and sq root is needed too
x14 = 0.5*x5
x15 = epsilon2*x11
x16 = x11_half*x9
x17 = x5*x6
dP_dT = R*x0 - da_alpha_dT*x1
dP_dV = x5*x7 - x3*x4
d2P_dT2 = -d2a_alpha_dT2*x1
d2P_dV2 = (x7 + x3*x4*x0 - a_alpha*x5*x17*x1)
d2P_dV2 += d2P_dV2
d2P_dTdV = da_alpha_dT*x17 - R*x4
H_dep = x12*(T*da_alpha_dT - a_alpha) - x3 + x8
t1 = (x3*x0/P)
S_dep = -R*log(t1) + da_alpha_dT*x12 # Consider Real part of the log only via log(x**2)/2 = Re(log(x))
# S_dep = -R_2*log(t1*t1) + da_alpha_dT*x12 # Consider Real part of the log only via log(x**2)/2 = Re(log(x))
x18 = x16 - x15
x19 = (x14 + x18)/(x14 - x18)
Cv_dep = T*d2a_alpha_dT2*x11*(log(x19)) # Consider Real part of the log only via log(x**2)/2 = Re(log(x))
return dP_dT, dP_dV, d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, Cv_dep
def main_derivatives_and_departures_VDW(T, P, V, b, delta, epsilon, a_alpha,
da_alpha_dT, d2a_alpha_dT2):
'''Re-implementation of derivatives and excess property calculations,
as ZeroDivisionError errors occur with the general solution. The
following derivation is the source of these formulas.
>>> from sympy import *
>>> P, T, V, R, b, a = symbols('P, T, V, R, b, a')
>>> P_vdw = R*T/(V-b) - a/(V*V)
>>> vdw = P_vdw - P
>>>
>>> dP_dT = diff(vdw, T)
>>> dP_dV = diff(vdw, V)
>>> d2P_dT2 = diff(vdw, T, 2)
>>> d2P_dV2 = diff(vdw, V, 2)
>>> d2P_dTdV = diff(vdw, T, V)
>>> H_dep = integrate(T*dP_dT - P_vdw, (V, oo, V))
>>> H_dep += P*V - R*T
>>> S_dep = integrate(dP_dT - R/V, (V,oo,V))
>>> S_dep += R*log(P*V/(R*T))
>>> Cv_dep = T*integrate(d2P_dT2, (V,oo,V))
>>>
>>> dP_dT, dP_dV, d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, Cv_dep
(R/(V - b), -R*T/(V - b)**2 + 2*a/V**3, 0, 2*(R*T/(V - b)**3 - 3*a/V**4), -R/(V - b)**2, P*V - R*T - a/V, R*(-log(V) + log(V - b)) + R*log(P*V/(R*T)), 0)
'''
V_inv = 1.0/V
V_inv2 = V_inv*V_inv
Vmb = V - b
Vmb_inv = 1.0/Vmb
dP_dT = R*Vmb_inv
dP_dV = -R*T*Vmb_inv*Vmb_inv + 2.0*a_alpha*V_inv*V_inv2
d2P_dT2 = 0.0
d2P_dV2 = 2.0*(R*T*Vmb_inv*Vmb_inv*Vmb_inv - 3.0*a_alpha*V_inv2*V_inv2) # Causes issues at low T when V fourth power fails
d2P_dTdV = -R*Vmb_inv*Vmb_inv
H_dep = P*V - R*T - a_alpha*V_inv
S_dep = R*(-log(V) + log(Vmb)) + R*log(P*V/(R*T))
Cv_dep = 0.0
return (dP_dT, dP_dV, d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, Cv_dep)
def eos_lnphi(T, P, V, b, delta, epsilon, a_alpha):
r'''Calculate the log fugacity coefficient of the general cubic equation
of state form.
.. math::
\ln \phi = \frac{P V}{R T} + \log{\left(V \right)} - \log{\left(\frac{P
V}{R T} \right)} - \log{\left(V - b \right)} - 1 - \frac{2 a {\alpha}
\operatorname{atanh}{\left(\frac{2 V}{\sqrt{\delta^{2} - 4 \epsilon}}
+ \frac{\delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right)}}
{R T \sqrt{\delta^{2} - 4 \epsilon}}
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
b : float
Coefficient calculated by EOS-specific method, [m^3/mol]
delta : float
Coefficient calculated by EOS-specific method, [m^3/mol]
epsilon : float
Coefficient calculated by EOS-specific method, [m^6/mol^2]
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
Returns
-------
lnphi : float
Log fugacity coefficient, [-]
Examples
--------
>>> eos_lnphi(299.0, 100000.0, 0.00013128, 0.000109389, 0.00021537, -1.1964711e-08, 3.8056296)
-1.560560970726
'''
RT = R*T
RT_inv = 1.0/RT
x0 = 1.0/sqrt(delta*delta - 4.0*epsilon)
arg = 2.0*V*x0 + delta*x0
# fancy = catanh(arg).real
fancy = 0.25*log1p(4.*arg/((1.0-arg)*(1.0-arg)))
return (P*V*RT_inv + log(RT/(P*(V-b))) - 1.0
- 2.0*a_alpha*fancy*RT_inv*x0)
def eos_G_dep(T, P, V, b, delta, epsilon, a_alpha):
lnphi = eos_lnphi(T, P, V, b, delta, epsilon, a_alpha)
return lnphi*R*T
[docs]class GCEOS:
r'''Class for solving a generic Pressure-explicit three-parameter cubic
equation of state. Does not implement any parameters itself; must be
subclassed by an equation of state class which uses it. Works for mixtures
or pure species for all properties except fugacity. All properties are
derived with the CAS SymPy, not relying on any derivations previously
published.
.. math::
P=\frac{RT}{V-b}-\frac{a\alpha(T)}{V^2 + \delta V + \epsilon}
The main methods (in order they are called) are :obj:`GCEOS.solve`, :obj:`GCEOS.set_from_PT`,
:obj:`GCEOS.volume_solutions`, and :obj:`GCEOS.set_properties_from_solution`.
:obj:`GCEOS.solve` calls :obj:`GCEOS.check_sufficient_inputs`, which checks if two of `T`, `P`,
and `V` were set. It then solves for the
remaining variable. If `T` is missing, method :obj:`GCEOS.solve_T` is used; it is
parameter specific, and so must be implemented in each specific EOS.
If `P` is missing, it is directly calculated. If `V` is missing, it
is calculated with the method :obj:`GCEOS.volume_solutions`. At this point, either
three possible volumes or one user specified volume are known. The
value of `a_alpha`, and its first and second temperature derivative are
calculated with the EOS-specific method :obj:`GCEOS.a_alpha_and_derivatives`.
If `V` is not provided, :obj:`GCEOS.volume_solutions` calculates the three
possible molar volumes which are solutions to the EOS; in the single-phase
region, only one solution is real and correct. In the two-phase region, all
volumes are real, but only the largest and smallest solution are physically
meaningful, with the largest being that of the gas and the smallest that of
the liquid.
:obj:`GCEOS.set_from_PT` is called to sort out the possible molar volumes. For the
case of a user-specified `V`, the possibility of there existing another
solution is ignored for speed. If there is only one real volume, the
method :obj:`GCEOS.set_properties_from_solution` is called with it. If there are
two real volumes, :obj:`GCEOS.set_properties_from_solution` is called once with each
volume. The phase is returned by :obj:`GCEOS.set_properties_from_solution`, and the
volumes is set to either :obj:`GCEOS.V_l` or :obj:`GCEOS.V_g` as appropriate.
:obj:`GCEOS.set_properties_from_solution` is a large function which calculates all relevant
partial derivatives and properties of the EOS. 17 derivatives and excess
enthalpy and entropy are calculated first.
Finally, it sets all these properties as attibutes for either
the liquid or gas phase with the convention of adding on `_l` or `_g` to
the variable names, respectively.
Attributes
----------
T : float
Temperature of cubic EOS state, [K]
P : float
Pressure of cubic EOS state, [Pa]
a : float
`a` parameter of cubic EOS; formulas vary with the EOS, [Pa*m^6/mol^2]
b : float
`b` parameter of cubic EOS; formulas vary with the EOS, [m^3/mol]
delta : float
Coefficient calculated by EOS-specific method, [m^3/mol]
epsilon : float
Coefficient calculated by EOS-specific method, [m^6/mol^2]
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of :math:`a \alpha` calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of :math:`a \alpha` calculated by
EOS-specific method, [J^2/mol^2/Pa/K**2]
Zc : float
Critical compressibility of cubic EOS state, [-]
phase : str
One of 'l', 'g', or 'l/g' to represent whether or not there is a
liquid-like solution, vapor-like solution, or both available, [-]
raw_volumes : list[(float, complex), 3]
Calculated molar volumes from the volume solver; depending on the state
and selected volume solver, imaginary volumes may be represented by 0
or -1j to save the time of actually calculating them, [m^3/mol]
V_l : float
Liquid phase molar volume, [m^3/mol]
V_g : float
Vapor phase molar volume, [m^3/mol]
V : float or None
Molar volume specified as input; otherwise None, [m^3/mol]
Z_l : float
Liquid phase compressibility, [-]
Z_g : float
Vapor phase compressibility, [-]
PIP_l : float
Liquid phase phase identification parameter, [-]
PIP_g : float
Vapor phase phase identification parameter, [-]
dP_dT_l : float
Liquid phase temperature derivative of pressure at constant volume,
[Pa/K].
.. math::
\left(\frac{\partial P}{\partial T}\right)_V = \frac{R}{V - b}
- \frac{a \frac{d \alpha{\left (T \right )}}{d T}}{V^{2} + V \delta
+ \epsilon}
dP_dT_g : float
Vapor phase temperature derivative of pressure at constant volume,
[Pa/K].
.. math::
\left(\frac{\partial P}{\partial T}\right)_V = \frac{R}{V - b}
- \frac{a \frac{d \alpha{\left (T \right )}}{d T}}{V^{2} + V \delta
+ \epsilon}
dP_dV_l : float
Liquid phase volume derivative of pressure at constant temperature,
[Pa*mol/m^3].
.. math::
\left(\frac{\partial P}{\partial V}\right)_T = - \frac{R T}{\left(
V - b\right)^{2}} - \frac{a \left(- 2 V - \delta\right) \alpha{
\left (T \right )}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}
dP_dV_g : float
Gas phase volume derivative of pressure at constant temperature,
[Pa*mol/m^3].
.. math::
\left(\frac{\partial P}{\partial V}\right)_T = - \frac{R T}{\left(
V - b\right)^{2}} - \frac{a \left(- 2 V - \delta\right) \alpha{
\left (T \right )}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}
dV_dT_l : float
Liquid phase temperature derivative of volume at constant pressure,
[m^3/(mol*K)].
.. math::
\left(\frac{\partial V}{\partial T}\right)_P =-\frac{
\left(\frac{\partial P}{\partial T}\right)_V}{
\left(\frac{\partial P}{\partial V}\right)_T}
dV_dT_g : float
Gas phase temperature derivative of volume at constant pressure,
[m^3/(mol*K)].
.. math::
\left(\frac{\partial V}{\partial T}\right)_P =-\frac{
\left(\frac{\partial P}{\partial T}\right)_V}{
\left(\frac{\partial P}{\partial V}\right)_T}
dV_dP_l : float
Liquid phase pressure derivative of volume at constant temperature,
[m^3/(mol*Pa)].
.. math::
\left(\frac{\partial V}{\partial P}\right)_T =-\frac{
\left(\frac{\partial V}{\partial T}\right)_P}{
\left(\frac{\partial P}{\partial T}\right)_V}
dV_dP_g : float
Gas phase pressure derivative of volume at constant temperature,
[m^3/(mol*Pa)].
.. math::
\left(\frac{\partial V}{\partial P}\right)_T =-\frac{
\left(\frac{\partial V}{\partial T}\right)_P}{
\left(\frac{\partial P}{\partial T}\right)_V}
dT_dV_l : float
Liquid phase volume derivative of temperature at constant pressure,
[K*mol/m^3].
.. math::
\left(\frac{\partial T}{\partial V}\right)_P = \frac{1}
{\left(\frac{\partial V}{\partial T}\right)_P}
dT_dV_g : float
Gas phase volume derivative of temperature at constant pressure,
[K*mol/m^3]. See :obj:`GCEOS.set_properties_from_solution` for
the formula.
dT_dP_l : float
Liquid phase pressure derivative of temperature at constant volume,
[K/Pa].
.. math::
\left(\frac{\partial T}{\partial P}\right)_V = \frac{1}
{\left(\frac{\partial P}{\partial T}\right)_V}
dT_dP_g : float
Gas phase pressure derivative of temperature at constant volume,
[K/Pa].
.. math::
\left(\frac{\partial T}{\partial P}\right)_V = \frac{1}
{\left(\frac{\partial P}{\partial T}\right)_V}
d2P_dT2_l : float
Liquid phase second derivative of pressure with respect to temperature
at constant volume, [Pa/K^2].
.. math::
\left(\frac{\partial^2 P}{\partial T^2}\right)_V = - \frac{a
\frac{d^{2} \alpha{\left (T \right )}}{d T^{2}}}{V^{2} + V \delta
+ \epsilon}
d2P_dT2_g : float
Gas phase second derivative of pressure with respect to temperature
at constant volume, [Pa/K^2].
.. math::
\left(\frac{\partial^2 P}{\partial T^2}\right)_V = - \frac{a
\frac{d^{2} \alpha{\left (T \right )}}{d T^{2}}}{V^{2} + V \delta
+ \epsilon}
d2P_dV2_l : float
Liquid phase second derivative of pressure with respect to volume
at constant temperature, [Pa*mol^2/m^6].
.. math::
\left(\frac{\partial^2 P}{\partial V^2}\right)_T = 2 \left(\frac{
R T}{\left(V - b\right)^{3}} - \frac{a \left(2 V + \delta\right)^{
2} \alpha{\left (T \right )}}{\left(V^{2} + V \delta + \epsilon
\right)^{3}} + \frac{a \alpha{\left (T \right )}}{\left(V^{2} + V
\delta + \epsilon\right)^{2}}\right)
d2P_dTdV_l : float
Liquid phase second derivative of pressure with respect to volume
and then temperature, [Pa*mol/(K*m^3)].
.. math::
\left(\frac{\partial^2 P}{\partial T \partial V}\right) = - \frac{
R}{\left(V - b\right)^{2}} + \frac{a \left(2 V + \delta\right)
\frac{d \alpha{\left (T \right )}}{d T}}{\left(V^{2} + V \delta
+ \epsilon\right)^{2}}
d2P_dTdV_g : float
Gas phase second derivative of pressure with respect to volume
and then temperature, [Pa*mol/(K*m^3)].
.. math::
\left(\frac{\partial^2 P}{\partial T \partial V}\right) = - \frac{
R}{\left(V - b\right)^{2}} + \frac{a \left(2 V + \delta\right)
\frac{d \alpha{\left (T \right )}}{d T}}{\left(V^{2} + V \delta
+ \epsilon\right)^{2}}
H_dep_l : float
Liquid phase departure enthalpy, [J/mol]. See
:obj:`GCEOS.set_properties_from_solution` for the formula.
H_dep_g : float
Gas phase departure enthalpy, [J/mol]. See
:obj:`GCEOS.set_properties_from_solution` for the formula.
S_dep_l : float
Liquid phase departure entropy, [J/(mol*K)]. See
:obj:`GCEOS.set_properties_from_solution` for the formula.
S_dep_g : float
Gas phase departure entropy, [J/(mol*K)]. See
:obj:`GCEOS.set_properties_from_solution` for the formula.
G_dep_l : float
Liquid phase departure Gibbs energy, [J/mol].
.. math::
G_{dep} = H_{dep} - T S_{dep}
G_dep_g : float
Gas phase departure Gibbs energy, [J/mol].
.. math::
G_{dep} = H_{dep} - T S_{dep}
Cp_dep_l : float
Liquid phase departure heat capacity, [J/(mol*K)]
.. math::
C_{p, dep} = (C_p-C_v)_{\text{from EOS}} + C_{v, dep} - R
Cp_dep_g : float
Gas phase departure heat capacity, [J/(mol*K)]
.. math::
C_{p, dep} = (C_p-C_v)_{\text{from EOS}} + C_{v, dep} - R
Cv_dep_l : float
Liquid phase departure constant volume heat capacity, [J/(mol*K)].
See :obj:`GCEOS.set_properties_from_solution` for
the formula.
Cv_dep_g : float
Gas phase departure constant volume heat capacity, [J/(mol*K)].
See :obj:`GCEOS.set_properties_from_solution` for
the formula.
c1 : float
Full value of the constant in the `a` parameter, set in some EOSs, [-]
c2 : float
Full value of the constant in the `b` parameter, set in some EOSs, [-]
A_dep_g
A_dep_l
beta_g
beta_l
Cp_minus_Cv_g
Cp_minus_Cv_l
d2a_alpha_dTdP_g_V
d2a_alpha_dTdP_l_V
d2H_dep_dT2_g
d2H_dep_dT2_g_P
d2H_dep_dT2_g_V
d2H_dep_dT2_l
d2H_dep_dT2_l_P
d2H_dep_dT2_l_V
d2H_dep_dTdP_g
d2H_dep_dTdP_l
d2P_drho2_g
d2P_drho2_l
d2P_dT2_PV_g
d2P_dT2_PV_l
d2P_dTdP_g
d2P_dTdP_l
d2P_dTdrho_g
d2P_dTdrho_l
d2P_dVdP_g
d2P_dVdP_l
d2P_dVdT_g
d2P_dVdT_l
d2P_dVdT_TP_g
d2P_dVdT_TP_l
d2rho_dP2_g
d2rho_dP2_l
d2rho_dPdT_g
d2rho_dPdT_l
d2rho_dT2_g
d2rho_dT2_l
d2S_dep_dT2_g
d2S_dep_dT2_g_V
d2S_dep_dT2_l
d2S_dep_dT2_l_V
d2S_dep_dTdP_g
d2S_dep_dTdP_l
d2T_dP2_g
d2T_dP2_l
d2T_dPdrho_g
d2T_dPdrho_l
d2T_dPdV_g
d2T_dPdV_l
d2T_drho2_g
d2T_drho2_l
d2T_dV2_g
d2T_dV2_l
d2T_dVdP_g
d2T_dVdP_l
d2V_dP2_g
d2V_dP2_l
d2V_dPdT_g
d2V_dPdT_l
d2V_dT2_g
d2V_dT2_l
d2V_dTdP_g
d2V_dTdP_l
d3a_alpha_dT3
da_alpha_dP_g_V
da_alpha_dP_l_V
dbeta_dP_g
dbeta_dP_l
dbeta_dT_g
dbeta_dT_l
dfugacity_dP_g
dfugacity_dP_l
dfugacity_dT_g
dfugacity_dT_l
dH_dep_dP_g
dH_dep_dP_g_V
dH_dep_dP_l
dH_dep_dP_l_V
dH_dep_dT_g
dH_dep_dT_g_V
dH_dep_dT_l
dH_dep_dT_l_V
dH_dep_dV_g_P
dH_dep_dV_g_T
dH_dep_dV_l_P
dH_dep_dV_l_T
dP_drho_g
dP_drho_l
dphi_dP_g
dphi_dP_l
dphi_dT_g
dphi_dT_l
drho_dP_g
drho_dP_l
drho_dT_g
drho_dT_l
dS_dep_dP_g
dS_dep_dP_g_V
dS_dep_dP_l
dS_dep_dP_l_V
dS_dep_dT_g
dS_dep_dT_g_V
dS_dep_dT_l
dS_dep_dT_l_V
dS_dep_dV_g_P
dS_dep_dV_g_T
dS_dep_dV_l_P
dS_dep_dV_l_T
dT_drho_g
dT_drho_l
dZ_dP_g
dZ_dP_l
dZ_dT_g
dZ_dT_l
fugacity_g
fugacity_l
kappa_g
kappa_l
lnphi_g
lnphi_l
more_stable_phase
mpmath_volume_ratios
mpmath_volumes
mpmath_volumes_float
phi_g
phi_l
rho_g
rho_l
sorted_volumes
state_specs
U_dep_g
U_dep_l
Vc
V_dep_g
V_dep_l
V_g_mpmath
V_l_mpmath
'''
# Slots does not help performance in either implementation
kwargs = {}
"""Dictionary which holds input parameters to an EOS which are non-standard;
this excludes `T`, `P`, `V`, `omega`, `Tc`, `Pc`, `Vc` but includes EOS
specific parameters like `S1` and `alpha_coeffs`.
"""
N = 1
"""The number of components in the EOS"""
scalar = True
multicomponent = False
"""Whether or not the EOS is multicomponent or not"""
_P_zero_l_cheb_coeffs = None
P_zero_l_cheb_limits = (0.0, 0.0)
_P_zero_g_cheb_coeffs = None
P_zero_g_cheb_limits = (0.0, 0.0)
Psat_cheb_range = (0.0, 0.0)
main_derivatives_and_departures = staticmethod(main_derivatives_and_departures)
c1 = None
"""Parameter used by some equations of state in the `a` calculation"""
c2 = None
"""Parameter used by some equations of state in the `b` calculation"""
nonstate_constants = ('Tc', 'Pc', 'omega', 'kwargs', 'a', 'b', 'delta', 'epsilon')
kwargs_keys = tuple()
if not is_micropython:
def __init_subclass__(cls):
cls.__full_path__ = f"{cls.__module__}.{cls.__qualname__}"
else:
__full_path__ = None
[docs] def state_hash(self):
r'''Basic method to calculate a hash of the state of the model and its
model parameters.
Note that the hashes should only be compared on the same system running
in the same process!
Returns
-------
state_hash : int
Hash of the object's model parameters and state, [-]
'''
if self.multicomponent:
comp = self.zs
else:
comp = 0
return hash_any_primitive((self.model_hash(), self.T, self.P, self.V, comp))
[docs] def model_hash(self):
r'''Basic method to calculate a hash of the non-state parts of the model
This is useful for comparing to models to
determine if they are the same, i.e. in a VLL flash it is important to
know if both liquids have the same model.
Note that the hashes should only be compared on the same system running
in the same process!
Returns
-------
model_hash : int
Hash of the object's model parameters, [-]
'''
try:
return self._model_hash
except AttributeError:
pass
h = hash(self.__class__.__name__)
for s in self.nonstate_constants:
try:
h = hash((h, s, hash_any_primitive(getattr(self, s))))
except AttributeError:
pass
self._model_hash = h
return h
def __hash__(self):
r'''Method to calculate and return a hash representing the exact state
of the object.
Returns
-------
hash : int
Hash of the object, [-]
'''
d = object_data(self)
ans = hash_any_primitive((self.__class__.__name__, d))
return ans
def __eq__(self, other):
return self.__hash__() == hash(other)
@property
def state_specs(self):
'''Convenience method to return the two specified state specs (`T`,
`P`, or `V`) as a dictionary.
Examples
--------
>>> PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=500.0, V=1.0).state_specs
{'T': 500.0, 'V': 1.0}
'''
d = {}
if hasattr(self, 'no_T_spec') and self.no_T_spec:
d['P'] = self.P
d['V'] = self.V
elif self.V is not None:
d['T'] = self.T
d['V'] = self.V
else:
d['T'] = self.T
d['P'] = self.P
return d
[docs] def __repr__(self):
'''Create a string representation of the EOS - by default, include
all parameters so as to make it easy to construct new instances from
states. Includes the two specified state variables, `Tc`, `Pc`, `omega`
and any `kwargs`.
Returns
-------
recreation : str
String which is valid Python and recreates the current state of
the object if ran, [-]
Examples
--------
>>> eos = PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=400.0, P=1e6)
>>> eos
PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=400.0, P=1000000.0)
'''
s = f'{self.__class__.__name__}(Tc={repr(self.Tc)}, Pc={repr(self.Pc)}, omega={repr(self.omega)}, '
for k, v in self.kwargs.items():
s += f'{k}={v}, '
if hasattr(self, 'no_T_spec') and self.no_T_spec:
s += f'P={self.P!r}, V={repr(self.V)}'
elif self.V is not None:
s += f'T={self.T!r}, V={repr(self.V)}'
else:
s += f'T={self.T!r}, P={repr(self.P)}'
s += ')'
return s
[docs] def as_json(self):
r'''Method to create a JSON-friendly serialization of the eos
which can be stored, and reloaded later.
Returns
-------
json_repr : dict
JSON-friendly representation, [-]
Notes
-----
Examples
--------
>>> import json
>>> eos = MSRKTranslated(Tc=507.6, Pc=3025000, omega=0.2975, c=22.0561E-6, M=0.7446, N=0.2476, T=250., P=1E6)
>>> assert eos == MSRKTranslated.from_json(json.loads(json.dumps(eos.as_json())))
'''
# vaguely jsonpickle compatible
d = object_data(self)
if not self.scalar:
d = serialize.arrays_to_lists(d)
# TODO: delete kwargs and reconstruct it
# Need to add all kwargs attributes
try:
del d['kwargs']
except:
pass
try:
del d['one_minus_kijs']
except:
pass
d["py/object"] = self.__full_path__
d['json_version'] = 1
return d
[docs] @classmethod
def from_json(cls, json_repr):
r'''Method to create a eos from a JSON
serialization of another eos.
Parameters
----------
json_repr : dict
JSON-friendly representation, [-]
Returns
-------
eos : :obj:`GCEOS`
Newly created object from the json serialization, [-]
Notes
-----
It is important that the input string be in the same format as that
created by :obj:`GCEOS.as_json`.
Examples
--------
>>> eos = MSRKTranslated(Tc=507.6, Pc=3025000, omega=0.2975, c=22.0561E-6, M=0.7446, N=0.2476, T=250., P=1E6)
>>> string = eos.as_json()
>>> new_eos = GCEOS.from_json(string)
>>> assert eos.__dict__ == new_eos.__dict__
'''
d = json_repr
eos_name = d['py/object']
del d['py/object']
del d['json_version']
try:
d['raw_volumes'] = tuple(d['raw_volumes'])
except:
pass
try:
d['alpha_coeffs'] = tuple(d['alpha_coeffs'])
except:
pass
eos = eos_full_path_dict[eos_name]
if eos.kwargs_keys:
d['kwargs'] = {k: d[k] for k in eos.kwargs_keys}
try:
d['kwargs']['alpha_coeffs'] = tuple(d['kwargs']['alpha_coeffs'])
except:
pass
new = eos.__new__(eos)
for k, v in d.items():
setattr(new, k, v)
# new.__dict__ = d
return new
[docs] def solve(self, pure_a_alphas=True, only_l=False, only_g=False, full_alphas=True):
'''First EOS-generic method; should be called by all specific EOSs.
For solving for `T`, the EOS must provide the method `solve_T`.
For all cases, the EOS must provide `a_alpha_and_derivatives`.
Calls `set_from_PT` once done.
'''
# self.check_sufficient_inputs()
if self.V is not None:
V = self.V
if self.P is not None:
solution = 'g' if (only_g and not only_l) else ('l' if only_l else None)
self.T = self.solve_T(self.P, V, solution=solution)
self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2 = self.a_alpha_and_derivatives(self.T, pure_a_alphas=pure_a_alphas)
elif self.T is not None:
self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2 = self.a_alpha_and_derivatives(self.T, pure_a_alphas=pure_a_alphas)
# Tested to change the result at the 7th decimal once
# V_r3 = V**(1.0/3.0)
# T, b, a_alpha, delta, epsilon = self.T, self.b, self.a_alpha, self.delta, self.epsilon
# P = R*T/(V-b) - a_alpha/((V_r3*V_r3)*(V_r3*(V+delta)) + epsilon)
#
# for _ in range(10):
# err = -T + (P*V**3 - P*V**2*b + P*V**2*delta - P*V*b*delta + P*V*epsilon - P*b*epsilon + V*a_alpha - a_alpha*b)/(R*(V**2 + V*delta + epsilon))
# derr = (V**3 - V**2*b + V**2*delta - V*b*delta + V*epsilon - b*epsilon)/(R*(V**2 + V*delta + epsilon))
# P = P - err/derr
# self.P = P
# Equation re-aranged to hopefully solve better
# Allow mpf multiple precision volume for flash initialization
# DO NOT TAKE OUT FLOAT CONVERSION!
T = self.T
if not isinstance(V, (float, int)):
import mpmath as mp
# mp.mp.dps = 50 # Do not need more decimal places than needed
# Need to complete the calculation with the RT term having higher precision as well
T = mp.mpf(T)
self.P = float(R*T/(V-self.b) - self.a_alpha/(V*V + self.delta*V + self.epsilon))
if self.P <= 0.0:
raise ValueError("TV inputs result in negative pressure of %f Pa" %(self.P))
# self.P = R*self.T/(V-self.b) - self.a_alpha/(V*(V + self.delta) + self.epsilon)
else:
raise ValueError("Two specs are required")
Vs = [V, 1.0j, 1.0j]
elif self.T is None or self.P is None:
raise ValueError("Two specs are required")
else:
if full_alphas:
self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2 = self.a_alpha_and_derivatives(self.T, pure_a_alphas=pure_a_alphas)
else:
self.a_alpha = self.a_alpha_and_derivatives(self.T, full=False, pure_a_alphas=pure_a_alphas)
self.da_alpha_dT, self.d2a_alpha_dT2 = -5e-3, 1.5e-5
self.raw_volumes = Vs = self.volume_solutions(self.T, self.P, self.b, self.delta, self.epsilon, self.a_alpha)
self.set_from_PT(Vs, only_l=only_l, only_g=only_g)
[docs] def resolve_full_alphas(self):
'''Generic method to resolve the eos with fully calculated alpha
derviatives. Re-calculates properties with the new alpha derivatives
for any previously solved roots.
'''
self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2 = self.a_alpha_and_derivatives(self.T, full=True, pure_a_alphas=False)
self.set_from_PT(self.raw_volumes, only_l=hasattr(self, 'V_l'), only_g=hasattr(self, 'V_g'))
[docs] def solve_missing_volumes(self):
r'''Generic method to ensure both volumes, if solutions are physical,
have calculated properties. This effectively un-does the optimization
of the `only_l` and `only_g` keywords.
'''
if self.phase == 'l/g':
try:
self.V_l
except:
self.set_from_PT(self.raw_volumes, only_l=True, only_g=False)
try:
self.V_g
except:
self.set_from_PT(self.raw_volumes, only_l=False, only_g=True)
[docs] def set_from_PT(self, Vs, only_l=False, only_g=False):
r'''Counts the number of real volumes in `Vs`, and determines what to do.
If there is only one real volume, the method
`set_properties_from_solution` is called with it. If there are
two real volumes, `set_properties_from_solution` is called once with
each volume. The phase is returned by `set_properties_from_solution`,
and the volumes is set to either `V_l` or `V_g` as appropriate.
Parameters
----------
Vs : list[float]
Three possible molar volumes, [m^3/mol]
only_l : bool
When true, if there is a liquid and a vapor root, only the liquid
root (and properties) will be set.
only_g : bool
When true, if there is a liquid and a vapor root, only the vapor
root (and properties) will be set.
Notes
-----
An optimization attempt was made to remove min() and max() from this
function; that is indeed possible, but the check for handling if there
are two or three roots makes it not worth it.
'''
# good_roots = [i.real for i in Vs if i.imag == 0.0 and i.real > 0.0]
# good_root_count = len(good_roots)
# All roots will have some imaginary component; ignore them if > 1E-9 (when using a solver that does not strip them)
b = self.b
# good_roots = [i.real for i in Vs if (i.real ==0 or abs(i.imag/i.real) < 1E-12) and i.real > 0.0]
# good_roots = [i.real for i in Vs if (i.real > b and
# (i.imag == 0.0 or abs(i.imag/i.real) < 1E-12))]
Vmin = 1e100
Vmax = -1e100
good_root_count = 0
for V in Vs:
if V.real == 0.0 or V.imag == 0.0 or abs(V.imag/V.real) < 1E-12:
V = V.real
if V <= b:
continue
good_root_count += 1
if V < Vmin:
Vmin = V
if V > Vmax:
Vmax = V
# Counter for the case of testing volume solutions that don't work
# good_roots = [i.real for i in Vs if (i.real > 0.0 and (i.real == 0.0 or abs(i.imag) < 1E-9))]
# good_root_count = len(good_roots)
if good_root_count == 1 or Vmin == Vmax:
V = Vmin if Vmin != 1e100 else Vmax
self.phase = self.set_properties_from_solution(self.T, self.P,
V, b,
self.delta, self.epsilon,
self.a_alpha, self.da_alpha_dT,
self.d2a_alpha_dT2)
if self.N == 1 and (
(self.multicomponent and (self.Tcs[0] == self.T and self.Pcs[0] == self.P))
or (not self.multicomponent and self.Tc == self.T and self.Pc == self.P)):
# Do not have any tests for this - not good!
force_l = not self.phase == 'l'
force_g = not self.phase == 'g'
V = Vmin if Vmin != 1e100 else Vmax
self.set_properties_from_solution(self.T, self.P,
V, b,
self.delta, self.epsilon,
self.a_alpha, self.da_alpha_dT,
self.d2a_alpha_dT2,
force_l=force_l,
force_g=force_g)
self.phase = 'l/g'
elif good_root_count > 1:
if not only_g:
self.set_properties_from_solution(self.T, self.P, Vmin, b,
self.delta, self.epsilon,
self.a_alpha, self.da_alpha_dT,
self.d2a_alpha_dT2,
force_l=True)
if not only_l:
self.set_properties_from_solution(self.T, self.P, Vmax, b,
self.delta, self.epsilon,
self.a_alpha, self.da_alpha_dT,
self.d2a_alpha_dT2, force_g=True)
self.phase = 'l/g'
else:
# Even in the case of three real roots, it is still the min/max that make sense
print([self.T, self.P, b, self.delta, self.epsilon, self.a_alpha, 'coordinates of failure'])
if self.multicomponent:
extra = ', zs is %s' %(self.zs)
else:
extra = ''
raise ValueError(f'No acceptable roots were found; the roots are {Vs!s}, T is {str(self.T)} K, P is {str(self.P)} Pa, a_alpha is {str([self.a_alpha])}, b is {str([self.b])}{extra}')
[docs] def set_properties_from_solution(self, T, P, V, b, delta, epsilon, a_alpha,
da_alpha_dT, d2a_alpha_dT2,
force_l=False, force_g=False):
r'''Sets all interesting properties which can be calculated from an
EOS alone. Determines which phase the fluid is on its own; for details,
see `phase_identification_parameter`.
The list of properties set is as follows, with all properties suffixed
with '_l' or '_g'.
dP_dT, dP_dV, dV_dT, dV_dP, dT_dV, dT_dP, d2P_dT2, d2P_dV2, d2V_dT2,
d2V_dP2, d2T_dV2, d2T_dP2, d2V_dPdT, d2P_dTdV, d2T_dPdV, H_dep, S_dep,
G_dep and PIP.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
b : float
Coefficient calculated by EOS-specific method, [m^3/mol]
delta : float
Coefficient calculated by EOS-specific method, [m^3/mol]
epsilon : float
Coefficient calculated by EOS-specific method, [m^6/mol^2]
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K**2]
Returns
-------
phase : str
Either 'l' or 'g'
Notes
-----
The individual formulas for the derivatives and excess properties are
as follows. For definitions of `beta`, see `isobaric_expansion`;
for `kappa`, see isothermal_compressibility; for `Cp_minus_Cv`, see
`Cp_minus_Cv`; for `phase_identification_parameter`, see
`phase_identification_parameter`.
First derivatives; in part using the Triple Product Rule [2]_, [3]_:
.. math::
\left(\frac{\partial P}{\partial T}\right)_V = \frac{R}{V - b}
- \frac{a \frac{d \alpha{\left (T \right )}}{d T}}{V^{2} + V \delta
+ \epsilon}
.. math::
\left(\frac{\partial P}{\partial V}\right)_T = - \frac{R T}{\left(
V - b\right)^{2}} - \frac{a \left(- 2 V - \delta\right) \alpha{
\left (T \right )}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}
.. math::
\left(\frac{\partial V}{\partial T}\right)_P =-\frac{
\left(\frac{\partial P}{\partial T}\right)_V}{
\left(\frac{\partial P}{\partial V}\right)_T}
.. math::
\left(\frac{\partial V}{\partial P}\right)_T =-\frac{
\left(\frac{\partial V}{\partial T}\right)_P}{
\left(\frac{\partial P}{\partial T}\right)_V}
.. math::
\left(\frac{\partial T}{\partial V}\right)_P = \frac{1}
{\left(\frac{\partial V}{\partial T}\right)_P}
.. math::
\left(\frac{\partial T}{\partial P}\right)_V = \frac{1}
{\left(\frac{\partial P}{\partial T}\right)_V}
Second derivatives with respect to one variable; those of `T` and `V`
use identities shown in [1]_ and verified numerically:
.. math::
\left(\frac{\partial^2 P}{\partial T^2}\right)_V = - \frac{a
\frac{d^{2} \alpha{\left (T \right )}}{d T^{2}}}{V^{2} + V \delta
+ \epsilon}
.. math::
\left(\frac{\partial^2 P}{\partial V^2}\right)_T = 2 \left(\frac{
R T}{\left(V - b\right)^{3}} - \frac{a \left(2 V + \delta\right)^{
2} \alpha{\left (T \right )}}{\left(V^{2} + V \delta + \epsilon
\right)^{3}} + \frac{a \alpha{\left (T \right )}}{\left(V^{2} + V
\delta + \epsilon\right)^{2}}\right)
Second derivatives with respect to the other two variables; those of
`T` and `V` use identities shown in [1]_ and verified numerically:
.. math::
\left(\frac{\partial^2 P}{\partial T \partial V}\right) = - \frac{
R}{\left(V - b\right)^{2}} + \frac{a \left(2 V + \delta\right)
\frac{d \alpha{\left (T \right )}}{d T}}{\left(V^{2} + V \delta
+ \epsilon\right)^{2}}
Excess properties
.. math::
H_{dep} = \int_{\infty}^V \left[T\frac{\partial P}{\partial T}_V
- P\right]dV + PV - RT= P V - R T + \frac{2}{\sqrt{
\delta^{2} - 4 \epsilon}} \left(T a \frac{d \alpha{\left (T \right
)}}{d T} - a \alpha{\left (T \right )}\right) \operatorname{atanh}
{\left (\frac{2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}}
\right)}
.. math::
S_{dep} = \int_{\infty}^V\left[\frac{\partial P}{\partial T}
- \frac{R}{V}\right] dV + R\ln\frac{PV}{RT} = - R \ln{\left (V
\right )} + R \ln{\left (\frac{P V}{R T} \right )} + R \ln{\left
(V - b \right )} + \frac{2 a \frac{d\alpha{\left (T \right )}}{d T}
}{\sqrt{\delta^{2} - 4 \epsilon}} \operatorname{atanh}{\left (\frac
{2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right )}
.. math::
G_{dep} = H_{dep} - T S_{dep}
.. math::
C_{v, dep} = T\int_\infty^V \left(\frac{\partial^2 P}{\partial
T^2}\right) dV = - T a \left(\sqrt{\frac{1}{\delta^{2} - 4
\epsilon}} \ln{\left (V - \frac{\delta^{2}}{2} \sqrt{\frac{1}{
\delta^{2} - 4 \epsilon}} + \frac{\delta}{2} + 2 \epsilon \sqrt{
\frac{1}{\delta^{2} - 4 \epsilon}} \right )} - \sqrt{\frac{1}{
\delta^{2} - 4 \epsilon}} \ln{\left (V + \frac{\delta^{2}}{2}
\sqrt{\frac{1}{\delta^{2} - 4 \epsilon}} + \frac{\delta}{2}
- 2 \epsilon \sqrt{\frac{1}{\delta^{2} - 4 \epsilon}} \right )}
\right) \frac{d^{2} \alpha{\left (T \right )} }{d T^{2}}
.. math::
C_{p, dep} = (C_p-C_v)_{\text{from EOS}} + C_{v, dep} - R
References
----------
.. [1] Thorade, Matthis, and Ali Saadat. "Partial Derivatives of
Thermodynamic State Properties for Dynamic Simulation."
Environmental Earth Sciences 70, no. 8 (April 10, 2013): 3497-3503.
doi:10.1007/s12665-013-2394-z.
.. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th
edition. New York: McGraw-Hill Professional, 2000.
.. [3] Walas, Stanley M. Phase Equilibria in Chemical Engineering.
Butterworth-Heinemann, 1985.
'''
dP_dT, dP_dV, d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, Cv_dep = (
self.main_derivatives_and_departures(T, P, V, b, delta, epsilon,
a_alpha, da_alpha_dT,
d2a_alpha_dT2))
try:
dV_dP = 1.0/dP_dV
except:
dV_dP = inf
dT_dP = 1./dP_dT
dV_dT = -dP_dT*dV_dP
dT_dV = 1./dV_dT
Z = P*V*R_inv/T
Cp_dep = T*dP_dT*dV_dT + Cv_dep - R
G_dep = H_dep - T*S_dep
PIP = V*(d2P_dTdV*dT_dP - d2P_dV2*dV_dP) # phase_identification_parameter(V, dP_dT, dP_dV, d2P_dV2, d2P_dTdV)
# 1 + 1e-14 - allow a few dozen unums of toleranve to keep ideal gas model a gas
if force_l or (not force_g and PIP > 1.00000000000001):
(self.V_l, self.Z_l, self.PIP_l, self.dP_dT_l, self.dP_dV_l,
self.dV_dT_l, self.dV_dP_l, self.dT_dV_l, self.dT_dP_l,
self.d2P_dT2_l, self.d2P_dV2_l, self.d2P_dTdV_l, self.H_dep_l,
self.S_dep_l, self.G_dep_l, self.Cp_dep_l, self.Cv_dep_l) = (
V, Z, PIP, dP_dT, dP_dV, dV_dT, dV_dP, dT_dV, dT_dP,
d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, G_dep, Cp_dep,
Cv_dep)
return 'l'
else:
(self.V_g, self.Z_g, self.PIP_g, self.dP_dT_g, self.dP_dV_g,
self.dV_dT_g, self.dV_dP_g, self.dT_dV_g, self.dT_dP_g,
self.d2P_dT2_g, self.d2P_dV2_g, self.d2P_dTdV_g, self.H_dep_g,
self.S_dep_g, self.G_dep_g, self.Cp_dep_g, self.Cv_dep_g) = (
V, Z, PIP, dP_dT, dP_dV, dV_dT, dV_dP, dT_dV, dT_dP,
d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, G_dep, Cp_dep,
Cv_dep)
return 'g'
[docs] def a_alpha_and_derivatives(self, T, full=True, quick=True,
pure_a_alphas=True):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives.
Parameters
----------
T : float
Temperature, [K]
full : bool, optional
If False, calculates and returns only `a_alpha`, [-]
quick : bool, optional
Legary parameter being phased out [-]
pure_a_alphas : bool, optional
Whether or not to recalculate the a_alpha terms of pure components
(for the case of mixtures only) which stay the same as the
composition changes (i.e in a PT flash); does nothing in the case
of pure EOSs [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
'''
if full:
return self.a_alpha_and_derivatives_pure(T=T)
return self.a_alpha_pure(T)
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Dummy method to calculate :math:`a \alpha` and its first and second
derivatives. Should be implemented with the same function signature in
each EOS variant; this only raises a NotImplemented Exception.
Should return 'a_alpha', 'da_alpha_dT', and 'd2a_alpha_dT2'.
Parameters
----------
T : float
Temperature, [K]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
'''
raise NotImplementedError('a_alpha and its first and second derivatives '
'should be calculated by this method, in a user subclass.')
@property
def d3a_alpha_dT3(self):
r'''Method to calculate the third temperature derivative of
:math:`a \alpha`, [J^2/mol^2/Pa/K^3]. This parameter is needed for
some higher derivatives that are needed in some flash calculations.
Returns
-------
d3a_alpha_dT3 : float
Third temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^3]
'''
try:
return self._d3a_alpha_dT3
except AttributeError:
pass
self._d3a_alpha_dT3 = self.d3a_alpha_dT3_pure(self.T)
return self._d3a_alpha_dT3
[docs] def a_alpha_plot(self, Tmin=1e-4, Tmax=None, pts=1000, plot=True,
show=True):
r'''Method to create a plot of the :math:`a \alpha` parameter and its
first two derivatives. This easily allows identification of EOSs which
are displaying inconsistent behavior.
Parameters
----------
Tmin : float
Minimum temperature of calculation, [K]
Tmax : float
Maximum temperature of calculation, [K]
pts : int, optional
The number of temperature points to include [-]
plot : bool
If False, the calculated values and temperatures are returned
without plotting the data, [-]
show : bool
Whether or not the plot should be rendered and shown; a handle to
it is returned if `plot` is True for other purposes such as saving
the plot to a file, [-]
Returns
-------
Ts : list[float]
Logarithmically spaced temperatures in specified range, [K]
a_alpha : list[float]
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : list[float]
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : list[float]
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
fig : matplotlib.figure.Figure
Plotted figure, only returned if `plot` is True, [-]
'''
if Tmax is None:
if self.multicomponent:
Tc = self.pseudo_Tc
else:
Tc = self.Tc
Tmax = Tc*10
Ts = logspace(log10(Tmin), log10(Tmax), pts)
a_alphas = []
da_alphas = []
d2a_alphas = []
for T in Ts:
v, d1, d2 = self.a_alpha_and_derivatives(T, full=True)
a_alphas.append(v)
da_alphas.append(d1)
d2a_alphas.append(d2)
if plot:
import matplotlib.pyplot as plt
fig = plt.figure()
ax1 = fig.add_subplot(111)
ax1.set_xlabel('Temperature [K]')
ln0 = ax1.plot(Ts, a_alphas, 'r', label=r'$a \alpha$ [J^2/mol^2/Pa]')
ln2 = ax1.plot(Ts, d2a_alphas, 'g', label='Second derivative [J^2/mol^2/Pa/K^2]')
ax1.set_yscale('log')
ax1.set_ylabel(r'$a \alpha$ and $\frac{\partial (a \alpha)^2}{\partial T^2}$')
ax2 = ax1.twinx()
ax2.set_yscale('symlog')
ln1 = ax2.plot(Ts, da_alphas, 'b', label='First derivative [J^2/mol^2/Pa/K]')
ax2.set_ylabel(r'$\frac{\partial a \alpha}{\partial T}$')
ax1.set_title(fr'$a \alpha$ vs temperature; range {max(a_alphas):.4g} to {min(a_alphas):.4g}')
lines = ln0 + ln1 + ln2
labels = [l.get_label() for l in lines]
ax1.legend(lines, labels, loc=9, bbox_to_anchor=(0.5,-0.18))
if show:
plt.show()
return Ts, a_alphas, da_alphas, d2a_alphas, fig
return Ts, a_alphas, da_alphas, d2a_alphas
[docs] def solve_T(self, P, V, solution=None):
'''Generic method to calculate `T` from a specified `P` and `V`.
Provides SciPy's `newton` solver, and iterates to solve the general
equation for `P`, recalculating `a_alpha` as a function of temperature
using `a_alpha_and_derivatives` each iteration.
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
'l' or 'g' to specify a liquid of vapor solution (if one exists);
if None, will select a solution more likely to be real (closer to
STP, attempting to avoid temperatures like 60000 K or 0.0001 K).
Returns
-------
T : float
Temperature, [K]
'''
high_prec = type(V) is not float
denominator_inv = 1.0/(V*V + self.delta*V + self.epsilon)
V_minus_b_inv = 1.0/(V-self.b)
self.no_T_spec = True
# dP_dT could be added to use a derivative-based method, however it is
# quite costly in comparison to the extra evaluations because it
# requires the temperature derivative of da_alpha_dT
def to_solve(T):
a_alpha = self.a_alpha_and_derivatives(T, full=False)
P_calc = R*T*V_minus_b_inv - a_alpha*denominator_inv
err = P_calc - P
return err
def to_solve_newton(T):
a_alpha, da_alpha_dT, _ = self.a_alpha_and_derivatives(T, full=True)
P_calc = R*T*V_minus_b_inv - a_alpha*denominator_inv
err = P_calc - P
derr_dT = R*V_minus_b_inv - denominator_inv*da_alpha_dT
return err, derr_dT
# import matplotlib.pyplot as plt
# xs = np.logspace(np.log10(1), np.log10(1e12), 15000)
# ys = np.abs([to_solve(T) for T in xs])
# plt.loglog(xs, ys)
# plt.show()
# max(ys), min(ys)
T_guess_ig = P*V*R_inv
T_guess_liq = P*V*R_inv*1000.0 # Compressibility factor of 0.001 for liquids
err_ig = to_solve(T_guess_ig)
err_liq = to_solve(T_guess_liq)
base_tol = 1e-12
if high_prec:
base_tol = 1e-18
T_brenth, T_secant = None, None
if err_ig*err_liq < 0.0 and T_guess_liq < 3e4:
try:
T_brenth = brenth(to_solve, T_guess_ig, T_guess_liq, xtol=base_tol,
fa=err_ig, fb=err_liq)
# Check the error
err = to_solve(T_brenth)
except:
pass
# if abs(err/P) < 1e-7:
# return T_brenth
if abs(err_ig) < abs(err_liq) or T_guess_liq > 20000 or solution == 'g':
T_guess = T_guess_ig
f0 = err_ig
else:
T_guess = T_guess_liq
f0 = err_liq
# T_guess = self.Tc*0.5
# ytol=T_guess*1e-9,
try:
T_secant = secant(to_solve, T_guess, low=1e-12, xtol=base_tol, same_tol=1e4, f0=f0)
except:
T_guess = T_guess_ig if T_guess != T_guess_ig else T_guess_liq
try:
T_secant = secant(to_solve, T_guess, low=1e-12, xtol=base_tol, same_tol=1e4, f0=f0)
except:
if T_brenth is None:
# Hardcoded limits, all the cleverness sometimes does not work
T_brenth = brenth(to_solve, 1e-3, 1e4, xtol=base_tol)
if solution is not None:
if T_brenth is None or (T_secant is not None and isclose(T_brenth, T_secant, rel_tol=1e-7)):
if T_secant is not None:
attempt_bounds = [(1e-3, T_secant-1e-5), (T_secant+1e-3, 1e4), (T_secant+1e-3, 1e5)]
else:
attempt_bounds = [(1e-3, 1e4), (1e4, 1e5)]
if T_guess_liq > 1e5:
attempt_bounds.append((1e4, T_guess_liq))
attempt_bounds.append((T_guess_liq, T_guess_liq*10))
for low, high in attempt_bounds:
try:
T_brenth = brenth(to_solve, low, high, xtol=base_tol)
break
except:
pass
if T_secant is None:
if T_secant is not None:
attempt_bounds = [(1e-3, T_brenth-1e-5), (T_brenth+1e-3, 1e4), (T_brenth+1e-3, 1e5)]
else:
attempt_bounds = [(1e4, 1e5), (1e-3, 1e4)]
if T_guess_liq > 1e5:
attempt_bounds.append((1e4, T_guess_liq))
attempt_bounds.append((T_guess_liq, T_guess_liq*10))
for low, high in attempt_bounds:
try:
T_secant = brenth(to_solve, low, high, xtol=base_tol)
break
except:
pass
try:
del self.a_alpha_ijs
del self.a_alpha_roots
del self.a_alpha_ij_roots_inv
except AttributeError:
pass
if T_secant is not None:
T_secant = float(T_secant)
if T_brenth is not None:
T_brenth = float(T_brenth)
if solution is not None:
if (T_secant is not None and T_brenth is not None):
if solution == 'g':
return max(T_brenth, T_secant)
else:
return min(T_brenth, T_secant)
if T_brenth is None:
return T_secant
elif T_brenth is not None and T_secant is not None and (abs(T_brenth - 298.15) < abs(T_secant - 298.15)):
return T_brenth
elif T_secant is not None:
return T_secant
return T_brenth
# return min(T_brenth, T_secant)
# Default method
# volume_solutions = volume_solutions_NR#volume_solutions_numpy#volume_solutions_NR
# volume_solutions = staticmethod(volume_solutions_numpy)
# volume_solutions = volume_solutions_fast
# volume_solutions = staticmethod(volume_solutions_Cardano)
volume_solutions = staticmethod(volume_solutions_halley)
# volume_solutions = staticmethod(volume_solutions_doubledouble_float)
volume_solutions_mp = staticmethod(volume_solutions_mpmath)
# Solver which actually has the roots
volume_solutions_full = staticmethod(volume_solutions_NR)
# volume_solutions = volume_solutions_mpmath_float
@property
def mpmath_volumes(self):
r'''Method to calculate to a high precision the exact roots to the
cubic equation, using `mpmath`.
Returns
-------
Vs : tuple[mpf]
3 Real or not real volumes as calculated by `mpmath`, [m^3/mol]
Notes
-----
Examples
--------
>>> eos = PRTranslatedTwu(T=300, P=1e5, Tc=512.5, Pc=8084000.0, omega=0.559, alpha_coeffs=(0.694911, 0.9199, 1.7), c=-1e-6)
>>> eos.mpmath_volumes
(mpf('0.0000489261705320261435106226558966745'), mpf('0.000541508154451321441068958547812526'), mpf('0.0243149463942697410611501615357228'))
'''
return volume_solutions_mpmath(self.T, self.P, self.b, self.delta, self.epsilon, self.a_alpha)
@property
def mpmath_volumes_float(self):
r'''Method to calculate real roots of a cubic equation, using `mpmath`,
but returned as floats.
Returns
-------
Vs : list[float]
All volumes calculated by `mpmath`, [m^3/mol]
Notes
-----
Examples
--------
>>> eos = PRTranslatedTwu(T=300, P=1e5, Tc=512.5, Pc=8084000.0, omega=0.559, alpha_coeffs=(0.694911, 0.9199, 1.7), c=-1e-6)
>>> eos.mpmath_volumes_float
((4.892617053202614e-05+0j), (0.0005415081544513214+0j), (0.024314946394269742+0j))
'''
return volume_solutions_mpmath_float(self.T, self.P, self.b, self.delta, self.epsilon, self.a_alpha)
@property
def mpmath_volume_ratios(self):
r'''Method to compare, as ratios, the volumes of the implemented
cubic solver versus those calculated using `mpmath`.
Returns
-------
ratios : list[mpc]
Either 1 or 3 volume ratios as calculated by `mpmath`, [-]
Notes
-----
Examples
--------
>>> eos = PRTranslatedTwu(T=300, P=1e5, Tc=512.5, Pc=8084000.0, omega=0.559, alpha_coeffs=(0.694911, 0.9199, 1.7), c=-1e-6)
>>> eos.mpmath_volume_ratios
(mpc(real='0.99999999999999995', imag='0.0'), mpc(real='0.999999999999999965', imag='0.0'), mpc(real='1.00000000000000005', imag='0.0'))
'''
return tuple(i/j for i, j in zip(self.sorted_volumes, self.mpmath_volumes))
[docs] def Vs_mpmath(self):
r'''Method to calculate real roots of a cubic equation, using `mpmath`.
Returns
-------
Vs : list[mpf]
Either 1 or 3 real volumes as calculated by `mpmath`, [m^3/mol]
Notes
-----
Examples
--------
>>> eos = PRTranslatedTwu(T=300, P=1e5, Tc=512.5, Pc=8084000.0, omega=0.559, alpha_coeffs=(0.694911, 0.9199, 1.7), c=-1e-6)
>>> eos.Vs_mpmath()
[mpf('0.0000489261705320261435106226558966745'), mpf('0.000541508154451321441068958547812526'), mpf('0.0243149463942697410611501615357228')]
'''
Vs = self.mpmath_volumes
good_roots = [i.real for i in Vs if (i.real > 0.0 and abs(i.imag/i.real) < 1E-12)]
good_roots.sort()
return good_roots
[docs] def volume_error(self):
r'''Method to calculate the relative absolute error in the calculated
molar volumes. This is computed with `mpmath`. If the number of real
roots is different between mpmath and the implemented solver, an
error of 1 is returned.
Parameters
----------
T : float
Temperature, [K]
Returns
-------
error : float
relative absolute error in molar volumes , [-]
Notes
-----
Examples
--------
>>> eos = PRTranslatedTwu(T=300, P=1e5, Tc=512.5, Pc=8084000.0, omega=0.559, alpha_coeffs=(0.694911, 0.9199, 1.7), c=-1e-6)
>>> eos.volume_error()
5.2192e-17
'''
# Vs_good, Vs = self.mpmath_volumes, self.sorted_volumes
# Compare the reals only if mpmath has the imaginary roots
Vs_good = self.volume_solutions_mp(self.T, self.P, self.b, self.delta, self.epsilon, self.a_alpha)
Vs_filtered = [i.real for i in Vs_good if (i.real ==0 or abs(i.imag/i.real) < 1E-20) and i.real > self.b]
if len(Vs_filtered) in (2, 3):
two_roots_mpmath = True
Vl_mpmath, Vg_mpmath = min(Vs_filtered), max(Vs_filtered)
else:
if hasattr(self, 'V_l') and hasattr(self, 'V_g'):
# Wrong number of roots!
return 1
elif hasattr(self, 'V_l'):
Vl_mpmath = Vs_filtered[0]
elif hasattr(self, 'V_g'):
Vg_mpmath = Vs_filtered[0]
two_roots_mpmath = False
err = 0
if two_roots_mpmath:
if (not hasattr(self, 'V_l') or not hasattr(self, 'V_g')):
return 1.0
# Important not to confuse the roots and also to not consider the third root
try:
Vl = self.V_l
err_i = abs((Vl - Vl_mpmath)/Vl_mpmath)
if err_i > err:
err = err_i
except:
pass
try:
Vg = self.V_g
err_i = abs((Vg - Vg_mpmath)/Vg_mpmath)
if err_i > err:
err = err_i
except:
pass
return float(err)
def _mpmath_volume_matching(self, V):
'''Helper method which, given one of the three molar volume solutions
of the EOS, returns the mpmath molar volume which is nearest it.
'''
Vs = self.mpmath_volumes
rel_diffs = []
for Vi in Vs:
err = abs(Vi.real - V.real) + abs(Vi.imag - V.imag)
rel_diffs.append(err)
return Vs[rel_diffs.index(min(rel_diffs))]
@property
def V_l_mpmath(self):
r'''The molar volume of the liquid phase calculated with `mpmath` to
a higher precision, [m^3/mol]. This is useful for validating the
cubic root solver(s). It is not quite a true arbitrary solution to the
EOS, because the constants `b`,`epsilon`, `delta` and `a_alpha` as well
as the input arguments `T` and `P` are not calculated with arbitrary
precision. This is a feature when comparing the volume solution
algorithms however as they work with the same finite-precision
variables.
'''
if not hasattr(self, 'V_l'):
raise ValueError("Not solved for that volume")
return self._mpmath_volume_matching(self.V_l)
@property
def V_g_mpmath(self):
r'''The molar volume of the gas phase calculated with `mpmath` to
a higher precision, [m^3/mol]. This is useful for validating the
cubic root solver(s). It is not quite a true arbitrary solution to the
EOS, because the constants `b`,`epsilon`, `delta` and `a_alpha` as well
as the input arguments `T` and `P` are not calculated with arbitrary
precision. This is a feature when comparing the volume solution
algorithms however as they work with the same finite-precision
variables.
'''
if not hasattr(self, 'V_g'):
raise ValueError("Not solved for that volume")
return self._mpmath_volume_matching(self.V_g)
# def fugacities_mpmath(self, dps=30):
# # At one point thought maybe the fugacity equation was the source of error.
# # No. always the volume equation.
# import mpmath as mp
# mp.mp.dps = dps
# R_mp = mp.mpf(R)
# b, T, P, epsilon, delta, a_alpha = self.b, self.T, self.P, self.epsilon, self.delta, self.a_alpha
# b, T, P, epsilon, delta, a_alpha = [mp.mpf(i) for i in [b, T, P, epsilon, delta, a_alpha]]
#
# Vs_good = volume_solutions_mpmath(self.T, self.P, self.b, self.delta, self.epsilon, self.a_alpha)
# Vs_filtered = [i.real for i in Vs_good if (i.real == 0 or abs(i.imag/i.real) < 1E-20) and i.real > self.b]
#
# if len(Vs_filtered) in (2, 3):
# Vs = min(Vs_filtered), max(Vs_filtered)
# else:
# if hasattr(self, 'V_l') and hasattr(self, 'V_g'):
# # Wrong number of roots!
# raise ValueError("Error")
# Vs = Vs_filtered
## elif hasattr(self, 'V_l'):
## Vs = Vs_filtered[0]
## elif hasattr(self, 'V_g'):
## Vg_mpmath = Vs_filtered[0]
#
# log, exp, atanh, sqrt = mp.log, mp.exp, mp.atanh, mp.sqrt
#
# return [P*exp((P*V + R_mp*T*log(V) - R_mp*T*log(P*V/(R_mp*T)) - R_mp*T*log(V - b)
# - R_mp*T - 2*a_alpha*atanh(2*V/sqrt(delta**2 - 4*epsilon)
# + delta/sqrt(delta**2 - 4*epsilon)).real/sqrt(delta**2 - 4*epsilon))/(R_mp*T))
# for V in Vs]
[docs] def volume_errors(self, Tmin=1e-4, Tmax=1e4, Pmin=1e-2, Pmax=1e9,
pts=50, plot=False, show=False, trunc_err_low=1e-18,
trunc_err_high=1.0, color_map=None, timing=False):
r'''Method to create a plot of the relative absolute error in the
cubic volume solution as compared to a higher-precision calculation.
This method is incredible valuable for the development of more reliable
floating-point based cubic solutions.
Parameters
----------
Tmin : float
Minimum temperature of calculation, [K]
Tmax : float
Maximum temperature of calculation, [K]
Pmin : float
Minimum pressure of calculation, [Pa]
Pmax : float
Maximum pressure of calculation, [Pa]
pts : int, optional
The number of points to include in both the `x` and `y` axis;
the validation calculation is slow, so increasing this too much
is not advisable, [-]
plot : bool
If False, the calculated errors are returned without plotting
the data, [-]
show : bool
Whether or not the plot should be rendered and shown; a handle to
it is returned if `plot` is True for other purposes such as saving
the plot to a file, [-]
trunc_err_low : float
Minimum plotted error; values under this are rounded to 0, [-]
trunc_err_high : float
Maximum plotted error; values above this are rounded to 1, [-]
color_map : matplotlib.cm.ListedColormap
Matplotlib colormap object, [-]
timing : bool
If True, plots the time taken by the volume root calculations
themselves; this can reveal whether the solvers are taking fast or
slow paths quickly, [-]
Returns
-------
errors : list[list[float]]
Relative absolute errors in the volume calculation (or timings in
seconds if `timing` is True), [-]
fig : matplotlib.figure.Figure
Plotted figure, only returned if `plot` is True, [-]
'''
if timing:
try:
from time import perf_counter
except:
from time import clock as perf_counter
Ts = logspace(log10(Tmin), log10(Tmax), pts)
Ps = logspace(log10(Pmin), log10(Pmax), pts)
kwargs = {}
if hasattr(self, 'zs'):
kwargs['zs'] = self.zs
kwargs['fugacities'] = False
errs = []
for T in Ts:
err_row = []
for P in Ps:
kwargs['T'] = T
kwargs['P'] = P
try:
obj = self.to(**kwargs)
except Exception as e:
print(f'Failed to go to point, kwargs={kwargs} with exception {e}')
# So bad we failed to calculate a real point
val = 1.0
if timing:
t0 = perf_counter()
obj.volume_solutions(obj.T, obj.P, obj.b, obj.delta, obj.epsilon, obj.a_alpha)
val = perf_counter() - t0
else:
val = float(obj.volume_error())
if val > 1e-7:
print([obj.T, obj.P, obj.b, obj.delta, obj.epsilon, obj.a_alpha, 'coordinates of failure', obj])
err_row.append(val)
errs.append(err_row)
if plot:
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.colors import LogNorm
X, Y = np.meshgrid(Ts, Ps)
z = np.array(errs).T
fig, ax = plt.subplots()
if not timing:
if trunc_err_low is not None:
z[np.where(abs(z) < trunc_err_low)] = trunc_err_low
if trunc_err_high is not None:
z[np.where(abs(z) > trunc_err_high)] = trunc_err_high
if color_map is None:
color_map = cm.viridis
if not timing:
norm = LogNorm(vmin=trunc_err_low, vmax=trunc_err_high)
else:
z *= 1e-6
norm = None
im = ax.pcolormesh(X, Y, z, cmap=color_map, norm=norm)
cbar = fig.colorbar(im, ax=ax)
if timing:
cbar.set_label('Time [us]')
else:
cbar.set_label('Relative error')
ax.set_yscale('log')
ax.set_xscale('log')
ax.set_xlabel('T [K]')
ax.set_ylabel('P [Pa]')
max_err = np.max(errs)
if trunc_err_low is not None and max_err < trunc_err_low:
max_err = 0
if trunc_err_high is not None and max_err > trunc_err_high:
max_err = trunc_err_high
if timing:
ax.set_title('Volume timings; max %.2e us' %(max_err*1e6))
else:
ax.set_title('Volume solution validation; max err %.4e' %(max_err))
if show:
plt.show()
return errs, fig
else:
return errs
[docs] def PT_surface_special(self, Tmin=1e-4, Tmax=1e4, Pmin=1e-2, Pmax=1e9,
pts=50, show=False, color_map=None,
mechanical=True, pseudo_critical=True, Psat=True,
determinant_zeros=True, phase_ID_transition=True,
base_property='V', base_min=None, base_max=None,
base_selection='Gmin'):
r'''Method to create a plot of the special curves of a pure fluid -
vapor pressure, determinant zeros, pseudo critical point,
and mechanical critical point.
The color background is a plot of the molar volume (by default) which
has the minimum Gibbs energy (by default). If shown with a sufficient
number of points, the curve between vapor and liquid should be shown
smoothly.
When called on a mixture, this method does not have physical
significance for the `Psat` term.
Parameters
----------
Tmin : float, optional
Minimum temperature of calculation, [K]
Tmax : float, optional
Maximum temperature of calculation, [K]
Pmin : float, optional
Minimum pressure of calculation, [Pa]
Pmax : float, optional
Maximum pressure of calculation, [Pa]
pts : int, optional
The number of points to include in both the `x` and `y` axis [-]
show : bool, optional
Whether or not the plot should be rendered and shown; a handle to
it is returned if `plot` is True for other purposes such as saving
the plot to a file, [-]
color_map : matplotlib.cm.ListedColormap, optional
Matplotlib colormap object, [-]
mechanical : bool, optional
Whether or not to include the mechanical critical point; this is
the same as the critical point for a pure compound but not for a
mixture, [-]
pseudo_critical : bool, optional
Whether or not to include the pseudo critical point; this is
the same as the critical point for a pure compound but not for a
mixture, [-]
Psat : bool, optional
Whether or not to include the vapor pressure curve; for mixtures
this is neither the bubble nor dew curve, but rather a hypothetical
one which uses the same equation as the pure components, [-]
determinant_zeros : bool, optional
Whether or not to include a curve showing when the EOS's
determinant hits zero, [-]
phase_ID_transition : bool, optional
Whether or not to show a curve of where the PIP hits 1 exactly, [-]
base_property : str, optional
The property which should be plotted; '_l' and '_g' are added
automatically according to the selected phase, [-]
base_min : float, optional
If specified, the `base` property will values will be limited to
this value at the minimum, [-]
base_max : float, optional
If specified, the `base` property will values will be limited to
this value at the maximum, [-]
base_selection : str, optional
For the base property, there are often two possible phases and but
only one value can be plotted; use 'l' to pefer liquid-like values,
'g' to prefer gas-like values, and 'Gmin' to prefer values of the
phase with the lowest Gibbs energy, [-]
Returns
-------
fig : matplotlib.figure.Figure
Plotted figure, only returned if `plot` is True, [-]
'''
Ts = logspace(log10(Tmin), log10(Tmax), pts)
Ps = logspace(log10(Pmin), log10(Pmax), pts)
kwargs = {}
if hasattr(self, 'zs'):
kwargs['zs'] = self.zs
l_prop = base_property + '_l'
g_prop = base_property + '_g'
base_positive = True
# Are we an ideal-gas or multicomponent?
if self.Zc == 1.:
phase_ID_transition = False
Psat = False
Vs = []
for T in Ts:
V_row = []
for P in Ps:
kwargs['T'] = T
kwargs['P'] = P
obj = self.to(**kwargs)
if obj.phase == 'l/g':
if base_selection == 'Gmin':
V = getattr(obj, l_prop) if obj.G_dep_l < obj.G_dep_g else getattr(obj, g_prop)
elif base_selection == 'l':
V = getattr(obj, l_prop)
elif base_selection == 'g':
V = getattr(obj, g_prop)
else:
raise ValueError("Unknown value for base_selection")
elif obj.phase == 'l':
V = getattr(obj, l_prop)
else:
V = getattr(obj, g_prop)
if base_max is not None and V > base_max: V = base_max
if base_min is not None and V < base_min: V = base_min
V_row.append(V)
base_positive = base_positive and V > 0.0
Vs.append(V_row)
if self.multicomponent:
Tc, Pc = self.pseudo_Tc, self.pseudo_Pc
else:
Tc, Pc = self.Tc, self.Pc
if Psat:
Pmax_Psat = min(Pc, Pmax)
Pmin_Psat = max(1e-20, Pmin)
Tmin_Psat, Tmax_Psat = self.Tsat(Pmin_Psat), self.Tsat(Pmax_Psat)
if Tmin_Psat < Tmin or Tmin_Psat > Tmax: Tmin_Psat = Tmin
if Tmax_Psat > Tmax or Tmax_Psat < Tmin: Tmax_Psat = Tmax
Ts_Psats = []
Psats = []
for T in linspace(Tmin_Psat, Tmax_Psat, pts):
P = self.Psat(T)
Ts_Psats.append(T)
Psats.append(P)
if phase_ID_transition:
Pmin_Psat = max(1e-20, Pmin)
Tmin_ID = self.Tsat(Pmin_Psat)
Tmax_ID = Tmax
phase_ID_Ts = linspace(Tmin_ID, Tmax_ID, pts)
low_P_limit = min(1e-4, Pmin)
phase_ID_Ps = [self.P_PIP_transition(T, low_P_limit=low_P_limit)
for T in phase_ID_Ts]
if mechanical:
if self.multicomponent:
TP_mechanical = self.mechanical_critical_point()
else:
TP_mechanical = (Tc, Pc)
if determinant_zeros:
lows_det_Ps, high_det_Ps, Ts_dets_low, Ts_dets_high = [], [], [], []
for T in Ts:
a_alpha = self.a_alpha_and_derivatives(T, full=False)
P_dets = self.P_discriminant_zeros_analytical(T=T, b=self.b, delta=self.delta,
epsilon=self.epsilon, a_alpha=a_alpha, valid=True)
if P_dets:
P_det_min = min(P_dets)
P_det_max = max(P_dets)
if Pmin <= P_det_min <= Pmax:
lows_det_Ps.append(P_det_min)
Ts_dets_low.append(T)
if Pmin <= P_det_max <= Pmax:
high_det_Ps.append(P_det_max)
Ts_dets_high.append(T)
# if plot:
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.colors import LogNorm
X, Y = np.meshgrid(Ts, Ps)
z = np.array(Vs).T
fig, ax = plt.subplots()
if color_map is None:
color_map = cm.viridis
norm = LogNorm() if base_positive else None
im = ax.pcolormesh(X, Y, z, cmap=color_map, norm=norm)
cbar = fig.colorbar(im, ax=ax)
cbar.set_label('%s' %base_property)
if Psat:
plt.plot(Ts_Psats, Psats, label='Psat')
if determinant_zeros:
plt.plot(Ts_dets_low, lows_det_Ps, label='Low trans')
plt.plot(Ts_dets_high, high_det_Ps, label='High trans')
if pseudo_critical:
plt.plot([Tc], [Pc], 'x', label='Pseudo crit')
if mechanical:
plt.plot([TP_mechanical[0]], [TP_mechanical[1]], 'o', label='Mechanical')
if phase_ID_transition:
plt.plot(phase_ID_Ts, phase_ID_Ps, label='PIP=1')
ax.set_yscale('log')
ax.set_xscale('log')
ax.set_xlabel('T [K]')
ax.set_ylabel('P [Pa]')
if (Psat or determinant_zeros or pseudo_critical or mechanical
or phase_ID_transition):
plt.legend()
ax.set_title('%s vs minimum Gibbs validation' %(base_property))
if show:
plt.show()
return fig
[docs] def saturation_prop_plot(self, prop, Tmin=None, Tmax=None, pts=100,
plot=False, show=False, both=False):
r'''Method to create a plot of a specified property of the EOS along
the (pure component) saturation line.
Parameters
----------
prop : str
Property to be used; such as 'H_dep_l' ( when `both` is False)
or 'H_dep' (when `both` is True), [-]
Tmin : float
Minimum temperature of calculation; if this is too low the
saturation routines will stop converging, [K]
Tmax : float
Maximum temperature of calculation; cannot be above the critical
temperature, [K]
pts : int, optional
The number of temperature points to include [-]
plot : bool
If False, the calculated values and temperatures are returned
without plotting the data, [-]
show : bool
Whether or not the plot should be rendered and shown; a handle to
it is returned if `plot` is True for other purposes such as saving
the plot to a file, [-]
both : bool
When true, append '_l' and '_g' and draw both the liquid and vapor
property specified and return two different sets of values.
Returns
-------
Ts : list[float]
Logarithmically spaced temperatures in specified range, [K]
props : list[float]
The property specified if `both` is False; otherwise, the liquid
properties, [various]
props_g : list[float]
The gas properties, only returned if `both` is True, [various]
fig : matplotlib.figure.Figure
Plotted figure, only returned if `plot` is True, [-]
'''
if Tmax is None:
if self.multicomponent:
Tmax = self.pseudo_Tc
else:
Tmax = self.Tc
if Tmin is None:
Tmin = self.Tsat(1e-5)
Ts = logspace(log10(Tmin), log10(Tmax), pts)
kwargs = {}
if hasattr(self, 'zs'):
kwargs['zs'] = self.zs
props = []
if both:
props2 = []
prop_l = prop + '_l'
prop_g = prop + '_g'
for T in Ts:
kwargs['T'] = T
kwargs['P'] = self.Psat(T)
obj = self.to(**kwargs)
if both:
v = getattr(obj, prop_l)
try:
v = v()
except:
pass
props.append(v)
v = getattr(obj, prop_g)
try:
v = v()
except:
pass
props2.append(v)
else:
v = getattr(obj, prop)
try:
v = v()
except:
pass
props.append(v)
if plot:
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
if both:
plt.plot(Ts, props, label='Liquid')
plt.plot(Ts, props2, label='Gas')
plt.legend()
else:
plt.plot(Ts, props)
ax.set_xlabel('Temperature [K]')
ax.set_ylabel(r'%s' %(prop))
ax.set_title(r'Saturation %s curve' %(prop))
if show:
plt.show()
if both:
return Ts, props, props2, fig
return Ts, props, fig
if both:
return Ts, props, props2
return Ts, props
[docs] def Psat_errors(self, Tmin=None, Tmax=None, pts=50, plot=False, show=False,
trunc_err_low=1e-18, trunc_err_high=1.0, Pmin=1e-100):
r'''Method to create a plot of vapor pressure and the relative error
of its calculation vs. the iterative `polish` approach.
Parameters
----------
Tmin : float
Minimum temperature of calculation; if this is too low the
saturation routines will stop converging, [K]
Tmax : float
Maximum temperature of calculation; cannot be above the critical
temperature, [K]
pts : int, optional
The number of temperature points to include [-]
plot : bool
If False, the solution is returned without plotting the data, [-]
show : bool
Whether or not the plot should be rendered and shown; a handle to
it is returned if `plot` is True for other purposes such as saving
the plot to a file, [-]
trunc_err_low : float
Minimum plotted error; values under this are rounded to 0, [-]
trunc_err_high : float
Maximum plotted error; values above this are rounded to 1, [-]
Pmin : float
Minimum pressure for the solution to work on, [Pa]
Returns
-------
errors : list[float]
Absolute relative errors, [-]
Psats_num : list[float]
Vapor pressures calculated to full precision, [Pa]
Psats_fit : list[float]
Vapor pressures calculated with the fast solution, [Pa]
fig : matplotlib.figure.Figure
Plotted figure, only returned if `plot` is True, [-]
'''
try:
Tc = self.Tc
except:
Tc = self.pseudo_Tc
if Tmax is None:
Tmax = Tc
if Tmin is None:
Tmin = .1*Tc
try:
# Can we get the direct temperature for Pmin
if Pmin is not None:
Tmin_Pmin = self.Tsat(P=Pmin, polish=True)
except:
Tmin_Pmin = None
if Tmin_Pmin is not None:
Tmin = max(Tmin, Tmin_Pmin)
Ts = logspace(log10(Tmin), log10(Tmax), int(pts/3))
Ts[-1] = Tmax
Ts_mid = linspace(Tmin, Tmax, int(pts/3))
Ts_high = linspace(Tmax*.99, Tmax, int(pts/3))
Ts = list(sorted(Ts_high + Ts + Ts_mid))
Ts_worked, Psats_num, Psats_fit = [], [], []
for T in Ts:
failed = False
try:
Psats_fit.append(self.Psat(T, polish=False))
except NoSolutionError:
# Trust the fit - do not continue if no good
continue
except Exception as e:
raise ValueError("Failed to converge at %.16f K with unexpected error" %(T), e, self)
try:
Psat_polished = self.Psat(T, polish=True)
Psats_num.append(Psat_polished)
except Exception as e:
failed = True
raise ValueError("Failed to converge at %.16f K with unexpected error" %(T), e, self)
Ts_worked.append(T)
Ts = Ts_worked
errs = np.array([abs(i-j)/i for i, j in zip(Psats_num, Psats_fit)])
if plot:
import matplotlib.pyplot as plt
fig, ax1 = plt.subplots()
ax2 = ax1.twinx()
if trunc_err_low is not None:
errs[np.where(abs(errs) < trunc_err_low)] = trunc_err_low
if trunc_err_high is not None:
errs[np.where(abs(errs) > trunc_err_high)] = trunc_err_high
Trs = np.array(Ts)/Tc
ax1.plot(Trs, errs)
ax2.plot(Trs, Psats_num)
ax2.plot(Trs, Psats_fit)
ax1.set_yscale('log')
ax1.set_xscale('log')
ax2.set_yscale('log')
ax2.set_xscale('log')
ax1.set_xlabel('Tr [-]')
ax1.set_ylabel('AARD [-]')
ax2.set_ylabel('Psat [Pa]')
max_err = np.max(errs)
if trunc_err_low is not None and max_err < trunc_err_low:
max_err = 0
if trunc_err_high is not None and max_err > trunc_err_high:
max_err = trunc_err_high
ax1.set_title('Vapor pressure validation; max rel err %.4e' %(max_err))
if show:
plt.show()
return errs, Psats_num, Psats_fit, fig
else:
return errs, Psats_num, Psats_fit
# def PIP_map(self, Tmin=1e-4, Tmax=1e4, Pmin=1e-2, Pmax=1e9,
# pts=50, plot=False, show=False, color_map=None):
# # TODO rename PIP_ID_map or add flag to change if it plots PIP or bools.
# # TODO add doc
# Ts = logspace(log10(Tmin), log10(Tmax), pts)
# Ps = logspace(log10(Pmin), log10(Pmax), pts)
# kwargs = {}
# if hasattr(self, 'zs'):
# kwargs['zs'] = self.zs
#
# PIPs = []
# for T in Ts:
# PIP_row = []
# for P in Ps:
# kwargs['T'] = T
# kwargs['P'] = P
# obj = self.to(**kwargs)
## v = obj.discriminant
## # need make negatives under 1, positive above 1
## if v > 0.0:
## v = (1.0 + (1e10 - 1.0)/(1.0 + trunc_exp(-v)))
## else:
## v = (1e-10 + (1.0 - 1e-10)/(1.0 + trunc_exp(-v)))
#
# if obj.phase == 'l/g':
# v = 1
# elif obj.phase == 'g':
# v = 0
# elif obj.phase == 'l':
# v = 2
# PIP_row.append(v)
# PIPs.append(PIP_row)
#
# if plot:
# import matplotlib.pyplot as plt
# from matplotlib import ticker, cm
# from matplotlib.colors import LogNorm
# X, Y = np.meshgrid(Ts, Ps)
# z = np.array(PIPs).T
# fig, ax = plt.subplots()
# if color_map is None:
# color_map = cm.viridis
#
# im = ax.pcolormesh(X, Y, z, cmap=color_map,
## norm=LogNorm(vmin=1e-10, vmax=1e10)
# )
# cbar = fig.colorbar(im, ax=ax)
# cbar.set_label('PIP')
#
# ax.set_yscale('log')
# ax.set_xscale('log')
# ax.set_xlabel('T [K]')
# ax.set_ylabel('P [Pa]')
#
#
# ax.set_title('Volume root/phase ID validation')
# if show:
# plt.show()
#
# return PIPs, fig
def derivatives_and_departures(self, T, P, V, b, delta, epsilon, a_alpha, da_alpha_dT, d2a_alpha_dT2, quick=True):
dP_dT, dP_dV, d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, Cv_dep = (
self.main_derivatives_and_departures(T, P, V, b, delta, epsilon,
a_alpha, da_alpha_dT,
d2a_alpha_dT2))
try:
dV_dP = 1.0/dP_dV
except:
dV_dP = inf
dT_dP = 1./dP_dT
dV_dT = -dP_dT*dV_dP
dT_dV = 1./dV_dT
dV_dP2 = dV_dP*dV_dP
dV_dP3 = dV_dP*dV_dP2
inverse_dP_dT2 = dT_dP*dT_dP
inverse_dP_dT3 = inverse_dP_dT2*dT_dP
d2V_dP2 = -d2P_dV2*dV_dP3 # unused
d2T_dP2 = -d2P_dT2*inverse_dP_dT3 # unused
d2T_dV2 = (-(d2P_dV2*dP_dT - dP_dV*d2P_dTdV)*inverse_dP_dT2
+(d2P_dTdV*dP_dT - dP_dV*d2P_dT2)*inverse_dP_dT3*dP_dV) # unused
d2V_dT2 = (-(d2P_dT2*dP_dV - dP_dT*d2P_dTdV)*dV_dP2 # unused
+(d2P_dTdV*dP_dV - dP_dT*d2P_dV2)*dV_dP3*dP_dT)
d2V_dPdT = -(d2P_dTdV*dP_dV - dP_dT*d2P_dV2)*dV_dP3 # unused
d2T_dPdV = -(d2P_dTdV*dP_dT - dP_dV*d2P_dT2)*inverse_dP_dT3 # unused
return (dP_dT, dP_dV, dV_dT, dV_dP, dT_dV, dT_dP,
d2P_dT2, d2P_dV2, d2V_dT2, d2V_dP2, d2T_dV2, d2T_dP2,
d2V_dPdT, d2P_dTdV, d2T_dPdV, # d2P_dTdV is used
H_dep, S_dep, Cv_dep)
@property
def sorted_volumes(self):
r'''List of lexicographically-sorted molar volumes available from the
root finding algorithm used to solve the PT point. The convention of
sorting lexicographically comes from numpy's handling of complex
numbers, which python does not define. This method was added to
facilitate testing, as the volume solution method changes over time
and the ordering does as well.
Examples
--------
>>> PR(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6).sorted_volumes
((0.000130222125139+0j), (0.00112363131346-0.00129269672343j), (0.00112363131346+0.00129269672343j))
'''
sort_fun = lambda x: (x.real, x.imag)
full_volumes = self.volume_solutions_full(self.T, self.P, self.b, self.delta, self.epsilon, self.a_alpha)
full_volumes = [i + 0.0j for i in full_volumes]
return tuple(sorted(full_volumes, key=sort_fun))
[docs] def Tsat(self, P, polish=False):
r'''Generic method to calculate the temperature for a specified
vapor pressure of the pure fluid.
This is simply a bounded solver running between `0.2Tc` and `Tc` on the
`Psat` method.
Parameters
----------
P : float
Vapor pressure, [Pa]
polish : bool, optional
Whether to attempt to use a numerical solver to make the solution
more precise or not
Returns
-------
Tsat : float
Temperature of saturation, [K]
Notes
-----
It is recommended not to run with `polish=True`, as that will make the
calculation much slower.
'''
fprime = False
global curr_err
def to_solve_newton(T):
global curr_err
assert T > 0.0
e = self.to_TP(T, P)
try:
fugacity_l = e.fugacity_l
except AttributeError as err:
raise err
try:
fugacity_g = e.fugacity_g
except AttributeError as err:
raise err
curr_err = fugacity_l - fugacity_g
if fprime:
d_err_d_T = e.dfugacity_dT_l - e.dfugacity_dT_g
return curr_err, d_err_d_T
# print('err', err, 'rel err', err/T, 'd_err_d_T', d_err_d_T, 'T', T)
return curr_err
logP = log(P)
def to_solve(T):
global curr_err
if fprime:
dPsat_dT, Psat = self.dPsat_dT(T, polish=polish, also_Psat=True)
curr_err = Psat - P
# Log translation - tends to save a few iterations
err_trans = log(Psat) - logP
return err_trans, dPsat_dT/Psat
# return curr_err, derr_dT
curr_err = self.Psat(T, polish=polish) - P
return curr_err#, derr_dT
# return copysign(log(abs(err)), err)
# Outstanding improvements to do: Better guess; get NR working;
# see if there is a general curve
try:
Tc, Pc = self.Tc, self.Pc
except:
Tc, Pc = self.pseudo_Tc, self.pseudo_Pc
guess = -5.4*Tc/(1.0*log(P/Pc) - 5.4)
high = guess*2.0
low = guess*0.5
# return newton(to_solve, guess, fprime=True, ytol=1e-6, high=self.Pc)
# return newton(to_solve, guess, ytol=1e-6, high=self.Pc)
# Methanol is a good example of why 1.5 is needed
low_hope, high_hope = max(guess*.5, 0.2*Tc), min(Tc, guess*1.5)
try:
err_low, err_high = to_solve(low_hope), to_solve(high_hope)
if err_low*err_high < 0.0:
if guess < low_hope or guess > high_hope:
guess = 0.5*(low_hope + high_hope)
fprime = True
Tsat = newton(to_solve, guess, xtol=1.48e-10,fprime=True, low=low_hope, high=high_hope, bisection=True)
# fprime = False
# Tsat = brenth(to_solve, low_hope, high_hope)
abs_rel_err = abs(curr_err)/P
if abs_rel_err < 1e-9:
return Tsat
elif abs_rel_err < 1e-2:
guess = Tsat
else:
try:
return brenth(to_solve, 0.2*Tc, Tc)
except:
try:
return brenth(to_solve, 0.2*Tc, Tc*1.5)
except:
pass
except:
pass
fprime = True
try:
try:
Tsat = newton(to_solve_newton, guess, fprime=True, maxiter=100,
xtol=4e-13, require_eval=False, damping=1.0, low=Tc*1e-5)
except:
try:
Tsat = newton(to_solve_newton, guess, fprime=True, maxiter=100,
xtol=4e-13, require_eval=False, damping=1.0, low=low, high=high)
assert Tsat != low and Tsat != high
except:
Tsat = newton(to_solve_newton, guess, fprime=True, maxiter=250, # the wider range can take more iterations
xtol=4e-13, require_eval=False, damping=1.0, low=low, high=high*2)
assert Tsat != low and Tsat != high*2
except:
# high = self.Tc
# try:
# high = min(high, self.T_discriminant_zero_l()*(1-1e-8))
# except:
# pass
# Does not seem to be working
try:
Tsat = None
Tsat = newton(to_solve_newton, guess, fprime=True, maxiter=200, high=high, low=low,
xtol=4e-13, require_eval=False, damping=1.0)
except:
pass
fprime = False
if Tsat is None or abs(to_solve_newton(Tsat)) == P:
Tsat = brenth(to_solve_newton, low, high)
return Tsat
[docs] def Psat(self, T, polish=False, guess=None):
r'''Generic method to calculate vapor pressure for a specified `T`.
From Tc to 0.32Tc, uses a 10th order polynomial of the following form:
.. math::
\ln\frac{P_r}{T_r} = \sum_{k=0}^{10} C_k\left(\frac{\alpha}{T_r}
-1\right)^{k}
If `polish` is True, SciPy's `newton` solver is launched with the
calculated vapor pressure as an initial guess in an attempt to get more
accuracy. This may not converge however.
Results above the critical temperature are meaningless. A first-order
polynomial is used to extrapolate under 0.32 Tc; however, there is
normally not a volume solution to the EOS which can produce that
low of a pressure.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to attempt to use a numerical solver to make the solution
more precise or not
Returns
-------
Psat : float
Vapor pressure, [Pa]
Notes
-----
EOSs sharing the same `b`, `delta`, and `epsilon` have the same
coefficient sets.
Form for the regression is inspired from [1]_.
No volume solution is needed when `polish=False`; the only external
call is for the value of `a_alpha`.
References
----------
.. [1] Soave, G. "Direct Calculation of Pure-Compound Vapour Pressures
through Cubic Equations of State." Fluid Phase Equilibria 31, no. 2
(January 1, 1986): 203-7. doi:10.1016/0378-3812(86)90013-0.
'''
Tc, Pc = self.Tc, self.Pc
if T == Tc:
return Pc
a_alpha = self.a_alpha_and_derivatives(T, full=False)
alpha = a_alpha/self.a
Tr = T/self.Tc
x = alpha/Tr - 1.
if Tr > 0.999 and not isinstance(self, RK):
y = horner(self.Psat_coeffs_critical, x)
Psat = y*Tr*Pc
if Psat > Pc and T < Tc:
Psat = Pc*(1.0 - 1e-14)
else:
# TWUPR/SRK TODO need to be prepared for x being way outside the range (in the weird direction - at the start)
Psat_ranges_low = self.Psat_ranges_low
if x > Psat_ranges_low[-1]:
if not polish:
raise NoSolutionError("T %.8f K is too low for equations to converge" %(T))
else:
# Needs to still be here for generating better data
x = Psat_ranges_low[-1]
polish = True
for i in range(len(Psat_ranges_low)):
if x < Psat_ranges_low[i]:
break
y = 0.0
for c in self.Psat_coeffs_low[i]:
y = y*x + c
try:
Psat = exp(y)*Tr*Pc
if Psat == 0.0:
if polish:
Psat = 1e-100
else:
raise NoSolutionError("T %.8f K is too low for equations to converge" %(T))
except OverflowError:
# coefficients sometimes overflow before T is lowered to 0.32Tr
# For
polish = True # There is no solution available to polish
Psat = 1
if polish:
if T > Tc:
raise ValueError("Cannot solve for equifugacity condition "
"beyond critical temperature")
if guess is not None:
Psat = guess
converged = False
def to_solve_newton(P):
# For use by newton. Only supports initialization with Tc, Pc and omega
# ~200x slower and not guaranteed to converge (primary issue is one phase)
# not existing
assert P > 0.0
e = self.to_TP(T, P)
# print(e.volume_error(), e)
try:
fugacity_l = e.fugacity_l
except AttributeError as err:
# return 1000, 1000
raise err
try:
fugacity_g = e.fugacity_g
except AttributeError as err:
# return 1000, 1000
raise err
err = fugacity_l - fugacity_g
d_err_d_P = e.dfugacity_dP_l - e.dfugacity_dP_g # -1 for low pressure
if isnan(d_err_d_P):
d_err_d_P = -1.0
# print('err', err, 'rel err', err/P, 'd_err_d_P', d_err_d_P, 'P', P)
# Clamp the derivative - if it will step to zero or negative, dampen to half the distance which gets to zero
if (P - err/d_err_d_P) <= 0.0: # This is the one matching newton
# if (P - err*d_err_d_P) <= 0.0:
d_err_d_P = -1.0001
return err, d_err_d_P
try:
try:
boundaries = GCEOS.P_discriminant_zeros_analytical(T, self.b, self.delta, self.epsilon, a_alpha, valid=True)
low, high = min(boundaries), max(boundaries)
except:
pass
try:
high = self.P_discriminant_zero()
except:
high = Pc
# def damping_func(p0, step, damping):
# if step == 1:
# damping = damping*0.5
# p = p0 + step * damping
# return p
Psat = newton(to_solve_newton, Psat, high=high, fprime=True, maxiter=100,
xtol=4e-13, require_eval=False, damping=1.0) # ,ytol=1e-6*Psat # damping_func=damping_func
# print(to_solve_newton(Psat), 'newton error')
converged = True
except:
pass
if not converged:
def to_solve_bisect(P):
e = self.to_TP(T, P)
# print(e.volume_error(), e)
try:
fugacity_l = e.fugacity_l
except AttributeError as err:
return 1e20
try:
fugacity_g = e.fugacity_g
except AttributeError as err:
return -1e20
err = fugacity_l - fugacity_g
# print(err, 'err', 'P', P)
return err
for low, high in zip([.98*Psat, 1, 1e-40, Pc*.9, Psat*.9999], [1.02*Psat, Pc, 1, Pc*1.000000001, Pc]):
try:
Psat = bisect(to_solve_bisect, low, high, ytol=1e-6*Psat, maxiter=128)
# print(to_solve_bisect(Psat), 'bisect error')
converged = True
break
except:
pass
# Last ditch attempt
if not converged:
# raise ValueError("Could not converge")
if Tr > 0.5:
# Near critical temperature issues
points = [Pc*f for f in linspace(1e-3, 1-1e-8, 50) + linspace(.9, 1-1e-8, 50)]
ytol = 1e-6*Psat
else:
# Low temperature issues
points = [Psat*f for f in logspace(-5.5, 5.5, 16)]
# points = [Psat*f for f in logspace(-2.5, 2.5, 100)]
ytol = None # Cryogenic point unlikely to work to desired tolerance
# Work on point closer to Psat first
points.sort(key=lambda x: abs(log10(x)))
low, high = None, None
for point in points:
try:
err = to_solve_newton(point)[0] # Do not use bisect function as it does not raise errors
if err > 0.0:
high = point
elif err < 0.0:
low = point
except:
pass
if low is not None and high is not None:
# print('reached bisection')
Psat = brenth(to_solve_bisect, low, high, ytol=ytol, maxiter=128)
# print(to_solve_bisect(Psat), 'bisect error')
converged = True
break
# print('tried all points')
# Check that the fugacity error vs. Psat is OK
if abs(to_solve_bisect(Psat)/Psat) > .0001:
converged = False
if not converged:
raise ValueError("Could not converge at T=%.6f K" %(T))
return Psat
[docs] def dPsat_dT(self, T, polish=False, also_Psat=False):
r'''Generic method to calculate the temperature derivative of vapor
pressure for a specified `T`. Implements the analytical derivative
of the three polynomials described in `Psat`.
As with `Psat`, results above the critical temperature are meaningless.
The first-order polynomial which is used to calculate it under 0.32 Tc
may not be physicall meaningful, due to there normally not being a
volume solution to the EOS which can produce that low of a pressure.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to attempt to use a numerical solver to make the solution
more precise or not
also_Psat : bool, optional
Calculating `dPsat_dT` necessarily involves calculating `Psat`;
when this is set to True, a second return value is added, whic is
the actual `Psat` value.
Returns
-------
dPsat_dT : float
Derivative of vapor pressure with respect to temperature, [Pa/K]
Psat : float, returned if `also_Psat` is `True`
Vapor pressure, [Pa]
Notes
-----
There is a small step change at 0.32 Tc for all EOS due to the two
switch between polynomials at that point.
Useful for calculating enthalpy of vaporization with the Clausius
Clapeyron Equation. Derived with SymPy's diff and cse.
'''
if polish:
# Calculate the derivative of saturation pressure analytically
Psat = self.Psat(T, polish=polish)
sat_eos = self.to(T=T, P=Psat)
dfg_T, dfl_T = sat_eos.dfugacity_dT_g, sat_eos.dfugacity_dT_l
dfg_P, dfl_P = sat_eos.dfugacity_dP_g, sat_eos.dfugacity_dP_l
dPsat_dT = (dfg_T - dfl_T)/(dfl_P - dfg_P)
if also_Psat:
return dPsat_dT, Psat
return dPsat_dT
a_alphas = self.a_alpha_and_derivatives(T)
a_inv = 1.0/self.a
try:
Tc, Pc = self.Tc, self.Pc
except:
Tc, Pc = self.pseudo_Tc, self.pseudo_Pc
alpha, d_alpha_dT = a_alphas[0]*a_inv, a_alphas[1]*a_inv
Tc_inv = 1.0/Tc
T_inv = 1.0/T
Tr = T*Tc_inv
# if Tr < 0.32 and not isinstance(self, PR):
# # Delete
# c = self.Psat_coeffs_limiting
# return self.Pc*T*c[0]*(self.Tc*d_alpha_dT/T - self.Tc*alpha/(T*T)
# )*exp(c[0]*(-1. + self.Tc*alpha/T) + c[1]
# )/self.Tc + self.Pc*exp(c[0]*(-1.
# + self.Tc*alpha/T) + c[1])/self.Tc
if Tr > 0.999 and not isinstance(self, RK):
# OK
x = alpha/Tr - 1.
y = horner(self.Psat_coeffs_critical, x)
dy_dT = T_inv*(Tc*d_alpha_dT - Tc*alpha*T_inv)*horner(self.Psat_coeffs_critical_der, x)
dPsat_dT = Pc*(T*dy_dT*Tc_inv + y*Tc_inv)
if also_Psat:
Psat = y*Tr*Pc
return dPsat_dT, Psat
return dPsat_dT
else:
Psat_coeffs_low = self.Psat_coeffs_low
Psat_ranges_low = self.Psat_ranges_low
x = alpha/Tr - 1.
if x > Psat_ranges_low[-1]:
raise NoSolutionError("T %.8f K is too low for equations to converge" %(T))
for i in range(len(Psat_ranges_low)):
if x < Psat_ranges_low[i]:
break
y, dy = 0.0, 0.0
for c in Psat_coeffs_low[i]:
dy = x*dy + y
y = x*y + c
exp_y = exp(y)
dy_dT = Tc*T_inv*(d_alpha_dT - alpha*T_inv)*dy#horner_and_der(Psat_coeffs_low[i], x)[1]
Psat = exp_y*Tr*Pc
dPsat_dT = (T*dy_dT + 1.0)*Pc*exp_y*Tc_inv
if also_Psat:
return dPsat_dT, Psat
return dPsat_dT
# # change chebval to horner, and get new derivative
# x = alpha/Tr - 1.
# arg = (self.Psat_cheb_constant_factor[1]*(x + self.Psat_cheb_constant_factor[0]))
# y = chebval(arg, self.Psat_cheb_coeffs)
#
# exp_y = exp(y)
# dy_dT = T_inv*(Tc*d_alpha_dT - Tc*alpha*T_inv)*chebval(arg,
# self.Psat_cheb_coeffs_der)*self.Psat_cheb_constant_factor[1]
# Psat = Pc*T*exp_y*dy_dT*Tc_inv + Pc*exp_y*Tc_inv
# return Psat
[docs] def phi_sat(self, T, polish=True):
r'''Method to calculate the saturation fugacity coefficient of the
compound. This does not require solving the EOS itself.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to perform a rigorous calculation or to use a polynomial
fit, [-]
Returns
-------
phi_sat : float
Fugacity coefficient along the liquid-vapor saturation line, [-]
Notes
-----
Accuracy is generally around 1e-7. If Tr is under 0.32, the rigorous
method is always used, but a solution may not exist if both phases
cannot coexist. If Tr is above 1, likewise a solution does not exist.
'''
Tr = T/self.Tc
if polish or not 0.32 <= Tr <= 1.0:
e = self.to_TP(T=T, P=self.Psat(T, polish=True)) # True
try:
return e.phi_l
except:
return e.phi_g
alpha = self.a_alpha_and_derivatives(T, full=False)/self.a
x = alpha/Tr - 1.
return horner(self.phi_sat_coeffs, x)
[docs] def dphi_sat_dT(self, T, polish=True):
r'''Method to calculate the temperature derivative of saturation
fugacity coefficient of the
compound. This does require solving the EOS itself.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to perform a rigorous calculation or to use a polynomial
fit, [-]
Returns
-------
dphi_sat_dT : float
First temperature derivative of fugacity coefficient along the
liquid-vapor saturation line, [1/K]
Notes
-----
'''
if T == self.Tc:
T = (self.Tc*(1.0 - 1e-15))
Psat = self.Psat(T, polish=polish)
sat_eos = self.to(T=T, P=Psat)
dfg_T, dfl_T = sat_eos.dfugacity_dT_g, sat_eos.dfugacity_dT_l
dfg_P, dfl_P = sat_eos.dfugacity_dP_g, sat_eos.dfugacity_dP_l
dPsat_dT = (dfg_T - dfl_T)/(dfl_P - dfg_P)
fugacity = sat_eos.fugacity_l
dfugacity_sat_dT = dPsat_dT*sat_eos.dfugacity_dP_l + sat_eos.dfugacity_dT_l
Psat_inv = 1.0/Psat
return (dfugacity_sat_dT - fugacity*dPsat_dT*Psat_inv)*Psat_inv
[docs] def d2phi_sat_dT2(self, T, polish=True):
r'''Method to calculate the second temperature derivative of saturation
fugacity coefficient of the
compound. This does require solving the EOS itself.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to perform a rigorous calculation or to use a polynomial
fit, [-]
Returns
-------
d2phi_sat_dT2 : float
Second temperature derivative of fugacity coefficient along the
liquid-vapor saturation line, [1/K^2]
Notes
-----
This is presently a numerical calculation.
'''
return derivative(lambda T: self.dphi_sat_dT(T, polish=polish), T,
dx=T*1e-7, upper_limit=self.Tc)
[docs] def V_l_sat(self, T):
r'''Method to calculate molar volume of the liquid phase along the
saturation line.
Parameters
----------
T : float
Temperature, [K]
Returns
-------
V_l_sat : float
Liquid molar volume along the saturation line, [m^3/mol]
Notes
-----
Computes `Psat`, and then uses `volume_solutions` to obtain the three
possible molar volumes. The lowest value is returned.
'''
Psat = self.Psat(T)
a_alpha = self.a_alpha_and_derivatives(T, full=False)
Vs = self.volume_solutions(T, Psat, self.b, self.delta, self.epsilon, a_alpha)
# Assume we can safely take the Vmax as gas, Vmin as l on the saturation line
return min([i.real for i in Vs if i.real > self.b])
[docs] def V_g_sat(self, T):
r'''Method to calculate molar volume of the vapor phase along the
saturation line.
Parameters
----------
T : float
Temperature, [K]
Returns
-------
V_g_sat : float
Gas molar volume along the saturation line, [m^3/mol]
Notes
-----
Computes `Psat`, and then uses `volume_solutions` to obtain the three
possible molar volumes. The highest value is returned.
'''
Psat = self.Psat(T)
a_alpha = self.a_alpha_and_derivatives(T, full=False)
Vs = self.volume_solutions(T, Psat, self.b, self.delta, self.epsilon, a_alpha)
# Assume we can safely take the Vmax as gas, Vmin as l on the saturation line
return max([i.real for i in Vs])
[docs] def Hvap(self, T):
r'''Method to calculate enthalpy of vaporization for a pure fluid from
an equation of state, without iteration.
.. math::
\frac{dP^{sat}}{dT}=\frac{\Delta H_{vap}}{T(V_g - V_l)}
Results above the critical temperature are meaningless. A first-order
polynomial is used to extrapolate under 0.32 Tc; however, there is
normally not a volume solution to the EOS which can produce that
low of a pressure.
Parameters
----------
T : float
Temperature, [K]
Returns
-------
Hvap : float
Increase in enthalpy needed for vaporization of liquid phase along
the saturation line, [J/mol]
Notes
-----
Calculates vapor pressure and its derivative with `Psat` and `dPsat_dT`
as well as molar volumes of the saturation liquid and vapor phase in
the process.
Very near the critical point this provides unrealistic results due to
`Psat`'s polynomials being insufficiently accurate.
References
----------
.. [1] Walas, Stanley M. Phase Equilibria in Chemical Engineering.
Butterworth-Heinemann, 1985.
'''
Psat = self.Psat(T)
dPsat_dT = self.dPsat_dT(T)
a_alpha = self.a_alpha_and_derivatives(T, full=False)
Vs = self.volume_solutions(T, Psat, self.b, self.delta, self.epsilon, a_alpha)
# Assume we can safely take the Vmax as gas, Vmin as l on the saturation line
Vs = [i.real for i in Vs]
V_l, V_g = min(Vs), max(Vs)
return dPsat_dT*T*(V_g - V_l)
[docs] def dH_dep_dT_sat_l(self, T, polish=False):
r'''Method to calculate and return the temperature derivative of
saturation liquid excess enthalpy.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to perform a rigorous calculation or to use a polynomial
fit, [-]
Returns
-------
dH_dep_dT_sat_l : float
Liquid phase temperature derivative of excess enthalpy along the
liquid-vapor saturation line, [J/mol/K]
Notes
-----
'''
sat_eos = self.to(T=T, P=self.Psat(T, polish=polish))
dfg_T, dfl_T = sat_eos.dfugacity_dT_g, sat_eos.dfugacity_dT_l
dfg_P, dfl_P = sat_eos.dfugacity_dP_g, sat_eos.dfugacity_dP_l
dPsat_dT = (dfg_T - dfl_T)/(dfl_P - dfg_P)
return dPsat_dT*sat_eos.dH_dep_dP_l + sat_eos.dH_dep_dT_l
[docs] def dH_dep_dT_sat_g(self, T, polish=False):
r'''Method to calculate and return the temperature derivative of
saturation vapor excess enthalpy.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to perform a rigorous calculation or to use a polynomial
fit, [-]
Returns
-------
dH_dep_dT_sat_g : float
Vapor phase temperature derivative of excess enthalpy along the
liquid-vapor saturation line, [J/mol/K]
Notes
-----
'''
sat_eos = self.to(T=T, P=self.Psat(T, polish=polish))
dfg_T, dfl_T = sat_eos.dfugacity_dT_g, sat_eos.dfugacity_dT_l
dfg_P, dfl_P = sat_eos.dfugacity_dP_g, sat_eos.dfugacity_dP_l
dPsat_dT = (dfg_T - dfl_T)/(dfl_P - dfg_P)
return dPsat_dT*sat_eos.dH_dep_dP_g + sat_eos.dH_dep_dT_g
[docs] def dS_dep_dT_sat_g(self, T, polish=False):
r'''Method to calculate and return the temperature derivative of
saturation vapor excess entropy.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to perform a rigorous calculation or to use a polynomial
fit, [-]
Returns
-------
dS_dep_dT_sat_g : float
Vapor phase temperature derivative of excess entropy along the
liquid-vapor saturation line, [J/mol/K^2]
Notes
-----
'''
sat_eos = self.to(T=T, P=self.Psat(T, polish=polish))
dfg_T, dfl_T = sat_eos.dfugacity_dT_g, sat_eos.dfugacity_dT_l
dfg_P, dfl_P = sat_eos.dfugacity_dP_g, sat_eos.dfugacity_dP_l
dPsat_dT = (dfg_T - dfl_T)/(dfl_P - dfg_P)
return dPsat_dT*sat_eos.dS_dep_dP_g + sat_eos.dS_dep_dT_g
[docs] def dS_dep_dT_sat_l(self, T, polish=False):
r'''Method to calculate and return the temperature derivative of
saturation liquid excess entropy.
Parameters
----------
T : float
Temperature, [K]
polish : bool, optional
Whether to perform a rigorous calculation or to use a polynomial
fit, [-]
Returns
-------
dS_dep_dT_sat_l : float
Liquid phase temperature derivative of excess entropy along the
liquid-vapor saturation line, [J/mol/K^2]
Notes
-----
'''
sat_eos = self.to(T=T, P=self.Psat(T, polish=polish))
dfg_T, dfl_T = sat_eos.dfugacity_dT_g, sat_eos.dfugacity_dT_l
dfg_P, dfl_P = sat_eos.dfugacity_dP_g, sat_eos.dfugacity_dP_l
dPsat_dT = (dfg_T - dfl_T)/(dfl_P - dfg_P)
return dPsat_dT*sat_eos.dS_dep_dP_l + sat_eos.dS_dep_dT_l
[docs] def a_alpha_for_V(self, T, P, V):
r'''Method to calculate which value of :math:`a \alpha` is required for
a given `T`, `P` pair to match a specified `V`. This is a
straightforward analytical equation.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
Returns
-------
a_alpha : float
Value calculated to match specified volume for the current EOS,
[J^2/mol^2/Pa]
Notes
-----
The derivation of the solution is as follows:
>>> from sympy import * # doctest:+SKIP
>>> P, T, V, R, b, a, delta, epsilon = symbols('P, T, V, R, b, a, delta, epsilon') # doctest:+SKIP
>>> a_alpha = symbols('a_alpha') # doctest:+SKIP
>>> CUBIC = R*T/(V-b) - a_alpha/(V*V + delta*V + epsilon) # doctest:+SKIP
>>> solve(Eq(CUBIC, P), a_alpha)# doctest:+SKIP
[(-P*V**3 + P*V**2*b - P*V**2*delta + P*V*b*delta - P*V*epsilon + P*b*epsilon + R*T*V**2 + R*T*V*delta + R*T*epsilon)/(V - b)]
'''
b, delta, epsilon = self.b, self.delta, self.epsilon
x0 = P*b
x1 = R*T
x2 = V*delta
x3 = V*V
x4 = x3*V
return ((-P*x4 - P*V*epsilon - P*delta*x3 + epsilon*x0 + epsilon*x1
+ x0*x2 + x0*x3 + x1*x2 + x1*x3)/(V - b))
[docs] def a_alpha_for_Psat(self, T, Psat, a_alpha_guess=None):
r'''Method to calculate which value of :math:`a \alpha` is required for
a given `T`, `Psat` pair. This is a numerical solution, but not a very
complicated one.
Parameters
----------
T : float
Temperature, [K]
Psat : float
Vapor pressure specified, [Pa]
a_alpha_guess : float
Optionally, an initial guess for the solver [J^2/mol^2/Pa]
Returns
-------
a_alpha : float
Value calculated to match specified volume for the current EOS,
[J^2/mol^2/Pa]
Notes
-----
The implementation of this function is a direct calculation of
departure gibbs energy, which is equal in both phases at saturation.
Examples
--------
>>> eos = PR(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.a_alpha_for_Psat(T=400, Psat=5e5)
3.1565798926
'''
P = Psat
b, delta, epsilon = self.b, self.delta, self.epsilon
RT = R*T
RT_inv = 1.0/RT
x0 = 1.0/sqrt(delta*delta - 4.0*epsilon)
x1 = delta*x0
x2 = 2.0*x0
def fug(V, a_alpha):
# Can simplify this to not use a function, avoid 1 log anywayS
G_dep = (P*V - RT - RT*log(P*RT_inv*(V-b))
- x2*a_alpha*catanh(2.0*V*x0 + x1).real)
return G_dep # No point going all the way to fugacity
def err(a_alpha):
# Needs some work right up to critical point
Vs = self.volume_solutions(T, P, b, delta, epsilon, a_alpha)
good_roots = [i.real for i in Vs if i.imag == 0.0 and i.real > 0.0]
good_root_count = len(good_roots)
if good_root_count == 1:
raise ValueError("Guess did not have two roots")
V_l, V_g = min(good_roots), max(good_roots)
# print(V_l, V_g, a_alpha)
return fug(V_l, a_alpha) - fug(V_g, a_alpha)
if a_alpha_guess is None:
try:
a_alpha_guess = self.a_alpha
except AttributeError:
a_alpha_guess = 0.002
try:
return secant(err, a_alpha_guess, xtol=1e-13)
except:
return secant(err, self.to(T=T, P=Psat).a_alpha, xtol=1e-13)
[docs] def to_TP(self, T, P):
r'''Method to construct a new EOS object at the spcified `T` and `P`.
In the event the `T` and `P` match the current object's `T` and `P`,
it will be returned unchanged.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
Returns
-------
obj : EOS
Pure component EOS at specified `T` and `P`, [-]
Notes
-----
Constructs the object with parameters `Tc`, `Pc`, `omega`, and
`kwargs`.
Examples
--------
>>> base = PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=500.0, P=1E6)
>>> new = base.to_TP(T=1.0, P=2.0)
>>> base.state_specs, new.state_specs
({'T': 500.0, 'P': 1000000.0}, {'T': 1.0, 'P': 2.0})
'''
if T != self.T or P != self.P:
return self.__class__(T=T, P=P, Tc=self.Tc, Pc=self.Pc, omega=self.omega, **self.kwargs)
else:
return self
[docs] def to_TV(self, T, V):
r'''Method to construct a new EOS object at the spcified `T` and `V`.
In the event the `T` and `V` match the current object's `T` and `V`,
it will be returned unchanged.
Parameters
----------
T : float
Temperature, [K]
V : float
Molar volume, [m^3/mol]
Returns
-------
obj : EOS
Pure component EOS at specified `T` and `V`, [-]
Notes
-----
Constructs the object with parameters `Tc`, `Pc`, `omega`, and
`kwargs`.
Examples
--------
>>> base = PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=500.0, P=1E6)
>>> new = base.to_TV(T=1000000.0, V=1.0)
>>> base.state_specs, new.state_specs
({'T': 500.0, 'P': 1000000.0}, {'T': 1000000.0, 'V': 1.0})
'''
if T != self.T or V != self.V:
# Only allow creation of new class if volume actually specified
# Ignores the posibility that V is V_l or V_g
return self.__class__(T=T, V=V, Tc=self.Tc, Pc=self.Pc, omega=self.omega, **self.kwargs)
else:
return self
[docs] def to_PV(self, P, V):
r'''Method to construct a new EOS object at the spcified `P` and `V`.
In the event the `P` and `V` match the current object's `P` and `V`,
it will be returned unchanged.
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
Returns
-------
obj : EOS
Pure component EOS at specified `P` and `V`, [-]
Notes
-----
Constructs the object with parameters `Tc`, `Pc`, `omega`, and
`kwargs`.
Examples
--------
>>> base = PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=500.0, P=1E6)
>>> new = base.to_PV(P=1000.0, V=1.0)
>>> base.state_specs, new.state_specs
({'T': 500.0, 'P': 1000000.0}, {'P': 1000.0, 'V': 1.0})
'''
if P != self.P or V != self.V:
return self.__class__(V=V, P=P, Tc=self.Tc, Pc=self.Pc, omega=self.omega, **self.kwargs)
else:
return self
[docs] def to(self, T=None, P=None, V=None):
r'''Method to construct a new EOS object at two of `T`, `P` or `V`.
In the event the specs match those of the current object, it will be
returned unchanged.
Parameters
----------
T : float or None, optional
Temperature, [K]
P : float or None, optional
Pressure, [Pa]
V : float or None, optional
Molar volume, [m^3/mol]
Returns
-------
obj : EOS
Pure component EOS at the two specified specs, [-]
Notes
-----
Constructs the object with parameters `Tc`, `Pc`, `omega`, and
`kwargs`.
Examples
--------
>>> base = PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=500.0, P=1E6)
>>> base.to(T=300.0, P=1e9).state_specs
{'T': 300.0, 'P': 1000000000.0}
>>> base.to(T=300.0, V=1.0).state_specs
{'T': 300.0, 'V': 1.0}
>>> base.to(P=1e5, V=1.0).state_specs
{'P': 100000.0, 'V': 1.0}
'''
if T is not None and P is not None:
return self.to_TP(T, P)
elif T is not None and V is not None:
return self.to_TV(T, V)
elif P is not None and V is not None:
return self.to_PV(P, V)
else:
# Error message
return self.__class__(T=T, V=V, P=P, Tc=self.Tc, Pc=self.Pc, omega=self.omega, **self.kwargs)
[docs] def T_min_at_V(self, V, Pmin=1e-15):
'''Returns the minimum temperature for the EOS to have the
volume as specified. Under this temperature, the pressure will go
negative (and the EOS will not solve).
'''
return self.solve_T(P=Pmin, V=V)
[docs] def T_max_at_V(self, V, Pmax=None):
r'''Method to calculate the maximum temperature the EOS can create at a
constant volume, if one exists; returns None otherwise.
Parameters
----------
V : float
Constant molar volume, [m^3/mol]
Pmax : float
Maximum possible isochoric pressure, if already known [Pa]
Returns
-------
T : float
Maximum possible temperature, [K]
Notes
-----
Examples
--------
>>> e = PR(P=1e5, V=0.0001437, Tc=512.5, Pc=8084000.0, omega=0.559)
>>> e.T_max_at_V(e.V)
431155.5
'''
if Pmax is None:
Pmax = self.P_max_at_V(V)
if Pmax is None:
return None
return self.solve_T(P=Pmax, V=V)
[docs] def P_max_at_V(self, V):
r'''Dummy method. The idea behind this method, which is implemented by some
subclasses, is to calculate the maximum pressure the EOS can create at a
constant volume, if one exists; returns None otherwise. This method,
as a dummy method, always returns None.
Parameters
----------
V : float
Constant molar volume, [m^3/mol]
Returns
-------
P : float
Maximum possible isochoric pressure, [Pa]
'''
return None
@property
def more_stable_phase(self):
r'''Checks the Gibbs energy of each possible phase, and returns
'l' if the liquid-like phase is more stable, and 'g' if the vapor-like
phase is more stable.
Examples
--------
>>> PR(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6).more_stable_phase
'l'
'''
try:
if self.G_dep_l < self.G_dep_g:
return 'l'
else:
return 'g'
except:
try:
self.Z_g
return 'g'
except:
return 'l'
[docs] def discriminant(self, T=None, P=None):
r'''Method to compute the discriminant of the cubic volume solution
with the current EOS parameters, optionally at the same (assumed) `T`,
and `P` or at different ones, if values are specified.
Parameters
----------
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
Returns
-------
discriminant : float
Discriminant, [-]
Notes
-----
This call is quite quick; only :math:`a \alpha` is needed and if `T` is
the same as the current object than it has already been computed.
The formula is as follows:
.. math::
\text{discriminant} = - \left(- \frac{27 P^{2} \epsilon \left(
\frac{P b}{R T} + 1\right)}{R^{2} T^{2}} - \frac{27 P^{2} b
\operatorname{a \alpha}{\left(T \right)}}{R^{3} T^{3}}\right)
\left(- \frac{P^{2} \epsilon \left(\frac{P b}{R T} + 1\right)}
{R^{2} T^{2}} - \frac{P^{2} b \operatorname{a \alpha}{\left(T
\right)}}{R^{3} T^{3}}\right) + \left(- \frac{P^{2} \epsilon \left(
\frac{P b}{R T} + 1\right)}{R^{2} T^{2}} - \frac{P^{2} b
\operatorname{a \alpha}{\left(T \right)}}{R^{3} T^{3}}\right)
\left(- \frac{18 P b}{R T} + \frac{18 P \delta}{R T} - 18\right)
\left(\frac{P^{2} \epsilon}{R^{2} T^{2}} - \frac{P \delta \left(
\frac{P b}{R T} + 1\right)}{R T} + \frac{P \operatorname{a \alpha}
{\left(T \right)}}{R^{2} T^{2}}\right) - \left(- \frac{P^{2}
\epsilon \left(\frac{P b}{R T} + 1\right)}{R^{2} T^{2}} - \frac{
P^{2} b \operatorname{a \alpha}{\left(T \right)}}{R^{3} T^{3}}
\right) \left(- \frac{4 P b}{R T} + \frac{4 P \delta}{R T}
- 4\right) \left(- \frac{P b}{R T} + \frac{P \delta}{R T}
- 1\right)^{2} + \left(- \frac{P b}{R T} + \frac{P \delta}{R T}
- 1\right)^{2} \left(\frac{P^{2} \epsilon}{R^{2} T^{2}} - \frac{P
\delta \left(\frac{P b}{R T} + 1\right)}{R T} + \frac{P
\operatorname{a \alpha}{\left(T \right)}}{R^{2} T^{2}}\right)^{2}
- \left(\frac{P^{2} \epsilon}{R^{2} T^{2}} - \frac{P \delta \left(
\frac{P b}{R T} + 1\right)}{R T} + \frac{P \operatorname{a \alpha}{
\left(T \right)}}{R^{2} T^{2}}\right)^{2} \left(\frac{4 P^{2}
\epsilon}{R^{2} T^{2}} - \frac{4 P \delta \left(\frac{P b}{R T}
+ 1\right)}{R T} + \frac{4 P \operatorname{a \alpha}{\left(T
\right)}}{R^{2} T^{2}}\right)
The formula is derived as follows:
>>> from sympy import *
>>> P, T, R, b = symbols('P, T, R, b')
>>> a_alpha = symbols(r'a\ \alpha', cls=Function)
>>> delta, epsilon = symbols('delta, epsilon')
>>> eta = b
>>> B = b*P/(R*T)
>>> deltas = delta*P/(R*T)
>>> thetas = a_alpha(T)*P/(R*T)**2
>>> epsilons = epsilon*(P/(R*T))**2
>>> etas = eta*P/(R*T)
>>> a = 1
>>> b = (deltas - B - 1)
>>> c = (thetas + epsilons - deltas*(B+1))
>>> d = -(epsilons*(B+1) + thetas*etas)
>>> disc = b*b*c*c - 4*a*c*c*c - 4*b*b*b*d - 27*a*a*d*d + 18*a*b*c*d
Examples
--------
>>> base = PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=500.0, P=1E6)
>>> base.discriminant()
-0.001026390999
>>> base.discriminant(T=400)
0.0010458828
>>> base.discriminant(T=400, P=1e9)
12584660355.4
'''
if P is None:
P = self.P
if T is None:
T = self.T
a_alpha = self.a_alpha
else:
a_alpha = self.a_alpha_and_derivatives(T, full=False)
RT = R*self.T
RT6 = RT*RT
RT6 *= RT6*RT6
x0 = P*P
x1 = P*self.b + RT
x2 = a_alpha*self.b + self.epsilon*x1
x3 = P*self.epsilon
x4 = self.delta*x1
x5 = -P*self.delta + x1
x6 = a_alpha + x3 - x4
x2_2 = x2*x2
x5_2 = x5*x5
x6_2 = x6*x6
x7 = (-a_alpha - x3 + x4)
return x0*(18.0*P*x2*x5*x6 + 4.0*P*x7*x7*x7
- 27.0*x0*x2_2 - 4.0*x2*x5_2*x5 + x5_2*x6_2)/RT6
def _discriminant_at_T_mp(self, P):
# Hopefully numerical difficulties can eventually be figured out to as to
# not need mpmath
import mpmath as mp
mp.mp.dps = 70
P, T, b, a_alpha, delta, epsilon, R_mp = (mp.mpf(i) for i in [P, self.T, self.b, self.a_alpha, self.delta, self.epsilon, R])
RT = R_mp*T
RT6 = RT**6
x0 = P*P
x1 = P*b + RT
x2 = a_alpha*b + epsilon*x1
x3 = P*epsilon
x4 = delta*x1
x5 = -P*delta + x1
x6 = a_alpha + x3 - x4
x2_2 = x2*x2
x5_2 = x5*x5
x6_2 = x6*x6
disc = (x0*(18.0*P*x2*x5*x6 + 4.0*P*(-a_alpha - x3 + x4)**3
- 27.0*x0*x2_2 - 4.0*x2*x5_2*x5 + x5_2*x6_2)/RT6)
return disc
[docs] def P_discriminant_zero_l(self):
r'''Method to calculate the pressure which zero the discriminant
function of the general cubic eos, and is likely to sit on a boundary
between not having a liquid-like volume; and having a liquid-like volume.
Returns
-------
P_discriminant_zero_l : float
Pressure which make the discriminants zero at the right condition,
[Pa]
Notes
-----
Examples
--------
>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> P_trans = eos.P_discriminant_zero_l()
>>> P_trans
478346.37289
In this case, the discriminant transition shows the change in roots:
>>> eos.to(T=eos.T, P=P_trans*.99999999).mpmath_volumes_float
((0.00013117994140177062+0j), (0.002479717165903531+0j), (0.002480236178570793+0j))
>>> eos.to(T=eos.T, P=P_trans*1.0000001).mpmath_volumes_float
((0.0001311799413872173+0j), (0.002479976386402769-8.206310112063695e-07j), (0.002479976386402769+8.206310112063695e-07j))
'''
return self._P_discriminant_zero(low=True)
[docs] def P_discriminant_zero_g(self):
r'''Method to calculate the pressure which zero the discriminant
function of the general cubic eos, and is likely to sit on a boundary
between not having a vapor-like volume; and having a vapor-like volume.
Returns
-------
P_discriminant_zero_g : float
Pressure which make the discriminants zero at the right condition,
[Pa]
Notes
-----
Examples
--------
>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> P_trans = eos.P_discriminant_zero_g()
>>> P_trans
149960391.7
In this case, the discriminant transition does not reveal a transition
to two roots being available, only negative roots becoming negative
and imaginary.
>>> eos.to(T=eos.T, P=P_trans*.99999999).mpmath_volumes_float
((-0.0001037013146195082-1.5043987866732543e-08j), (-0.0001037013146195082+1.5043987866732543e-08j), (0.00011799201928619508+0j))
>>> eos.to(T=eos.T, P=P_trans*1.0000001).mpmath_volumes_float
((-0.00010374888853182635+0j), (-0.00010365374200380354+0j), (0.00011799201875924273+0j))
'''
return self._P_discriminant_zero(low=False)
[docs] def P_discriminant_zeros(self):
r'''Method to calculate the pressures which zero the discriminant
function of the general cubic eos, at the current temperature.
Returns
-------
P_discriminant_zeros : list[float]
Pressures which make the discriminants zero, [Pa]
Notes
-----
Examples
--------
>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.P_discriminant_zeros()
[478346.3, 149960391.7]
'''
return GCEOS.P_discriminant_zeros_analytical(self.T, self.b, self.delta, self.epsilon, self.a_alpha, valid=True)
[docs] @staticmethod
def P_discriminant_zeros_analytical(T, b, delta, epsilon, a_alpha, valid=False):
r'''Method to calculate the pressures which zero the discriminant
function of the general cubic eos. This is a quartic function
solved analytically.
Parameters
----------
T : float
Temperature, [K]
b : float
Coefficient calculated by EOS-specific method, [m^3/mol]
delta : float
Coefficient calculated by EOS-specific method, [m^3/mol]
epsilon : float
Coefficient calculated by EOS-specific method, [m^6/mol^2]
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
valid : bool
Whether to filter the calculated pressures so that they are all
real, and positive only, [-]
Returns
-------
P_discriminant_zeros : float
Pressures which make the discriminants zero, [Pa]
Notes
-----
Calculated analytically. Derived as follows.
>>> from sympy import *
>>> P, T, V, R, b, a, delta, epsilon = symbols('P, T, V, R, b, a, delta, epsilon')
>>> eta = b
>>> B = b*P/(R*T)
>>> deltas = delta*P/(R*T)
>>> thetas = a*P/(R*T)**2
>>> epsilons = epsilon*(P/(R*T))**2
>>> etas = eta*P/(R*T)
>>> a_coeff = 1
>>> b_coeff = (deltas - B - 1)
>>> c = (thetas + epsilons - deltas*(B+1))
>>> d = -(epsilons*(B+1) + thetas*etas)
>>> disc = b_coeff*b_coeff*c*c - 4*a_coeff*c*c*c - 4*b_coeff*b_coeff*b_coeff*d - 27*a_coeff*a_coeff*d*d + 18*a_coeff*b_coeff*c*d
>>> base = -(expand(disc/P**2*R**3*T**3))
>>> sln = collect(base, P)
'''
# Can also have one at g
# T, a_alpha = self.T, self.a_alpha
a = a_alpha
# b, epsilon, delta = self.b, self.epsilon, self.delta
T_inv = 1.0/T
# TODO cse
x0 = 4.0*a
x1 = b*x0
x2 = a+a
x3 = delta*x2
x4 = R*T
x5 = 4.0*epsilon
x6 = delta*delta
x7 = a*a
x8 = T_inv*R_inv
x9 = 8.0*epsilon
x10 = b*x9
x11 = 4.0*delta
x12 = delta*x6
x13 = 2.0*x6
x14 = b*x13
x15 = a*x8
x16 = epsilon*x15
x20 = x8*x8
x17 = x20*x8
x18 = b*delta
x19 = 6.0*x15
x21 = x20*x7
x22 = 10.0*b
x23 = b*b
x24 = 6.0*x23
x25 = x0*x8
x26 = x6*x6
x27 = epsilon*epsilon
x28 = 8.0*x27
x29 = 24.0*epsilon
x30 = b*x12
x31 = epsilon*x13
x32 = epsilon*x8
x33 = 12.0*epsilon
x34 = b*x23
x35 = x2*x8
x36 = 8.0*x21
x37 = x15*x6
x38 = delta*x23
x39 = b*x28
x40 = x34*x9
x41 = epsilon*x12
x42 = x23*x23
e = x1 + x3 + x4*x5 - x4*x6 - x7*x8
d = (4.0*x7*a*x17 - 10.0*delta*x21 + 2.0*(epsilon*x11 + x10 - x12
- x14 + x15*x24 + x18*x19 - x21*x22 + x25*x6) - 20.0*x16)
c = x8*(-x1*x32 + x12*x35 + x15*(12.0*x34 + 18.0*x38) + x18*(x29 + x36)
+ x21*(x33 - x6) + x22*x37 + x23*(x29 + x36) - x24*x6 - x26
+ x28 - x3*x32 - 6.0*x30 + x31)
b_coeff = (2.0*x20*(-b*x26 + delta*(x10*x15 + x25*x34) + epsilon*x14
+ x23*(x15*x9 - 3.0*x12 + x37) - x13*x34 - x15*x30
-x16*x6 + x27*(x19 + x11) + x33*x38 + x35*x42
+ x39 + x40 - x41))
a_coeff = x17*(-2.0*b*x41 + delta*(x39 + x40)
+ x27*(4.0*epsilon - x6)
- 2.0*x12*x34 + x23*(x28 + x31 - x26)
+ x42*(x5 - x6))
# e = (2*a*delta + 4*a*b -R*T*delta**2 - a**2/(R*T) + 4*R*T*epsilon)
# d = (-4*b*delta**2 + 16*b*epsilon - 2*delta**3 + 8*delta*epsilon + 12*a*b**2/(R*T) + 12*a*b*delta/(R*T) + 8*a*delta**2/(R*T) - 20*a*epsilon/(R*T) - 20*a**2*b/(R**2*T**2) - 10*a**2*delta/(R**2*T**2) + 4*a**3/(R**3*T**3))
# c = (-6*b**2*delta**2/(R*T) + 24*b**2*epsilon/(R*T) - 6*b*delta**3/(R*T) + 24*b*delta*epsilon/(R*T) - delta**4/(R*T) + 2*delta**2*epsilon/(R*T) + 8*epsilon**2/(R*T) + 12*a*b**3/(R**2*T**2) + 18*a*b**2*delta/(R**2*T**2) + 10*a*b*delta**2/(R**2*T**2) - 4*a*b*epsilon/(R**2*T**2) + 2*a*delta**3/(R**2*T**2) - 2*a*delta*epsilon/(R**2*T**2) + 8*a**2*b**2/(R**3*T**3) + 8*a**2*b*delta/(R**3*T**3) - a**2*delta**2/(R**3*T**3) + 12*a**2*epsilon/(R**3*T**3))
# b_coeff = (-4*b**3*delta**2/(R**2*T**2) + 16*b**3*epsilon/(R**2*T**2) - 6*b**2*delta**3/(R**2*T**2) + 24*b**2*delta*epsilon/(R**2*T**2) - 2*b*delta**4/(R**2*T**2) + 4*b*delta**2*epsilon/(R**2*T**2) + 16*b*epsilon**2/(R**2*T**2) - 2*delta**3*epsilon/(R**2*T**2) + 8*delta*epsilon**2/(R**2*T**2) + 4*a*b**4/(R**3*T**3) + 8*a*b**3*delta/(R**3*T**3) + 2*a*b**2*delta**2/(R**3*T**3) + 16*a*b**2*epsilon/(R**3*T**3) - 2*a*b*delta**3/(R**3*T**3) + 16*a*b*delta*epsilon/(R**3*T**3) - 2*a*delta**2*epsilon/(R**3*T**3) + 12*a*epsilon**2/(R**3*T**3))
# a_coeff = (-b**4*delta**2/(R**3*T**3) + 4*b**4*epsilon/(R**3*T**3) - 2*b**3*delta**3/(R**3*T**3) + 8*b**3*delta*epsilon/(R**3*T**3) - b**2*delta**4/(R**3*T**3) + 2*b**2*delta**2*epsilon/(R**3*T**3) + 8*b**2*epsilon**2/(R**3*T**3) - 2*b*delta**3*epsilon/(R**3*T**3) + 8*b*delta*epsilon**2/(R**3*T**3) - delta**2*epsilon**2/(R**3*T**3) + 4*epsilon**3/(R**3*T**3))
roots = roots_quartic(a_coeff, b_coeff, c, d, e)
# roots = np.roots([a_coeff, b_coeff, c, d, e]).tolist()
if valid:
# TODO - only include ones when switching phases from l/g to either g/l
# Do not know how to handle
roots = [r.real for r in roots if (r.real >= 0.0)]
roots.sort()
return roots
def _P_discriminant_zero(self, low):
# Can also have one at g
T, a_alpha = self.T, self.a_alpha
b, epsilon, delta = self.b, self.epsilon, self.delta
global niter
niter = 0
RT = R*T
x13 = RT**-6.0
x14 = b*epsilon
x15 = -b*delta + epsilon
x18 = b - delta
def discriminant_fun(P):
if P < 0:
raise ValueError("Will not converge")
global niter
niter += 1
x0 = P*P
x1 = P*epsilon
x2 = P*b + RT
x3 = a_alpha - delta*x2 + x1
x3_x3 = x3*x3
x4 = x3*x3_x3
x5 = a_alpha*b + epsilon*x2
x6 = 27.0*x5*x5
x7 = -P*delta + x2
x9 = x7*x7
x8 = x7*x9
x11 = x3*x5*x7
x12 = -18.0*P*x11 + 4.0*(P*x4 +x5*x8) + x0*x6 - x3_x3*x9
x16 = P*x15
x17 = 9.0*x3
x19 = x18*x5
# 26 mult so far
err = -x0*x12*x13
fprime = (-2.0*P*x13*(P*(-P*x17*x19 + P*x6 - b*x1*x17*x7
+ 27.0*x0*x14*x5 + 6.0*x3_x3*x16 - x3_x3*x18*x7
- 9.0*x11 + 2.0*x14*x8 - x15*x3*x9 - 9.0*x16*x5*x7 + 6.0*x19*x9 + 2.0*x4) + x12))
if niter > 3 and (.40 < (err/(P*fprime)) < 0.55):
raise ValueError("Not going to work")
# a = (err/fprime)/P
# print('low probably kill point')
return err, fprime
# New answer: Above critical T only high P result
# Ps = logspace(log10(1), log10(1e10), 40000)
# errs = []
# for P in Ps:
# erri = self.discriminant(P=P)
# if erri < 0:
# erri = -log10(abs(erri))
# else:
# erri = log10(erri)
# errs.append(erri)
# import matplotlib.pyplot as plt
# plt.semilogx(Ps, errs, 'x')
# # plt.ylim((-1e-3, 1e-3))
# plt.show()
# Checked once
# def damping_func(p0, step, damping):
# if p0 + step < 0.0:
# return 0.9*p0
# # while p0 + step < 1e3:
# # if p0 + step < 1e3:
# # step = 0.5*step
# return p0 + step
#low=1,damping_func=damping_func
# 5e7
try:
Tc = self.Tc
except:
Tc = self.pseudo_Tc
guesses = [1e5, 1e6, 1e7, 1e8, 1e9, .5, 1e-4, 1e-8, 1e-12, 1e-16, 1e-20]
if not low:
guesses = [1e9, 1e10, 1e10, 5e10, 2e10, 5e9, 5e8, 1e8]
if self.N == 1 and low:
try:
try:
Tc, Pc, omega = self.Tc, self.Pc, self.omega
except:
Tc, Pc, omega = self.Tcs[0], self.Pcs[0], self.omegas[0]
guesses.append(Pc*.99999999)
assert T/Tc > .3
P_wilson = Wilson_K_value(self.T, self.P, Tc, Pc, omega)*self.P
guesses.insert(0, P_wilson*3)
except:
pass
if low:
coeffs = self._P_zero_l_cheb_coeffs
coeffs_low, coeffs_high = self.P_zero_l_cheb_limits
else:
coeffs = self._P_zero_g_cheb_coeffs
coeffs_low, coeffs_high = self.P_zero_g_cheb_limits
if coeffs is not None:
try:
a = self.a
except:
a = self.pseudo_a
alpha = self.a_alpha/a
try:
Pc = self.Pc
except:
Pc = self.pseudo_Pc
Tr = self.T/Tc
alpha_Tr = alpha/(Tr)
x = alpha_Tr - 1.0
if coeffs_low < x < coeffs_high:
constant = 0.5*(-coeffs_low - coeffs_high)
factor = 2.0/(coeffs_high - coeffs_low)
y = chebval(factor*(x + constant), coeffs)
P_trans = y*Tr*Pc
guesses.insert(0, P_trans)
global_iter = 0
for P in guesses:
try:
global_iter += niter
niter = 0
# try:
# P_disc = newton(discriminant_fun, P, fprime=True, xtol=1e-16, low=1, maxiter=200, bisection=False, damping=1)
# except:
# high = None
# if self.N == 1:
# try:
# high = self.Pc
# except:
# high = self.Pcs[0]
# high *= (1+1e-11)
if not low and T < Tc:
low_bound = 1e8
else:
if Tr > .3:
low_bound = 1.0
else:
low_bound = None
P_disc = newton(discriminant_fun, P, fprime=True, xtol=4e-12, low=low_bound,
maxiter=80, bisection=False, damping=1)
assert P_disc > 0 and not P_disc == 1
if not low:
assert P_disc > low_bound
break
except:
pass
if not low:
assert P_disc > low_bound
global_iter += niter
# for i in range(1000):
# a = 1
if 0:
try:
P_disc = bisect(self._discriminant_at_T_mp, P_disc*(1-1e-8), P_disc*(1+1e-8), xtol=1e-18)
except:
try:
P_disc = bisect(self._discriminant_at_T_mp, P_disc*(1-1e-5), P_disc*(1+1e-5), xtol=1e-18)
except:
try:
P_disc = bisect(self._discriminant_at_T_mp, P_disc*(1-1e-2), P_disc*(1+1e-2))
except:
pass
# if not low:
# P_disc_base = None
# try:
# if T < Tc:
# P_disc_base = self._P_discriminant_zero(True)
# except:
# pass
# if P_disc_base is not None:
# # pass
# if isclose(P_disc_base, P_disc, rel_tol=1e-4):
# raise ValueError("Converged to wrong solution")
return float(P_disc)
# Can take a while to converge
P_disc = secant(lambda P: self.discriminant(P=P), self.P, xtol=1e-7, low=1e-12, maxiter=200, bisection=True)
if P_disc <= 0.0:
P_disc = secant(lambda P: self.discriminant(P=P), self.P*100, xtol=1e-7, maxiter=200)
# P_max = self.P*1000
# P_disc = brenth(lambda P: self.discriminant(P=P), self.P*1e-3, P_max, rtol=1e-7, maxiter=200)
return P_disc
def _plot_T_discriminant_zero(self):
Ts = logspace(log10(1), log10(1e4), 10000)
errs = []
for T in Ts:
erri = self.discriminant(T=T)
# if erri < 0:
# erri = -log10(abs(erri))
# else:
# erri = log10(erri)
errs.append(erri)
import matplotlib.pyplot as plt
plt.semilogx(Ts, errs, 'x')
# plt.ylim((-1e-3, 1e-3))
plt.show()
[docs] def T_discriminant_zero_l(self, T_guess=None):
r'''Method to calculate the temperature which zeros the discriminant
function of the general cubic eos, and is likely to sit on a boundary
between not having a liquid-like volume; and having a liquid-like volume.
Parameters
----------
T_guess : float, optional
Temperature guess, [K]
Returns
-------
T_discriminant_zero_l : float
Temperature which make the discriminants zero at the right condition,
[K]
Notes
-----
Significant numerical issues remain in improving this method.
Examples
--------
>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> T_trans = eos.T_discriminant_zero_l()
>>> T_trans
644.3023307
In this case, the discriminant transition does not reveal a transition
to two roots being available, only to there being a double (imaginary)
root.
>>> eos.to(P=eos.P, T=T_trans).mpmath_volumes_float
((9.309597822372529e-05-0.00015876248805149625j), (9.309597822372529e-05+0.00015876248805149625j), (0.005064847204219234+0j))
'''
# Can also have one at g
global niter
niter = 0
guesses = [100, 150, 200, 250, 300, 350, 400, 450]
if T_guess is not None:
guesses.append(T_guess)
if self.N == 1:
pass
global_iter = 0
for T in guesses:
try:
global_iter += niter
niter = 0
T_disc = secant(lambda T: self.discriminant(T=T), T, xtol=1e-10, low=1, maxiter=60, bisection=False, damping=1)
assert T_disc > 0 and not T_disc == 1
break
except:
pass
global_iter += niter
return T_disc
[docs] def T_discriminant_zero_g(self, T_guess=None):
r'''Method to calculate the temperature which zeros the discriminant
function of the general cubic eos, and is likely to sit on a boundary
between not having a vapor-like volume; and having a vapor-like volume.
Parameters
----------
T_guess : float, optional
Temperature guess, [K]
Returns
-------
T_discriminant_zero_g : float
Temperature which make the discriminants zero at the right condition,
[K]
Notes
-----
Significant numerical issues remain in improving this method.
Examples
--------
>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> T_trans = eos.T_discriminant_zero_g()
>>> T_trans
644.3023307
In this case, the discriminant transition does not reveal a transition
to two roots being available, only to there being a double (imaginary)
root.
>>> eos.to(P=eos.P, T=T_trans).mpmath_volumes_float
((9.309597822372529e-05-0.00015876248805149625j), (9.309597822372529e-05+0.00015876248805149625j), (0.005064847204219234+0j))
'''
global niter
niter = 0
guesses = [700, 600, 500, 400, 300, 200]
if T_guess is not None:
guesses.append(T_guess)
if self.N == 1:
pass
global_iter = 0
for T in guesses:
try:
global_iter += niter
niter = 0
T_disc = secant(lambda T: self.discriminant(T=T), T, xtol=1e-10, low=1, maxiter=60, bisection=False, damping=1)
assert T_disc > 0 and not T_disc == 1
break
except:
pass
global_iter += niter
return T_disc
[docs] def P_PIP_transition(self, T, low_P_limit=0.0):
r'''Method to calculate the pressure which makes the phase
identification parameter exactly 1. There are three regions for this
calculation:
* subcritical - PIP = 1 for the gas-like phase at P = 0
* initially supercritical - PIP = 1 on a curve starting at the
critical point, increasing for a while, decreasing for a while,
and then curving sharply back to a zero pressure.
* later supercritical - PIP = 1 for the liquid-like phase at P = 0
Parameters
----------
T : float
Temperature for the calculation, [K]
low_P_limit : float
What value to return for the subcritical and later region, [Pa]
Returns
-------
P : float
Pressure which makes the PIP = 1, [Pa]
Notes
-----
The transition between the region where this function returns values
and the high temperature region that doesn't is the Joule-Thomson
inversion point at a pressure of zero and can be directly solved for.
Examples
--------
>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.P_PIP_transition(100)
0.0
>>> low_T = eos.to(T=100.0, P=eos.P_PIP_transition(100, low_P_limit=1e-5))
>>> low_T.PIP_l, low_T.PIP_g
(45.778088191, 0.9999999997903)
>>> initial_super = eos.to(T=600.0, P=eos.P_PIP_transition(600))
>>> initial_super.P, initial_super.PIP_g
(6456282.17132, 0.999999999999)
>>> high_T = eos.to(T=900.0, P=eos.P_PIP_transition(900, low_P_limit=1e-5))
>>> high_T.P, high_T.PIP_g
(12536704.763, 0.9999999999)
'''
try:
subcritical = T < self.Tc
except:
subcritical = T < self.pseudo_Tc
if subcritical:
return low_P_limit
else:
def to_solve(P):
e = self.to(T=T, P=P)
# TODO: as all a_alpha is the same for all conditions, should be
# able to derive a direct expression for this from the EOS which
# only uses a volume solution
# TODO: should be able to get the derivative of PIP w.r.t. pressure
if hasattr(e, 'V_l'):
return e.PIP_l-1.0
else:
return e.PIP_g-1.0
try:
# Near the critical point these equations turn extremely nasty!
# bisection is the most reliable solver
if subcritical:
Psat = self.Psat(T)
low, high = 10.0*Psat, Psat
else:
low, high = 1e-3, 1e11
P = bisect(to_solve, low, high)
return P
except:
err_low = to_solve(low_P_limit)
if abs(err_low) < 1e-9:
# Well above the critical point all solutions except the
# zero-pressure limit have PIP values above 1
# This corresponds to the JT inversion temperature at a
# pressure of zero.
return low_P_limit
raise ValueError("Could not converge")
def _V_g_extrapolated(self):
P_pseudo_mc = sum([self.Pcs[i]*self.zs[i] for i in self.cmps])
T_pseudo_mc = sum([(self.Tcs[i]*self.Tcs[j])**0.5*self.zs[j]*self.zs[i]
for i in self.cmps for j in self.cmps])
V_pseudo_mc = (self.Zc*R*T_pseudo_mc)/P_pseudo_mc
rho_pseudo_mc = 1.0/V_pseudo_mc
P_disc = self.P_discriminant_zero_l()
try:
P_low = max(P_disc - 10.0, 1e-3)
eos_low = self.to_TP_zs(T=self.T, P=P_low, zs=self.zs)
rho_low = 1.0/eos_low.V_g
except:
P_low = max(P_disc + 10.0, 1e-3)
eos_low = self.to_TP_zs(T=self.T, P=P_low, zs=self.zs)
rho_low = 1.0/eos_low.V_g
rho0 = (rho_low + 1.4*rho_pseudo_mc)*0.5
dP_drho = eos_low.dP_drho_g
rho1 = P_low*((rho_low - 1.4*rho_pseudo_mc) + P_low/dP_drho)
rho2 = -P_low*P_low*((rho_low - 1.4*rho_pseudo_mc)*0.5 + P_low/dP_drho)
rho_ans = rho0 + rho1/eos_low.P + rho2/(eos_low.P*eos_low.P)
return 1.0/rho_ans
@property
def fugacity_l(self):
r'''Fugacity for the liquid phase, [Pa].
.. math::
\text{fugacity} = P\exp\left(\frac{G_{dep}}{RT}\right)
'''
arg = self.G_dep_l*R_inv/self.T
try:
return self.P*exp(arg)
except:
return self.P*1e308
@property
def fugacity_g(self):
r'''Fugacity for the gas phase, [Pa].
.. math::
\text{fugacity} = P\exp\left(\frac{G_{dep}}{RT}\right)
'''
arg = self.G_dep_g*R_inv/self.T
try:
return self.P*exp(arg)
except:
return self.P*1e308
@property
def phi_l(self):
r'''Fugacity coefficient for the liquid phase, [Pa].
.. math::
\phi = \frac{\text{fugacity}}{P}
'''
arg = self.G_dep_l*R_inv/self.T
try:
return exp(arg)
except:
return 1e308
@property
def phi_g(self):
r'''Fugacity coefficient for the gas phase, [Pa].
.. math::
\phi = \frac{\text{fugacity}}{P}
'''
arg = self.G_dep_g*R_inv/self.T
try:
return exp(arg)
except:
return 1e308
@property
def Cp_minus_Cv_l(self):
r'''Cp - Cv for the liquid phase, [J/mol/K].
.. math::
C_p - C_v = -T\left(\frac{\partial P}{\partial T}\right)_V^2/
\left(\frac{\partial P}{\partial V}\right)_T
'''
return -self.T*self.dP_dT_l*self.dP_dT_l*self.dV_dP_l
@property
def Cp_minus_Cv_g(self):
r'''Cp - Cv for the gas phase, [J/mol/K].
.. math::
C_p - C_v = -T\left(\frac{\partial P}{\partial T}\right)_V^2/
\left(\frac{\partial P}{\partial V}\right)_T
'''
return -self.T*self.dP_dT_g*self.dP_dT_g*self.dV_dP_g
@property
def beta_l(self):
r'''Isobaric (constant-pressure) expansion coefficient for the liquid
phase, [1/K].
.. math::
\beta = \frac{1}{V}\frac{\partial V}{\partial T}
'''
return self.dV_dT_l/self.V_l
@property
def beta_g(self):
r'''Isobaric (constant-pressure) expansion coefficient for the gas
phase, [1/K].
.. math::
\beta = \frac{1}{V}\frac{\partial V}{\partial T}
'''
return self.dV_dT_g/self.V_g
@property
def kappa_l(self):
r'''Isothermal (constant-temperature) expansion coefficient for the liquid
phase, [1/Pa].
.. math::
\kappa = \frac{-1}{V}\frac{\partial V}{\partial P}
'''
return -self.dV_dP_l/self.V_l
@property
def kappa_g(self):
r'''Isothermal (constant-temperature) expansion coefficient for the gas
phase, [1/Pa].
.. math::
\kappa = \frac{-1}{V}\frac{\partial V}{\partial P}
'''
return -self.dV_dP_g/self.V_g
@property
def V_dep_l(self):
r'''Departure molar volume from ideal gas behavior for the liquid phase,
[m^3/mol].
.. math::
V_{dep} = V - \frac{RT}{P}
'''
return self.V_l - self.T*R/self.P
@property
def V_dep_g(self):
r'''Departure molar volume from ideal gas behavior for the gas phase,
[m^3/mol].
.. math::
V_{dep} = V - \frac{RT}{P}
'''
return self.V_g - self.T*R/self.P
@property
def U_dep_l(self):
r'''Departure molar internal energy from ideal gas behavior for the
liquid phase, [J/mol].
.. math::
U_{dep} = H_{dep} - P V_{dep}
'''
return self.H_dep_l - self.P*(self.V_l - self.T*R/self.P)
@property
def U_dep_g(self):
r'''Departure molar internal energy from ideal gas behavior for the
gas phase, [J/mol].
.. math::
U_{dep} = H_{dep} - P V_{dep}
'''
return self.H_dep_g - self.P*(self.V_g - self.T*R/self.P)
@property
def A_dep_l(self):
r'''Departure molar Helmholtz energy from ideal gas behavior for the
liquid phase, [J/mol].
.. math::
A_{dep} = U_{dep} - T S_{dep}
'''
return self.H_dep_l - self.P*(self.V_l - self.T*R/self.P) - self.T*self.S_dep_l
@property
def A_dep_g(self):
r'''Departure molar Helmholtz energy from ideal gas behavior for the
gas phase, [J/mol].
.. math::
A_{dep} = U_{dep} - T S_{dep}
'''
return self.H_dep_g - self.P*(self.V_g - self.T*R/self.P) - self.T*self.S_dep_g
@property
def d2T_dPdV_l(self):
r'''Second partial derivative of temperature with respect to
pressure (constant volume) and then volume (constant pressure)
for the liquid phase, [K*mol/(Pa*m^3)].
.. math::
\left(\frac{\partial^2 T}{\partial P\partial V}\right) =
- \left[\left(\frac{\partial^2 P}{\partial T \partial V}\right)
\left(\frac{\partial P}{\partial T}\right)_V
- \left(\frac{\partial P}{\partial V}\right)_T
\left(\frac{\partial^2 P}{\partial T^2}\right)_V
\right]\left(\frac{\partial P}{\partial T}\right)_V^{-3}
'''
inverse_dP_dT2 = self.dT_dP_l*self.dT_dP_l
inverse_dP_dT3 = inverse_dP_dT2*self.dT_dP_l
d2T_dPdV = -(self.d2P_dTdV_l*self.dP_dT_l - self.dP_dV_l*self.d2P_dT2_l)*inverse_dP_dT3
return d2T_dPdV
@property
def d2T_dPdV_g(self):
r'''Second partial derivative of temperature with respect to
pressure (constant volume) and then volume (constant pressure)
for the gas phase, [K*mol/(Pa*m^3)].
.. math::
\left(\frac{\partial^2 T}{\partial P\partial V}\right) =
- \left[\left(\frac{\partial^2 P}{\partial T \partial V}\right)
\left(\frac{\partial P}{\partial T}\right)_V
- \left(\frac{\partial P}{\partial V}\right)_T
\left(\frac{\partial^2 P}{\partial T^2}\right)_V
\right]\left(\frac{\partial P}{\partial T}\right)_V^{-3}
'''
inverse_dP_dT2 = self.dT_dP_g*self.dT_dP_g
inverse_dP_dT3 = inverse_dP_dT2*self.dT_dP_g
d2T_dPdV = -(self.d2P_dTdV_g*self.dP_dT_g - self.dP_dV_g*self.d2P_dT2_g)*inverse_dP_dT3
return d2T_dPdV
@property
def d2V_dPdT_l(self):
r'''Second partial derivative of volume with respect to
pressure (constant temperature) and then presssure (constant temperature)
for the liquid phase, [m^3/(K*Pa*mol)].
.. math::
\left(\frac{\partial^2 V}{\partial T\partial P}\right) =
- \left[\left(\frac{\partial^2 P}{\partial T \partial V}\right)
\left(\frac{\partial P}{\partial V}\right)_T
- \left(\frac{\partial P}{\partial T}\right)_V
\left(\frac{\partial^2 P}{\partial V^2}\right)_T
\right]\left(\frac{\partial P}{\partial V}\right)_T^{-3}
'''
dV_dP = self.dV_dP_l
return -(self.d2P_dTdV_l*self.dP_dV_l - self.dP_dT_l*self.d2P_dV2_l)*dV_dP*dV_dP*dV_dP
@property
def d2V_dPdT_g(self):
r'''Second partial derivative of volume with respect to
pressure (constant temperature) and then presssure (constant temperature)
for the gas phase, [m^3/(K*Pa*mol)].
.. math::
\left(\frac{\partial^2 V}{\partial T\partial P}\right) =
- \left[\left(\frac{\partial^2 P}{\partial T \partial V}\right)
\left(\frac{\partial P}{\partial V}\right)_T
- \left(\frac{\partial P}{\partial T}\right)_V
\left(\frac{\partial^2 P}{\partial V^2}\right)_T
\right]\left(\frac{\partial P}{\partial V}\right)_T^{-3}
'''
dV_dP = self.dV_dP_g
return -(self.d2P_dTdV_g*self.dP_dV_g - self.dP_dT_g*self.d2P_dV2_g)*dV_dP*dV_dP*dV_dP
@property
def d2T_dP2_l(self):
r'''Second partial derivative of temperature with respect to
pressure (constant temperature) for the liquid phase, [K/Pa^2].
.. math::
\left(\frac{\partial^2 T}{\partial P^2}\right)_V = -\left(\frac{
\partial^2 P}{\partial T^2}\right)_V \left(\frac{\partial P}{
\partial T}\right)^{-3}_V
'''
dT_dP = self.dT_dP_l
return -self.d2P_dT2_l*dT_dP*dT_dP*dT_dP # unused
@property
def d2T_dP2_g(self):
r'''Second partial derivative of temperature with respect to
pressure (constant volume) for the gas phase, [K/Pa^2].
.. math::
\left(\frac{\partial^2 T}{\partial P^2}\right)_V = -\left(\frac{
\partial^2 P}{\partial T^2}\right)_V \left(\frac{\partial P}{
\partial T}\right)^{-3}_V
'''
dT_dP = self.dT_dP_g
return -self.d2P_dT2_g*dT_dP*dT_dP*dT_dP # unused
@property
def d2V_dP2_l(self):
r'''Second partial derivative of volume with respect to
pressure (constant temperature) for the liquid phase, [m^3/(Pa^2*mol)].
.. math::
\left(\frac{\partial^2 V}{\partial P^2}\right)_T = -\left(\frac{
\partial^2 P}{\partial V^2}\right)_T \left(\frac{\partial P}{
\partial V}\right)^{-3}_T
'''
dV_dP = self.dV_dP_l
return -self.d2P_dV2_l*dV_dP*dV_dP*dV_dP
@property
def d2V_dP2_g(self):
r'''Second partial derivative of volume with respect to
pressure (constant temperature) for the gas phase, [m^3/(Pa^2*mol)].
.. math::
\left(\frac{\partial^2 V}{\partial P^2}\right)_T = -\left(\frac{
\partial^2 P}{\partial V^2}\right)_T \left(\frac{\partial P}{
\partial V}\right)^{-3}_T
'''
dV_dP = self.dV_dP_g
return -self.d2P_dV2_g*dV_dP*dV_dP*dV_dP
@property
def d2T_dV2_l(self):
r'''Second partial derivative of temperature with respect to
volume (constant pressure) for the liquid phase, [K*mol^2/m^6].
.. math::
\left(\frac{\partial^2 T}{\partial V^2}\right)_P = -\left[
\left(\frac{\partial^2 P}{\partial V^2}\right)_T
\left(\frac{\partial P}{\partial T}\right)_V
- \left(\frac{\partial P}{\partial V}\right)_T
\left(\frac{\partial^2 P}{\partial T \partial V}\right) \right]
\left(\frac{\partial P}{\partial T}\right)^{-2}_V
+ \left[\left(\frac{\partial^2 P}{\partial T\partial V}\right)
\left(\frac{\partial P}{\partial T}\right)_V
- \left(\frac{\partial P}{\partial V}\right)_T
\left(\frac{\partial^2 P}{\partial T^2}\right)_V\right]
\left(\frac{\partial P}{\partial T}\right)_V^{-3}
\left(\frac{\partial P}{\partial V}\right)_T
'''
dT_dP = self.dT_dP_l
dT_dP2 = dT_dP*dT_dP
d2T_dV2 = dT_dP2*(-(self.d2P_dV2_l*self.dP_dT_l - self.dP_dV_l*self.d2P_dTdV_l)
+(self.d2P_dTdV_l*self.dP_dT_l - self.dP_dV_l*self.d2P_dT2_l)*dT_dP*self.dP_dV_l)
return d2T_dV2
@property
def d2T_dV2_g(self):
r'''Second partial derivative of temperature with respect to
volume (constant pressure) for the gas phase, [K*mol^2/m^6].
.. math::
\left(\frac{\partial^2 T}{\partial V^2}\right)_P = -\left[
\left(\frac{\partial^2 P}{\partial V^2}\right)_T
\left(\frac{\partial P}{\partial T}\right)_V
- \left(\frac{\partial P}{\partial V}\right)_T
\left(\frac{\partial^2 P}{\partial T \partial V}\right) \right]
\left(\frac{\partial P}{\partial T}\right)^{-2}_V
+ \left[\left(\frac{\partial^2 P}{\partial T\partial V}\right)
\left(\frac{\partial P}{\partial T}\right)_V
- \left(\frac{\partial P}{\partial V}\right)_T
\left(\frac{\partial^2 P}{\partial T^2}\right)_V\right]
\left(\frac{\partial P}{\partial T}\right)_V^{-3}
\left(\frac{\partial P}{\partial V}\right)_T
'''
dT_dP = self.dT_dP_g
dT_dP2 = dT_dP*dT_dP
d2T_dV2 = dT_dP2*(-(self.d2P_dV2_g*self.dP_dT_g - self.dP_dV_g*self.d2P_dTdV_g)
+(self.d2P_dTdV_g*self.dP_dT_g - self.dP_dV_g*self.d2P_dT2_g)*dT_dP*self.dP_dV_g)
return d2T_dV2
@property
def d2V_dT2_l(self):
r'''Second partial derivative of volume with respect to
temperature (constant pressure) for the liquid phase, [m^3/(mol*K^2)].
.. math::
\left(\frac{\partial^2 V}{\partial T^2}\right)_P = -\left[
\left(\frac{\partial^2 P}{\partial T^2}\right)_V
\left(\frac{\partial P}{\partial V}\right)_T
- \left(\frac{\partial P}{\partial T}\right)_V
\left(\frac{\partial^2 P}{\partial T \partial V}\right) \right]
\left(\frac{\partial P}{\partial V}\right)^{-2}_T
+ \left[\left(\frac{\partial^2 P}{\partial T\partial V}\right)
\left(\frac{\partial P}{\partial V}\right)_T
- \left(\frac{\partial P}{\partial T}\right)_V
\left(\frac{\partial^2 P}{\partial V^2}\right)_T\right]
\left(\frac{\partial P}{\partial V}\right)_T^{-3}
\left(\frac{\partial P}{\partial T}\right)_V
'''
dV_dP = self.dV_dP_l
dP_dV = self.dP_dV_l
d2P_dTdV = self.d2P_dTdV_l
dP_dT = self.dP_dT_l
d2V_dT2 = dV_dP*dV_dP*(-(self.d2P_dT2_l*dP_dV - dP_dT*d2P_dTdV) # unused
+(d2P_dTdV*dP_dV - dP_dT*self.d2P_dV2_l)*dV_dP*dP_dT)
return d2V_dT2
@property
def d2V_dT2_g(self):
r'''Second partial derivative of volume with respect to
temperature (constant pressure) for the gas phase, [m^3/(mol*K^2)].
.. math::
\left(\frac{\partial^2 V}{\partial T^2}\right)_P = -\left[
\left(\frac{\partial^2 P}{\partial T^2}\right)_V
\left(\frac{\partial P}{\partial V}\right)_T
- \left(\frac{\partial P}{\partial T}\right)_V
\left(\frac{\partial^2 P}{\partial T \partial V}\right) \right]
\left(\frac{\partial P}{\partial V}\right)^{-2}_T
+ \left[\left(\frac{\partial^2 P}{\partial T\partial V}\right)
\left(\frac{\partial P}{\partial V}\right)_T
- \left(\frac{\partial P}{\partial T}\right)_V
\left(\frac{\partial^2 P}{\partial V^2}\right)_T\right]
\left(\frac{\partial P}{\partial V}\right)_T^{-3}
\left(\frac{\partial P}{\partial T}\right)_V
'''
dV_dP = self.dV_dP_g
dP_dV = self.dP_dV_g
d2P_dTdV = self.d2P_dTdV_g
dP_dT = self.dP_dT_g
d2V_dT2 = dV_dP*dV_dP*(-(self.d2P_dT2_g*dP_dV - dP_dT*d2P_dTdV) # unused
+(d2P_dTdV*dP_dV - dP_dT*self.d2P_dV2_g)*dV_dP*dP_dT)
return d2V_dT2
@property
def Vc(self):
r'''Critical volume, [m^3/mol].
.. math::
V_c = \frac{Z_c R T_c}{P_c}
'''
return self.Zc*R*self.Tc/self.Pc
@property
def rho_l(self):
r'''Liquid molar density, [mol/m^3].
.. math::
\rho_l = \frac{1}{V_l}
'''
return 1.0/self.V_l
@property
def rho_g(self):
r'''Gas molar density, [mol/m^3].
.. math::
\rho_g = \frac{1}{V_g}
'''
return 1.0/self.V_g
@property
def dZ_dT_l(self):
r'''Derivative of compressibility factor with respect to temperature
for the liquid phase, [1/K].
.. math::
\frac{\partial Z}{\partial T} = \frac{P}{RT}\left(
\frac{\partial V}{\partial T} - \frac{V}{T}
\right)
'''
T_inv = 1.0/self.T
return self.P*R_inv*T_inv*(self.dV_dT_l - self.V_l*T_inv)
@property
def dZ_dT_g(self):
r'''Derivative of compressibility factor with respect to temperature
for the gas phase, [1/K].
.. math::
\frac{\partial Z}{\partial T} = \frac{P}{RT}\left(
\frac{\partial V}{\partial T} - \frac{V}{T}
\right)
'''
T_inv = 1.0/self.T
return self.P*R_inv*T_inv*(self.dV_dT_g - self.V_g*T_inv)
@property
def dZ_dP_l(self):
r'''Derivative of compressibility factor with respect to pressure
for the liquid phase, [1/Pa].
.. math::
\frac{\partial Z}{\partial P} = \frac{1}{RT}\left(
V - \frac{\partial V}{\partial P}
\right)
'''
return (self.V_l + self.P*self.dV_dP_l)/(self.T*R)
@property
def dZ_dP_g(self):
r'''Derivative of compressibility factor with respect to pressure
for the gas phase, [1/Pa].
.. math::
\frac{\partial Z}{\partial P} = \frac{1}{RT}\left(
V - \frac{\partial V}{\partial P}
\right)
'''
return (self.V_g + self.P*self.dV_dP_g)/(self.T*R)
d2V_dTdP_l = d2V_dPdT_l
d2V_dTdP_g = d2V_dPdT_g
d2T_dVdP_l = d2T_dPdV_l
d2T_dVdP_g = d2T_dPdV_g
@property
def d2P_dVdT_l(self):
'''Alias of :obj:`GCEOS.d2P_dTdV_l`'''
return self.d2P_dTdV_l
@property
def d2P_dVdT_g(self):
'''Alias of :obj:`GCEOS.d2P_dTdV_g`'''
return self.d2P_dTdV_g
@property
def dP_drho_l(self):
r'''Derivative of pressure with respect to molar density for the liquid
phase, [Pa/(mol/m^3)].
.. math::
\frac{\partial P}{\partial \rho} = -V^2 \frac{\partial P}{\partial V}
'''
return -self.V_l*self.V_l*self.dP_dV_l
@property
def dP_drho_g(self):
r'''Derivative of pressure with respect to molar density for the gas
phase, [Pa/(mol/m^3)].
.. math::
\frac{\partial P}{\partial \rho} = -V^2 \frac{\partial P}{\partial V}
'''
return -self.V_g*self.V_g*self.dP_dV_g
@property
def drho_dP_l(self):
r'''Derivative of molar density with respect to pressure for the liquid
phase, [(mol/m^3)/Pa].
.. math::
\frac{\partial \rho}{\partial P} = \frac{-1}{V^2} \frac{\partial V}{\partial P}
'''
return -self.dV_dP_l/(self.V_l*self.V_l)
@property
def drho_dP_g(self):
r'''Derivative of molar density with respect to pressure for the gas
phase, [(mol/m^3)/Pa].
.. math::
\frac{\partial \rho}{\partial P} = \frac{-1}{V^2} \frac{\partial V}{\partial P}
'''
return -self.dV_dP_g/(self.V_g*self.V_g)
@property
def d2P_drho2_l(self):
r'''Second derivative of pressure with respect to molar density for the
liquid phase, [Pa/(mol/m^3)^2].
.. math::
\frac{\partial^2 P}{\partial \rho^2} = -V^2\left(
-V^2\frac{\partial^2 P}{\partial V^2} - 2V \frac{\partial P}{\partial V}
\right)
'''
return -self.V_l**2*(-self.V_l**2*self.d2P_dV2_l - 2*self.V_l*self.dP_dV_l)
@property
def d2P_drho2_g(self):
r'''Second derivative of pressure with respect to molar density for the
gas phase, [Pa/(mol/m^3)^2].
.. math::
\frac{\partial^2 P}{\partial \rho^2} = -V^2\left(
-V^2\frac{\partial^2 P}{\partial V^2} - 2V \frac{\partial P}{\partial V}
\right)
'''
return -self.V_g**2*(-self.V_g**2*self.d2P_dV2_g - 2*self.V_g*self.dP_dV_g)
@property
def d2rho_dP2_l(self):
r'''Second derivative of molar density with respect to pressure for the
liquid phase, [(mol/m^3)/Pa^2].
.. math::
\frac{\partial^2 \rho}{\partial P^2} =
-\frac{\partial^2 V}{\partial P^2}\frac{1}{V^2}
+ 2 \left(\frac{\partial V}{\partial P}\right)^2\frac{1}{V^3}
'''
return -self.d2V_dP2_l/self.V_l**2 + 2*self.dV_dP_l**2/self.V_l**3
@property
def d2rho_dP2_g(self):
r'''Second derivative of molar density with respect to pressure for the
gas phase, [(mol/m^3)/Pa^2].
.. math::
\frac{\partial^2 \rho}{\partial P^2} =
-\frac{\partial^2 V}{\partial P^2}\frac{1}{V^2}
+ 2 \left(\frac{\partial V}{\partial P}\right)^2\frac{1}{V^3}
'''
return -self.d2V_dP2_g/self.V_g**2 + 2*self.dV_dP_g**2/self.V_g**3
@property
def dT_drho_l(self):
r'''Derivative of temperature with respect to molar density for the
liquid phase, [K/(mol/m^3)].
.. math::
\frac{\partial T}{\partial \rho} = V^2 \frac{\partial T}{\partial V}
'''
return -self.V_l*self.V_l*self.dT_dV_l
@property
def dT_drho_g(self):
r'''Derivative of temperature with respect to molar density for the
gas phase, [K/(mol/m^3)].
.. math::
\frac{\partial T}{\partial \rho} = V^2 \frac{\partial T}{\partial V}
'''
return -self.V_g*self.V_g*self.dT_dV_g
@property
def d2T_drho2_l(self):
r'''Second derivative of temperature with respect to molar density for
the liquid phase, [K/(mol/m^3)^2].
.. math::
\frac{\partial^2 T}{\partial \rho^2} =
-V^2(-V^2 \frac{\partial^2 T}{\partial V^2} -2V \frac{\partial T}{\partial V} )
'''
return -self.V_l**2*(-self.V_l**2*self.d2T_dV2_l - 2*self.V_l*self.dT_dV_l)
@property
def d2T_drho2_g(self):
r'''Second derivative of temperature with respect to molar density for
the gas phase, [K/(mol/m^3)^2].
.. math::
\frac{\partial^2 T}{\partial \rho^2} =
-V^2(-V^2 \frac{\partial^2 T}{\partial V^2} -2V \frac{\partial T}{\partial V} )
'''
return -self.V_g**2*(-self.V_g**2*self.d2T_dV2_g - 2*self.V_g*self.dT_dV_g)
@property
def drho_dT_l(self):
r'''Derivative of molar density with respect to temperature for the
liquid phase, [(mol/m^3)/K].
.. math::
\frac{\partial \rho}{\partial T} = - \frac{1}{V^2}
\frac{\partial V}{\partial T}
'''
return -self.dV_dT_l/(self.V_l*self.V_l)
@property
def drho_dT_g(self):
r'''Derivative of molar density with respect to temperature for the
gas phase, [(mol/m^3)/K].
.. math::
\frac{\partial \rho}{\partial T} = - \frac{1}{V^2}
\frac{\partial V}{\partial T}
'''
return -self.dV_dT_g/(self.V_g*self.V_g)
@property
def d2rho_dT2_l(self):
r'''Second derivative of molar density with respect to temperature for
the liquid phase, [(mol/m^3)/K^2].
.. math::
\frac{\partial^2 \rho}{\partial T^2} =
-\frac{\partial^2 V}{\partial T^2}\frac{1}{V^2}
+ 2 \left(\frac{\partial V}{\partial T}\right)^2\frac{1}{V^3}
'''
return -self.d2V_dT2_l/self.V_l**2 + 2*self.dV_dT_l**2/self.V_l**3
@property
def d2rho_dT2_g(self):
r'''Second derivative of molar density with respect to temperature for
the gas phase, [(mol/m^3)/K^2].
.. math::
\frac{\partial^2 \rho}{\partial T^2} =
-\frac{\partial^2 V}{\partial T^2}\frac{1}{V^2}
+ 2 \left(\frac{\partial V}{\partial T}\right)^2\frac{1}{V^3}
'''
return -self.d2V_dT2_g/self.V_g**2 + 2*self.dV_dT_g**2/self.V_g**3
@property
def d2P_dTdrho_l(self):
r'''Derivative of pressure with respect to molar density, and
temperature for the liquid phase, [Pa/(K*mol/m^3)].
.. math::
\frac{\partial^2 P}{\partial \rho\partial T}
= -V^2 \frac{\partial^2 P}{\partial T \partial V}
'''
return -(self.V_l*self.V_l)*self.d2P_dTdV_l
@property
def d2P_dTdrho_g(self):
r'''Derivative of pressure with respect to molar density, and
temperature for the gas phase, [Pa/(K*mol/m^3)].
.. math::
\frac{\partial^2 P}{\partial \rho\partial T}
= -V^2 \frac{\partial^2 P}{\partial T \partial V}
'''
return -(self.V_g*self.V_g)*self.d2P_dTdV_g
@property
def d2T_dPdrho_l(self):
r'''Derivative of temperature with respect to molar density, and
pressure for the liquid phase, [K/(Pa*mol/m^3)].
.. math::
\frac{\partial^2 T}{\partial \rho\partial P}
= -V^2 \frac{\partial^2 T}{\partial P \partial V}
'''
return -(self.V_l*self.V_l)*self.d2T_dPdV_l
@property
def d2T_dPdrho_g(self):
r'''Derivative of temperature with respect to molar density, and
pressure for the gas phase, [K/(Pa*mol/m^3)].
.. math::
\frac{\partial^2 T}{\partial \rho\partial P}
= -V^2 \frac{\partial^2 T}{\partial P \partial V}
'''
return -(self.V_g*self.V_g)*self.d2T_dPdV_g
@property
def d2rho_dPdT_l(self):
r'''Second derivative of molar density with respect to pressure
and temperature for the liquid phase, [(mol/m^3)/(K*Pa)].
.. math::
\frac{\partial^2 \rho}{\partial T \partial P} =
-\frac{\partial^2 V}{\partial T \partial P}\frac{1}{V^2}
+ 2 \left(\frac{\partial V}{\partial T}\right)
\left(\frac{\partial V}{\partial P}\right)
\frac{1}{V^3}
'''
return -self.d2V_dPdT_l/self.V_l**2 + 2*self.dV_dT_l*self.dV_dP_l/self.V_l**3
@property
def d2rho_dPdT_g(self):
r'''Second derivative of molar density with respect to pressure
and temperature for the gas phase, [(mol/m^3)/(K*Pa)].
.. math::
\frac{\partial^2 \rho}{\partial T \partial P} =
-\frac{\partial^2 V}{\partial T \partial P}\frac{1}{V^2}
+ 2 \left(\frac{\partial V}{\partial T}\right)
\left(\frac{\partial V}{\partial P}\right)
\frac{1}{V^3}
'''
return -self.d2V_dPdT_g/self.V_g**2 + 2*self.dV_dT_g*self.dV_dP_g/self.V_g**3
@property
def dH_dep_dT_l(self):
r'''Derivative of departure enthalpy with respect to
temperature for the liquid phase, [(J/mol)/K].
.. math::
\frac{\partial H_{dep, l}}{\partial T} = P \frac{d}{d T} V{\left (T
\right )} - R + \frac{2 T}{\sqrt{\delta^{2} - 4 \epsilon}}
\operatorname{atanh}{\left (\frac{\delta + 2 V{\left (T \right
)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right )} \frac{d^{2}}{d
T^{2}} \operatorname{a \alpha}{\left (T \right )} + \frac{4
\left(T \frac{d}{d T} \operatorname{a \alpha}{\left (T \right
)} - \operatorname{a \alpha}{\left (T \right )}\right) \frac{d}
{d T} V{\left (T \right )}}{\left(\delta^{2} - 4 \epsilon
\right) \left(- \frac{\left(\delta + 2 V{\left (T \right )}
\right)^{2}}{\delta^{2} - 4 \epsilon} + 1\right)}
'''
x0 = self.V_l
x1 = self.dV_dT_l
x2 = self.a_alpha
x3 = self.delta*self.delta - 4.0*self.epsilon
if x3 == 0.0:
x3 = 1e-100
x4 = x3**-0.5
x5 = self.delta + x0 + x0
x6 = 1.0/x3
return (self.P*x1 - R + 2.0*self.T*x4*catanh(x4*x5).real*self.d2a_alpha_dT2
- 4.0*x1*x6*(self.T*self.da_alpha_dT - x2)/(x5*x5*x6 - 1.0))
@property
def dH_dep_dT_g(self):
r'''Derivative of departure enthalpy with respect to
temperature for the gas phase, [(J/mol)/K].
.. math::
\frac{\partial H_{dep, g}}{\partial T} = P \frac{d}{d T} V{\left (T
\right )} - R + \frac{2 T}{\sqrt{\delta^{2} - 4 \epsilon}}
\operatorname{atanh}{\left (\frac{\delta + 2 V{\left (T \right
)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right )} \frac{d^{2}}{d
T^{2}} \operatorname{a \alpha}{\left (T \right )} + \frac{4
\left(T \frac{d}{d T} \operatorname{a \alpha}{\left (T \right
)} - \operatorname{a \alpha}{\left (T \right )}\right) \frac{d}
{d T} V{\left (T \right )}}{\left(\delta^{2} - 4 \epsilon
\right) \left(- \frac{\left(\delta + 2 V{\left (T \right )}
\right)^{2}}{\delta^{2} - 4 \epsilon} + 1\right)}
'''
x0 = self.V_g
x1 = self.dV_dT_g
if x0 > 1e50:
if isinf(self.dV_dT_g) or self.H_dep_g == 0.0:
return 0.0
x2 = self.a_alpha
x3 = self.delta*self.delta - 4.0*self.epsilon
if x3 == 0.0:
x3 = 1e-100
x4 = x3**-0.5
x5 = self.delta + x0 + x0
x6 = 1.0/x3
return (self.P*x1 - R + 2.0*self.T*x4*catanh(x4*x5).real*self.d2a_alpha_dT2
- 4.0*x1*x6*(self.T*self.da_alpha_dT - x2)/(x5*x5*x6 - 1.0))
@property
def dH_dep_dT_l_V(self):
r'''Derivative of departure enthalpy with respect to
temperature at constant volume for the liquid phase, [(J/mol)/K].
.. math::
\left(\frac{\partial H_{dep, l}}{\partial T}\right)_{V} =
- R + \frac{2 T
\operatorname{atanh}{\left(\frac{2 V_l + \delta}{\sqrt{\delta^{2}
- 4 \epsilon}} \right)} \frac{d^{2}}{d T^{2}} \operatorname{
a_{\alpha}}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}}
+ V_l \frac{\partial}{\partial T} P{\left(T,V \right)}
'''
T = self.T
delta, epsilon = self.delta, self.epsilon
V = self.V_l
dP_dT = self.dP_dT_l
try:
x0 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x0 = 1e100
return -R + 2.0*T*x0*catanh(x0*(V + V + delta)).real*self.d2a_alpha_dT2 + V*dP_dT
@property
def dH_dep_dT_g_V(self):
r'''Derivative of departure enthalpy with respect to
temperature at constant volume for the gas phase, [(J/mol)/K].
.. math::
\left(\frac{\partial H_{dep, g}}{\partial T}\right)_{V} =
- R + \frac{2 T
\operatorname{atanh}{\left(\frac{2 V_g + \delta}{\sqrt{\delta^{2}
- 4 \epsilon}} \right)} \frac{d^{2}}{d T^{2}} \operatorname{
a_{\alpha}}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}}
+ V_g \frac{\partial}{\partial T} P{\left(T,V \right)}
'''
T = self.T
delta, epsilon = self.delta, self.epsilon
V = self.V_g
dP_dT = self.dP_dT_g
try:
x0 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x0 = 1e100
return -R + 2.0*T*x0*catanh(x0*(V + V + delta)).real*self.d2a_alpha_dT2 + V*dP_dT
@property
def dH_dep_dP_l(self):
r'''Derivative of departure enthalpy with respect to
pressure for the liquid phase, [(J/mol)/Pa].
.. math::
\frac{\partial H_{dep, l}}{\partial P} = P \frac{d}{d P} V{\left (P
\right )} + V{\left (P \right )} + \frac{4 \left(T \frac{d}{d T}
\operatorname{a \alpha}{\left (T \right )} - \operatorname{a
\alpha}{\left (T \right )}\right) \frac{d}{d P} V{\left (P \right
)}}{\left(\delta^{2} - 4 \epsilon\right) \left(- \frac{\left(\delta
+ 2 V{\left (P \right )}\right)^{2}}{\delta^{2} - 4 \epsilon}
+ 1\right)}
'''
delta = self.delta
x0 = self.V_l
x2 = delta*delta - 4.0*self.epsilon
x4 = (delta + x0 + x0)
return (x0 + self.dV_dP_l*(self.P - 4.0*(self.T*self.da_alpha_dT
- self.a_alpha)/(x4*x4 - x2)))
@property
def dH_dep_dP_g(self):
r'''Derivative of departure enthalpy with respect to
pressure for the gas phase, [(J/mol)/Pa].
.. math::
\frac{\partial H_{dep, g}}{\partial P} = P \frac{d}{d P} V{\left (P
\right )} + V{\left (P \right )} + \frac{4 \left(T \frac{d}{d T}
\operatorname{a \alpha}{\left (T \right )} - \operatorname{a
\alpha}{\left (T \right )}\right) \frac{d}{d P} V{\left (P \right
)}}{\left(\delta^{2} - 4 \epsilon\right) \left(- \frac{\left(\delta
+ 2 V{\left (P \right )}\right)^{2}}{\delta^{2} - 4 \epsilon}
+ 1\right)}
'''
delta = self.delta
x0 = self.V_g
x2 = delta*delta - 4.0*self.epsilon
x4 = (delta + x0 + x0)
# if isinf(self.dV_dP_g):
# This does not appear to be correct
# return 0.0
return (x0 + self.dV_dP_g*(self.P - 4.0*(self.T*self.da_alpha_dT
- self.a_alpha)/(x4*x4 - x2)))
@property
def dH_dep_dP_l_V(self):
r'''Derivative of departure enthalpy with respect to
pressure at constant volume for the gas phase, [(J/mol)/Pa].
.. math::
\left(\frac{\partial H_{dep, g}}{\partial P}\right)_{V} =
- R \left(\frac{\partial T}{\partial P}\right)_V + V + \frac{2 \left(T
\left(\frac{\partial \left(\frac{\partial a \alpha}{\partial T}
\right)_P}{\partial P}\right)_{V}
+ \left(\frac{\partial a \alpha}{\partial T}\right)_P
\left(\frac{\partial T}{\partial P}\right)_V - \left(\frac{
\partial a \alpha}{\partial P}\right)_{V} \right)
\operatorname{atanh}{\left(\frac{2 V + \delta}
{\sqrt{\delta^{2} - 4 \epsilon}} \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}}
'''
T, V, delta, epsilon = self.T, self.V_l, self.delta, self.epsilon
da_alpha_dT, d2a_alpha_dT2 = self.da_alpha_dT, self.d2a_alpha_dT2
dT_dP = self.dT_dP_l
d2a_alpha_dTdP_V = d2a_alpha_dT2*dT_dP
da_alpha_dP_V = da_alpha_dT*dT_dP
try:
x0 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x0 = 1e100
return (-R*dT_dP + V + 2.0*x0*(
T*d2a_alpha_dTdP_V + dT_dP*da_alpha_dT - da_alpha_dP_V)
*catanh(x0*(V + V + delta)).real)
@property
def dH_dep_dP_g_V(self):
r'''Derivative of departure enthalpy with respect to
pressure at constant volume for the liquid phase, [(J/mol)/Pa].
.. math::
\left(\frac{\partial H_{dep, g}}{\partial P}\right)_{V} =
- R \left(\frac{\partial T}{\partial P}\right)_V + V + \frac{2 \left(T
\left(\frac{\partial \left(\frac{\partial a \alpha}{\partial T}
\right)_P}{\partial P}\right)_{V}
+ \left(\frac{\partial a \alpha}{\partial T}\right)_P
\left(\frac{\partial T}{\partial P}\right)_V - \left(\frac{
\partial a \alpha}{\partial P}\right)_{V} \right)
\operatorname{atanh}{\left(\frac{2 V + \delta}
{\sqrt{\delta^{2} - 4 \epsilon}} \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}}
'''
T, V, delta, epsilon = self.T, self.V_g, self.delta, self.epsilon
da_alpha_dT, d2a_alpha_dT2 = self.da_alpha_dT, self.d2a_alpha_dT2
dT_dP = self.dT_dP_g
d2a_alpha_dTdP_V = d2a_alpha_dT2*dT_dP
da_alpha_dP_V = da_alpha_dT*dT_dP
try:
x0 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x0 = 1e100
return (-R*dT_dP + V + 2.0*x0*(
T*d2a_alpha_dTdP_V + dT_dP*da_alpha_dT - da_alpha_dP_V)
*catanh(x0*(V + V + delta)).real)
@property
def dH_dep_dV_g_T(self):
r'''Derivative of departure enthalpy with respect to
volume at constant temperature for the gas phase, [J/m^3].
.. math::
\left(\frac{\partial H_{dep, g}}{\partial V}\right)_{T} =
\left(\frac{\partial H_{dep, g}}{\partial P}\right)_{T} \cdot
\left(\frac{\partial P}{\partial V}\right)_{T}
'''
return self.dH_dep_dP_g*self.dP_dV_g
@property
def dH_dep_dV_l_T(self):
r'''Derivative of departure enthalpy with respect to
volume at constant temperature for the gas phase, [J/m^3].
.. math::
\left(\frac{\partial H_{dep, l}}{\partial V}\right)_{T} =
\left(\frac{\partial H_{dep, l}}{\partial P}\right)_{T} \cdot
\left(\frac{\partial P}{\partial V}\right)_{T}
'''
return self.dH_dep_dP_l*self.dP_dV_l
@property
def dH_dep_dV_g_P(self):
r'''Derivative of departure enthalpy with respect to
volume at constant pressure for the gas phase, [J/m^3].
.. math::
\left(\frac{\partial H_{dep, g}}{\partial V}\right)_{P} =
\left(\frac{\partial H_{dep, g}}{\partial T}\right)_{P} \cdot
\left(\frac{\partial T}{\partial V}\right)_{P}
'''
return self.dH_dep_dT_g*self.dT_dV_g
@property
def dH_dep_dV_l_P(self):
r'''Derivative of departure enthalpy with respect to
volume at constant pressure for the liquid phase, [J/m^3].
.. math::
\left(\frac{\partial H_{dep, l}}{\partial V}\right)_{P} =
\left(\frac{\partial H_{dep, l}}{\partial T}\right)_{P} \cdot
\left(\frac{\partial T}{\partial V}\right)_{P}
'''
return self.dH_dep_dT_l*self.dT_dV_l
@property
def dS_dep_dT_l(self):
r'''Derivative of departure entropy with respect to
temperature for the liquid phase, [(J/mol)/K^2].
.. math::
\frac{\partial S_{dep, l}}{\partial T} = - \frac{R \frac{d}{d T}
V{\left (T \right )}}{V{\left (T \right )}} + \frac{R \frac{d}{d T}
V{\left (T \right )}}{- b + V{\left (T \right )}} + \frac{4
\frac{d}{d T} V{\left (T \right )} \frac{d}{d T} \operatorname{a
\alpha}{\left (T \right )}}{\left(\delta^{2} - 4 \epsilon\right)
\left(- \frac{\left(\delta + 2 V{\left (T \right )}\right)^{2}}
{\delta^{2} - 4 \epsilon} + 1\right)} + \frac{2 \frac{d^{2}}{d
T^{2}} \operatorname{a \alpha}{\left (T \right )}}
{\sqrt{\delta^{2} - 4 \epsilon}} \operatorname{atanh}{\left (\frac{
\delta + 2 V{\left (T \right )}}{\sqrt{\delta^{2} - 4 \epsilon}}
\right )} + \frac{R^{2} T}{P V{\left (T \right )}} \left(\frac{P}
{R T} \frac{d}{d T} V{\left (T \right )} - \frac{P}{R T^{2}}
V{\left (T \right )}\right)
'''
x0 = self.V_l
x1 = 1./x0
x2 = self.dV_dT_l
x3 = R*x2
x4 = self.a_alpha
x5 = self.delta*self.delta - 4.0*self.epsilon
if x5 == 0.0:
x5 = 1e-100
x6 = x5**-0.5
x7 = self.delta + 2.0*x0
x8 = 1.0/x5
return (R*x1*(x2 - x0/self.T) - x1*x3 - 4.0*x2*x8*self.da_alpha_dT
/(x7*x7*x8 - 1.0) - x3/(self.b - x0)
+ 2.0*x6*catanh(x6*x7).real*self.d2a_alpha_dT2)
@property
def dS_dep_dT_g(self):
r'''Derivative of departure entropy with respect to
temperature for the gas phase, [(J/mol)/K^2].
.. math::
\frac{\partial S_{dep, g}}{\partial T} = - \frac{R \frac{d}{d T}
V{\left (T \right )}}{V{\left (T \right )}} + \frac{R \frac{d}{d T}
V{\left (T \right )}}{- b + V{\left (T \right )}} + \frac{4
\frac{d}{d T} V{\left (T \right )} \frac{d}{d T} \operatorname{a
\alpha}{\left (T \right )}}{\left(\delta^{2} - 4 \epsilon\right)
\left(- \frac{\left(\delta + 2 V{\left (T \right )}\right)^{2}}
{\delta^{2} - 4 \epsilon} + 1\right)} + \frac{2 \frac{d^{2}}{d
T^{2}} \operatorname{a \alpha}{\left (T \right )}}
{\sqrt{\delta^{2} - 4 \epsilon}} \operatorname{atanh}{\left (\frac{
\delta + 2 V{\left (T \right )}}{\sqrt{\delta^{2} - 4 \epsilon}}
\right )} + \frac{R^{2} T}{P V{\left (T \right )}} \left(\frac{P}
{R T} \frac{d}{d T} V{\left (T \right )} - \frac{P}{R T^{2}}
V{\left (T \right )}\right)
'''
x0 = self.V_g
if x0 > 1e50:
if self.S_dep_g == 0.0:
return 0.0
x1 = 1./x0
x2 = self.dV_dT_g
if isinf(x2):
return 0.0
x3 = R*x2
x4 = self.a_alpha
x5 = self.delta*self.delta - 4.0*self.epsilon
if x5 == 0.0:
x5 = 1e-100
x6 = x5**-0.5
x7 = self.delta + 2.0*x0
x8 = 1.0/x5
return (R*x1*(x2 - x0/self.T) - x1*x3 - 4.0*x2*x8*self.da_alpha_dT
/(x7*x7*x8 - 1.0) - x3/(self.b - x0)
+ 2.0*x6*catanh(x6*x7).real*self.d2a_alpha_dT2)
@property
def dS_dep_dT_l_V(self):
r'''Derivative of departure entropy with respect to
temperature at constant volume for the liquid phase, [(J/mol)/K^2].
.. math::
\left(\frac{\partial S_{dep, l}}{\partial T}\right)_{V} =
\frac{R^{2} T \left(\frac{V \frac{\partial}{\partial T} P{\left(T,V
\right)}}{R T} - \frac{V P{\left(T,V \right)}}{R T^{2}}\right)}{
V P{\left(T,V \right)}} + \frac{2 \operatorname{atanh}{\left(
\frac{2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right)}
\frac{d^{2}}{d T^{2}} \operatorname{a \alpha}{\left(T \right)}}
{\sqrt{\delta^{2} - 4 \epsilon}}
'''
T, P = self.T, self.P
delta, epsilon = self.delta, self.epsilon
V = self.V_l
dP_dT = self.dP_dT_l
try:
x1 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x1 = 1e100
return (R*(dP_dT/P - 1.0/T) + 2.0*x1*catanh(x1*(V + V + delta)).real*self.d2a_alpha_dT2)
@property
def dS_dep_dT_g_V(self):
r'''Derivative of departure entropy with respect to
temperature at constant volume for the gas phase, [(J/mol)/K^2].
.. math::
\left(\frac{\partial S_{dep, g}}{\partial T}\right)_{V} =
\frac{R^{2} T \left(\frac{V \frac{\partial}{\partial T} P{\left(T,V
\right)}}{R T} - \frac{V P{\left(T,V \right)}}{R T^{2}}\right)}{
V P{\left(T,V \right)}} + \frac{2 \operatorname{atanh}{\left(
\frac{2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right)}
\frac{d^{2}}{d T^{2}} \operatorname{a \alpha}{\left(T \right)}}
{\sqrt{\delta^{2} - 4 \epsilon}}
'''
T, P = self.T, self.P
delta, epsilon = self.delta, self.epsilon
V = self.V_g
dP_dT = self.dP_dT_g
try:
x1 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x1 = 1e100
return (R*(dP_dT/P - 1.0/T) + 2.0*x1*catanh(x1*(V + V + delta)).real*self.d2a_alpha_dT2)
@property
def dS_dep_dP_l(self):
r'''Derivative of departure entropy with respect to
pressure for the liquid phase, [(J/mol)/K/Pa].
.. math::
\frac{\partial S_{dep, l}}{\partial P} = - \frac{R \frac{d}{d P}
V{\left (P \right )}}{V{\left (P \right )}} + \frac{R \frac{d}{d P}
V{\left (P \right )}}{- b + V{\left (P \right )}} + \frac{4 \frac{
d}{d P} V{\left (P \right )} \frac{d}{d T} \operatorname{a \alpha}
{\left (T \right )}}{\left(\delta^{2} - 4 \epsilon\right) \left(
- \frac{\left(\delta + 2 V{\left (P \right )}\right)^{2}}{
\delta^{2} - 4 \epsilon} + 1\right)} + \frac{R^{2} T}{P V{\left (P
\right )}} \left(\frac{P}{R T} \frac{d}{d P} V{\left (P \right )}
+ \frac{V{\left (P \right )}}{R T}\right)
'''
x0 = self.V_l
x1 = 1.0/x0
x2 = self.dV_dP_l
x3 = R*x2
try:
x4 = 1.0/(self.delta*self.delta - 4.0*self.epsilon)
except ZeroDivisionError:
x4 = 1e50
return (-x1*x3 - 4.0*x2*x4*self.da_alpha_dT/(x4*(self.delta + 2*x0)**2
- 1) - x3/(self.b - x0) + R*x1*(self.P*x2 + x0)/self.P)
@property
def dS_dep_dP_g(self):
r'''Derivative of departure entropy with respect to
pressure for the gas phase, [(J/mol)/K/Pa].
.. math::
\frac{\partial S_{dep, g}}{\partial P} = - \frac{R \frac{d}{d P}
V{\left (P \right )}}{V{\left (P \right )}} + \frac{R \frac{d}{d P}
V{\left (P \right )}}{- b + V{\left (P \right )}} + \frac{4 \frac{
d}{d P} V{\left (P \right )} \frac{d}{d T} \operatorname{a \alpha}
{\left (T \right )}}{\left(\delta^{2} - 4 \epsilon\right) \left(
- \frac{\left(\delta + 2 V{\left (P \right )}\right)^{2}}{
\delta^{2} - 4 \epsilon} + 1\right)} + \frac{R^{2} T}{P V{\left (P
\right )}} \left(\frac{P}{R T} \frac{d}{d P} V{\left (P \right )}
+ \frac{V{\left (P \right )}}{R T}\right)
'''
x0 = self.V_g
x1 = 1.0/x0
x2 = self.dV_dP_g
x3 = R*x2
try:
x4 = 1.0/(self.delta*self.delta - 4.0*self.epsilon)
except ZeroDivisionError:
x4 = 1e200
ans = (-x1*x3 - 4.0*x2*x4*self.da_alpha_dT/(x4*(self.delta + 2*x0)**2
- 1) - x3/(self.b - x0) + R*x1*(self.P*x2 + x0)/self.P)
return ans
@property
def dS_dep_dP_g_V(self):
r'''Derivative of departure entropy with respect to
pressure at constant volume for the gas phase, [(J/mol)/K/Pa].
.. math::
\left(\frac{\partial S_{dep, g}}{\partial P}\right)_{V} =
\frac{2 \operatorname{atanh}{\left(\frac{2 V + \delta}{
\sqrt{\delta^{2} - 4 \epsilon}} \right)}
\left(\frac{\partial \left(\frac{\partial a \alpha}{\partial T}
\right)_P}{\partial P}\right)_{V}}{\sqrt{\delta^{2} - 4 \epsilon}}
+ \frac{R^{2} \left(- \frac{P V \frac{d}{d P} T{\left(P \right)}}
{R T^{2}{\left(P \right)}}
+ \frac{V}{R T{\left(P \right)}}\right) T{\left(P \right)}}{P V}
'''
T, P, delta, epsilon = self.T, self.P, self.delta, self.epsilon
d2a_alpha_dT2 = self.d2a_alpha_dT2
V, dT_dP = self.V_g, self.dT_dP_g
d2a_alpha_dTdP_V = d2a_alpha_dT2*dT_dP
try:
x0 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x0 = 1e100
return (2.0*x0*catanh(x0*(V + V + delta)).real*d2a_alpha_dTdP_V
- R*(P*dT_dP/T - 1.0)/P)
@property
def dS_dep_dP_l_V(self):
r'''Derivative of departure entropy with respect to
pressure at constant volume for the liquid phase, [(J/mol)/K/Pa].
.. math::
\left(\frac{\partial S_{dep, l}}{\partial P}\right)_{V} =
\frac{2 \operatorname{atanh}{\left(\frac{2 V + \delta}{
\sqrt{\delta^{2} - 4 \epsilon}} \right)}
\left(\frac{\partial \left(\frac{\partial a \alpha}{\partial T}
\right)_P}{\partial P}\right)_{V}}{\sqrt{\delta^{2} - 4 \epsilon}}
+ \frac{R^{2} \left(- \frac{P V \frac{d}{d P} T{\left(P \right)}}
{R T^{2}{\left(P \right)}}
+ \frac{V}{R T{\left(P \right)}}\right) T{\left(P \right)}}{P V}
'''
T, P, delta, epsilon = self.T, self.P, self.delta, self.epsilon
d2a_alpha_dT2 = self.d2a_alpha_dT2
V, dT_dP = self.V_l, self.dT_dP_l
d2a_alpha_dTdP_V = d2a_alpha_dT2*dT_dP
try:
x0 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x0 = 1e100
return (2.0*x0*catanh(x0*(V + V + delta)).real*d2a_alpha_dTdP_V
- R*(P*dT_dP/T - 1.0)/P)
@property
def dS_dep_dV_g_T(self):
r'''Derivative of departure entropy with respect to
volume at constant temperature for the gas phase, [J/K/m^3].
.. math::
\left(\frac{\partial S_{dep, g}}{\partial V}\right)_{T} =
\left(\frac{\partial S_{dep, g}}{\partial P}\right)_{T} \cdot
\left(\frac{\partial P}{\partial V}\right)_{T}
'''
return self.dS_dep_dP_g*self.dP_dV_g
@property
def dS_dep_dV_l_T(self):
r'''Derivative of departure entropy with respect to
volume at constant temperature for the gas phase, [J/K/m^3].
.. math::
\left(\frac{\partial S_{dep, l}}{\partial V}\right)_{T} =
\left(\frac{\partial S_{dep, l}}{\partial P}\right)_{T} \cdot
\left(\frac{\partial P}{\partial V}\right)_{T}
'''
return self.dS_dep_dP_l*self.dP_dV_l
@property
def dS_dep_dV_g_P(self):
r'''Derivative of departure entropy with respect to
volume at constant pressure for the gas phase, [J/K/m^3].
.. math::
\left(\frac{\partial S_{dep, g}}{\partial V}\right)_{P} =
\left(\frac{\partial S_{dep, g}}{\partial T}\right)_{P} \cdot
\left(\frac{\partial T}{\partial V}\right)_{P}
'''
return self.dS_dep_dT_g*self.dT_dV_g
@property
def dS_dep_dV_l_P(self):
r'''Derivative of departure entropy with respect to
volume at constant pressure for the liquid phase, [J/K/m^3].
.. math::
\left(\frac{\partial S_{dep, l}}{\partial V}\right)_{P} =
\left(\frac{\partial S_{dep, l}}{\partial T}\right)_{P} \cdot
\left(\frac{\partial T}{\partial V}\right)_{P}
'''
return self.dS_dep_dT_l*self.dT_dV_l
@property
def d2H_dep_dT2_g(self):
r'''Second temperature derivative of departure enthalpy with respect to
temperature for the gas phase, [(J/mol)/K^2].
.. math::
\frac{\partial^2 H_{dep, g}}{\partial T^2} =
P \frac{d^{2}}{d T^{2}} V{\left(T \right)} - \frac{8 T \frac{d}{d T}
V{\left(T \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}
{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{
\left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2}
- 4 \epsilon} - 1\right)} + \frac{2 T \operatorname{atanh}{\left(
\frac{\delta + 2 V{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}}
\operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}} + \frac{16 \left(\delta + 2 V{\left(T \right)}
\right) \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T
\right)} - \operatorname{a\alpha}{\left(T \right)}\right) \left(
\frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(\delta^{2}
- 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T
\right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}}
- \frac{4 \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T
\right)} - \operatorname{a\alpha}{\left(T \right)}\right)
\frac{d^{2}}{d T^{2}} V{\left(T \right)}}{\left(\delta^{2}
- 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)}
\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2
\operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T \right)}}
{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}}{d T^{2}}
\operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}}
'''
T, P, delta, epsilon = self.T, self.P, self.delta, self.epsilon
x0 = self.V_g
x1 = self.d2V_dT2_g
x2 = self.a_alpha
x3 = self.d2a_alpha_dT2
x4 = delta*delta - 4.0*epsilon
try:
x5 = x4**-0.5
except:
x5 = 1e100
x6 = delta + x0 + x0
x7 = 2.0*x5*catanh(x5*x6).real
x8 = self.dV_dT_g
x9 = x5*x5
x10 = x6*x6*x9 - 1.0
x11 = x9/x10
x12 = T*self.da_alpha_dT - x2
x50 = self.d3a_alpha_dT3
return (P*x1 + x3*x7 + T*x7*x50- 4.0*x1*x11*x12 - 8.0*T*x11*x3*x8 + 16.0*x12*x6*x8*x8*x11*x11)
d2H_dep_dT2_g_P = d2H_dep_dT2_g
@property
def d2H_dep_dT2_l(self):
r'''Second temperature derivative of departure enthalpy with respect to
temperature for the liquid phase, [(J/mol)/K^2].
.. math::
\frac{\partial^2 H_{dep, l}}{\partial T^2} =
P \frac{d^{2}}{d T^{2}} V{\left(T \right)} - \frac{8 T \frac{d}{d T}
V{\left(T \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}
{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{
\left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2}
- 4 \epsilon} - 1\right)} + \frac{2 T \operatorname{atanh}{\left(
\frac{\delta + 2 V{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}}
\operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}} + \frac{16 \left(\delta + 2 V{\left(T \right)}
\right) \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T
\right)} - \operatorname{a\alpha}{\left(T \right)}\right) \left(
\frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(\delta^{2}
- 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T
\right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}}
- \frac{4 \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T
\right)} - \operatorname{a\alpha}{\left(T \right)}\right)
\frac{d^{2}}{d T^{2}} V{\left(T \right)}}{\left(\delta^{2}
- 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)}
\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2
\operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T \right)}}
{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}}{d T^{2}}
\operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}}
'''
T, P, delta, epsilon = self.T, self.P, self.delta, self.epsilon
x0 = self.V_l
x1 = self.d2V_dT2_l
x2 = self.a_alpha
x3 = self.d2a_alpha_dT2
x4 = delta*delta - 4.0*epsilon
try:
x5 = x4**-0.5
except:
x5 = 1e100
x6 = delta + x0 + x0
x7 = 2.0*x5*catanh(x5*x6).real
x8 = self.dV_dT_l
x9 = x5*x5
x10 = x6*x6*x9 - 1.0
x11 = x9/x10
x12 = T*self.da_alpha_dT - x2
x50 = self.d3a_alpha_dT3
return (P*x1 + x3*x7 + T*x7*x50- 4.0*x1*x11*x12 - 8.0*T*x11*x3*x8 + 16.0*x12*x6*x8*x8*x11*x11)
d2H_dep_dT2_l_P = d2H_dep_dT2_l
@property
def d2S_dep_dT2_g(self):
r'''Second temperature derivative of departure entropy with respect to
temperature for the gas phase, [(J/mol)/K^3].
.. math::
\frac{\partial^2 S_{dep, g}}{\partial T^2} = - \frac{R \left(
\frac{d}{d T} V{\left(T \right)} - \frac{V{\left(T \right)}}{T}
\right) \frac{d}{d T} V{\left(T \right)}}{V^{2}{\left(T \right)}}
+ \frac{R \left(\frac{d^{2}}{d T^{2}} V{\left(T \right)}
- \frac{2 \frac{d}{d T} V{\left(T \right)}}{T} + \frac{2
V{\left(T \right)}}{T^{2}}\right)}{V{\left(T \right)}}
- \frac{R \frac{d^{2}}{d T^{2}} V{\left(T \right)}}{V{\left(T
\right)}} + \frac{R \left(\frac{d}{d T} V{\left(T \right)}
\right)^{2}}{V^{2}{\left(T \right)}} - \frac{R \frac{d^{2}}{dT^{2}}
V{\left(T \right)}}{b - V{\left(T \right)}} - \frac{R \left(
\frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(b - V{\left(T
\right)}\right)^{2}} + \frac{R \left(\frac{d}{d T} V{\left(T
\right)} - \frac{V{\left(T \right)}}{T}\right)}{T V{\left(T
\right)}} + \frac{16 \left(\delta + 2 V{\left(T \right)}\right)
\left(\frac{d}{d T} V{\left(T \right)}\right)^{2} \frac{d}{d T}
\operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2}
- 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T
\right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}}
- \frac{8 \frac{d}{d T} V{\left(T \right)} \frac{d^{2}}{d T^{2}}
\operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2}
- 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)}
\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} - \frac{4
\frac{d^{2}}{d T^{2}} V{\left(T \right)} \frac{d}{d T}
\operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2}
- 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)}
\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2
\operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T
\right)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}
{d T^{3}} \operatorname{a\alpha}{\left(T \right)}}
{\sqrt{\delta^{2} - 4 \epsilon}}
'''
T, P, b, delta, epsilon = self.T, self.P, self.b, self.delta, self.epsilon
V = x0 = self.V_g
V_inv = 1.0/V
x1 = self.d2V_dT2_g
x2 = R*V_inv
x3 = V_inv*V_inv
x4 = self.dV_dT_g
x5 = x4*x4
x6 = R*x5
x7 = b - x0
x8 = 1.0/T
x9 = -x0*x8 + x4
x10 = x0 + x0
x11 = self.a_alpha
x12 = delta*delta - 4.0*epsilon
try:
x13 = x12**-0.5
except ZeroDivisionError:
x13 = 1e100
x14 = delta + x10
x15 = x13*x13
x16 = x14*x14*x15 - 1.0
x51 = 1.0/x16
x17 = x15*x51
x18 = self.da_alpha_dT
x50 = 1.0/x7
d2a_alpha_dT2 = self.d2a_alpha_dT2
d3a_alpha_dT3 = self.d3a_alpha_dT3
return (-R*x1*x50 - R*x3*x4*x9 - 4.0*x1*x17*x18 - x1*x2
+ 2.0*x13*catanh(x13*x14).real*d3a_alpha_dT3
- 8.0*x17*x4*d2a_alpha_dT2 + x2*x8*x9
+ x2*(x1 - 2.0*x4*x8 + x10*x8*x8) + x3*x6 - x6*x50*x50
+ 16.0*x14*x18*x5*x51*x51*x15*x15)
@property
def d2S_dep_dT2_l(self):
r'''Second temperature derivative of departure entropy with respect to
temperature for the liquid phase, [(J/mol)/K^3].
.. math::
\frac{\partial^2 S_{dep, l}}{\partial T^2} = - \frac{R \left(
\frac{d}{d T} V{\left(T \right)} - \frac{V{\left(T \right)}}{T}
\right) \frac{d}{d T} V{\left(T \right)}}{V^{2}{\left(T \right)}}
+ \frac{R \left(\frac{d^{2}}{d T^{2}} V{\left(T \right)}
- \frac{2 \frac{d}{d T} V{\left(T \right)}}{T} + \frac{2
V{\left(T \right)}}{T^{2}}\right)}{V{\left(T \right)}}
- \frac{R \frac{d^{2}}{d T^{2}} V{\left(T \right)}}{V{\left(T
\right)}} + \frac{R \left(\frac{d}{d T} V{\left(T \right)}
\right)^{2}}{V^{2}{\left(T \right)}} - \frac{R \frac{d^{2}}{dT^{2}}
V{\left(T \right)}}{b - V{\left(T \right)}} - \frac{R \left(
\frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(b - V{\left(T
\right)}\right)^{2}} + \frac{R \left(\frac{d}{d T} V{\left(T
\right)} - \frac{V{\left(T \right)}}{T}\right)}{T V{\left(T
\right)}} + \frac{16 \left(\delta + 2 V{\left(T \right)}\right)
\left(\frac{d}{d T} V{\left(T \right)}\right)^{2} \frac{d}{d T}
\operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2}
- 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T
\right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}}
- \frac{8 \frac{d}{d T} V{\left(T \right)} \frac{d^{2}}{d T^{2}}
\operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2}
- 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)}
\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} - \frac{4
\frac{d^{2}}{d T^{2}} V{\left(T \right)} \frac{d}{d T}
\operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2}
- 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)}
\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2
\operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T
\right)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}
{d T^{3}} \operatorname{a\alpha}{\left(T \right)}}
{\sqrt{\delta^{2} - 4 \epsilon}}
'''
T, P, b, delta, epsilon = self.T, self.P, self.b, self.delta, self.epsilon
V = x0 = self.V_l
V_inv = 1.0/V
x1 = self.d2V_dT2_l
x2 = R*V_inv
x3 = V_inv*V_inv
x4 = self.dV_dT_l
x5 = x4*x4
x6 = R*x5
x7 = b - x0
x8 = 1.0/T
x9 = -x0*x8 + x4
x10 = x0 + x0
x11 = self.a_alpha
x12 = delta*delta - 4.0*epsilon
try:
x13 = x12**-0.5
except ZeroDivisionError:
x13 = 1e100
x14 = delta + x10
x15 = x13*x13
x16 = x14*x14*x15 - 1.0
x51 = 1.0/x16
x17 = x15*x51
x18 = self.da_alpha_dT
x50 = 1.0/x7
d2a_alpha_dT2 = self.d2a_alpha_dT2
d3a_alpha_dT3 = self.d3a_alpha_dT3
return (-R*x1*x50 - R*x3*x4*x9 - 4.0*x1*x17*x18 - x1*x2
+ 2.0*x13*catanh(x13*x14).real*d3a_alpha_dT3
- 8.0*x17*x4*d2a_alpha_dT2 + x2*x8*x9
+ x2*(x1 - 2.0*x4*x8 + x10*x8*x8) + x3*x6 - x6*x50*x50
+ 16.0*x14*x18*x5*x51*x51*x15*x15)
@property
def d2H_dep_dT2_g_V(self):
r'''Second temperature derivative of departure enthalpy with respect to
temperature at constant volume for the gas phase, [(J/mol)/K^2].
.. math::
\left(\frac{\partial^2 H_{dep, g}}{\partial T^2}\right)_V =
\frac{2 T \operatorname{atanh}{\left(\frac{2 V + \delta}{\sqrt{
\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}}
\operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}} + V \frac{\partial^{2}}{\partial T^{2}}
P{\left(V,T \right)} + \frac{2 \operatorname{atanh}{\left(\frac{
2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}}
{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}}
'''
V, T, delta, epsilon = self.V_g, self.T, self.delta, self.epsilon
x51 = delta*delta - 4.0*epsilon
d2a_alpha_dT2 = self.d2a_alpha_dT2
d3a_alpha_dT3 = self.d3a_alpha_dT3
d2P_dT2 = self.d2P_dT2_g
try:
x1 = x51**-0.5
except ZeroDivisionError:
x1 = 1e100
x2 = 2.0*x1*catanh(x1*(V + V + delta)).real
return T*x2*d3a_alpha_dT3 + V*d2P_dT2 + x2*d2a_alpha_dT2
@property
def d2H_dep_dT2_l_V(self):
r'''Second temperature derivative of departure enthalpy with respect to
temperature at constant volume for the liquid phase, [(J/mol)/K^2].
.. math::
\left(\frac{\partial^2 H_{dep, l}}{\partial T^2}\right)_V =
\frac{2 T \operatorname{atanh}{\left(\frac{2 V + \delta}{\sqrt{
\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}}
\operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}} + V \frac{\partial^{2}}{\partial T^{2}}
P{\left(V,T \right)} + \frac{2 \operatorname{atanh}{\left(\frac{
2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}}
{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}}
'''
V, T, delta, epsilon = self.V_l, self.T, self.delta, self.epsilon
x51 = delta*delta - 4.0*epsilon
d2a_alpha_dT2 = self.d2a_alpha_dT2
d3a_alpha_dT3 = self.d3a_alpha_dT3
d2P_dT2 = self.d2P_dT2_l
try:
x1 = x51**-0.5
except ZeroDivisionError:
x1 = 1e100
x2 = 2.0*x1*catanh(x1*(V + V + delta)).real
return T*x2*d3a_alpha_dT3 + V*d2P_dT2 + x2*d2a_alpha_dT2
@property
def d2S_dep_dT2_g_V(self):
r'''Second temperature derivative of departure entropy with respect to
temperature at constant volume for the gas phase, [(J/mol)/K^3].
.. math::
\left(\frac{\partial^2 S_{dep, g}}{\partial T^2}\right)_V =
- \frac{R \left(\frac{\partial}{\partial T} P{\left(V,T \right)}
- \frac{P{\left(V,T \right)}}{T}\right) \frac{\partial}{\partial T}
P{\left(V,T \right)}}{P^{2}{\left(V,T \right)}} + \frac{R \left(
\frac{\partial^{2}}{\partial T^{2}} P{\left(V,T \right)} - \frac{2
\frac{\partial}{\partial T} P{\left(V,T \right)}}{T} + \frac{2
P{\left(V,T \right)}}{T^{2}}\right)}{P{\left(V,T \right)}}
+ \frac{R \left(\frac{\partial}{\partial T} P{\left(V,T \right)}
- \frac{P{\left(V,T \right)}}{T}\right)}{T P{\left(V,T \right)}}
+ \frac{2 \operatorname{atanh}{\left(\frac{2 V + \delta}{\sqrt{
\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}}
\operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}}
'''
V, T, delta, epsilon = self.V_g, self.T, self.delta, self.epsilon
d2a_alpha_dT2 = self.d2a_alpha_dT2
d3a_alpha_dT3 = self.d3a_alpha_dT3
d2P_dT2 = self.d2P_dT2_g
x0 = 1.0/T
x1 = self.P
P_inv = 1.0/x1
x2 = self.dP_dT_g
x3 = -x0*x1 + x2
x4 = R*P_inv
try:
x5 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x5 = 1e100
return (-R*x2*x3*P_inv*P_inv + x0*x3*x4 + x4*(d2P_dT2 - 2.0*x0*x2
+ 2.0*x1*x0*x0) + 2.0*x5*catanh(x5*(V + V + delta)
).real*d3a_alpha_dT3)
@property
def d2S_dep_dT2_l_V(self):
r'''Second temperature derivative of departure entropy with respect to
temperature at constant volume for the liquid phase, [(J/mol)/K^3].
.. math::
\left(\frac{\partial^2 S_{dep, l}}{\partial T^2}\right)_V =
- \frac{R \left(\frac{\partial}{\partial T} P{\left(V,T \right)}
- \frac{P{\left(V,T \right)}}{T}\right) \frac{\partial}{\partial T}
P{\left(V,T \right)}}{P^{2}{\left(V,T \right)}} + \frac{R \left(
\frac{\partial^{2}}{\partial T^{2}} P{\left(V,T \right)} - \frac{2
\frac{\partial}{\partial T} P{\left(V,T \right)}}{T} + \frac{2
P{\left(V,T \right)}}{T^{2}}\right)}{P{\left(V,T \right)}}
+ \frac{R \left(\frac{\partial}{\partial T} P{\left(V,T \right)}
- \frac{P{\left(V,T \right)}}{T}\right)}{T P{\left(V,T \right)}}
+ \frac{2 \operatorname{atanh}{\left(\frac{2 V + \delta}{\sqrt{
\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}}
\operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2}
- 4 \epsilon}}
'''
V, T, delta, epsilon = self.V_l, self.T, self.delta, self.epsilon
d2a_alpha_dT2 = self.d2a_alpha_dT2
d3a_alpha_dT3 = self.d3a_alpha_dT3
d2P_dT2 = self.d2P_dT2_l
x0 = 1.0/T
x1 = self.P
P_inv = 1.0/x1
x2 = self.dP_dT_l
x3 = -x0*x1 + x2
x4 = R*P_inv
try:
x5 = (delta*delta - 4.0*epsilon)**-0.5
except ZeroDivisionError:
x5 = 1e100
return (-R*x2*x3*P_inv*P_inv + x0*x3*x4 + x4*(d2P_dT2 - 2.0*x0*x2
+ 2.0*x1*x0*x0) + 2.0*x5*catanh(x5*(V + V + delta)
).real*d3a_alpha_dT3)
@property
def d2H_dep_dTdP_g(self):
r'''Temperature and pressure derivative of departure enthalpy
at constant pressure then temperature for the gas phase,
[(J/mol)/K/Pa].
.. math::
\left(\frac{\partial^2 H_{dep, g}}{\partial T \partial P}\right)_{T, P}
= P \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P \right)}
- \frac{4 T \frac{\partial}{\partial P} V{\left(T,P \right)}
\frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}
{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta
+ 2 V{\left(T,P \right)}\right)^{2}}{\delta^{2} - 4 \epsilon}
- 1\right)} + \frac{16 \left(\delta + 2 V{\left(T,P \right)}\right)
\left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}
- \operatorname{a\alpha}{\left(T \right)}\right) \frac{\partial}
{\partial P} V{\left(T,P \right)} \frac{\partial}{\partial T}
V{\left(T,P \right)}}{\left(\delta^{2} - 4 \epsilon\right)^{2}
\left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}}
{\delta^{2} - 4 \epsilon} - 1\right)^{2}} + \frac{\partial}
{\partial T} V{\left(T,P \right)} - \frac{4 \left(T \frac{d}{d T}
\operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}
{\left(T \right)}\right) \frac{\partial^{2}}{\partial T\partial P}
V{\left(T,P \right)}}{\left(\delta^{2} - 4 \epsilon\right)
\left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}}
{\delta^{2} - 4 \epsilon} - 1\right)}
'''
V, T, P, delta, epsilon = self.V_g, self.T, self.P, self.delta, self.epsilon
dV_dT = self.dV_dT_g
d2V_dTdP = self.d2V_dTdP_g
dV_dP = self.dV_dP_g
a_alpha = self.a_alpha
d2a_alpha_dT2 = self.d2a_alpha_dT2
x5 = delta*delta - 4.0*epsilon
try:
x6 = 1.0/x5
except ZeroDivisionError:
x6 = 1e100
x7 = delta + V + V
x8 = x6*x7*x7 - 1.0
x8_inv = 1.0/x8
x9 = 4.0*x6*x8_inv
x10 = T*self.da_alpha_dT - a_alpha
return (P*d2V_dTdP - T*dV_dP*x9*d2a_alpha_dT2
+ 16.0*dV_dT*x10*dV_dP*x7*x6*x6*x8_inv*x8_inv
+ dV_dT - x10*d2V_dTdP*x9)
@property
def d2H_dep_dTdP_l(self):
r'''Temperature and pressure derivative of departure enthalpy
at constant pressure then temperature for the liquid phase,
[(J/mol)/K/Pa].
.. math::
\left(\frac{\partial^2 H_{dep, l}}{\partial T \partial P}\right)_V
= P \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P \right)}
- \frac{4 T \frac{\partial}{\partial P} V{\left(T,P \right)}
\frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}
{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta
+ 2 V{\left(T,P \right)}\right)^{2}}{\delta^{2} - 4 \epsilon}
- 1\right)} + \frac{16 \left(\delta + 2 V{\left(T,P \right)}\right)
\left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}
- \operatorname{a\alpha}{\left(T \right)}\right) \frac{\partial}
{\partial P} V{\left(T,P \right)} \frac{\partial}{\partial T}
V{\left(T,P \right)}}{\left(\delta^{2} - 4 \epsilon\right)^{2}
\left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}}
{\delta^{2} - 4 \epsilon} - 1\right)^{2}} + \frac{\partial}
{\partial T} V{\left(T,P \right)} - \frac{4 \left(T \frac{d}{d T}
\operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}
{\left(T \right)}\right) \frac{\partial^{2}}{\partial T\partial P}
V{\left(T,P \right)}}{\left(\delta^{2} - 4 \epsilon\right)
\left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}}
{\delta^{2} - 4 \epsilon} - 1\right)}
'''
V, T, P, delta, epsilon = self.V_l, self.T, self.P, self.delta, self.epsilon
dV_dT = self.dV_dT_l
d2V_dTdP = self.d2V_dTdP_l
dV_dP = self.dV_dP_l
a_alpha = self.a_alpha
d2a_alpha_dT2 = self.d2a_alpha_dT2
x5 = delta*delta - 4.0*epsilon
try:
x6 = 1.0/x5
except ZeroDivisionError:
x6 = 1e100
x7 = delta + V + V
x8 = x6*x7*x7 - 1.0
x8_inv = 1.0/x8
x9 = 4.0*x6*x8_inv
x10 = T*self.da_alpha_dT - a_alpha
return (P*d2V_dTdP - T*dV_dP*x9*d2a_alpha_dT2
+ 16.0*dV_dT*x10*dV_dP*x7*x6*x6*x8_inv*x8_inv
+ dV_dT - x10*d2V_dTdP*x9)
@property
def d2S_dep_dTdP_g(self):
r'''Temperature and pressure derivative of departure entropy
at constant pressure then temperature for the gas phase,
[(J/mol)/K^2/Pa].
.. math::
\left(\frac{\partial^2 S_{dep, g}}{\partial T \partial P}\right)_{T, P}
= - \frac{R \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P
\right)}}{V{\left(T,P \right)}} + \frac{R \frac{\partial}{\partial
P} V{\left(T,P \right)} \frac{\partial}{\partial T} V{\left(T,P
\right)}}{V^{2}{\left(T,P \right)}} - \frac{R \frac{\partial^{2}}
{\partial T\partial P} V{\left(T,P \right)}}{b - V{\left(T,P
\right)}} - \frac{R \frac{\partial}{\partial P} V{\left(T,P
\right)} \frac{\partial}{\partial T} V{\left(T,P \right)}}{\left(b
- V{\left(T,P \right)}\right)^{2}} + \frac{16 \left(\delta
+ 2 V{\left(T,P \right)}\right) \frac{\partial}{\partial P}
V{\left(T,P \right)} \frac{\partial}{\partial T} V{\left(T,P
\right)} \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}}
{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta
+ 2 V{\left(T,P \right)}\right)^{2}}{\delta^{2} - 4 \epsilon}
- 1\right)^{2}} - \frac{4 \frac{\partial}{\partial P} V{\left(T,P
\right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T
\right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(
\delta + 2 V{\left(T,P \right)}\right)^{2}}{\delta^{2}
- 4 \epsilon} - 1\right)} - \frac{4 \frac{d}{d T}
\operatorname{a\alpha}{\left(T \right)} \frac{\partial^{2}}
{\partial T\partial P} V{\left(T,P \right)}}{\left(\delta^{2}
- 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T,P
\right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)}
- \frac{R \left(P \frac{\partial}{\partial P} V{\left(T,P \right)}
+ V{\left(T,P \right)}\right) \frac{\partial}{\partial T}
V{\left(T,P \right)}}{P V^{2}{\left(T,P \right)}} + \frac{R
\left(P \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P
\right)} - \frac{P \frac{\partial}{\partial P} V{\left(T,P
\right)}}{T} + \frac{\partial}{\partial T} V{\left(T,P \right)}
- \frac{V{\left(T,P \right)}}{T}\right)}{P V{\left(T,P \right)}}
+ \frac{R \left(P \frac{\partial}{\partial P} V{\left(T,P \right)}
+ V{\left(T,P \right)}\right)}{P T V{\left(T,P \right)}}
'''
V, T, P, b, delta, epsilon = self.V_g, self.T, self.P, self.b, self.delta, self.epsilon
dV_dT = self.dV_dT_g
d2V_dTdP = self.d2V_dTdP_g
dV_dP = self.dV_dP_g
x0 = V
V_inv = 1.0/V
x2 = d2V_dTdP
x3 = R*x2
x4 = dV_dT
x5 = x4*V_inv*V_inv
x6 = dV_dP
x7 = R*x6
x8 = b - V
x8_inv = 1.0/x8
x9 = 1.0/T
x10 = P*x6
x11 = V + x10
x12 = R/P
x13 = V_inv*x12
x14 = self.a_alpha
x15 = delta*delta - 4.0*epsilon
try:
x16 = 1.0/x15
except ZeroDivisionError:
x16 = 1e100
x17 = delta + V + V
x18 = x16*x17*x17 - 1.0
x50 = 1.0/x18
x19 = 4.0*x16*x50
x20 = self.da_alpha_dT
return (-V_inv*x3 - x11*x12*x5 + x11*x13*x9 + x13*(P*x2 - V*x9 - x10*x9
+ x4) - x19*x2*x20 - x19*x6*self.d2a_alpha_dT2 - x3*x8_inv
- x4*x7*x8_inv*x8_inv + x5*x7
+ 16.0*x17*x20*x4*x6*x16*x16*x50*x50)
@property
def d2S_dep_dTdP_l(self):
r'''Temperature and pressure derivative of departure entropy
at constant pressure then temperature for the liquid phase,
[(J/mol)/K^2/Pa].
.. math::
\left(\frac{\partial^2 S_{dep, l}}{\partial T \partial P}\right)_{T, P}
= - \frac{R \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P
\right)}}{V{\left(T,P \right)}} + \frac{R \frac{\partial}{\partial
P} V{\left(T,P \right)} \frac{\partial}{\partial T} V{\left(T,P
\right)}}{V^{2}{\left(T,P \right)}} - \frac{R \frac{\partial^{2}}
{\partial T\partial P} V{\left(T,P \right)}}{b - V{\left(T,P
\right)}} - \frac{R \frac{\partial}{\partial P} V{\left(T,P
\right)} \frac{\partial}{\partial T} V{\left(T,P \right)}}{\left(b
- V{\left(T,P \right)}\right)^{2}} + \frac{16 \left(\delta
+ 2 V{\left(T,P \right)}\right) \frac{\partial}{\partial P}
V{\left(T,P \right)} \frac{\partial}{\partial T} V{\left(T,P
\right)} \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}}
{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta
+ 2 V{\left(T,P \right)}\right)^{2}}{\delta^{2} - 4 \epsilon}
- 1\right)^{2}} - \frac{4 \frac{\partial}{\partial P} V{\left(T,P
\right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T
\right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(
\delta + 2 V{\left(T,P \right)}\right)^{2}}{\delta^{2}
- 4 \epsilon} - 1\right)} - \frac{4 \frac{d}{d T}
\operatorname{a\alpha}{\left(T \right)} \frac{\partial^{2}}
{\partial T\partial P} V{\left(T,P \right)}}{\left(\delta^{2}
- 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T,P
\right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)}
- \frac{R \left(P \frac{\partial}{\partial P} V{\left(T,P \right)}
+ V{\left(T,P \right)}\right) \frac{\partial}{\partial T}
V{\left(T,P \right)}}{P V^{2}{\left(T,P \right)}} + \frac{R
\left(P \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P
\right)} - \frac{P \frac{\partial}{\partial P} V{\left(T,P
\right)}}{T} + \frac{\partial}{\partial T} V{\left(T,P \right)}
- \frac{V{\left(T,P \right)}}{T}\right)}{P V{\left(T,P \right)}}
+ \frac{R \left(P \frac{\partial}{\partial P} V{\left(T,P \right)}
+ V{\left(T,P \right)}\right)}{P T V{\left(T,P \right)}}
'''
V, T, P, b, delta, epsilon = self.V_l, self.T, self.P, self.b, self.delta, self.epsilon
dV_dT = self.dV_dT_l
d2V_dTdP = self.d2V_dTdP_l
dV_dP = self.dV_dP_l
x0 = V
V_inv = 1.0/V
x2 = d2V_dTdP
x3 = R*x2
x4 = dV_dT
x5 = x4*V_inv*V_inv
x6 = dV_dP
x7 = R*x6
x8 = b - V
x8_inv = 1.0/x8
x9 = 1.0/T
x10 = P*x6
x11 = V + x10
x12 = R/P
x13 = V_inv*x12
x14 = self.a_alpha
x15 = delta*delta - 4.0*epsilon
try:
x16 = 1.0/x15
except ZeroDivisionError:
x16 = 1e100
x17 = delta + V + V
x18 = x16*x17*x17 - 1.0
x50 = 1.0/x18
x19 = 4.0*x16*x50
x20 = self.da_alpha_dT
return (-V_inv*x3 - x11*x12*x5 + x11*x13*x9 + x13*(P*x2 - V*x9 - x10*x9
+ x4) - x19*x2*x20 - x19*x6*self.d2a_alpha_dT2 - x3*x8_inv
- x4*x7*x8_inv*x8_inv + x5*x7
+ 16.0*x17*x20*x4*x6*x16*x16*x50*x50)
@property
def dfugacity_dT_l(self):
r'''Derivative of fugacity with respect to temperature for the liquid
phase, [Pa/K].
.. math::
\frac{\partial (\text{fugacity})_{l}}{\partial T} = P \left(\frac{1}
{R T} \left(- T \frac{\partial}{\partial T} \operatorname{S_{dep}}
{\left (T,P \right )} - \operatorname{S_{dep}}{\left (T,P \right )}
+ \frac{\partial}{\partial T} \operatorname{H_{dep}}{\left (T,P
\right )}\right) - \frac{1}{R T^{2}} \left(- T \operatorname{
S_{dep}}{\left (T,P \right )} + \operatorname{H_{dep}}{\left
(T,P \right )}\right)\right) e^{\frac{1}{R T} \left(- T
\operatorname{S_{dep}}{\left (T,P \right )} + \operatorname
{H_{dep}}{\left (T,P \right )}\right)}
'''
T, P = self.T, self.P
T_inv = 1.0/T
S_dep_l = self.S_dep_l
x4 = R_inv*(self.H_dep_l - T*S_dep_l)
return P*(T_inv*R_inv*(self.dH_dep_dT_l - T*self.dS_dep_dT_l - S_dep_l)
- x4*T_inv*T_inv)*exp(T_inv*x4)
@property
def dfugacity_dT_g(self):
r'''Derivative of fugacity with respect to temperature for the gas
phase, [Pa/K].
.. math::
\frac{\partial (\text{fugacity})_{g}}{\partial T} = P \left(\frac{1}
{R T} \left(- T \frac{\partial}{\partial T} \operatorname{S_{dep}}
{\left (T,P \right )} - \operatorname{S_{dep}}{\left (T,P \right )}
+ \frac{\partial}{\partial T} \operatorname{H_{dep}}{\left (T,P
\right )}\right) - \frac{1}{R T^{2}} \left(- T \operatorname{
S_{dep}}{\left (T,P \right )} + \operatorname{H_{dep}}{\left
(T,P \right )}\right)\right) e^{\frac{1}{R T} \left(- T
\operatorname{S_{dep}}{\left (T,P \right )} + \operatorname
{H_{dep}}{\left (T,P \right )}\right)}
'''
T, P = self.T, self.P
T_inv = 1.0/T
S_dep_g = self.S_dep_g
x4 = R_inv*(self.H_dep_g - T*S_dep_g)
return P*(T_inv*R_inv*(self.dH_dep_dT_g - T*self.dS_dep_dT_g - S_dep_g)
- x4*T_inv*T_inv)*exp(T_inv*x4)
@property
def dfugacity_dP_l(self):
r'''Derivative of fugacity with respect to pressure for the liquid
phase, [-].
.. math::
\frac{\partial (\text{fugacity})_{l}}{\partial P} = \frac{P}{R T}
\left(- T \frac{\partial}{\partial P} \operatorname{S_{dep}}{\left
(T,P \right )} + \frac{\partial}{\partial P} \operatorname{H_{dep}}
{\left (T,P \right )}\right) e^{\frac{1}{R T} \left(- T
\operatorname{S_{dep}}{\left (T,P \right )} + \operatorname{
H_{dep}}{\left (T,P \right )}\right)} + e^{\frac{1}{R T}
\left(- T \operatorname{S_{dep}}{\left (T,P \right )}
+ \operatorname{H_{dep}}{\left (T,P \right )}\right)}
'''
T, P = self.T, self.P
x0 = 1.0/(R*T)
return (1.0 - P*x0*(T*self.dS_dep_dP_l - self.dH_dep_dP_l))*exp(
-x0*(T*self.S_dep_l - self.H_dep_l))
@property
def dfugacity_dP_g(self):
r'''Derivative of fugacity with respect to pressure for the gas
phase, [-].
.. math::
\frac{\partial (\text{fugacity})_{g}}{\partial P} = \frac{P}{R T}
\left(- T \frac{\partial}{\partial P} \operatorname{S_{dep}}{\left
(T,P \right )} + \frac{\partial}{\partial P} \operatorname{H_{dep}}
{\left (T,P \right )}\right) e^{\frac{1}{R T} \left(- T
\operatorname{S_{dep}}{\left (T,P \right )} + \operatorname{
H_{dep}}{\left (T,P \right )}\right)} + e^{\frac{1}{R T}
\left(- T \operatorname{S_{dep}}{\left (T,P \right )}
+ \operatorname{H_{dep}}{\left (T,P \right )}\right)}
'''
T, P = self.T, self.P
x0 = 1.0/(R*T)
try:
ans = (1.0 - P*x0*(T*self.dS_dep_dP_g - self.dH_dep_dP_g))*exp(
-x0*(T*self.S_dep_g - self.H_dep_g))
if isinf(ans) or isnan(ans):
return 1.0
return ans
except Exception as e:
if P < 1e-50:
# Applies to gas phase only!
return 1.0
else:
raise e
@property
def dphi_dT_l(self):
r'''Derivative of fugacity coefficient with respect to temperature for
the liquid phase, [1/K].
.. math::
\frac{\partial \phi}{\partial T} = \left(\frac{- T \frac{\partial}
{\partial T} \operatorname{S_{dep}}{\left(T,P \right)}
- \operatorname{S_{dep}}{\left(T,P \right)} + \frac{\partial}
{\partial T} \operatorname{H_{dep}}{\left(T,P \right)}}{R T}
- \frac{- T \operatorname{S_{dep}}{\left(T,P \right)}
+ \operatorname{H_{dep}}{\left(T,P \right)}}{R T^{2}}\right)
e^{\frac{- T \operatorname{S_{dep}}{\left(T,P \right)}
+ \operatorname{H_{dep}}{\left(T,P \right)}}{R T}}
'''
T, P = self.T, self.P
T_inv = 1.0/T
x4 = T_inv*(T*self.S_dep_l - self.H_dep_l)
return (-R_inv*T_inv*(T*self.dS_dep_dT_l + self.S_dep_l - x4
- self.dH_dep_dT_l)*exp(-R_inv*x4))
@property
def dphi_dT_g(self):
r'''Derivative of fugacity coefficient with respect to temperature for
the gas phase, [1/K].
.. math::
\frac{\partial \phi}{\partial T} = \left(\frac{- T \frac{\partial}
{\partial T} \operatorname{S_{dep}}{\left(T,P \right)}
- \operatorname{S_{dep}}{\left(T,P \right)} + \frac{\partial}
{\partial T} \operatorname{H_{dep}}{\left(T,P \right)}}{R T}
- \frac{- T \operatorname{S_{dep}}{\left(T,P \right)}
+ \operatorname{H_{dep}}{\left(T,P \right)}}{R T^{2}}\right)
e^{\frac{- T \operatorname{S_{dep}}{\left(T,P \right)}
+ \operatorname{H_{dep}}{\left(T,P \right)}}{R T}}
'''
T, P = self.T, self.P
T_inv = 1.0/T
x4 = T_inv*(T*self.S_dep_g - self.H_dep_g)
return (-R_inv*T_inv*(T*self.dS_dep_dT_g + self.S_dep_g - x4
- self.dH_dep_dT_g)*exp(-R_inv*x4))
@property
def dphi_dP_l(self):
r'''Derivative of fugacity coefficient with respect to pressure for
the liquid phase, [1/Pa].
.. math::
\frac{\partial \phi}{\partial P} = \frac{\left(- T \frac{\partial}
{\partial P} \operatorname{S_{dep}}{\left(T,P \right)}
+ \frac{\partial}{\partial P} \operatorname{H_{dep}}{\left(T,P
\right)}\right) e^{\frac{- T \operatorname{S_{dep}}{\left(T,P
\right)} + \operatorname{H_{dep}}{\left(T,P \right)}}{R T}}}{R T}
'''
T = self.T
x0 = self.S_dep_l
x1 = self.H_dep_l
x2 = 1.0/(R*T)
return -x2*(T*self.dS_dep_dP_l - self.dH_dep_dP_l)*exp(-x2*(T*x0 - x1))
@property
def dphi_dP_g(self):
r'''Derivative of fugacity coefficient with respect to pressure for
the gas phase, [1/Pa].
.. math::
\frac{\partial \phi}{\partial P} = \frac{\left(- T \frac{\partial}
{\partial P} \operatorname{S_{dep}}{\left(T,P \right)}
+ \frac{\partial}{\partial P} \operatorname{H_{dep}}{\left(T,P
\right)}\right) e^{\frac{- T \operatorname{S_{dep}}{\left(T,P
\right)} + \operatorname{H_{dep}}{\left(T,P \right)}}{R T}}}{R T}
'''
T = self.T
x0 = self.S_dep_g
x1 = self.H_dep_g
x2 = 1.0/(R*T)
return -x2*(T*self.dS_dep_dP_g - self.dH_dep_dP_g)*exp(-x2*(T*x0 - x1))
@property
def dbeta_dT_g(self):
r'''Derivative of isobaric expansion coefficient with respect to
temperature for the gas phase, [1/K^2].
.. math::
\frac{\partial \beta_g}{\partial T} = \frac{\frac{\partial^{2}}
{\partial T^{2}} V{\left (T,P \right )_g}}{V{\left (T,P \right )_g}} -
\frac{\left(\frac{\partial}{\partial T} V{\left (T,P \right )_g}
\right)^{2}}{V^{2}{\left (T,P \right )_g}}
'''
V_inv = 1.0/self.V_g
dV_dT = self.dV_dT_g
return V_inv*(self.d2V_dT2_g - dV_dT*dV_dT*V_inv)
@property
def dbeta_dT_l(self):
r'''Derivative of isobaric expansion coefficient with respect to
temperature for the liquid phase, [1/K^2].
.. math::
\frac{\partial \beta_l}{\partial T} = \frac{\frac{\partial^{2}}
{\partial T^{2}} V{\left (T,P \right )_l}}{V{\left (T,P \right )_l}} -
\frac{\left(\frac{\partial}{\partial T} V{\left (T,P \right )_l}
\right)^{2}}{V^{2}{\left (T,P \right )_l}}
'''
V_inv = 1.0/self.V_l
dV_dT = self.dV_dT_l
return V_inv*(self.d2V_dT2_l - dV_dT*dV_dT*V_inv)
@property
def dbeta_dP_g(self):
r'''Derivative of isobaric expansion coefficient with respect to
pressure for the gas phase, [1/(Pa*K)].
.. math::
\frac{\partial \beta_g}{\partial P} = \frac{\frac{\partial^{2}}
{\partial T\partial P} V{\left (T,P \right )_g}}{V{\left (T,
P \right )_g}} - \frac{\frac{\partial}{\partial P} V{\left (T,P
\right )_g} \frac{\partial}{\partial T} V{\left (T,P \right )_g}}
{V^{2}{\left (T,P \right )_g}}
'''
V_inv = 1.0/self.V_g
dV_dT = self.dV_dT_g
dV_dP = self.dV_dP_g
return V_inv*(self.d2V_dTdP_g - dV_dT*dV_dP*V_inv)
@property
def dbeta_dP_l(self):
r'''Derivative of isobaric expansion coefficient with respect to
pressure for the liquid phase, [1/(Pa*K)].
.. math::
\frac{\partial \beta_g}{\partial P} = \frac{\frac{\partial^{2}}
{\partial T\partial P} V{\left (T,P \right )_l}}{V{\left (T,
P \right )_l}} - \frac{\frac{\partial}{\partial P} V{\left (T,P
\right )_l} \frac{\partial}{\partial T} V{\left (T,P \right )_l}}
{V^{2}{\left (T,P \right )_l}}
'''
V_inv = 1.0/self.V_l
dV_dT = self.dV_dT_l
dV_dP = self.dV_dP_l
return V_inv*(self.d2V_dTdP_l - dV_dT*dV_dP*V_inv)
@property
def da_alpha_dP_g_V(self):
r'''Derivative of the `a_alpha` with respect to
pressure at constant volume (varying T) for the gas phase,
[J^2/mol^2/Pa^2].
.. math::
\left(\frac{\partial a \alpha}{\partial P}\right)_{V}
= \left(\frac{\partial a \alpha}{\partial T}\right)_{P}
\cdot\left( \frac{\partial T}{\partial P}\right)_V
'''
return self.da_alpha_dT*self.dT_dP_g
@property
def da_alpha_dP_l_V(self):
r'''Derivative of the `a_alpha` with respect to
pressure at constant volume (varying T) for the liquid phase,
[J^2/mol^2/Pa^2].
.. math::
\left(\frac{\partial a \alpha}{\partial P}\right)_{V}
= \left(\frac{\partial a \alpha}{\partial T}\right)_{P}
\cdot\left( \frac{\partial T}{\partial P}\right)_V
'''
return self.da_alpha_dT*self.dT_dP_l
@property
def d2a_alpha_dTdP_g_V(self):
r'''Derivative of the temperature derivative of `a_alpha` with respect
to pressure at constant volume (varying T) for the gas phase,
[J^2/mol^2/Pa^2/K].
.. math::
\left(\frac{\partial \left(\frac{\partial a \alpha}{\partial T}
\right)_P}{\partial P}\right)_{V}
= \left(\frac{\partial^2 a \alpha}{\partial T^2}\right)_{P}
\cdot\left( \frac{\partial T}{\partial P}\right)_V
'''
return self.d2a_alpha_dT2*self.dT_dP_g
@property
def d2a_alpha_dTdP_l_V(self):
r'''Derivative of the temperature derivative of `a_alpha` with respect
to pressure at constant volume (varying T) for the liquid phase,
[J^2/mol^2/Pa^2/K].
.. math::
\left(\frac{\partial \left(\frac{\partial a \alpha}{\partial T}
\right)_P}{\partial P}\right)_{V}
= \left(\frac{\partial^2 a \alpha}{\partial T^2}\right)_{P}
\cdot\left( \frac{\partial T}{\partial P}\right)_V
'''
return self.d2a_alpha_dT2*self.dT_dP_l
@property
def d2P_dVdP_g(self):
r'''Second derivative of pressure with respect to molar volume and
then pressure for the gas phase, [mol/m^3].
.. math::
\frac{\partial^2 P}{\partial V \partial P} =
\frac{2 R T \frac{d}{d P} V{\left(P \right)}}{\left(- b + V{\left(P
\right)}\right)^{3}} - \frac{\left(- \delta - 2 V{\left(P \right)}
\right) \left(- 2 \delta \frac{d}{d P} V{\left(P \right)}
- 4 V{\left(P \right)} \frac{d}{d P} V{\left(P \right)}\right)
\operatorname{a\alpha}{\left(T \right)}}{\left(\delta V{\left(P
\right)} + \epsilon + V^{2}{\left(P \right)}\right)^{3}} + \frac{2
\operatorname{a\alpha}{\left(T \right)} \frac{d}{d P} V{\left(P
\right)}}{\left(\delta V{\left(P \right)} + \epsilon + V^{2}
{\left(P \right)}\right)^{2}}
'''
r"""Feels like a really strange derivative. Have not been able to construct
it from others yet. Value is Symmetric - can calculate it both ways.
Still feels like there should be a general method for obtaining these derivatives.
from sympy import *
P, T, R, b, delta, epsilon = symbols('P, T, R, b, delta, epsilon')
a_alpha, V = symbols(r'a\alpha, V', cls=Function)
dP_dV = 1/(1/(-R*T/(V(P) - b)**2 - a_alpha(T)*(-2*V(P) - delta)/(V(P)**2 + V(P)*delta + epsilon)**2))
cse(diff(dP_dV, P), optimizations='basic')
"""
T, P, b, delta, epsilon = self.T, self.P, self.b, self.delta, self.epsilon
x0 = self.V_g
x1 = self.a_alpha
x2 = delta*x0 + epsilon + x0*x0
x50 = self.dV_dP_g
x51 = x0 + x0 + delta
x52 = 1.0/(b - x0)
x2_inv = 1.0/x2
return 2.0*(-R*T*x52*x52*x52 + x1*x2_inv*x2_inv*(1.0 - x51*x51*x2_inv))*x50
@property
def d2P_dVdP_l(self):
r'''Second derivative of pressure with respect to molar volume and
then pressure for the liquid phase, [mol/m^3].
.. math::
\frac{\partial^2 P}{\partial V \partial P} =
\frac{2 R T \frac{d}{d P} V{\left(P \right)}}{\left(- b + V{\left(P
\right)}\right)^{3}} - \frac{\left(- \delta - 2 V{\left(P \right)}
\right) \left(- 2 \delta \frac{d}{d P} V{\left(P \right)}
- 4 V{\left(P \right)} \frac{d}{d P} V{\left(P \right)}\right)
\operatorname{a\alpha}{\left(T \right)}}{\left(\delta V{\left(P
\right)} + \epsilon + V^{2}{\left(P \right)}\right)^{3}} + \frac{2
\operatorname{a\alpha}{\left(T \right)} \frac{d}{d P} V{\left(P
\right)}}{\left(\delta V{\left(P \right)} + \epsilon + V^{2}
{\left(P \right)}\right)^{2}}
'''
T, b, delta, epsilon = self.T, self.b, self.delta, self.epsilon
x0 = self.V_l
x1 = self.a_alpha
x2 = delta*x0 + epsilon + x0*x0
x50 = self.dV_dP_l
x51 = x0 + x0 + delta
x52 = 1.0/(b - x0)
x2_inv = 1.0/x2
return 2.0*(-R*T*x52*x52*x52 + x1*x2_inv*x2_inv*(1.0 - x51*x51*x2_inv))*x50
@property
def d2P_dVdT_TP_g(self):
r'''Second derivative of pressure with respect to molar volume and
then temperature at constant temperature then pressure for the gas
phase, [Pa*mol/m^3/K].
.. math::
\left(\frac{\partial^2 P}{\partial V \partial T}\right)_{T,P} =
\frac{2 R T \frac{d}{d T} V{\left(T \right)}}{\left(- b + V{\left(T
\right)}\right)^{3}} - \frac{R}{\left(- b + V{\left(T \right)}
\right)^{2}} - \frac{\left(- \delta - 2 V{\left(T \right)}\right)
\left(- 2 \delta \frac{d}{d T} V{\left(T \right)} - 4 V{\left(T
\right)} \frac{d}{d T} V{\left(T \right)}\right) \operatorname{
a\alpha}{\left(T \right)}}{\left(\delta V{\left(T \right)}
+ \epsilon + V^{2}{\left(T \right)}\right)^{3}} - \frac{\left(
- \delta - 2 V{\left(T \right)}\right) \frac{d}{d T} \operatorname{
a\alpha}{\left(T \right)}}{\left(\delta V{\left(T \right)}
+ \epsilon + V^{2}{\left(T \right)}\right)^{2}} + \frac{2
\operatorname{a\alpha}{\left(T \right)} \frac{d}{d T} V{\left(T
\right)}}{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(
T \right)}\right)^{2}}
'''
T, b, delta, epsilon = self.T, self.b, self.delta, self.epsilon
x0 = self.V_g
x2 = 2.0*self.dV_dT_g
x1 = self.b - x0
x1_inv = 1.0/x1
x3 = delta*x0 + epsilon + x0*x0
x3_inv = 1.0/x3
x4 = x3_inv*x3_inv
x5 = self.a_alpha
x6 = x2*x5
x7 = delta + x0 + x0
return (-x1_inv*x1_inv*R*(T*x2*x1_inv + 1.0) + x4*x6
+ x4*x7*(self.da_alpha_dT - x6*x7*x3_inv))
@property
def d2P_dVdT_TP_l(self):
r'''Second derivative of pressure with respect to molar volume and
then temperature at constant temperature then pressure for the liquid
phase, [Pa*mol/m^3/K].
.. math::
\left(\frac{\partial^2 P}{\partial V \partial T}\right)_{T,P} =
\frac{2 R T \frac{d}{d T} V{\left(T \right)}}{\left(- b + V{\left(T
\right)}\right)^{3}} - \frac{R}{\left(- b + V{\left(T \right)}
\right)^{2}} - \frac{\left(- \delta - 2 V{\left(T \right)}\right)
\left(- 2 \delta \frac{d}{d T} V{\left(T \right)} - 4 V{\left(T
\right)} \frac{d}{d T} V{\left(T \right)}\right) \operatorname{
a\alpha}{\left(T \right)}}{\left(\delta V{\left(T \right)}
+ \epsilon + V^{2}{\left(T \right)}\right)^{3}} - \frac{\left(
- \delta - 2 V{\left(T \right)}\right) \frac{d}{d T} \operatorname{
a\alpha}{\left(T \right)}}{\left(\delta V{\left(T \right)}
+ \epsilon + V^{2}{\left(T \right)}\right)^{2}} + \frac{2
\operatorname{a\alpha}{\left(T \right)} \frac{d}{d T} V{\left(T
\right)}}{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(
T \right)}\right)^{2}}
'''
T, b, delta, epsilon = self.T, self.b, self.delta, self.epsilon
x0 = self.V_l
x2 = 2.0*self.dV_dT_l
x1 = self.b - x0
x1_inv = 1.0/x1
x3 = delta*x0 + epsilon + x0*x0
x3_inv = 1.0/x3
x4 = x3_inv*x3_inv
x5 = self.a_alpha
x6 = x2*x5
x7 = delta + x0 + x0
return (-x1_inv*x1_inv*R*(T*x2*x1_inv + 1.0) + x4*x6
+ x4*x7*(self.da_alpha_dT - x6*x7*x3_inv))
@property
def d2P_dT2_PV_g(self):
r'''Second derivative of pressure with respect to temperature twice,
but with pressure held constant the first time and volume held
constant the second time for the gas phase, [Pa/K^2].
.. math::
\left(\frac{\partial^2 P}{\partial T \partial T}\right)_{P,V} =
- \frac{R \frac{d}{d T} V{\left(T \right)}}{\left(- b + V{\left(T
\right)}\right)^{2}} - \frac{\left(- \delta \frac{d}{d T} V{\left(T
\right)} - 2 V{\left(T \right)} \frac{d}{d T} V{\left(T \right)}
\right) \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}}
{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T
\right)}\right)^{2}} - \frac{\frac{d^{2}}{d T^{2}}
\operatorname{a\alpha}{\left(T \right)}}{\delta V{\left(T \right)}
+ \epsilon + V^{2}{\left(T \right)}}
'''
T, b, delta, epsilon = self.T, self.b, self.delta, self.epsilon
V = self.V_g
dV_dT = self.dV_dT_g
x2 = self.a_alpha
x0 = V
x1 = dV_dT
x3 = delta*x0 + epsilon + x0*x0
x3_inv = 1.0/x3
x50 = 1.0/(b - x0)
return (-R*x1*x50*x50 + x1*(delta + x0 + x0)*self.da_alpha_dT*x3_inv*x3_inv - self.d2a_alpha_dT2*x3_inv)
@property
def d2P_dT2_PV_l(self):
r'''Second derivative of pressure with respect to temperature twice,
but with pressure held constant the first time and volume held
constant the second time for the liquid phase, [Pa/K^2].
.. math::
\left(\frac{\partial^2 P}{\partial T \partial T}\right)_{P,V} =
- \frac{R \frac{d}{d T} V{\left(T \right)}}{\left(- b + V{\left(T
\right)}\right)^{2}} - \frac{\left(- \delta \frac{d}{d T} V{\left(T
\right)} - 2 V{\left(T \right)} \frac{d}{d T} V{\left(T \right)}
\right) \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}}
{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T
\right)}\right)^{2}} - \frac{\frac{d^{2}}{d T^{2}}
\operatorname{a\alpha}{\left(T \right)}}{\delta V{\left(T \right)}
+ \epsilon + V^{2}{\left(T \right)}}
'''
T, b, delta, epsilon = self.T, self.b, self.delta, self.epsilon
V = self.V_l
dV_dT = self.dV_dT_l
x0 = V
x1 = dV_dT
x2 = self.a_alpha
x3 = delta*x0 + epsilon + x0*x0
x3_inv = 1.0/x3
x50 = 1.0/(b - x0)
return (-R*x1*x50*x50 + x1*(delta + x0 + x0)*self.da_alpha_dT*x3_inv*x3_inv - self.d2a_alpha_dT2*x3_inv)
@property
def d2P_dTdP_g(self):
r'''Second derivative of pressure with respect to temperature and,
then pressure; and with volume held constant at first, then temperature,
for the gas phase, [1/K].
.. math::
\left(\frac{\partial^2 P}{\partial T \partial P}\right)_{V, T} =
- \frac{R \frac{d}{d P} V{\left(P \right)}}{\left(- b + V{\left(P
\right)}\right)^{2}} - \frac{\left(- \delta \frac{d}{d P} V{\left(P
\right)} - 2 V{\left(P \right)} \frac{d}{d P} V{\left(P \right)}
\right) \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}}
{\left(\delta V{\left(P \right)} + \epsilon + V^{2}{\left(P
\right)}\right)^{2}}
'''
V = self.V_g
dV_dP = self.dV_dP_g
T, b, delta, epsilon = self.T, self.b, self.delta, self.epsilon
da_alpha_dT = self.da_alpha_dT
x0 = V - b
x1 = delta*V + epsilon + V*V
return (-R*dV_dP/(x0*x0) - (-delta*dV_dP - 2.0*V*dV_dP)*da_alpha_dT/(x1*x1))
@property
def d2P_dTdP_l(self):
r'''Second derivative of pressure with respect to temperature and,
then pressure; and with volume held constant at first, then temperature,
for the liquid phase, [1/K].
.. math::
\left(\frac{\partial^2 P}{\partial T \partial P}\right)_{V, T} =
- \frac{R \frac{d}{d P} V{\left(P \right)}}{\left(- b + V{\left(P
\right)}\right)^{2}} - \frac{\left(- \delta \frac{d}{d P} V{\left(P
\right)} - 2 V{\left(P \right)} \frac{d}{d P} V{\left(P \right)}
\right) \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}}
{\left(\delta V{\left(P \right)} + \epsilon + V^{2}{\left(P
\right)}\right)^{2}}
'''
V = self.V_l
dV_dP = self.dV_dP_l
T, b, delta, epsilon = self.T, self.b, self.delta, self.epsilon
da_alpha_dT = self.da_alpha_dT
x0 = V - b
x1 = delta*V + epsilon + V*V
return (-R*dV_dP/(x0*x0) - (-delta*dV_dP - 2.0*V*dV_dP)*da_alpha_dT/(x1*x1))
@property
def lnphi_l(self):
r'''The natural logarithm of the fugacity coefficient for
the liquid phase, [-].
'''
return self.G_dep_l*R_inv/self.T
@property
def lnphi_g(self):
r'''The natural logarithm of the fugacity coefficient for
the gas phase, [-].
'''
return log(self.phi_g)
[docs]class IG(GCEOS):
r'''Class for solving the ideal gas equation in the `GCEOS` framework.
This provides access to a number of derivatives and properties easily.
It also keeps a common interface for all gas models. However, it is
somewhat slow.
Subclasses :obj:`GCEOS`, which
provides the methods for solving the EOS and calculating its assorted
relevant thermodynamic properties. Solves the EOS on initialization.
Two of `T`, `P`, and `V` are needed to solve the EOS; values for `Tc` and
`Pc` and `omega`, which are not used in the calculates, are set to those of
methane by default to allow use without specifying them.
.. math::
P = \frac{RT}{V}
Parameters
----------
Tc : float, optional
Critical temperature, [K]
Pc : float, optional
Critical pressure, [Pa]
omega : float, optional
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
T-P initialization, and exploring each phase's properties:
>>> eos = IG(T=400., P=1E6)
>>> eos.V_g, eos.phase
(0.003325785047261296, 'g')
>>> eos.H_dep_g, eos.S_dep_g, eos.U_dep_g, eos.G_dep_g, eos.A_dep_g
(0.0, 0.0, 0.0, 0.0, 0.0)
>>> eos.beta_g, eos.kappa_g, eos.Cp_dep_g, eos.Cv_dep_g
(0.0025, 1e-06, 0.0, 0.0)
>>> eos.fugacity_g, eos.PIP_g, eos.Z_g, eos.dP_dT_g
(1000000.0, 0.9999999999999999, 1.0, 2500.0)
Notes
-----
References
----------
.. [1] Smith, J. M, H. C Van Ness, and Michael M Abbott. Introduction to
Chemical Engineering Thermodynamics. Boston: McGraw-Hill, 2005.
'''
Zc = 1.0
"""float: Critical compressibility for an ideal gas is 1"""
a = 0.0
"""float: `a` parameter for an ideal gas is 0"""
b = 0.0
"""float: `b` parameter for an ideal gas is 0"""
delta = 0.0
"""float: `delta` parameter for an ideal gas is 0"""
epsilon = 0.0
"""float: `epsilon` parameter for an ideal gas is 0"""
volume_solutions = staticmethod(volume_solutions_ideal)
# Handle the properties where numerical error puts values - but they should
# be zero. Not all of them are non-zero all the time - but some times
# they are
def _zero(self): return 0.0
def _set_nothing(self, thing): return
try:
try:
doc = GCEOS.d2T_dV2_g.__doc__
except:
doc = ''
d2T_dV2_g = property(_zero, fset=_set_nothing, doc=doc)
try:
doc = GCEOS.d2V_dT2_g.__doc__
except:
doc = ''
d2V_dT2_g = property(_zero, fset=_set_nothing, doc=doc)
try:
doc = GCEOS.U_dep_g.__doc__
except:
doc = ''
U_dep_g = property(_zero, fset=_set_nothing, doc=doc)
try:
doc = GCEOS.A_dep_g.__doc__
except:
doc = ''
A_dep_g = property(_zero, fset=_set_nothing, doc=doc)
try:
doc = GCEOS.V_dep_g.__doc__
except:
doc = ''
V_dep_g = property(_zero, fset=_set_nothing, doc=doc)
G_dep_g = property(_zero, fset=_set_nothing, doc='Departure Gibbs free energy of an ideal gas is zero, [J/(mol)]')
H_dep_g = property(_zero, fset=_set_nothing, doc='Departure enthalpy of an ideal gas is zero, [J/(mol)]')
S_dep_g = property(_zero, fset=_set_nothing, doc='Departure entropy of an ideal gas is zero, [J/(mol*K)]')
Cp_dep_g = property(_zero, fset=_set_nothing, doc='Departure heat capacity of an ideal gas is zero, [J/(mol*K)]')
# Replace methods
dH_dep_dP_g = property(_zero, doc=GCEOS.dH_dep_dP_g.__doc__)
dH_dep_dT_g = property(_zero, doc=GCEOS.dH_dep_dT_g.__doc__)
dS_dep_dP_g = property(_zero, doc=GCEOS.dS_dep_dP_g.__doc__)
dS_dep_dT_g = property(_zero, doc=GCEOS.dS_dep_dT_g.__doc__)
dfugacity_dT_g = property(_zero, doc=GCEOS.dfugacity_dT_g.__doc__)
dphi_dP_g = property(_zero, doc=GCEOS.dphi_dP_g.__doc__)
dphi_dT_g = property(_zero, doc=GCEOS.dphi_dT_g.__doc__)
except:
pass
def __init__(self, Tc=None, Pc=None, omega=None, T=None, P=None,
V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
self.solve()
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives for this EOS. All values are zero.
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
'''
return (0.0, 0.0, 0.0)
[docs] def a_alpha_pure(self, T):
r'''Method to calculate :math:`a \alpha` for the ideal gas law, which
is zero.
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
'''
return 0.0
[docs] def solve_T(self, P, V, solution=None):
r'''Method to calculate `T` from a specified `P` and `V` for the
ideal gas equation of state.
.. math::
T = \frac{PV}{R}
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
Not used, [-]
Returns
-------
T : float
Temperature, [K]
Notes
-----
'''
self.no_T_spec = True
return P*V*R_inv
def phi_sat(self, T, polish=True):
return 1.0
def dphi_sat_dT(self, T, polish=True):
return 0.0
def d2phi_sat_dT2(self, T, polish=True):
return 0.0
[docs]class PR(GCEOS):
r'''Class for solving the Peng-Robinson [1]_ [2]_ cubic
equation of state for a pure compound. Subclasses :obj:`GCEOS`, which
provides the methods for solving the EOS and calculating its assorted
relevant thermodynamic properties. Solves the EOS on initialization.
The main methods here are :obj:`PR.a_alpha_and_derivatives_pure`, which calculates
:math:`a \alpha` and its first and second derivatives, and :obj:`PR.solve_T`, which from a
specified `P` and `V` obtains `T`.
Two of (`T`, `P`, `V`) are needed to solve the EOS.
.. math::
P = \frac{RT}{v-b}-\frac{a\alpha(T)}{v(v+b)+b(v-b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha(T)=[1+\kappa(1-\sqrt{T_r})]^2
.. math::
\kappa=0.37464+1.54226\omega-0.26992\omega^2
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
T-P initialization, and exploring each phase's properties:
>>> eos = PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=400., P=1E6)
>>> eos.V_l, eos.V_g
(0.000156073184785, 0.0021418768167)
>>> eos.phase
'l/g'
>>> eos.H_dep_l, eos.H_dep_g
(-26111.8775716, -3549.30057795)
>>> eos.S_dep_l, eos.S_dep_g
(-58.098447843, -6.4394518931)
>>> eos.U_dep_l, eos.U_dep_g
(-22942.1657091, -2365.3923474)
>>> eos.G_dep_l, eos.G_dep_g
(-2872.49843435, -973.51982071)
>>> eos.A_dep_l, eos.A_dep_g
(297.21342811, 210.38840980)
>>> eos.beta_l, eos.beta_g
(0.00269337091778, 0.0101232239111)
>>> eos.kappa_l, eos.kappa_g
(9.3357215438e-09, 1.97106698097e-06)
>>> eos.Cp_minus_Cv_l, eos.Cp_minus_Cv_g
(48.510162249, 44.544161128)
>>> eos.Cv_dep_l, eos.Cp_dep_l
(18.8921126734, 59.0878123050)
P-T initialization, liquid phase, and round robin trip:
>>> eos = PR(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.000130222125139, -31134.75084, -72.47561931)
T-V initialization, liquid phase:
>>> eos2 = PR(Tc=507.6, Pc=3025000, omega=0.2975, T=299., V=eos.V_l)
>>> eos2.P, eos2.phase
(1000000.00, 'l')
P-V initialization at same state:
>>> eos3 = PR(Tc=507.6, Pc=3025000, omega=0.2975, V=eos.V_l, P=1E6)
>>> eos3.T, eos3.phase
(299.0000000000, 'l')
Notes
-----
The constants in the expresions for `a` and `b` are given to full precision
in the actual code, as derived in [3]_.
The full expression for critical compressibility is:
.. math::
Z_c = \frac{1}{32} \left(\sqrt[3]{16 \sqrt{2}-13}-\frac{7}{\sqrt[3]
{16 \sqrt{2}-13}}+11\right)
References
----------
.. [1] Peng, Ding-Yu, and Donald B. Robinson. "A New Two-Constant Equation
of State." Industrial & Engineering Chemistry Fundamentals 15, no. 1
(February 1, 1976): 59-64. doi:10.1021/i160057a011.
.. [2] Robinson, Donald B., Ding-Yu Peng, and Samuel Y-K Chung. "The
Development of the Peng - Robinson Equation and Its Application to Phase
Equilibrium in a System Containing Methanol." Fluid Phase Equilibria 24,
no. 1 (January 1, 1985): 25-41. doi:10.1016/0378-3812(85)87035-7.
.. [3] Privat, R., and J.-N. Jaubert. "PPR78, a Thermodynamic Model for the
Prediction of Petroleum Fluid-Phase Behaviour," 11. EDP Sciences, 2011.
doi:10.1051/jeep/201100011.
'''
# constant part of `a`,
# X = (-1 + (6*sqrt(2)+8)**Rational(1,3) - (6*sqrt(2)-8)**Rational(1,3))/3
# (8*(5*X+1)/(49-37*X)).evalf(40)
c1 = 0.4572355289213821893834601962251837888504
"""Full value of the constant in the `a` parameter"""
c1R2 = c1*R2
# Constant part of `b`, (X/(X+3)).evalf(40)
c2 = 0.0777960739038884559718447100373331839711
"""Full value of the constant in the `b` parameter"""
c2R = c2*R
c1R2_c2R = c1R2/c2R
# c1, c2 = 0.45724, 0.07780
# Zc is the mechanical compressibility for mixtures as well.
Zc = 0.3074013086987038480093850966542222720096
"""Mechanical compressibility of Peng-Robinson EOS"""
Psat_coeffs_limiting = [-3.4758880164801873, 0.7675486448347723]
Psat_coeffs_critical = [13.906174756604267, -8.978515559640332,
6.191494729386664, -3.3553014047359286,
1.0000000000011509]
Psat_cheb_coeffs = [-7.693430141477579, -7.792157693145173, -0.12584439451814622, 0.0045868660863990305,
0.011902728116315585, -0.00809984848593371, 0.0035807374586641324, -0.001285457896498948,
0.0004379441379448949, -0.0001701325511665626, 7.889450459420399e-05, -3.842330780886875e-05,
1.7884847876342805e-05, -7.9432179091441e-06, 3.51726370898656e-06, -1.6108797741557683e-06,
7.625638345550717e-07, -3.6453554523813245e-07, 1.732454904858089e-07, -8.195124459058523e-08,
3.8929380082904216e-08, -1.8668536344161905e-08, 9.021955971552252e-09, -4.374277331168795e-09,
2.122697092724708e-09, -1.0315557015083254e-09, 5.027805333255708e-10, -2.4590905784642285e-10,
1.206301486380689e-10, -5.932583414867791e-11, 2.9274476912683964e-11, -1.4591650777202522e-11,
7.533835507484918e-12, -4.377200831613345e-12, 1.7413208326438542e-12]
# below - down to .14 Tr
# Psat_cheb_coeffs = [-69.78144560030312, -70.82020621910401, -0.5505993362058134, 0.262763240774557, -0.13586962327984622, 0.07091484524874882, -0.03531507189835045, 0.015348266653126313, -0.004290800414097142, -0.0015192254949775404, 0.004230003950690049, -0.005148646330256051, 0.005067979846360524, -0.004463618393006094, 0.0036338412594165456, -0.002781745442601943, 0.0020410583004693912, -0.0014675469823800154, 0.001041797382518202, -0.0007085008245359792, 0.0004341450533632967, -0.00023059133991796472, 0.00012404966848973944, -0.00010575986390189084, 0.00011927874294723816, -0.00010216011382070127, 4.142986825089964e-05, 1.6994654942134455e-05, -2.0393896226146606e-05, -3.05495184394464e-05, 7.840494892004187e-05, -6.715144915784917e-05, 1.9360256298218764e-06, 5.342823303794287e-05, -4.2445268102696054e-05, -2.258059184830652e-05, 7.156133295478447e-05, -5.0419963297068014e-05, -2.1185333936025785e-05, 6.945722167248469e-05, -4.3468774802286496e-05, -3.0211658906858938e-05, 7.396450066832002e-05, -4.0987041756199036e-05, -3.4507186813052766e-05, 3.6619358939125855e-05]
# down to .05 Tr
# Psat_cheb_coeffs = [-71.62442148475718, -72.67946752713178, -0.5550432977559888, 0.2662527679044299, -0.13858385912471755, 0.07300013042829502, -0.03688566755461173, 0.01648745160444604, -0.005061858504315144, -0.0010519595693067093, 0.0039868988560367085, -0.005045456840770146, 0.00504419254495023, -0.0044982000664379905, 0.003727506855649437, -0.002922838794275898, 0.0021888012528213734, -0.0015735578492615076, 0.0010897606359061226, -0.0007293553555925913, 0.0004738606767778966, -0.00030120118607927907, 0.00018992197213856394, -0.00012147385378832608, 8.113736696036817e-05, -5.806550163389163e-05, 4.4822397778703055e-05, -3.669084579413651e-05, 3.0945466319478186e-05, -2.62003968013127e-05, 2.1885122184587654e-05, -1.786717828032663e-05, 1.420082721312861e-05, -1.0981475209780111e-05, 8.276527284992199e-06, -6.100440122314813e-06, 4.420342273408809e-06, -3.171239452318529e-06, 2.2718591475182304e-06, -1.641149583754854e-06, 1.2061284404980935e-06, -9.067266070702959e-07, 6.985214276328142e-07, -5.490755862981909e-07, 4.372991567070929e-07, -3.504743494298746e-07, 2.8019662848682576e-07, -2.2266768846404626e-07, 1.7533403880408145e-07, -1.3630227589226426e-07, 1.0510214144142285e-07, -8.02098792008235e-08, 6.073935683412093e-08, -4.6105511380996746e-08, 3.478599121821662e-08, -2.648029023793574e-08, 2.041302301328165e-08, -1.5671212844805128e-08, 1.2440282394539782e-08, -9.871977759603047e-09, 7.912503992331811e-09, -6.6888910721434e-09, 5.534654087073205e-09, -4.92019981055108e-09, 4.589363968756223e-09, -2.151778718334702e-09]
# down to .05 Tr polishing
# Psat_cheb_coeffs = [-73.9119088855554, -74.98674794418481, -0.5603678572345178, 0.2704608002227193, -0.1418754021264281, 0.07553218818095526, -0.03878657980070652, 0.017866520164384912, -0.0060152224341743525, -0.0004382750653244775, 0.003635841462596336, -0.004888955750612924, 0.005023631814771542, -0.004564880757514128, 0.003842769402817585, -0.0030577040987875793, 0.0023231191552369407, -0.001694755295849508, 0.0011913577693282759, -0.0008093955530850967, 0.0005334402485338361, -0.0003431831424850387, 0.00021792836239828482, -0.00013916167527852, 9.174638441139245e-05, -6.419699908390207e-05, 4.838277855408256e-05, -3.895686370452493e-05, 3.267491660000825e-05, -2.7780478658642705e-05, 2.3455257030895833e-05, -1.943068869205973e-05, 1.5702249378726904e-05, -1.2352834841441616e-05, 9.468188716352547e-06, -7.086815965689662e-06, 5.202794456673999e-06, -3.7660662091643354e-06, 2.710802447723022e-06, -1.9547001517481854e-06, 1.4269579917305496e-06, -1.0627333211922062e-06, 8.086972219940435e-07, -6.313736088052035e-07, 5.002098614800398e-07, -4.014517222719182e-07, 3.222357369727768e-07, -2.591706410738203e-07, 2.0546606649125658e-07, -1.6215902481453263e-07, 1.2645321295092458e-07, -9.678506993483597e-08, 7.52490799383037e-08, -5.60685972986457e-08, 4.3358661542007224e-08, -3.2329350971261814e-08, 2.5091238603112617e-08, -1.8903964302567286e-08, 1.4892047699817043e-08, -1.1705624527623068e-08, 8.603302527636011e-09, -7.628847828412486e-09, 5.0543164590698825e-09, -5.102159698856454e-09, 3.0709992836479988e-09, -2.972533529000884e-09, 2.0494601230946347e-09, -1.626141536313283e-09, 1.6617716853181003e-09, -6.470653307871083e-10, 1.1333690091031717e-09, -1.2451614782651999e-10, 1.098942683163892e-09, 9.673645066411718e-11, 6.206934530152836e-10, -1.1913910201270805e-10, 3.559906774745769e-11, -5.419942764994107e-10, -2.372580701782284e-10, -5.785415972247437e-10, -1.789757696430208e-10]
# down to .05 with lots of failures C40 only
# Psat_cheb_coeffs = [-186.30264784196294, -188.01235085131194, -0.6975588305160902, 0.38422679790906106, -0.2358303051434559, 0.15258449381119304, -0.101338177792044, 0.0679573457611134, -0.045425247476661136, 0.029879338234709937, -0.019024330378443737, 0.011418999154577504, -0.006113230472632388, 0.00246054797767154, -4.3960533109688155e-06, -0.0015825897164979809, 0.002540504992834563, -0.003046881596822211, 0.0032353807402903272, -0.0032061955400497044, 0.0030337264005811464, -0.0027744314554593126, 0.002469806934918433, -0.002149376765619085, 0.001833408492489406, -0.00153552022142691, 0.0012645817528752557, -0.0010249792000921317, 0.0008181632585418055, -0.0006436998283177283, 0.0004995903113614604, -0.0003828408287994695, 0.0002896812774307662, -0.00021674416012176133, 0.00016131784370737042, -0.00012009195488808489, 8.966908457382076e-05, -6.764450681363164e-05, 5.209192773849304e-05, -4.1139971086693995e-05, 3.3476318185800505e-05, -2.8412997762476805e-05, 2.513421113263226e-05, -2.2567508719078435e-05, 2.0188809493379843e-05, -1.810962700274516e-05, 1.643508229137845e-05, -1.503569055933669e-05, 1.3622272823701577e-05, -1.2076671646564277e-05, 1.054271875585668e-05, -9.007273271254411e-06, 7.523720857264602e-06, -6.424404525130439e-06, 5.652203861001342e-06, -4.7755499168431625e-06, 3.7604252783225858e-06, -2.92395389072605e-06, 2.3520802660480336e-06, -1.9209673206999083e-06, 1.6125790706312328e-06, -1.4083468032508143e-06, 1.1777450938630518e-06, -8.636616122606049e-07, 5.749905340593687e-07, -4.644992178826096e-07, 5.109912172256424e-07, -5.285927442208997e-07, 4.4610491153173465e-07, -3.3435155715273366e-07, 2.2022096388817243e-07, -1.3138808837994352e-07, 1.5788807254228123e-07, -2.6570415873228444e-07, 2.820563887584985e-07, -1.6783703722562406e-07, 4.477559158897425e-08, -2.4698813388799755e-09, 5.082691394016857e-08, -1.364026020206371e-07, 1.6850593650100272e-07, -1.0443374638586546e-07, -6.029473813268628e-10, 5.105380858617091e-08, -1.5066843023282578e-08, -5.630921379297198e-08, 9.561766786891034e-08, -8.044216329068123e-08, 3.359993333902796e-08, 1.692366968619578e-08, -2.021364343358841e-08]
# down to .03, plenty of failures
# Psat_cheb_coeffs = [-188.50329975567104, -190.22994960376462, -0.6992886012204886, 0.3856961269737735, -0.23707446208582353, 0.15363415372584763, -0.10221883018831106, 0.06869084576669, -0.046030774233320346, 0.03037297246598552, -0.019421744608583133, 0.011732910491046633, -0.006355800820106353, 0.0026413894471214202, -0.0001333621829559692, -0.0014967435287118152, 0.002489721202961943, -0.00302447283347462, 0.0032350727289014642, -0.0032223921492743357, 0.0030622558268892, -0.0028113049747675455, 0.002511348612059362, -0.002192644454555338, 0.0018764599744331163, -0.0015770771123065552, 0.0013034116032509804, -0.0010603100672178776, 0.00084960767850329, -0.0006709816561447436, 0.0005226330473731801, -0.0004018349441941878, 0.0003053468509191052, -0.00022974201509485604, 0.00017163053097478257, -0.0001278303586505278, 9.545950876002835e-05, -7.200007894259846e-05, 5.5312909934416405e-05, -4.3632781581719854e-05, 3.554641644507928e-05, -2.99488097950353e-05, 2.6011962388807256e-05, -2.3127603908643427e-05, 2.0875472981740965e-05, -1.8975408339047864e-05, 1.7255291079923385e-05, -1.562250114123633e-05, 1.4033483268247027e-05, -1.2483202707948607e-05, 1.0981181475278024e-05, -9.547990214685254e-06, 8.20534723265339e-06, -6.970215811404035e-06, 5.857096216944197e-06, -4.8714713996210945e-06, 4.015088107327757e-06, -3.2837642912761844e-06, 2.6688332761922373e-06, -2.1605704853781956e-06, 1.745415965345872e-06, -1.4112782858614675e-06, 1.1450344603347899e-06, -9.34468189749192e-07, 7.693687927218034e-07, -6.395653830685742e-07, 5.378418354520407e-07, -4.570688107726579e-07, 3.922470141699613e-07, -3.396066879296283e-07, 2.9547505651179775e-07, -2.5824629138078686e-07, 2.259435099158857e-07, -1.9759059073588738e-07, 1.7245665023281603e-07, -1.499107122703144e-07, 1.2993920706246258e-07, -1.1188458371271578e-07, 9.59786582193289e-08, -8.193904465038978e-08, 6.951736088200208e-08, -5.883242593822998e-08, 4.953479013200448e-08, -4.159778119910192e-08, 3.4903544554923914e-08, -2.9199660726126307e-08, 2.4491065764276586e-08, -2.0543807377807442e-08, 1.716620639244989e-08, -1.4598093803545008e-08, 1.2247184453541803e-08, -1.0378062685590349e-08, 8.941636289359033e-09, -7.547512972569913e-09, 6.5406029883590885e-09, -5.55017639345453e-09, 4.857924129262302e-09, -4.170327848134446e-09, 3.5473818590708514e-09, -3.1820101162273115e-09, 2.634813506155291e-09, -2.3186710334946806e-09, 1.9854991410760484e-09, -1.698026932061246e-09, 1.4939355398374196e-09, -1.2257013267845049e-09, 1.1034926144506615e-09, -8.867213325365261e-10, 7.759313594207437e-10, -6.85530513757325e-10, 5.315937675947832e-10, -5.001264119638624e-10, 4.2230130059116994e-10, -3.259379961024697e-10, 2.8696408042785254e-10, -2.654348289559891e-10, 2.240260857681517e-10, -1.5881755448515084e-10, 1.7089871651079086e-10, -1.743032336304004e-10, 5.736029218880029e-11, -9.974594793790009e-11, 1.2854164813721342e-10, -5.569999528883679e-11, 5.432760350528726e-11, -5.900487596351839e-11, 7.348655484042815e-11, 1.9834070367000245e-12, 3.887800704201888e-11, -6.528210426664377e-11, 6.144420801150463e-12, -2.0697350409069892e-11, 9.512216860539657e-12, -4.439607915237426e-11, -1.6185927706642567e-11, -2.8071628138323645e-12, 6.158579755107668e-11, 2.148407244207534e-11, 5.277970985609337e-13, -9.859059640730805e-12, 4.1564767036192385e-12, -1.5577673049063656e-11, -1.2654069415571345e-12, -1.9761710714008562e-12, 9.40276686806768e-12, 4.583732482119074e-13, -1.8523582732792032e-11, -1.7428972653131536e-11, 2.334371921024897e-11, 1.2661569384099514e-11, -2.4431492094169338e-11, -2.720598171659233e-11, 1.579179961710281e-11, 4.682966091729829e-11, 2.026395923889618e-11, -4.163510324266956e-11, -2.7091399111035808e-11, 3.978859743850732e-11, 3.993365393136633e-11, -2.4706365750991333e-11, -2.8201589338545247e-11]
# Psat_cheb_coeffs = [-188.81248459710693, -190.53226813843213, -0.6992718797266877, 0.3857083557782601, -0.23710917890714395, 0.15368561772753983, -0.10228211161653594, 0.06876166878498034, -0.046105558737181966, 0.030448740221432544, -0.019496099441454324, 0.01180400058944964, -0.006422229275450882, 0.002702227307086234, -0.00018800410519084597, -0.0014485238631714243, 0.0024479474900583895, -0.002988894024752606, 0.0032053382330997785, -0.003197984048551589, 0.0030426262430619812, -0.0027958384579597137, 0.0024994432437511482, -0.00218371114178375, 0.0018699437151919942, -0.0015724843629802854, 0.0013002928376298992, -0.0010582955457831876, 0.0008483768179051751, -0.0006702845742590901, 0.0005222702922150421, -0.0004016564112164708, 0.0003052504825598366, -0.00022965330503168022, 0.00017151209256412164, -0.00012765639237664444, 9.522751362437718e-05, -7.17145087909031e-05, 5.498576051758942e-05, -4.328024825801364e-05, 3.518008638334846e-05, -2.9585552080573432e-05, 2.5660899927246663e-05, -2.2801213593209296e-05, 2.0579135430209277e-05, -1.871227629774825e-05, 1.702697381072197e-05, -1.5427107330232484e-05, 1.3871955438611369e-05, -1.235063269577285e-05, 1.087503047126396e-05, -9.463372111120008e-06, 8.138409928400627e-06, -6.918751587310431e-06, 5.817036690746729e-06, -4.841268302762132e-06, 3.990762592248579e-06, -3.264055878954419e-06, 2.6526744772618845e-06, -2.146826614278467e-06, 1.7339220505229884e-06, -1.4002686597492801e-06, 1.1352817872143799e-06, -9.252727697582733e-07, 7.610055457905131e-07, -6.319237506120556e-07, 5.30160897737689e-07, -4.5034836164150563e-07, 3.8588236023116243e-07, -3.345288398991865e-07, 2.910099599025734e-07, -2.538502269447694e-07, 2.2221275929649412e-07, -1.9404386102611735e-07, 1.7012903413041972e-07, -1.4791267614537682e-07, 1.281131161442957e-07, -1.1035351009983888e-07, 9.412216917920838e-08, -8.103521480312085e-08, 6.889862034626618e-08, -5.823229805384481e-08, 4.888865274151847e-08, -4.0647361572055817e-08, 3.461181492625629e-08, -2.890818104595808e-08, 2.4189127295759093e-08, -2.036506388954876e-08, 1.6621054692260028e-08, -1.4376599744841544e-08, 1.2262293144383739e-08, -1.0166543599991339e-08, 8.776172074614484e-09, -7.244748882363349e-09, 6.552057774765062e-09, -5.655401910624057e-09, 4.4124427509814644e-09, -4.138406545361605e-09, 3.4155934985322144e-09, -3.1467981765942498e-09, 3.138041596064127e-09, -2.097881746535653e-09, 1.6538597491971884e-09, -1.4302796654967797e-09, 1.3958696624380472e-09, -1.6941697510614072e-09, 1.1559050790778446e-09, -8.424336557798272e-10, 7.445069759938515e-10, -3.8008350586066653e-10, 6.681447868524303e-10, -5.609484209193093e-10, 1.1709177677205352e-10, -5.781259004102078e-10, 5.45265361901197e-10, -1.3987335287680026e-10, 1.7128157135074418e-10, 1.0377866018526204e-10, 1.449451573983006e-10, -4.977625195297418e-10, 1.7368603686632612e-10, -3.571321706516851e-11, -1.6249813391308165e-10, 4.6148221569532015e-11, 3.9554757121876716e-10, -1.0268016727946628e-10, -7.436027752479989e-11, -1.6876374859490107e-10, -4.24547853876368e-11, 9.538626006134858e-12, 1.5150070863903953e-10, 2.7005277922459003e-10, -1.6342760518896042e-11, -4.572503911555491e-10, 4.922727672815753e-11, 9.160300994028991e-11, -7.120976338703244e-11, 2.164872706420613e-10, 1.1646536920908047e-10, -2.7132159904485077e-10, -9.18445653054099e-11, 1.1410414945528784e-10, 1.1967624164073171e-10, -5.5743966043066313e-11, 3.9042323803713426e-11, 4.316392256370049e-11, -1.8428367625021157e-10, -9.040283123061977e-11, 1.857434297108983e-10, 1.592233467198178e-11, -1.173771592481677e-10, 1.1665496090537252e-10, 1.2886364193873557e-10, -2.1093389704449506e-10, -2.4675247129314452e-11, 1.515767676711589e-10, -1.2689980450730342e-10, -4.2776899169681866e-11, 1.6317818359826586e-10, -1.4821901477978135e-11, -5.8141610036405774e-11]
Psat_cheb_coeffs_der = chebder(Psat_cheb_coeffs)
Psat_coeffs_critical_der = polyder(Psat_coeffs_critical[::-1])[::-1]
Psat_cheb_constant_factor = (-2.355355160853182, 0.42489124941587103)
# Psat_cheb_constant_factor = (-19.744219083323905, 0.050649991923423815) # down to .14 Tr
# Psat_cheb_constant_factor = (-20.25334447874608, 0.049376705093756613) # down to .05
# Psat_cheb_constant_factor = (-20.88507690836272, 0.0478830941599295) # down to .05 repolishing
# Psat_cheb_constant_factor = (-51.789209241068214, 0.019310239068163836) # down to .05 with lots of failures C40 only
# Psat_cheb_constant_factor = (-52.392851049631986, 0.01908689378961204) # down to .03, plenty of failures
# Psat_cheb_constant_factor = (-52.47770345042524, 0.01905687810661655)
Psat_cheb_range = (0.003211332390446207, 104.95219556846003)
phi_sat_coeffs = [4.040440857039882e-09, -1.512382901024055e-07, 2.5363900091436416e-06,
-2.4959001060510725e-05, 0.00015714708105355206, -0.0006312347348814933,
0.0013488647482434379, 0.0008510254890166079, -0.017614759099592196,
0.06640627813169839, -0.13427456425899886, 0.1172205279608668,
0.13594473870160448, -0.5560225934266592, 0.7087599054079694,
0.6426353018023558]
_P_zero_l_cheb_coeffs = [0.13358936990391557, -0.20047353906149878, 0.15101308518135467, -0.11422662323168498, 0.08677799907222833, -0.06622719396774103, 0.05078577177767531, -0.03913992025038471, 0.030322206247168845, -0.023618484941949063, 0.018500212460075605, -0.014575143278285305, 0.011551352410948363, -0.00921093058565245, 0.007390713292456164, -0.005968132800177682, 0.00485080886172241, -0.003968872414987763, 0.003269291360484698, -0.002711665819666899, 0.0022651044970457743, -0.0019058978265104418, 0.0016157801830935644, -0.0013806283122768208, 0.0011894838915417153, -0.0010338173333182162, 0.0009069721482541163, -0.0008037443041438563, 0.0007200633946601682, -0.0006527508698173454, 0.0005993365082194993, -0.0005579199462298259, 0.0005270668422661141, -0.0005057321913053223, 0.0004932057251527365, -0.00024453764761005106]
P_zero_l_cheb_limits = (0.002068158270122966, 27.87515959722943)
def __init__(self, Tc, Pc, omega, T=None, P=None, V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
self.b = b = self.c2R*Tc/Pc
self.a = b*Tc*self.c1R2_c2R
self.kappa = omega*(-0.26992*omega + 1.54226) + 0.37464
self.delta, self.epsilon = 2.0*b, -b*b
self.solve()
[docs] def a_alpha_pure(self, T):
r'''Method to calculate :math:`a \alpha` for this EOS. Uses the set values of
`Tc`, `kappa`, and `a`.
.. math::
a\alpha = a \left(\kappa \left(- \frac{T^{0.5}}{Tc^{0.5}}
+ 1\right) + 1\right)^{2}
Parameters
----------
T : float
Temperature at which to calculate the value, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
Notes
-----
This method does not alter the object's state and the temperature
provided can be a different than that of the object.
Examples
--------
Dodecane at 250 K:
>>> eos = PR(Tc=658.0, Pc=1820000.0, omega=0.562, T=500., P=1e5)
>>> eos.a_alpha_pure(250.0)
15.66839156301
'''
x0 = (1.0 + self.kappa*(1.0 - sqrt(T/self.Tc)))
return self.a*x0*x0
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives for this EOS. Uses the set values of `Tc`, `kappa`, and `a`.
.. math::
a\alpha = a \left(\kappa \left(- \frac{T^{0.5}}{Tc^{0.5}}
+ 1\right) + 1\right)^{2}
.. math::
\frac{d a\alpha}{dT} = - \frac{1.0 a \kappa}{T^{0.5} Tc^{0.5}}
\left(\kappa \left(- \frac{T^{0.5}}{Tc^{0.5}} + 1\right) + 1\right)
.. math::
\frac{d^2 a\alpha}{dT^2} = 0.5 a \kappa \left(- \frac{1}{T^{1.5}
Tc^{0.5}} \left(\kappa \left(\frac{T^{0.5}}{Tc^{0.5}} - 1\right)
- 1\right) + \frac{\kappa}{T^{1.0} Tc^{1.0}}\right)
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
Notes
-----
This method does not alter the object's state and the temperature
provided can be a different than that of the object.
Examples
--------
Dodecane at 250 K:
>>> eos = PR(Tc=658.0, Pc=1820000.0, omega=0.562, T=500., P=1e5)
>>> eos.a_alpha_and_derivatives_pure(250.0)
(15.66839156301, -0.03094091246957, 9.243186769880e-05)
'''
# TODO custom water a_alpha?
# Peng, DY, and DB Robinson. "Two-and Three-Phase Equilibrium Calculations
# for Coal Gasification and Related Processes,", 1980
# Thermodynamics of aqueous systems with industrial applications 133 (1980): 393-414.
# Applies up to Tr .85.
# Suggested in Equations of State And PVT Analysis.
Tc, kappa, a = self.Tc, self.kappa, self.a
x0 = sqrt(T)
x1 = 1.0/sqrt(Tc)
x2 = kappa*(x0*x1 - 1.) - 1.
x3 = a*kappa
x4 = x1*x2/x0
a_alpha = a*x2*x2
da_alpha_dT = x4*x3
d2a_alpha_dT2 = 0.5*x3*(kappa*x1*x1 - x4)/T
return a_alpha, da_alpha_dT, d2a_alpha_dT2
[docs] def d3a_alpha_dT3_pure(self, T):
r'''Method to calculate the third temperature derivative of `a_alpha`.
Uses the set values of `Tc`, `kappa`, and `a`. This property is not
normally needed.
.. math::
\frac{d^3 a\alpha}{dT^3} = \frac{3 a\kappa \left(- \frac{\kappa}
{T_{c}} + \frac{\sqrt{\frac{T}{T_{c}}} \left(\kappa \left(\sqrt{\frac{T}
{T_{c}}} - 1\right) - 1\right)}{T}\right)}{4 T^{2}}
Parameters
----------
T : float
Temperature at which to calculate the derivative, [-]
Returns
-------
d3a_alpha_dT3 : float
Third temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^3]
Notes
-----
This method does not alter the object's state and the temperature
provided can be a different than that of the object.
Examples
--------
Dodecane at 500 K:
>>> eos = PR(Tc=658.0, Pc=1820000.0, omega=0.562, T=500., P=1e5)
>>> eos.d3a_alpha_dT3_pure(500.0)
-9.8038800671e-08
'''
kappa = self.kappa
x0 = 1.0/self.Tc
T_inv = 1.0/T
x1 = sqrt(T*x0)
return -self.a*0.75*kappa*(kappa*x0 - x1*(kappa*(x1 - 1.0) - 1.0)*T_inv)*T_inv*T_inv
[docs] def P_max_at_V(self, V):
r'''Method to calculate the maximum pressure the EOS can create at a
constant volume, if one exists; returns None otherwise.
Parameters
----------
V : float
Constant molar volume, [m^3/mol]
Returns
-------
P : float
Maximum possible isochoric pressure, [Pa]
Notes
-----
The analytical determination of this formula involved some part of the
discriminant, and much black magic.
Examples
--------
>>> e = PR(P=1e5, V=0.0001437, Tc=512.5, Pc=8084000.0, omega=0.559)
>>> e.P_max_at_V(e.V)
2247886208.7
'''
"""# Partial notes on how this was determined.
from sympy import *
P, T, V = symbols('P, T, V', positive=True)
Tc, Pc, omega = symbols('Tc, Pc, omega', positive=True)
R, a, b, kappa = symbols('R, a, b, kappa')
main = P*R*Tc*V**2 + 2*P*R*Tc*V*b - P*R*Tc*b**2 - P*V*a*kappa**2 + P*a*b*kappa**2 + R*Tc*a*kappa**2 + 2*R*Tc*a*kappa + R*Tc*a
to_subs = {b: thing.b,
kappa: thing.kappa,
a: thing.a, R: thermo.eos.R, Tc: thing.Tc, V: thing.V, Tc: thing.Tc, omega: thing.omega}
solve(Eq(main, 0), P)[0].subs(to_subs)
"""
try:
Tc, a, b, kappa = self.Tc, self.a, self.b, self.kappa
except:
Tc, a, b, kappa = self.Tcs[0], self.ais[0], self.bs[0], self.kappas[0]
P_max = (-R*Tc*a*(kappa**2 + 2*kappa + 1)/(R*Tc*V**2 + 2*R*Tc*V*b - R*Tc*b**2 - V*a*kappa**2 + a*b*kappa**2))
if P_max < 0.0:
# No positive pressure - it's negative
return None
return P_max
# (V - b)**3*(V**2 + 2*V*b - b**2)*(P*R*Tc*V**2 + 2*P*R*Tc*V*b - P*R*Tc*b**2 - P*V*a*kappa**2 + P*a*b*kappa**2 + R*Tc*a*kappa**2 + 2*R*Tc*a*kappa + R*Tc*a)
[docs] def solve_T(self, P, V, solution=None):
r'''Method to calculate `T` from a specified `P` and `V` for the PR
EOS. Uses `Tc`, `a`, `b`, and `kappa` as well, obtained from the
class's namespace.
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
'l' or 'g' to specify a liquid of vapor solution (if one exists);
if None, will select a solution more likely to be real (closer to
STP, attempting to avoid temperatures like 60000 K or 0.0001 K).
Returns
-------
T : float
Temperature, [K]
Notes
-----
The exact solution can be derived as follows, and is excluded for
breviety.
>>> from sympy import *
>>> P, T, V = symbols('P, T, V')
>>> Tc, Pc, omega = symbols('Tc, Pc, omega')
>>> R, a, b, kappa = symbols('R, a, b, kappa')
>>> a_alpha = a*(1 + kappa*(1-sqrt(T/Tc)))**2
>>> PR_formula = R*T/(V-b) - a_alpha/(V*(V+b)+b*(V-b)) - P
>>> #solve(PR_formula, T)
After careful evaluation of the results of the analytical formula,
it was discovered, that numerical precision issues required several
NR refinement iterations; at at times, when the analytical value is
extremely erroneous, a call to a full numerical solver not using the
analytical solution at all is required.
Examples
--------
>>> eos = PR(Tc=658.0, Pc=1820000.0, omega=0.562, T=500., P=1e5)
>>> eos.solve_T(P=eos.P, V=eos.V_g)
500.0000000
'''
self.no_T_spec = True
Tc, a, b, kappa = self.Tc, self.a, self.b, self.kappa
# Needs to be improved to do a NR or two at the end!
x0 = V*V
x1 = R*Tc
x2 = x0*x1
x3 = kappa*kappa
x4 = a*x3
x5 = b*x4
x6 = 2.*V*b
x7 = x1*x6
x8 = b*b
x9 = x1*x8
x10 = V*x4
thing = (x2 - x10 + x5 + x7 - x9)
x11 = thing*thing
x12 = x0*x0
x13 = R*R
x14 = Tc*Tc
x15 = x13*x14
x16 = x8*x8
x17 = a*a
x18 = x3*x3
x19 = x17*x18
x20 = x0*V
x21 = 2.*R*Tc*a*x3
x22 = x8*b
x23 = 4.*V*x22
x24 = 4.*b*x20
x25 = a*x1
x26 = x25*x8
x27 = x26*x3
x28 = x0*x25
x29 = x28*x3
x30 = 2.*x8
x31 = (6.*V*x27 - 2.*b*x29 + x0*x13*x14*x30 + x0*x19 + x12*x15
+ x15*x16 - x15*x23 + x15*x24 - x19*x6 + x19*x8 - x20*x21
- x21*x22)
V_m_b = V - b
x33 = 2.*(R*Tc*a*kappa)
x34 = P*x2
x35 = P*x5
x36 = x25*x3
x37 = P*x10
x38 = P*R*Tc
x39 = V*x17
x40 = 2.*kappa*x3
x41 = b*x17
x42 = P*a*x3
# 2.*a*kappa - add a negative sign to get the high temperature solution
# sometimes it is complex!
# try:
root_term = sqrt(V_m_b**3*(x0 + x6 - x8)*(P*x7 -
P*x9 + x25 + x33 + x34 + x35
+ x36 - x37))
# except ValueError:
# # negative number in sqrt
# return super(PR, self).solve_T(P, V)
x100 = 2.*a*kappa*x11*(root_term*(kappa + 1.))
x101 = (x31*V_m_b*((4.*V)*(R*Tc*a*b*kappa) + x0*x33 - x0*x35 + x12*x38
+ x16*x38 + x18*x39 - x18*x41 - x20*x42 - x22*x42
- x23*x38 + x24*x38 + x25*x6 - x26 - x27 + x28 + x29
+ x3*x39 - x3*x41 + x30*x34 - x33*x8 + x36*x6
+ 3*x37*x8 + x39*x40 - x40*x41))
x102 = -Tc/(x11*x31)
T_calc = (x102*(x100 - x101)) # Normally the correct root
if T_calc < 0.0:
# Ruined, call the numerical method; sometimes it happens
return super().solve_T(P, V, solution=solution)
Tc_inv = 1.0/Tc
T_calc_high = (x102*(-x100 - x101))
if solution is not None and solution == 'g':
T_calc = T_calc_high
if True:
c1, c2 = R/(V_m_b), a/(V*(V+b) + b*V_m_b)
rt = (T_calc*Tc_inv)**0.5
alpha_root = (1.0 + kappa*(1.0-rt))
err = c1*T_calc - alpha_root*alpha_root*c2 - P
if abs(err/P) > 1e-2:
# Numerical issue - such a bad solution we cannot converge
return super().solve_T(P, V, solution=solution)
# Newton step - might as well compute it
derr = c1 + c2*kappa*rt*(kappa*(1.0 -rt) + 1.0)/T_calc
if derr == 0.0:
return T_calc
T_calc = T_calc - err/derr
# Step 2 - cannot find occasion to need more steps, most of the time
# this does nothing!
rt = (T_calc*Tc_inv)**0.5
alpha_root = (1.0 + kappa*(1.0-rt))
err = c1*T_calc - alpha_root*alpha_root*c2 - P
derr = c1 + c2*kappa*rt*(kappa*(1.0 -rt) + 1.0)/T_calc
T_calc = T_calc - err/derr
return T_calc
c1, c2 = R/(V_m_b), a/(V*(V+b) + b*V_m_b)
rt = (T_calc_high*Tc_inv)**0.5
alpha_root = (1.0 + kappa*(1.0-rt))
err = c1*T_calc_high - alpha_root*alpha_root*c2 - P
# Newton step - might as well compute it
derr = c1 + c2*kappa*rt*(kappa*(1.0 -rt) + 1.0)/T_calc_high
T_calc_high = T_calc_high - err/derr
# Step 2 - cannot find occasion to need more steps, most of the time
# this does nothing!
rt = (T_calc_high*Tc_inv)**0.5
alpha_root = (1.0 + kappa*(1.0-rt))
err = c1*T_calc_high - alpha_root*alpha_root*c2 - P
derr = c1 + c2*kappa*rt*(kappa*(1.0 -rt) + 1.0)/T_calc_high
T_calc_high = T_calc_high - err/derr
delta, epsilon = self.delta, self.epsilon
w0 = 1.0*(delta*delta - 4.0*epsilon)**-0.5
w1 = delta*w0
w2 = 2.0*w0
# print(T_calc, T_calc_high)
a_alpha_low = a*(1.0 + kappa*(1.0-(T_calc/Tc)**0.5))**2.0
a_alpha_high = a*(1.0 + kappa*(1.0-(T_calc_high/Tc)**0.5))**2.0
err_low = abs(R*T_calc/(V-b) - a_alpha_low/(V*V + delta*V + epsilon) - P)
err_high = abs(R*T_calc_high/(V-b) - a_alpha_high/(V*V + delta*V + epsilon) - P)
# print(err_low, err_high, T_calc, T_calc_high, a_alpha_low, a_alpha_high)
RT_low = R*T_calc
G_dep_low = (P*V - RT_low - RT_low*clog(P/RT_low*(V-b)).real
- w2*a_alpha_low*catanh(2.0*V*w0 + w1).real)
RT_high = R*T_calc_high
G_dep_high = (P*V - RT_high - RT_high*clog(P/RT_high*(V-b)).real
- w2*a_alpha_high*catanh(2.0*V*w0 + w1).real)
# print(G_dep_low, G_dep_high)
# ((err_low > err_high*2)) and
if (T_calc.imag != 0.0 and T_calc_high.imag == 0.0) or (G_dep_high < G_dep_low and (err_high < err_low)):
T_calc = T_calc_high
return T_calc
# if err_high < err_low:
# T_calc = T_calc_high
# for Ti in (T_calc, T_calc_high):
# a_alpha = a*(1.0 + kappa*(1.0-(Ti/Tc)**0.5))**2.0
#
#
# # Compute P, and the difference?
# self.P = float(R*self.T/(V-self.b) - self.a_alpha/(V*V + self.delta*V + self.epsilon)
#
#
#
# RT = R*Ti
# print(RT, V-b, P/RT*(V-b))
# G_dep = (P*V - RT - RT*log(P/RT*(V-b))
# - w2*a_alpha*catanh(2.0*V*w0 + w1).real)
# print(G_dep)
# if G_dep < G_dep_base:
# T = Ti
# G_dep_base = G_dep
# T_calc = T
# print(T_calc, T_calc_high)
# T_calc = (-Tc*(2.*a*kappa*x11*sqrt(V_m_b**3*(x0 + x6 - x8)*(P*x7 -
# P*x9 + x25 + x33 + x34 + x35
# + x36 - x37))*(kappa + 1.) -
# x31*V_m_b*((4.*V)*(R*Tc*a*b*kappa) + x0*x33 - x0*x35 + x12*x38
# + x16*x38 + x18*x39 - x18*x41 - x20*x42 - x22*x42
# - x23*x38 + x24*x38 + x25*x6 - x26 - x27 + x28 + x29
# + x3*x39 - x3*x41 + x30*x34 - x33*x8 + x36*x6
# + 3*x37*x8 + x39*x40 - x40*x41))/(x11*x31))
# print(T_calc2/T_calc)
# Validation code - although the solution is analytical some issues
# with floating points can still occur
# Although 99.9 % of points anyone would likely want are plenty good,
# there are some edge cases as P approaches T or goes under it.
# c1, c2 = R/(V_m_b), a/(V*(V+b) + b*V_m_b)
#
# rt = (T_calc*Tc_inv)**0.5
# alpha_root = (1.0 + kappa*(1.0-rt))
# err = c1*T_calc - alpha_root*alpha_root*c2 - P
#
# # Newton step - might as well compute it
# derr = c1 + c2*kappa*rt*(kappa*(1.0 -rt) + 1.0)/T_calc
# T_calc = T_calc - err/derr
#
# # Step 2 - cannot find occasion to need more steps, most of the time
# # this does nothing!
# rt = (T_calc*Tc_inv)**0.5
# alpha_root = (1.0 + kappa*(1.0-rt))
# err = c1*T_calc - alpha_root*alpha_root*c2 - P
# derr = c1 + c2*kappa*rt*(kappa*(1.0 -rt) + 1.0)/T_calc
# T_calc = T_calc - err/derr
## print(T_calc)
# return T_calc
# P_inv = 1.0/P
# if abs(err/P) < 1e-6:
# return T_calc
## print(abs(err/P))
## return GCEOS.solve_T(self, P, V)
# for i in range(7):
# rt = (T_calc*Tc_inv)**0.5
# alpha_root = (1.0 + kappa*(1.0-rt))
# err = c1*T_calc - alpha_root*alpha_root*c2 - P
# derr = c1 + c2*kappa*rt*(kappa*(1.0 -rt) + 1.0)/T_calc
#
# T_calc = T_calc - err/derr
# print(err/P, T_calc, derr)
# if abs(err/P) < 1e-12:
# return T_calc
# return T_calc
# starts at 0.0008793111898930736
# Psat_ranges_low = (0.011527649224138653,
# 0.15177700441811506, 0.7883172905889053, 2.035659276638337,
# 4.53501754500169, 10.745446771738406, 22.67639480888016,
# 50.03388490796283, 104.02786866285064)
# 2019 Nov
# Psat_ranges_low = (0.15674244743681393, 0.8119861320343748, 2.094720219302703, 4.960845727141835, 11.067460617890934, 25.621853405705796, 43.198888850643804, 104.02786866285064)
# Psat_coeffs_low = [[-227953.8193412378, 222859.8202525231, -94946.0644714779, 22988.662866916213, -3436.218010266234, 314.10561626462993, -12.536721169650086, -2.392026378146748, 1.7425442228873158, -1.2062891595039678, 0.9256591091303878, -0.7876053099939332, 0.5624587154041579, -3.3553013976814365, 5.4012350148013866e-14], [0.017979999443171253, -0.1407329351142875, 0.5157655870958351, -1.1824391743389553, 1.9175463304080598, -2.370060249233812, 2.3671981077067543, -2.0211919069051754, 1.5662532616167582, -1.1752554496422438, 0.9211423805826566, -0.7870983088912286, 0.5624192663836626, -3.3552995268181935, -4.056076807756881e-08], [2.3465238783212443e-06, -5.1803023754491137e-05, 0.0005331498955415226, -0.0034021195248914006, 0.015107808977575897, -0.04968952806811015, 0.12578046832772882, -0.25143473221174495, 0.40552536074726614, -0.5443994966086247, 0.6434269285808626, -0.6923484892423339, 0.5390886452491613, -3.3516377955152628, -0.0002734868035272342], [-4.149916661961022e-10, 2.1845922714910234e-08, -5.293093383029167e-07, 7.799519138713084e-06, -7.769053551547911e-05, 0.0005486109959120195, -0.0027872878510967723, 0.010013711509364028, -0.023484350891214936, 0.024784713187904924, 0.04189568427991252, -0.2040017547275196, 0.25395831370937016, -3.2456178797446413, -0.01903130694686439], [5.244405747881219e-16, -1.5454390343008565e-14, -2.0604241377631507e-12, 1.8208689279561933e-10, -7.250743412052849e-09, 1.8247981842001254e-07, -3.226779942705286e-06, 4.21332816427672e-05, -0.00041707954900317614, 0.003173654759907457, -0.01868692125208627, 0.0855653889368932, -0.31035507126284995, -2.6634237299183328, -0.2800897855694018], [-2.1214680302656463e-19, 5.783021422459962e-17, -7.315923275334905e-15, 5.698692571821259e-13, -3.0576045765082714e-11, 1.1975824393534794e-09, -3.540115921441331e-08, 8.052781011110919e-07, -1.424237637885889e-05, 0.00019659116938228988, -0.0021156267397923314, 0.017700252965885416, -0.11593142002481696, -3.013661988282298, 0.01996154251720128], [-2.8970166603270677e-23, 1.694610551839978e-20, -4.467776279776866e-18, 7.096773522723984e-16, -7.632413053542317e-14, 5.906374821509563e-12, -3.4056397726361876e-10, 1.4928364875485495e-08, -5.025465019680778e-07, 1.3027126331371714e-05, -0.00025915855275578494, 0.003928557567224198, -0.04532442889219183, -3.235941699431832, 0.33934709098936366], [-1.0487638177712636e-27, 1.1588074100262264e-24, -5.933272229330526e-22, 1.8676144445612704e-19, -4.0425091708892395e-17, 6.37584823835825e-15, -7.573969719222655e-13, 6.907076002118451e-11, -4.883344880881757e-09, 2.6844313931168583e-07, -1.1443544240867529e-05, 0.0003760349651708502, -0.009520080664949915, -3.464433298845877, 1.0399494170785033]]
# 2019 Dec 08 #1
# Psat_ranges_low = ([0.1566663623710075, 0.8122712349481437, 2.0945197784666294, 4.961535043425216, 11.064718660459363, 25.62532893636351, 43.17405809523583, 85.5638421625653, 169.8222874125952)
# Psat_coeffs_low = [[-6.364470992262544e-23, 1.5661396802352383e-19, -1.788719435685493e-16, 1.2567790299823932e-13, -6.068855158259506e-11, 2.130642024043302e-08, -5.608337854780211e-06, 0.0011243910475529856, -0.17253439771817053, 20.164796917496496, -1766.983966143576, 112571.42973915562, -4928969.89775339, 132767165.35442507, -1659856970.7084315], [-6.755028337063007e-31, 1.2373135465776702e-27, -1.0534911582623026e-24, 5.532082037130418e-22, -2.0042818462405888e-19, 5.3092667094437664e-17, -1.0629813459498251e-14, 1.6396189295145161e-12, -1.9677160870915945e-10, 1.8425759971191095e-08, -1.3425348946576017e-06, 7.562661739651473e-05, -0.0032885862389808195, -3.5452990752336735, 1.5360178058346605], [-5.909795950371768e-27, 5.645060782013921e-24, -2.5062698828832408e-21, 6.861883492029141e-19, -1.2960098086863643e-16, 1.7893963536931406e-14, -1.8669999568680822e-12, 1.5005071785133313e-10, -9.381783948347974e-09, 4.576967837674971e-07, -1.7378660968493725e-05, 0.0005105597560223805, -0.011603105202254462, -3.4447117223858394, 0.9538198797898474], [-2.8780483706946006e-23, 1.4693097909367858e-20, -3.492711723365092e-18, 5.129438453755985e-16, -5.2066819983096923e-14, 3.87131295903126e-12, -2.1797843188384387e-10, 9.475510493050094e-09, -3.212229879279181e-07, 8.520129885652724e-06, -0.00017645941977890718, 0.0028397690069188186, -0.035584878748907235, -3.2889972189483, 0.47227047696507896], [-2.133647784270567e-19, 5.813855761166538e-17, -7.351939324704256e-15, 5.724415520048679e-13, -3.0701524683808055e-11, 1.2020043191332715e-09, -3.5517231986184477e-08, 8.075833591581873e-07, -1.4277180602174389e-05, 0.0001969886336996064, -0.0021190060629508248, 0.017720993486168023, -0.11601827744842373, -3.0134398433062954, 0.019699769017179847], [5.217055552725474e-16, -1.561972494582649e-14, -2.027739589933126e-12, 1.8030004183143271e-10, -7.1961213928967356e-09, 1.8138160781745565e-07, -3.2112101506231723e-06, 4.197218861582643e-05, -0.00041584453068251905, 0.0031666287443832307, -0.018657602063128432, 0.08547811393673718, -0.31017952035114504, -2.6636376461277504, -0.27997050354186115], [-4.1558987320232216e-10, 2.1874838982254277e-08, -5.299524926441045e-07, 7.808241563359814e-06, -7.777110034030892e-05, 0.0005491470176474339, -0.002789936581283384, 0.010023585334231266, -0.023512249664927133, 0.02484416646533969, 0.04180162903589153, -0.20389464760201653, 0.25387532037317434, -3.245578712101638, -0.01903980099778657], [2.3320945490434305e-06, -5.15194336734163e-05, 0.0005305911686609431, -0.003388078003236081, 0.015055473744080193, -0.049549442201717114, 0.12550289037335455, -0.251021291476035, 0.40506041321992375, -0.5440068047537978, 0.6431818377117259, -0.6922389245218481, 0.5390554975784367, -3.3516317236219626, -0.00027399457467680577], [0.017760683349597454, -0.1392342452029993, 0.5111179189769633, -1.1737814955588932, 1.9067391494716879, -2.3605113086814407, 2.361048334775187, -2.0182633656154794, 1.5652184041682835, -1.1749857171593956, 0.92109138142958, -0.7870915307971148, 0.5624186680171368, -3.3552994954150326, -4.130013597780646e-08], [1842638.012244339, -2064103.5077599594, 1029111.4284441478, -300839.92590603326, 57174.96949130112, -7405.305505076668, 668.4504791023379, -43.94219790319933, 3.4634979070792977, -1.2528527563309222, 0.9264289045482768, -0.787612207652486, 0.5624587411994793, -3.3553013976928456, 4.846123502488808e-14]]
# 2019 Dec 08 #2
# Psat_ranges_low = (0.15674244743681393, 0.8119861320343748, 2.094720219302703, 4.961535043425216, 11.064718660459363, 25.62532893636351, 43.17405809523583, 85.5638421625653, 169.8222874125952, 192.707581659434)
# Psat_coeffs_low = [[-393279.9328001248, 414920.88015712175, -194956.1186003408, 53799.692378381624, -9679.442200674115, 1189.1133946984114, -99.38789237175924, 3.7558250389696366, 1.4341105372610397, -1.195532646019414, 0.9254075742030472, -0.7876016031722438, 0.5624586846061402, -3.355301397567417, -2.475797344914099e-14], [0.018200741617324958, -0.14216111513088853, 0.5199706046777292, -1.1898993034816217, 1.9264460624802726, -2.377604380463091, 2.3718790446551283, -2.0233492715449346, 1.5669946704278936, -1.175444344921655, 0.9211774746760774, -0.787102916441927, 0.5624196703434721, -3.3552995479850125, -4.006059328709455e-08], [2.362594082154845e-06, -5.213477214805086e-05, 0.0005363047209564668, -0.0034204334370065157, 0.015180294585886198, -0.04989640532490752, 0.1262194343941631, -0.252138050376706, 0.4063802322466773, -0.5451837881722801, 0.643961448026334, -0.6926108644042617, 0.5391763183580807, -3.3516556444811516, -0.00027181665396192045], [-4.1566510211197074e-10, 2.1878563345656593e-08, -5.30037387599558e-07, 7.809422248533072e-06, -7.77822904769859e-05, 0.0005492234565335112, -0.002790324592151159, 0.010025071882175543, -0.023516568419967406, 0.024853633218471893, 0.04178621870041742, -0.20387658476895476, 0.2538609101701838, -3.2455717084245443, -0.019041365569938407], [5.952860605957254e-16, -2.3560872386568428e-14, -1.6328974906691505e-12, 1.6831386671561567e-10, -6.947967158882692e-09, 1.77675502929117e-07, -3.170039732850266e-06, 4.162662881336586e-05, -0.0004136425496617131, 0.0031560285189308705, -0.018619655683130842, 0.085380163769752, -0.3100071777702119, -2.6638226631426187, -0.279879068340815], [-2.1336825570293267e-19, 5.813946215182557e-17, -7.352047876443287e-15, 5.724495165386215e-13, -3.0701923762367554e-11, 1.2020187632285275e-09, -3.5517621350872006e-08, 8.075912994222895e-07, -1.4277303680626562e-05, 0.00019699007656794466, -0.00211901865445771, 0.01772107279538477, -0.1160186182468458, -3.0134389491023668, 0.019698688209032866], [-2.8780483706946006e-23, 1.4693097909367858e-20, -3.492711723365092e-18, 5.129438453755985e-16, -5.2066819983096923e-14, 3.87131295903126e-12, -2.1797843188384387e-10, 9.475510493050094e-09, -3.212229879279181e-07, 8.520129885652724e-06, -0.00017645941977890718, 0.0028397690069188186, -0.035584878748907235, -3.2889972189483, 0.47227047696507896], [-5.909795950371768e-27, 5.645060782013921e-24, -2.5062698828832408e-21, 6.861883492029141e-19, -1.2960098086863643e-16, 1.7893963536931406e-14, -1.8669999568680822e-12, 1.5005071785133313e-10, -9.381783948347974e-09, 4.576967837674971e-07, -1.7378660968493725e-05, 0.0005105597560223805, -0.011603105202254462, -3.4447117223858394, 0.9538198797898474], [-6.755028337063007e-31, 1.2373135465776702e-27, -1.0534911582623026e-24, 5.532082037130418e-22, -2.0042818462405888e-19, 5.3092667094437664e-17, -1.0629813459498251e-14, 1.6396189295145161e-12, -1.9677160870915945e-10, 1.8425759971191095e-08, -1.3425348946576017e-06, 7.562661739651473e-05, -0.0032885862389808195, -3.5452990752336735, 1.5360178058346605], [-6.364470992262544e-23, 1.5661396802352383e-19, -1.788719435685493e-16, 1.2567790299823932e-13, -6.068855158259506e-11, 2.130642024043302e-08, -5.608337854780211e-06, 0.0011243910475529856, -0.17253439771817053, 20.164796917496496, -1766.983966143576, 112571.42973915562, -4928969.89775339, 132767165.35442507, -1659856970.7084315]]
# 2019 Dec 08 #3
Psat_ranges_low = (0.038515189998761204, 0.6472853332269844, 2.0945197784666294, 4.961232873814024, 11.067553885784903, 25.624838497870584, 43.20169529076582, 85.5588271726612, 192.72834691988226)
Psat_coeffs_low = [[2338676895826482.5, -736415034973095.6, 105113277697825.1, -8995168780410.754, 514360029044.81494, -20734723655.83978, 605871516.8891307, -12994014.122638363, 204831.11357912835, -2351.9913154464143, 18.149657683324232, 0.8151930684866298, -0.7871881357728392, 0.5624577476810062, -3.35530139647672, -4.836964162535651e-13], [-0.13805715433070773, 0.8489231609102119, -2.450329797856018, 4.447856574793218, -5.767299107094559, 5.794674157897756, -4.825296555657044, 3.5520183799445926, -2.4600869594916634, 1.6909163275418595, -1.2021498414235525, 0.9254639369127162, -0.7875982246546266, 0.5624585116206676, -3.3553013938160787, -3.331224185387782e-11], [-2.3814071133383825e-06, 5.318261908739265e-05, -0.0005538990617858645, 0.0035761255785055936, -0.016054997425247523, 0.05333504500541739, -0.13636391080337568, 0.27593424749870343, -0.4517901507372948, 0.6114112167354924, -0.7059858408782421, 0.7385376731146207, -0.7329884294338728, 0.5509890744823249, -3.353773232516225, -9.646546737407391e-05], [2.6058661808460023e-11, -1.75914103924121e-09, 5.396299167286894e-08, -1.0007922530068192e-06, 1.2554484077194732e-05, -0.0001125821062183067, 0.0007410322067253991, -0.0035992993229111833, 0.012657105041028169, -0.030121969848977304, 0.03753504314148813, 0.02349666014556937, -0.18469580367455368, 0.24005237728233714, -3.239469690554324, -0.020289142467969867], [-1.082394018559102e-15, 1.2914854481231322e-13, -7.104839518580019e-12, 2.3832489222439473e-10, -5.425087002560749e-09, 8.804418548276272e-08, -1.0364065054630989e-06, 8.719985338278278e-06, -4.8325538208084174e-05, 0.00011200959608941485, 0.0008028675551716892, -0.010695106054891056, 0.06594801536296582, -0.27725262867260253, -2.6977571369079514, -0.2635895959694814], [1.1488824622125947e-20, -3.331154652317046e-18, 4.503372697637035e-16, -3.7684497582121125e-14, 2.1852058912840643e-12, -9.313780852814459e-11, 3.019939074381905e-09, -7.605074783395472e-08, 1.5052679183948458e-06, -2.354701523431422e-05, 0.00029127690705745875, -0.0028399757838276493, 0.02173245057169364, -0.13135011490812692, -2.9774476427885146, -0.01942256817236654], [1.0436558787976772e-24, -5.473723131383567e-22, 1.3452696879486453e-19, -2.0573736968717295e-17, 2.1924486360657888e-15, -1.7272619586846295e-13, 1.0413985148866247e-11, -4.906312890258065e-10, 1.8279149292524938e-08, -5.414588408693672e-07, 1.275367009914141e-05, -0.00023786604002741, 0.0034903075344121025, -0.04033658323380905, -3.2676007023496245, 0.42749816097639837], [9.060766533667912e-29, -9.196819760777788e-26, 4.3601925662975664e-23, -1.2818245897574232e-20, 2.615903295904718e-18, -3.930631843509798e-16, 4.500311702777485e-14, -4.007582103109645e-12, 2.808196479352211e-10, -1.5562164421777763e-08, 6.818206236433737e-07, -2.350273523243411e-05, 0.0006326097721162514, -0.013277937187152783, -3.4305615375066876, 0.8983326523220114], [1.1247677438654667e-33, -2.4697583969349065e-30, 2.5286510080356973e-27, -1.6024926981128421e-24, 7.03655740810716e-22, -2.2705238015446456e-19, 5.57121222696514e-17, -1.0609879702627998e-14, 1.5863699537553053e-12, -1.8713657213281574e-10, 1.7407548458856668e-08, -1.2702047168798462e-06, 7.210856106809965e-05, -0.0031754110755806966, -3.5474790036315795, 1.555110704923493]]
[docs]class PR78(PR):
r'''Class for solving the Peng-Robinson cubic
equation of state for a pure compound according to the 1978 variant [1]_ [2]_.
Subclasses :obj:`PR`, which provides everything except the variable `kappa`.
Solves the EOS on initialization. See :obj:`PR` for further documentation.
.. math::
P = \frac{RT}{v-b}-\frac{a\alpha(T)}{v(v+b)+b(v-b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha(T)=[1+\kappa(1-\sqrt{T_r})]^2
.. math::
\kappa_i = 0.37464+1.54226\omega_i-0.26992\omega_i^2 \text{ if } \omega_i
\le 0.491
.. math::
\kappa_i = 0.379642 + 1.48503 \omega_i - 0.164423\omega_i^2 + 0.016666
\omega_i^3 \text{ if } \omega_i > 0.491
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization (furfuryl alcohol), liquid phase:
>>> eos = PR78(Tc=632, Pc=5350000, omega=0.734, T=299., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 8.3519628969e-05, -63764.671093, -130.737153225)
Notes
-----
This variant is recommended over the original.
References
----------
.. [1] Robinson, Donald B, and Ding-Yu Peng. The Characterization of the
Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs.
Tulsa, Okla.: Gas Processors Association, 1978.
.. [2] Robinson, Donald B., Ding-Yu Peng, and Samuel Y-K Chung. "The
Development of the Peng - Robinson Equation and Its Application to Phase
Equilibrium in a System Containing Methanol." Fluid Phase Equilibria 24,
no. 1 (January 1, 1985): 25-41. doi:10.1016/0378-3812(85)87035-7.
'''
low_omega_constants = (0.37464, 1.54226, -0.26992)
"""Constants for the `kappa` formula for the low-omega region."""
high_omega_constants = (0.379642, 1.48503, -0.164423, 0.016666)
"""Constants for the `kappa` formula for the high-omega region."""
def __init__(self, Tc, Pc, omega, T=None, P=None, V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
self.a = self.c1R2*Tc*Tc/Pc
self.b = b = self.c2R*Tc/Pc
self.delta, self.epsilon = 2.0*b, -b*b
if omega <= 0.491:
self.kappa = 0.37464 + omega*(1.54226 - 0.26992*omega)
else:
self.kappa = omega*(omega*(0.016666*omega - 0.164423) + 1.48503) + 0.379642
self.solve()
[docs]class PRTranslated(PR):
r'''Class for solving the volume translated Peng-Robinson equation of state.
Subclasses :obj:`PR`. Solves the EOS on initialization.
This is intended as a base class for all translated variants of the
Peng-Robinson EOS.
.. math::
P = \frac{RT}{v + c - b} - \frac{a\alpha(T)}{(v+c)(v + c + b)+b(v
+ c - b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha(T)=[1+\kappa(1-\sqrt{T_r})]^2
.. math::
\kappa=0.37464+1.54226\omega-0.26992\omega^2
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
alpha_coeffs : tuple or None
Coefficients which may be specified by subclasses; set to None to use
the original Peng-Robinson alpha function, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization:
>>> eos = PRTranslated(T=305, P=1.1e5, Tc=512.5, Pc=8084000.0, omega=0.559, c=-1e-6)
>>> eos.phase, eos.V_l, eos.V_g
('l/g', 4.90798083711e-05, 0.0224350982488)
Notes
-----
References
----------
.. [1] Gmehling, Jürgen, Michael Kleiber, Bärbel Kolbe, and Jürgen Rarey.
Chemical Thermodynamics for Process Simulation. John Wiley & Sons, 2019.
'''
solve_T = GCEOS.solve_T
P_max_at_V = GCEOS.P_max_at_V
kwargs_keys = ('c', 'alpha_coeffs')
def __init__(self, Tc, Pc, omega, alpha_coeffs=None, c=0.0, T=None, P=None,
V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
Pc_inv = 1.0/Pc
self.a = self.c1R2*Tc*Tc*Pc_inv
self.c = c
if alpha_coeffs is None:
self.kappa = omega*(-0.26992*omega + 1.54226) + 0.37464
# Does not have an impact on phase equilibria
self.alpha_coeffs = alpha_coeffs
self.kwargs = {'c': c, 'alpha_coeffs': alpha_coeffs}
# self.C0, self.C1, self.C2 = Twu_coeffs
b0 = self.c2*R*Tc*Pc_inv
self.b = b = b0 - c
# Cannot reference b directly
self.delta = 2.0*(c + b0)
self.epsilon = -b0*b0 + c*c + 2.0*c*b0
# C**2 + 2*C*b + V**2 + V*(2*C + 2*b) - b**2
self.solve()
[docs]class PRTranslatedPPJP(PRTranslated):
r'''Class for solving the volume translated Pina-Martinez, Privat, Jaubert,
and Peng revision of the Peng-Robinson equation of state
for a pure compound according to [1]_.
Subclasses :obj:`PRTranslated`, which provides everything except the variable `kappa`.
Solves the EOS on initialization. See :obj:`PRTranslated` for further documentation.
.. math::
P = \frac{RT}{v + c - b} - \frac{a\alpha(T)}{(v+c)(v + c + b)+b(v
+ c - b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha(T)=[1+\kappa(1-\sqrt{T_r})]^2
.. math::
\kappa = 0.3919 + 1.4996 \omega - 0.2721\omega^2 + 0.1063\omega^3
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization (methanol), liquid phase:
>>> eos = PRTranslatedPPJP(Tc=507.6, Pc=3025000, omega=0.2975, c=0.6390E-6, T=250., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.0001229231238092, -33466.2428296, -80.75610242427)
Notes
-----
This variant offers incremental improvements in accuracy only, but those
can be fairly substantial for some substances.
References
----------
.. [1] Pina-Martinez, Andrés, Romain Privat, Jean-Noël Jaubert, and
Ding-Yu Peng. "Updated Versions of the Generalized Soave a-Function
Suitable for the Redlich-Kwong and Peng-Robinson Equations of State."
Fluid Phase Equilibria, December 7, 2018.
https://doi.org/10.1016/j.fluid.2018.12.007.
'''
# Direct solver for T could be implemented but cannot use the PR one
kwargs_keys = ('c',)
def __init__(self, Tc, Pc, omega, c=0.0, T=None, P=None, V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
Pc_inv = 1.0/Pc
self.a = self.c1*R2*Tc*Tc*Pc_inv
self.c = c
# 0.3919 + 1.4996*omega - 0.2721*omega**2+0.1063*omega**3
self.kappa = omega*(omega*(0.1063*omega - 0.2721) + 1.4996) + 0.3919
self.kwargs = {'c': c}
b0 = self.c2*R*Tc*Pc_inv
self.b = b = b0 - c
self.delta = 2.0*(c + b0)
self.epsilon = -b0*b0 + c*c + 2.0*c*b0
self.solve()
def P_max_at_V(self, V):
if self.c == 0.0:
return PR.P_max_at_V(self, V)
return None
[docs]class PRTranslatedPoly(Poly_a_alpha, PRTranslated):
r'''Class for solving the volume translated Peng-Robinson equation of state
with a polynomial alpha function. With the right coefficients, this model
can reproduce any property incredibly well.
Subclasses :obj:`PRTranslated`. Solves the EOS on initialization.
This is intended as a base class for all translated variants of the
Peng-Robinson EOS.
.. math::
P = \frac{RT}{v + c - b} - \frac{a\alpha(T)}{(v+c)(v + c + b)+b(v
+ c - b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha(T)=f(T)
.. math::
\kappa=0.37464+1.54226\omega-0.26992\omega^2
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
alpha_coeffs : tuple or None
Coefficients which may be specified by subclasses; set to None to use
the original Peng-Robinson alpha function, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
Methanol, with alpha functions reproducing CoolProp's implementation of
its vapor pressure (up to 13 coefficients)
>>> alpha_coeffs_exact = [9.645280470011588e-32, -4.362226651748652e-28, 9.034194757823037e-25, -1.1343330204981244e-21, 9.632898335494218e-19, -5.841502902171077e-16, 2.601801729901228e-13, -8.615431349241052e-11, 2.1202999753932622e-08, -3.829144045293198e-06, 0.0004930777289075716, -0.04285337965522619, 2.2473964123842705, -51.13852710672087]
>>> kwargs = dict(Tc=512.5, Pc=8084000.0, omega=0.559, alpha_coeffs=alpha_coeffs_exact, c=1.557458e-05)
>>> eos = PRTranslatedPoly(T=300, P=1e5, **kwargs)
>>> eos.Psat(500)/PropsSI("P", 'T', 500.0, 'Q', 0, 'methanol') # doctest:+SKIP
1.0000112765
Notes
-----
'''
class PRTranslatedMathiasCopeman(Mathias_Copeman_poly_a_alpha, PRTranslated):
pass
class PRTranslatedCoqueletChapoyRichon(PRTranslatedMathiasCopeman):
kwargs_keys = ('c', 'alpha_coeffs')
def __init__(self, Tc, Pc, omega, c=0.0, alpha_coeffs=None, T=None, P=None, V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
Pc_inv = 1.0/Pc
self.a = self.c1*R2*Tc*Tc*Pc_inv
self.c = c
if alpha_coeffs is None:
c1 = omega*(0.1316*omega + 1.4031) + 0.3906
c2 = omega*(-1.3127*omega + 0.3015) - 0.1213
c3 = 0.7661*omega + 0.3041
alpha_coeffs = [c3, c2, c1, 1.0]
elif alpha_coeffs[-1] != 1.0:
alpha_coeffs = list(alpha_coeffs)
alpha_coeffs.append(1.0)
self.kwargs = {'c': c, 'alpha_coeffs': alpha_coeffs}
self.alpha_coeffs = alpha_coeffs
b0 = self.c2*R*Tc*Pc_inv
self.b = b = b0 - c
self.delta = 2.0*(c + b0)
self.epsilon = -b0*b0 + c*c + 2.0*c*b0
self.solve()
[docs]class PRTranslatedTwu(Twu91_a_alpha, PRTranslated):
r'''Class for solving the volume translated Peng-Robinson equation of state
with the Twu (1991) [1]_ alpha function.
Subclasses :obj:`thermo.eos_alpha_functions.Twu91_a_alpha` and :obj:`PRTranslated`.
Solves the EOS on initialization.
.. math::
P = \frac{RT}{v + c - b} - \frac{a\alpha(T)}{(v+c)(v + c + b)+b(v
+ c - b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha = \left(\frac{T}{T_{c}}\right)^{c_{3} \left(c_{2}
- 1\right)} e^{c_{1} \left(- \left(\frac{T}{T_{c}}
\right)^{c_{2} c_{3}} + 1\right)}
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
alpha_coeffs : tuple(float[3])
Coefficients L, M, N (also called C1, C2, C3) of TWU 1991 form, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization:
>>> alpha_coeffs = (0.694911381318495, 0.919907783415812, 1.70412689631515)
>>> kwargs = dict(Tc=512.5, Pc=8084000.0, omega=0.559, alpha_coeffs=alpha_coeffs, c=-1e-6)
>>> eos = PRTranslatedTwu(T=300, P=1e5, **kwargs)
>>> eos.phase, eos.V_l, eos.V_g
('l/g', 4.8918748906e-05, 0.024314406330)
Notes
-----
This variant offers substantial improvements to the PR-type EOSs - likely
getting about as accurate as this form of cubic equation can get.
References
----------
.. [1] Twu, Chorng H., David Bluck, John R. Cunningham, and John E.
Coon. "A Cubic Equation of State with a New Alpha Function and a
New Mixing Rule." Fluid Phase Equilibria 69 (December 10, 1991):
33-50. doi:10.1016/0378-3812(91)90024-2.
'''
[docs]class PRTranslatedConsistent(PRTranslatedTwu):
r'''Class for solving the volume translated Le Guennec, Privat, and Jaubert
revision of the Peng-Robinson equation of state
for a pure compound according to [1]_.
Subclasses :obj:`PRTranslatedTwu`, which provides everything except the
estimation of `c` and the alpha coefficients. This model's `alpha` is based
on the TWU 1991 model; when estimating, `N` is set to 2.
Solves the EOS on initialization. See :obj:`PRTranslated` for further documentation.
.. math::
P = \frac{RT}{v + c - b} - \frac{a\alpha(T)}{(v+c)(v + c + b)+b(v
+ c - b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha = \left(\frac{T}{T_{c}}\right)^{c_{3} \left(c_{2}
- 1\right)} e^{c_{1} \left(- \left(\frac{T}{T_{c}}
\right)^{c_{2} c_{3}} + 1\right)}
If `c` is not provided, it is estimated as:
.. math::
c =\frac{R T_c}{P_c}(0.0198\omega - 0.0065)
If `alpha_coeffs` is not provided, the parameters `L` and `M` are estimated
from the acentric factor as follows:
.. math::
L = 0.1290\omega^2 + 0.6039\omega + 0.0877
.. math::
M = 0.1760\omega^2 - 0.2600\omega + 0.8884
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
alpha_coeffs : tuple(float[3]), optional
Coefficients L, M, N (also called C1, C2, C3) of TWU 1991 form, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization (methanol), liquid phase:
>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=250., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.000124374813374486, -34155.16119794619, -83.34913258614345)
Notes
-----
This variant offers substantial improvements to the PR-type EOSs - likely
getting about as accurate as this form of cubic equation can get.
References
----------
.. [1] Le Guennec, Yohann, Romain Privat, and Jean-Noël Jaubert.
"Development of the Translated-Consistent Tc-PR and Tc-RK Cubic
Equations of State for a Safe and Accurate Prediction of Volumetric,
Energetic and Saturation Properties of Pure Compounds in the Sub- and
Super-Critical Domains." Fluid Phase Equilibria 429 (December 15, 2016):
301-12. https://doi.org/10.1016/j.fluid.2016.09.003.
'''
kwargs_keys = ('c', 'alpha_coeffs')
def __init__(self, Tc, Pc, omega, alpha_coeffs=None, c=None, T=None,
P=None, V=None):
# estimates volume translation and alpha function parameters
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
Pc_inv = 1.0/Pc
# Limit the fitting omega to a little under the range reported
o = min(max(omega, -0.01), 1.48)
if c is None:
c = R*Tc*Pc_inv*(0.0198*o - 0.0065)
if alpha_coeffs is None:
L = o*(0.1290*o + 0.6039) + 0.0877
M = o*(0.1760*o - 0.2600) + 0.8884
N = 2.0
alpha_coeffs = (L, M, N)
self.c = c
self.alpha_coeffs = alpha_coeffs
self.kwargs = {'c': c, 'alpha_coeffs': alpha_coeffs}
self.a = self.c1*R2*Tc*Tc*Pc_inv
b0 = self.c2*R*Tc*Pc_inv
self.b = b = b0 - c
self.delta = 2.0*(c + b0)
self.epsilon = -b0*b0 + c*c + 2.0*c*b0
self.solve()
[docs]class PRSV(PR):
r'''Class for solving the Peng-Robinson-Stryjek-Vera equations of state for
a pure compound as given in [1]_. The same as the Peng-Robinson EOS,
except with a different `kappa` formula and with an optional fit parameter.
Subclasses :obj:`PR`, which provides only several constants. See :obj:`PR` for
further documentation and examples.
.. math::
P = \frac{RT}{v-b}-\frac{a\alpha(T)}{v(v+b)+b(v-b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha(T)=[1+\kappa(1-\sqrt{T_r})]^2
.. math::
\kappa = \kappa_0 + \kappa_1(1 + T_r^{0.5})(0.7 - T_r)
.. math::
\kappa_0 = 0.378893 + 1.4897153\omega - 0.17131848\omega^2
+ 0.0196554\omega^3
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
kappa1 : float, optional
Fit parameter; available in [1]_ for over 90 compounds, [-]
Examples
--------
P-T initialization (hexane, with fit parameter in [1]_), liquid phase:
>>> eos = PRSV(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6, kappa1=0.05104)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.000130126913554, -31698.926746, -74.16751538)
Notes
-----
[1]_ recommends that `kappa1` be set to 0 for Tr > 0.7. This is not done by
default; the class boolean `kappa1_Tr_limit` may be set to True and the
problem re-solved with that specified if desired. `kappa1_Tr_limit` is not
supported for P-V inputs.
Solutions for P-V solve for `T` with SciPy's `newton` solver, as there is no
analytical solution for `T`
[2]_ and [3]_ are two more resources documenting the PRSV EOS. [4]_ lists
`kappa` values for 69 additional compounds. See also `PRSV2`. Note that
tabulated `kappa` values should be used with the critical parameters used
in their fits. Both [1]_ and [4]_ only considered vapor pressure in fitting
the parameter.
References
----------
.. [1] Stryjek, R., and J. H. Vera. "PRSV: An Improved Peng-Robinson
Equation of State for Pure Compounds and Mixtures." The Canadian Journal
of Chemical Engineering 64, no. 2 (April 1, 1986): 323-33.
doi:10.1002/cjce.5450640224.
.. [2] Stryjek, R., and J. H. Vera. "PRSV - An Improved Peng-Robinson
Equation of State with New Mixing Rules for Strongly Nonideal Mixtures."
The Canadian Journal of Chemical Engineering 64, no. 2 (April 1, 1986):
334-40. doi:10.1002/cjce.5450640225.
.. [3] Stryjek, R., and J. H. Vera. "Vapor-liquid Equilibrium of
Hydrochloric Acid Solutions with the PRSV Equation of State." Fluid
Phase Equilibria 25, no. 3 (January 1, 1986): 279-90.
doi:10.1016/0378-3812(86)80004-8.
.. [4] Proust, P., and J. H. Vera. "PRSV: The Stryjek-Vera Modification of
the Peng-Robinson Equation of State. Parameters for Other Pure Compounds
of Industrial Interest." The Canadian Journal of Chemical Engineering
67, no. 1 (February 1, 1989): 170-73. doi:10.1002/cjce.5450670125.
'''
kappa1_Tr_limit = False
kwargs_keys = ('kappa1',)
def __init__(self, Tc, Pc, omega, T=None, P=None, V=None, kappa1=None):
self.T, self.P, self.V, self.omega, self.Tc, self.Pc = T, P, V, omega, Tc, Pc
if kappa1 is None:
kappa1 = 0.0
self.kwargs = {'kappa1': kappa1}
self.b = b = self.c2R*Tc/Pc
self.a = self.c1R2_c2R*Tc*b
self.delta = 2.0*b
self.epsilon = -b*b
self.kappa0 = omega*(omega*(0.0196554*omega - 0.17131848) + 1.4897153) + 0.378893
self.check_sufficient_inputs()
if V and P:
# Deal with T-solution here; does NOT support kappa1_Tr_limit.
self.kappa1 = kappa1
self.T = T = self.solve_T(P, V)
Tr = T/Tc
else:
Tr = T/Tc
if self.kappa1_Tr_limit and Tr > 0.7:
self.kappa1 = 0.0
else:
self.kappa1 = kappa1
self.kappa = self.kappa0 + self.kappa1*(1.0 + sqrt(Tr))*(0.7 - Tr)
self.solve()
[docs] def solve_T(self, P, V, solution=None):
r'''Method to calculate `T` from a specified `P` and `V` for the PRSV
EOS. Uses `Tc`, `a`, `b`, `kappa0` and `kappa` as well, obtained from
the class's namespace.
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
'l' or 'g' to specify a liquid of vapor solution (if one exists);
if None, will select a solution more likely to be real (closer to
STP, attempting to avoid temperatures like 60000 K or 0.0001 K).
Returns
-------
T : float
Temperature, [K]
Notes
-----
Not guaranteed to produce a solution. There are actually two solution,
one much higher than normally desired; it is possible the solver could
converge on this.
'''
Tc, a, b, kappa0, kappa1 = self.Tc, self.a, self.b, self.kappa0, self.kappa1
self.no_T_spec = True
x0 = V - b
R_x0 = R/x0
x3_inv = 1.0/(100.*(V*(V + b) + b*x0))
x4 = 10.*kappa0
kappa110 = kappa1*10.
kappa17 = kappa1*7.
Tc_inv = 1.0/Tc
x51 = x3_inv*a
def to_solve(T):
x1 = T*Tc_inv
x2 = x1**0.5
x10 = ((x4 - (kappa110*x1 - kappa17)*(x2 + 1.))*(x2 - 1.) - 10.)
x11 = T*R_x0 - P
return x11 - x10*x10*x51
if solution is None:
try:
return newton(to_solve, Tc*0.5)
except:
pass
# The above method handles fewer cases, but the below is less optimized
return GCEOS.solve_T(self, P, V, solution=solution)
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives for this EOS. Uses the set values of `Tc`, `kappa0`,
`kappa1`, and `a`.
The `a_alpha` function is shown below; the first and second derivatives
are not shown for brevity.
.. math::
a\alpha = a \left(\left(\kappa_{0} + \kappa_{1} \left(\sqrt{\frac{
T}{T_{c}}} + 1\right) \left(- \frac{T}{T_{c}} + \frac{7}{10}\right)
\right) \left(- \sqrt{\frac{T}{T_{c}}} + 1\right) + 1\right)^{2}
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
Notes
-----
This method does not alter the object's state and the temperature
provided can be a different than that of the object.
The expressions can be derived as follows:
>>> from sympy import *
>>> P, T, V = symbols('P, T, V')
>>> Tc, Pc, omega = symbols('Tc, Pc, omega')
>>> R, a, b, kappa0, kappa1 = symbols('R, a, b, kappa0, kappa1')
>>> kappa = kappa0 + kappa1*(1 + sqrt(T/Tc))*(Rational(7, 10)-T/Tc)
>>> a_alpha = a*(1 + kappa*(1-sqrt(T/Tc)))**2
>>> # diff(a_alpha, T)
>>> # diff(a_alpha, T, 2)
Examples
--------
>>> eos = PRSV(Tc=507.6, Pc=3025000, omega=0.2975, T=406.08, P=1E6, kappa1=0.05104)
>>> eos.a_alpha_and_derivatives_pure(185.0)
(4.76865472591, -0.0101408587212, 3.9138298092e-05)
'''
Tc, a, kappa0, kappa1 = self.Tc, self.a, self.kappa0, self.kappa1
x1 = T/Tc
T_inv = 1.0/T
x2 = sqrt(x1)
x3 = x2 - 1.
x4 = 10.*x1 - 7.
x5 = x2 + 1.
x6 = 10.*kappa0 - kappa1*x4*x5
x7 = x3*x6
x8 = x7*0.1 - 1.
x10 = x6*T_inv
x11 = kappa1*x3
x12 = x4*T_inv
x13 = 20./Tc*x5 + x12*x2
x14 = -x10*x2 + x11*x13
a_alpha = a*x8*x8
da_alpha_dT = -a*x14*x8*0.1
d2a_alpha_dT2 = a*(x14*x14 - x2*T_inv*(x7 - 10.)*(2.*kappa1*x13 + x10 + x11*(40.*x1*T_inv - x12)))*(1.0/200.)
return a_alpha, da_alpha_dT, d2a_alpha_dT2
[docs] def a_alpha_pure(self, T):
r'''Method to calculate :math:`a \alpha` for this EOS. Uses the set values of `Tc`, `kappa0`,
`kappa1`, and `a`.
.. math::
a\alpha = a \left(\left(\kappa_{0} + \kappa_{1} \left(\sqrt{\frac{
T}{T_{c}}} + 1\right) \left(- \frac{T}{T_{c}} + \frac{7}{10}\right)
\right) \left(- \sqrt{\frac{T}{T_{c}}} + 1\right) + 1\right)^{2}
Parameters
----------
T : float
Temperature at which to calculate the value, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
Notes
-----
This method does not alter the object's state and the temperature
provided can be a different than that of the object.
Examples
--------
>>> eos = PRSV(Tc=507.6, Pc=3025000, omega=0.2975, T=406.08, P=1E6, kappa1=0.05104)
>>> eos.a_alpha_pure(185.0)
4.7686547259
'''
Tc, a, kappa0, kappa1 = self.Tc, self.a, self.kappa0, self.kappa1
Tr = T/Tc
sqrtTr = sqrt(Tr)
v = ((kappa0 + kappa1*(sqrtTr + 1.0)*(-Tr + 0.7))*(-sqrtTr + 1.0) + 1.0)
return a*v*v
[docs]class PRSV2(PR):
r'''Class for solving the Peng-Robinson-Stryjek-Vera 2 equations of state
for a pure compound as given in [1]_. The same as the Peng-Robinson EOS,
except with a different `kappa` formula and with three fit parameters.
Subclasses :obj:`PR`, which provides only several constants. See :obj:`PR` for
further documentation and examples.
.. math::
P = \frac{RT}{v-b}-\frac{a\alpha(T)}{v(v+b)+b(v-b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha(T)=[1+\kappa(1-\sqrt{T_r})]^2
.. math::
\kappa = \kappa_0 + [\kappa_1 + \kappa_2(\kappa_3 - T_r)(1-T_r^{0.5})]
(1 + T_r^{0.5})(0.7 - T_r)
.. math::
\kappa_0 = 0.378893 + 1.4897153\omega - 0.17131848\omega^2
+ 0.0196554\omega^3
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
kappa1 : float, optional
Fit parameter; available in [1]_ for over 90 compounds, [-]
kappa2 : float, optional
Fit parameter; available in [1]_ for over 90 compounds, [-]
kappa : float, optional
Fit parameter; available in [1]_ for over 90 compounds, [-]
Examples
--------
P-T initialization (hexane, with fit parameter in [1]_), liquid phase:
>>> eos = PRSV2(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6, kappa1=0.05104, kappa2=0.8634, kappa3=0.460)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.000130188257591, -31496.1841687, -73.615282963)
Notes
-----
Note that tabulated `kappa` values should be used with the critical
parameters used in their fits. [1]_ considered only vapor
pressure in fitting the parameter.
References
----------
.. [1] Stryjek, R., and J. H. Vera. "PRSV2: A Cubic Equation of State for
Accurate Vapor-liquid Equilibria Calculations." The Canadian Journal of
Chemical Engineering 64, no. 5 (October 1, 1986): 820-26.
doi:10.1002/cjce.5450640516.
'''
kwargs_keys = ('kappa1', 'kappa2', 'kappa3')
def __init__(self, Tc, Pc, omega, T=None, P=None, V=None, kappa1=0, kappa2=0, kappa3=0):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
self.check_sufficient_inputs()
self.kwargs = {'kappa1': kappa1, 'kappa2': kappa2, 'kappa3': kappa3}
self.a = self.c1*R*R*Tc*Tc/Pc
self.b = self.c2*R*Tc/Pc
self.delta = 2*self.b
self.epsilon = -self.b*self.b
self.kappa0 = kappa0 = omega*(omega*(0.0196554*omega - 0.17131848) + 1.4897153) + 0.378893
self.kappa1, self.kappa2, self.kappa3 = kappa1, kappa2, kappa3
if self.V and self.P:
# Deal with T-solution here
self.T = T = self.solve_T(self.P, self.V)
Tr = T/Tc
sqrtTr = sqrt(Tr)
self.kappa = kappa0 + ((kappa1 + kappa2*(kappa3 - Tr)*(1.0 -sqrtTr))*(1.0 + sqrtTr)*(0.7 - Tr))
self.solve()
[docs] def solve_T(self, P, V, solution=None):
r'''Method to calculate `T` from a specified `P` and `V` for the PRSV2
EOS. Uses `Tc`, `a`, `b`, `kappa0`, `kappa1`, `kappa2`, and `kappa3`
as well, obtained from the class's namespace.
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
'l' or 'g' to specify a liquid of vapor solution (if one exists);
if None, will select a solution more likely to be real (closer to
STP, attempting to avoid temperatures like 60000 K or 0.0001 K).
Returns
-------
T : float
Temperature, [K]
Notes
-----
Not guaranteed to produce a solution. There are actually 8 solutions,
six with an imaginary component at a tested point. The two temperature
solutions are quite far apart, with one much higher than the other;
it is possible the solver could converge on the higher solution, so use
`T` inputs with care. This extra solution is a perfectly valid one
however. The secant method is implemented at present.
Examples
--------
>>> eos = PRSV2(Tc=507.6, Pc=3025000, omega=0.2975, T=400., P=1E6, kappa1=0.05104, kappa2=0.8634, kappa3=0.460)
>>> eos.solve_T(P=eos.P, V=eos.V_g)
400.0
'''
self.no_T_spec = True
if solution is None:
Tc, a, b, kappa0, kappa1, kappa2, kappa3 = self.Tc, self.a, self.b, self.kappa0, self.kappa1, self.kappa2, self.kappa3
x0 = V - b
R_x0 = R/x0
x5_inv = 1.0/(100.*(V*(V + b) + b*x0))
x4 = 10.*kappa0
def to_solve(T):
x1 = T/Tc
x2 = sqrt(x1)
x3 = x2 - 1.
x100 = (x3*(x4 - (kappa1 + kappa2*x3*(-kappa3 + x1))*(10.*x1 - 7.)*(x2 + 1.)) - 10.)
return (R_x0*T - a*x100*x100*x5_inv) - P
try:
return newton(to_solve, Tc*0.5)
except:
pass
# The above method handles fewer cases, but the below is less optimized
return GCEOS.solve_T(self, P, V, solution=solution)
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives for this EOS. Uses the set values of `Tc`, `kappa0`, `kappa1`,
`kappa2`, `kappa3`, and `a`.
.. math::
\alpha(T)=[1+\kappa(1-\sqrt{T_r})]^2
.. math::
\kappa = \kappa_0 + [\kappa_1 + \kappa_2(\kappa_3 - T_r)(1-T_r^{0.5})]
(1 + T_r^{0.5})(0.7 - T_r)
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
Notes
-----
The first and second derivatives of `a_alpha` are available through the
following SymPy expression.
>>> from sympy import * # doctest:+SKIP
>>> P, T, V = symbols('P, T, V') # doctest:+SKIP
>>> Tc, Pc, omega = symbols('Tc, Pc, omega') # doctest:+SKIP
>>> R, a, b, kappa0, kappa1, kappa2, kappa3 = symbols('R, a, b, kappa0, kappa1, kappa2, kappa3') # doctest:+SKIP
>>> Tr = T/Tc # doctest:+SKIP
>>> kappa = kappa0 + (kappa1 + kappa2*(kappa3-Tr)*(1-sqrt(Tr)))*(1+sqrt(Tr))*(Rational('0.7')-Tr) # doctest:+SKIP
>>> a_alpha = a*(1 + kappa*(1-sqrt(T/Tc)))**2 # doctest:+SKIP
>>> diff(a_alpha, T) # doctest:+SKIP
>>> diff(a_alpha, T, 2) # doctest:+SKIP
Examples
--------
>>> eos = PRSV2(Tc=507.6, Pc=3025000, omega=0.2975, T=400., P=1E6, kappa1=0.05104, kappa2=0.8634, kappa3=0.460)
>>> eos.a_alpha_and_derivatives_pure(311.0)
(3.7245418495, -0.0066115440470, 2.05871011677e-05)
'''
Tc, a, kappa0, kappa1, kappa2, kappa3 = self.Tc, self.a, self.kappa0, self.kappa1, self.kappa2, self.kappa3
Tc_inv = 1.0/Tc
T_inv = 1.0/T
x1 = T*Tc_inv
x2 = sqrt(x1)
x3 = x2 - 1.
x4 = x2 + 1.
x5 = 10.*x1 - 7.
x6 = -kappa3 + x1
x7 = kappa1 + kappa2*x3*x6
x8 = x5*x7
x9 = 10.*kappa0 - x4*x8
x10 = x3*x9
x11 = x10*0.1 - 1.
x13 = x2*T_inv
x14 = x7*Tc_inv
x15 = kappa2*x4*x5
x16 = 2.*(-x2 + 1.)*Tc_inv + x13*(kappa3 - x1)
x17 = -x13*x8 - x14*(20.*x2 + 20.) + x15*x16
x18 = x13*x9 + x17*x3
x19 = x2*T_inv*T_inv
x20 = 2.*x2*T_inv
a_alpha = a*x11*x11
da_alpha_dT = a*x11*x18*0.1
d2a_alpha_dT2 = a*(x18*x18 + (x10 - 10.)*(x17*x20 - x19*x9
+ x3*(40.*kappa2*Tc_inv*x16*x4 + kappa2*x16*x20*x5
- 40.*T_inv*x14*x2 - x15*T_inv*x2*(4.*Tc_inv - x6*T_inv) + x19*x8)))*(1.0/200.)
return a_alpha, da_alpha_dT, d2a_alpha_dT2
[docs] def a_alpha_pure(self, T):
r'''Method to calculate :math:`a \alpha` for this EOS. Uses the set values of `Tc`, `kappa0`, `kappa1`,
`kappa2`, `kappa3`, and `a`.
.. math::
\alpha(T)=[1+\kappa(1-\sqrt{T_r})]^2
.. math::
\kappa = \kappa_0 + [\kappa_1 + \kappa_2(\kappa_3 - T_r)(1-T_r^{0.5})]
(1 + T_r^{0.5})(0.7 - T_r)
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
Examples
--------
>>> eos = PRSV2(Tc=507.6, Pc=3025000, omega=0.2975, T=400., P=1E6, kappa1=0.05104, kappa2=0.8634, kappa3=0.460)
>>> eos.a_alpha_pure(1276.0)
33.321674050
'''
Tc, a, kappa0, kappa1, kappa2, kappa3 = self.Tc, self.a, self.kappa0, self.kappa1, self.kappa2, self.kappa3
Tr = T/Tc
sqrtTr = sqrt(Tr)
kappa = kappa0 + ((kappa1 + kappa2*(kappa3 - Tr)*(1.0 -sqrtTr))*(1.0 + sqrtTr)*(0.7 - Tr))
x0 = (1.0 + kappa*(1.0 - sqrtTr))
return a*x0*x0
[docs]class VDW(GCEOS):
r'''Class for solving the Van der Waals [1]_ [2]_ cubic
equation of state for a pure compound. Subclasses :obj:`GCEOS`, which
provides the methods for solving the EOS and calculating its assorted
relevant thermodynamic properties. Solves the EOS on initialization.
Two of `T`, `P`, and `V` are needed to solve the EOS.
.. math::
P=\frac{RT}{V-b}-\frac{a}{V^2}
.. math::
a=\frac{27}{64}\frac{(RT_c)^2}{P_c}
.. math::
b=\frac{RT_c}{8P_c}
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
omega : float, optional
Acentric factor - not used in equation of state!, [-]
Examples
--------
>>> eos = VDW(Tc=507.6, Pc=3025000, T=299., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.000223329856081, -13385.7273746, -32.65923125)
Notes
-----
`omega` is allowed as an input for compatibility with the other EOS forms,
but is not used.
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th
edition. New York: McGraw-Hill Professional, 2000.
.. [2] Walas, Stanley M. Phase Equilibria in Chemical Engineering.
Butterworth-Heinemann, 1985.
'''
delta = 0.0
"""`delta` is always zero for the :obj:`VDW` EOS"""
epsilon = 0.0
"""`epsilon` is always zero for the :obj:`VDW` EOS"""
omega = None
"""`omega` has no impact on the :obj:`VDW` EOS"""
Zc = 3.0/8.
"""Mechanical compressibility of :obj:`VDW` EOS"""
c1 = 27.0/64.0
c2 = 1.0/8.0
c1R2 = c1*R2
c2R = c2*R
c1R2_c2R = c1R2/c2R
Psat_coeffs_limiting = [-3.0232164484175756, 0.20980668241160666]
Psat_coeffs_critical = [9.575399398167086, -5.742004486758378,
4.8000085098196745, -3.000000002903554,
1.0000000000002651]
Psat_cheb_coeffs = [-3.0938407448693392, -3.095844800654779, -0.01852425171597184, -0.009132810281704463,
0.0034478548769173167, -0.0007513250489879469, 0.0001425235859202672, -3.18455900032599e-05,
8.318773833859442e-06, -2.125810773856036e-06, 5.171012493290658e-07, -1.2777009201877978e-07,
3.285945705657834e-08, -8.532047244427343e-09, 2.196978792832582e-09, -5.667409821199761e-10,
1.4779624173003134e-10, -3.878590467732996e-11, 1.0181633097391951e-11, -2.67662653922595e-12,
7.053635426397184e-13, -1.872821965868618e-13, 4.9443291800198297e-14, -1.2936198878592264e-14,
2.9072203628840998e-15, -4.935694864968698e-16, 2.4160767787481663e-15, 8.615748088927622e-16,
-5.198342841253312e-16, -2.19739320055784e-15, -1.0876309618559898e-15, 7.727786509661994e-16,
7.958450521858285e-16, 2.088444434750203e-17, -1.3864912907016191e-16]
Psat_cheb_constant_factor = (-1.0630005005005003, 0.9416200294550813)
Psat_cheb_coeffs_der = chebder(Psat_cheb_coeffs)
Psat_coeffs_critical_der = polyder(Psat_coeffs_critical[::-1])[::-1]
# old
# Psat_ranges_low = (0.01718036869384043, 0.17410832232527684, 0.8905407354904298, 2.2829574334301284, 4.426814725758547, 8.56993985840095, 17.61166533755772, 34.56469477688733, 77.37121667459365, 153.79268928425782, 211.1818863599383)
# Psat_coeffs_low = [[7.337186734125958e+20, -1.0310276898768364e+20, 6.571811243740342e+18, -2.5142563688011328e+17, 6438369447367660.0, -116506416837877.06, 1533164370692.7314, -14873856417.118628, 106700642.42147608, -562656.3974485798, 2148.713955632803, -5.548751336002541, -0.32195156938958336, 0.29998758814658794, -2.9999999917461664, -2.349558048120315e-12], [-47716.736350665735, 77971.24589571664, -57734.17539993985, 25680.2304910714, -7665.471399138683, 1624.1156139380123, -251.90520898330854, 29.14383349387213, -2.6381384873389155, 0.32059956364530534, -0.19926642267720887, 0.24817720022049916, -0.33257749010194143, 0.3000000932130581, -3.000000000833664, 3.2607250233240848e-12], [-0.00016633444687699065, 0.0015485952911635057, -0.006863235318198338, 0.019451003485722398, -0.04015910528306605, 0.06568824742274948, -0.09110903840180812, 0.11396786842400031, -0.13566008631414134, 0.15955575257376992, -0.19156365559810004, 0.2479007648557237, -0.33256946631897144, 0.29999987151721086, -2.9999999954189773, -5.750377951585506e-11], [-7.428009121019926e-08, 1.9162063229590812e-06, -2.309905985537932e-05, 0.00017322131856139868, -0.0009091957309876683, 0.003571436963791831, -0.010987705972491674, 0.027369774602560334, -0.05652165868271822, 0.0989022663430445, -0.15341525362487263, 0.22884492561559433, -0.325345424979866, 0.2980576936663475, -2.999671413421381, -2.6222517302443293e-05], [-1.304829383752042e-10, 6.461311349903354e-09, -1.4697523398500654e-07, 2.0279063155086473e-06, -1.8858104539128372e-05, 0.0001240696514816671, -0.0005894769787540043, 0.0020347845952011574, -0.0051863170297969646, 0.010970032738000915, -0.02673039729181951, 0.07989719021025658, -0.18981116875502171, 0.20975049244314053, -2.9633925585336254, -0.007040461060364933], [-8.483418232487059e-14, 8.63974862373075e-12, -4.1017440987789216e-10, 1.2043221191702449e-08, -2.4458795830937423e-07, 3.6395727498191334e-06, -4.098703864306506e-05, 0.00035554065503799216, -0.0023925268788656615, 0.012458721418882563, -0.04951152909450557, 0.14533920657540916, -0.29136837208180943, 0.2957408216961629, -2.992226481317782, -0.010497873866540886], [1.2057846439922303e-18, -2.498037369038981e-16, 2.4169298089982375e-14, -1.4499679111714204e-12, 6.038719260446175e-11, -1.8521190625043091e-09, 4.3303351110208387e-08, -7.880798028324914e-07, 1.130024120583253e-05, -0.00012841965962337936, 0.0011579326786125476, -0.008265573217317893, 0.04660472417726327, -0.20986415552448967, -2.53897638918461, -0.1886467797596083], [5.456456120655508e-23, -2.2443095704330367e-20, 4.312575509674247e-18, -5.139829986632021e-16, 4.2535953008708246e-14, -2.592784941046291e-12, 1.2047971820045452e-10, -4.356922932676532e-09, 1.2408018207471443e-07, -2.797822690789934e-06, 4.99619390186859e-05, -0.0007038763395268029, 0.007780538926245783, -0.06771345498616513, -2.871932127187338, 0.18812475659373717], [7.867249251522955e-28, -6.9880843771805455e-25, 2.8943351684355176e-22, -7.420582747372289e-20, 1.3183352580324197e-17, -1.7213805273492628e-15, 1.7095155009610707e-13, -1.3180402544717915e-11, 7.98160114045682e-10, -3.815575213074619e-08, 1.4395507736418072e-06, -4.266216879490852e-05, 0.0009859688747049194, -0.01775904876058947, -3.107864564311215, 0.729035082405801], [1.8311839997067204e-31, -3.0736833050041296e-28, 2.3992665338781007e-25, -1.1556016072304783e-22, 3.842081607645237e-20, -9.344614680779128e-18, 1.7187598144123416e-15, -2.436914034605369e-13, 2.6895257030219836e-11, -2.3163949881188256e-09, 1.5507887324406105e-07, -7.988762893413175e-06, 0.0003116704944009503, -0.00906717310261831, -3.1702256862598945, 0.8792501521348868], [3.527506950185549e-29, -9.295645359865239e-26, 1.1416182235481363e-22, -8.667770574363085e-20, 4.550011954816112e-17, -1.7491901974552787e-14, 5.087496940001709e-12, -1.1399408706170077e-09, 1.9839477686722418e-07, -2.6819268483650753e-05, 0.0027930054114231415, -0.2200566323239571, 12.697254920421862, -506.5185701848038, 12488.177914134796, -143563.720494474]]
Psat_ranges_low = (0.01718036869384043, 0.17410832232527684, 0.8905407354904298, 2.2829574334301284, 4.588735762374165, 9.198213969343113, 19.164801360905702, 39.162202265367675, 89.80614296441635, 211.1818863599383)
Psat_coeffs_low = [[3.709170427241228e+20, -5.369876399773602e+19, 3.55437646625649e+18, -1.4237055014239443e+17, 3848808938963197.0, -74140204939074.31, 1047156764817.9473, -10990806592.937004, 85953150.78472495, -497625.69096407224, 2099.583641752129, -6.04136142777935, -0.31974206497045565, 0.29998332413453277, -2.9999999877560994, -3.795894154556834e-12], [202181.2230940579, -296048.9678516585, 197754.0893494042, -79779.91915062582, 21690.65516277411, -4199.021355925494, 596.0913050938211, -62.88287531914928, 4.838615877346316, -0.13231119871015717, -0.17907064524060426, 0.24752941358391634, -0.33256309490225905, 0.29999988493075114, -2.9999999990847974, -3.155253835984695e-12], [-0.00014029558609732686, 0.0013389844121899472, -0.0060888948581993155, 0.017710934821923482, -0.0375010580271008, 0.06276682660210708, -0.08872419862950512, 0.11249655170661062, -0.1349689265044028, 0.15930871464919516, -0.19149706848521642, 0.24788748254751106, -0.3325675697571886, 0.2999996886491002, -2.999999984780528, -3.387914393471192e-10], [-7.63500937520021e-08, 1.963108844693943e-06, -2.3590460584275485e-05, 0.0001763784708554673, -0.0009231029234653729, 0.0036159164438775227, -0.011094383271741722, 0.027565091035355725, -0.05679683707503139, 0.09920052620961826, -0.1536618676986315, 0.22899765112464673, -0.32541398139525823, 0.29807874689899555, -2.9996753674171845, -2.5880253545551568e-05], [-8.042041394684212e-11, 3.985029597822566e-09, -9.007790594461407e-08, 1.222381124984873e-06, -1.1000101355126748e-05, 6.812347222100813e-05, -0.00028917548838158866, 0.0007973445988826675, -0.0012398033018025893, 0.0012288503460726383, -0.008274005514143725, 0.05353750365067098, -0.16234102442586626, 0.19003067203938392, -2.9546730846617515, -0.008830685622417178], [-3.611616220031973e-14, 3.8767347748860204e-12, -1.9376800248066728e-10, 5.982204949605256e-09, -1.2756854237545315e-07, 1.9899440138492874e-06, -2.344690467724149e-05, 0.00021230626049088423, -0.0014868552467794268, 0.008024812789608007, -0.03284208875861816, 0.0980794205896591, -0.19356262181013717, 0.15625529853158554, -2.8696503839999488, -0.06053293499127932], [3.9101454054983654e-19, -8.77263958995087e-17, 9.190024742411764e-15, -5.968146499787873e-13, 2.6900134213207918e-11, -8.926852543915742e-10, 2.2576199533792054e-08, -4.44287432728095e-07, 6.886340930232061e-06, -8.455663245991894e-05, 0.0008233162588686339, -0.006341223591956234, 0.038529175147467405, -0.1865188282734164, -2.580546988555211, -0.15427433578447847], [1.0947478032468857e-23, -5.042777995894199e-21, 1.0846467451595606e-18, -1.4462299284667368e-16, 1.3382745001929549e-14, -9.116107354161303e-13, 4.730996512803351e-11, -1.9095977742126322e-09, 6.06591823816572e-08, -1.524502829215733e-06, 3.0318007813133075e-05, -0.00047520009323191216, 0.005836174792523882, -0.056313989030004126, -2.913137841802409, 0.2573512083085632], [1.0417003605891542e-28, -1.0617427986062235e-25, 5.045404400073696e-23, -1.4839108957248214e-20, 3.023754129891556e-18, -4.527590970541259e-16, 5.155153568793495e-14, -4.555877665139844e-12, 3.161476174754809e-10, -1.731323722560287e-08, 7.479908191797176e-07, -2.537158469008294e-05, 0.0006706500253454564, -0.013799582450476145, -3.1384761956985416, 0.8388793704120587], [3.619666517093681e-34, -8.603334584486169e-31, 9.530335406766774e-28, -6.531547415539122e-25, 3.100047792152191e-22, -1.0806960083892972e-19, 2.863332612760165e-17, -5.885004890375537e-15, 9.491165245604907e-13, -1.207013136689613e-10, 1.2097185039766346e-08, -9.505275874736017e-07, 5.807225072909577e-05, -0.002750509744231165, -3.2675197703421506, 1.578136241005211]]
phi_sat_coeffs = [-4.703247660146169e-06, 7.276853488756492e-05, -0.0005008397610615123,
0.0019560274384829595, -0.004249875101260566, 0.001839985687730564,
0.02021191780955066, -0.07056928933569773, 0.09941120467466309,
0.021295687530901747, -0.32582447905247514, 0.521321793740683,
0.6950957738017804]
_P_zero_l_cheb_coeffs = [0.23949680596158576, -0.28552048884377407, 0.17223773827357045, -0.10535895068953466, 0.06539081523178862, -0.04127943642449526, 0.02647106353835149, -0.017260750015435533, 0.011558172064668568, -0.007830624115831804, 0.005422844032253547, -0.00383463423135285, 0.0027718803475398936, -0.0020570084561681613, 0.0015155074622906842, -0.0011495238177958583, 0.000904782154904249, -0.000683347677699564, 0.0005800187592994201, -0.0004529246894177611, 0.00032901743817593566, -0.0002990561659229427, 0.00023524411148843384, -0.00019464055011993858, 0.0001441665975916752, -0.00013106835607900116, 9.72812311007959e-05, -7.611327134024459e-05, 5.240433315348986e-05, -3.6415012576658176e-05, 3.89310794418167e-05, -2.2160354688301534e-05, 2.7908599229672926e-05, 1.6405692108915904e-05, -1.3931165551671343e-06, -4.80770003354232e-06]
P_zero_l_cheb_limits = (0.002354706203222534, 9.0)
main_derivatives_and_departures = staticmethod(main_derivatives_and_departures_VDW)
def __init__(self, Tc, Pc, T=None, P=None, V=None, omega=None):
self.Tc = Tc
self.Pc = Pc
self.T = T
self.P = P
self.V = V
Tc_Pc = Tc/Pc
self.a = (27.0/64.0)*R2*Tc*Tc_Pc
self.b = 0.125*R*Tc_Pc
self.solve()
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives for this EOS. Uses the set values of `a`.
.. math::
a\alpha = a
.. math::
\frac{d a\alpha}{dT} = 0
.. math::
\frac{d^2 a\alpha}{dT^2} = 0
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
'''
return self.a, 0.0, 0.0
[docs] def a_alpha_pure(self, T):
r'''Method to calculate :math:`a \alpha`. Uses the set values of `a`.
.. math::
a\alpha = a
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
'''
return self.a
[docs] def solve_T(self, P, V, solution=None):
r'''Method to calculate `T` from a specified `P` and `V` for the :obj:`VDW`
EOS. Uses `a`, and `b`, obtained from the class's namespace.
.. math::
T = \frac{1}{R V^{2}} \left(P V^{2} \left(V - b\right)
+ V a - a b\right)
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
'l' or 'g' to specify a liquid of vapor solution (if one exists);
if None, will select a solution more likely to be real (closer to
STP, attempting to avoid temperatures like 60000 K or 0.0001 K).
Returns
-------
T : float
Temperature, [K]
'''
self.no_T_spec = True
return (P*V**2*(V - self.b) + V*self.a - self.a*self.b)/(R*V**2)
[docs] def T_discriminant_zeros_analytical(self, valid=False):
r'''Method to calculate the temperatures which zero the discriminant
function of the :obj:`VDW` eos. This is an analytical cubic function solved
analytically.
Parameters
----------
valid : bool
Whether to filter the calculated temperatures so that they are all
real, and positive only, [-]
Returns
-------
T_discriminant_zeros : list[float]
Temperatures which make the discriminant zero, [K]
Notes
-----
Calculated analytically. Derived as follows. Has multiple solutions.
>>> from sympy import * # doctest:+SKIP
>>> P, T, V, R, b, a = symbols('P, T, V, R, b, a') # doctest:+SKIP
>>> delta, epsilon = 0, 0 # doctest:+SKIP
>>> eta = b # doctest:+SKIP
>>> B = b*P/(R*T) # doctest:+SKIP
>>> deltas = delta*P/(R*T) # doctest:+SKIP
>>> thetas = a*P/(R*T)**2 # doctest:+SKIP
>>> epsilons = epsilon*(P/(R*T))**2 # doctest:+SKIP
>>> etas = eta*P/(R*T) # doctest:+SKIP
>>> a_coeff = 1 # doctest:+SKIP
>>> b_coeff = (deltas - B - 1) # doctest:+SKIP
>>> c = (thetas + epsilons - deltas*(B+1)) # doctest:+SKIP
>>> d = -(epsilons*(B+1) + thetas*etas) # doctest:+SKIP
>>> disc = b_coeff*b_coeff*c*c - 4*a_coeff*c*c*c - 4*b_coeff*b_coeff*b_coeff*d - 27*a_coeff*a_coeff*d*d + 18*a_coeff*b_coeff*c*d # doctest:+SKIP
>>> base = -(expand(disc/P**2*R**3*T**3/a)) # doctest:+SKIP
>>> base_T = simplify(base*T**3) # doctest:+SKIP
>>> sln = collect(expand(base_T), T).args # doctest:+SKIP
'''
P, a, b = self.P, self.a_alpha, self.b
b2 = b*b
x1 = P*b2
x0 = 12.0*x1
d = 4.0*P*R_inv2*R_inv*(a*a + x1*(x1 + 2.0*a))
c = (x0 - 20.0*a)*R_inv2*b*P
b_coeff = (x0 - a)*R_inv
a_coeff = 4.0*b
roots = roots_cubic(a_coeff, b_coeff, c, d)
# roots = np.roots([a_coeff, b_coeff, c, d]).tolist()
if valid:
# TODO - only include ones when switching phases from l/g to either g/l
# Do not know how to handle
roots = [r.real for r in roots if (r.real >= 0.0 and (abs(r.imag) <= 1e-12))]
roots.sort()
return roots
[docs] @staticmethod
def P_discriminant_zeros_analytical(T, b, delta, epsilon, a_alpha, valid=False):
r'''Method to calculate the pressures which zero the discriminant
function of the :obj:`VDW` eos. This is an cubic function solved
analytically.
Parameters
----------
T : float
Temperature, [K]
b : float
Coefficient calculated by EOS-specific method, [m^3/mol]
delta : float
Coefficient calculated by EOS-specific method, [m^3/mol]
epsilon : float
Coefficient calculated by EOS-specific method, [m^6/mol^2]
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
valid : bool
Whether to filter the calculated pressures so that they are all
real, and positive only, [-]
Returns
-------
P_discriminant_zeros : tuple[float]
Pressures which make the discriminant zero, [Pa]
Notes
-----
Calculated analytically. Derived as follows. Has multiple solutions.
>>> from sympy import * # doctest:+SKIP
>>> P, T, V, R, b, a = symbols('P, T, V, R, b, a') # doctest:+SKIP
>>> P_vdw = R*T/(V-b) - a/(V*V) # doctest:+SKIP
>>> delta, epsilon = 0, 0 # doctest:+SKIP
>>> eta = b # doctest:+SKIP
>>> B = b*P/(R*T) # doctest:+SKIP
>>> deltas = delta*P/(R*T) # doctest:+SKIP
>>> thetas = a*P/(R*T)**2 # doctest:+SKIP
>>> epsilons = epsilon*(P/(R*T))**2 # doctest:+SKIP
>>> etas = eta*P/(R*T) # doctest:+SKIP
>>> a_coeff = 1 # doctest:+SKIP
>>> b_coeff = (deltas - B - 1) # doctest:+SKIP
>>> c = (thetas + epsilons - deltas*(B+1)) # doctest:+SKIP
>>> d = -(epsilons*(B+1) + thetas*etas) # doctest:+SKIP
>>> disc = b_coeff*b_coeff*c*c - 4*a_coeff*c*c*c - 4*b_coeff*b_coeff*b_coeff*d - 27*a_coeff*a_coeff*d*d + 18*a_coeff*b_coeff*c*d # doctest:+SKIP
>>> base = -(expand(disc/P**2*R**3*T**3/a)) # doctest:+SKIP
>>> collect(base, P).args # doctest:+SKIP
'''
# T, a_alpha = self.T, self.a_alpha
# a = a_alpha
# b, epsilon, delta = self.b, self.epsilon, self.delta
a = a_alpha
d = 4*b - a/(R*T)
c = (12*b**2/(R*T) - 20*a*b/(R**2*T**2) + 4*a**2/(R**3*T**3))
b_coeff = (12*b**3/(R**2*T**2) + 8*a*b**2/(R**3*T**3))
a_coeff = 4*b**4/(R**3*T**3)
roots = roots_cubic(a_coeff, b_coeff, c, d)
# roots = np.roots([a_coeff, b_coeff, c, d]).tolist()
return roots
[docs]class RK(GCEOS):
r'''Class for solving the Redlich-Kwong [1]_ [2]_ [3]_ cubic
equation of state for a pure compound. Subclasses :obj:`GCEOS`, which
provides the methods for solving the EOS and calculating its assorted
relevant thermodynamic properties. Solves the EOS on initialization.
Two of `T`, `P`, and `V` are needed to solve the EOS.
.. math::
P =\frac{RT}{V-b}-\frac{a}{V\sqrt{\frac{T}{T_{c}}}(V+b)}
.. math::
a=\left(\frac{R^2(T_c)^{2}}{9(\sqrt[3]{2}-1)P_c} \right)
=\frac{0.42748\cdot R^2(T_c)^{2.5}}{P_c}
.. math::
b=\left( \frac{(\sqrt[3]{2}-1)}{3}\right)\frac{RT_c}{P_c}
=\frac{0.08664\cdot R T_c}{P_c}
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
>>> eos = RK(Tc=507.6, Pc=3025000, T=299., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.000151893468781, -26160.8424877, -63.013137852)
Notes
-----
`omega` is allowed as an input for compatibility with the other EOS forms,
but is not used.
References
----------
.. [1] Redlich, Otto., and J. N. S. Kwong. "On the Thermodynamics of
Solutions. V. An Equation of State. Fugacities of Gaseous Solutions."
Chemical Reviews 44, no. 1 (February 1, 1949): 233-44.
doi:10.1021/cr60137a013.
.. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th
edition. New York: McGraw-Hill Professional, 2000.
.. [3] Walas, Stanley M. Phase Equilibria in Chemical Engineering.
Butterworth-Heinemann, 1985.
'''
c1 = 0.4274802335403414043909906940611707345513 # 1/(9*(2**(1/3.)-1))
"""Full value of the constant in the `a` parameter"""
c2 = 0.08664034996495772158907020242607611685675 # (2**(1/3.)-1)/3
"""Full value of the constant in the `b` parameter"""
epsilon = 0.0
"""`epsilon` is always zero for the :obj:`RK` EOS"""
omega = None
"""`omega` has no impact on the :obj:`RK` EOS"""
Zc = 1.0/3.
"""Mechanical compressibility of :obj:`RK` EOS"""
c1R2 = c1*R2
c2R = c2*R
c1R2_c2R = c1R2/c2R
Psat_coeffs_limiting = [-72.700288369511583, -68.76714163049]
Psat_coeffs_critical = [1129250.3276866912, 4246321.053155941,
5988691.4873851035, 3754317.4112657467,
882716.2189281426]
Psat_cheb_coeffs = [-6.8488798834192215, -6.93992806360099, -0.11216113842675507, 0.0022494496508455135,
0.00995148012561513, -0.005789786392208277, 0.0021454644555051177, -0.0006192510387981658,
0.00016870584348326536, -5.828094356536212e-05, 2.5829410448955883e-05, -1.1312372380559225e-05,
4.374040785359406e-06, -1.5546789700246184e-06, 5.666723613325655e-07, -2.2701147218271074e-07,
9.561199996134724e-08, -3.934646467524511e-08, 1.55272396700466e-08, -6.061097474369418e-09,
2.4289648176102022e-09, -1.0031987621530753e-09, 4.168016003137324e-10, -1.7100917451312765e-10,
6.949731049432813e-11, -2.8377758503521713e-11, 1.1741734564892428e-11, -4.891469634936765e-12,
2.0373765879672795e-12, -8.507821454718095e-13, 3.4975627537410514e-13, -1.4468659018281038e-13,
6.536766028637786e-14, -2.7636123641275323e-14, 1.105377996166862e-14]
Psat_cheb_constant_factor = (0.8551757791729341, 9.962912449541513)
Psat_cheb_coeffs_der = chebder(Psat_cheb_coeffs)
Psat_coeffs_critical_der = polyder(Psat_coeffs_critical[::-1])[::-1]
phi_sat_coeffs = [156707085.9178746, 1313005585.0874271, 4947242291.244957,
11038959845.808495, 16153986262.1129, 16199294577.496677,
11273931409.81048, 5376831929.990161, 1681814895.2875218,
311544335.80653775, 25954329.68176187]
Psat_ranges_low = (0.033797068457719265, 0.06786604965443845, 0.1712297613585108, 0.34622987689428786, 0.7712336381743264, 1.745621379817678, 3.7256294306343207, 6.581228646986647, 12.781884795234102, 24.412307224840184, 48.39951213433041, 99.16043966361465, 206.52538850089107)
Psat_coeffs_low = [[-2.0473027805583304e+16, 5590407450630671.0, -691651500345804.2, 51280971870837.32, -2539543204646.707, 88630534161.11136, -2241691916.726171, 41616814.12629884, -568152.2176538995, 5661.177783316078, -40.73060128671707, 0.5116910477178825, -0.3837083115168163, 0.33323887045969375, -3.0536215774324824, 7.04644675941779e-13], [308999097192961.8, -225585388170583.75, 76474322668841.14, -15968075011367.047, 2296463426010.7324, -240935052543.90527, 19048380996.66752, -1155476440.462194, 54215636.4938359, -1967442.1357566533, 54762.443278119186, -1147.7177667787553, 17.16085684519316, 0.14875852741749432, -3.052428192332575, -3.5796888761263634e-06], [-19650136.181735344, 31781193.928728923, -23482166.19636667, 10469551.38856498, -3130203.8712914013, 658159.0021417511, -98797.59681465346, 10408.710407311624, -708.5507506253232, 20.511506773041855, 1.1894690300458857, 0.09491188992340026, -0.3699746434334233, 0.3327794679246402, -3.053612527869441, -7.854416195218761e-08], [27999.20768241787, -105403.37695226824, 184231.72791343703, -198316.18594155577, 147020.31646897402, -79504.54402254449, 32396.282402624805, -10128.009032305941, 2449.0150598681134, -457.86581824588563, 65.46517369332473, -6.79444639388651, 0.17688177142370798, 0.30286151881162965, -3.052607289294409, -1.571427065993891e-05], [0.03260876969558781, -0.2687907725537127, 1.0267017754905683, -2.4094567772527298, 3.8817887528347166, -4.539672024261187, 3.9650465848385767, -2.6039495096751812, 1.2459350220734762, -0.3527513106804435, -0.07989160047420704, 0.2668158546701413, -0.37571189865522364, 0.33238476075275014, -3.053560614808337, -2.0178433899342707e-06], [-7.375386804046186e-07, 1.5495867993995408e-05, -0.00015284972568903337, 0.0009416918743616606, -0.0040706288679345, 0.013170886849436726, -0.033323844522302436, 0.0682465756200053, -0.11659032663823216, 0.171245314502395, -0.22682939669727753, 0.29437853298508776, -0.3782483397093579, 0.33220274751099305, -3.053481229409229, -8.962708854642898e-06], [-9.64878085834743e-10, 4.3676512898669364e-08, -9.195173654927243e-07, 1.192726889996796e-05, -0.00010641091957706412, 0.0006901346742808183, -0.0033536294555478845, 0.012421045099127406, -0.03550565516993044, 0.07992816727224494, -0.14853460918439382, 0.24510479474467387, -0.35702531869477633, 0.32684341660780225, -3.053040405250035, 6.241135226403571e-05], [-3.5002452124187664e-13, 3.2833906636665076e-11, -1.430637752387875e-09, 3.84703168582193e-08, -7.149328354678022e-07, 9.737245185617094e-06, -0.0001004961143222046, 0.0008008011389293154, -0.004967550562579805, 0.023964447789498418, -0.08887464138262652, 0.24644247973322814, -0.4795516986170165, 0.5340043615968529, -3.218003440328873, 0.0551398944675352], [5.034255484339691e-17, -7.864397963490581e-15, 5.752225082446006e-13, -2.615885881066589e-11, 8.282570031686524e-10, -1.9374016535411608e-08, 3.4664702879779335e-07, -4.845857364906252e-06, 5.359278262994369e-05, -0.00047191509173119853, 0.003314508017194061, -0.018546171084714562, 0.08264133469800018, -0.2976511343203724, -2.45051660976546, -0.2780886842259136], [6.989268516115097e-21, -2.0569549793133175e-18, 2.829181210013469e-16, -2.4145085922204994e-14, 1.4314693471489465e-12, -6.253869824774858e-11, 2.0839676060467857e-09, -5.407898264555903e-08, 1.10600368057607e-06, -1.7926789523973723e-05, 0.00023042013161443632, -0.0023410716986771423, 0.018721566225139197, -0.11858986125950052, -2.7694948645429545, -0.005601354706335826], [4.123334769968695e-25, -2.3690486736737077e-22, 6.357338194540137e-20, -1.0578385702299804e-17, 1.2218888207640753e-15, -1.0392124936479462e-13, 6.735204427884249e-12, -3.395649622455689e-10, 1.3474573409568848e-08, -4.230554509480506e-07, 1.0508993131776807e-05, -0.000205653116778391, 0.0031500689366463458, -0.037812816408928154, -3.0367716832005502, 0.42028748728880316], [1.2637873346500257e-29, -1.4691821304898006e-26, 7.97371063405237e-24, -2.6821456640173466e-21, 6.259659753789673e-19, -1.0750710034717391e-16, 1.4061365634529063e-14, -1.4296683760114274e-12, 1.1431380830397977e-10, -7.224377488638715e-09, 3.607342375851496e-07, -1.4162123664055414e-05, 0.0004338140901526731, -0.010352095429488306, -3.214333239715912, 0.9750918877217316], [2.4467625180409936e-34, -5.900060775579966e-31, 6.640727340409195e-28, -4.631434540342247e-25, 2.240558311318526e-22, -7.974393923385324e-20, 2.1607792871336745e-17, -4.549744513625842e-15, 7.53070203647074e-13, -9.846740932533425e-11, 1.0165520378636594e-08, -8.242924637302648e-07, 5.2066757251344576e-05, -0.002554238879420431, -3.3164175313132174, 1.6226038152294109]]
# Thought the below was better - is failing. Need better ranges
# Psat_ranges_low = (0.002864867449609054, 0.00841672469500749, 0.016876032463065772, 0.0338307436429719, 0.06789926110832244, 0.17126106018604287, 0.346311983890616, 0.7714158394657995, 1.7457236753368228, 3.726128003826199, 6.581228646986647, 12.781884795234102, 24.412307224840184, 48.39951213433041, 99.16043966361465, 206.52538850089107)
# Psat_coeffs_low = [[7.4050168086686e+36, -2.3656234050215595e+35, 3.5140849350729625e+33, -3.219960263668302e+31, 2.0353831678446376e+29, -9.401903615685883e+26, 3.27870193930672e+24, -8.790141973297096e+21, 1.8267653379529515e+19, -2.9430079135183156e+16, 36458127001690.625, -34108081817.37184, 23328876.278645795, -11013.595381204012, 0.15611893268569021, -0.0004353146166747694], [-9.948446571395775e+25, 5.19510926774992e+24, -7.189006307350884e+22, -1.5680848799655576e+21, 8.249811116671015e+19, -1.6950901342526528e+18, 2.1845526238020284e+16, -197095048940187.34, 1299650648245.581, -6369933293.417471, 23232428.90696315, -62270.805040856954, 118.78599182230242, 0.17914187929689024, -3.053500975368179, -4.3120679728281264e-08], [9.397084120968658e+23, -1.7568386783383563e+23, 1.5270752841173062e+22, -8.18635952561484e+20, 3.0268596563705795e+19, -8.176367091459369e+17, 1.6669079239736548e+16, -261159410245086.28, 3170226264128.2866, -29816091359.74791, 215481434.94837964, -1175107.961691127, 4680.2781732486055, -12.521703234052518, -3.031856323244789, -1.7125428611007576e-05], [-5.837633410530431e+19, 2.2354665725528674e+19, -3.9748648336736783e+18, 4.353010078685106e+17, -3.2833626753213436e+16, 1806702598986387.0, -74919223763900.97, 2383883719613.177, -58681349911.8085, 1117424654.4682977, -16325312.825120728, 179698.66647387162, -1442.9221906440623, 8.305809571517658, -3.0807467744908252, 4.282801743092646e-05], [371247743335500.9, -296343269154610.4, 109587709190787.98, -24907918727301.76, 3891709522603.097, -442798937995.7918, 37904104192.54725, -2485814525.910429, 125929932.89424846, -4928093.141384071, 147763.74249085123, -3333.473113107637, 54.40288611988696, -0.28587642030919863, -3.04931950649037, -1.3857488372237547e-05], [16208032.322457436, -28812339.698805887, 23769768.592785712, -12069159.698844183, 4216830.274106061, -1073538.4841311441, 205658.8661178185, -30177.62398799642, 3418.24878855674, -298.57690432008536, 19.71936439114255, -0.6937801749294776, -0.34639354927988253, 0.3323199955834364, -3.053607492200311, -9.988880111944098e-08], [-27604.735354758857, 105025.68737778495, -185570.40811520678, 201981.46688618566, -151443.86118322334, 82852.55333788031, -34164.546515476206, 10812.149811177835, -2647.7278495315263, 501.83786547752425, -73.19000295074032, 8.299579502519556, -1.0215340085992137, 0.3683751980772054, -3.0548122319685604, 1.8727318915723323e-05], [0.005501339854516574, -0.030871868660992247, 0.06262306950800559, -0.016054714694926787, -0.19049678051599286, 0.4916342706149181, -0.6988226755673006, 0.6995056451067202, -0.5569157797560214, 0.40541813786380926, -0.32359936610250595, 0.32562413248075583, -0.38602496065455755, 0.33362571128345786, -3.0536522381393123, 1.1116248239684268e-06], [-7.625288226006888e-07, 1.592958164613657e-05, -0.00015633640254263914, 0.0009589163584223709, -0.004129113991499528, 0.01331550052681755, -0.033592935849933975, 0.06863043023079776, -0.11701377319077316, 0.17160678539192847, -0.22706640260572322, 0.2944958500921784, -0.3782908182523437, 0.3322133797180262, -3.0534828760507775, -8.843634685451462e-06], [-9.602485478694793e-10, 4.349693466931302e-08, -9.162924468728032e-07, 1.1891710154307035e-05, -0.00010614176529656917, 0.0006886536353876484, -0.003347511076288892, 0.012401728251598975, -0.03545868060099854, 0.07984021871367283, -0.1484089324448234, 0.24497026761102456, -0.35692097886460605, 0.32678811005407216, -3.0530225111683147, 5.975140457969985e-05], [-3.271879015155258e-13, 3.106593030234596e-11, -1.3669954450963802e-09, 3.7057309629853086e-08, -6.932939676452128e-07, 9.49513835382941e-06, -9.845166709551053e-05, 0.000787533270198052, -0.0049008339165666475, 0.023704510936127153, -0.08809636022958753, 0.24468388333303692, -0.4766488164995153, 0.5306997327949121, -3.2156835292074972, 0.05438278235662608], [5.027674758216368e-17, -7.856316772563329e-15, 5.74771486417108e-13, -2.6143808092714456e-11, 8.279258549554745e-10, -1.9369063289334646e-08, 3.4659817641020265e-07, -4.8455984691411336e-06, 5.3593276238540865e-05, -0.0004719384663395697, 0.003314740905347161, -0.018547549367299895, 0.08264669293617401, -0.29766465209956755, -2.450496401425025, -0.2781023348227336], [7.050221439220754e-21, -2.0731943309577094e-18, 2.84928020400626e-16, -2.429837955412963e-14, 1.4395267045494148e-12, -6.284786016386046e-11, 2.092913781550848e-09, -5.427778893624791e-08, 1.1094245992741828e-06, -1.7972372175390485e-05, 0.0002308866520395049, -0.002344673517182507, 0.018741874333835694, -0.11866881228864186, -2.7693056026459097, -0.00581227798597439], [4.1192237682040195e-25, -2.366064116476803e-22, 6.347935162866163e-20, -1.0560893898667048e-17, 1.2197114527874372e-15, -1.037276215071595e-13, 6.722432950732127e-12, -3.389265643942945e-10, 1.3450132487487853e-08, -4.2233752380466274e-07, 1.0492923968957467e-05, -0.00020538369710127645, 0.003146790904310898, -0.0377854748223825, -3.036911550333188, 0.4206184380126672], [1.2604142367817428e-29, -1.4652181650108445e-26, 7.952241970757233e-24, -2.6750298800006056e-21, 6.243504642033308e-19, -1.0724081450429274e-16, 1.4028428417591862e-14, -1.4265539327144317e-12, 1.1408674781257439e-10, -7.211610181866389e-09, 3.6018494473345496e-07, -1.4144362093631258e-05, 0.0004333961723540267, -0.010345339188037806, -3.2144003543293014, 0.9754007550013739], [2.4330594298876834e-34, -5.870171131499116e-31, 6.610469013522427e-28, -4.6125772590980585e-25, 2.232467903153395e-22, -7.949083788375843e-20, 2.1548151171435654e-17, -4.538965432777591e-15, 7.515638568810398e-13, -9.83046463909207e-11, 1.0152034186437456e-08, -8.234510061656029e-07, 5.202848938898853e-05, -0.002553041403736697, -3.316440584285066, 1.6228096260313123]]
def __init__(self, Tc, Pc, T=None, P=None, V=None, omega=None):
self.Tc = Tc
self.Pc = Pc
self.T = T
self.P = P
self.V = V
self.omega = omega
self.a = self.c1R2*Tc*Tc/Pc
self.b = self.delta = self.c2R*Tc/Pc
self.solve()
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives for this EOS. Uses the set values of `a`.
.. math::
a\alpha = \frac{a}{\sqrt{\frac{T}{T_{c}}}}
.. math::
\frac{d a\alpha}{dT} = - \frac{a}{2 T\sqrt{\frac{T}{T_{c}}}}
.. math::
\frac{d^2 a\alpha}{dT^2} = \frac{3 a}{4 T^{2}\sqrt{\frac{T}{T_{c}}}}
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
'''
Tc = self.Tc
sqrt_Tr_inv = (T/Tc)**-0.5
a_alpha = self.a*sqrt_Tr_inv
T_inv = 1.0/T
da_alpha_dT = -0.5*self.a*T_inv*sqrt_Tr_inv
d2a_alpha_dT2 = 0.75*self.a*T_inv*T_inv*sqrt_Tr_inv
return a_alpha, da_alpha_dT, d2a_alpha_dT2
[docs] def a_alpha_pure(self, T):
r'''Method to calculate :math:`a \alpha` for this EOS. Uses the set
values of `a`.
.. math::
a\alpha = \frac{a}{\sqrt{\frac{T}{T_{c}}}}
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
'''
Tc = self.Tc
sqrt_Tr_inv = sqrt(Tc/T)
return self.a*sqrt_Tr_inv
[docs] def solve_T(self, P, V, solution=None):
r'''Method to calculate `T` from a specified `P` and `V` for the RK
EOS. Uses `a`, and `b`, obtained from the class's namespace.
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
'l' or 'g' to specify a liquid of vapor solution (if one exists);
if None, will select a solution more likely to be real (closer to
STP, attempting to avoid temperatures like 60000 K or 0.0001 K).
Returns
-------
T : float
Temperature, [K]
Notes
-----
The exact solution can be derived as follows; it is excluded for
breviety.
>>> from sympy import * # doctest:+SKIP
>>> P, T, V, R = symbols('P, T, V, R') # doctest:+SKIP
>>> Tc, Pc = symbols('Tc, Pc') # doctest:+SKIP
>>> a, b = symbols('a, b') # doctest:+SKIP
>>> RK = Eq(P, R*T/(V-b) - a/sqrt(T)/(V*V + b*V)) # doctest:+SKIP
>>> solve(RK, T) # doctest:+SKIP
'''
a, b = self.a, self.b
a = a*self.Tc**0.5
# print([R, V, b, P, a])
if solution is None:
# COnfirmed with mpmath - has numerical issues
x0 = 3**0.5
x1 = 1j*x0
x2 = x1 + 1.0
x3 = V + b
x4 = V - b
x5 = x4/R
x6 = (x0*(x4**2*(-4*P**3*x5 + 27*a**2/(V**2*x3**2))/R**2+0.0j)**0.5 - 9*a*x5/(V*x3))**0.333333333333333
x7 = 0.190785707092222*x6
x8 = P*x5/x6
x9 =1.7471609294726*x8
x10 = 1.0 - x1
slns = [(x2*x7 + x9/x2)**2,
(x10*x7 + x9/x10)**2,
(0.381571414184444*x6 + 0.873580464736299*x8)**2]
try:
self.no_T_spec = True
x1 = -1.j*1.7320508075688772 + 1.
x2 = V - b
x3 = x2/R
x4 = V + b
x5 = (1.7320508075688772*(x2*x2*(-4.*P*P*P*x3 + 27.*a*a/(V*V*x4*x4))/(R*R))**0.5 - 9.*a*x3/(V*x4) +0j)**(1./3.)
T_sln = (3.3019272488946263*(11.537996562459266*P*x3/(x1*x5) + 1.2599210498948732*x1*x5)**2/144.0).real
if T_sln > 1e-3:
return T_sln
except:
pass
# Turns out the above solution does not cover all cases
return super().solve_T(P, V, solution=solution)
[docs] def T_discriminant_zeros_analytical(self, valid=False):
r'''Method to calculate the temperatures which zero the discriminant
function of the `RK` eos. This is an analytical function with an
11-coefficient polynomial which is solved with `numpy`.
Parameters
----------
valid : bool
Whether to filter the calculated temperatures so that they are all
real, and positive only, [-]
Returns
-------
T_discriminant_zeros : float
Temperatures which make the discriminant zero, [K]
Notes
-----
Calculated analytically. Derived as follows. Has multiple solutions.
>>> from sympy import * # doctest:+SKIP
>>> P, T, V, R, b, a, Troot = symbols('P, T, V, R, b, a, Troot') # doctest:+SKIP
>>> a_alpha = a/sqrt(T) # doctest:+SKIP
>>> delta, epsilon = b, 0 # doctest:+SKIP
>>> eta = b # doctest:+SKIP
>>> B = b*P/(R*T) # doctest:+SKIP
>>> deltas = delta*P/(R*T) # doctest:+SKIP
>>> thetas = a_alpha*P/(R*T)**2 # doctest:+SKIP
>>> epsilons = epsilon*(P/(R*T))**2 # doctest:+SKIP
>>> etas = eta*P/(R*T) # doctest:+SKIP
>>> a_coeff = 1 # doctest:+SKIP
>>> b_coeff = (deltas - B - 1) # doctest:+SKIP
>>> c = (thetas + epsilons - deltas*(B+1)) # doctest:+SKIP
>>> d = -(epsilons*(B+1) + thetas*etas) # doctest:+SKIP
>>> disc = b_coeff*b_coeff*c*c - 4*a_coeff*c*c*c - 4*b_coeff*b_coeff*b_coeff*d - 27*a_coeff*a_coeff*d*d + 18*a_coeff*b_coeff*c*d # doctest:+SKIP
>>> new_disc = disc.subs(sqrt(T), Troot) # doctest:+SKIP
>>> new_T_base = expand(expand(new_disc)*Troot**15) # doctest:+SKIP
>>> ans = collect(new_T_base, Troot).args # doctest:+SKIP
'''
P, a, b, epsilon, delta = self.P, self.a, self.b, self.epsilon, self.delta
a *= self.Tc**0.5 # pre-dates change in alpha definition
P2 = P*P
P3 = P2*P
P4 = P2*P2
R_inv4 = R_inv2*R_inv2
R_inv5 = R_inv4*R_inv
R_inv6 = R_inv4*R_inv2
b2 = b*b
b3 = b2*b
b4 = b2*b2
a2 = a*a
x5 = 15.0*a2
x8 = 2.0*P3
x9 = b4*b*R_inv
x13 = 6.0*R_inv*R_inv2
coeffs = [P2*b2*R_inv2,
0.0,
P3*b3*x13,
-a*b*P2*x13,
13.0*R_inv4*P4*b4,
-32.0*a*P3*R_inv4*b2,
P2*R_inv4*(12.0*P3*x9 + a2),
-42.0*a*b3*P4*R_inv5,
b*R_inv5*x8*(x5 + x8*x9),
-12.0*P2*P3*a*R_inv6*b4,
-R_inv6*b2*P4*x5,
-4.0*a2*a*P3*R_inv6]
roots = np.roots(coeffs).tolist()
roots = [i*i for i in roots]
if valid:
# TODO - only include ones when switching phases from l/g to either g/l
# Do not know how to handle
roots = [r.real for r in roots if (r.real >= 0.0 and (abs(r.imag) <= 1e-12))]
roots.sort()
return roots
[docs]class SRK(GCEOS):
r'''Class for solving the Soave-Redlich-Kwong [1]_ [2]_ [3]_ cubic
equation of state for a pure compound. Subclasses :obj:`GCEOS`, which
provides the methods for solving the EOS and calculating its assorted
relevant thermodynamic properties. Solves the EOS on initialization.
Two of `T`, `P`, and `V` are needed to solve the EOS.
.. math::
P = \frac{RT}{V-b} - \frac{a\alpha(T)}{V(V+b)}
.. math::
a=\left(\frac{R^2(T_c)^{2}}{9(\sqrt[3]{2}-1)P_c} \right)
=\frac{0.42748\cdot R^2(T_c)^{2}}{P_c}
.. math::
b=\left( \frac{(\sqrt[3]{2}-1)}{3}\right)\frac{RT_c}{P_c}
=\frac{0.08664\cdot R T_c}{P_c}
.. math::
\alpha(T) = \left[1 + m\left(1 - \sqrt{\frac{T}{T_c}}\right)\right]^2
.. math::
m = 0.480 + 1.574\omega - 0.176\omega^2
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
>>> eos = SRK(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.000146821077354, -31754.663859, -74.373272044)
References
----------
.. [1] Soave, Giorgio. "Equilibrium Constants from a Modified Redlich-Kwong
Equation of State." Chemical Engineering Science 27, no. 6 (June 1972):
1197-1203. doi:10.1016/0009-2509(72)80096-4.
.. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th
edition. New York: McGraw-Hill Professional, 2000.
.. [3] Walas, Stanley M. Phase Equilibria in Chemical Engineering.
Butterworth-Heinemann, 1985.
'''
c1 = 0.4274802335403414043909906940611707345513 # 1/(9*(2**(1/3.)-1))
"""Full value of the constant in the `a` parameter"""
c2 = 0.08664034996495772158907020242607611685675 # (2**(1/3.)-1)/3
"""Full value of the constant in the `b` parameter"""
c1R2 = c1*R2
c2R = c2*R
c1R2_c2R = c1R2/c2R
epsilon = 0.0
"""`epsilon` is always zero for the :obj:`SRK` EOS"""
Zc = 1/3.
"""Mechanical compressibility of :obj:`SRK` EOS"""
Psat_coeffs_limiting = [-3.2308843103522107, 0.7210534170705403]
Psat_coeffs_critical = [9.374273428735918, -6.15924292062784,
4.995561268009732, -3.0536215892966374,
1.0000000000023588]
Psat_cheb_coeffs = [-7.871741490227961, -7.989748461289071, -0.1356344797770207, 0.009506579247579184,
0.009624489219138763, -0.007066708482598217, 0.003075503887853841, -0.001012177935988426,
0.00028619693856193646, -8.960150789432905e-05, 3.8678642545223406e-05, -1.903594210476056e-05,
8.531492278109217e-06, -3.345456890803595e-06, 1.2311165149343946e-06, -4.784033464026011e-07,
2.0716513992539553e-07, -9.365210448247373e-08, 4.088078067054522e-08, -1.6950725229317957e-08,
6.9147476960875615e-09, -2.9036036947212296e-09, 1.2683728020787197e-09, -5.610046772833513e-10,
2.444858416194781e-10, -1.0465240317131946e-10, 4.472305869824417e-11, -1.9380782026977295e-11,
8.525075935982007e-12, -3.770209730351304e-12, 1.6512636527230007e-12, -7.22057288092548e-13,
3.2921267708457824e-13, -1.616661448808343e-13, 6.227456701354828e-14]
Psat_cheb_constant_factor = (-2.5857326352412238, 0.38702722494279784)
Psat_cheb_coeffs_der = chebder(Psat_cheb_coeffs)
Psat_coeffs_critical_der = polyder(Psat_coeffs_critical[::-1])[::-1]
phi_sat_coeffs = [4.883976406433718e-10, -2.00532968010467e-08, 3.647765457046907e-07,
-3.794073186960753e-06, 2.358762477641146e-05, -7.18419726211543e-05,
-0.00013493130050539593, 0.002716443506003684, -0.015404883730347763,
0.05251643616017714, -0.11346125895127993, 0.12885073074459652,
0.0403144920149403, -0.39801902918654086, 0.5962308106352003,
0.6656153310272716]
_P_zero_l_cheb_coeffs = [0.08380676900731782, -0.14019219743961803, 0.11742103327156811, -0.09849160801348428, 0.08273868596563422, -0.0696144897386927, 0.05866765693877264, -0.04952599518184439, 0.04188237509387957, -0.03548315864697149, 0.03011872010893725, -0.02561566850666065, 0.021830462208254395, -0.018644172238802145, 0.015958169671823057, -0.013690592984703707, 0.011773427351986342, -0.01015011684404267, 0.00877358083868034, -0.007604596012758029, 0.006610446573984768, -0.005763823407070205, 0.005041905306198975, -0.004425605918781876, 0.003898948480582476, -0.003448548272342346, 0.0030631866753461218, -0.002733454718851159, 0.0024514621141303247, -0.0022105921907339815, 0.002005302198095145, -0.0018309561248985515, 0.0016836870172771135, -0.0015602844635190134, 0.0014581002673540663, -0.0013749738886825284, 0.0013091699610779176, -0.001259330218826276, 0.001224435336044407, -0.0012037764696538264, 0.0005984681105455358]
P_zero_l_cheb_limits = (0.0009838646849082977, 77.36362033836788)
_P_zero_g_cheb_coeffs = [4074.379698522392, 4074.0787931079158, -0.011974050537509407, 0.011278738948946121, -0.010623695898806596, 0.010006612855718989, -0.00942531345107397, 0.008877745971729046, -0.008361976307962505, 0.007876181274528127, -0.007418642356788098, 0.006987739799033855, -0.006581946966943887, 0.006199825106351055, -0.00584001837117817, 0.005501249138059018, -0.0051823135959307666, 0.004882077534036101, -0.004599472449233056, 0.004333491845900562, -0.004083187738391304, 0.00384766734038441, -0.0036260899632846967, 0.00341766412351482, -0.003221644783037071, 0.003037330724301647, -0.0028640621256170408, 0.0027012182634600047, -0.0025482153614670667, 0.00240450452795168, -0.0022695698397005816, 0.002142926545674241, -0.0020241193744505405, 0.0019127209575542047, -0.0018083302956923518, 0.0017105713491642703, -0.0016190917803071369, 0.0015335616642137794, -0.0014536723452853405, 0.001379135339081262, -0.0013096813358352787, 0.001245059275896766, -0.0011850354079059, 0.001129392510023498, -0.0010779290997626433, 0.0010304587604847658, -0.0009868094600730913, 0.0009468229117978965, -0.0009103540735282088, 0.0008772706097128445, -0.0008474524304184726, 0.0008207912556403528, -0.0007971902270068286, 0.000776563594266667, -0.0007588363976502542, 0.0007439441894165576, -0.0007318328255327643, 0.0007224582401159317, -0.0007157863644244543, 0.0007117929301416425, -0.0003552316997513632]
P_zero_g_cheb_limits = (-0.9648141211597231, 34.80547339996925)
# Nov 2019
# Psat_ranges_low = (0.016623281310365744, 0.1712825877822172, 0.8775228637034642, 2.4185778704993384, 4.999300695376596, 10.621733701210832, 21.924686089216046, 46.23658652939059, 111.97303237634476)
# Psat_coeffs_low = [[3.3680989254730784e+17, -4.074818435768317e+16, 2209815483748018.2, -70915386325767.22, 1497265032843.7883, -21873765985.033226, 226390057.35154417, -1671116.2651395416, 8737.327630885395, -31.38783252903762, -0.3062576872041857, 0.33312130842499577, -3.053621478895965, -3.31001545617049e-11], [-1277.9939354449687, 1593.2536430725866, -891.7360022419283, 295.8342513857935, -64.78832619622327, 9.999684056700664, -1.2867875988497843, 0.32779561506053606, -0.2700702867816281, 0.3102313474312917, -0.38304293907646136, 0.3332375577689779, -3.0536215764869326, 2.360556194958008e-12], [-0.000830505972635258, 0.006865390327553869, -0.026817234829898506, 0.0665672622542815, -0.119964606281312, 0.17169598695361063, -0.2106764625423519, 0.23752105248153738, -0.2630589319705226, 0.3095504696125893, -0.3829670684832488, 0.3332307853291866, -3.053621198743048, -9.605050754757372e-09], [-3.9351433749518387e-07, 9.598894454703375e-06, -0.00010967371180484353, 0.0007796748366203162, -0.0038566383026144148, 0.014042499802344673, -0.03890674173205208, 0.08460429046640522, -0.15233989442440943, 0.24596202389042182, -0.3552486542779753, 0.32467374082763434, -3.0519642694194045, -0.00015063394168279842], [9.630665339915374e-10, -4.7371315246695036e-08, 1.058979499331494e-06, -1.4148499852908107e-05, 0.00012453404851616215, -0.000744874809594022, 0.00295128240256063, -0.006592268033281193, -0.00018461593083082123, 0.055334589912369295, -0.18248894355952128, 0.21859711935534465, -3.0129799097945456, -0.006486114455004355], [-2.710468940879406e-14, 2.361990546087112e-12, -8.567303244166706e-11, 1.4893663407003366e-09, -3.858548795875803e-09, -4.2380960424066427e-07, 1.1242926127857602e-05, -0.0001632710637823122, 0.001612992494042694, -0.011564861884906304, 0.06197160418123528, -0.25590435995049254, -2.502008242669764, -0.2488201810754127], [1.6083147737513614e-17, -3.625948333600919e-15, 3.779796571460543e-13, -2.4156008540207418e-11, 1.0579233657428334e-09, -3.361617809293566e-08, 8.003083689291334e-07, -1.4534087164009375e-05, 0.0002031152121569259, -0.002189767259610491, 0.018210271943770506, -0.11793369058835379, -2.7679151499088324, -0.010786625844602327], [1.4110844119182223e-21, -6.673466228406512e-19, 1.458414381897635e-16, -1.9526180721541405e-14, 1.790075947985849e-12, -1.1894965358505194e-10, 5.91471378248656e-09, -2.2398516390225003e-07, 6.512345505221323e-06, -0.0001455617494641103, 0.002494987494838556, -0.03292429235639192, -3.0591018122950038, 0.4673525314587721], [2.1123051961710074e-25, -2.1083091388936946e-22, 9.662063972407386e-20, -2.693978918168614e-17, 5.1041762501040065e-15, -6.950692277142413e-13, 7.0161397235176e-11, -5.335818543025505e-09, 3.076755677498343e-07, -1.3436008106355354e-05, 0.00044163897730386165, -0.010903751691783911, -3.2044489982966082, 0.9087101274749898]]
# Dec 2019
Psat_ranges_low = (0.016623281310365744, 0.1712825877822172, 0.8775228637034642, 2.4185778704993384, 5.001795116965993, 9.695206781787403, 21.057832476172877, 46.27931918489475, 106.60008206151481, 206.46094535380982)
Psat_coeffs_low = [[-3.021742864809473e+22, 4.214239383836392e+21, -2.6741136978422124e+20, 1.0217841941834105e+19, -2.622411357075726e+17, 4774407979621216.0, -63484876960438.69, 625375442527.9032, -4581025644.8943615, 24826982.029340215, -98159.10795313062, 276.50340512054163, -0.9143654631342611, 0.3338916360475657, -3.0536220324119654, 1.3475048507571863e-10], [1851732.9501797194, -2605860.7399044684, 1659613.6490526376, -632650.6574176748, 160893.98711246205, -28808.208934565508, 3736.231492867314, -355.6971459537775, 24.760714782747538, -1.0423063013028406, -0.21733901552867047, 0.3087713311546577, -0.3830146794101142, 0.3332371947330795, -3.05362157370627, -7.227607401461e-12], [-0.0004521121892877062, 0.004090442697504045, -0.0175578643282661, 0.04794251207356016, -0.09461038038864959, 0.14623339766010854, -0.18881543977182574, 0.216237584862916, -0.23240996967465383, 0.2455135904651031, -0.2652544858737314, 0.30999258403096164, -0.383030205401439, 0.33323681959713436, -3.053621543830392, -7.017832981404126e-10], [-7.808701919427604e-08, 2.077661560061249e-06, -2.5817479151146433e-05, 0.00019946831868046186, -0.0010777009706245766, 0.004349178573005843, -0.01369170969849733, 0.03466227625915358, -0.07207265601431376, 0.12553031872708703, -0.19076746372442283, 0.2729239080553058, -0.36893226041645655, 0.32941620336187777, -3.0529679799065943, -5.2835266906470224e-05], [1.4671478312986887e-11, -1.0110442264467632e-09, 3.1974970384785367e-08, -6.163209773850742e-07, 8.094505145684214e-06, -7.656745582686593e-05, 0.0005363374762358416, -0.002806916403123579, 0.010866043155195763, -0.029908987582571975, 0.052158093336682546, -0.032663621693629505, -0.0751716186255902, 0.12892284191276268, -2.967051324985992, -0.017359415309005755], [-1.091394193997453e-14, 1.2357554812857725e-12, -6.508815052182767e-11, 2.1148805902814486e-09, -4.738775276800668e-08, 7.750511363006775e-07, -9.547217015303917e-06, 9.00036272090859e-05, -0.0006521700537524791, 0.0036045130640996606, -0.014814913109019256, 0.04242862345426275, -0.06750190169757553, -0.04208608128453949, -2.7194330511309253, -0.14620471607422303], [1.2884100931052691e-19, -3.1352234465014476e-17, 3.5604267936540494e-15, -2.505139585733175e-13, 1.222651372599914e-11, -4.3907452596568276e-10, 1.200890247630985e-08, -2.5540606353043675e-07, 4.275108728223701e-06, -5.664187536164573e-05, 0.0005945229499256919, -0.004930179620565587, 0.03219833012293544, -0.16707167951374932, -2.661698607705941, -0.11728474036871717], [1.4718778168288712e-24, -7.827527144097033e-22, 1.9419598473231654e-19, -2.9838110847734e-17, 3.17855768668334e-15, -2.4899868244476545e-13, 1.4844855897901513e-11, -6.8756271353964e-10, 2.503217441471856e-08, -7.201291894218883e-07, 1.637034511468395e-05, -0.0002928284260408074, 0.004096142697714258, -0.0448856246444143, -3.0042011777749145, 0.3506385318223977], [8.493893312557766e-30, -1.0255827364680145e-26, 5.772959502607649e-24, -2.0110473459289428e-21, 4.853221287871829e-19, -8.60543209118676e-17, 1.1601546784661254e-14, -1.2138195783117383e-12, 9.970282232925484e-11, -6.461623477876083e-09, 3.3028403185783974e-07, -1.3249473054302956e-05, 0.00041394079374352697, -0.010055381746092074, -3.217048228957832, 0.986564340774919], [5.909724328094521e-34, -1.373533672302407e-30, 1.4873473750776103e-27, -9.960045776404399e-25, 4.616398303867023e-22, -1.5703667768168258e-19, 4.056122702342784e-17, -8.116898226364067e-15, 1.2725759075121173e-12, -1.5701193776679807e-10, 1.5228962066971988e-08, -1.1543563076442494e-06, 6.776387262920499e-05, -0.0030684174426771718, -3.30604432857107, 1.5254388255202684]]
def __init__(self, Tc, Pc, omega, T=None, P=None, V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
self.a = self.c1*R*R*Tc*Tc/Pc
self.b = self.c2*R*Tc/Pc
self.m = 0.480 + 1.574*omega - 0.176*omega*omega
self.delta = self.b
self.solve()
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives for this EOS. Uses the set values of `Tc`, `m`, and `a`.
.. math::
a\alpha = a \left(m \left(- \sqrt{\frac{T}{T_{c}}} + 1\right)
+ 1\right)^{2}
.. math::
\frac{d a\alpha}{dT} = \frac{a m}{T} \sqrt{\frac{T}{T_{c}}} \left(m
\left(\sqrt{\frac{T}{T_{c}}} - 1\right) - 1\right)
.. math::
\frac{d^2 a\alpha}{dT^2} = \frac{a m \sqrt{\frac{T}{T_{c}}}}{2 T^{2}}
\left(m + 1\right)
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
da_alpha_dT : float
Temperature derivative of coefficient calculated by EOS-specific
method, [J^2/mol^2/Pa/K]
d2a_alpha_dT2 : float
Second temperature derivative of coefficient calculated by
EOS-specific method, [J^2/mol^2/Pa/K^2]
'''
a, Tc, m = self.a, self.Tc, self.m
sqTr = (T/Tc)**0.5
a_alpha = a*(m*(1. - sqTr) + 1.)**2
da_alpha_dT = -a*m*sqTr*(m*(-sqTr + 1.) + 1.)/T
d2a_alpha_dT2 = a*m*sqTr*(m + 1.)/(2.*T*T)
return a_alpha, da_alpha_dT, d2a_alpha_dT2
[docs] def a_alpha_pure(self, T):
r'''Method to calculate :math:`a \alpha` for this EOS. Uses the set
values of `Tc`, `m`, and `a`.
.. math::
a\alpha = a \left(m \left(- \sqrt{\frac{T}{T_{c}}} + 1\right)
+ 1\right)^{2}
Parameters
----------
T : float
Temperature at which to calculate the values, [-]
Returns
-------
a_alpha : float
Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
'''
a, Tc, m = self.a, self.Tc, self.m
sqTr = sqrt(T/Tc)
x0 = (m*(1. - sqTr) + 1.)
return a*x0*x0
[docs] def P_max_at_V(self, V):
r'''Method to calculate the maximum pressure the EOS can create at a
constant volume, if one exists; returns None otherwise.
Parameters
----------
V : float
Constant molar volume, [m^3/mol]
Returns
-------
P : float
Maximum possible isochoric pressure, [Pa]
Notes
-----
The analytical determination of this formula involved some part of the
discriminant, and much black magic.
Examples
--------
>>> e = SRK(P=1e5, V=0.0001437, Tc=512.5, Pc=8084000.0, omega=0.559)
>>> e.P_max_at_V(e.V)
490523786.2
'''
"""
from sympy import *
# Solve for when T equal
P, T, V, R, a, b, m = symbols('P, T, V, R, a, b, m')
Tc, Pc, omega = symbols('Tc, Pc, omega')
# from the T solution, get the square root part, find when it hits zero
# to_zero = sqrt(Tc**2*V*a**2*m**2*(V - b)**3*(V + b)*(m + 1)**2*(P*R*Tc*V**2 + P*R*Tc*V*b - P*V*a*m**2 + P*a*b*m**2 + R*Tc*a*m**2 + 2*R*Tc*a*m + R*Tc*a))
lhs = P*R*Tc*V**2 + P*R*Tc*V*b - P*V*a*m**2 + P*a*b*m**2
rhs = R*Tc*a*m**2 + 2*R*Tc*a*m + R*Tc*a
hit = solve(Eq(lhs, rhs), P)
"""
# grows unbounded for all mixture EOS?
try:
Tc, a, m, b = self.Tc, self.a, self.m, self.b
except:
Tc, a, m, b = self.Tcs[0], self.ais[0], self.ms[0], self.bs[0]
P_max = -R*Tc*a*(m**2 + 2*m + 1)/(R*Tc*V**2 + R*Tc*V*b - V*a*m**2 + a*b*m**2)
if P_max < 0.0:
return None
return P_max
[docs] def solve_T(self, P, V, solution=None):
r'''Method to calculate `T` from a specified `P` and `V` for the SRK
EOS. Uses `a`, `b`, and `Tc` obtained from the class's namespace.
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
'l' or 'g' to specify a liquid of vapor solution (if one exists);
if None, will select a solution more likely to be real (closer to
STP, attempting to avoid temperatures like 60000 K or 0.0001 K).
Returns
-------
T : float
Temperature, [K]
Notes
-----
The exact solution can be derived as follows; it is excluded for
breviety.
>>> from sympy import * # doctest:+SKIP
>>> P, T, V, R, a, b, m = symbols('P, T, V, R, a, b, m') # doctest:+SKIP
>>> Tc, Pc, omega = symbols('Tc, Pc, omega') # doctest:+SKIP
>>> a_alpha = a*(1 + m*(1-sqrt(T/Tc)))**2 # doctest:+SKIP
>>> SRK = R*T/(V-b) - a_alpha/(V*(V+b)) - P # doctest:+SKIP
>>> solve(SRK, T) # doctest:+SKIP
'''
# Takes like half an hour to be derived, saved here for convenience
# ([(Tc*(V - b)*(R**2*Tc**2*V**4 + 2*R**2*Tc**2*V**3*b + R**2*Tc**2*V**2*b**2
# - 2*R*Tc*V**3*a*m**2 + 2*R*Tc*V*a*b**2*m**2 + V**2*a**2*m**4 - 2*V*a**2*b*m**4
# + a**2*b**2*m**4)*(P*R*Tc*V**4 + 2*P*R*Tc*V**3*b + P*R*Tc*V**2*b**2
# - P*V**3*a*m**2 + P*V*a*b**2*m**2 + R*Tc*V**2*a*m**2
# + 2*R*Tc*V**2*a*m + R*Tc*V**2*a + R*Tc*V*a*b*m**2
# + 2*R*Tc*V*a*b*m + R*Tc*V*a*b + V*a**2*m**4 + 2*V*a**2*m**3
# + V*a**2*m**2 - a**2*b*m**4 - 2*a**2*b*m**3 - a**2*b*m**2)
# - 2*sqrt(Tc**2*V*a**2*m**2*(V - b)**3*(V + b)*(m + 1)**2*(P*R*Tc*V**2
# + P*R*Tc*V*b - P*V*a*m**2 + P*a*b*m**2 + R*Tc*a*m**2 + 2*R*Tc*a*m + R*Tc*a))*(R*Tc*V**2 + R*Tc*V*b - V*a*m**2 + a*b*m**2)**2)/((R*Tc*V**2 + R*Tc*V*b - V*a*m**2 + a*b*m**2)**2*(R**2*Tc**2*V**4 + 2*R**2*Tc**2*V**3*b + R**2*Tc**2*V**2*b**2 - 2*R*Tc*V**3*a*m**2 + 2*R*Tc*V*a*b**2*m**2 + V**2*a**2*m**4 - 2*V*a**2*b*m**4 + a**2*b**2*m**4)),
# (Tc*(V - b)*(R**2*Tc**2*V**4 + 2*R**2*Tc**2*V**3*b + R**2*Tc**2*V**2*b**2
# - 2*R*Tc*V**3*a*m**2 + 2*R*Tc*V*a*b**2*m**2 + V**2*a**2*m**4 - 2*V*a**2*b*m**4
# + a**2*b**2*m**4)*(P*R*Tc*V**4 + 2*P*R*Tc*V**3*b + P*R*Tc*V**2*b**2
# - P*V**3*a*m**2 + P*V*a*b**2*m**2 + R*Tc*V**2*a*m**2
# + 2*R*Tc*V**2*a*m + R*Tc*V**2*a + R*Tc*V*a*b*m**2
# + 2*R*Tc*V*a*b*m + R*Tc*V*a*b + V*a**2*m**4 + 2*V*a**2*m**3
# + V*a**2*m**2 - a**2*b*m**4 - 2*a**2*b*m**3 - a**2*b*m**2)
# + 2*sqrt(Tc**2*V*a**2*m**2*(V - b)**3*(V + b)*(m + 1)**2*(P*R*Tc*V**2
# + P*R*Tc*V*b - P*V*a*m**2 + P*a*b*m**2 + R*Tc*a*m**2
# + 2*R*Tc*a*m + R*Tc*a))*(R*Tc*V**2 + R*Tc*V*b - V*a*m**2
# + a*b*m**2)**2)/((R*Tc*V**2 + R*Tc*V*b - V*a*m**2 + a*b*m**2
# )**2*(R**2*Tc**2*V**4 + 2*R**2*Tc**2*V**3*b
# + R**2*Tc**2*V**2*b**2 - 2*R*Tc*V**3*a*m**2
# + 2*R*Tc*V*a*b**2*m**2 + V**2*a**2*m**4
# - 2*V*a**2*b*m**4 + a**2*b**2*m**4))])
self.no_T_spec = True
a, b, Tc, m = self.a, self.b, self.Tc, self.m
if True:
x0 = R*Tc
x1 = V*b
x2 = x0*x1
x3 = V*V
x4 = x0*x3
x5 = m*m
x6 = a*x5
x7 = b*x6
x8 = V*x6
x9 = (x2 + x4 + x7 - x8)**2
x10 = x3*x3
x11 = R*R*Tc*Tc
x12 = a*a
x13 = x5*x5
x14 = x12*x13
x15 = b*b
x16 = x3*V
x17 = a*x0
x18 = x17*x5
x19 = 2.*b*x16
x20 = -2.*V*b*x14 + 2.*V*x15*x18 + x10*x11 + x11*x15*x3 + x11*x19 + x14*x15 + x14*x3 - 2*x16*x18
x21 = V - b
x22 = 2*m*x17
x23 = P*x4
x24 = P*x8
x25 = x1*x17
x26 = P*R*Tc
x27 = x17*x3
x28 = V*x12
x29 = 2.*m*m*m
x30 = b*x12
T_calc = -Tc*(2.*a*m*x9*(V*x21*x21*x21*(V + b)*(P*x2 + P*x7 + x17 + x18 + x22 + x23 - x24))**0.5*(m + 1.) - x20*x21*(-P*x16*x6 + x1*x22 + x10*x26 + x13*x28 - x13*x30 + x15*x23 + x15*x24 + x19*x26 + x22*x3 + x25*x5 + x25 + x27*x5 + x27 + x28*x29 + x28*x5 - x29*x30 - x30*x5))/(x20*x9)
if abs(T_calc.imag) > 1e-12:
raise ValueError("Calculated imaginary temperature %s" %(T_calc))
return T_calc
else:
return Tc*(-2*a*m*sqrt(V*(V - b)**3*(V + b)*(P*R*Tc*V**2 + P*R*Tc*V*b - P*V*a*m**2 + P*a*b*m**2 + R*Tc*a*m**2 + 2*R*Tc*a*m + R*Tc*a))*(m + 1)*(R*Tc*V**2 + R*Tc*V*b - V*a*m**2 + a*b*m**2)**2 + (V - b)*(R**2*Tc**2*V**4 + 2*R**2*Tc**2*V**3*b + R**2*Tc**2*V**2*b**2 - 2*R*Tc*V**3*a*m**2 + 2*R*Tc*V*a*b**2*m**2 + V**2*a**2*m**4 - 2*V*a**2*b*m**4 + a**2*b**2*m**4)*(P*R*Tc*V**4 + 2*P*R*Tc*V**3*b + P*R*Tc*V**2*b**2 - P*V**3*a*m**2 + P*V*a*b**2*m**2 + R*Tc*V**2*a*m**2 + 2*R*Tc*V**2*a*m + R*Tc*V**2*a + R*Tc*V*a*b*m**2 + 2*R*Tc*V*a*b*m + R*Tc*V*a*b + V*a**2*m**4 + 2*V*a**2*m**3 + V*a**2*m**2 - a**2*b*m**4 - 2*a**2*b*m**3 - a**2*b*m**2))/((R*Tc*V**2 + R*Tc*V*b - V*a*m**2 + a*b*m**2)**2*(R**2*Tc**2*V**4 + 2*R**2*Tc**2*V**3*b + R**2*Tc**2*V**2*b**2 - 2*R*Tc*V**3*a*m**2 + 2*R*Tc*V*a*b**2*m**2 + V**2*a**2*m**4 - 2*V*a**2*b*m**4 + a**2*b**2*m**4))
[docs]class SRKTranslated(SRK):
r'''Class for solving the volume translated Peng-Robinson equation of state.
Subclasses :obj:`SRK`. Solves the EOS on initialization.
This is intended as a base class for all translated variants of the
SRK EOS.
.. math::
P = \frac{RT}{V + c - b} - \frac{a\alpha(T)}{(V + c)(V + c + b)}
.. math::
a=\left(\frac{R^2(T_c)^{2}}{9(\sqrt[3]{2}-1)P_c} \right)
=\frac{0.42748\cdot R^2(T_c)^{2}}{P_c}
.. math::
b=\left( \frac{(\sqrt[3]{2}-1)}{3}\right)\frac{RT_c}{P_c}
=\frac{0.08664\cdot R T_c}{P_c}
.. math::
\alpha(T) = \left[1 + m\left(1 - \sqrt{\frac{T}{T_c}}\right)\right]^2
.. math::
m = 0.480 + 1.574\omega - 0.176\omega^2
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
alpha_coeffs : tuple or None
Coefficients which may be specified by subclasses; set to None to use
the original Peng-Robinson alpha function, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization:
>>> eos = SRKTranslated(T=305, P=1.1e5, Tc=512.5, Pc=8084000.0, omega=0.559, c=-1e-6)
>>> eos.phase, eos.V_l, eos.V_g
('l/g', 5.5131657318e-05, 0.022447661363)
Notes
-----
References
----------
.. [1] Gmehling, Jürgen, Michael Kleiber, Bärbel Kolbe, and Jürgen Rarey.
Chemical Thermodynamics for Process Simulation. John Wiley & Sons, 2019.
'''
solve_T = GCEOS.solve_T
P_max_at_V = GCEOS.P_max_at_V
kwargs_keys = ('c', 'alpha_coeffs')
def __init__(self, Tc, Pc, omega, alpha_coeffs=None, c=0.0, T=None, P=None,
V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
Pc_inv = 1.0/Pc
self.a = self.c1*R*R*Tc*Tc*Pc_inv
self.c = c
if alpha_coeffs is None:
self.m = 0.480 + 1.574*omega - 0.176*omega*omega
self.alpha_coeffs = alpha_coeffs
self.kwargs = {'c': c, 'alpha_coeffs': alpha_coeffs}
b0 = self.c2*R*Tc*Pc_inv
self.b = b0 - c
### from sympy.abc import V, c, b, epsilon, delta
### expand((V+c)*((V+c)+b))
# delta = (b + 2*c)
self.delta = c + c + b0
# epsilon = b*c + c*c
self.epsilon = c*(b0 + c)
self.solve()
[docs]class MSRKTranslated(Soave_1979_a_alpha, SRKTranslated):
r'''Class for solving the volume translated Soave (1980) alpha function,
revision of the Soave-Redlich-Kwong equation of state
for a pure compound according to [1]_. Uses two fitting parameters `N` and
`M` to more accurately fit the vapor pressure of pure species.
Subclasses `SRKTranslated`.
Solves the EOS on initialization. See `SRKTranslated` for further
documentation.
.. math::
P = \frac{RT}{V + c - b} - \frac{a\alpha(T)}{(V + c)(V + c + b)}
.. math::
a=\left(\frac{R^2(T_c)^{2}}{9(\sqrt[3]{2}-1)P_c} \right)
=\frac{0.42748\cdot R^2(T_c)^{2}}{P_c}
.. math::
b=\left( \frac{(\sqrt[3]{2}-1)}{3}\right)\frac{RT_c}{P_c}
=\frac{0.08664\cdot R T_c}{P_c}
.. math::
\alpha(T) = 1 + (1 - T_r)(M + \frac{N}{T_r})
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
alpha_coeffs : tuple(float[3]), optional
Coefficients M, N of this EOS's alpha function, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization (hexane), liquid phase:
>>> eos = MSRKTranslated(Tc=507.6, Pc=3025000, omega=0.2975, c=22.0561E-6, M=0.7446, N=0.2476, T=250., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.0001169276461322, -34571.6862673, -84.757900348)
Notes
-----
This is an older correlation that offers lower accuracy on many properties
which were sacrificed to obtain the vapor pressure accuracy. The alpha
function of this EOS does not meet any of the consistency requriements for
alpha functions.
Coefficients can be found in [2]_, or estimated with the method in [3]_.
The estimation method in [3]_ works as follows, using the acentric factor
and true critical compressibility:
.. math::
M = 0.4745 + 2.7349(\omega Z_c) + 6.0984(\omega Z_c)^2
.. math::
N = 0.0674 + 2.1031(\omega Z_c) + 3.9512(\omega Z_c)^2
An alternate estimation scheme is provided in [1]_, which provides
analytical solutions to calculate the parameters `M` and `N` from two
points on the vapor pressure curve, suggested as 10 mmHg and 1 atm.
This is used as an estimation method here if the parameters are not
provided, and the two vapor pressure points are obtained from the original
SRK equation of state.
References
----------
.. [1] Soave, G. "Rigorous and Simplified Procedures for Determining
the Pure-Component Parameters in the Redlich—Kwong—Soave Equation of
State." Chemical Engineering Science 35, no. 8 (January 1, 1980):
1725-30. https://doi.org/10.1016/0009-2509(80)85007-X.
.. [2] Sandarusi, Jamal A., Arthur J. Kidnay, and Victor F. Yesavage.
"Compilation of Parameters for a Polar Fluid Soave-Redlich-Kwong
Equation of State." Industrial & Engineering Chemistry Process Design
and Development 25, no. 4 (October 1, 1986): 957-63.
https://doi.org/10.1021/i200035a020.
.. [3] Valderrama, Jose O., Héctor De la Puente, and Ahmed A. Ibrahim.
"Generalization of a Polar-Fluid Soave-Redlich-Kwong Equation of State."
Fluid Phase Equilibria 93 (February 11, 1994): 377-83.
https://doi.org/10.1016/0378-3812(94)87021-7.
'''
kwargs_keys = ('c', 'alpha_coeffs')
def __init__(self, Tc, Pc, omega, M=None, N=None, alpha_coeffs=None, c=0.0,
T=None, P=None, V=None):
# Ready for mixture class implemenentation
# M, N may be specified instead of alpha_coeffs currently only
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
Pc_inv = 1.0/Pc
self.a = self.c1*R*R*Tc*Tc*Pc_inv
self.c = c
b0 = self.c2*R*Tc*Pc_inv
self.b = b0 - c
self.delta = c + c + b0
self.epsilon = c*(b0 + c)
if alpha_coeffs is None and (M is None or N is None):
alpha_coeffs = MSRKTranslated.estimate_MN(Tc, Pc, omega, c)
if M is not None and N is not None:
alpha_coeffs = (M, N)
self.alpha_coeffs = alpha_coeffs
self.kwargs = {'c': c, 'alpha_coeffs': alpha_coeffs}
self.solve()
[docs] @staticmethod
def estimate_MN(Tc, Pc, omega, c=0.0):
r'''Calculate the alpha values for the MSRK equation to match two pressure
points, and solve analytically for the M, N required to match exactly that.
Since no experimental data is available, make it up with the original
SRK EOS.
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
Returns
-------
M : float
M parameter, [-]
N : float
N parameter, [-]
Examples
--------
>>> from sympy import * # doctest:+SKIP
>>> Tc, m, n = symbols('Tc, m, n') # doctest:+SKIP
>>> T0, T1 = symbols('T_10, T_760') # doctest:+SKIP
>>> alpha0, alpha1 = symbols('alpha_10, alpha_760') # doctest:+SKIP
>>> Eqs = [Eq(alpha0, 1 + (1 - T0/Tc)*(m + n/(T0/Tc))), Eq(alpha1, 1 + (1 - T1/Tc)*(m + n/(T1/Tc)))] # doctest:+SKIP
>>> solve(Eqs, [n, m]) # doctest:+SKIP
'''
SRK_base = SRKTranslated(T=Tc*0.5, P=Pc*0.5, c=c, Tc=Tc, Pc=Pc, omega=omega)
# Temperatures at 10 mmHg, 760 mmHg
P_10, P_760 = 10.0*mmHg, 760.0*mmHg
T_10 = SRK_base.Tsat(P_10)
T_760 = SRK_base.Tsat(P_760)
alpha_10 = SRK_base.a_alpha_and_derivatives(T=T_10, full=False)/SRK_base.a
alpha_760 = SRK_base.a_alpha_and_derivatives(T=T_760, full=False)/SRK_base.a
N = T_10*T_760*(-(T_10 - Tc)*(alpha_760 - 1) + (T_760 - Tc)*(alpha_10 - 1))/((T_10 - T_760)*(T_10 - Tc)*(T_760 - Tc))
M = Tc*(-T_10*(T_760 - Tc)*(alpha_10 - 1) + T_760*(T_10 - Tc)*(alpha_760 - 1))/((T_10 - T_760)*(T_10 - Tc)*(T_760 - Tc))
return (M, N)
[docs]class SRKTranslatedPPJP(SRK):
r'''Class for solving the volume translated Pina-Martinez, Privat, Jaubert,
and Peng revision of the Soave-Redlich-Kwong equation of state
for a pure compound according to [1]_.
Subclasses `SRK`, which provides everything except the variable `kappa`.
Solves the EOS on initialization. See `SRK` for further documentation.
.. math::
P = \frac{RT}{V + c - b} - \frac{a\alpha(T)}{(V + c)(V + c + b)}
.. math::
a=\left(\frac{R^2(T_c)^{2}}{9(\sqrt[3]{2}-1)P_c} \right)
=\frac{0.42748\cdot R^2(T_c)^{2}}{P_c}
.. math::
b=\left( \frac{(\sqrt[3]{2}-1)}{3}\right)\frac{RT_c}{P_c}
=\frac{0.08664\cdot R T_c}{P_c}
.. math::
\alpha(T) = \left[1 + m\left(1 - \sqrt{\frac{T}{T_c}}\right)\right]^2
.. math::
m = 0.4810 + 1.5963 \omega - 0.2963\omega^2 + 0.1223\omega^3
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization (hexane), liquid phase:
>>> eos = SRKTranslatedPPJP(Tc=507.6, Pc=3025000, omega=0.2975, c=22.3098E-6, T=250., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.00011666322408111662, -34158.934132722185, -83.06507748137201)
Notes
-----
This variant offers incremental improvements in accuracy only, but those
can be fairly substantial for some substances.
References
----------
.. [1] Pina-Martinez, Andrés, Romain Privat, Jean-Noël Jaubert, and
Ding-Yu Peng. "Updated Versions of the Generalized Soave a-Function
Suitable for the Redlich-Kwong and Peng-Robinson Equations of State."
Fluid Phase Equilibria, December 7, 2018.
https://doi.org/10.1016/j.fluid.2018.12.007.
'''
kwargs_keys = ('c',)
# No point in subclassing SRKTranslated - just disables direct solver for T
def __init__(self, Tc, Pc, omega, c=0.0, T=None, P=None, V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
Pc_inv = 1.0/Pc
self.a = self.c1*R2*Tc*Tc*Pc_inv
self.c = c
self.m = omega*(omega*(0.1223*omega - 0.2963) + 1.5963) + 0.4810
self.kwargs = {'c': c}
b0 = self.c2*R*Tc*Pc_inv
self.b = b0 - c
self.delta = c + c + b0
self.epsilon = c*(b0 + c)
self.solve()
class SRKTranslatedTwu(Twu91_a_alpha, SRKTranslated): pass
[docs]class SRKTranslatedConsistent(Twu91_a_alpha, SRKTranslated):
r'''Class for solving the volume translated Le Guennec, Privat, and Jaubert
revision of the SRK equation of state
for a pure compound according to [1]_.
This model's `alpha` is based
on the TWU 1991 model; when estimating, `N` is set to 2.
Solves the EOS on initialization. See `SRK` for further documentation.
.. math::
P = \frac{RT}{V + c - b} - \frac{a\alpha(T)}{(V + c)(V + c + b)}
.. math::
a=\left(\frac{R^2(T_c)^{2}}{9(\sqrt[3]{2}-1)P_c} \right)
=\frac{0.42748\cdot R^2(T_c)^{2}}{P_c}
.. math::
b=\left( \frac{(\sqrt[3]{2}-1)}{3}\right)\frac{RT_c}{P_c}
=\frac{0.08664\cdot R T_c}{P_c}
.. math::
\alpha = \left(\frac{T}{T_{c}}\right)^{c_{3} \left(c_{2}
- 1\right)} e^{c_{1} \left(- \left(\frac{T}{T_{c}}
\right)^{c_{2} c_{3}} + 1\right)}
If `c` is not provided, it is estimated as:
.. math::
c =\frac{R T_c}{P_c}(0.0172\omega - 0.0096)
If `alpha_coeffs` is not provided, the parameters `L` and `M` are estimated
from the acentric factor as follows:
.. math::
L = 0.0947\omega^2 + 0.6871\omega + 0.1508
.. math::
M = 0.1615\omega^2 - 0.2349\omega + 0.8876
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
alpha_coeffs : tuple(float[3]), optional
Coefficients L, M, N (also called C1, C2, C3) of TWU 1991 form, [-]
c : float, optional
Volume translation parameter, [m^3/mol]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
P-T initialization (methanol), liquid phase:
>>> eos = SRKTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=250., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.00011846802568940222, -34324.05211005662, -83.83861726864234)
Notes
-----
This variant offers substantial improvements to the SRK-type EOSs - likely
getting about as accurate as this form of cubic equation can get.
References
----------
.. [1] Le Guennec, Yohann, Romain Privat, and Jean-Noël Jaubert.
"Development of the Translated-Consistent Tc-PR and Tc-RK Cubic
Equations of State for a Safe and Accurate Prediction of Volumetric,
Energetic and Saturation Properties of Pure Compounds in the Sub- and
Super-Critical Domains." Fluid Phase Equilibria 429 (December 15, 2016):
301-12. https://doi.org/10.1016/j.fluid.2016.09.003.
'''
kwargs_keys = ('c', 'alpha_coeffs')
def __init__(self, Tc, Pc, omega, alpha_coeffs=None, c=None, T=None,
P=None, V=None):
# estimates volume translation and alpha function parameters
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
Pc_inv = 1.0/Pc
# limit oemga to 0.01 under the eos limit 1.47 for the estimation
o = min(max(omega, -0.01), 1.46)
if c is None:
c = R*Tc*Pc_inv*(0.0172*o + 0.0096)
if alpha_coeffs is None:
L = o*(0.0947*o + 0.6871) + 0.1508
M = o*(0.1615*o - 0.2349) + 0.8876
N = 2.0
alpha_coeffs = (L, M, N)
self.c = c
self.alpha_coeffs = alpha_coeffs
self.kwargs = {'c': c, 'alpha_coeffs': alpha_coeffs}
self.a = self.c1*R2*Tc*Tc*Pc_inv
b0 = self.c2*R*Tc*Pc_inv
self.b = b = b0 - c
self.delta = c + c + b0
self.epsilon = c*(b0 + c)
self.solve()
[docs]class APISRK(SRK):
r'''Class for solving the Refinery Soave-Redlich-Kwong cubic
equation of state for a pure compound shown in the API Databook [1]_.
Subclasses :obj:`GCEOS`, which
provides the methods for solving the EOS and calculating its assorted
relevant thermodynamic properties. Solves the EOS on initialization.
Implemented methods here are `a_alpha_and_derivatives`, which sets
:math:`a \alpha` and its first and second derivatives, and `solve_T`, which from a
specified `P` and `V` obtains `T`. Two fit constants are used in this
expresion, with an estimation scheme for the first if unavailable and the
second may be set to zero.
Two of `T`, `P`, and `V` are needed to solve the EOS.
.. math::
P = \frac{RT}{V-b} - \frac{a\alpha(T)}{V(V+b)}
.. math::
a=\left(\frac{R^2(T_c)^{2}}{9(\sqrt[3]{2}-1)P_c} \right)
=\frac{0.42748\cdot R^2(T_c)^{2}}{P_c}
.. math::
b=\left( \frac{(\sqrt[3]{2}-1)}{3}\right)\frac{RT_c}{P_c}
=\frac{0.08664\cdot R T_c}{P_c}
.. math::
\alpha(T) = \left[1 + S_1\left(1-\sqrt{T_r}\right) + S_2\frac{1
- \sqrt{T_r}}{\sqrt{T_r}}\right]^2
.. math::
S_1 = 0.48508 + 1.55171\omega - 0.15613\omega^2 \text{ if S1 is not tabulated }
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float, optional
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
S1 : float, optional
Fit constant or estimated from acentric factor if not provided [-]
S2 : float, optional
Fit constant or 0 if not provided [-]
Examples
--------
>>> eos = APISRK(Tc=514.0, Pc=6137000.0, S1=1.678665, S2=-0.216396, P=1E6, T=299)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 7.0456950702e-05, -42826.286146, -103.626979037)
References
----------
.. [1] API Technical Data Book: General Properties & Characterization.
American Petroleum Institute, 7E, 2005.
'''
kwargs_keys = ('S1', 'S2')
def __init__(self, Tc, Pc, omega=None, T=None, P=None, V=None, S1=None,
S2=0):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
self.check_sufficient_inputs()
if S1 is None and omega is None:
raise Exception('Either acentric factor of S1 is required')
if S1 is None:
self.S1 = S1 = 0.48508 + 1.55171*omega - 0.15613*omega*omega
else:
self.S1 = S1
self.S2 = S2
self.kwargs = {'S1': S1, 'S2': S2}
self.a = self.c1*R*R*Tc*Tc/Pc
self.b = self.c2*R*Tc/Pc
self.delta = self.b
self.solve()
[docs] def a_alpha_and_derivatives_pure(self, T):
r'''Method to calculate :math:`a \alpha` and its first and second
derivatives for this EOS. Returns `a_alpha`, `da_alpha_dT`, and
`d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more
documentation. Uses the set values of `Tc`, `a`, `S1`, and `S2`.
.. math::
a\alpha(T) = a\left[1 + S_1\left(1-\sqrt{T_r}\right) + S_2\frac{1
- \sqrt{T_r}}{\sqrt{T_r}}\right]^2
.. math::
\frac{d a\alpha}{dT} = a\frac{T_{c}}{T^{2}} \left(- S_{2} \left(\sqrt{
\frac{T}{T_{c}}} - 1\right) + \sqrt{\frac{T}{T_{c}}} \left(S_{1} \sqrt{
\frac{T}{T_{c}}} + S_{2}\right)\right) \left(S_{2} \left(\sqrt{\frac{
T}{T_{c}}} - 1\right) + \sqrt{\frac{T}{T_{c}}} \left(S_{1} \left(\sqrt{
\frac{T}{T_{c}}} - 1\right) - 1\right)\right)
.. math::
\frac{d^2 a\alpha}{dT^2} = a\frac{1}{2 T^{3}} \left(S_{1}^{2} T
\sqrt{\frac{T}{T_{c}}} - S_{1} S_{2} T \sqrt{\frac{T}{T_{c}}} + 3 S_{1}
S_{2} Tc \sqrt{\frac{T}{T_{c}}} + S_{1} T \sqrt{\frac{T}{T_{c}}}
- 3 S_{2}^{2} Tc \sqrt{\frac{T}{T_{c}}} + 4 S_{2}^{2} Tc + 3 S_{2}
Tc \sqrt{\frac{T}{T_{c}}}\right)
'''
# possible TODO: custom hydrogen a_alpha from
# Graboski, Michael S., and Thomas E. Daubert. "A Modified Soave Equation
# of State for Phase Equilibrium Calculations. 3. Systems Containing
# Hydrogen." Industrial & Engineering Chemistry Process Design and
# Development 18, no. 2 (April 1, 1979): 300-306. https://doi.org/10.1021/i260070a022.
# 1.202*exp(-.30228Tr)
# Will require CAss in kwargs, is_hydrogen array (or skip vectorized approach)
a, Tc, S1, S2 = self.a, self.Tc, self.S1, self.S2
x0 = (T/Tc)**0.5
x1 = x0 - 1.
x2 = x1/x0
x3 = S2*x2
x4 = S1*x1 + x3 - 1.
x5 = S1*x0
x6 = S2 - x3 + x5
x7 = 3.*S2
a_alpha = a*x4*x4
da_alpha_dT = a*x4*x6/T
d2a_alpha_dT2 = a*(-x4*(-x2*x7 + x5 + x7) + x6*x6)/(2.*T*T)
return a_alpha, da_alpha_dT, d2a_alpha_dT2
[docs] def a_alpha_pure(self, T):
a, Tc, S1, S2 = self.a, self.Tc, self.S1, self.S2
return a*(S1*(-(T/Tc)**0.5 + 1.) + S2*(-(T/Tc)**0.5 + 1)*(T/Tc)**-0.5 + 1)**2
[docs] def solve_T(self, P, V, solution=None):
r'''Method to calculate `T` from a specified `P` and `V` for the API
SRK EOS. Uses `a`, `b`, and `Tc` obtained from the class's namespace.
Parameters
----------
P : float
Pressure, [Pa]
V : float
Molar volume, [m^3/mol]
solution : str or None, optional
'l' or 'g' to specify a liquid of vapor solution (if one exists);
if None, will select a solution more likely to be real (closer to
STP, attempting to avoid temperatures like 60000 K or 0.0001 K).
Returns
-------
T : float
Temperature, [K]
Notes
-----
If S2 is set to 0, the solution is the same as in the SRK EOS, and that
is used. Otherwise, newton's method must be used to solve for `T`.
There are 8 roots of T in that case, six of them real. No guarantee can
be made regarding which root will be obtained.
'''
self.no_T_spec = True
if self.S2 == 0:
self.m = self.S1
return SRK.solve_T(self, P, V, solution=solution)
else:
# Previously coded method is 63 microseconds vs 47 here
# return super(SRK, self).solve_T(P, V)
Tc, a, b, S1, S2 = self.Tc, self.a, self.b, self.S1, self.S2
x2 = R/(V-b)
x3 = (V*(V + b))
def to_solve(T):
x0 = (T/Tc)**0.5
x1 = x0 - 1.
return (x2*T - a*(S1*x1 + S2*x1/x0 - 1.)**2/x3) - P
if solution is None:
try:
return newton(to_solve, Tc*0.5)
except:
pass
return GCEOS.solve_T(self, P, V, solution=solution)
def P_max_at_V(self, V):
if self.S2 == 0:
self.m = self.S1
return SRK.P_max_at_V(self, V)
return GCEOS.P_max_at_V(self, V)
[docs]class TWUPR(TwuPR95_a_alpha, PR):
r'''Class for solving the Twu (1995) [1]_ variant of the Peng-Robinson cubic
equation of state for a pure compound. Subclasses :obj:`PR`, which
provides the methods for solving the EOS and calculating its assorted
relevant thermodynamic properties. Solves the EOS on initialization.
The main implemented method here is :obj:`a_alpha_and_derivatives_pure`,
which sets :math:`a \alpha` and its first and second derivatives.
Two of `T`, `P`, and `V` are needed to solve the EOS.
.. math::
P = \frac{RT}{v-b}-\frac{a\alpha(T)}{v(v+b)+b(v-b)}
.. math::
a=0.45724\frac{R^2T_c^2}{P_c}
.. math::
b=0.07780\frac{RT_c}{P_c}
.. math::
\alpha = \alpha^{(0)} + \omega(\alpha^{(1)}-\alpha^{(0)})
.. math::
\alpha^{(i)} = T_r^{N(M-1)}\exp[L(1-T_r^{NM})]
For sub-critical conditions:
L0, M0, N0 = 0.125283, 0.911807, 1.948150;
L1, M1, N1 = 0.511614, 0.784054, 2.812520
For supercritical conditions:
L0, M0, N0 = 0.401219, 4.963070, -0.2;
L1, M1, N1 = 0.024955, 1.248089, -8.
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
>>> eos = TWUPR(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.V_l, eos.H_dep_l, eos.S_dep_l
(0.00013017554170, -31652.73712, -74.112850429)
Notes
-----
Claimed to be more accurate than the PR, PR78 and PRSV equations.
There is no analytical solution for `T`. There are multiple possible
solutions for `T` under certain conditions; no guaranteed are provided
regarding which solution is obtained.
References
----------
.. [1] Twu, Chorng H., John E. Coon, and John R. Cunningham. "A New
Generalized Alpha Function for a Cubic Equation of State Part 1.
Peng-Robinson Equation." Fluid Phase Equilibria 105, no. 1 (March 15,
1995): 49-59. doi:10.1016/0378-3812(94)02601-V.
'''
P_max_at_V = GCEOS.P_max_at_V
solve_T = GCEOS.solve_T
def __init__(self, Tc, Pc, omega, T=None, P=None, V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
self.a = self.c1*R*R*Tc*Tc/Pc
self.b = self.c2*R*Tc/Pc
self.delta = 2.*self.b
self.epsilon = -self.b*self.b
self.check_sufficient_inputs()
self.solve()
[docs]class TWUSRK(TwuSRK95_a_alpha, SRK):
r'''Class for solving the Soave-Redlich-Kwong cubic
equation of state for a pure compound. Subclasses :obj:`GCEOS`, which
provides the methods for solving the EOS and calculating its assorted
relevant thermodynamic properties. Solves the EOS on initialization.
The main implemented method here is :obj:`a_alpha_and_derivatives_pure`,
which sets :math:`a \alpha` and its first and second derivatives.
Two of `T`, `P`, and `V` are needed to solve the EOS.
.. math::
P = \frac{RT}{V-b} - \frac{a\alpha(T)}{V(V+b)}
.. math::
a=\left(\frac{R^2(T_c)^{2}}{9(\sqrt[3]{2}-1)P_c} \right)
=\frac{0.42748\cdot R^2(T_c)^{2}}{P_c}
.. math::
b=\left( \frac{(\sqrt[3]{2}-1)}{3}\right)\frac{RT_c}{P_c}
=\frac{0.08664\cdot R T_c}{P_c}
.. math::
\alpha = \alpha^{(0)} + \omega(\alpha^{(1)}-\alpha^{(0)})
.. math::
\alpha^{(i)} = T_r^{N(M-1)}\exp[L(1-T_r^{NM})]
For sub-critical conditions:
L0, M0, N0 = 0.141599, 0.919422, 2.496441
L1, M1, N1 = 0.500315, 0.799457, 3.291790
For supercritical conditions:
L0, M0, N0 = 0.441411, 6.500018, -0.20
L1, M1, N1 = 0.032580, 1.289098, -8.0
Parameters
----------
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
omega : float
Acentric factor, [-]
T : float, optional
Temperature, [K]
P : float, optional
Pressure, [Pa]
V : float, optional
Molar volume, [m^3/mol]
Examples
--------
>>> eos = TWUSRK(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.phase, eos.V_l, eos.H_dep_l, eos.S_dep_l
('l', 0.000146892222966, -31612.6025870, -74.022966093)
Notes
-----
There is no analytical solution for `T`. There are multiple possible
solutions for `T` under certain conditions; no guaranteed are provided
regarding which solution is obtained.
References
----------
.. [1] Twu, Chorng H., John E. Coon, and John R. Cunningham. "A New
Generalized Alpha Function for a Cubic Equation of State Part 2.
Redlich-Kwong Equation." Fluid Phase Equilibria 105, no. 1 (March 15,
1995): 61-69. doi:10.1016/0378-3812(94)02602-W.
'''
P_max_at_V = GCEOS.P_max_at_V
solve_T = GCEOS.solve_T
def __init__(self, Tc, Pc, omega, T=None, P=None, V=None):
self.Tc = Tc
self.Pc = Pc
self.omega = omega
self.T = T
self.P = P
self.V = V
self.a = self.c1*R*R*Tc*Tc/Pc
self.b = self.c2*R*Tc/Pc
self.delta = self.b
self.check_sufficient_inputs()
self.solve()
eos_list = [IG, PR, PR78, PRSV, PRSV2, VDW, RK, SRK, APISRK, TWUPR, TWUSRK,
PRTranslatedPPJP, SRKTranslatedPPJP, MSRKTranslated,
PRTranslatedConsistent, SRKTranslatedConsistent]
"""list : List of all cubic equation of state classes.
"""
eos_2P_list = list(eos_list)
"""list : List of all cubic equation of state classes that can represent
multiple phases.
"""
eos_2P_list.remove(IG)
eos_dict = {c.__name__: c for c in eos_list}
"""dict : Dict of all cubic equation of state classes, indexed by their class name.
"""
eos_full_path_dict = {c.__full_path__: c for c in eos_list}
"""dict : Dict of all cubic equation of state classes, indexed by their module path and class name.
"""